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--git a/09FRT4oBgHgl3EQflTce/content/tmp_files/2301.13598v1.pdf.txt b/09FRT4oBgHgl3EQflTce/content/tmp_files/2301.13598v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ec5c529e0a179c6d20330791c56b4e1785d4de4 --- /dev/null +++ b/09FRT4oBgHgl3EQflTce/content/tmp_files/2301.13598v1.pdf.txt @@ -0,0 +1,900 @@ +Economic Predictive Control with Periodic +Horizon for Water Distribution Networks +Mirhan ¨Urkmez ∗ Carsten Kallesøe ∗ Jan Dimon Bendtsen ∗ +John Leth ∗ +∗ Aalborg University, Fredrik Bajers Vej 7c, DK-9220 Aalborg, +Denmark +(e-mail: {mu,csk,dimon,jjl}@es.aau.dk) +Abstract: This paper deals with the control of pumps in large-scale water distribution networks +with the aim of minimizing economic costs while satisfying operational constraints. Finding a +control algorithm in combination with a model that can be applied in real-time is a challenging +problem due to the nonlinearities presented by the pipes and the network sizes. We propose +a predictive control algorithm with a periodic horizon. The method provides a way for the +economic operation of large water networks with a small linear model. Economic Predictive +control with a periodic horizon and a terminal state constraint is constructed to keep the state +trajectories close to an optimal periodic trajectory. Barrier terms are also included in the cost +function to prevent constraint violations. The proposed method is tested on the EPANET +implementation of the water network of a medium size Danish town (Randers) and shown to +perform as intended under varying conditions. +Keywords: Water distribution networks, Pump Scheduling, Predictive control, Periodic +horizon, Economic model predictive control +1. INTRODUCTION +Water distribution networks (WDNs) deliver drinkable +water from water sources to consumers using elements +such as pumps, pipes, tanks etc. About 7% − 8% of the +world’s energy is used for water production and distribu- +tion (Sharif et al., 2019). Water pumps account for a sig- +nificant part of the energy required for water distribution +with their percentage ranging from 90% to 95% of the +total (Abdelsalam and Gabbar, 2021). There have been +many works to schedule the operation of the pumps in +WDNs with proper methods so as to reduce energy costs. +However, pump scheduling is not an easy task because +of the nonlinearities governing the network elements. The +problem gets complicated with increasing network size. +Also, there are constraints to be satisfied such as limits +on tank levels. +In the literature, WDNs with both constant and variable +speed pumps are studied extensively. The control input is +turning on and off the pump for constant speed pumps. In +Lindell Ormsbee (2009), a constant-speed pump schedul- +ing problem is posed as an optimization problem in which +the decision variables are the operation times of the pumps +and the objective is the energy cost. After observing that +the optimal solution would be not running the pumps at +all without the constraints, the authors try to find the +solution closest to the origin that also complies with the +constraints. The proposed way to find such a solution is +⋆ This work is funded by Independent Research Fund Denmark +(DFF). We acknowledge Verdo company, Peter Nordahn, and Steffen +Schmidt for providing us with the EPANET model and the network +information. +using a Genetic Algorithm (GA). In Bagirov et al. (2013), +the Hooke-Jeeves method is used for finding optimal pump +operating times for a similar problem. Then, network sim- +ulation algorithms are used to check if the constraints are +satisfied. In Castro-Gama et al. (2017), binary decision +variables are used to represent the opening and the closing +of each pump. The feasibility of the solution found with +GA is checked with EPANET, a WDN modeling software, +and a high cost is assigned to the infeasible solutions. +The number of open pumps is also taken as the input +to the system in some works, e.g., Wang et al. (2021). +The problem is then solved using mixed-integer nonlinear +programming. In Berkel et al. (2018), a network in which +pressure zones are connected via constant speed pumps +is considered. Each pressure zone is treated as a subsys- +tem and distributed model predictive control (DMPC) is +applied. +The flow rate of the pumps should be determined for +networks with variable speed pumps. In Pour et al. (2019), +Linear Parameter Varying (LPV) system modeling is used +to replace the nonlinear part of the network, and an +Economic Model Predictive Control (EMPC) is applied +on top of the LPV system to find the optimal flow rates. +In Kallesøe et al. (2017), a network structure with an +elevated reservoir is considered. Available data is used +for the identification of a reduced system model. Then, +EMPC is applied to the model. In the EMPC formulation, +node pressures are not constrained. It is assumed that the +pressures would be in the accepted range because there is +an elevated reservoir. Relaxation of the original problem +into a simpler one is commonly used because of the large +network sizes. The relaxation is generally achieved by +arXiv:2301.13598v1 [eess.SY] 31 Jan 2023 + +approximating the nonlinear pipe equations with some +sort of linear equations or inequalities. In Baunsgaard +et al. (2016), pipe equations are linearized around an +operating point, and model predictive control (MPC) is +applied. In Wang et al. (2018), an EMPC is applied to a +network where the nonlinear pipe equations are relaxed +into a set of linear inequalities. Before simplifying the +system model, the network structure is also simplified in +Fiedler et al. (2020). A hierarchical clustering method is +used to represent the original network with a smaller one +which originally had 378 junctions. A system model is +derived from the simplified structure using a Deep Neural +Network (DNN) structure. Lagrangian relaxation is used +to approximate the original problem in Ghaddar et al. +(2015). +In this paper, a way for optimal pump scheduling of large- +scale WDNs is presented. To control the pumps, a linear +model of the system is derived. Then, a predictive control +method with a periodic horizon is constructed. Barrier +functions are utilized to prevent constraint violation due +to the model-plant mismatch. With the introduction of +the periodic horizon and the terminal state constraint, +the chance of finding a feasible solution is increased by +keeping trajectories close to an optimal periodic trajectory. +The method is applied to a medium-sized Danish town’s +network (Randers). +The outline of the rest of the paper is as follows. The +network model is given in Section 2. The proposed control +method is explained in Section 3. The experimental results +are presented in Section 4. The paper is concluded with +final remarks in Section 5. +2. NETWORK MODEL +A typical water distribution network consists of pipes, +pumps, tanks, junction nodes and reservoirs. Water in +the network flows from high hydraulic head to low head. +Hydraulic head is a measure of the fluid pressure and is +equal to the height of a fluid in a static column at a point. +Hydraulic head loss occurring in a pipe can be approxi- +mated by the Hazen-Williams Equation as +∆h = h1 − h2 = Kq|q|0.852 +(1) +where K is the pipe resistance that depends on the physical +features of a pipe such as diameter and length, q is the flow +rate, and h1 and h2 are the heads at the two ends of the +pipe. +At each node j, the mass conservation law is satisfied. It +can be expressed as +� +i∈Nj +qij = dj +(2) +where qij is the flow entering the node j from node i and +dj is the demand at node j, which is the water requested +by the user at node j. The symbol Nj denotes the set of +neighbor nodes of node j. Note that qij is positive if the +flow is from node i to the neighbor node j and negative +vice versa. +Tanks are storage elements that provide water to the users. +In the network, tanks are elevated so that water can be +pressurized enough to be delivered to the consumers. The +change in the water level of a tank is dependent on the +flow coming from neighbor nodes and can be written for +the tank j as +Aj ˙hj = +� +i∈Nj +qij +(3) +where Aj is the cross-sectional area, hj is the level of the +tank. Tank levels change according to the flow passing +through the pipes connected to the tanks. Those flows +are determined by a set of pipe head loss equations (1), +and mass balance equations (2) throughout the whole +network. As Equation (1) is nonlinear, flow through pipes +connected to the tanks are nonlinear functions fi of the +demand at each node, tank levels, and the amount of water +coming from the pumps. Explicit forms of those nonlinear +functions could be derived if the vector d = [d1, d2...]T +containing the demands of all the nodes is known, which is +not possible unless demand data for all nodes are available. +In our work, we assume that the total demand of the zones +that are supplied by the pumps can be estimated through +available data with time series analysis methods, but not +require d vector to be known. Since fi functions can not be +found without d vector, we approximate them using linear +models and write tank level change equations as +˙h(t) = Ah(t) + B1u(t) + B2da(t) +(4) +where h(t) ∈ Rn includes tank levels, A ∈ Rn×n, B1 ∈ +Rn×m, B2 ∈ Rn×1 are constant system matrices and da(t) +is the aggregated demand of controlled zone at time t, +u(t) ∈ Rm is the input containing pump flows. The reason +we chose a linear model is to increase the chance of finding +a feasible solution for the controller which is posed as +an optimization problem in the next section. Although +capturing the full dynamics of a large-scale network is not +possible with a linear model, the proposed control method +is designed to compensate for model inaccuracies and we +have observed that it was enough to control the system +while satisfying the constraints. +3. PERIODIC HORIZON CONTROL +In this section, a predictive control algorithm for pump +scheduling is presented to minimize the economical costs. +The aim is to run the pumps when the electricity price is +low and let tanks deliver water when the price is high while +also satisfying input and output constraints. The problem +at time t is posed as +min +ut +0,ut +1···ut +N(t)−1 +N(t)−1 +� +j=0 +J(ht +j, ut +j) +(5a) +ht +j = Adht +j−1 + Bd1ut +j−1 + Bd2da(j − 1) +(5b) +ht +0 = h(t) +(5c) +ut +j ∈ U ⊆ Rm +(5d) +ht +j ∈ H ⊆ Rn +(5e) +ht +N(t) ∈ Htf ⊆ Rn +(5f) +where J(ht +j, ut +j) is the economic cost function, ht += +[ht +1 · · · ht +N(t)] ∈ Rn×N(t) is the predicted future states, ut +j is +the input vector, N(t) is the prediction horizon, U ⊆ Rm +and H ⊆ Rn denotes the input and state constraints +respectively and Htf ⊆ Rn is the terminal state set. The +continuous system (4) is discretized and (5b) is obtained. + +The optimization problem (5) is solved at every time step +separated by ∆t and the first term ut +0 of the optimal input +sequence ut = [ut +0 · · · ut +N(t)−1] ∈ Rm×N(t) is applied to the +system. +Input constraints come from the physical limitations and +working principles of the pumps. A pump can not provide +water in the opposite direction and it can deliver a maxi- +mum amount of water per unit of time. These conditions +are expressed as +U = {[u1, · · · um] ∈ Rm | 0 ≤ u1 ≤ u1, · · · 0 ≤ um ≤ um} +(6) +where u1 · · · um are upper flow limits. Tank levels are also +constrained so that there is always enough water in the +tanks in case of an emergency and there is no overflow of +water. The set H can be defined as +H = {[h1, · · · h2] ∈ Rn | ˜h1 ≤ h1 ≤ h1, · · · ˜hn ≤ hn ≤ hn} +(7) +The cost function includes the electricity costs of the +pumps. The power provided to the network by the pump +i is equal to qpi(pout +i +− pin +i ), where qpi is the pump flow, +pi +out and pi +in are the outlet and inlet pressures of the pump +i. The inlet pressures pin = [pin +1 , pin +2 ] are the pressures of +the related reservoirs and are assumed to be constant. The +outlet pressures pout = [pout +1 +, pout +2 +] are given as the output +of the linear model +pout(t) = Aph(t) + Bpu(t) +(8) +where Ap and Bp are found using system identification +on data generated by the EPANET model. Electricity +cost at time t is then found by multiplying total power +consumption u(t)T (pout(t) − pin(t)) with the electricity +price c(t). +We acknowledge a certain degree of model-plant mismatch +by using a linear model (4) to represent the whole network. +This causes actual states h(t) to be different than the +predicted states ht. We know that the predicted states +satisfy the state constraints (7) since they are the solution +to the optimization problem 5, but the actual states +might violate them. To ensure the satisfaction of the state +constraints with the model-plant mismatch, we introduce +new terms to the cost function. First, we rewrite state +constraints (7) as +Ci(h) ≤ 0, +i = 0, 1, · · · 2 × n − 1 +(9) +where C0(h) = ˜h1 −h1 and the rest of the Ci functions are +chosen in a similar manner. The cost function terms are +then defined as +Jhi(h) = eai(Ci(h)+bi) +i = 0, 1, · · · 2 × n − 1 +(10) +where ai, bi ∈ R>0. This can be seen as an exponential +barrier function. The parameters ai, bi determine a danger- +ous region close to the boundaries of the state constraints +where cost function Jhi attains high values. The predicted +optimal state trajectories ht do not enter the dangerous +region if possible because of the high cost values in the +dangerous region. Then, the actual states h(t) do not +violate the state constraints (7) assuming the difference +between the predicted state and the actual state is small. +If the state trajectory enters one of the dangerous regions +at any step due to the model-plant mismatch, then the +cost function will try to drive the trajectory out of the +region. +∆t +N(t)∆t +N(t + ∆t)∆t +h(t) +h(t + ∆t) +ht +1 +ut +0 +ut+∆t +0 +ht+∆t +ht +Br(h∗ +Tday/∆t) +Fig. 1. Predicted state trajectories ht, ht+∆t at times +t, t + ∆t. Sampling time ∆t, prediction horizons +N(t), N(t + ∆t) and the applied inputs ut +0, ut+∆t +0 +are +shown. The true state h(t + ∆t) and the predicted +state ht +1 are indicated to emphasize the deviation from +the prediction. The terminal set Br(h∗ +Tday/∆t) is also +illustrated. +The overall cost function includes both the electricity +expense term and the constraint barrier functions and it +can be expressed as +J(h(t), u(t)) = c(t)u(t)T (pout(t)−pin(t))+ +2×n−1 +� +i=0 +Jhi(h(t)) +(11) +Both electricity price c(t) and total water demand da(t) +signals can be viewed as consisting of a periodic signal +with a period of 1 day and a relatively small deviation +signal. This can be leveraged to find a feasible controller. +Suppose a sequence of inputs can be found for some +initial tank levels such that levels after 1 day are equal +to initial levels. In that case, the problem after 1 day is +the same as in the beginning assuming deviation signals +of the electricity price and the demand are zero, hence +they are periodic. Then, the input sequence from the +previous day could be applied and produce the same +path for tank levels. Taking into account the deviation +signals and supposing that a solution exists such that +levels after 1 day are close to initial levels, the input +sequence from the previous day could be a good point of +start to search for a feasible solution if the map from the +initial conditions and the demand profile to the optimal +input sequences is continuous. Therefore, we choose a +terminal state constraint for the end of each day to +increase the chance of finding a feasible solution. Now, the +remaining problem is to decide which tank levels should +the trajectories turn back to at the end of each day. We +define the optimal periodic trajectory of the system as the +solution of +(u∗, h∗) = arg min +ui,hi +(Tday/∆t)−1 +� +i=0 +J(hi, ui) +(12a) +hi = Adhi−1 + Bd1ui−1 + Bd2d∗ +a(i − 1) +(12b) +ui ∈ U ⊆ Rm +(12c) +hi ∈ H ⊆ Rn +(12d) +h0 = hTday/∆t +(12e) +where Tday is the duration of a whole day, d∗ +a is the +average daily demand profile obtained from the past +measurements. The resulting state trajectory h∗ += +[h∗ +0 · · · h∗ +Tday/∆t] ∈ Rn×(Tday/∆t+1) is the optimal periodic +trajectory because of the constraint (12e). The terminal +set Htf and the prediction horizon N(t) is chosen to make +tank levels at the end of each day close to h∗ +Tday/∆t. At + +High Zone +Low Zone +Fig. 2. Water Distribution Network of Randers. The pump- +ing stations to be controlled are shown in red. Tanks +are shown with a ’T’ shaped symbol in yellow. +any time t, t + N(t)∆t should be equal to the end of the +day. Htf and N(t) could be written as +Htf = Br(h∗ +Tday/∆t) +(13a) +N(t) = (Tday − t mod Tday)/∆t +(13b) +where Br(h∗ +Tday/∆t) is the open ball centered at h∗ +Tday/∆t +with radius r. Note that N(t) changes so that t + N(t)∆t +is the end of the day for all t. With these definitions, the +condition (13a) will translate to tank levels at the end of +the day being close to the final point in optimal periodic +trajectory h∗ +Tday/∆t as shown in Figure 1. Therefore, not +only chance of finding a feasible solution is increased but +also the solutions are kept around the optimal periodic +trajectory h∗. If the problem (5) becomes infeasible at +any time step t, we apply the second term of the input +sequence from the previous step ut−∆t +1 +. The reason behind +this choice is as follows: If we apply the optimal control +input ut−∆t +0 +to the network model (4) at time t − ∆t, +then the optimal sequence in the next time step will be +ut = [ut−∆t +1 +· · · ut−∆t +N(t−∆t)−1] following Bellman’s principle +of optimality. Then, at time t, ut−∆t +1 +will be applied to +the system as calculated at t − ∆t. Assuming the model- +plant mismatch is small enough, ut−∆t +1 +is still a good input +candidate if the problem is infeasible at time t. +4. APPLICATION +The presented method is applied to WDN of Randers, a +Danish city, which is shown in Figure 2. The network con- +sists of 4549 nodes and 4905 links connecting them. There +are 8 pumping stations in the network, 6 of which are +shown in the figure whereas the other 2 are stationed where +tanks are placed. The goal is to derive the schedules for +2 of the pumping stations while other pumps are already +working according to some predetermined strategies. The +stations to be controlled are shown in red in the figure. +Their task is to deliver water mostly to the High Zone (HZ) +and Low Zone (LZ). However, connections exist between +HZ-LZ and the rest of the city, so we can not think of the +system as composed of isolated networks entirely. There +are also 3 tanks in the HZ. While 2 of them are directly +connected via pipes, the third one stands alone as shown +in the figure. +The overall structure of the Randers WDN with tanks and +pumps to be controlled are given in Figure 3. There are 3 +water tanks in the network, 2 of which have been connected +Fig. 3. Structure of the WDN. +with a pipe directly. The tank level changes can be written +by applying the mass conservation law (3) to the tanks in +Figure 3 as +A1 ˙h1 = q1down + q1up + qinter +(14a) += f1(h1, h2, h3, qp1, qp2, d), +A2 ˙h2 = q2down + q2up − qinter +(14b) += f2(h1, h2, h3, qp1, qp2, d), +A3 ˙h3 = q3 = f3(h1, h2, h3, qp1, qp2, d), +(14c) +where d is the vector containing the demands of all the +nodes, qp1, qp2 are the pump flows, A1, A2, A3 are the cross +sectional areas of the tanks and f1, f2, f3 are nonlinear +flow functions. Water levels at the two connected tanks are +almost equal h1 ≈ h2 all the time since the pipe connecting +respective tanks is big enough to oppose the water flows +coming from neighbor nodes. That enables us to consider +h1, h2 together as +(A1 + A2)˙h1,2 ≈ q1down + q2down + qup = f1 + f2. +(15) +We have used the EPANET model of the network to +generate the data required for approximating f1 + f2 +and f3. The model is simulated with various tank level +initial conditions and flow rates of 2 pumping stations +to be controlled. The control laws for the remaining +pumping stations are already defined in the EPANET +model. Then, the linear model (4) is fitted to simulation +data using least squares. The state variables for the model +are h(t) = [h1,2(t), h3(t)] ∈ R2 and the inputs are u(t) = +[qp1(t), qp2(t)] ∈ R2. The total demand of High and Low +Zone is used as aggregated demand da in the model since +mainly those areas are supplied by the controlled pumps. +4.1 Simulation Results +The proposed control method is tested on EPANET model +of Randers water network. Epanet-Matlab toolkit Eliades +et al. (2016) is used to set the flow of the 2 pumps at +each time step and simulate the network. The remaining +pumps are controlled with rule-based control laws that are +previously defined on EPANET. +The parameters of exponential barrier functions Jhi are +chosen as ai = 80, bi = 0.3 for all i. It is assumed +that the electricity prices are known in advance during +the test. Tank levels h1, h2 have a maximum value of 3m +while h3 has 2.8m. Tanks are required to be at least half +full. Maximum pump flows are set to 100. Sampling time +∆t is set to 1 hour in the experiments, so the control +input is recalculated at each hour. We assume that total +demand da(t) of HZ and LZ can be estimated up to 1 day +from available data. Although we do not have historical + +qup +qiup +q2up +h1 +h2 +h3 +qinter +q1down +2down +q3 +Pump 1 +Pump 2 +9p1 +qp2data on the demand, we imitate this behaviour by using +a slightly perturbed version of the real demand used +in EPANET simulation during MPC calculations. The +perturbations are adapted from a real demand data set of +a small Danish facility. Normalized difference between the +average demand and the demand of a random day in data +set is added to EPANET demand to replicate estimated +demand. In each experiment a different day from the data +set is used, so the assumed estimated demand is different +each time. +The simulation results when the presented method is +applied to the EPANET model are given in Figure 4. The +initial tank levels are equal to h∗ +Tday/∆t in the simulation. +The top plot shows the evolution of tank levels along +with the upper and lower thresholds. It is seen that the +thresholds are not violated and moreover tank levels are +not getting too close to them, which was the idea behind +exponential barrier functions. Both the real demand and +the assumed estimated demand of HZ and LZ are in the +figure below. Total applied pump flows and electricity +prices are in the following figures. The expected result is +pump flows being higher when electricity prices are low, +and lower when they are high, which seems to be the case +as can be seen in the plot. Pump flows drop significantly +when prices are at the peak and they reach their highest +value at the end of the day when prices are low. A more +aggressive controller can be obtained by picking a smaller +bi value for barrier functions at the expense of risking +constraint violation. In Figure 5, the tank level simulation +results and control inputs for different initial conditions +and different assumed estimated demands are given. The +electricity price profile is the same as before. It is seen that +the algorithm is able to control the network on various +cases while satisfying the constraints. Regardless of initial +tank levels, the pumping profiles have a similar profile: +high pump flows close to midnight and in the middle of +the day. The only exception is the bottom plot. In the +beginning, prices are low but pump flows are not high. +This can be attributed to water levels h1, h2 being close to +the upper thresholds and water demand being low in the +beginning. +The assumption that the optimal input sequences U(t) +would not diverge a lot from the one found in previous +step U(t − ∆t) is the reason we apply ut−∆t +1 +at time t if +the problem (5) is infeasible at time t. This assumption is +tested with initial conditions h1,2,3 = h∗ +Tday/∆t. In figure +6, total pump flow [1, 1]T ut +i, i = 0 · · · N(t) − 1 of the +found optimal input sequences U(t), t = 0, ∆t · · · Tday−∆t, +except when the problem were infeasible, are given. It can +be seen that ut−∆t +1 +is close to the ut +0 for all t, which shows +that our assumption is valid at least for this experiment. +Finally, the ability of the algorithm to decrease economic +costs is tested with various initial conditions. For each +case, a demand follower pumping strategy is used as a +benchmark. The flow of the 2 pumps is adjusted with trial +and error for each demand follower such that the total flow +of the 2 pumps is equal to water demand at each time step +and tank levels satisfy the terminal constraint (13a). The +demand follower is a natural candidate to be a benchmark +method since providing as much water as demand is an +intuitive idea and the constraints in (5) can be satisfied +(a) +(b) +(c) +(d) +Fig. 4. Sample simulation. (a) evolution of tank levels +through 1 day with upper and lower level thresholds; +(b) real total demand of HZ and LZ used in EPANET +simulation and the demand used in MPC calculations; +(c) total flow provided by the 2 pumps; (d) electricity +price. +Proposed Method +Demand Follower +0.5967 +1 +0.5745 +1 +0.5826 +1 +0.5558 +1 +Table 1. Relative economic costs of the pro- +posed method and demand follower strategy +for various demand profiles +with manual adjustments of pump flows. The economic +costs are presented relatively in Table 1 As it is seen, the +proposed algorithm saves between 40% and 45% of the +cost with different demand profiles. +5. CONCLUSION +We have presented a predictive control algorithm with a +periodic horizon for WDNs. The aim is to minimize the + +3.5 +h1 +h2 +upper threshold +3 +h3 +3 upper threshold +Tank Levels +1.5 +1 +0 +5 +10 +15 +20 +25140 +120 +100 +80 +60 +40 +Real Demand +Known Demand +20 +0 +5 +10 +15 +20 +25200 +Total Pump Flow +150 +100 +50 +0 +0 +5 +10 +15 +20 +251.2 +Price +0.8 +lectricity +0.6 +0.4 +E +0.2 +0 +0 +5 +10 +15 +20 +25 +HoursFig. 5. Tank levels and pump flows for different initial +conditions +Fig. 6. Evolution of found input sequences U(t) through 1 +day. It can be seen that the solutions remain close to +the initial optimal sequence U(0). +economic cost and satisfy the operational constraints. A +linear model is used to represent Randers WDN to increase +the chance of finding a solution to the problem (5) at +expense of a model-plant mismatch. Periodic horizon is +introduced to the predictive control formulation to keep +the resulting state trajectories around the optimal periodic +trajectory. Barrier functions are used to prevent constraint +violation since there is a model-plant mismatch. +The presented algorithm is tested on Randers WDN using +EPANET. It is shown in various situations that the +method is able to find an economic solution where pump +flows are adjusted according to electricity prices. Also, +it is shown that the system trajectories do not enter +dangerous zones introduced by barrier functions as long +as the predicted demand and the actual demand are +somewhat close. +As future work, we plan to work on theoretical guarantees +of the existence of solutions to the proposed method. Also, +the robustness of periodic horizon control of periodical +systems with barrier functions will be investigated. +REFERENCES +Abdelsalam, A.A. and Gabbar, H.A. (2021). Energy saving +and management of water pumping networks. Heliyon, +7(8), e07820. doi:https://doi.org/10.1016/j.heliyon.20 +21.e07820. +Bagirov, A.M., Barton, A., Mala-Jetmarova, H., Nuaimat, +A.A., Ahmed, S.T., Sultanova, N., and Yearwood, J. +(2013). An algorithm for minimization of pumping costs +in water distribution systems using a novel approach to +pump scheduling. Math. Comput. Model., 57, 873–886. +Baunsgaard, K.M.H., Ravn, O., Kallesøe, C.S., and +Poulsen, N.K. (2016). +Mpc control of water supply +networks. +2016 European Control Conference (ECC), +1770–1775. +Berkel, F., Caba, S., Bleich, J., and Liu, S. (2018). +A +modeling and distributed mpc approach for water dis- +tribution networks. Control Engineering Practice. +Castro-Gama, M.E., Pan, Q., Lanfranchi, E.A., Jonoski, +A., and Solomatine, D.P. (2017). Pump scheduling for a +large water distribution network. milan, italy. Procedia +Engineering, 186, 436–443. +Eliades, D.G., Kyriakou, M., Vrachimis, S., and Polycar- +pou, M.M. (2016). +Epanet-matlab toolkit: An open- +source software for interfacing epanet with matlab. In +Proc. 14th International Conference on Computing and +Control for the Water Industry (CCWI), 8. The Nether- +lands. doi:10.5281/zenodo.831493. +Fiedler, F., Cominola, A., and Lucia, S. (2020). +Eco- +nomic nonlinear predictive control of water distribution +networks based on surrogate modeling and automatic +clustering. IFAC-PapersOnLine, 53, 16636–16643. +Ghaddar, B., Naoum-Sawaya, J., Kishimoto, A., Taheri, +N., and Eck, B. (2015). +A lagrangian decomposition +approach for the pump scheduling problem in water +networks. Eur. J. Oper. Res., 241, 490–501. +Kallesøe, C.S., Jensen, T.N., and Bendtsen, J.D. (2017). +Plug-and-play model predictive control for water supply +networks with storage. IFAC-PapersOnLine, 50, 6582– +6587. +Lindell Ormsbee, Srini Lingireddy, D.C. (2009). Optimal +pump scheduling for water distribution systems. URL +http://www.uky.edu/WDST/PDFs/[73.3]%20Ormsbee% +20Optimal%20Pump%20Scheduling%20Paper.pdf. +Pour, F.K., Puig, V., and Cembra˜no, G. (2019). Economic +mpc-lpv control for the operational management of +water distribution networks. IFAC-PapersOnLine. +Sharif, N., Haider, H., Farahat, A., Hewage, K., and Sadiq, +R. (2019). Water energy nexus for water distribution +systems: A literature review. Environmental Reviews, +27. doi:10.1139/er-2018-0106. +Wang, Y., Alamo, T., Puig, V., and Cembra˜no, G. (2018). +Economic model predictive control with nonlinear con- +straint relaxation for the operational management of +water distribution networks. Energies, 11, 991. +Wang, Y., Yok, K.T., Wu, W., Simpson, A.R., Weyer, E., +and Manzie, C. (2021). +Minimizing pumping energy +cost in real-time operations of water distribution sys- +tems using economic model predictive control. ArXiv, +abs/2010.07477. + +200 +Total Pump Flow +150 +100 +50 +0 +0 +5 +10 +15 +20 +253.5 +h1 +h2 +upperthreshold +h3 +3 upper threshold +Tank Levels +2.5 +1.5 +1 +0 +5 +10 +15 +20 +25250 +Total Pump Flow +200 +150 +100 +50 +0 +0 +5 +10 +15 +20 +253.5 +h1 +h2 +upper threshold +3 +h3 + upper threshold +Tank Levels +1.5 +1 +0 +5 +10 +15 +20 +25200 +Total Pump Flow +150 +100 +50 +0 +0 +5 +10 +15 +20 +25200 +U(0) +150 +Flow +100 +U(1) +50 +0 +0 +5 +10 +15 +20 +25 +Hours3.5 +h1 +h2 +upperthreshold +3 +h3 +3 upper threshold +Tank Levels +2.5 +1.5 +1 +0 +5 +10 +15 +20 +25250 +Total Pump Flow +200 +150 +100 +50 +0 +0 +5 +10 +15 +20 +253.5 +h1 +h2 +upperthreshold +3 +h3 +3 upper threshold +Tank Levels +1.5 +1 +0 +5 +10 +15 +20 +25 \ No newline at end of file diff --git a/09FRT4oBgHgl3EQflTce/content/tmp_files/load_file.txt b/09FRT4oBgHgl3EQflTce/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a135dcffec3705113d785aa12c6dfcc9e3cb6a73 --- /dev/null +++ b/09FRT4oBgHgl3EQflTce/content/tmp_files/load_file.txt @@ -0,0 +1,420 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf,len=419 +page_content='Economic Predictive Control with Periodic Horizon for Water Distribution Networks Mirhan ¨Urkmez ∗ Carsten Kallesøe ∗ Jan Dimon Bendtsen ∗ John Leth ∗ ∗ Aalborg University, Fredrik Bajers Vej 7c, DK-9220 Aalborg, Denmark (e-mail: {mu,csk,dimon,jjl}@es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='aau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='dk) Abstract: This paper deals with the control of pumps in large-scale water distribution networks with the aim of minimizing economic costs while satisfying operational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Finding a control algorithm in combination with a model that can be applied in real-time is a challenging problem due to the nonlinearities presented by the pipes and the network sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' We propose a predictive control algorithm with a periodic horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The method provides a way for the economic operation of large water networks with a small linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Economic Predictive control with a periodic horizon and a terminal state constraint is constructed to keep the state trajectories close to an optimal periodic trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Barrier terms are also included in the cost function to prevent constraint violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The proposed method is tested on the EPANET implementation of the water network of a medium size Danish town (Randers) and shown to perform as intended under varying conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Keywords: Water distribution networks, Pump Scheduling, Predictive control, Periodic horizon, Economic model predictive control 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' INTRODUCTION Water distribution networks (WDNs) deliver drinkable water from water sources to consumers using elements such as pumps, pipes, tanks etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' About 7% − 8% of the world’s energy is used for water production and distribu- tion (Sharif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Water pumps account for a sig- nificant part of the energy required for water distribution with their percentage ranging from 90% to 95% of the total (Abdelsalam and Gabbar, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' There have been many works to schedule the operation of the pumps in WDNs with proper methods so as to reduce energy costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' However, pump scheduling is not an easy task because of the nonlinearities governing the network elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The problem gets complicated with increasing network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Also, there are constraints to be satisfied such as limits on tank levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In the literature, WDNs with both constant and variable speed pumps are studied extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The control input is turning on and off the pump for constant speed pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In Lindell Ormsbee (2009), a constant-speed pump schedul- ing problem is posed as an optimization problem in which the decision variables are the operation times of the pumps and the objective is the energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' After observing that the optimal solution would be not running the pumps at all without the constraints, the authors try to find the solution closest to the origin that also complies with the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The proposed way to find such a solution is ⋆ This work is funded by Independent Research Fund Denmark (DFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' We acknowledge Verdo company, Peter Nordahn, and Steffen Schmidt for providing us with the EPANET model and the network information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' using a Genetic Algorithm (GA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In Bagirov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2013), the Hooke-Jeeves method is used for finding optimal pump operating times for a similar problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Then, network sim- ulation algorithms are used to check if the constraints are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In Castro-Gama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2017), binary decision variables are used to represent the opening and the closing of each pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The feasibility of the solution found with GA is checked with EPANET, a WDN modeling software, and a high cost is assigned to the infeasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The number of open pumps is also taken as the input to the system in some works, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=', Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The problem is then solved using mixed-integer nonlinear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In Berkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2018), a network in which pressure zones are connected via constant speed pumps is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Each pressure zone is treated as a subsys- tem and distributed model predictive control (DMPC) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The flow rate of the pumps should be determined for networks with variable speed pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In Pour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2019), Linear Parameter Varying (LPV) system modeling is used to replace the nonlinear part of the network, and an Economic Model Predictive Control (EMPC) is applied on top of the LPV system to find the optimal flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In Kallesøe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2017), a network structure with an elevated reservoir is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Available data is used for the identification of a reduced system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Then, EMPC is applied to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In the EMPC formulation, node pressures are not constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' It is assumed that the pressures would be in the accepted range because there is an elevated reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Relaxation of the original problem into a simpler one is commonly used because of the large network sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The relaxation is generally achieved by arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='13598v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='SY] 31 Jan 2023 approximating the nonlinear pipe equations with some sort of linear equations or inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In Baunsgaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2016), pipe equations are linearized around an operating point, and model predictive control (MPC) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2018), an EMPC is applied to a network where the nonlinear pipe equations are relaxed into a set of linear inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Before simplifying the system model, the network structure is also simplified in Fiedler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' A hierarchical clustering method is used to represent the original network with a smaller one which originally had 378 junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' A system model is derived from the simplified structure using a Deep Neural Network (DNN) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Lagrangian relaxation is used to approximate the original problem in Ghaddar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In this paper, a way for optimal pump scheduling of large- scale WDNs is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' To control the pumps, a linear model of the system is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Then, a predictive control method with a periodic horizon is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Barrier functions are utilized to prevent constraint violation due to the model-plant mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' With the introduction of the periodic horizon and the terminal state constraint, the chance of finding a feasible solution is increased by keeping trajectories close to an optimal periodic trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The method is applied to a medium-sized Danish town’s network (Randers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The outline of the rest of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The network model is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The proposed control method is explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The experimental results are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The paper is concluded with final remarks in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' NETWORK MODEL A typical water distribution network consists of pipes, pumps, tanks, junction nodes and reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Water in the network flows from high hydraulic head to low head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Hydraulic head is a measure of the fluid pressure and is equal to the height of a fluid in a static column at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Hydraulic head loss occurring in a pipe can be approxi- mated by the Hazen-Williams Equation as ∆h = h1 − h2 = Kq|q|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='852 (1) where K is the pipe resistance that depends on the physical features of a pipe such as diameter and length, q is the flow rate, and h1 and h2 are the heads at the two ends of the pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' At each node j, the mass conservation law is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' It can be expressed as � i∈Nj qij = dj (2) where qij is the flow entering the node j from node i and dj is the demand at node j, which is the water requested by the user at node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The symbol Nj denotes the set of neighbor nodes of node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Note that qij is positive if the flow is from node i to the neighbor node j and negative vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Tanks are storage elements that provide water to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In the network, tanks are elevated so that water can be pressurized enough to be delivered to the consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The change in the water level of a tank is dependent on the flow coming from neighbor nodes and can be written for the tank j as Aj ˙hj = � i∈Nj qij (3) where Aj is the cross-sectional area, hj is the level of the tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Tank levels change according to the flow passing through the pipes connected to the tanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Those flows are determined by a set of pipe head loss equations (1), and mass balance equations (2) throughout the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' As Equation (1) is nonlinear, flow through pipes connected to the tanks are nonlinear functions fi of the demand at each node, tank levels, and the amount of water coming from the pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Explicit forms of those nonlinear functions could be derived if the vector d = [d1, d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=']T containing the demands of all the nodes is known, which is not possible unless demand data for all nodes are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In our work, we assume that the total demand of the zones that are supplied by the pumps can be estimated through available data with time series analysis methods, but not require d vector to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Since fi functions can not be found without d vector, we approximate them using linear models and write tank level change equations as ˙h(t) = Ah(t) + B1u(t) + B2da(t) (4) where h(t) ∈ Rn includes tank levels, A ∈ Rn×n, B1 ∈ Rn×m, B2 ∈ Rn×1 are constant system matrices and da(t) is the aggregated demand of controlled zone at time t, u(t) ∈ Rm is the input containing pump flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The reason we chose a linear model is to increase the chance of finding a feasible solution for the controller which is posed as an optimization problem in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Although capturing the full dynamics of a large-scale network is not possible with a linear model, the proposed control method is designed to compensate for model inaccuracies and we have observed that it was enough to control the system while satisfying the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' PERIODIC HORIZON CONTROL In this section, a predictive control algorithm for pump scheduling is presented to minimize the economical costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The aim is to run the pumps when the electricity price is low and let tanks deliver water when the price is high while also satisfying input and output constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The problem at time t is posed as min ut 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='ut 1···ut N(t)−1 N(t)−1 � j=0 J(ht j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' ut j) (5a) ht j = Adht j−1 + Bd1ut j−1 + Bd2da(j − 1) (5b) ht 0 = h(t) (5c) ut j ∈ U ⊆ Rm (5d) ht j ∈ H ⊆ Rn (5e) ht N(t) ∈ Htf ⊆ Rn (5f) where J(ht j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' ut j) is the economic cost function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' ht = [ht 1 · · · ht N(t)] ∈ Rn×N(t) is the predicted future states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' ut j is the input vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' N(t) is the prediction horizon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' U ⊆ Rm and H ⊆ Rn denotes the input and state constraints respectively and Htf ⊆ Rn is the terminal state set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The continuous system (4) is discretized and (5b) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The optimization problem (5) is solved at every time step separated by ∆t and the first term ut 0 of the optimal input sequence ut = [ut 0 · · · ut N(t)−1] ∈ Rm×N(t) is applied to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Input constraints come from the physical limitations and working principles of the pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' A pump can not provide water in the opposite direction and it can deliver a maxi- mum amount of water per unit of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' These conditions are expressed as U = {[u1, · · · um] ∈ Rm | 0 ≤ u1 ≤ u1, · · · 0 ≤ um ≤ um} (6) where u1 · · · um are upper flow limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Tank levels are also constrained so that there is always enough water in the tanks in case of an emergency and there is no overflow of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The set H can be defined as H = {[h1, · · · h2] ∈ Rn | ˜h1 ≤ h1 ≤ h1, · · · ˜hn ≤ hn ≤ hn} (7) The cost function includes the electricity costs of the pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The power provided to the network by the pump i is equal to qpi(pout i − pin i ), where qpi is the pump flow, pi out and pi in are the outlet and inlet pressures of the pump i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The inlet pressures pin = [pin 1 , pin 2 ] are the pressures of the related reservoirs and are assumed to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The outlet pressures pout = [pout 1 , pout 2 ] are given as the output of the linear model pout(t) = Aph(t) + Bpu(t) (8) where Ap and Bp are found using system identification on data generated by the EPANET model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Electricity cost at time t is then found by multiplying total power consumption u(t)T (pout(t) − pin(t)) with the electricity price c(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' We acknowledge a certain degree of model-plant mismatch by using a linear model (4) to represent the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' This causes actual states h(t) to be different than the predicted states ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' We know that the predicted states satisfy the state constraints (7) since they are the solution to the optimization problem 5, but the actual states might violate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' To ensure the satisfaction of the state constraints with the model-plant mismatch, we introduce new terms to the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' First, we rewrite state constraints (7) as Ci(h) ≤ 0, i = 0, 1, · · · 2 × n − 1 (9) where C0(h) = ˜h1 −h1 and the rest of the Ci functions are chosen in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The cost function terms are then defined as Jhi(h) = eai(Ci(h)+bi) i = 0, 1, · · · 2 × n − 1 (10) where ai, bi ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' This can be seen as an exponential barrier function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The parameters ai, bi determine a danger- ous region close to the boundaries of the state constraints where cost function Jhi attains high values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The predicted optimal state trajectories ht do not enter the dangerous region if possible because of the high cost values in the dangerous region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Then, the actual states h(t) do not violate the state constraints (7) assuming the difference between the predicted state and the actual state is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' If the state trajectory enters one of the dangerous regions at any step due to the model-plant mismatch, then the cost function will try to drive the trajectory out of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' ∆t N(t)∆t N(t + ∆t)∆t h(t) h(t + ∆t) ht 1 ut 0 ut+∆t 0 ht+∆t ht Br(h∗ Tday/∆t) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Predicted state trajectories ht, ht+∆t at times t, t + ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Sampling time ∆t, prediction horizons N(t), N(t + ∆t) and the applied inputs ut 0, ut+∆t 0 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The true state h(t + ∆t) and the predicted state ht 1 are indicated to emphasize the deviation from the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The terminal set Br(h∗ Tday/∆t) is also illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The overall cost function includes both the electricity expense term and the constraint barrier functions and it can be expressed as J(h(t), u(t)) = c(t)u(t)T (pout(t)−pin(t))+ 2×n−1 � i=0 Jhi(h(t)) (11) Both electricity price c(t) and total water demand da(t) signals can be viewed as consisting of a periodic signal with a period of 1 day and a relatively small deviation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' This can be leveraged to find a feasible controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Suppose a sequence of inputs can be found for some initial tank levels such that levels after 1 day are equal to initial levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In that case, the problem after 1 day is the same as in the beginning assuming deviation signals of the electricity price and the demand are zero, hence they are periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Then, the input sequence from the previous day could be applied and produce the same path for tank levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Taking into account the deviation signals and supposing that a solution exists such that levels after 1 day are close to initial levels, the input sequence from the previous day could be a good point of start to search for a feasible solution if the map from the initial conditions and the demand profile to the optimal input sequences is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Therefore, we choose a terminal state constraint for the end of each day to increase the chance of finding a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Now, the remaining problem is to decide which tank levels should the trajectories turn back to at the end of each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' We define the optimal periodic trajectory of the system as the solution of (u∗, h∗) = arg min ui,hi (Tday/∆t)−1 � i=0 J(hi, ui) (12a) hi = Adhi−1 + Bd1ui−1 + Bd2d∗ a(i − 1) (12b) ui ∈ U ⊆ Rm (12c) hi ∈ H ⊆ Rn (12d) h0 = hTday/∆t (12e) where Tday is the duration of a whole day, d∗ a is the average daily demand profile obtained from the past measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The resulting state trajectory h∗ = [h∗ 0 · · · h∗ Tday/∆t] ∈ Rn×(Tday/∆t+1) is the optimal periodic trajectory because of the constraint (12e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The terminal set Htf and the prediction horizon N(t) is chosen to make tank levels at the end of each day close to h∗ Tday/∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' At High Zone Low Zone Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Water Distribution Network of Randers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The pump- ing stations to be controlled are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Tanks are shown with a ’T’ shaped symbol in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' any time t, t + N(t)∆t should be equal to the end of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Htf and N(t) could be written as Htf = Br(h∗ Tday/∆t) (13a) N(t) = (Tday − t mod Tday)/∆t (13b) where Br(h∗ Tday/∆t) is the open ball centered at h∗ Tday/∆t with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Note that N(t) changes so that t + N(t)∆t is the end of the day for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' With these definitions, the condition (13a) will translate to tank levels at the end of the day being close to the final point in optimal periodic trajectory h∗ Tday/∆t as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Therefore, not only chance of finding a feasible solution is increased but also the solutions are kept around the optimal periodic trajectory h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' If the problem (5) becomes infeasible at any time step t, we apply the second term of the input sequence from the previous step ut−∆t 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The reason behind this choice is as follows: If we apply the optimal control input ut−∆t 0 to the network model (4) at time t − ∆t, then the optimal sequence in the next time step will be ut = [ut−∆t 1 · · ut−∆t N(t−∆t)−1] following Bellman’s principle of optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Then, at time t, ut−∆t 1 will be applied to the system as calculated at t − ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Assuming the model- plant mismatch is small enough, ut−∆t 1 is still a good input candidate if the problem is infeasible at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' APPLICATION The presented method is applied to WDN of Randers, a Danish city, which is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The network con- sists of 4549 nodes and 4905 links connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' There are 8 pumping stations in the network, 6 of which are shown in the figure whereas the other 2 are stationed where tanks are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The goal is to derive the schedules for 2 of the pumping stations while other pumps are already working according to some predetermined strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The stations to be controlled are shown in red in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Their task is to deliver water mostly to the High Zone (HZ) and Low Zone (LZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' However, connections exist between HZ-LZ and the rest of the city, so we can not think of the system as composed of isolated networks entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' There are also 3 tanks in the HZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' While 2 of them are directly connected via pipes, the third one stands alone as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The overall structure of the Randers WDN with tanks and pumps to be controlled are given in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' There are 3 water tanks in the network, 2 of which have been connected Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Structure of the WDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' with a pipe directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The tank level changes can be written by applying the mass conservation law (3) to the tanks in Figure 3 as A1 ˙h1 = q1down + q1up + qinter (14a) = f1(h1, h2, h3, qp1, qp2, d), A2 ˙h2 = q2down + q2up − qinter (14b) = f2(h1, h2, h3, qp1, qp2, d), A3 ˙h3 = q3 = f3(h1, h2, h3, qp1, qp2, d), (14c) where d is the vector containing the demands of all the nodes, qp1, qp2 are the pump flows, A1, A2, A3 are the cross sectional areas of the tanks and f1, f2, f3 are nonlinear flow functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Water levels at the two connected tanks are almost equal h1 ≈ h2 all the time since the pipe connecting respective tanks is big enough to oppose the water flows coming from neighbor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' That enables us to consider h1, h2 together as (A1 + A2)˙h1,2 ≈ q1down + q2down + qup = f1 + f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (15) We have used the EPANET model of the network to generate the data required for approximating f1 + f2 and f3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The model is simulated with various tank level initial conditions and flow rates of 2 pumping stations to be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The control laws for the remaining pumping stations are already defined in the EPANET model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Then, the linear model (4) is fitted to simulation data using least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The state variables for the model are h(t) = [h1,2(t), h3(t)] ∈ R2 and the inputs are u(t) = [qp1(t), qp2(t)] ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The total demand of High and Low Zone is used as aggregated demand da in the model since mainly those areas are supplied by the controlled pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='1 Simulation Results The proposed control method is tested on EPANET model of Randers water network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Epanet-Matlab toolkit Eliades et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2016) is used to set the flow of the 2 pumps at each time step and simulate the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The remaining pumps are controlled with rule-based control laws that are previously defined on EPANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The parameters of exponential barrier functions Jhi are chosen as ai = 80, bi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='3 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' It is assumed that the electricity prices are known in advance during the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Tank levels h1, h2 have a maximum value of 3m while h3 has 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='8m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Tanks are required to be at least half full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Maximum pump flows are set to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Sampling time ∆t is set to 1 hour in the experiments, so the control input is recalculated at each hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' We assume that total demand da(t) of HZ and LZ can be estimated up to 1 day from available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Although we do not have historical qup qiup q2up h1 h2 h3 qinter q1down 2down q3 Pump 1 Pump 2 9p1 qp2data on the demand, we imitate this behaviour by using a slightly perturbed version of the real demand used in EPANET simulation during MPC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The perturbations are adapted from a real demand data set of a small Danish facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Normalized difference between the average demand and the demand of a random day in data set is added to EPANET demand to replicate estimated demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In each experiment a different day from the data set is used, so the assumed estimated demand is different each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The simulation results when the presented method is applied to the EPANET model are given in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The initial tank levels are equal to h∗ Tday/∆t in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The top plot shows the evolution of tank levels along with the upper and lower thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' It is seen that the thresholds are not violated and moreover tank levels are not getting too close to them, which was the idea behind exponential barrier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Both the real demand and the assumed estimated demand of HZ and LZ are in the figure below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Total applied pump flows and electricity prices are in the following figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The expected result is pump flows being higher when electricity prices are low, and lower when they are high, which seems to be the case as can be seen in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Pump flows drop significantly when prices are at the peak and they reach their highest value at the end of the day when prices are low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' A more aggressive controller can be obtained by picking a smaller bi value for barrier functions at the expense of risking constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In Figure 5, the tank level simulation results and control inputs for different initial conditions and different assumed estimated demands are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The electricity price profile is the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' It is seen that the algorithm is able to control the network on various cases while satisfying the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Regardless of initial tank levels, the pumping profiles have a similar profile: high pump flows close to midnight and in the middle of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The only exception is the bottom plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In the beginning, prices are low but pump flows are not high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' This can be attributed to water levels h1, h2 being close to the upper thresholds and water demand being low in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The assumption that the optimal input sequences U(t) would not diverge a lot from the one found in previous step U(t − ∆t) is the reason we apply ut−∆t 1 at time t if the problem (5) is infeasible at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' This assumption is tested with initial conditions h1,2,3 = h∗ Tday/∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' In figure 6, total pump flow [1, 1]T ut i, i = 0 · · · N(t) − 1 of the found optimal input sequences U(t), t = 0, ∆t · · · Tday−∆t, except when the problem were infeasible, are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' It can be seen that ut−∆t 1 is close to the ut 0 for all t, which shows that our assumption is valid at least for this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Finally, the ability of the algorithm to decrease economic costs is tested with various initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' For each case, a demand follower pumping strategy is used as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The flow of the 2 pumps is adjusted with trial and error for each demand follower such that the total flow of the 2 pumps is equal to water demand at each time step and tank levels satisfy the terminal constraint (13a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The demand follower is a natural candidate to be a benchmark method since providing as much water as demand is an intuitive idea and the constraints in (5) can be satisfied (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Sample simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (a) evolution of tank levels through 1 day with upper and lower level thresholds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (b) real total demand of HZ and LZ used in EPANET simulation and the demand used in MPC calculations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (c) total flow provided by the 2 pumps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (d) electricity price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Proposed Method Demand Follower 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5967 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5745 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5826 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5558 1 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Relative economic costs of the pro- posed method and demand follower strategy for various demand profiles with manual adjustments of pump flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The economic costs are presented relatively in Table 1 As it is seen, the proposed algorithm saves between 40% and 45% of the cost with different demand profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' CONCLUSION We have presented a predictive control algorithm with a periodic horizon for WDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The aim is to minimize the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5 h1 h2 upper threshold 3 h3 3 upper threshold Tank Levels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5 1 0 5 10 15 20 25140 120 100 80 60 40 Real Demand Known Demand 20 0 5 10 15 20 25200 Total Pump Flow 150 100 50 0 0 5 10 15 20 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='2 Price 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='8 lectricity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='4 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='2 0 0 5 10 15 20 25 HoursFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Tank levels and pump flows for different initial conditions Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Evolution of found input sequences U(t) through 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' It can be seen that the solutions remain close to the initial optimal sequence U(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' economic cost and satisfy the operational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' A linear model is used to represent Randers WDN to increase the chance of finding a solution to the problem (5) at expense of a model-plant mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Periodic horizon is introduced to the predictive control formulation to keep the resulting state trajectories around the optimal periodic trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Barrier functions are used to prevent constraint violation since there is a model-plant mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' The presented algorithm is tested on Randers WDN using EPANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' It is shown in various situations that the method is able to find an economic solution where pump flows are adjusted according to electricity prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Also, it is shown that the system trajectories do not enter dangerous zones introduced by barrier functions as long as the predicted demand and the actual demand are somewhat close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' As future work, we plan to work on theoretical guarantees of the existence of solutions to the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Also, the robustness of periodic horizon control of periodical systems with barrier functions will be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' REFERENCES Abdelsalam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=', Wu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=', Simpson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=', Weyer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=', and Manzie, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' Minimizing pumping energy cost in real-time operations of water distribution sys- tems using economic model predictive control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' ArXiv, abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='07477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content=' 200 Total Pump Flow 150 100 50 0 0 5 10 15 20 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5 h1 h2 upperthreshold h3 3 upper threshold Tank Levels 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5 1 0 5 10 15 20 25250 Total Pump Flow 200 150 100 50 0 0 5 10 15 20 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5 h1 h2 upper threshold 3 h3 upper threshold Tank Levels 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5 1 0 5 10 15 20 25200 Total Pump Flow 150 100 50 0 0 5 10 15 20 25200 U(0) 150 Flow 100 U(1) 50 0 0 5 10 15 20 25 Hours3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FRT4oBgHgl3EQflTce/content/2301.13598v1.pdf'} +page_content='5 h1 h2 upperthreshold 3 h3 3 upper threshold Tank Levels 2.' metadata={'source': 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b/2tFRT4oBgHgl3EQfnjcF/content/tmp_files/2301.13605v1.pdf.txt @@ -0,0 +1,1888 @@ +IMSc/2023/02 +Aspects of the map from Exact RG to Holographic RG in +AdS and dS +Pavan Dharanipragada ∗1, 2, Semanti Dutta †3, and B. Sathiapalan ‡1,2 +1Institute of Mathematical Sciences,CIT Campus, Tharamani, Chennai +600113, India +2Homi Bhabha National Institute, Training School Complex, Anushakti +Nagar, Mumbai 400085, India +3Centre for High Energy Physics, Indian Institute of Science, C.V. Raman +Avenue, Bangalore 560012, India +February 1, 2023 +Abstract +In earlier work the evolution operator for the exact RG equation was mapped to a +field theory in Euclidean AdS. This gives a simple way of understanding AdS/CFT. We +explore aspects of this map by studying a simple example of a Schroedinger equation for +a free particle with time dependent mass. This is an analytic continuation of an ERG +like equation. We show for instance that it can be mapped to a harmonic oscillator. We +show that the same techniques can lead to an understanding of dS/CFT too. +Contents +1 +Introduction +3 +2 +Mapping Free Particle with Time Dependent Mass to a Harmonic Oscillator +3 +2.1 +Mapping Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.1.1 +Lorentzian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.1.2 +Euclidean Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.2 +Mapping Schrodinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.2.1 +Lorentzian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.2.2 +Euclidean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.2.3 +Analytic Continuation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.3 +Semiclassical Treatment +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.3.1 +Using Harmonic Oscillator Formulation . . . . . . . . . . . . . . . . . . . +8 +2.3.2 +Using ERG formulation +. . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +∗pavand@imsc.res.in +†semantidutta@iisc.ac.in +‡bala@imsc.res.in +1 +arXiv:2301.13605v1 [hep-th] 31 Jan 2023 + +3 +ERG to field theory in dS +10 +3.1 +Analytic Continuation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +3.1.1 +Analytic Continuation of the Action +. . . . . . . . . . . . . . . . . . . . +10 +3.2 +Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +3.2.1 +Mapping from Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . +11 +3.2.2 +Mapping from ERG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +3.3 +Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +3.4 +dS-CFT correspondence +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +4 +Obtaining Bulk field from ERG +16 +5 +Summary and Conclusions +20 +2 + +1 +Introduction +It has been recognized from the early days of the AdS/CFT correspondence [1, 2, 3, 4] that +the radial coordinate of the AdS space behaves like a scale for the boundary field theory. This +observation follows directly from the form of the AdS metric in Poincare coordinates: +ds2 = R2dz2 + dxµdxµ +z2 +(1.1) +This leads naturally to the idea of the “Holographic” renormalization group: If the AdS/CFT +conjecture is correct then radial evolution in the bulk must correspond to RG evolution in the +boundary theory [[9]-[25]]. +In [5, 6, 7] a mathematically precise connection was made between the exact RG (ERG) +equation of a boundary theory and holographic RG equations of a bulk theory in Euclidean +AdS (EAdS) space. It was shown that the ERG evolution operator of the boundary theory +can be mapped by a field redefinition to a functional integral of a field theory in the bulk +AdS space. This guarantees the existence of an EAdS bulk dual of a boundary CFT without +invoking the AdS/CFT conjecture 1 +Given that the crucial ingredient in this connection with ERG is the form of the metric +(1.1) with the factor z2 in the denominator, one is naturally led to ask if similar mappings can +be done for the dS metric +ds2 = L2−dη2 + dxµdxµ +η2 +(1.2) +It too has a scaling form. The difference is that the scale is a time like coordinate - so RG +evolution seems to be related to a real time evolution. In fact this metric is related to the +EAdS metric by an analytic continuation: iη = z, iL = R. Thus real time evolution should be +related to RG evolution by analytic continuation. These points have been discussed in many +of the early papers on de Sitter holography [[30]-[43]], (see also [44] for more recent work and +further references.) +This paper is an attempt to address the question of whether the mapping of [5] can be +generalised to include for instance dS-CFT. One is also led to explore other kinds of mapping +in an effort to understand the nature of this map better. In [5] the map was first introduced in +the case of 0-dimensional field theory in the boundary, which gave a one dimensional bulk field +theory or equivalently a point particle quantum mechanical system. In this paper therefore we +start by exploring maps for point particle quantum mechanical systems. In Section 2 we show +that the dynamics of a free particle with a time dependent mass can be mapped to a harmonic +oscillator. The Euclidean version of this is relevant for the ERG equation. In Section 3 the case +of mapping a field theory ERG equation to de Sitter space is considered by starting with the +analytically continued form. This complements the discussion of earlier papers where dS-CFT +is described as an analytic continuation of EAdS-CFT. In Section 4 we give some examples +of two point functions obtained using the techniques of [5] being analytically continued to dS +space. Section 5 contains a summary and conclusions. +2 +Mapping Free Particle with Time Dependent Mass to +a Harmonic Oscillator +In this section we reconsider the construction of [5] where the action for a free field theory +in D + 1 dimension with a non standard kinetic term was mapped to a free field in AdSD+1. +1There is still the open question of the locality properties of interaction terms in this bulk field theory. For +the case of the O(N) model some aspects of this issue were discussed in [7]. +3 + +When D = 0 this is just a particle: we will map a free particle with time dependent mass to a +harmonic oscillator. +2.1 +Mapping Actions +2.1.1 +Lorentzian Case +Consider the following action. It defines an evolution operator for free particle (with time +dependent mass) wave function. +S = 1 +2 +� tf +ti +dt M(t) ˙x2 +(2.3) +Ψ(x,t) = +� +dxi +� +x(ti) += +xi +x(t) += +x +Dx ei 1 +2 +� t +ti M(t′) ˙x2dt′Ψ(xi, ti) +(2.4) +Let x(t) = f(t)y(t) with f 2(t) = +1 +M(t). Substitute this in (2.3). +S = 1 +2 +� +dt ( ˙y2 + ( +˙f +f )2y2 + 2 +˙f +f ˙yy) += 1 +2 +� +dt [ ˙y2 + (d ln f +dt )2y2 − ( d2 +dt2 ln f)y2] + 1 +2 +� +dt d +dt(d ln f +dt y2) +Thus, upto the boundary term, the action is +S = 1 +2 +� +dt [ ˙y2 + eln f( d2 +dt2e− ln f)y2] +(2.5) +Now choose +eln f( d2 +dt2e− ln f) = −ω2 +0 +(2.6) +and we get +¯S = 1 +2 +� +dt [ ˙y2 − ω2 +0y2] +(2.7) +which is the action for a harmonic oscillator. And we define ¯Ψ by absorbing the contribution +from the boundary term: +e− 1 +2 i d ln f(t) +dt +y2(t)Ψ(f(t)y, t) +� +�� +� +¯Ψ(y,t) += +� +dyi +� +y(ti) += +yi +y(t) += +y +Dy ei 1 +2 +� t +ti[ ˙y2−ω2 +0y2]dt′ e− 1 +2 i d ln f(ti) +dt +y2(ti)Ψ(f(ti)yi, ti) +� +�� +� +¯Ψ(yi,ti) +(2.8) +¯S thus defines an evolution operator for the harmonic oscillator wave function ¯Ψ. f satisfies +d2 +dt2 +1 +f = −ω2 +0 +1 +f +(2.9) +y obeys the same equation. +Thus we can take +1 +f = a cos ω0(t − t0) +(2.10) +4 + +which requires +M(t) = a2cos2ω0(t − t0) +Note that one can do more general cases if one is willing to reparametrize time [26, 27]. +Thus let +dτ = +dt +Mf 2 +(2.11) +Then one gets (2.7), (2.9) and (2.10) with τ replacing t. In terms of t, (2.9) becomes +d +dt(M ˙f) = +ω2 +0 +Mf 3 +(2.12) +Very interestingly, as pointed out in [26], it is clear from (2.7) that the energy of the +harmonic oscillator given by +E = 1 +2( ˙y2 + ω2 +0y2) +is a conerved quantity. In terms of the original variables this is +E = 1 +2(( ˙xf − x ˙f +f 2 +)2 + ω2 +0(x +f )2) +These are known as Ermakov-Lewis invariants - see [26] for references to the literature on these +invariants - and we see a nice interpretation for them. +2.1.2 +Euclidean Case +In the Euclidean case the functional integral is +Ψ(x,τ) = +� +dxi +� +x(τi) += +xi +x(τ) += +x +Dx e− 1 +2 +� τ +τi M(τ ′) ˙x2dτ ′Ψ(xi, τi) +(2.13) +Ψ in this case is not a wave function. It was shown in [5] that the evolution operator for +a D-dimensional Euclidean field theory is of this form if we take ME(τ) = − +1 +˙G(τ) and D = 0. +In this case Ψ can be taken to be e−H[xi,τi] where H is a Hamiltonian or Euclideanized action. +Alternatively (depending on what ME(τ) is) it can also be eW[J] - a generating functional or +partition function. +Setting x = fy with f 2 = +1 +ME(τ), one goes through the same manipulations but replacing +(2.6) by +eln f( d2 +dτ 2e− ln f) = +ω2 +0 +(2.14) +and (2.7),(2.8) and (2.9) are replaced by +¯S = 1 +2 +� +dτ [ ˙y2 + ω2 +0y2] +(2.15) +¯Ψ(y, τ) = +� +dyi +� +y(τi) += +yi +y(τ) += +y +Dy e− 1 +2 +� τ +τi[ ˙y2+ω2 +0y2]dτ ′ ¯Ψ(yi, τi) +(2.16) +and +d2 +dτ 2 +1 +f = ω2 +0 +1 +f +(2.17) +5 + +The solutions are of the form +f = A sech ω0(τ − τ0) +(2.18) +which means ME(τ) = +1 +A2cosh2ω0(τ − τ0). +(2.16) has a τ independent action. In this case there are well known physical interpretations +for the Euclidean theory. The evolution operator, K(y, τ; yi, 0), where +K(y, τ; yi, 0) = +� +y(0) += +yi +y(τ) += +y +Dy e− 1 +2 +� τ +0 [ ˙y2+ω2 +0y2]dτ ′ +(2.19) +is the density operator of a QM harmonic oscillator in equilibrium at temperature specified by +β = τ. +Less well known is that the evolution operator of the Fokker-Planck equation in stochastic +quantization can be written in the form given in (2.16). ¯Ψ is then related to the probability +function (see, for instance, [29] for a nice discussion). +In the next section we discuss the mappings directly for the Schroedinger equation, rather +than its evolution operator. +2.2 +Mapping Schrodinger Equations +2.2.1 +Lorentzian +Let us consider the same mapping from the point of view of the Schroedinger equation for the +free particle wave function. +Schrodinger’s equation for the free particle is +i∂Ψ(x, t) +∂t += − +1 +2M(t) +∂2Ψ(x, t) +∂x2 +(2.20) +Ψ given by (2.4) obeys this equation. +We make a coordinate transformation and a wave function redefinition. Both can be un- +derstood as canonical transformations [28]. +Let x = f(t)y with f 2 = +1 +M(t). We take f, M to be dimensionless. We treat this as a 0 + 1 +dimensional field theory where x has the canonical dimension of − 1 +2. So x = L +1 +2X would define +a dimensionless X. L is some length scale. +∂Ψ(x, t) +∂t += ∂Ψ(f(t)y, t) +∂t +− +˙fy +f +∂Ψ(f(t)y, t) +∂y +Let +Ψ(f(t)y, t) = e− 1 +2 αy2 ¯Ψ(y, t) +∂Ψ +∂t = e− 1 +2 αy2(−1 +2 ˙αy2 + ∂ +∂t)¯Ψ(y, t) +−i +˙fy +f +∂Ψ(f(t)y, t) +∂y += ie− 1 +2 αy2(α +˙f +f y2 − +˙f +f y ∂ +∂y)¯Ψ(y, t) +1 +M +1 +2 +∂2 +∂x2Ψ = 1 +2 +∂2 +∂y2e− 1 +2 αy2 ¯Ψ = (1 +2e− 1 +2 αy2(α2y2 − 2αy ∂ +∂y − α + ∂2 +∂y2)¯Ψ) +Collecting all the terms one finds that (2.20) becomes: +i∂ ¯Ψ +∂t = (1 +2i ˙α − iα +˙f +f − 1 +2α2)y2 ¯Ψ + (i +˙f +f y ∂ +∂y + αy ∂ +∂y)¯Ψ + 1 +2αΨ − 1 +2 +∂2 +∂y2 ¯Ψ +(2.21) +6 + +We choose α = −i +˙f +f to get rid of the second term on the RHS. We get +i∂ ¯Ψ +∂t = [(1 +2 +d2 ln f +dt2 +− 1 +2(d ln f +dt )2)y2 + 1 +2α − 1 +2 +∂2 +∂y2]¯Ψ +As before it can be rewritten as +i∂ ¯Ψ +∂t = 1 +2[−eln f( d2 +dt2e− ln f)y2 − ∂2 +∂y2 + α]¯Ψ +(2.22) +Set +d2 +dt2 +1 +f = −ω2 +0 +1 +f +again as before to get +i∂ ¯Ψ +∂t = 1 +2[− ∂2 +∂y2 + ω2 +0y2 + α]¯Ψ +(2.23) +The term 1 +2α generates a scale transformation e− 1 +2 ln f(t) +f(ti) for ¯Ψ. +2.2.2 +Euclidean +The Euclidean version is +∂Ψ(x, τ) +∂τ += +1 +2ME(τ) +∂2Ψ(x, τ) +∂x2 +(2.24) +As mentioned above, this is of the form of a Polchinski ERG equation (with +1 +2ME(τ) = − ˙G(τ)) +for H defined by Ψ ≡ e−H. Going through the same steps one finds, with f 2 = +1 +ME(τ), +∂ ¯Ψ +∂τ = (1 +2 ˙α − α +˙f +f + 1 +2α2)y2 ¯Ψ + ( +˙f +f y ∂ +∂y − αy ∂ +∂y)¯Ψ − 1 +2αΨ + 1 +2 +∂2 +∂y2 ¯Ψ +(2.25) +the condition α = +˙f +f and the equation becomes +∂ ¯Ψ +∂t = 1 +2[− eln f( d2 +dt2e− ln f) +� +�� +� += ω2 +0 +y2 + ∂2 +∂y2 − α]¯Ψ +(2.26) +Thus +∂ ¯Ψ +∂τ = 1 +2[ ∂2 +∂y2 − ω2 +0y2 − α]¯Ψ +(2.27) +And f obeys +d2 +dt2 +1 +f = ω2 +0 +1 +f +(2.28) +This is a Euclidean harmonic oscillator equation. +Various physical interpretations of this +equation were given in the last section. The term α in (2.27) provides a multiplicative scaling +e− 1 +2 +� t +ti dt′ ∂t′ ln f = ( f(ti) +f(t) ) +1 +2 of ¯Ψ. +2.2.3 +Analytic Continuation +If we set it = τ, (2.20) becomes (2.24) provided M(−iτ) = ME(τ). Similarly (2.23) becomes +(2.27). Note that in (2.23) α = −i +˙f +f . This analytically continues to +˙f +f as required. +7 + +2.3 +Semiclassical Treatment +Most of the AdS/CFT calculations invoke large N to do a semiclassical treatment of the bulk +theory- one can evaluate boundary Green’s function. The analysis in [5, 7] did this for the +ERG treatment - the evolution of the Wilson action/Generating functional were calculated. In +[32] a semiclassical treatment was used to obtain the ground state wave function in dS space. +For completeness we do the same for the simple systems discussed in this paper. This +illustrates the connection between ERG and dS. +2.3.1 +Using Harmonic Oscillator Formulation +Since +Ψ(x, t) = +� +dxi +� +x(ti) += +xi +x(t) += +x +Dx ei +� t +ti L(x(t′), ˙x(t′),t′)dt′Ψ(xi, ti) +(2.29) +solves Schroedinger’s equation. For the Harmonic Oscillator +L = 1 +2( ˙x2 − ω0x2) +(2.30) +for the Lorentzian version. +One can evaluate the path integral semiclassically by plugging in a classical solution with +some regular boundary condition. We choose x = 0 at t = −∞. The initial state wave function +is thus a delta function. Classical solution of the EOM is of the form +x(t) = ae−iω0t + a∗eiω0t +Since a should annihilate the vacuum state in the far past we would like the solution to look +like +x(t) → eiω0t +in order to ensure that we are in the ground state. +x(t) = xfe−iω0(tf−t) +(2.31) +At t = −∞ we assume that the solution vanishes. This is justified by an infinitesimal rotation +t → t + iϵt. Evaluated on this solution, the action becomes +Sclassical = 1 +2x(t) ˙x(t)| +tf +−∞ +We get +Sclassical = 1 +2iω0x2 +f +(2.32) +Plugging (2.31) into (2.29) we obtain +Ψ(xf) ≈ e− 1 +2 ω0x2 +f +(2.33) +If we repeat this for the free field in dS space we get the ground state wave functional [32]. +8 + +2.3.2 +Using ERG formulation +For the Euclidean version, we set it = τ and we write +Ψ(x, τ) = +� +dxi +� +x(τi) += +xi +x(τ) += +x +Dx e− +� τ +τi LE(x(τ ′), ˙x(τ ′),τ ′)dτ ′Ψ(xi, τi) +(2.34) +It is well known that if one does the semiclassical analysis for the Euclidean case with general +boundary condition one recovers the thermal density matrix. This is for the time independent +Hamiltonian - such as the harmonic oscillator. We will not do this here. Instead we proceed +directly to the ERG interpretation of the calculation. Here the Hamiltonian is time dependent. +In [5] the analysis given below was applied to W[J]. We repeat it here for the Wilson action. +Our starting action in this case is (Note ˙G < 0): +S = −1 +2 +� τf +τi +˙x2 +˙G +(2.35) +EOM is given by, +∂τ( ˙x +˙G +) = 0 +˙x +˙G += b =⇒ x = bG + c +We choose G so that it vanishes at τ = ∞ . +For the Euclidean Harmonic oscillator case G has then to be +G = − 1 +ω0 +(tanh ω(τ − τi) − 1) +Also x → 0 as τ → ∞. So c = 0. +x = bG +(2.36) +x(τ) = − b +ω0 +(tanh ω(τ − τi) − 1) +On shell +S = −1 +2 +� τf +τi +dτ +d +dτ (x ˙x +G ) += 1 +2(x(τf) − x(τi))b = 1 +2[x(τf)x(τf) +G(τf) +− x(τi)x(τi) +G(τi) +] +If we add this change to the initial Wilson action 1 +2 +x(τi)x(τi) +G(τi) +we get the final Wilson action +Hf = 1 +2 +x(τf)x(τf) +G(τf) +If, for instance, we are interested in evaluating H semiclassically at τ = τi. +x(τi) = b +ω0 +=⇒ b = x(τi)ω0 +x(τ) = −x(0)(tanh ω(τ − τi) − 1) +˙x(τ) = −x(0)ω0sech2ω0(τ − τi) +9 + +The classical action is +Sclassical = 1 +2ω0x(τi)2 +Thus since G(τi) = +1 +ω0, H evaluated semiclassically is: +H[x, τi] ≈ 1 +2ω0x(τi)2 +(2.37) +Then +Ψ = e−H[x,τi] = e−ω0x(τi)2 +which coincides with the ground state wave function of the harmonic oscillator. This is essen- +tially the Hartle Hawking prescription [45]. This also motivates the dS-CFT correspondence +statement [30, 31, 32] that ΨdS = ZCFT +This concludes the discussion of the mapping of ERG equation to a Euclidean harmonic +oscillator. In higher dimensions this gives free field theory in flat space. We now return to the +case of interest, namely dS space. +3 +ERG to field theory in dS +We first map the system to Euclidean AdS. Then analytically continue and obtain dS results. +Alternatively, one can analytically continue the ERG equation to the Schroedinger equation +(when D = 0 this is a free particle with a time dependent mass) and then map to de Sitter +space. This is all exactly as was done for the harmonic oscillator. +3.1 +Analytic Continuation +The EAdS metric in Poincare coordinates is +ds2 = R2[dxidxi + dz2 +z2 +] +(3.38) +The dS metric in Poincare coordinates is: +ds2 = L2[dxidxi − dη2 +η2 +] +(3.39) +The metrics are related by analytic continuation: +iη = z, +iL = R +3.1.1 +Analytic Continuation of the Action +The action generically is +S = −1 +2 +� +dD+1x√g[gµν∂µφ∂νφ + m2φ2] +(3.40) +10 + +de Sitter +In this case we write √−g since g is negative: g = −( L2 +η2 )D+1. Also g00 = − η2 +L2 +and gij = δij η2 +L2. +Thus +SdS = +� +dDx +� ∞ +0 +dη (L +η )D+1[ η2 +L2∂ηφ∂ηφ − η2 +L2∂iφ∂iφ − m2φ2] +(3.41) +In momentum space: +SdS = +� +dDp +(2π)D +� ∞ +0 +dη (L +η )D+1[ η2 +L2∂ηφ(p)∂ηφ(−p) − ( η2 +L2p2 + m2)φ(p)φ(−p)] +(3.42) +The functional integral description of the quantum mechanical evolution operator for the +wave functional of the fields in dS space-time is +¯Ψ[φ(p), t] = +� +dφi(p) +� +φ(p, ti) += +φi(p) +φ(p, t) += +φ(p) +Dφ(p, t) ei 1 +2 +� t +ti[ ˙φ(p,t′)2−ω2 +0φ(p,t′)2]dt′ ¯Ψ[φi(p), ti] +(3.43) +Euclidean Anti de Sitter +g = ( R2 +z2 )D+1. Also g00 = z2 +R2 and gij = δij z2 +R2. +SEAdS = +� +dDx +� ∞ +0 +dz (R +z )D+1[ z2 +R2∂zφ∂zφ + z2 +R2∂iφ∂iφ + m2φ2] +(3.44) +In momentum space +SEAdS = +� +dDp +(2π)D +� ∞ +0 +dz (R +z )D+1[ z2 +R2∂zφ(p)∂zφ(−p) + ( z2 +R2p2 + m2)φ(p)φ(−p)] +(3.45) +If we set iη = z and iL = R we see that the functional integral (3.43) becomes +¯Ψ[φ(p), t] = +� +dφi(p) +� +φ(p, ti) += +φi(p) +φ(p, t) += +φ(p) +Dφ(p, t) e− 1 +2 +� t +ti[ ˙φ(p,t′)2+ω2 +0φ(p,t′)2]dt′ ¯Ψ[φi(p), ti] (3.46) +In holograhic RG this is interpreted as a Euclidean functional integral giving the evolution in +the radial direction. ¯Ψ is to be interpreted as e−SI[φ(p),t] where SI is the Wilson action. It was +shown in [5] (see below) that this can be obtained by mapping an ERG evolution operator. +The dS functional integral (3.43) above is thus an analytically continued version of this. +3.2 +Mapping +3.2.1 +Mapping from Quantum Mechanics +Let us go back to Section (2.1) and consider the mapping from the Quantum Mechanics of a +free particle with time dependent mass. We think of it as a 0 + 1 dimensional field theory. +M(t) is taken to be dimensionless and x has canonical dimensions of − 1 +2. +S = 1 +2 +� +dt M(t) ˙x2 +(3.47) +(In the ERG version M(t) = 1 +˙G) +The path integral is +� +Dx eiS +(3.48) +11 + +As before x(t) = f(t)y(t) with f 2(t) = +1 +M(t). Substitute this in (3.47) and go through the +same steps to obtain: +S = 1 +2 +� +dt [ ˙y2 + eln f( d2 +dt2e− ln f)y2] +(3.49) +Now choose +eln f( d2 +dt2e− ln f) = −( η2 +L2p2 + m2) +(3.50) +where η = Le +t +L. to obtain SdS +SdS = 1 +2 +� +dt [ ˙y2 − ( η2 +L2p2 + m2)y2] += 1 +2 +� +dη (L +η )[ η2 +L2∂ηy∂ηy − ( η2 +L2p2 + m2)y2] +(3.51) +p, m here are just some parameters. When D > 0 they will stand for momentum and mass +of the field respectively. So starting from a free particle with time dependent mass we obtain +the free field action in de Sitter space dSD+1 with D = 0. +Schroedinger Equation: +i∂Ψ(x, t) +∂t += − +1 +2M(t) +∂2Ψ(x, t) +∂x2 +(3.52) +Using the same mapping as in Section (2.2.1), x = fy +Ψ(f(t)y, t) = e− 1 +2 αy2 ¯Ψ(y, t) +with α = −i +˙f +f one obtains +i∂ ¯Ψ +∂t = [(1 +2 +d2 ln f +dt2 +− 1 +2(d ln f +dt )2)y2 + 1 +2α − 1 +2 +∂2 +∂y2]¯Ψ +Using (3.50) this becomes +i η +L +∂ ¯Ψ +∂η = [−1 +2 +∂2 +∂y2 + 1 +2( η2 +L2p2 + m2)y2 + 1 +2α]¯Ψ +(3.53) +If we construct the Schroedinger equation corresponding to the action (3.51) one obtains +i η +L +∂ ¯Ψ +∂η = [−1 +2 +∂2 +∂y2 + 1 +2( η2 +L2p2 + m2)y2]¯Ψ +(3.54) +which barring the field independent term α is exactly the same as (3.53). This term as we +have seen provides an overall field independent scaling for all wave functions. It is a consequence +of the ordering ambiguity in going from classical to quantum treatment. (3.54) (or its extension +to D > 0) describes the quantum mechanical time evolution of the matter field wave functional +in de Sitter space. +12 + +3.2.2 +Mapping from ERG +Action +We now consider the Euclidean version of (3.47), which is the Polchinski ERG +equation. This is what was done in [5]. Thus we replace M(t) by − 1 +˙G. +S = −1 +2 +� +dτ ˙x2 +˙G +(3.55) +The path integral is ( ˙G < 0) +� +Dx e +1 +2 +� +dτ +˙x2 +˙G +(3.56) +which can be obtained from (3.52) by setting it = τ. We take z = Re +τ +R If we let iη = z, iL = +R, it = τ then this can be obtained from the corresponding Minkowski case. +As before x(τ) = f(τ)y(τ) with f 2(τ) = ˙G. Substitute this in (3.55) and go through the +same steps to obtain: +S = 1 +2 +� +dτ [ ˙y2 + eln f( d2 +dτ 2e− ln f)y2] +(3.57) +Now choose +eln f( d2 +dτ 2e− ln f) = ( z2 +R2p2 + m2) +(3.58) +where z = Re +τ +R. to obtain SEAdS +SEAdS = +� +dz (R +z )[ z2 +R2∂zy∂zy + ( z2 +R2p2 + m2)y2] +(3.59) +ERG Equation +By analogy with the Schroedinger equation we can see that (3.56) is the +evolution operator corresponding to the ERG equation +∂Ψ(x, τ) +∂τ += −1 +2 +˙G∂2Ψ(x, τ) +∂x2 +(3.60) +By the same series of transformations as in the de Sitter case, but using (3.58), one obtains +z +R +∂ ¯Ψ +∂z = [1 +2 +∂2 +∂y2 − ( z2 +R2p2 + m2)y2 − 1 +2α]¯Ψ +(3.61) +with α = +˙f +f generating an overall scale transformation for ¯Ψ. +In the ERG context ¯Ψ +represents eW[J] upto a quadratic term. This equation is the holographic RG equation in the +AdS/CFT correspondence for an elementary scalar field [5]. +3.3 +Connections +Let us summarize the various connections obtained above. +• We start with the quantum mechanics of a free particle having a time dependent mass. +The Schroedinger equation (SE) for this is (2.20). Analytical continuation of this equation +(generalized to higher dimensions) gives the Polchinski ERG equation (2.24). +• The free particle SE (2.20) can be mapped to a SE for a harmonic oscillator (2.23). The +ERG equation (2.24) can similarly be mapped to a Euclidean harmonic oscillator (2.27)- +analytically continued version of (2.23). +13 + +• The evolution operators for the above equations are defined in terms of path integrals +over some actions. The same mapping function f maps the corresponding actions to each +other. Thus the evolution operator for the free particle Schroedinger equation is given by +the action in (2.3) which is mapped to a harmonic oscillator action (2.7). The analytical +continuation of these are the Euclidean ERG evolution operator (2.13) mapped to a +harmonic oscillator Hamiltonian (2.16). These steps are summarized in the flow diagram +in Figure 1. +• The mapping function f was originally chosen in [5] to map the free particle ERG action +(3.55) to an action for free fields in EAdS0+1 given in (3.60). The analytical continuation +of this problem to real time gives us an action in dS0+1 (3.51). +• One can also repeat these steps for the corresponding “wave” equations. The Polchinski +ERG equation for eW[J] gets mapped to an equation in EAdS for eW[J] which is nothing but +the holographic RG equations. Analytically continuing this, the Schroedinger equation +for a wave functional is mapped to a Schroedinger equation for wave functionals of fields +in dS. +These are summarized in the figure below (Fig.2). The analytic continuation can be done +before the map with f is applied or after as shown in the figure. It can be done both for the +actions as well as for the equations. +ERG +Equation +Holographic RG: +Radial evolution +in EAdS +Schroedinger +Equation +Real time QM +evolution +In dS +Map “f” +Map “f” +Analytic +Continuation +Analytic +Continuation +ERG +Equation +Evolution equation +For Euclidean +Harmonic Oscillator +QM Schroedinger +Equation +Real time QM +Schroedinger +Equation for + Harmonic Oscillator +Map “f” +Map “f” +Analytic +Continuation +Analytic +Continuation +ERG +Equation +Holographic RG: +Radial evolution +in EAdS +Schroedinger +Equation +Real time QM +evolution +In dS +Map “f” +Map “f” +Analytic +Continuation +Analytic +Continuation +Euclidean Action +For ERG evolution +By Feynman +Path Integral +Euclidean action for +Harmonic Oscillator +Path Integral +Lorentzian +Action for QM +Evolution by +Path Integral +QM evolution: + Action for +Harmonic Oscillator +Path Integral +Map “f” +Map “f” +Analytic +Continuation +Analytic +Continuation +Flow of equations-Harmonic Oscillator +Flow of actions – Harmonic Oscillator +Figure 1: Mapping ERG to Harmonic Oscillator +3.4 +dS-CFT correspondence +The connections with ERG mentioned above should, if pursued, provide some insights into +dS-CFT correspondence. We restrict ourselves to some preliminary observations in this paper. +14 + +ERG +Equation +Holographic RG: +Radial evolution +in EAdS +Schroedinger +Equation +Real time QM +evolution +In dS +Map “f” +Map “f” +Analytic +Continuation +Analytic +Continuation +ERG +Equation +Holographic RG: +Radial evolution +equation in EAdS +QM Schroedinger +Equation +Real time QM +Schroedinger +equation +in dS +Map “f” +Map “f” +Analytic +Continuation +Analytic +Continuation +ERG +Equation +Holographic RG: +Radial evolution +in EAdS +Schroedinger +Equation +Real time QM +evolution +In dS +Map “f” +Map “f” +Analytic +Continuation +Analytic +Continuation +Euclidean Action +For ERG evolution +by functional integral +Holographic RG: + Action in EadS +for functional +integral +Lorentzian +Action for QM +evolution +QM evolution by +Functional integral: + Action in dS +Map “f” +Map “f” +Analytic +Continuation +Analytic +Continuation +Flow of equations +Flow of actions +Figure 2: Mapping ERG to Holographic RG +The idea of dS-CFT correspondence was suggested in [30, 31, 32]. This has been investigated +further by many authors, e.g. [33, 34, 38, 39, 35, 37, 36]. +What we see from the above analysis is that considering the relation between the evolution +equations, one can say that +Ψ[φ, J]wave−functional in dS = {Z[φ, J]CFT}analytically continued +(3.62) +Thus we see that the dS-CFT correspondence suggested by this analysis is one between an +ERG equation for a CFT generating functional and a real time quantum mechanical evolution +of a wave functional in dS space time. +The LHS of (3.62) is a QM wave functional of fields on a D-dimensional spatial slice of +a D + 1 dimensional dS spacetime. The RHS is the analytically continued partition function +of a D-dimensional Euclidean CFT - more precisely, either eWΛ[J] or e−SI,Λ[φ]. The precise +statement has to involve some statement of the boundary conditions. In the next section we +give a concrete example with boundary conditions specified. +Note that the LHS is a complex probability amplitude. Expectation values will involve Ψ∗Ψ +and were calculated first in [30, 31, 32]. +One can proceed to ask whether the expectations on the spatial slice calculated using Ψ∗Ψ +also correspond to some other Euclidean CFT on the spatial slice. This was explored further +in [38]. We do not address this question here. +In the next section we give some examples that explicitly illustrate the connection made by +(3.62). +15 + +4 +Obtaining Bulk field from ERG +The ERG formulation stated in this paper starts with the boundary fields. The evolution +operator for this involves bulk fields but with a non standard action. When this action is +mapped to EAdS action one can interpret the newly mapped field as the EAdS bulk field. This +analysis for Euclidean AdS is well defined and has been done in [5, 7]. However, this treatment +does not have a natural interpretation in the context in dS space. We have elaborated that in +this section. +Bulk scalar field in Euclidean AdS and dS +There are conceptual barriers if one tries to do similar analysis to map the ERG evolution +operator directly to Lorentzian dS. First of all, it is not clear as in EAdS whether the function +G(t) a.k.a f 2(t) = ˙G(t) is the Green’s function of the dual field theory of dS. It has an oscillatory +cutoff function. Therefore we analytically continue the ERG action to a Lorentzian action first, +and then do the mapping. +The result thus obtained (4.74) matches with the value found in [39] where the authors have +found the bulk field in semicalssical approximation from dS bulk action. For the Lorentzian dS +analysis presented here the RG interpretation is not clearly understood - except as an anlytic +continuation. We have presented it here for sake of completeness. +Euclidean AdS +The Euclidean action of the ERG evolution operator in momentum space, +S = −1 +2 +� +dτ +� +p +˙φ2 +˙G +(4.63) +is mapped to +SEAdS = +� +dDp +(2π)D +� ∞ +ϵEAdS +dz (R +z )d+1[ z2 +R2∂zyEAdS(p)∂zyEAdS(−p)+( z2 +R2p2+m2)yEAdS(p)yEAdS(−p)] +(4.64) +with z = Re +τ +R as described in [5]. We have rescaled the field as φ = fyEAdS where f is +related to the boundary Green’s function G as f 2 = − +� z +R +�−d ˙G. +The constraint on 1 +f is given by, +∂ +∂z{ +� z +R +�−d+1 ∂ +∂z +1 +f } = +� z +R +�−d+1 � +p2 + m2R2 +z2 +� 1 +f +(4.65) +The solutions are zd/2Kα(pz) and zd/2Iα(pz) where α2 = m2R2 + d2 +4 . +So 1 +f can be taken as, +1 +f(p, z) = (z)d/2 (AKα(pz) + BIα(pz)) +(4.66) +The Green’s function is +G(p, z) = CKα(pz) + DIα(pz) +AKα(pz) + BIα(pz) +(4.67) +The large argument asymptotic form of the Modified Bessel function Iα(z) and Kα(z) are +given by, +Iα(z) ∼ +ez +√ +2πz +� +1 + O(1 +z) +� +for |arg z| < π +2 +16 + +Kα(z) ∼ +� π +2ze−z +� +1 + O(1 +z) +� +for |arg z| < 3π +2 +Putting two constraints on G- i)G(pz → ∞) = 0 ii)G(pz → 0) = γEAdS p−2α, we get, +D = 0; C(p) = γEAdS p−α; B(p) = − +1 +γEAdS +pα +In semiclassical approximation the bulk field yEAdS = bEAdS +G +f . If yEAdS satisfies yEAdS +0 +the +bulk field is given by, +yEAdS = yEAdS +0 +zd/2 +ϵd/2 +Kα(pz) +Kα(pϵ) +(4.68) +Now let’s check by analytic continuation iη = z and iL = R. First of all, α becomes ν. ϵ +is replaced by iϵ. We get, +yEAdS|z=iη, R=iL = yEAdS +0 +|z=iη, R=iL +(iη)d/2 +(iϵ)d/2 +Kν(ipη) +Kν(ipϵ) +(4.69) +As, +yEAdS +0 += bEAdS ϵd/2 +EAdS +γEAdS Kα(pϵ) +pα +(4.70) +de Sitter +We would like to do the same analysis as above for the Lorentzian case. +The Lorentzian action obtained from (4.63) by analytic continuation, in momentum space, +S = − +� +dt +� +dDp +(2π)D +1 +2 ˙G(p) +˙φ(p) ˙φ(−p) +and needs to be mapped to += 1 +2 +� ∞ +ϵdS +dη +� +dDp +(2π)D +��L +η +�D−1 +{(∂ηydS)2 − p2ydS2 − m2L2 +η2 +ydS2} +� +Here η = Le +t +L. We do the field redefinition of boundary field +φ = fydS +f is a scale dependent quantity which is related to Green’s function G as f 2 = − +� η +L +�−D ˙G. +Performing the same manipulations as in [5], one can get the constraint on f as, +� η +L +�d−1 �� η +L +�−d+1 d +dη +�2 +e− ln f = +� η +L +�−d+1 � +−p2 − m2L2 +η2 +� +e− ln f +−d + 1 +η +∂ +∂η +1 +f + ∂2 +∂η2 +1 +f = +� +−p2 − m2L2 +η2 +� 1 +f +The solutions are +� η +L +�d/2 H(1) +ν (pη) and +� η +L +�d/2 H(2) +ν (pη) with ν2 = d2 +4 − m2L2. +The 1 +f can be written in general as( note f is dimensionless), +1 +f(p, η) = +� η +L +�d/2 � +AH(1) +ν (pη) + BH(2) +ν (pη) +� +(4.71) +17 + +and the Green’s function is 2 +G(pη) = CH(1) +ν (pη) + DH(2) +ν (pη) +AH(1) +ν (pη) + BH(2) +ν (pη) +Physically one can expect G(pη → ∞) = 0 which yields, +CH(1) +ν (pη) + DH(2) +ν (pη) = 0 +(4.72) +The asymptotic forms of Hankel functions of both kind for large arguments are, +H(1) +ν (z) ∼ +� +2 +πzei(z− νπ +2 − π +4 ) +− π < arg z < 2π +H(2) +ν (z) ∼ +� +2 +πze−i(z− νπ +2 − π +4 ) +− 2π < arg z < π +The presence of the oscillatory functions will not let eq.4.72 to be satisfied. +Hence we +analytically continue the argument of Green’s function G. The choice of direction of the analytic +continuation is based on the anticipation that the bulk field will have positive frequency. Hence +we take +η = −iz +(4.73) +which prompts us to make C = 0. Also, from the constraint AD − BC = 1 we get A = 1 +D. +Hence the Green’s function now takes the form, +G(pz) = +DH(2) +ν (ipz) +1 +DH(1) +ν (ipz) + BH(2) +ν (ipz) +Next another constraint will come from the fact that boundary Green’s function is γdS p−2ν. +So in the limit of z → 0 using the formulae, +H(1) +ν (z) = iYν(z); H(2) +ν (z) = −iYν(z); Yν(z) = −Γ(ν) +π +�2 +z +�ν +One can get, +−iD +i +D − iB = γdS p−2ν +On the other side, f should become a p independent constant at boundary x = 0 so that +it does not modify the boundary Green’s function, also ydS and f should become same field in +boundary field theory. This gives, +i +D − iB = pν +Finally we get, +D = iγdS p−ν ; B = i +� +1 − 1 +γdS +� +pν +The bulk field ydS is given by, +2We use the term Green function by analogy with the EAdS case, where G is the propagator of the boundary +CFT. Also see for instance [39]. +18 + +ydS = bdS +G +f = bdS(iγp−ν) 1 +Ld/2xd/2H(2) +ν (ipx) +If we analytically continue back to η we get, +ydS = bdS(iγp−ν) 1 +Ld/2(−iη)d/2H(2) +ν (pη) +If the field ydS satisfies ydS +0 +at η = ϵdS then, +ydS = ydS +0 +ηd/2 +ϵd/2 +dS +H(2) +ν (pη) +H(2) +ν (pϵdS) +(4.74) +ydS satisfies Bunch-Davies condition. +Relation between bulk fields in EAdS and dS +The bulk field in EAdS space is given +by, +yEAdS = yEAdS +0 +zd/2 +ϵd/2 +Kα(pz) +Kα(pϵ) +(4.75) +Let’s apply the analytic continuation continuation iη = z and iL = R. First of all, α becomes +ν. ϵ is replaced by iϵ. We get, +yEAdS|z=iη, R=iL = yEAdS +0 +|z=iη, R=iL +(iη)d/2 +(iϵ)d/2 +Kν(ipη) +Kν(ipϵ) +(4.76) +As, +yEAdS +0 += bEAdS ϵd/2 +EAdS +γEAdS Kα(pϵ) +pα +(4.77) +Using the relation between Kα(x) and Hα(x), +Kα(x) = π +2 iα+1H(1) +α (ix); − π < arg x ≤ π +2 += π +2 (−i)α+1H(2) +α (−ix); − π +2 < arg x ≤ π +(4.78) +Here also we want to ensure the bulk field to be of positive frequency, hence choosing +H(2)(x). +yEAdS +0 +|z=iη, R=iL = π +2 (i)d/2+α+1bEAdSϵd/2γEAdS +H(2) +α (pϵ) +pα += bEAdS +bdS +γEAdS +γdS +π +2 (i)d/2+α+1ydS +0 +Hence, +yEAdS|z=iη, R=iL =bEAdS +bdS +γEAdS +γdS +π +2 (i)d/2+α+1ydS +0 +ηd/2 +ϵd/2 +H(2) +α (pη) +H(2) +α (pϵ) += bEAdS +bdS +γEAdS +γdS +π +2 (i)d/2+α+1ydS +(4.79) +Upto various normalization constants we see that they agree. +19 + +5 +Summary and Conclusions +In [5, 6] an evolution operator for an ERG equation of a perturbed D-dimensional free field +theory in flat space was mapped to a field theory action in AdSD+1. Similar mappings were done +subsequently for the interacting O(N) model at both the free fixed point and at the Wilson- +Fisher fixed point [7]. The main aim of this paper was to understand better the mapping used +in these papers and to see if there are other examples. A related question was that of analytic +continuation of these theories. These questions can posed, both for the ERG equation and its +evolution operator. +It was shown that a mapping of this type can map the ERG evolution operator of a (zero- +dimensional) field theory to the action of a Euclidean harmonic oscillator. Furthermore the +analytic continuation of the ERG evolution operator action gives the path integral for a free +particle with a time dependent mass. A similar mapping takes this to a harmonic oscillator. +This method also gives new way of obtaining the Ermakov-Lewis invariants for the original +theory. +The analytically continued ERG equation is a Schroedinger like equation for a free field +theory wave functional. This gets mapped to the Schroedinger equation for a wave functional +of a free field theory in de Sitter space. These are summarized in Figures 1,2. This is one +version of the dS-CFT correspondence. From this point of view, the QM evolution of dS field +theory is also an ERG evolution of a field theory, but accompanied by an analytic continuation. +An example was worked out to illustrate this correspondence. +To understand these issues further it would be useful to apply these techniques to the O(N) +model ERG equation written in [7]. 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Danielsson, “Inflation, holography, and the choice of vacuum in de Sitter space,” +JHEP 07, 040 (2002) doi:10.1088/1126-6708/2002/07/040 [arXiv:hep-th/0205227 [hep- +th]]. +24 + diff --git a/2tFRT4oBgHgl3EQfnjcF/content/tmp_files/load_file.txt b/2tFRT4oBgHgl3EQfnjcF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..815d0e32274d59371aaa70d86d84f9b050365d41 --- /dev/null +++ b/2tFRT4oBgHgl3EQfnjcF/content/tmp_files/load_file.txt @@ -0,0 +1,1448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf,len=1447 +page_content='IMSc/2023/02 Aspects of the map from Exact RG to Holographic RG in AdS and dS Pavan Dharanipragada ∗1, 2, Semanti Dutta †3, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Sathiapalan ‡1,2 1Institute of Mathematical Sciences,CIT Campus, Tharamani, Chennai 600113, India 2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India 3Centre for High Energy Physics, Indian Institute of Science, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Raman Avenue, Bangalore 560012, India February 1, 2023 Abstract In earlier work the evolution operator for the exact RG equation was mapped to a field theory in Euclidean AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This gives a simple way of understanding AdS/CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We explore aspects of this map by studying a simple example of a Schroedinger equation for a free particle with time dependent mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This is an analytic continuation of an ERG like equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We show for instance that it can be mapped to a harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We show that the same techniques can lead to an understanding of dS/CFT too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Contents 1 Introduction 3 2 Mapping Free Particle with Time Dependent Mass to a Harmonic Oscillator 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1 Mapping Actions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='3 Analytic Continuation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='4 dS-CFT correspondence .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 14 4 Obtaining Bulk field from ERG 16 5 Summary and Conclusions 20 2 1 Introduction It has been recognized from the early days of the AdS/CFT correspondence [1, 2, 3, 4] that the radial coordinate of the AdS space behaves like a scale for the boundary field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This observation follows directly from the form of the AdS metric in Poincare coordinates: ds2 = R2dz2 + dxµdxµ z2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1) This leads naturally to the idea of the “Holographic” renormalization group: If the AdS/CFT conjecture is correct then radial evolution in the bulk must correspond to RG evolution in the boundary theory [[9]-[25]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In [5, 6, 7] a mathematically precise connection was made between the exact RG (ERG) equation of a boundary theory and holographic RG equations of a bulk theory in Euclidean AdS (EAdS) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' It was shown that the ERG evolution operator of the boundary theory can be mapped by a field redefinition to a functional integral of a field theory in the bulk AdS space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This guarantees the existence of an EAdS bulk dual of a boundary CFT without invoking the AdS/CFT conjecture 1 Given that the crucial ingredient in this connection with ERG is the form of the metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1) with the factor z2 in the denominator, one is naturally led to ask if similar mappings can be done for the dS metric ds2 = L2−dη2 + dxµdxµ η2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2) It too has a scaling form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The difference is that the scale is a time like coordinate - so RG evolution seems to be related to a real time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In fact this metric is related to the EAdS metric by an analytic continuation: iη = z, iL = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Thus real time evolution should be related to RG evolution by analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' These points have been discussed in many of the early papers on de Sitter holography [[30]-[43]], (see also [44] for more recent work and further references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=') This paper is an attempt to address the question of whether the mapping of [5] can be generalised to include for instance dS-CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' One is also led to explore other kinds of mapping in an effort to understand the nature of this map better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In [5] the map was first introduced in the case of 0-dimensional field theory in the boundary, which gave a one dimensional bulk field theory or equivalently a point particle quantum mechanical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In this paper therefore we start by exploring maps for point particle quantum mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In Section 2 we show that the dynamics of a free particle with a time dependent mass can be mapped to a harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The Euclidean version of this is relevant for the ERG equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In Section 3 the case of mapping a field theory ERG equation to de Sitter space is considered by starting with the analytically continued form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This complements the discussion of earlier papers where dS-CFT is described as an analytic continuation of EAdS-CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In Section 4 we give some examples of two point functions obtained using the techniques of [5] being analytically continued to dS space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Section 5 contains a summary and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 2 Mapping Free Particle with Time Dependent Mass to a Harmonic Oscillator In this section we reconsider the construction of [5] where the action for a free field theory in D + 1 dimension with a non standard kinetic term was mapped to a free field in AdSD+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 1There is still the open question of the locality properties of interaction terms in this bulk field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' For the case of the O(N) model some aspects of this issue were discussed in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 3 When D = 0 this is just a particle: we will map a free particle with time dependent mass to a harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1 Mapping Actions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1 Lorentzian Case Consider the following action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' It defines an evolution operator for free particle (with time dependent mass) wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' S = 1 2 � tf ti dt M(t) ˙x2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='3) Ψ(x,t) = � dxi � x(ti) = xi x(t) = x Dx ei 1 2 � t ti M(t′) ˙x2dt′Ψ(xi, ti) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='4) Let x(t) = f(t)y(t) with f 2(t) = 1 M(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Substitute this in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' S = 1 2 � dt ( ˙y2 + ( ˙f f )2y2 + 2 ˙f f ˙yy) = 1 2 � dt [ ˙y2 + (d ln f dt )2y2 − ( d2 dt2 ln f)y2] + 1 2 � dt d dt(d ln f dt y2) Thus, upto the boundary term, the action is S = 1 2 � dt [ ˙y2 + eln f( d2 dt2e− ln f)y2] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='5) Now choose eln f( d2 dt2e− ln f) = −ω2 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='6) and we get ¯S = 1 2 � dt [ ˙y2 − ω2 0y2] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='7) which is the action for a harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' And we define ¯Ψ by absorbing the contribution from the boundary term: e− 1 2 i d ln f(t) dt y2(t)Ψ(f(t)y, t) � �� � ¯Ψ(y,t) = � dyi � y(ti) = yi y(t) = y Dy ei 1 2 � t ti[ ˙y2−ω2 0y2]dt′ e− 1 2 i d ln f(ti) dt y2(ti)Ψ(f(ti)yi, ti) � �� � ¯Ψ(yi,ti) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='8) ¯S thus defines an evolution operator for the harmonic oscillator wave function ¯Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' f satisfies d2 dt2 1 f = −ω2 0 1 f (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='9) y obeys the same equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Thus we can take 1 f = a cos ω0(t − t0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='10) 4 which requires M(t) = a2cos2ω0(t − t0) Note that one can do more general cases if one is willing to reparametrize time [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Thus let dτ = dt Mf 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='11) Then one gets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='10) with τ replacing t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In terms of t, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='9) becomes d dt(M ˙f) = ω2 0 Mf 3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='12) Very interestingly, as pointed out in [26], it is clear from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='7) that the energy of the harmonic oscillator given by E = 1 2( ˙y2 + ω2 0y2) is a conerved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In terms of the original variables this is E = 1 2(( ˙xf − x ˙f f 2 )2 + ω2 0(x f )2) These are known as Ermakov-Lewis invariants - see [26] for references to the literature on these invariants - and we see a nice interpretation for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2 Euclidean Case In the Euclidean case the functional integral is Ψ(x,τ) = � dxi � x(τi) = xi x(τ) = x Dx e− 1 2 � τ τi M(τ ′) ˙x2dτ ′Ψ(xi, τi) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='13) Ψ in this case is not a wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' It was shown in [5] that the evolution operator for a D-dimensional Euclidean field theory is of this form if we take ME(τ) = − 1 ˙G(τ) and D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In this case Ψ can be taken to be e−H[xi,τi] where H is a Hamiltonian or Euclideanized action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Alternatively (depending on what ME(τ) is) it can also be eW[J] - a generating functional or partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Setting x = fy with f 2 = 1 ME(τ), one goes through the same manipulations but replacing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='6) by eln f( d2 dτ 2e− ln f) = +ω2 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='7),(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='9) are replaced by ¯S = 1 2 � dτ [ ˙y2 + ω2 0y2] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='15) ¯Ψ(y, τ) = � dyi � y(τi) = yi y(τ) = y Dy e− 1 2 � τ τi[ ˙y2+ω2 0y2]dτ ′ ¯Ψ(yi, τi) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='16) and d2 dτ 2 1 f = ω2 0 1 f (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='17) 5 The solutions are of the form f = A sech ω0(τ − τ0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='18) which means ME(τ) = 1 A2cosh2ω0(τ − τ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='16) has a τ independent action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In this case there are well known physical interpretations for the Euclidean theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The evolution operator, K(y, τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' yi, 0), where K(y, τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' yi, 0) = � y(0) = yi y(τ) = y Dy e− 1 2 � τ 0 [ ˙y2+ω2 0y2]dτ ′ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='19) is the density operator of a QM harmonic oscillator in equilibrium at temperature specified by β = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Less well known is that the evolution operator of the Fokker-Planck equation in stochastic quantization can be written in the form given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' ¯Ψ is then related to the probability function (see, for instance, [29] for a nice discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In the next section we discuss the mappings directly for the Schroedinger equation, rather than its evolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2 Mapping Schrodinger Equations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1 Lorentzian Let us consider the same mapping from the point of view of the Schroedinger equation for the free particle wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Schrodinger’s equation for the free particle is i∂Ψ(x, t) ∂t = − 1 2M(t) ∂2Ψ(x, t) ∂x2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='20) Ψ given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='4) obeys this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We make a coordinate transformation and a wave function redefinition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Both can be un- derstood as canonical transformations [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Let x = f(t)y with f 2 = 1 M(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We take f, M to be dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We treat this as a 0 + 1 dimensional field theory where x has the canonical dimension of − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' So x = L 1 2X would define a dimensionless X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' L is some length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' ∂Ψ(x, t) ∂t = ∂Ψ(f(t)y, t) ∂t − ˙fy f ∂Ψ(f(t)y, t) ∂y Let Ψ(f(t)y, t) = e− 1 2 αy2 ¯Ψ(y, t) ∂Ψ ∂t = e− 1 2 αy2(−1 2 ˙αy2 + ∂ ∂t)¯Ψ(y, t) −i ˙fy f ∂Ψ(f(t)y, t) ∂y = ie− 1 2 αy2(α ˙f f y2 − ˙f f y ∂ ∂y)¯Ψ(y, t) 1 M 1 2 ∂2 ∂x2Ψ = 1 2 ∂2 ∂y2e− 1 2 αy2 ¯Ψ = (1 2e− 1 2 αy2(α2y2 − 2αy ∂ ∂y − α + ∂2 ∂y2)¯Ψ) Collecting all the terms one finds that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='20) becomes: i∂ ¯Ψ ∂t = (1 2i ˙α − iα ˙f f − 1 2α2)y2 ¯Ψ + (i ˙f f y ∂ ∂y + αy ∂ ∂y)¯Ψ + 1 2αΨ − 1 2 ∂2 ∂y2 ¯Ψ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='21) 6 We choose α = −i ˙f f to get rid of the second term on the RHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We get i∂ ¯Ψ ∂t = [(1 2 d2 ln f dt2 − 1 2(d ln f dt )2)y2 + 1 2α − 1 2 ∂2 ∂y2]¯Ψ As before it can be rewritten as i∂ ¯Ψ ∂t = 1 2[−eln f( d2 dt2e− ln f)y2 − ∂2 ∂y2 + α]¯Ψ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='22) Set d2 dt2 1 f = −ω2 0 1 f again as before to get i∂ ¯Ψ ∂t = 1 2[− ∂2 ∂y2 + ω2 0y2 + α]¯Ψ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='23) The term 1 2α generates a scale transformation e− 1 2 ln f(t) f(ti) for ¯Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2 Euclidean The Euclidean version is ∂Ψ(x, τ) ∂τ = 1 2ME(τ) ∂2Ψ(x, τ) ∂x2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='24) As mentioned above, this is of the form of a Polchinski ERG equation (with 1 2ME(τ) = − ˙G(τ)) for H defined by Ψ ≡ e−H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Going through the same steps one finds, with f 2 = 1 ME(τ), ∂ ¯Ψ ∂τ = (1 2 ˙α − α ˙f f + 1 2α2)y2 ¯Ψ + ( ˙f f y ∂ ∂y − αy ∂ ∂y)¯Ψ − 1 2αΨ + 1 2 ∂2 ∂y2 ¯Ψ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='25) the condition α = ˙f f and the equation becomes ∂ ¯Ψ ∂t = 1 2[− eln f( d2 dt2e− ln f) � �� � = ω2 0 y2 + ∂2 ∂y2 − α]¯Ψ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='26) Thus ∂ ¯Ψ ∂τ = 1 2[ ∂2 ∂y2 − ω2 0y2 − α]¯Ψ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='27) And f obeys d2 dt2 1 f = ω2 0 1 f (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='28) This is a Euclidean harmonic oscillator equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Various physical interpretations of this equation were given in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The term α in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='27) provides a multiplicative scaling e− 1 2 � t ti dt′ ∂t′ ln f = ( f(ti) f(t) ) 1 2 of ¯Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='3 Analytic Continuation If we set it = τ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='20) becomes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='24) provided M(−iτ) = ME(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Similarly (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='23) becomes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Note that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='23) α = −i ˙f f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This analytically continues to ˙f f as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='3 Semiclassical Treatment Most of the AdS/CFT calculations invoke large N to do a semiclassical treatment of the bulk theory- one can evaluate boundary Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The analysis in [5, 7] did this for the ERG treatment - the evolution of the Wilson action/Generating functional were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In [32] a semiclassical treatment was used to obtain the ground state wave function in dS space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' For completeness we do the same for the simple systems discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This illustrates the connection between ERG and dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1 Using Harmonic Oscillator Formulation Since Ψ(x, t) = � dxi � x(ti) = xi x(t) = x Dx ei � t ti L(x(t′), ˙x(t′),t′)dt′Ψ(xi, ti) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='29) solves Schroedinger’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' For the Harmonic Oscillator L = 1 2( ˙x2 − ω0x2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='30) for the Lorentzian version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' One can evaluate the path integral semiclassically by plugging in a classical solution with some regular boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We choose x = 0 at t = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The initial state wave function is thus a delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Classical solution of the EOM is of the form x(t) = ae−iω0t + a∗eiω0t Since a should annihilate the vacuum state in the far past we would like the solution to look like x(t) → eiω0t in order to ensure that we are in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' x(t) = xfe−iω0(tf−t) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='31) At t = −∞ we assume that the solution vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This is justified by an infinitesimal rotation t → t + iϵt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Evaluated on this solution, the action becomes Sclassical = 1 2x(t) ˙x(t)| tf −∞ We get Sclassical = 1 2iω0x2 f (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='32) Plugging (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='31) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='29) we obtain Ψ(xf) ≈ e− 1 2 ω0x2 f (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='33) If we repeat this for the free field in dS space we get the ground state wave functional [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2 Using ERG formulation For the Euclidean version, we set it = τ and we write Ψ(x, τ) = � dxi � x(τi) = xi x(τ) = x Dx e− � τ τi LE(x(τ ′), ˙x(τ ′),τ ′)dτ ′Ψ(xi, τi) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='34) It is well known that if one does the semiclassical analysis for the Euclidean case with general boundary condition one recovers the thermal density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This is for the time independent Hamiltonian - such as the harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We will not do this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Instead we proceed directly to the ERG interpretation of the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Here the Hamiltonian is time dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In [5] the analysis given below was applied to W[J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We repeat it here for the Wilson action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Our starting action in this case is (Note ˙G < 0): S = −1 2 � τf τi ˙x2 ˙G (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='35) EOM is given by, ∂τ( ˙x ˙G ) = 0 ˙x ˙G = b =⇒ x = bG + c We choose G so that it vanishes at τ = ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' For the Euclidean Harmonic oscillator case G has then to be G = − 1 ω0 (tanh ω(τ − τi) − 1) Also x → 0 as τ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' So c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' x = bG (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='36) x(τ) = − b ω0 (tanh ω(τ − τi) − 1) On shell S = −1 2 � τf τi dτ d dτ (x ˙x G ) = 1 2(x(τf) − x(τi))b = 1 2[x(τf)x(τf) G(τf) − x(τi)x(τi) G(τi) ] If we add this change to the initial Wilson action 1 2 x(τi)x(τi) G(τi) we get the final Wilson action Hf = 1 2 x(τf)x(τf) G(τf) If, for instance, we are interested in evaluating H semiclassically at τ = τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' x(τi) = b ω0 =⇒ b = x(τi)ω0 x(τ) = −x(0)(tanh ω(τ − τi) − 1) ˙x(τ) = −x(0)ω0sech2ω0(τ − τi) 9 The classical action is Sclassical = 1 2ω0x(τi)2 Thus since G(τi) = 1 ω0, H evaluated semiclassically is: H[x, τi] ≈ 1 2ω0x(τi)2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='37) Then Ψ = e−H[x,τi] = e−ω0x(τi)2 which coincides with the ground state wave function of the harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This is essen- tially the Hartle Hawking prescription [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This also motivates the dS-CFT correspondence statement [30, 31, 32] that ΨdS = ZCFT This concludes the discussion of the mapping of ERG equation to a Euclidean harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In higher dimensions this gives free field theory in flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We now return to the case of interest, namely dS space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 3 ERG to field theory in dS We first map the system to Euclidean AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Then analytically continue and obtain dS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Alternatively, one can analytically continue the ERG equation to the Schroedinger equation (when D = 0 this is a free particle with a time dependent mass) and then map to de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This is all exactly as was done for the harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1 Analytic Continuation The EAdS metric in Poincare coordinates is ds2 = R2[dxidxi + dz2 z2 ] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='38) The dS metric in Poincare coordinates is: ds2 = L2[dxidxi − dη2 η2 ] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='39) The metrics are related by analytic continuation: iη = z, iL = R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1 Analytic Continuation of the Action The action generically is S = −1 2 � dD+1x√g[gµν∂µφ∂νφ + m2φ2] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='40) 10 de Sitter In this case we write √−g since g is negative: g = −( L2 η2 )D+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Also g00 = − η2 L2 and gij = δij η2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Thus SdS = � dDx � ∞ 0 dη (L η )D+1[ η2 L2∂ηφ∂ηφ − η2 L2∂iφ∂iφ − m2φ2] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='41) In momentum space: SdS = � dDp (2π)D � ∞ 0 dη (L η )D+1[ η2 L2∂ηφ(p)∂ηφ(−p) − ( η2 L2p2 + m2)φ(p)φ(−p)] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='42) The functional integral description of the quantum mechanical evolution operator for the wave functional of the fields in dS space-time is ¯Ψ[φ(p), t] = � dφi(p) � φ(p, ti) = φi(p) φ(p, t) = φ(p) Dφ(p, t) ei 1 2 � t ti[ ˙φ(p,t′)2−ω2 0φ(p,t′)2]dt′ ¯Ψ[φi(p), ti] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='43) Euclidean Anti de Sitter g = ( R2 z2 )D+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Also g00 = z2 R2 and gij = δij z2 R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' SEAdS = � dDx � ∞ 0 dz (R z )D+1[ z2 R2∂zφ∂zφ + z2 R2∂iφ∂iφ + m2φ2] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='44) In momentum space SEAdS = � dDp (2π)D � ∞ 0 dz (R z )D+1[ z2 R2∂zφ(p)∂zφ(−p) + ( z2 R2p2 + m2)φ(p)φ(−p)] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='45) If we set iη = z and iL = R we see that the functional integral (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='43) becomes ¯Ψ[φ(p), t] = � dφi(p) � φ(p, ti) = φi(p) φ(p, t) = φ(p) Dφ(p, t) e− 1 2 � t ti[ ˙φ(p,t′)2+ω2 0φ(p,t′)2]dt′ ¯Ψ[φi(p), ti] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='46) In holograhic RG this is interpreted as a Euclidean functional integral giving the evolution in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' ¯Ψ is to be interpreted as e−SI[φ(p),t] where SI is the Wilson action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' It was shown in [5] (see below) that this can be obtained by mapping an ERG evolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The dS functional integral (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='43) above is thus an analytically continued version of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2 Mapping 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1 Mapping from Quantum Mechanics Let us go back to Section (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1) and consider the mapping from the Quantum Mechanics of a free particle with time dependent mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We think of it as a 0 + 1 dimensional field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' M(t) is taken to be dimensionless and x has canonical dimensions of − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' S = 1 2 � dt M(t) ˙x2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='47) (In the ERG version M(t) = 1 ˙G) The path integral is � Dx eiS (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='48) 11 As before x(t) = f(t)y(t) with f 2(t) = 1 M(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Substitute this in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='47) and go through the same steps to obtain: S = 1 2 � dt [ ˙y2 + eln f( d2 dt2e− ln f)y2] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='49) Now choose eln f( d2 dt2e− ln f) = −( η2 L2p2 + m2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='50) where η = Le t L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' to obtain SdS SdS = 1 2 � dt [ ˙y2 − ( η2 L2p2 + m2)y2] = 1 2 � dη (L η )[ η2 L2∂ηy∂ηy − ( η2 L2p2 + m2)y2] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='51) p, m here are just some parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' When D > 0 they will stand for momentum and mass of the field respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' So starting from a free particle with time dependent mass we obtain the free field action in de Sitter space dSD+1 with D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Schroedinger Equation: i∂Ψ(x, t) ∂t = − 1 2M(t) ∂2Ψ(x, t) ∂x2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='52) Using the same mapping as in Section (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='1), x = fy Ψ(f(t)y, t) = e− 1 2 αy2 ¯Ψ(y, t) with α = −i ˙f f one obtains i∂ ¯Ψ ∂t = [(1 2 d2 ln f dt2 − 1 2(d ln f dt )2)y2 + 1 2α − 1 2 ∂2 ∂y2]¯Ψ Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='50) this becomes i η L ∂ ¯Ψ ∂η = [−1 2 ∂2 ∂y2 + 1 2( η2 L2p2 + m2)y2 + 1 2α]¯Ψ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='53) If we construct the Schroedinger equation corresponding to the action (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='51) one obtains i η L ∂ ¯Ψ ∂η = [−1 2 ∂2 ∂y2 + 1 2( η2 L2p2 + m2)y2]¯Ψ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='54) which barring the field independent term α is exactly the same as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This term as we have seen provides an overall field independent scaling for all wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' It is a consequence of the ordering ambiguity in going from classical to quantum treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='54) (or its extension to D > 0) describes the quantum mechanical time evolution of the matter field wave functional in de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2 Mapping from ERG Action We now consider the Euclidean version of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='47), which is the Polchinski ERG equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This is what was done in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Thus we replace M(t) by − 1 ˙G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' S = −1 2 � dτ ˙x2 ˙G (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='55) The path integral is ( ˙G < 0) � Dx e 1 2 � dτ ˙x2 ˙G (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='56) which can be obtained from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='52) by setting it = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We take z = Re τ R If we let iη = z, iL = R, it = τ then this can be obtained from the corresponding Minkowski case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' As before x(τ) = f(τ)y(τ) with f 2(τ) = ˙G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Substitute this in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='55) and go through the same steps to obtain: S = 1 2 � dτ [ ˙y2 + eln f( d2 dτ 2e− ln f)y2] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='57) Now choose eln f( d2 dτ 2e− ln f) = ( z2 R2p2 + m2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='58) where z = Re τ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' to obtain SEAdS SEAdS = � dz (R z )[ z2 R2∂zy∂zy + ( z2 R2p2 + m2)y2] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='59) ERG Equation By analogy with the Schroedinger equation we can see that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='56) is the evolution operator corresponding to the ERG equation ∂Ψ(x, τ) ∂τ = −1 2 ˙G∂2Ψ(x, τ) ∂x2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='60) By the same series of transformations as in the de Sitter case, but using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='58), one obtains z R ∂ ¯Ψ ∂z = [1 2 ∂2 ∂y2 − ( z2 R2p2 + m2)y2 − 1 2α]¯Ψ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='61) with α = ˙f f generating an overall scale transformation for ¯Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In the ERG context ¯Ψ represents eW[J] upto a quadratic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This equation is the holographic RG equation in the AdS/CFT correspondence for an elementary scalar field [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='3 Connections Let us summarize the various connections obtained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We start with the quantum mechanics of a free particle having a time dependent mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The Schroedinger equation (SE) for this is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Analytical continuation of this equation (generalized to higher dimensions) gives the Polchinski ERG equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The free particle SE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='20) can be mapped to a SE for a harmonic oscillator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The ERG equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='24) can similarly be mapped to a Euclidean harmonic oscillator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='27)- analytically continued version of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 13 The evolution operators for the above equations are defined in terms of path integrals over some actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The same mapping function f maps the corresponding actions to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Thus the evolution operator for the free particle Schroedinger equation is given by the action in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='3) which is mapped to a harmonic oscillator action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The analytical continuation of these are the Euclidean ERG evolution operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='13) mapped to a harmonic oscillator Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' These steps are summarized in the flow diagram in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The mapping function f was originally chosen in [5] to map the free particle ERG action (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='55) to an action for free fields in EAdS0+1 given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The analytical continuation of this problem to real time gives us an action in dS0+1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' One can also repeat these steps for the corresponding “wave” equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The Polchinski ERG equation for eW[J] gets mapped to an equation in EAdS for eW[J] which is nothing but the holographic RG equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Analytically continuing this, the Schroedinger equation for a wave functional is mapped to a Schroedinger equation for wave functionals of fields in dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' These are summarized in the figure below (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The analytic continuation can be done before the map with f is applied or after as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' It can be done both for the actions as well as for the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='ERG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Holographic RG: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Radial evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='in EAdS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Schroedinger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Real time QM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='In dS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='ERG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Evolution equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='For Euclidean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Harmonic Oscillator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='QM Schroedinger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Real time QM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Schroedinger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Harmonic Oscillator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='ERG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Holographic RG: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Radial evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='in EAdS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Schroedinger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Real time QM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='In dS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Euclidean Action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='For ERG evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='By Feynman ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Path Integral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Euclidean action for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Harmonic Oscillator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Path Integral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Lorentzian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Action for QM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Evolution by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Path Integral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='QM evolution: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Action for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Harmonic Oscillator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Path Integral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Flow of equations-Harmonic Oscillator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Flow of actions – Harmonic Oscillator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Figure 1: Mapping ERG to Harmonic Oscillator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='4 dS-CFT correspondence The connections with ERG mentioned above should, if pursued, provide some insights into dS-CFT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We restrict ourselves to some preliminary observations in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='ERG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Holographic RG: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Radial evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='in EAdS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Schroedinger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Real time QM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='In dS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='ERG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Holographic RG: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Radial evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='equation in EAdS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='QM Schroedinger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Real time QM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Schroedinger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='in dS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='ERG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Holographic RG: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Radial evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='in EAdS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Schroedinger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Real time QM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='In dS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Euclidean Action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='For ERG evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='by functional integral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Holographic RG: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='in EadS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='for functional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='integral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Lorentzian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Action for QM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='evolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='QM evolution by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Functional integral: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Action in dS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Map “f” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Analytic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Continuation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Flow of equations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Flow of actions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='Figure 2: Mapping ERG to Holographic RG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='The idea of dS-CFT correspondence was suggested in [30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This has been investigated further by many authors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' [33, 34, 38, 39, 35, 37, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' What we see from the above analysis is that considering the relation between the evolution equations, one can say that Ψ[φ, J]wave−functional in dS = {Z[φ, J]CFT}analytically continued (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='62) Thus we see that the dS-CFT correspondence suggested by this analysis is one between an ERG equation for a CFT generating functional and a real time quantum mechanical evolution of a wave functional in dS space time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The LHS of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='62) is a QM wave functional of fields on a D-dimensional spatial slice of a D + 1 dimensional dS spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The RHS is the analytically continued partition function of a D-dimensional Euclidean CFT - more precisely, either eWΛ[J] or e−SI,Λ[φ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The precise statement has to involve some statement of the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In the next section we give a concrete example with boundary conditions specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Note that the LHS is a complex probability amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Expectation values will involve Ψ∗Ψ and were calculated first in [30, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' One can proceed to ask whether the expectations on the spatial slice calculated using Ψ∗Ψ also correspond to some other Euclidean CFT on the spatial slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This was explored further in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We do not address this question here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' In the next section we give some examples that explicitly illustrate the connection made by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 15 4 Obtaining Bulk field from ERG The ERG formulation stated in this paper starts with the boundary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The evolution operator for this involves bulk fields but with a non standard action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' When this action is mapped to EAdS action one can interpret the newly mapped field as the EAdS bulk field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This analysis for Euclidean AdS is well defined and has been done in [5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' However, this treatment does not have a natural interpretation in the context in dS space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We have elaborated that in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Bulk scalar field in Euclidean AdS and dS There are conceptual barriers if one tries to do similar analysis to map the ERG evolution operator directly to Lorentzian dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' First of all, it is not clear as in EAdS whether the function G(t) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='a f 2(t) = ˙G(t) is the Green’s function of the dual field theory of dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' It has an oscillatory cutoff function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Therefore we analytically continue the ERG action to a Lorentzian action first, and then do the mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The result thus obtained (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='74) matches with the value found in [39] where the authors have found the bulk field in semicalssical approximation from dS bulk action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' For the Lorentzian dS analysis presented here the RG interpretation is not clearly understood - except as an anlytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We have presented it here for sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Euclidean AdS The Euclidean action of the ERG evolution operator in momentum space, S = −1 2 � dτ � p ˙φ2 ˙G (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='63) is mapped to SEAdS = � dDp (2π)D � ∞ ϵEAdS dz (R z )d+1[ z2 R2∂zyEAdS(p)∂zyEAdS(−p)+( z2 R2p2+m2)yEAdS(p)yEAdS(−p)] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='64) with z = Re τ R as described in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We have rescaled the field as φ = fyEAdS where f is related to the boundary Green’s function G as f 2 = − � z R �−d ˙G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The constraint on 1 f is given by, ∂ ∂z{ � z R �−d+1 ∂ ∂z 1 f } = � z R �−d+1 � p2 + m2R2 z2 � 1 f (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='65) The solutions are zd/2Kα(pz) and zd/2Iα(pz) where α2 = m2R2 + d2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' So 1 f can be taken as, 1 f(p, z) = (z)d/2 (AKα(pz) + BIα(pz)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='66) The Green’s function is G(p, z) = CKα(pz) + DIα(pz) AKα(pz) + BIα(pz) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='67) The large argument asymptotic form of the Modified Bessel function Iα(z) and Kα(z) are given by, Iα(z) ∼ ez √ 2πz � 1 + O(1 z) � for |arg z| < π 2 16 Kα(z) ∼ � π 2ze−z � 1 + O(1 z) � for |arg z| < 3π 2 Putting two constraints on G- i)G(pz → ∞) = 0 ii)G(pz → 0) = γEAdS p−2α, we get, D = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' C(p) = γEAdS p−α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' B(p) = − 1 γEAdS pα In semiclassical approximation the bulk field yEAdS = bEAdS G f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' If yEAdS satisfies yEAdS 0 the bulk field is given by, yEAdS = yEAdS 0 zd/2 ϵd/2 Kα(pz) Kα(pϵ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='68) Now let’s check by analytic continuation iη = z and iL = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' First of all, α becomes ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' ϵ is replaced by iϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We get, yEAdS|z=iη, R=iL = yEAdS 0 |z=iη, R=iL (iη)d/2 (iϵ)d/2 Kν(ipη) Kν(ipϵ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='69) As, yEAdS 0 = bEAdS ϵd/2 EAdS γEAdS Kα(pϵ) pα (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='70) de Sitter We would like to do the same analysis as above for the Lorentzian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The Lorentzian action obtained from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='63) by analytic continuation, in momentum space, S = − � dt � dDp (2π)D 1 2 ˙G(p) ˙φ(p) ˙φ(−p) and needs to be mapped to = 1 2 � ∞ ϵdS dη � dDp (2π)D ��L η �D−1 {(∂ηydS)2 − p2ydS2 − m2L2 η2 ydS2} � Here η = Le t L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We do the field redefinition of boundary field φ = fydS f is a scale dependent quantity which is related to Green’s function G as f 2 = − � η L �−D ˙G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Performing the same manipulations as in [5], one can get the constraint on f as, � η L �d−1 �� η L �−d+1 d dη �2 e− ln f = � η L �−d+1 � −p2 − m2L2 η2 � e− ln f −d + 1 η ∂ ∂η 1 f + ∂2 ∂η2 1 f = � −p2 − m2L2 η2 � 1 f The solutions are � η L �d/2 H(1) ν (pη) and � η L �d/2 H(2) ν (pη) with ν2 = d2 4 − m2L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The 1 f can be written in general as( note f is dimensionless), 1 f(p, η) = � η L �d/2 � AH(1) ν (pη) + BH(2) ν (pη) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='71) 17 and the Green’s function is 2 G(pη) = CH(1) ν (pη) + DH(2) ν (pη) AH(1) ν (pη) + BH(2) ν (pη) Physically one can expect G(pη → ∞) = 0 which yields, CH(1) ν (pη) + DH(2) ν (pη) = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='72) The asymptotic forms of Hankel functions of both kind for large arguments are, H(1) ν (z) ∼ � 2 πzei(z− νπ 2 − π 4 ) − π < arg z < 2π H(2) ν (z) ∼ � 2 πze−i(z− νπ 2 − π 4 ) − 2π < arg z < π The presence of the oscillatory functions will not let eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='72 to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Hence we analytically continue the argument of Green’s function G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The choice of direction of the analytic continuation is based on the anticipation that the bulk field will have positive frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Hence we take η = −iz (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='73) which prompts us to make C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Also, from the constraint AD − BC = 1 we get A = 1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Hence the Green’s function now takes the form, G(pz) = DH(2) ν (ipz) 1 DH(1) ν (ipz) + BH(2) ν (ipz) Next another constraint will come from the fact that boundary Green’s function is γdS p−2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' So in the limit of z → 0 using the formulae, H(1) ν (z) = iYν(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' H(2) ν (z) = −iYν(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Yν(z) = −Γ(ν) π �2 z �ν One can get, −iD i D − iB = γdS p−2ν On the other side, f should become a p independent constant at boundary x = 0 so that it does not modify the boundary Green’s function, also ydS and f should become same field in boundary field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This gives, i D − iB = pν Finally we get, D = iγdS p−ν ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' B = i � 1 − 1 γdS � pν The bulk field ydS is given by, 2We use the term Green function by analogy with the EAdS case, where G is the propagator of the boundary CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Also see for instance [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 18 ydS = bdS G f = bdS(iγp−ν) 1 Ld/2xd/2H(2) ν (ipx) If we analytically continue back to η we get, ydS = bdS(iγp−ν) 1 Ld/2(−iη)d/2H(2) ν (pη) If the field ydS satisfies ydS 0 at η = ϵdS then, ydS = ydS 0 ηd/2 ϵd/2 dS H(2) ν (pη) H(2) ν (pϵdS) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='74) ydS satisfies Bunch-Davies condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Relation between bulk fields in EAdS and dS The bulk field in EAdS space is given by, yEAdS = yEAdS 0 zd/2 ϵd/2 Kα(pz) Kα(pϵ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='75) Let’s apply the analytic continuation continuation iη = z and iL = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' First of all, α becomes ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' ϵ is replaced by iϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' We get, yEAdS|z=iη, R=iL = yEAdS 0 |z=iη, R=iL (iη)d/2 (iϵ)d/2 Kν(ipη) Kν(ipϵ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='76) As, yEAdS 0 = bEAdS ϵd/2 EAdS γEAdS Kα(pϵ) pα (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='77) Using the relation between Kα(x) and Hα(x), Kα(x) = π 2 iα+1H(1) α (ix);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' − π < arg x ≤ π 2 = π 2 (−i)α+1H(2) α (−ix);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' − π 2 < arg x ≤ π (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='78) Here also we want to ensure the bulk field to be of positive frequency, hence choosing H(2)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' yEAdS 0 |z=iη, R=iL = π 2 (i)d/2+α+1bEAdSϵd/2γEAdS H(2) α (pϵ) pα = bEAdS bdS γEAdS γdS π 2 (i)d/2+α+1ydS 0 Hence, yEAdS|z=iη, R=iL =bEAdS bdS γEAdS γdS π 2 (i)d/2+α+1ydS 0 ηd/2 ϵd/2 H(2) α (pη) H(2) α (pϵ) = bEAdS bdS γEAdS γdS π 2 (i)d/2+α+1ydS (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content='79) Upto various normalization constants we see that they agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 19 5 Summary and Conclusions In [5, 6] an evolution operator for an ERG equation of a perturbed D-dimensional free field theory in flat space was mapped to a field theory action in AdSD+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Similar mappings were done subsequently for the interacting O(N) model at both the free fixed point and at the Wilson- Fisher fixed point [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The main aim of this paper was to understand better the mapping used in these papers and to see if there are other examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' A related question was that of analytic continuation of these theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' These questions can posed, both for the ERG equation and its evolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' It was shown that a mapping of this type can map the ERG evolution operator of a (zero- dimensional) field theory to the action of a Euclidean harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Furthermore the analytic continuation of the ERG evolution operator action gives the path integral for a free particle with a time dependent mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' A similar mapping takes this to a harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This method also gives new way of obtaining the Ermakov-Lewis invariants for the original theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' The analytically continued ERG equation is a Schroedinger like equation for a free field theory wave functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This gets mapped to the Schroedinger equation for a wave functional of a free field theory in de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' These are summarized in Figures 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This is one version of the dS-CFT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' From this point of view, the QM evolution of dS field theory is also an ERG evolution of a field theory, but accompanied by an analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' An example was worked out to illustrate this correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' To understand these issues further it would be useful to apply these techniques to the O(N) model ERG equation written in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' This ERG equation has extra terms and thus the theory naturally has interaction terms in the EAdS bulk action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Similarly it would be interesting to study the connection between bulk Green functions and the QM correlation functions on the space-like time slice of these theories, as considered originally in [30, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' Acknowledgements SD would like to thank IMSc,Chennai where part of the work was done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tFRT4oBgHgl3EQfnjcF/content/2301.13605v1.pdf'} +page_content=' 20 References [1] J.' metadata={'source': 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machinery +Article Type: +Original Research +Keywords: +The benefits of noise, weak signature enhancement, fault identification, fault diagnosis +Corresponding Author: +Zijian Qiao, Ph.D. +Ningbo University +Ningbo, CHINA +Corresponding Author Secondary +Information: +Corresponding Author's Institution: +Ningbo University +Corresponding Author's Secondary +Institution: +First Author: +Zijian Qiao, Ph.D. +First Author Secondary Information: +Order of Authors: +Zijian Qiao, Ph.D. +Shuai Chen +Zhihui Lai +Shengtong Zhou +Miguel A. F. Sanjuán +Order of Authors Secondary Information: +Funding Information: +Foundation of the State Key Laboratory of +Performance Monitoring and Protecting of +Rail Transit Infrastructure of East China +Jiaotong University +(HJGZ2021114) +Dr. Zijian Qiao +Zhejiang Provincial Natural Science +Foundation of China +(LQ22E050003) +Dr. Zijian Qiao +National Natural Science Foundation of +China +(62001210, 51905349) +Dr. Zhihui Lai +The Spanish State Research Agency +(AEI) and the European Regional +Development Fund (ERDF) +(PID2019-105554GB-I00) +Dr. Miguel A. F. Sanjuán +Abstract: +Noise would give rise to incorrect filtering frequency-band selection in signal filtering- +based methods including fast kurtogram, teager energy operators and wavelet packet +transform filters and meanwhile would result in incorrect selection of useful +components and even mode mixing, end effects and etc. in signal decomposition- +based methods including empirical mode decomposition, singular value decomposition +and local mean decomposition. On the contrary, noise in stochastic resonance (SR) is +beneficial to enhance weak signals of interest embedded in signals with strong +background noise. Taking into account that nonlinear systems are crucial ingredients +to activate the SR, here we investigate the SR in the cases of overdamped and +underdamped harmonic-Gaussian double-well potential systems subjected to noise +and a periodic signal. We derive and measure the analytic expression of the output +signal-to-noise ratio (SNR) and the steady-state probability density (SPD) function +Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation + +under approximate adiabatic conditions. When the harmonic-Gaussian double-well +potential loses its stability, we can observe the antiresonance phenomenon, whereas +adding the damped factor into the overdamped system can change the stability of the +harmonic-Gaussian double-well potential, resulting that the antiresonance behavior +disappears in the underdamped system. Then, we use the overdamped and +underdamped harmonic-Gaussian double-well potential SR to enhance weak useful +characteristics for diagnosing incipient rotating machinery failures. Theoretical and +experimental results show that adjusting both noise intensity and system parameters +can activate overdamped and underdamped harmonic-Gaussian double-well potential +SR in which there is a bell-shaped peak for the SNR. Additionally, the underdamped +harmonic-Gaussian double-well potential SR is independent of frequency-shifted and +rescaling transform to process large machine parameter signals and outperforms the +overdamped one. Finally, comparing the advanced robust local mean decomposition +(RLMD) method based on signal decomposition and the wavelet transform method +based on noise cancellation or infogram method based on signal filtering, the +overdamped or underdamped harmonic-Gaussian double-well potential SR methods +characterize a better performance to detect a weak signal. Fault characteristics in the +early stage of failures are successful in improving the incipient fault characteristic +identification of rolling element bearings. +Response to Reviewers: +Please see the attached "response to reviewers". +Order of Authors (with Contributor Roles): Zijian Qiao, Ph.D. (Funding acquisition: Supporting; Validation: Lead; Writing – original +draft: Lead) +Shuai Chen (Data curation: Equal; Visualization: Lead) +Zhihui Lai (Investigation: Equal; Visualization: Equal; Writing – review & editing: +Supporting) +Shengtong Zhou (Data curation: Lead; Formal analysis: Equal; Investigation: Equal; +Writing – review & editing: Equal) +Miguel A. F. Sanjuán (Formal analysis: Equal; Funding acquisition: Equal; +Methodology: Equal; Writing – review & editing: Lead) +Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation + + 1 / 27 + +Harmonic-Gaussian double-well potential stochastic resonance with +its application to enhance weak fault characteristics of machinery + +Zijian Qiao1,2,3,4, , Shuai Chen1, Zhihui Lai5, Shengtong Zhou2, Miguel A. F. Sanjuán6 +1 School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo 315211, China +2. State Key Laboratory of Performance Monitoring and Protecting of Rail Transit Infrastructure, East China +Jiaotong University, Nanchang 330013, China +3. Laboratory of Yangjiang Offshore Wind Power, Yangjiang 529599, Guangdong, China +4. Zhejiang Provincial Key Laboratory of Part Rolling Technology, Ningbo 315211, China +5. Guangdong Provincial Key Laboratory of Micro/Nano Optomechatronics Engineering, College of Mechatronics +and Control Engineering, Shenzhen University, Shenzhen 518060, China +6. Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan +Carlos, Tulipán s/n, Móstoles, 28933, Madrid, Spain + +Abstract: +Noise would give rise to incorrect filtering frequency-band selection in signal +filtering-based methods including fast kurtogram, teager energy operators and wavelet +packet transform filters and meanwhile would result in incorrect selection of useful +components +and +even +mode +mixing, +end +effects +and +etc. +in +signal +decomposition-based methods including empirical mode decomposition, singular +value decomposition and local mean decomposition. On the contrary, noise in +stochastic resonance (SR) is beneficial to enhance weak signals of interest embedded +in signals with strong background noise. Taking into account that nonlinear systems +are crucial ingredients to activate the SR, here we investigate the SR in the cases of +overdamped and underdamped harmonic-Gaussian double-well potential systems +subjected to noise and a periodic signal. We derive and measure the analytic +expression of the output signal-to-noise ratio (SNR) and the steady-state probability + +Corresponding author. +E-mail address: zijianqiao@hotmail.com, qiaozijian@nbu.edu.cn (Z. Qiao). +Manuscript +Click here to access/download;Manuscript;Manuscript.docx +Click here to view linked References + + 2 / 27 + +density (SPD) function under approximate adiabatic conditions. When the +harmonic-Gaussian double-well potential loses its stability, we can observe the +antiresonance phenomenon, whereas adding the damped factor into the overdamped +system can change the stability of the harmonic-Gaussian double-well potential, +resulting that the antiresonance behavior disappears in the underdamped system. Then, +we use the overdamped and underdamped harmonic-Gaussian double-well potential +SR to enhance weak useful characteristics for diagnosing incipient rotating machinery +failures. Theoretical and experimental results show that adjusting both noise intensity +and +system +parameters +can +activate +overdamped +and +underdamped +harmonic-Gaussian double-well potential SR in which there is a bell-shaped peak for +the SNR. Additionally, the underdamped harmonic-Gaussian double-well potential SR +is independent of frequency-shifted and rescaling transform to process large machine +parameter signals and outperforms the overdamped one. Finally, comparing the +advanced robust local mean decomposition (RLMD) method based on signal +decomposition and the wavelet transform method based on noise cancellation or +infogram method based on signal filtering, the overdamped or underdamped +harmonic-Gaussian double-well potential SR methods characterize a better +performance to detect a weak signal. Fault characteristics in the early stage of failures +are successful in improving the incipient fault characteristic identification of rolling +element bearings. + +Key words: The benefits of noise, weak signature enhancement, fault identification, +fault diagnosis + +1. Introduction +Noise is ubiquitous but unwanted in detecting weak signals [1], but noise in +biological systems can be used to amplify weak signals embedded by a strong +background noise. Such an ingenious phenomenon is observed in a bistable nonlinear +system, namely stochastic resonance (SR) [2]. SR is a kind of synchronization +mechanism among the nonlinear systems, noise and a weak periodic signal, which + + 3 / 27 + +takes place to activate the SR for amplifying weak useful signals [3]. +SR has been investigated from theory to engineering application widely [4-6]. +Among three ingredients for activating SR including noise, nonlinear systems and +weak useful signals, nonlinear systems are crucial ingredients for extracting weak +useful signals and moreover can harvest the energy of noise located at the whole +frequency band of a noisy signal to enhance or amplify a weak useful signal. For this +purpose, most of scholars pay attention to exploring the behaviors of SR in novel +nonlinear systems from bistable [7] to multistable ones [8-10], from overdamped [11] +and underdamped [12] to fractional-order [13] ones, and even from cascaded [14] and +coupled [15, 16] to time-delayed feedback [17] ones and biological systems [18, 19]. +Because the bistable system is most classical among them, it has been investigated, +such as classical bistable potential overdamped systems, noisy confined bistable +potential overdamped systems [20], asymmetric bistable potential overdamped +systems [21], classical bistable potential underdamped systems, noisy bistable +potential fractional-order systems [22] and E-exponential potential underdamped +systems [23, 24]. The E-exponential potential named by the references [23, 24] is a +narrow version of the harmonic-Gaussian double-well potential. The references above +show that overdamped and underdamped harmonic-Gaussian double-well potential +SR has not been studied systematically in theory and further applied to enhance +incipient fault identification of machinery for providing a tutorial of other readers and +researchers on the SR in the overdamped and underdamped systems with novel +generalized double-well potentials yet. Even, the comparison between overdamped +and underdamped harmonic-Gaussian double-well potential SR has not been made in +theory and engineering applications. Therefore, this paper attempts to investigate the +SR in the overdamped and underdamped harmonic-Gaussian double-well potential +systems theoretically and then apply it to enhance weak fault characteristics and +diagnose incipient faults of machinery. Additionally, some comparisons with other +advanced signal processing techniques including signal decomposition-based and +noise cancellation or signal filtering-based methods for enhancing weak fault +characteristics of machinery are given. + + 4 / 27 + +The remainder of this paper is organized as follows. Section 2 and Section 3 +investigate the overdamped and underdamped harmonic-Gaussian double-well +potential SR by deriving the analytic expressions of signal-to-noise ratio (SNR) and +steady-state probability density (SPD) functions, respectively. In Section 4, we apply +the overdamped and underdamped harmonic-Gaussian double-well potential SR to +enhance weak fault characteristics and incipient fault identification of rolling element +bearings. Finally, conclusions are drawn in Section 5. + +2. Overdamped harmonic-Gaussian double-well potential SR +The overdamped Langevin equation driven by a harmonic-Gaussian double-well +potential under the action of random noise and a periodic signal can be described as +[25] +d𝑦 +d𝑥 = − +𝜕𝑈(𝑥) +𝜕𝑥 ++ 𝐴 cos(𝜔0𝑡) + 𝜀(𝑡) (1) +where 𝐴 and ω0 are the amplitude and angular frequency of the periodic signal +respectively, and 𝜀(𝑡) is the Gaussian white noise with mean zero and variance 𝐷 +i.e. noise intensity. +The harmonic-Gaussian double-well potential which is a variant of a double-well +potential can be expressed as [26] +𝑈(𝑥) = +𝑘 +2 𝑥2 + 𝛼exp(−𝛽𝑥2) (2) +where two stable states and one unstable state are located at 𝑥± = ±√ln(2𝛼𝛽 𝑘 +⁄ ) 𝛽 +⁄ +and +𝑥𝑢 = 0 + respectively, +and +the +barrier +height +is +∆𝑈 = α − +𝑘[1 + ln(2𝛼𝛽 𝑘 +⁄ )] (2𝛽) +⁄ +. To ensure the stability of the harmonic-Gaussian +double-well potential, this condition ln(2𝛼𝛽 𝑘 +⁄ ) > 0 must be satisfied, further 𝑘 < +2αβ. When 𝑘 = 1 , Fig. 1(a) shows the harmonic-Gaussian double-well potential +under different system parameter sets (𝛼, 𝛽), while Fig. 1(b) depicts those with +varying 𝑘. It is seen from Fig. 1(a) that adjusting the system parameter 𝛽 controls +the potential-well width whereas the potential-barrier height nearly keeps unchanged, +but varying 𝛼 changes the potential-barrier height whereas the potential-well width +nearly remains unchanged. Such a behavior is helpful to tune the potential-well width + + 5 / 27 + +and depth individually to activate the optimal harmonic-Gaussian double-well +potential SR. Meanwhile, it is found from Fig. 1(b) that adjusting 𝑘 can also change +the slope of the harmonic-Gaussian double-well potential. + +Fig. 1 Harmonic-Gaussian double-well potentials under different parameter sets (a) +(𝛼, 𝛽) and (b) (𝛼, 𝛽, 𝑘). +The Langevin equation in Eq. (1) can be transformed as further [27] +∂ρ(𝑥,𝑡) +∂𝑡 += − +𝜕 +𝜕𝑥 [−𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + 𝐴 cos(𝜔0𝑡)]𝜌(𝑥, 𝑡) + 𝐷 +𝜕2 +𝜕𝑥2 𝜌(𝑥, 𝑡) (3) +where 𝜌(𝑥, 𝑡) is the probability density function (PDF) of the stochastic process +𝑥(𝑡) which denotes the transition trajectory of Brownian particles in the +harmonic-Gaussian double-well potential as time varies. The corresponding SPD +function can be denoted as +𝜌𝑠(𝑥, 𝑡) = +𝑁(𝑡) +√𝐷 exp [− +∅(𝑥,𝑡) +𝐷 ] (4) +where 𝑁(𝑡) is the normalization constant and 𝑁(𝑡) = √𝐷 ∫ +exp[−∅(𝑥, 𝑡) 𝐷 +⁄ ]d𝑥 +∞ +−∞ +⁄ +, +and ∅(𝑥, 𝑡) is the generalized potential +∅(𝑥, 𝑡) = 𝑈(𝑥) − 𝑥𝐴 cos(𝜔0𝑡) . (5) +Assuming that the periodic signal 𝐴 cos(𝜔0𝑡) can satisfy the requirement of small +parameters under approximate adiabatic conditions, i.e., 𝜔0 is larger than the +characteristic relaxation time in double potential wells [28]. Then, the transition rates +between the two stable states are given by the Kramers-like formulas [29] +𝑊±(𝑥, 𝑡) = +√|𝑈′′(𝑥±,𝑡)𝑈′′(𝑥𝑢,𝑡)| +2π +exp [ +∅(𝑥±,𝑡)−∅(𝑥𝑢,𝑡) +𝐷 +] (6) +where the notation | ∙ | denotes the absolute value and +(a) +(b) + + 6 / 27 + +𝑈′′(𝑥, 𝑡) = 𝑘 − 2𝛼𝛽exp(−𝛽𝑥2)(1 − 2𝛽𝑥2) +𝑈′′(𝑥𝑢, 𝑡) = 𝑘 − 2𝛼𝛽 +𝑈′′(𝑥±, 𝑡) = 2𝑘ln ( +2𝛼𝛽 +𝑘 ) (7) +∅(𝑥𝑢, 𝑡) = 𝑈(𝑥𝑢, 𝑡) − 𝑥𝑢𝐴 cos(𝜔0𝑡) = 𝛼 +∅(𝑥±, 𝑡) = 𝑈(𝑥±, 𝑡)−𝑥±𝐴 cos(𝜔0𝑡) = 𝑘 +2𝛽 (1 + ln 2𝛼𝛽 +𝑘 ) ∓ 𝐴 cos(𝜔0𝑡) √ln(2𝛼𝛽 𝑘 +⁄ ) +𝛽 + +When we introduce Eq. (7) into Eq. (6), we can obtain +𝑊±(𝑥, 𝑡) = √𝑘(2𝛼𝛽 − 𝑘)ln(2𝛼𝛽 𝑘 +⁄ ) +√2π + +× exp [− +𝛼 +𝐷 + +𝑘(1+ln(2𝛼𝛽 𝑘 +⁄ )) +2𝛽𝐷 +∓ 𝐴 cos(𝜔0𝑡) √ +ln(2𝛼𝛽 𝑘 +⁄ ) +𝛽𝐷2 +] (8) +Furthermore, Eq. (8) can be transformed as +𝑊±(𝑥, 𝑡) = 𝑓(𝜇 ± 𝜂0 cos(𝜔0𝑡)) (9) +where +𝜇 = +𝛼 +𝐷 − +𝑘 +2𝛽𝐷 (1 + ln +2𝛼𝛽 +𝑘 ) (10) +𝜂0 = +𝐴 +𝐷 √ +ln(2𝛼𝛽 𝑘 +⁄ ) +𝛽 + (11) +Thus, we can simplify Eq. (8) as +𝑊±(𝑥, 𝑡) = 𝑓(𝜇 ± 𝜂0 cos(𝜔0𝑡)) = +√2𝑘(2𝛼𝛽−𝑘)ln2𝛼𝛽 +𝑘 +2π +exp[−(𝜇 ± 𝜂0 cos(𝜔0𝑡))] (12) +The response of the nonlinear system in Eq. (1) can be quantified using a classical +measure, i.e., SNR [30]. To derive its analytic expression, the power spectral density +of the system response can be described as +𝑆(Ω) = [1 − +𝛼12𝜂02 +2(𝛼02+𝜔02)] ( +4𝑐2𝛼0 +𝛼02+𝜔02) + +π𝑐2𝜂02𝛼12 +𝛼02+Ω2 [𝛿(Ω − 𝜔0) + 𝛿(Ω + 𝜔0)] (13) +where +𝑐 = √ +ln(2𝛼𝛽 𝑘 +⁄ ) +𝛽 + (14) +𝛼1 = 𝛼0 = +√2𝑘ln(2𝛼𝛽 𝑘 +⁄ )(2𝛼𝛽−𝑘) +π +exp(−𝜇) (15) +Finally, the output SNR of the response of the overdamped harmonic-Gaussian +double-well potential system can be derived as + + 7 / 27 + +SNR = +π𝑐2𝛼12𝜂02 +𝛼02+Ω2 |Ω=𝜔0 × +𝛼02+𝜔02 +4𝑐2𝛼0 [1 − +𝛼12𝜂02 +2(𝛼02+𝜔02)] +−1 += +π𝛼1𝜂02 +4 +[1 − +𝛼12𝜂02 +2(𝛼02+𝜔02)] +−1 + (16) +Therefore, we can analyze the function between the output SNR and system +parameters using the analytic expression in Eq. (16). Figure 2 shows the output SNR +of overdamped harmonic-Gaussian double-well potential SR as system parameters +and noise intensity vary. It can be seen from Fig. 2(a) that the output SNR is a +nonmonotonic function of noise intensity 𝐷 under different 𝑘 and the peak value of +output SNR increases when 𝑘 raises, suggesting that adjusting 𝑘 is able to activate +the SR in the overdamped harmonic-Gaussian double-well potential system for +improving the output SNR. Similarly, adjusting 𝛼 and 𝛽 can also maximize the +output SNR, and the peak value of the output SNR declines as 𝛼 or 𝛽 increases but +the resonant noise intensity at the peak value becomes larger, as shown in Fig. 2(b) +and Fig. 2(c), respectively. We visualize the two-dimensional function among SNR +and two of system parameters (𝛼, 𝛽, 𝑘), as shown in Fig. 2(d), Fig. 2(c) and Fig. 2(d). +One can observe from Fig. 2(d) that a moderate parameter set (𝛼, 𝑘) can improve the +SNR of a given signal, whereas there exists a negative output SNR because the +harmonic-Gaussian double-well potential loses its stability when 𝑘 ≥ 2𝛼𝛽, resulting +in an antiresonance phenomenon. Meanwhile, we fix 𝑘 to express the output SNR as +a function of (𝛼, 𝛽) in Fig. 2(e), indicating that only an optimal matching between 𝛼 +and 𝛽 can activate the overdamped harmonic-Gaussian double-well potential SR to +enhance the weak periodic signal embedded by a strong background noise. Similarly, +Fig. 2(f) also demonstrates that such a parameter matching is necessary to activate the +overdamped harmonic-Gaussian double-well potential SR. When 𝑘 ≥ 2𝛼𝛽, one can +also see the antiresonance from Fig. 2(e) and 2(f), respectively. The above results +demonstrate that the optimal parameter matching among 𝑘, 𝛼 and 𝛽 is able to +maximize the SR. + + 8 / 27 + + +Fig. 2 SNR of overdamped harmonic-Gaussian double-well potential SR varies with +system parameters and noise intensity: SNR as a function of noise intensity under +different 𝑘 in (a), 𝛽 in (b) and 𝛼 in (c); SNR as a two-dimensional function of +(𝑘, 𝛼) in (d), (𝛽, 𝛼) in (e) and (𝑘, 𝛽) in (f). +Figure 3 depicts the SPD function and the corresponding system responses. The +SPD indicates the probability of Brownian particles to reside in double potential wells. +It is found from Fig. 3(a) that when 𝐷 = 0.3 the particles oscillate at the right +potential well located at 𝑥+ = √ln(2𝛼𝛽 𝑘 +⁄ ) 𝛽 +⁄ for activating intra-well SR, which is +demonstrated by the system response in Fig. 3(b) further. When we increase the noise +intensity 𝐷, the particles can jump across the potential barrier to go back and forth in +double wells for activating the inter-well SR marked in red in Fig. 3(a), whose system +response characterizes the eye-catching period marked in red in Fig. 3(b). When the +noise intensity is fixed as 𝐷 = 3, two peaks of SPD decline and the corresponding +(a) +(b) +(c) +(d) +(e) +(f) + + 9 / 27 + +system response marked in green becomes noisy. + +Fig. 3 SPD functions and the corresponding system responses of overdamped +harmonic-Gaussian double-well potential SR under different noise intensity: (a) the +SPD functions and (b) the corresponding system responses. + +3. Underdamped harmonic-Gaussian double-well potential SR +The underdamped harmonic-Gaussian double-well potential system subjected to a +periodic signal and noise can be described as [31] +d2𝑥 +d𝑡2 + 𝛾 +d𝑥 +d𝑡 = − +𝜕𝑈(𝑥) +𝜕𝑥 ++ 𝐴 cos(𝜔0𝑡) + 𝜉(𝑡) (17) +where 𝛾 is the damped factor and 𝛾 > 0. Equation (17) can be transformed as [32] +{ +d𝑥 +d𝑡 = 𝑦 +d𝑦 +d𝑡 = −𝛾𝑦 − 𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + 𝐴 cos(𝜔0𝑡) + 𝜉(𝑡) + (18) +Supposing that 𝐴 = 0, 𝐷 = 0, d𝑥 d𝑡 +⁄ += 0 and d𝑥 d𝑡 = 0 +⁄ +, we can obtain three +singular points +(𝑥± +𝑦±) = (±√ +ln(2𝛼𝛽 𝑘 +⁄ ) +𝛽 +0 +) , (𝑥𝑢 +𝑦𝑢) = (0 +0) (19) +Let +𝜕𝑈(𝑥, 𝑦) 𝜕𝑥 +⁄ + and +𝜕𝑈(𝑥, 𝑦) 𝜕𝑦 +⁄ + mark +as +𝑈𝑥(𝑥, 𝑦) and +𝑈𝑦(𝑥, 𝑦) +respectively, and then Eq. (18) can be rewritten as +{ +𝑈𝑥(𝑥, 𝑦) = 𝑦 +𝑈𝑦(𝑥, 𝑦) = −𝛾𝑦 − 𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + 𝐴 cos(𝜔0𝑡) + 𝜉(𝑡) (20) +The linearization matrix of Eq. (18) can be calculated as +(𝑈𝑥𝑥(𝑥, 𝑦) +𝑈𝑥𝑦(𝑥, 𝑦) +𝑈𝑦𝑥(𝑥, 𝑦) +𝑈𝑦𝑦(𝑥, 𝑦)) = ( +0 +1 +−𝑘 + 2𝛼𝛽exp(−𝛽𝑥2)[1 − 2𝛽𝑥2exp(−𝛽𝑥2)] +−𝛾) +(a) +(b) + + 10 / 27 + +(21) +Further, the linearization matrix at the singular points (±√ln(2𝛼𝛽 𝑘 +⁄ ) 𝛽 +⁄ +, 0) can +be denoted as +(𝑈𝑥𝑥(𝑥, 𝑦) +𝑈𝑥𝑦(𝑥, 𝑦) +𝑈𝑦𝑥(𝑥, 𝑦) +𝑈𝑦𝑦(𝑥, 𝑦)) | +𝑥=±√ln(2𝛼𝛽 𝑘 +⁄ ) +𝛽 +,𝑦=0 = ( +0 +1 +− +𝑘2 +𝛼𝛽 ln ( +2𝛼𝛽 +𝑘 ) +−𝛾) (22) +By solving Eq. (22), the corresponding eigenvalues are calculated as +𝛽1,2 = +−𝛾±√𝛾2−4𝑘2 +𝛼𝛽 ln(2𝛼𝛽 +𝑘 ) +2 + (23) +Similarly, the linearization matrix at the singular point (0,0) is +(𝑈𝑥𝑥(𝑥, 𝑦) +𝑈𝑥𝑦(𝑥, 𝑦) +𝑈𝑦𝑥(𝑥, 𝑦) +𝑈𝑦𝑦(𝑥, 𝑦)) |𝑥=0,𝑦=0 = ( +0 +1 +−𝑘 + 2𝛼𝛽 +−𝛾) (24) +The corresponding eigenvalues to the linearization matrix in Eq. (24) are +𝜆1,2 = +−𝛾±√𝛾2+4(2𝛼𝛽−𝑘) +2 + (25) +Assuming that 𝜌(𝑥, 𝑦, 𝑡) is the PDF of the stochastic process in Eq. (18), the +corresponding the Fokker-Planck equation is [33] +𝜕𝜌(𝑥,𝑦,𝑡) +𝜕𝑡 += − +𝜕𝑦 +𝜕𝑥 𝜌(𝑥, 𝑦, 𝑡) − +𝜕 +𝜕𝑦 [−𝛾𝑦 − 𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + +𝐴 cos(𝜔0𝑡)]𝜌(𝑥, 𝑦, 𝑡) + 𝛾𝐷 +𝜕2 +𝜕𝑦2 𝜌(𝑥, 𝑦, 𝑡) (26) +Then, the corresponding SPD function to Eq. (18) can be denoted as +𝜌s(𝑥, 𝑦, 𝑡) = 𝑁(𝑡)exp [− +1 +𝐷 ( +1 +2 𝑦2 + +𝑘 +2 𝑥2 + 𝛼exp(−𝛽𝑥2) − 𝑥𝐴 cos(𝜔0𝑡))] (27) +where 𝑁(𝑡) stands for the normalization constant [34] +𝑁(𝑡) = +1 +∫ +∫ +exp[−𝑈̂(𝑥,𝑦,𝑡) +𝐷 +]d𝑥d𝑦 +∞ +−∞ +∞ +−∞ + (28) +in which 𝑈̂(𝑥, 𝑦, 𝑡) denotes the generalized potential +𝑈̂(𝑥, 𝑦, 𝑡) = +1 +2 𝑦2 + +𝑘 +2 𝑥2 + 𝛼exp(−𝛽𝑥2) − 𝑥𝐴 cos(𝜔0𝑡) (29) +The +transition +rates +of +particles +at +the +singular +points +(𝑥±, 𝑦±) = +(±√ln(2𝛼𝛽 𝑘 +⁄ ) 𝛽 +⁄ +, 0) can be calculated as [35] +𝑊±(𝑡) = +√𝛽1𝛽2 +2π +√− +𝜆1 +𝜆2 exp [ +1 +𝐷 ( +𝑘 +2𝛽 (1 + ln ( +2𝛼𝛽 +𝑘 )) − 𝛼 ∓ √ +ln(2𝛼𝛽 𝑘 +⁄ ) +𝛽 +𝐴 cos(𝜔0𝑡))] +(30) + + 11 / 27 + +Finally, the analytic expression of the output SNR of the response of the +underdamped harmonic-Gaussian double-well potential system in Eq. (17) is derived +as +SNR = +π𝑐2𝛼12𝜂02 +𝛼02+Ω2 |Ω=𝜔0 × +𝛼02+𝜔02 +4𝑐2𝛼0 [1 − +𝛼12𝜂02 +2(𝛼02+𝜔02)] +−1 += +π𝛼1𝜂02 +4 +[1 − +𝛼12𝜂02 +2(𝛼02+𝜔02)] +−1 + (31) +where +𝛼1 = 𝛼0 = +√𝛽1𝛽2 +π +√− +𝜆1 +𝜆2 exp(−𝑢) (32) +Figures 4(a)-4(d) show the output SNR as noise intensity 𝐷 varies under different +system parameters. It is found from Fig. 4(a) that the output SNR increases and then +decreases as noise intensity increases, suggesting that a noise-induced underdamped +harmonic-Gaussian double-well potential SR happens. Moreover, increasing 𝑘 can +maximize the output SNR. Like this, adjusting 𝛾, 𝛼 and 𝛽 can also improve the +output SNR as shown in Fig. 4(b), Fig. 4(c) and Fig. 4(d) respectively, where the peak +value of output SNR and the resonant noise intensity are changed. Figures 4(e)-4(h) +show the output SNR as the function of system parameters for a given signal. +Adjusting the system parameters can activate the underdamped harmonic-Gaussian +double-well potential SR, as shown in Fig. 4(e)-4(h). Different from the overdamped +harmonic-Gaussian double-well potential SR, it is noticed from Fig. 4(e)-4(h) that the +antiresonance disappears in the underdamped one. That is because the damped factor +changes the stability of the nonlinear system. + + 12 / 27 + + +Fig. 4 SNR of underdamped harmonic-Gaussian double-well potential SR varies with +system parameters and noise intensity: SNR as a function of noise intensity under +different 𝑘 in (a), 𝛾 in (b) and 𝛼 in (c), 𝛽 in (d); SNR as a two-dimensional +function of (𝛽, 𝛼) in (e), (𝛽, 𝑘) in (f), (𝛾, 𝑘) in (g) and (𝛾, 𝛼) in (h). +Figure 5 shows the SPD functions and the corresponding system responses. In Fig. +5(a), the SPD functions vary from asymmetrical peaks into two symmetrical ones as +noise intensity raises, suggesting that the underdamped harmonic-Gaussian +double-well potential SR changes from intra-well SR into inter-well one. In Fig. 5(b), +a weak period occurs when intra-well SR happens, and then the system response +(a) +(b) +(c) +(e) +(f) +(g) +(d) +(h) + + 13 / 27 + +becomes chaotic when the particles jump randomly between double wells and finally +is periodic when the inter-well SR takes place. + +Fig. 5 SPD functions and the corresponding system responses of underdamped +harmonic-Gaussian double-well potential SR under different noise intensity: (a) the +SPD functions and (b) the corresponding system responses. + +4. Application of harmonic-Gaussian double-well potential SR to enhance weak +fault characteristics of machinery +Rotating components of machinery including bearings, gears and rotors are more +prone to failures than fixed components due to contact fatigue, uneven lubrication, +misalignment and so on [36-38]. Therefore, how to detect weak fault characteristics of +rotating components in the early stage becomes a challenge [39]. Lots of scholars +have attempted to cancel or suppress the noise embedded in a signal to extract weak +fault characteristics further [40, 41]. On the contrary, we would apply +harmonic-Gaussian double-well potential SR to enhance weak fault characteristics of +machinery by using noise. +Four Rexnord ZA-2115 double row bearing run-to-failure experiments under the +rotating speed 2000 rpm and radial load 6000 lbs were performed to acquire the +bearing failure data by using accelerometers and a data acquisition card. The bearing +parameters are listed as below: the ball number 16, the pitch diameter 2.815 inches, +the contact angle 15.17 degrees and rolling element diameter 0.331 inches. The +bearing experimental rig is shown in Fig. 6(a) and the corresponding sensor +placement is illustrated in Fig. 6(b). This experimental rig is composed of four tested +bearings, an AC motor and rub belts [42]. In the bearing run-to-failure experiment, the +(a) +(b) + + 14 / 27 + +sampling frequency is 20 kHz and the sampling time is 1.024 seconds. + +Fig. 6 Bearing test rigs and sensor placement illustration: (a) bearing test rigs and (b) +sensor placement illustration. +All failures occurred after exceeding designed life time of the bearing which is +more than 100 million revolutions. The data set describes a test-to-failure experiment +and consists of individual files that are 1.024-seconds vibration signal snapshots +recorded at 10-minutes intervals. The recording duration is from February 12, 2004 +10:32:39 to February 19, 2004 06:22:39. At the end of the test-to-failure experiment, +outer race failure occurred in the tested bearing 1. The root mean square (RMS), an +effective health indicator, is often used to reflect the vibration intensity and monitor +the health state of bearings further. Therefore, RMS of bearing run-to-failure +experimental vibration data is calculated and depicted in Fig. 7 to observe the +degradation trend of the tested bearing 1. The degradation trend changes slowly with +slight fluctuation in the range of 0~88 hours and then raises into the larger RMS for +degradation marked in red dot in the zoomed RMS plot, suggesting that a tiny outer +race failure occurs in the early stage of the tested bearing 1. As time went on, it can be +seen from Fig. 7 that RMS keeps increasing, indicating that the outer race failure +becomes more and more severe. Finally, this test-to-failure experiment was stopped +because of strong vibration. In the test-to-failure experiment, the health state of the +tested bearing varies from normal to the early failure to severe failure to end of life, +which is consistent with the degradation trend reflected by RMS. +(a) +Accelerometers +Bearing 1 +Bearing 2 +Bearing 3 +Bearing 4 +Motor +Rub belts +(b) + + 15 / 27 + + +Fig. 7 RMS of the bearing run-to-failure vibration signals. +The raw vibration signal at the 88.83th hour marked in red dot and its frequency +and envelope spectrum are depicted in Fig. 8. We cannot observe the eye-catching +spectral peaks at the theoretical outer race/inner race/roller/cage fault characteristic +frequency and its harmonics from both the frequency spectrum in Fig. 8(b) and the +zoomed envelope spectrum in Fig. 8(c), which are submerged by other spectral peaks +from background noise and excited by other normal components. Although we have +completed the bearing run-to-failure experiment and observed that a failure occurred +at the outer race of the tested bearing 1 by disassembling four tested bearings, we +cannot judge what time a tiny failure occurs at the outer race of the tested bearing 1 +by virtue of the raw vibration signal and its spectrum in Fig. 8, which is very +important for early fault diagnosis and remaining useful life prediction. + + 16 / 27 + +Fig. 8 The vibration signal and its spectrum of outer race failure bearing at the early +stage: (a) the raw signal, (b) its frequency spectrum and (c) zoomed envelope +spectrum. +We apply the overdamped harmonic-Gaussian double-well potential SR to enhance +the weak fault characteristics in the early stage of the tested bearing 1. Figure 9 shows +the enhanced results of weak fault characteristics embedded in the raw vibration +signal, where the system parameters are given as 𝑘 = 1.1, 𝛼 = 1.2, 𝛽 = 0.24 and +the integral step is ℎ=0.035. The overdamped SR cannot be used to process +large-parameter signals directly and frequency-shifted and rescaling transform is +widely to solve it. Three key parameters of frequency-shifted and rescaling transform +in the overdamped harmonic-Gaussian double-well potential SR are given as below +by virtue of the theoretical outer race fault characteristic frequency 236.4 Hz that can +be calculated according to the structural parameters and rotating speed of the tested +bearing 1: the pass-band cut-off frequency 220 Hz, the stop-band cut-off frequency +200 Hz and the carrier frequency 200 Hz. These parameters in the frequency-shifted +and rescaling transform could be selected according to the reference [43]. One can + + 17 / 27 + +observe from Fig. 9 that the enhanced signal characterizes strong impacts and +dominant spectral peaks are at the outer race fault characteristic frequency and its +second harmonic of the tested bearing 1, suggesting that a tiny failure occurs at the +outer race of the tested bearing 1. However, the overdamped harmonic-Gaussian +double-well potential SR depends on the high-pass filter to perform the +frequency-shifted and rescaling transform, whose parameters are given artificially. +In the overdamped harmonic-Gaussian double-well potential SR-based enhanced +results, the low-frequency components of the raw vibration signal (<200Hz) have +been removed by using the frequency-shifted and rescaling transform. Moreover, the +overdamped harmonic-Gaussian double-well potential SR method would suppress the +components beyond the nonlinear filtering frequency band of overdamped +harmonic-Gaussian +double-well +potential +SR. +Although +overdamped +harmonic-Gaussian double-well potential SR method is able to utilize the noise +located in the nonlinear filtering frequency band of overdamped harmonic-Gaussian +double-well potential SR for enhancing weak fault characteristics, a part of noise is +removed. Therefore, the amplitude of the detected result in Fig. 9 is smaller than that +in Fig. 8. + +Fig. 9 Overdamped harmonic-Gaussian double-well potential SR-based enhanced +results: (a) the enhanced signal and (b) its zoomed frequency spectrum. +fouter +2fouter + + 18 / 27 + +Further, we apply the underdamped harmonic-Gaussian double-well potential SR to +enhance weak fault characteristics embedded in the raw vibration signal, as shown in +Fig. 10 whose system parameters are given as 𝑘=1.2, 𝛼=1.1, 𝛽=0.24, 𝛾=0.33 and +ℎ =0.035. There are obvious repetitive transients in the enhanced signal and +eye-catching spectral peaks at the outer race fault characteristic frequency and its +second harmonic in the zoomed frequency spectrum as shown in Fig. 10(b). +Compared with the overdamped harmonic-Gaussian double-well potential SR-based +results, the underdamped one characterizes the higher spectral peaks at the outer race +fault characteristic frequency and its second harmonic in the zoomed frequency +spectrum. + +Fig. 10 Underdamped harmonic-Gaussian double-well potential SR-based enhanced +results: (a) the enhanced signal and (b) its zoomed frequency spectrum. +For a comparison, we use the advanced robust local mean decomposition (RLMD) +[44, 45] to decompose the raw vibration signal of the tested bearing 1 into the product +functions (PFs) and a residual component (Res) for extracting weak fault +characteristics. The product functions and their zoomed envelope spectrum are shown +in Fig. 11(a) and Fig. 11(b), respectively. One cannot observe the obvious spectral +peaks at the outer race fault characteristic frequency and its harmonics from the +zoomed envelope spectrum. +fouter +2fouter +Rotating frequency +and its harmonics + + 19 / 27 + + +Fig. 11 RLMD-based results: (a) product functions and (b) their zoomed envelope +spectrum. +In addition to signal decomposition methods, signal denoising or signal filtering +methods also have been widely applied to extract weak fault characteristics of +machinery. Among them, wavelet transform [46, 47] is typical to obtain a denoised +version of the raw signal by thresholding the wavelet coefficients. Here, the maximal +overlap discrete wavelet transform is used to denoise the signal with soft thresholding, +level 3 and db4 wavelet. The denoised signal and its zoomed envelope spectrum are +shown in Fig. 12 and Fig. 13, respectively. It is found from Fig. 12 that the wavelet +transform can cancel strong background noise, but we cannot see any fault +characteristics at the first sight from the zoomed envelope spectrum in Fig. 13. +(a) +(b) +PF1 +PF2 +PF3 +PF4 +PF5 +Res +PF1 +PF2 +PF3 +PF4 +PF5 +Res +Time [s] +Frequency [Hz] +Amplitude [g] + +0.2 +h +0 +0.2 +0 +0.2 +0.4 +0.6 +0.8 +0.1 +0 +-0.1 +0 +0.2 +0.4 +0.6 +0.8 +1 +0.05 +0.05 +0 +0.2 +0.4 +0.6 +0.8 +0.02 +-0.02 +0 +0.2 +0.4 +0.6 +0.8 +×10-3 +?>> +-5 +-10 +0 +0.2 +0.4 +0.6 +0.8 +X10-3 +5 +0 +-5 +0 +0.2 +0.4 +0.6 +0.8 20 / 27 + + +Fig. 12 Undecimated wavelet transform-based denoised signals. + +Fig. 13 The zoomed envelope spectrum of undecimated wavelet transform-based +denoised signals. +A classical symptom of rotating machines failures in vibration signals is the +presence of repetitive transients. Antoni [48] proposed an infogram method to capture +the signature of repetitive transients in time domain, which is the variant of classical +fast kurtogram method. This method is used to process the raw vibration signal for +extracting repetitive transients in time domain. The corresponding results are shown +in Fig. 14. Although it can see the slight repetitive transients from the filtered signal in +Fig. 14(b), it is difficult for us to identify the period of repetitive transients because of +strong background noise and other normal vibration components. The above +conclusion could be further confirmed by the squared envelope amplitude sepctrum of + + 21 / 27 + +the filtered signal in Fig. 14(b), in which we cannot see the eye-catching spectral +peaks at the outer race fault characteristic frequency and its harmonics. + +Fig. 14 The detected results using infogram: (a) infogram and (b) the filtered signal +and its squared envelope amplitude sepctrum. + +5. Conclusions +The overdamped and underdamped harmonic-Gaussian double-well potential SR +are investigated by deriving the output SNR and SPD functions. It is found that both +noise-induced SR and parameter-induced SR can be activated in the overdamped and +underdamped harmonic-Gaussian double-well potential systems. Moreover, since the +harmonic-Gaussian double-well potential in the range of 𝑘 ≥ 2𝛼𝛽 loses the stability, +we can observe the antiresonance, whereas adding the damped factor into the +overdamped harmonic-Gaussian double-well potential system can change the stability, +resulting that the antiresonance disappears. Above conclusion is applicable under all +parameters. +Finally, +we +apply +both +the +overdamped +and +underdamped +harmonic-Gaussian double-well potential SR to enhance weak fault characteristics of +bearings for incipient fault identification, where the corresponding parameters would +be adjusted or optimized instead of all parameters are applicable to activate the +optimal SR. The weak fault characteristics are enhanced successfully to identify the +early failure of bearings, which somewhat outperforms to the RLMD, wavelet +transform and infogram-based results. But the SR-based methods depend on the prior +knowledge of the signals to be detected or structural parameters and rotating speeds of +bearings, and cannot detect unknown multiple-frequency and multiple-component +(a) +(b) + + 22 / 27 + +coupled signals without any prior knowledge. Therefore, we would study the +SR-based signal decomposition method by using noise to decouple and detect +unknown +multiple-frequency +and +multiple-component +signals, +especially +time-varying nonstationary signals in the future. + +Acknowledgments +This research was supported by Foundation of the State Key Laboratory of +Performance Monitoring and Protecting of Rail Transit Infrastructure of East China +Jiaotong University (HJGZ2021114), Laboratory of Yangjiang Offshore Wind Power +(YJOFWD-OF-2022A08), Zhejiang Provincial Natural Science Foundation of China +(LQ22E050003), National Natural Science Foundation of China (52205569), Ningbo +Science and Technology Major Project (2020Z110, 2022Z057, 2022Z002), National +Natural Science Foundation of China (51905349, 62001210, U2013603), Natural +Science Foundation of Guangdong Province (2022A1515010126, 2020A1515011509), +Ningbo Natural Science Foundation (2022J098) and also sponsored by K.C. Wong +Magna Fund in Ningbo University. The Spanish State Research Agency (AEI) and the +European +Regional +Development +Fund +(ERDF) +under +Project +No. +PID2019-105554GB-I00 is also aknowledged. + +Conflicct of Interest +The authors declare that they have no conflict of interest. + +Data availability +The datasets generated during and/or analysed during the current study are +available from the corresponding author on reasonable request. + +References +[1] Rai A, Upadhyay S H. A review on signal processing techniques utilized in the +fault diagnosis of rolling element bearings, Tribology International, 2016, 96: +289-306. + + 23 / 27 + +[2] Benzi R, Sutera A, Vulpiani A. The mechanism of stochastic resonance, Journal +of Physics A: Mathematical and General, 1981, 14(11): L453-L457. +[3] Gammaitoni L, Hanggi P, Jung P, et al. Stochastic resonance, Reviews of Modern +Physics, 1998, 70(1): 223-287. +[4] Qiao Z, Lei Y, Li N. 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Wavelet transform based on inner product in fault +diagnosis of rotating machinery: A review, Mechanical Systems and Signal +Processing, 2016, 70: 1-35. +[47] Abbasion S, Rafsanjani A, Farshidianfar A, et al. Rolling element bearings +multi-fault classification based on the wavelet denoising and support vector +machine, Mechanical Systems and Signal Processing, 2007, 21(7): 2933-2945. +[48] Antoni J. The infogram: Entropic evidence of the signature of repetitive transients, +Mechanical Systems and Signal Processing, 2016, 74: 73-94. + + + + + + + + +Zhejiang Provincial Key Laboratory of Part Rolling Technology + +School of Mechanical Engineering and Mechanics • Ningbo University + +Ningbo University + + +July 24, 2022 + +RE: “Harmonic-Gaussian double-well potential stochastic resonance with its +application to enhance weak fault characteristics of machinery” by Zijian Qiao, Shuai +Chen, Zhihui Lai, Shengtong Zhou and Miguel A. F. Sanjuán (Manuscript Number: +NODY-D-22-01167) + + + +Dear Editor, + +We have carefully revised our paper taking into account your suggestions and the +comments of the reviewers. We have uploaded the revised version and the revision +notes. Thank you very much for processing our paper. + +We appreciate very much the constructive comments and suggestions provided by the +reviewers. They have been incorporated in the revised version of this paper. Major +changes made in the paper are marked in blue. The following summarizes our +response to each point raised by each reviewer. + +We would like to thank the three reviewers for their valuable comments and +constructive suggestions to improve the quality of this paper. We have fully +considered their comments and suggestions and made revisions accordingly. The +major revisions are highlighted by BLUE color. The point-to-point explanations and +revisions are listed as follow. + +We have taken into full consideration all comments of the three referees and made a +thorough revision of the paper. + + + +Sincerely yours, + +Zijian Qiao Ph.D +Shuai Chen M.S. +Zhihui Lai Ph.D +Shengtong Zhou Ph.D +Miguel A. F. Sanjuán Ph.D +Cover Letter +Click here to access/download;attachment to +manuscript;Cover Letter.doc +Click here to view linked References + +波 +大 +漢 +Page 1 of 1 +Highlights + +RE: “Harmonic-Gaussian double-well potential stochastic resonance with its application to +enhance weak fault characteristics of machinery” by Zijian Qiao, Shuai Chen, Zhihui Lai, +Shengtong Zhou and Miguel A. F. Sanjuán + + Harmonic-Gaussian double-well potential SR is investigated by deriving and measuring +the output SNR. + Steady-state probability density functions are used to evaluate the transition rates of +particles in the harmonic-Gaussian double-well potential. + Parameter-induced SR, noise-induced SR and antiresonance are observed by analyzing +the output SNR. + Harmonic-Gaussian double-well potential SR is applied to enhance weak fault +characteristics of machinery successfully. +Highlights +Click here to access/download;attachment to +manuscript;Highlights.doc +Click here to view linked References + +Declaration of Interest Statement +The authors declare that they have no conflict of interest. +Declaration of Interest Statement +Click here to access/download;attachment to +manuscript;Declaration of Interest Statement.docx +Click here to view linked References + +Manuscript Number: NODY-D-22-01167R1 +Title: Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance +weak fault characteristics of machinery +Response to Editor +There are still some minor comments raised by one of the reviewers needed to be addressed. A minor +revision is recommended. +Response: We appreciate the constructive comments from two reviewers. According to their +comments and suggestions, we have made a thorough revision for the manuscript and have addressed +all points raised by each reviewer. The major changes made in the manuscript are marked in BLUE +color. We also include the major changes of the manuscript into the response point by point. For +convenient review, the page numbers or paragraph numbers of the revision in the manuscript are +cited below. +We hope that this revised submission is satisfactory. The authors thank editors and anonymous +reviewers for their valuable and helpful comments to revise and improve our manuscript. +Compressed File +Click here to access/download;Compressed File;Response to +Reviewers.docx + +Manuscript Number: NODY-D-22-01167R1 +Title: Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance +weak fault characteristics of machinery +Response to Reviewer #3 +The authors have correctly taken into consideration the reviewers comments. +Response: Thanks for your recommendation. + + +Manuscript Number: NODY-D-22-01167R1 +Title: Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance +weak fault characteristics of machinery +Response to Reviewer #5 +The paper presents that the overdamped or underdamped harmonic Gaussian double-well potential +SR methods characterize a better performance to detect a weak signal. The work is organized in a +clear form, but the technical content looks not high and there are important aspects that are not +discussed. +1. Whether the analysis conclusion obtained is the conclusion under these special parameters or +whether all parameters are applicable? Please give some explanations. +Response: According to the comments of the reviewer #5, we think that two analysis conclusion +obtained could be illustrated whether under these special parameters or all parameters. +In Sections 2 and 3: Overdamped and underdamped harmonic-Gaussian double-well +potential SR +We investigate the SR in the cases of overdamped and underdamped harmonic-Gaussian +double-well potential systems subjected to noise and a periodic signal. We derive and measure the +analytic expression of the output signal-to-noise ratio (SNR) and the steady-state probability density +(SPD) function under approximate adiabatic conditions. When the harmonic-Gaussian double-well +potential loses its stability, we can observe the antiresonance phenomenon, whereas adding the +damped factor into the overdamped system can change the stability of the harmonic-Gaussian +double-well potential, resulting that the antiresonance behavior disappears in the underdamped +system. Although above analysis conclusion is obtained under these special parameters, other +parameters would depict the same findings. As a result, the analysis conclusion obtained in two + +sections is applicable under all parameters. +In Section 4: Application of harmonic-Gaussian double-well potential SR to enhance weak +fault characteristics of machinery +Harmonic-Gaussian double-well potential stochastic resonance is a typical nonlinear filter with the +adjusting parameters in which the noise embedded in a signal is able to be utilized to enhance weak +useful information by activating the stochastic resonance phenomenon. The stochastic resonance +phenomenon could be activated when the optimal matching among the weak useful information, +noise and these parameters of stochastic resonance. For a different signal, therefore, these parameters +of the harmonic-Gaussian double-well potential stochastic resonance must be tuned to activate the +stochastic resonance phenomenon for enhancing weak useful information by using noise. As a result, +applying harmonic-Gaussian double-well potential stochastic resonance to enhance weak fault +characteristics of machinery, these parameters of harmonic-Gaussian double-well potential stochastic +resonance would be adjusted or optimized instead of all parameters are applicable to activate the +optimal stochastic resonance phenomenon. (See the conclusion in Section 5 page 21, which is +marked in BLUE) + +2. "Noise is ubiquitous and unwanted in detecting weak signals", This sentence is repeated and can +be deleted. +Response: Thanks for your suggestions. We have deleted it in Abstract. (See the abstract in page 1, +which is marked in BLUE) + + +3 "Key words" write too long. +Response: Thanks for your suggestions. We have reduced the key words as below: The benefits of +noise, weak signature enhancement, fault identification, fault diagnosis. (See the key words in page 2, +which is marked in BLUE) + +4 "The recording duration is from February 12, 2004 10:32:39 to February 19, 2004 06:22:39". +Why was it 18 years ago？ +Response: That is because Four Rexnord ZA-2115 double row bearing run-to-failure experiments +under the rotating speed 2000 rpm and radial load 6000 lbs were performed in 2004 year. In future +work, we would perform and conduct new bearing run-to-failure experiments. Now, our team is +designing the new experimental platform and project to acquire new bearing and gear vibration data. +Thanks for your understanding. + + diff --git a/49E2T4oBgHgl3EQfOgYr/content/tmp_files/load_file.txt b/49E2T4oBgHgl3EQfOgYr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d3736149c37fc08df272eb8cc9a597368685991 --- /dev/null +++ b/49E2T4oBgHgl3EQfOgYr/content/tmp_files/load_file.txt @@ -0,0 +1,561 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf,len=560 +page_content='Nonlinear Dynamics Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance weak fault characteristics of machinery --Manuscript Draft-- Manuscript Number: NODY-D-22-01167R2 Full Title: Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance weak fault characteristics of machinery Article Type: Original Research Keywords: The benefits of noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' weak signature enhancement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' fault identification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' fault diagnosis Corresponding Author: Zijian Qiao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=" Ningbo University Ningbo, CHINA Corresponding Author Secondary Information: Corresponding Author's Institution: Ningbo University Corresponding Author's Secondary Institution: First Author: Zijian Qiao, Ph." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' First Author Secondary Information: Order of Authors: Zijian Qiao, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Shuai Chen Zhihui Lai Shengtong Zhou Miguel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Sanjuán Order of Authors Secondary Information: Funding Information: Foundation of the State Key Laboratory of Performance Monitoring and Protecting of Rail Transit Infrastructure of East China Jiaotong University (HJGZ2021114) Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Zijian Qiao Zhejiang Provincial Natural Science Foundation of China (LQ22E050003) Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Zijian Qiao National Natural Science Foundation of China (62001210, 51905349) Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Zhihui Lai The Spanish State Research Agency (AEI) and the European Regional Development Fund (ERDF) (PID2019-105554GB-I00) Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Miguel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Sanjuán Abstract: Noise would give rise to incorrect filtering frequency-band selection in signal filtering- based methods including fast kurtogram, teager energy operators and wavelet packet transform filters and meanwhile would result in incorrect selection of useful components and even mode mixing, end effects and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' in signal decomposition- based methods including empirical mode decomposition, singular value decomposition and local mean decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' On the contrary, noise in stochastic resonance (SR) is beneficial to enhance weak signals of interest embedded in signals with strong background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Taking into account that nonlinear systems are crucial ingredients to activate the SR, here we investigate the SR in the cases of overdamped and underdamped harmonic-Gaussian double-well potential systems subjected to noise and a periodic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We derive and measure the analytic expression of the output signal-to-noise ratio (SNR) and the steady-state probability density (SPD) function Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation under approximate adiabatic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' When the harmonic-Gaussian double-well potential loses its stability, we can observe the antiresonance phenomenon, whereas adding the damped factor into the overdamped system can change the stability of the harmonic-Gaussian double-well potential, resulting that the antiresonance behavior disappears in the underdamped system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Then, we use the overdamped and underdamped harmonic-Gaussian double-well potential SR to enhance weak useful characteristics for diagnosing incipient rotating machinery failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Theoretical and experimental results show that adjusting both noise intensity and system parameters can activate overdamped and underdamped harmonic-Gaussian double-well potential SR in which there is a bell-shaped peak for the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Additionally, the underdamped harmonic-Gaussian double-well potential SR is independent of frequency-shifted and rescaling transform to process large machine parameter signals and outperforms the overdamped one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Finally, comparing the advanced robust local mean decomposition (RLMD) method based on signal decomposition and the wavelet transform method based on noise cancellation or infogram method based on signal filtering, the overdamped or underdamped harmonic-Gaussian double-well potential SR methods characterize a better performance to detect a weak signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Fault characteristics in the early stage of failures are successful in improving the incipient fault characteristic identification of rolling element bearings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Response to Reviewers: Please see the attached "response to reviewers".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Order of Authors (with Contributor Roles): Zijian Qiao, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (Funding acquisition: Supporting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Validation: Lead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Writing – original draft: Lead) Shuai Chen (Data curation: Equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Visualization: Lead) Zhihui Lai (Investigation: Equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Visualization: Equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Writing – review & editing: Supporting) Shengtong Zhou (Data curation: Lead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Formal analysis: Equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Investigation: Equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Writing – review & editing: Equal) Miguel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Sanjuán (Formal analysis: Equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Funding acquisition: Equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Methodology: Equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Writing – review & editing: Lead) Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation 1 / 27 Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance weak fault characteristics of machinery Zijian Qiao1,2,3,4, \uf02a, Shuai Chen1, Zhihui Lai5, Shengtong Zhou2, Miguel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Sanjuán6 1 School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo 315211, China 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' State Key Laboratory of Performance Monitoring and Protecting of Rail Transit Infrastructure, East China Jiaotong University, Nanchang 330013, China 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Laboratory of Yangjiang Offshore Wind Power, Yangjiang 529599, Guangdong, China 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Zhejiang Provincial Key Laboratory of Part Rolling Technology, Ningbo 315211, China 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Guangdong Provincial Key Laboratory of Micro/Nano Optomechatronics Engineering, College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen 518060, China 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, Móstoles, 28933, Madrid, Spain Abstract: Noise would give rise to incorrect filtering frequency-band selection in signal filtering-based methods including fast kurtogram, teager energy operators and wavelet packet transform filters and meanwhile would result in incorrect selection of useful components and even mode mixing, end effects and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' in signal decomposition-based methods including empirical mode decomposition, singular value decomposition and local mean decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' On the contrary, noise in stochastic resonance (SR) is beneficial to enhance weak signals of interest embedded in signals with strong background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Taking into account that nonlinear systems are crucial ingredients to activate the SR, here we investigate the SR in the cases of overdamped and underdamped harmonic-Gaussian double-well potential systems subjected to noise and a periodic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We derive and measure the analytic expression of the output signal-to-noise ratio (SNR) and the steady-state probability \uf02aCorresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' E-mail address: zijianqiao@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='com, qiaozijian@nbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='cn (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Qiao).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Manuscript Click here to access/download;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='Manuscript;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='Manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='docx Click here to view linked References 2 / 27 density (SPD) function under approximate adiabatic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' When the harmonic-Gaussian double-well potential loses its stability, we can observe the antiresonance phenomenon, whereas adding the damped factor into the overdamped system can change the stability of the harmonic-Gaussian double-well potential, resulting that the antiresonance behavior disappears in the underdamped system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Then, we use the overdamped and underdamped harmonic-Gaussian double-well potential SR to enhance weak useful characteristics for diagnosing incipient rotating machinery failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Theoretical and experimental results show that adjusting both noise intensity and system parameters can activate overdamped and underdamped harmonic-Gaussian double-well potential SR in which there is a bell-shaped peak for the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Additionally, the underdamped harmonic-Gaussian double-well potential SR is independent of frequency-shifted and rescaling transform to process large machine parameter signals and outperforms the overdamped one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Finally, comparing the advanced robust local mean decomposition (RLMD) method based on signal decomposition and the wavelet transform method based on noise cancellation or infogram method based on signal filtering, the overdamped or underdamped harmonic-Gaussian double-well potential SR methods characterize a better performance to detect a weak signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Fault characteristics in the early stage of failures are successful in improving the incipient fault characteristic identification of rolling element bearings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Key words: The benefits of noise, weak signature enhancement, fault identification, fault diagnosis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Introduction Noise is ubiquitous but unwanted in detecting weak signals [1], but noise in biological systems can be used to amplify weak signals embedded by a strong background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Such an ingenious phenomenon is observed in a bistable nonlinear system, namely stochastic resonance (SR) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' SR is a kind of synchronization mechanism among the nonlinear systems, noise and a weak periodic signal, which 3 / 27 takes place to activate the SR for amplifying weak useful signals [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' SR has been investigated from theory to engineering application widely [4-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Among three ingredients for activating SR including noise, nonlinear systems and weak useful signals, nonlinear systems are crucial ingredients for extracting weak useful signals and moreover can harvest the energy of noise located at the whole frequency band of a noisy signal to enhance or amplify a weak useful signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' For this purpose, most of scholars pay attention to exploring the behaviors of SR in novel nonlinear systems from bistable [7] to multistable ones [8-10], from overdamped [11] and underdamped [12] to fractional-order [13] ones, and even from cascaded [14] and coupled [15, 16] to time-delayed feedback [17] ones and biological systems [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Because the bistable system is most classical among them, it has been investigated, such as classical bistable potential overdamped systems, noisy confined bistable potential overdamped systems [20], asymmetric bistable potential overdamped systems [21], classical bistable potential underdamped systems, noisy bistable potential fractional-order systems [22] and E-exponential potential underdamped systems [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The E-exponential potential named by the references [23, 24] is a narrow version of the harmonic-Gaussian double-well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The references above show that overdamped and underdamped harmonic-Gaussian double-well potential SR has not been studied systematically in theory and further applied to enhance incipient fault identification of machinery for providing a tutorial of other readers and researchers on the SR in the overdamped and underdamped systems with novel generalized double-well potentials yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Even, the comparison between overdamped and underdamped harmonic-Gaussian double-well potential SR has not been made in theory and engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Therefore, this paper attempts to investigate the SR in the overdamped and underdamped harmonic-Gaussian double-well potential systems theoretically and then apply it to enhance weak fault characteristics and diagnose incipient faults of machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Additionally, some comparisons with other advanced signal processing techniques including signal decomposition-based and noise cancellation or signal filtering-based methods for enhancing weak fault characteristics of machinery are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 4 / 27 The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Section 2 and Section 3 investigate the overdamped and underdamped harmonic-Gaussian double-well potential SR by deriving the analytic expressions of signal-to-noise ratio (SNR) and steady-state probability density (SPD) functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' In Section 4, we apply the overdamped and underdamped harmonic-Gaussian double-well potential SR to enhance weak fault characteristics and incipient fault identification of rolling element bearings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Finally, conclusions are drawn in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Overdamped harmonic-Gaussian double-well potential SR The overdamped Langevin equation driven by a harmonic-Gaussian double-well potential under the action of random noise and a periodic signal can be described as [25] d𝑦 d𝑥 = − 𝜕𝑈(𝑥) 𝜕𝑥 + 𝐴 cos(𝜔0𝑡) + 𝜀(𝑡) (1) where 𝐴 and ω0 are the amplitude and angular frequency of the periodic signal respectively, and 𝜀(𝑡) is the Gaussian white noise with mean zero and variance 𝐷 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' noise intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The harmonic-Gaussian double-well potential which is a variant of a double-well potential can be expressed as [26] 𝑈(𝑥) = 𝑘 2 𝑥2 + 𝛼exp(−𝛽𝑥2) (2) where two stable states and one unstable state are located at 𝑥± = ±√ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽 ⁄ and 𝑥𝑢 = 0 respectively, and the barrier height is ∆𝑈 = α − 𝑘[1 + ln(2𝛼𝛽 𝑘 ⁄ )] (2𝛽) ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' To ensure the stability of the harmonic-Gaussian double-well potential, this condition ln(2𝛼𝛽 𝑘 ⁄ ) > 0 must be satisfied, further 𝑘 < 2αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' When 𝑘 = 1 , Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 1(a) shows the harmonic-Gaussian double-well potential under different system parameter sets (𝛼, 𝛽), while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 1(b) depicts those with varying 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' It is seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 1(a) that adjusting the system parameter 𝛽 controls the potential-well width whereas the potential-barrier height nearly keeps unchanged, but varying 𝛼 changes the potential-barrier height whereas the potential-well width nearly remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Such a behavior is helpful to tune the potential-well width 5 / 27 and depth individually to activate the optimal harmonic-Gaussian double-well potential SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Meanwhile, it is found from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 1(b) that adjusting 𝑘 can also change the slope of the harmonic-Gaussian double-well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 1 Harmonic-Gaussian double-well potentials under different parameter sets (a) (𝛼, 𝛽) and (b) (𝛼, 𝛽, 𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The Langevin equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (1) can be transformed as further [27] ∂ρ(𝑥,𝑡) ∂𝑡 = − 𝜕 𝜕𝑥 [−𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + 𝐴 cos(𝜔0𝑡)]𝜌(𝑥, 𝑡) + 𝐷 𝜕2 𝜕𝑥2 𝜌(𝑥, 𝑡) (3) where 𝜌(𝑥, 𝑡) is the probability density function (PDF) of the stochastic process 𝑥(𝑡) which denotes the transition trajectory of Brownian particles in the harmonic-Gaussian double-well potential as time varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The corresponding SPD function can be denoted as 𝜌𝑠(𝑥, 𝑡) = 𝑁(𝑡) √𝐷 exp [− ∅(𝑥,𝑡) 𝐷 ] (4) where 𝑁(𝑡) is the normalization constant and 𝑁(𝑡) = √𝐷 ∫ exp[−∅(𝑥, 𝑡) 𝐷 ⁄ ]d𝑥 ∞ −∞ ⁄ , and ∅(𝑥, 𝑡) is the generalized potential ∅(𝑥, 𝑡) = 𝑈(𝑥) − 𝑥𝐴 cos(𝜔0𝑡) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (5) Assuming that the periodic signal 𝐴 cos(𝜔0𝑡) can satisfy the requirement of small parameters under approximate adiabatic conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=', 𝜔0 is larger than the characteristic relaxation time in double potential wells [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' the transition rates between the two stable states are given by the Kramers-like formulas [29] 𝑊±(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡) = √|𝑈′′(𝑥±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='𝑡)𝑈′′(𝑥𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='𝑡)| 2π exp [ ∅(𝑥±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='𝑡)−∅(𝑥𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='𝑡) 𝐷 ] (6) where the notation | ∙ | denotes the absolute value and (a) (b) 6 / 27 𝑈′′(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡) = 𝑘 − 2𝛼𝛽exp(−𝛽𝑥2)(1 − 2𝛽𝑥2) 𝑈′′(𝑥𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡) = 𝑘 − 2𝛼𝛽 𝑈′′(𝑥±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡) = 2𝑘ln ( 2𝛼𝛽 𝑘 ) (7) ∅(𝑥𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡) = 𝑈(𝑥𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡) − 𝑥𝑢𝐴 cos(𝜔0𝑡) = 𝛼 ∅(𝑥±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡) = 𝑈(𝑥±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡)−𝑥±𝐴 cos(𝜔0𝑡) = 𝑘 2𝛽 (1 + ln 2𝛼𝛽 𝑘 ) ∓ 𝐴 cos(𝜔0𝑡) √ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽 When we introduce Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (7) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (6), we can obtain 𝑊±(𝑥, 𝑡) = √𝑘(2𝛼𝛽 − 𝑘)ln(2𝛼𝛽 𝑘 ⁄ ) √2π × exp [− 𝛼 𝐷 + 𝑘(1+ln(2𝛼𝛽 𝑘 ⁄ )) 2𝛽𝐷 ∓ 𝐴 cos(𝜔0𝑡) √ ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽𝐷2 ] (8) Furthermore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (8) can be transformed as 𝑊±(𝑥, 𝑡) = 𝑓(𝜇 ± 𝜂0 cos(𝜔0𝑡)) (9) where 𝜇 = 𝛼 𝐷 − 𝑘 2𝛽𝐷 (1 + ln 2𝛼𝛽 𝑘 ) (10) 𝜂0 = 𝐴 𝐷 √ ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽 (11) Thus, we can simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (8) as 𝑊±(𝑥, 𝑡) = 𝑓(𝜇 ± 𝜂0 cos(𝜔0𝑡)) = √2𝑘(2𝛼𝛽−𝑘)ln2𝛼𝛽 𝑘 2π exp[−(𝜇 ± 𝜂0 cos(𝜔0𝑡))] (12) The response of the nonlinear system in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (1) can be quantified using a classical measure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=', SNR [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' To derive its analytic expression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' the power spectral density of the system response can be described as 𝑆(Ω) = [1 − 𝛼12𝜂02 2(𝛼02+𝜔02)] ( 4𝑐2𝛼0 𝛼02+𝜔02) + π𝑐2𝜂02𝛼12 𝛼02+Ω2 [𝛿(Ω − 𝜔0) + 𝛿(Ω + 𝜔0)] (13) where 𝑐 = √ ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽 (14) 𝛼1 = 𝛼0 = √2𝑘ln(2𝛼𝛽 𝑘 ⁄ )(2𝛼𝛽−𝑘) π exp(−𝜇) (15) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' the output SNR of the response of the overdamped harmonic-Gaussian double-well potential system can be derived as 7 / 27 SNR = π𝑐2𝛼12𝜂02 𝛼02+Ω2 |Ω=𝜔0 × 𝛼02+𝜔02 4𝑐2𝛼0 [1 − 𝛼12𝜂02 2(𝛼02+𝜔02)] −1 = π𝛼1𝜂02 4 [1 − 𝛼12𝜂02 2(𝛼02+𝜔02)] −1 (16) Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' we can analyze the function between the output SNR and system parameters using the analytic expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Figure 2 shows the output SNR of overdamped harmonic-Gaussian double-well potential SR as system parameters and noise intensity vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2(a) that the output SNR is a nonmonotonic function of noise intensity 𝐷 under different 𝑘 and the peak value of output SNR increases when 𝑘 raises, suggesting that adjusting 𝑘 is able to activate the SR in the overdamped harmonic-Gaussian double-well potential system for improving the output SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Similarly, adjusting 𝛼 and 𝛽 can also maximize the output SNR, and the peak value of the output SNR declines as 𝛼 or 𝛽 increases but the resonant noise intensity at the peak value becomes larger, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2(c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We visualize the two-dimensional function among SNR and two of system parameters (𝛼, 𝛽, 𝑘), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2(d), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' One can observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2(d) that a moderate parameter set (𝛼, 𝑘) can improve the SNR of a given signal, whereas there exists a negative output SNR because the harmonic-Gaussian double-well potential loses its stability when 𝑘 ≥ 2𝛼𝛽, resulting in an antiresonance phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Meanwhile, we fix 𝑘 to express the output SNR as a function of (𝛼, 𝛽) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2(e), indicating that only an optimal matching between 𝛼 and 𝛽 can activate the overdamped harmonic-Gaussian double-well potential SR to enhance the weak periodic signal embedded by a strong background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Similarly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2(f) also demonstrates that such a parameter matching is necessary to activate the overdamped harmonic-Gaussian double-well potential SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' When 𝑘 ≥ 2𝛼𝛽, one can also see the antiresonance from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2(e) and 2(f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The above results demonstrate that the optimal parameter matching among 𝑘, 𝛼 and 𝛽 is able to maximize the SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 8 / 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2 SNR of overdamped harmonic-Gaussian double-well potential SR varies with system parameters and noise intensity: SNR as a function of noise intensity under different 𝑘 in (a), 𝛽 in (b) and 𝛼 in (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' SNR as a two-dimensional function of (𝑘, 𝛼) in (d), (𝛽, 𝛼) in (e) and (𝑘, 𝛽) in (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Figure 3 depicts the SPD function and the corresponding system responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The SPD indicates the probability of Brownian particles to reside in double potential wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' It is found from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 3(a) that when 𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='3 the particles oscillate at the right potential well located at 𝑥+ = √ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽 ⁄ for activating intra-well SR, which is demonstrated by the system response in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 3(b) further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' When we increase the noise intensity 𝐷, the particles can jump across the potential barrier to go back and forth in double wells for activating the inter-well SR marked in red in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 3(a), whose system response characterizes the eye-catching period marked in red in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' When the noise intensity is fixed as 𝐷 = 3, two peaks of SPD decline and the corresponding (a) (b) (c) (d) (e) (f) 9 / 27 system response marked in green becomes noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 3 SPD functions and the corresponding system responses of overdamped harmonic-Gaussian double-well potential SR under different noise intensity: (a) the SPD functions and (b) the corresponding system responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Underdamped harmonic-Gaussian double-well potential SR The underdamped harmonic-Gaussian double-well potential system subjected to a periodic signal and noise can be described as [31] d2𝑥 d𝑡2 + 𝛾 d𝑥 d𝑡 = − 𝜕𝑈(𝑥) 𝜕𝑥 + 𝐴 cos(𝜔0𝑡) + 𝜉(𝑡) (17) where 𝛾 is the damped factor and 𝛾 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Equation (17) can be transformed as [32] { d𝑥 d𝑡 = 𝑦 d𝑦 d𝑡 = −𝛾𝑦 − 𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + 𝐴 cos(𝜔0𝑡) + 𝜉(𝑡) (18) Supposing that 𝐴 = 0, 𝐷 = 0, d𝑥 d𝑡 ⁄ = 0 and d𝑥 d𝑡 = 0 ⁄ , we can obtain three singular points (𝑥± 𝑦±) = (±√ ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽 0 ) , (𝑥𝑢 𝑦𝑢) = (0 0) (19) Let 𝜕𝑈(𝑥, 𝑦) 𝜕𝑥 ⁄ and 𝜕𝑈(𝑥, 𝑦) 𝜕𝑦 ⁄ mark as 𝑈𝑥(𝑥, 𝑦) and 𝑈𝑦(𝑥, 𝑦) respectively, and then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (18) can be rewritten as { 𝑈𝑥(𝑥, 𝑦) = 𝑦 𝑈𝑦(𝑥, 𝑦) = −𝛾𝑦 − 𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + 𝐴 cos(𝜔0𝑡) + 𝜉(𝑡) (20) The linearization matrix of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (18) can be calculated as (𝑈𝑥𝑥(𝑥, 𝑦) 𝑈𝑥𝑦(𝑥, 𝑦) 𝑈𝑦𝑥(𝑥, 𝑦) 𝑈𝑦𝑦(𝑥, 𝑦)) = ( 0 1 −𝑘 + 2𝛼𝛽exp(−𝛽𝑥2)[1 − 2𝛽𝑥2exp(−𝛽𝑥2)] −𝛾) (a) (b) 10 / 27 (21) Further, the linearization matrix at the singular points (±√ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽 ⁄ , 0) can be denoted as (𝑈𝑥𝑥(𝑥, 𝑦) 𝑈𝑥𝑦(𝑥, 𝑦) 𝑈𝑦𝑥(𝑥, 𝑦) 𝑈𝑦𝑦(𝑥, 𝑦)) | 𝑥=±√ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽 ,𝑦=0 = ( 0 1 − 𝑘2 𝛼𝛽 ln ( 2𝛼𝛽 𝑘 ) −𝛾) (22) By solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (22), the corresponding eigenvalues are calculated as 𝛽1,2 = −𝛾±√𝛾2−4𝑘2 𝛼𝛽 ln(2𝛼𝛽 𝑘 ) 2 (23) Similarly, the linearization matrix at the singular point (0,0) is (𝑈𝑥𝑥(𝑥, 𝑦) 𝑈𝑥𝑦(𝑥, 𝑦) 𝑈𝑦𝑥(𝑥, 𝑦) 𝑈𝑦𝑦(𝑥, 𝑦)) |𝑥=0,𝑦=0 = ( 0 1 −𝑘 + 2𝛼𝛽 −𝛾) (24) The corresponding eigenvalues to the linearization matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (24) are 𝜆1,2 = −𝛾±√𝛾2+4(2𝛼𝛽−𝑘) 2 (25) Assuming that 𝜌(𝑥, 𝑦, 𝑡) is the PDF of the stochastic process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (18), the corresponding the Fokker-Planck equation is [33] 𝜕𝜌(𝑥,𝑦,𝑡) 𝜕𝑡 = − 𝜕𝑦 𝜕𝑥 𝜌(𝑥, 𝑦, 𝑡) − 𝜕 𝜕𝑦 [−𝛾𝑦 − 𝑘𝑥 + 2𝛼𝛽𝑥exp(−𝛽𝑥2) + 𝐴 cos(𝜔0𝑡)]𝜌(𝑥, 𝑦, 𝑡) + 𝛾𝐷 𝜕2 𝜕𝑦2 𝜌(𝑥, 𝑦, 𝑡) (26) Then, the corresponding SPD function to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (18) can be denoted as 𝜌s(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡) = 𝑁(𝑡)exp [− 1 𝐷 ( 1 2 𝑦2 + 𝑘 2 𝑥2 + 𝛼exp(−𝛽𝑥2) − 𝑥𝐴 cos(𝜔0𝑡))] (27) where 𝑁(𝑡) stands for the normalization constant [34] 𝑁(𝑡) = 1 ∫ ∫ exp[−𝑈̂(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='𝑡) 𝐷 ]d𝑥d𝑦 ∞ −∞ ∞ −∞ (28) in which 𝑈̂(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡) denotes the generalized potential 𝑈̂(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑡) = 1 2 𝑦2 + 𝑘 2 𝑥2 + 𝛼exp(−𝛽𝑥2) − 𝑥𝐴 cos(𝜔0𝑡) (29) The transition rates of particles at the singular points (𝑥±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 𝑦±) = (±√ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽 ⁄ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 0) can be calculated as [35] 𝑊±(𝑡) = √𝛽1𝛽2 2π √− 𝜆1 𝜆2 exp [ 1 𝐷 ( 𝑘 2𝛽 (1 + ln ( 2𝛼𝛽 𝑘 )) − 𝛼 ∓ √ ln(2𝛼𝛽 𝑘 ⁄ ) 𝛽 𝐴 cos(𝜔0𝑡))] (30) 11 / 27 Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' the analytic expression of the output SNR of the response of the underdamped harmonic-Gaussian double-well potential system in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (17) is derived as SNR = π𝑐2𝛼12𝜂02 𝛼02+Ω2 |Ω=𝜔0 × 𝛼02+𝜔02 4𝑐2𝛼0 [1 − 𝛼12𝜂02 2(𝛼02+𝜔02)] −1 = π𝛼1𝜂02 4 [1 − 𝛼12𝜂02 2(𝛼02+𝜔02)] −1 (31) where 𝛼1 = 𝛼0 = √𝛽1𝛽2 π √− 𝜆1 𝜆2 exp(−𝑢) (32) Figures 4(a)-4(d) show the output SNR as noise intensity 𝐷 varies under different system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' It is found from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 4(a) that the output SNR increases and then decreases as noise intensity increases, suggesting that a noise-induced underdamped harmonic-Gaussian double-well potential SR happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Moreover, increasing 𝑘 can maximize the output SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Like this, adjusting 𝛾, 𝛼 and 𝛽 can also improve the output SNR as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 4(b), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 4(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 4(d) respectively, where the peak value of output SNR and the resonant noise intensity are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Figures 4(e)-4(h) show the output SNR as the function of system parameters for a given signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Adjusting the system parameters can activate the underdamped harmonic-Gaussian double-well potential SR, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 4(e)-4(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Different from the overdamped harmonic-Gaussian double-well potential SR, it is noticed from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 4(e)-4(h) that the antiresonance disappears in the underdamped one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' That is because the damped factor changes the stability of the nonlinear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 12 / 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 4 SNR of underdamped harmonic-Gaussian double-well potential SR varies with system parameters and noise intensity: SNR as a function of noise intensity under different 𝑘 in (a), 𝛾 in (b) and 𝛼 in (c), 𝛽 in (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' SNR as a two-dimensional function of (𝛽, 𝛼) in (e), (𝛽, 𝑘) in (f), (𝛾, 𝑘) in (g) and (𝛾, 𝛼) in (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Figure 5 shows the SPD functions and the corresponding system responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 5(a), the SPD functions vary from asymmetrical peaks into two symmetrical ones as noise intensity raises, suggesting that the underdamped harmonic-Gaussian double-well potential SR changes from intra-well SR into inter-well one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 5(b), a weak period occurs when intra-well SR happens, and then the system response (a) (b) (c) (e) (f) (g) (d) (h) 13 / 27 becomes chaotic when the particles jump randomly between double wells and finally is periodic when the inter-well SR takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 5 SPD functions and the corresponding system responses of underdamped harmonic-Gaussian double-well potential SR under different noise intensity: (a) the SPD functions and (b) the corresponding system responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Application of harmonic-Gaussian double-well potential SR to enhance weak fault characteristics of machinery Rotating components of machinery including bearings, gears and rotors are more prone to failures than fixed components due to contact fatigue, uneven lubrication, misalignment and so on [36-38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Therefore, how to detect weak fault characteristics of rotating components in the early stage becomes a challenge [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Lots of scholars have attempted to cancel or suppress the noise embedded in a signal to extract weak fault characteristics further [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' On the contrary, we would apply harmonic-Gaussian double-well potential SR to enhance weak fault characteristics of machinery by using noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Four Rexnord ZA-2115 double row bearing run-to-failure experiments under the rotating speed 2000 rpm and radial load 6000 lbs were performed to acquire the bearing failure data by using accelerometers and a data acquisition card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The bearing parameters are listed as below: the ball number 16, the pitch diameter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='815 inches, the contact angle 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='17 degrees and rolling element diameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='331 inches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The bearing experimental rig is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 6(a) and the corresponding sensor placement is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' This experimental rig is composed of four tested bearings, an AC motor and rub belts [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' In the bearing run-to-failure experiment, the (a) (b) 14 / 27 sampling frequency is 20 kHz and the sampling time is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='024 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 6 Bearing test rigs and sensor placement illustration: (a) bearing test rigs and (b) sensor placement illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' All failures occurred after exceeding designed life time of the bearing which is more than 100 million revolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The data set describes a test-to-failure experiment and consists of individual files that are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='024-seconds vibration signal snapshots recorded at 10-minutes intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The recording duration is from February 12, 2004 10:32:39 to February 19, 2004 06:22:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' At the end of the test-to-failure experiment, outer race failure occurred in the tested bearing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The root mean square (RMS), an effective health indicator, is often used to reflect the vibration intensity and monitor the health state of bearings further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Therefore, RMS of bearing run-to-failure experimental vibration data is calculated and depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 7 to observe the degradation trend of the tested bearing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The degradation trend changes slowly with slight fluctuation in the range of 0~88 hours and then raises into the larger RMS for degradation marked in red dot in the zoomed RMS plot, suggesting that a tiny outer race failure occurs in the early stage of the tested bearing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' As time went on, it can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 7 that RMS keeps increasing, indicating that the outer race failure becomes more and more severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Finally, this test-to-failure experiment was stopped because of strong vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' In the test-to-failure experiment, the health state of the tested bearing varies from normal to the early failure to severe failure to end of life, which is consistent with the degradation trend reflected by RMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (a) Accelerometers Bearing 1 Bearing 2 Bearing 3 Bearing 4 Motor Rub belts (b) 15 / 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 7 RMS of the bearing run-to-failure vibration signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The raw vibration signal at the 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='83th hour marked in red dot and its frequency and envelope spectrum are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We cannot observe the eye-catching spectral peaks at the theoretical outer race/inner race/roller/cage fault characteristic frequency and its harmonics from both the frequency spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 8(b) and the zoomed envelope spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 8(c), which are submerged by other spectral peaks from background noise and excited by other normal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Although we have completed the bearing run-to-failure experiment and observed that a failure occurred at the outer race of the tested bearing 1 by disassembling four tested bearings, we cannot judge what time a tiny failure occurs at the outer race of the tested bearing 1 by virtue of the raw vibration signal and its spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 8, which is very important for early fault diagnosis and remaining useful life prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 16 / 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 8 The vibration signal and its spectrum of outer race failure bearing at the early stage: (a) the raw signal, (b) its frequency spectrum and (c) zoomed envelope spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We apply the overdamped harmonic-Gaussian double-well potential SR to enhance the weak fault characteristics in the early stage of the tested bearing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Figure 9 shows the enhanced results of weak fault characteristics embedded in the raw vibration signal, where the system parameters are given as 𝑘 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='1, 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='2, 𝛽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='24 and the integral step is ℎ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The overdamped SR cannot be used to process large-parameter signals directly and frequency-shifted and rescaling transform is widely to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Three key parameters of frequency-shifted and rescaling transform in the overdamped harmonic-Gaussian double-well potential SR are given as below by virtue of the theoretical outer race fault characteristic frequency 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='4 Hz that can be calculated according to the structural parameters and rotating speed of the tested bearing 1: the pass-band cut-off frequency 220 Hz, the stop-band cut-off frequency 200 Hz and the carrier frequency 200 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' These parameters in the frequency-shifted and rescaling transform could be selected according to the reference [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' One can 17 / 27 observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 9 that the enhanced signal characterizes strong impacts and dominant spectral peaks are at the outer race fault characteristic frequency and its second harmonic of the tested bearing 1, suggesting that a tiny failure occurs at the outer race of the tested bearing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' However, the overdamped harmonic-Gaussian double-well potential SR depends on the high-pass filter to perform the frequency-shifted and rescaling transform, whose parameters are given artificially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' In the overdamped harmonic-Gaussian double-well potential SR-based enhanced results, the low-frequency components of the raw vibration signal (<200Hz) have been removed by using the frequency-shifted and rescaling transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Moreover, the overdamped harmonic-Gaussian double-well potential SR method would suppress the components beyond the nonlinear filtering frequency band of overdamped harmonic-Gaussian double-well potential SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Although overdamped harmonic-Gaussian double-well potential SR method is able to utilize the noise located in the nonlinear filtering frequency band of overdamped harmonic-Gaussian double-well potential SR for enhancing weak fault characteristics, a part of noise is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Therefore, the amplitude of the detected result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 9 is smaller than that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 9 Overdamped harmonic-Gaussian double-well potential SR-based enhanced results: (a) the enhanced signal and (b) its zoomed frequency spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' fouter 2fouter 18 / 27 Further, we apply the underdamped harmonic-Gaussian double-well potential SR to enhance weak fault characteristics embedded in the raw vibration signal, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 10 whose system parameters are given as 𝑘=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='2, 𝛼=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='1, 𝛽=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='24, 𝛾=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='33 and ℎ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' There are obvious repetitive transients in the enhanced signal and eye-catching spectral peaks at the outer race fault characteristic frequency and its second harmonic in the zoomed frequency spectrum as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 10(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Compared with the overdamped harmonic-Gaussian double-well potential SR-based results, the underdamped one characterizes the higher spectral peaks at the outer race fault characteristic frequency and its second harmonic in the zoomed frequency spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 10 Underdamped harmonic-Gaussian double-well potential SR-based enhanced results: (a) the enhanced signal and (b) its zoomed frequency spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' For a comparison, we use the advanced robust local mean decomposition (RLMD) [44, 45] to decompose the raw vibration signal of the tested bearing 1 into the product functions (PFs) and a residual component (Res) for extracting weak fault characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The product functions and their zoomed envelope spectrum are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 11(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 11(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' One cannot observe the obvious spectral peaks at the outer race fault characteristic frequency and its harmonics from the zoomed envelope spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' fouter 2fouter Rotating frequency and its harmonics 19 / 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 11 RLMD-based results: (a) product functions and (b) their zoomed envelope spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' In addition to signal decomposition methods, signal denoising or signal filtering methods also have been widely applied to extract weak fault characteristics of machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Among them, wavelet transform [46, 47] is typical to obtain a denoised version of the raw signal by thresholding the wavelet coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Here, the maximal overlap discrete wavelet transform is used to denoise the signal with soft thresholding, level 3 and db4 wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The denoised signal and its zoomed envelope spectrum are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 12 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 13, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' It is found from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 12 that the wavelet transform can cancel strong background noise, but we cannot see any fault characteristics at the first sight from the zoomed envelope spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (a) (b) PF1 PF2 PF3 PF4 PF5 Res PF1 PF2 PF3 PF4 PF5 Res Time [s] Frequency [Hz] Amplitude [g] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='2 h 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='8 20 / 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 12 Undecimated wavelet transform-based denoised signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 13 The zoomed envelope spectrum of undecimated wavelet transform-based denoised signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' A classical symptom of rotating machines failures in vibration signals is the presence of repetitive transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Antoni [48] proposed an infogram method to capture the signature of repetitive transients in time domain, which is the variant of classical fast kurtogram method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' This method is used to process the raw vibration signal for extracting repetitive transients in time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The corresponding results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Although it can see the slight repetitive transients from the filtered signal in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 14(b), it is difficult for us to identify the period of repetitive transients because of strong background noise and other normal vibration components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The above conclusion could be further confirmed by the squared envelope amplitude sepctrum of 21 / 27 the filtered signal in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 14(b), in which we cannot see the eye-catching spectral peaks at the outer race fault characteristic frequency and its harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 14 The detected results using infogram: (a) infogram and (b) the filtered signal and its squared envelope amplitude sepctrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Conclusions The overdamped and underdamped harmonic-Gaussian double-well potential SR are investigated by deriving the output SNR and SPD functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' It is found that both noise-induced SR and parameter-induced SR can be activated in the overdamped and underdamped harmonic-Gaussian double-well potential systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Moreover, since the harmonic-Gaussian double-well potential in the range of 𝑘 ≥ 2𝛼𝛽 loses the stability, we can observe the antiresonance, whereas adding the damped factor into the overdamped harmonic-Gaussian double-well potential system can change the stability, resulting that the antiresonance disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Above conclusion is applicable under all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Finally, we apply both the overdamped and underdamped harmonic-Gaussian double-well potential SR to enhance weak fault characteristics of bearings for incipient fault identification, where the corresponding parameters would be adjusted or optimized instead of all parameters are applicable to activate the optimal SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The weak fault characteristics are enhanced successfully to identify the early failure of bearings, which somewhat outperforms to the RLMD, wavelet transform and infogram-based results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' But the SR-based methods depend on the prior knowledge of the signals to be detected or structural parameters and rotating speeds of bearings, and cannot detect unknown multiple-frequency and multiple-component (a) (b) 22 / 27 coupled signals without any prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Therefore, we would study the SR-based signal decomposition method by using noise to decouple and detect unknown multiple-frequency and multiple-component signals, especially time-varying nonstationary signals in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Acknowledgments This research was supported by Foundation of the State Key Laboratory of Performance Monitoring and Protecting of Rail Transit Infrastructure of East China Jiaotong University (HJGZ2021114),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Laboratory of Yangjiang Offshore Wind Power (YJOFWD-OF-2022A08),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Zhejiang Provincial Natural Science Foundation of China (LQ22E050003),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' National Natural Science Foundation of China (52205569),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Ningbo Science and Technology Major Project (2020Z110,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2022Z057,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2022Z002),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' National Natural Science Foundation of China (51905349,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 62001210,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' U2013603),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Natural Science Foundation of Guangdong Province (2022A1515010126,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 2020A1515011509),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Ningbo Natural Science Foundation (2022J098) and also sponsored by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Wong Magna Fund in Ningbo University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The Spanish State Research Agency (AEI) and the European Regional Development Fund (ERDF) under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' PID2019-105554GB-I00 is also aknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Conflicct of Interest The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Data availability The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' [48] Antoni J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The infogram: Entropic evidence of the signature of repetitive transients, Mechanical Systems and Signal Processing, 2016, 74: 73-94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Zhejiang Provincial Key Laboratory of Part Rolling Technology School of Mechanical Engineering and Mechanics • Ningbo University Ningbo University July 24, 2022 RE: “Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance weak fault characteristics of machinery” by Zijian Qiao, Shuai Chen, Zhihui Lai, Shengtong Zhou and Miguel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Sanjuán (Manuscript Number: NODY-D-22-01167) Dear Editor, We have carefully revised our paper taking into account your suggestions and the comments of the reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We have uploaded the revised version and the revision notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Thank you very much for processing our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We appreciate very much the constructive comments and suggestions provided by the reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' They have been incorporated in the revised version of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Major changes made in the paper are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The following summarizes our response to each point raised by each reviewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We would like to thank the three reviewers for their valuable comments and constructive suggestions to improve the quality of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We have fully considered their comments and suggestions and made revisions accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The major revisions are highlighted by BLUE color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The point-to-point explanations and revisions are listed as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We have taken into full consideration all comments of the three referees and made a thorough revision of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Sincerely yours, Zijian Qiao Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='D Shuai Chen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Zhihui Lai Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='D Shengtong Zhou Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='D Miguel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Sanjuán Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='D Cover Letter Click here to access/download;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='attachment to manuscript;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='Cover Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='doc Click here to view linked References 波 大 漢 Page 1 of 1 Highlights RE: “Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance weak fault characteristics of machinery” by Zijian Qiao, Shuai Chen, Zhihui Lai, Shengtong Zhou and Miguel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Sanjuán \uf0d8 Harmonic-Gaussian double-well potential SR is investigated by deriving and measuring the output SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' \uf0d8 Steady-state probability density functions are used to evaluate the transition rates of particles in the harmonic-Gaussian double-well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' \uf0d8 Parameter-induced SR, noise-induced SR and antiresonance are observed by analyzing the output SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' \uf0d8 Harmonic-Gaussian double-well potential SR is applied to enhance weak fault characteristics of machinery successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Highlights Click here to access/download;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='attachment to manuscript;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='Highlights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='doc Click here to view linked References Declaration of Interest Statement The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Declaration of Interest Statement Click here to access/download;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='attachment to manuscript;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='Declaration of Interest Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='docx Click here to view linked References Manuscript Number: NODY-D-22-01167R1 Title: Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance weak fault characteristics of machinery Response to Editor There are still some minor comments raised by one of the reviewers needed to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' A minor revision is recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Response: We appreciate the constructive comments from two reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' According to their comments and suggestions, we have made a thorough revision for the manuscript and have addressed all points raised by each reviewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The major changes made in the manuscript are marked in BLUE color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We also include the major changes of the manuscript into the response point by point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' For convenient review, the page numbers or paragraph numbers of the revision in the manuscript are cited below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We hope that this revised submission is satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The authors thank editors and anonymous reviewers for their valuable and helpful comments to revise and improve our manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Compressed File Click here to access/download;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='Compressed File;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='Response to Reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content='docx Manuscript Number: NODY-D-22-01167R1 Title: Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance weak fault characteristics of machinery Response to Reviewer #3 The authors have correctly taken into consideration the reviewers comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Response: Thanks for your recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Manuscript Number: NODY-D-22-01167R1 Title: Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance weak fault characteristics of machinery Response to Reviewer #5 The paper presents that the overdamped or underdamped harmonic Gaussian double-well potential SR methods characterize a better performance to detect a weak signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The work is organized in a clear form, but the technical content looks not high and there are important aspects that are not discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Whether the analysis conclusion obtained is the conclusion under these special parameters or whether all parameters are applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Please give some explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Response: According to the comments of the reviewer #5, we think that two analysis conclusion obtained could be illustrated whether under these special parameters or all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' In Sections 2 and 3: Overdamped and underdamped harmonic-Gaussian double-well potential SR We investigate the SR in the cases of overdamped and underdamped harmonic-Gaussian double-well potential systems subjected to noise and a periodic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We derive and measure the analytic expression of the output signal-to-noise ratio (SNR) and the steady-state probability density (SPD) function under approximate adiabatic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' When the harmonic-Gaussian double-well potential loses its stability, we can observe the antiresonance phenomenon, whereas adding the damped factor into the overdamped system can change the stability of the harmonic-Gaussian double-well potential, resulting that the antiresonance behavior disappears in the underdamped system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Although above analysis conclusion is obtained under these special parameters, other parameters would depict the same findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' As a result, the analysis conclusion obtained in two sections is applicable under all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' In Section 4: Application of harmonic-Gaussian double-well potential SR to enhance weak fault characteristics of machinery Harmonic-Gaussian double-well potential stochastic resonance is a typical nonlinear filter with the adjusting parameters in which the noise embedded in a signal is able to be utilized to enhance weak useful information by activating the stochastic resonance phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' The stochastic resonance phenomenon could be activated when the optimal matching among the weak useful information, noise and these parameters of stochastic resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' For a different signal, therefore, these parameters of the harmonic-Gaussian double-well potential stochastic resonance must be tuned to activate the stochastic resonance phenomenon for enhancing weak useful information by using noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' As a result, applying harmonic-Gaussian double-well potential stochastic resonance to enhance weak fault characteristics of machinery, these parameters of harmonic-Gaussian double-well potential stochastic resonance would be adjusted or optimized instead of all parameters are applicable to activate the optimal stochastic resonance phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (See the conclusion in Section 5 page 21, which is marked in BLUE) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' "Noise is ubiquitous and unwanted in detecting weak signals", This sentence is repeated and can be deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Response: Thanks for your suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We have deleted it in Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (See the abstract in page 1, which is marked in BLUE) 3 "Key words" write too long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Response: Thanks for your suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' We have reduced the key words as below: The benefits of noise, weak signature enhancement, fault identification, fault diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' (See the key words in page 2, which is marked in BLUE) 4 "The recording duration is from February 12, 2004 10:32:39 to February 19, 2004 06:22:39".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Why was it 18 years ago？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Response: That is because Four Rexnord ZA-2115 double row bearing run-to-failure experiments under the rotating speed 2000 rpm and radial load 6000 lbs were performed in 2004 year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' In future work, we would perform and conduct new bearing run-to-failure experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Now, our team is designing the new experimental platform and project to acquire new bearing and gear vibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} +page_content=' Thanks for your understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfOgYr/content/2301.03748v1.pdf'} diff --git 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b/5dE2T4oBgHgl3EQf6wiO/content/tmp_files/2301.04203v1.pdf.txt @@ -0,0 +1,945 @@ +arXiv:2301.04203v1 [math.CV] 10 Jan 2023 +ZERO DISTRIBUTION OF RANDOM BERNOULLI POLYNOMIAL MAPPINGS +TURGAY BAYRAKTAR & C¸˙I˘GDEM C¸EL˙IK +ABSTRACT. In this note, we study asymptotic zero distribution of multivariable full sys- +tem of random polynomials with independent Bernoulli coefficients. We prove that with +overwhelming probability their simultaneous zeros sets are discrete and the associated +normalized empirical measure of zeros asymptotic to the Haar measure on the unit torus. +1. INTRODUCTION +A random Kac polynomial on the complex plane is of the form +(1.1) +fd(z) = +d +� +j=0 +ajzj +where the coefficients aj are independent copies of the (real or complex) standard Gauss- +ian. A classical result due to Kac, Hammersley and Shepp & Vanderbei [20, 16, 23] asserts +that almost surely the normalized empirical measure of zeros δZ(fd) := 1 +d +� +fd(ζ)=0 δζ, con- +verges to normalized arc length measure on S1 := {|z| = 1} as d → ∞. Asymptotic +zero distribution of Kac polynomials with i.i.d. discrete random coefficients have also +been studied extensively (see eg. [22, 14]). More recently, Ibragimov and Zaporozhets +[19] proved that the empirical measure of zeros δZ(fd) almost surely converges to the the +normalized arc length measure if and only if the moment condition E[log(1 + |ai|)] < ∞ +holds. This property can be considered as a global universality property of the zeros of +random polynomials (see also [27] for a local version). +Building upon the work of Shiffman and Zelditch [26], equilibrium distribution of +random systems of polynomials with Gaussian coefficients was obtained by Bloom & +Shiffman [9] and Shiffman [24]. More recently, these results were generalized for inde- +pendent identically distributed (i.i.d.) random coefficients with bounded density [1, 2]. +We refer the reader to the survey [4] and references therein for the state of the art. On +the other hand, asymptotic zero distribution of random polynomial mappings with dis- +crete random coefficients remained open (cf. [3, 8, 5]). In this note, we study asymptotic +zero distribution of multivariable full system of random polynomials with independent +Bernoulli coefficients. +1.1. Statement of the results. A random Bernoulli polynomial is of the form +fd,i(x) = +� +|J|≤d +αi,JxJ ∈ C [x1, . . . , xn] +T.B. and C¸.C¸. are partially supported by T¨UB˙ITAK grant ARDEB-1001/119F184. +1 + +where xJ = xj1 +1 . . . xjn +n and αi,J are ±1 Bernoulli random variables. Throughout this work, +we consider systems (fd,1, . . . , fd,n) of random Bernoulli polynomials with independent +coefficients. We write f d = (fd,1, . . . , fd,n) for short. We denote the collection of all +systems of polynomials in n variables and of degree d by Polyn,d that is endowed with +the product probability measure Probd. +Theorem 1.1. Let f d = (fd,1, . . . , fd,n) be a system of random polynomials with independent +±1 valued Bernoulli coefficients. Then there exists a dimensional constant K = K(n) > +0 and an exceptional set En,d ⊂ Polyn,d such that Probd(En,d) ≤ K/d and for all f d ∈ +Polyn,d(A)\En,d the simultaneous solutions of the system f d are isolated with #Z(f d) = dn. +For a system f d ∈ Polyn,d, if the simultaneous zeros Z(f d) are isolated we denote +the corresponding normalized empirical measure by δZ(f d). That is δZ(fd) is a probabil- +ity measure supported on the isolated zeros. We also let νHaar denote the Haar measure +of (S1)n of total mass 1. As an application of Theorem 1.1 together with a determinis- +tic equidistribution result [13, Theorem 1.7], we obtain asymptotic zero distribution of +random Bernoulli polynomial mappings: +Corollary 1.2. Let f d = (fd,1, . . . , fd,n) be system of random polynomials with independent +±1 valued Bernoulli coefficients and En,d ⊂ Polyn,d be as in Theorem 1.1. Then for each +sequence f d ∈ Polyn,d \ En,d we have +lim +d→∞ δZ(fd) = νHaar. +in weak topology. In particular, δZ(f d) → νHaar in probability Probd as d → ∞. +Finally, we consider the measure valued random variables +(1.2) +�Z(f d) = +�� +ξi∈Z(f d) δ(ξi) +for f d ∈ Polyn,d \ En,d +0 +otherwise. +and define the expected zero measure by +(1.3) +� +E[ �Z(f d)], ϕ +� += +� +P olyn,d\En,d +� +ξi∈Z(f d) +ϕ(ξi) dProbd(f d) +where ϕ is a continuous function with compact support in Cn and En,d denote the excep- +tional set given by Theorem 1.1. +Theorem 1.3. Let f d = (fd,1, . . . , fd,n) be a system of random polynomials with independent +±1 valued Bernoulli coefficients. Then +lim +d→∞ d−nE[Z(f d)] = νHaar +in weak topology. +The outline of this work as follows. In §2, we introduce some algebraic background +on resultants. In particular, we recall multi-polynomial resultant and sparse resultant for +polynomial systems [15, 11] as well as directional resultant [12]. In §3, we prove the +main result Theorem 1.1. Finally, in §4 we prove Theorem 1.3. +2 + +2. PRELIMINARIES +In this section, we collect some basic facts and algebraic background related to our +results. More precisely, we discuss the multi-homogenous (classical) resultant and the +sparse eliminant as well as the relation of these two notions. For a detailed account of the +subject and proofs we refer the reader to [15, 11]. We also discuss the sparse resultant +introduced by D’Andrea and Sombra, and corresponding directional sparse resultants +[13, 12]. +2.1. Lattice points, polytopes. For a nonempty subset P ⊂ Rn, we denote its convex +hull in Rn by conv(P). For two nonempty convex sets Q1, Q2, their Minkowski sum is +defined as +Q1 + Q2 := {q1 + q2 : q1 ∈ Q1, q2 ∈ Q2} +and for λ ∈ R, the scaled polytope is of the form +λQ := {λq : q ∈ Q}. +It is well known that V oln(d1Q1 + . . . + dnQn) is a homogenous polynomial of degree n in +the variables d1, . . . , dn ∈ Z+ where V oln denotes the normalized volume of the subsets +in Rn with respect to the Lebesgue measure. The coefficient of the monomial d1 . . . dn +is called the mixed volume of Q1, . . . , Qn and denoted by MV (Q1, . . . , Qn). One can use +the polarization formula to compute the mixed volume of the convex sets Q1, . . . , Qn. +Namely, +MVn(Q1, . . . , Qn) = +n +� +k=1 +� +1≤j1≤...≤jk≤n +(−1)n−kV oln(Qj1 + . . . + Qjk). +In particular, if Q = Q1 = . . . = Qn then +MVn(Q) := MVn(Q, . . . , Q) = n!V oln(Q). +For a convex set Q ⊂ Rn its support function sQ : Rn → R is defined by +(2.1) +sQ(v) := inf +q∈Q ⟨q, v⟩ +where ⟨·, ·⟩ represents the Euclidean inner product of Rn. Then the equation +⟨q, v⟩ = sQ(v) +defines supporting hyperplane of Q and v is called an inward pointing normal. The inter- +section of Q with the supporting hyperplane in the direction v ∈ Rn is denoted by +(2.2) +Qv := {q ∈ Q : ⟨q, v⟩ = sQ(v)}. +Qv is called the face of Q determined by v. If Qv has codimension 1, it is called a facet of +Q. +3 + +2.2. Resultant of polynomial systems. +2.2.1. Multipolynomial Resultant. We consider homogenous polynomials of degree di +Fi(t0, . . . , tn) = +� +|J|=di +ui,JtJ +for i = 0, . . . , n where J is a multi-index (j0, . . . , jn) and tJ indicates the monomial +tj0 +0 · · · tjn +n which is of degree |J| = �n +i=0 ji. One can see that the homogenous polynomi- +als of degree di form an affine space by identifying � +|J|=di ui,JtJ with point (ui,J)|J|=di ∈ +CN(di), where N(di) = +�n+di−1 +n−1 +� +. +Letting N := �n +i=0 N(di), recall that the incidence variety is defined by +W = +� +(a, t) ∈ CN × Pn : F0(a0, t) = · · · = Fn(an, t) = 0 +� +. +We let π : CN × Pn → CN be the projection onto first coordinate. By Projective Extension +Theorem (see eg. [11]) the image π(W) forms a variety in the affine space CN. +Definition 2.1. The multipolynomial resultant Resd0,...,dn is defined as the irreducible unique +(up to a sign) polynomial in Z[a0, . . . , an] which is the defining equation of the variety π(W). +The resultant of the homogeneous polynomials F0, . . . , Fn is the evaluation of Resd0,...,dn at +the coefficients of F0, . . . , Fn and it is denoted by Resd0,...,dn(F0, . . . , Fn). +Note that if d0 = . . . = dn = 1, then the evaluation of multipolynomial resultant +Resd0,...,dn at the coefficients of F0, . . . , Fn is the determinant of the coefficient matrix. For +more general cases, we have the following result: +Theorem 2.2 ([15],[11]). Let F0, . . . , Fn ∈ C[x0, . . . , xn] be homogenous polynomials of +positive total degrees d0, . . . , dn. Then the system F0 = . . . = Fn = 0 has a solution in the +complex projective space Pn if and only if Resd0...,dn(F0, . . . , Fn) = 0. +Theorem 2.2 gives a characterization to determine the existence of nontrivial solutions +for the systems of homogenous polynomials based on the coefficients of the polynomials +in the system. However, not all the systems of equations are homogenous, and in the +power series expansions not all the monomial terms appear. Hence, we need to introduce +a more general version of the multi-homogenous resultant. +2.2.2. Sparse Eliminant. Following [15], we will recall the definition of sparse resultant. +Let A0, . . . , An be a non-empty finite subsets of Zn, and let ui = {ui,a}a∈Ai be a group of +#Ai variables, i = 0, . . . , n and set u = {u0, . . . , un} . For each i, the general Laurent +polynomial fi with support supp(fi) = Ai given by +f(x) = +� +a∈Ai +ui,axa ∈ C[u][x±1 +1 , . . . , x±1 +n ]. +We let A = (A0, . . . , An) and consider the incidence variety in this setting defined by +(2.3) +WA = +� +(u, x) ∈ +n +� +i=0 +P(CAi) × (C∗)n : f0(ui, x) = · · · = fn(un, x) = 0 +� +. +4 + +Consider the canonical projection on the first coordinate +πA : +n +� +i=0 +P(CAi) × (C∗)n → +n +� +i=0 +P(CAi) +and let πA(WA) denote the Zariski closure of WA under the projection π. +Definition 2.3. The sparse eliminant, denoted by ResA, is defined as follows: if the variety +πA(WA) has codimension 1, then the sparse eliminant is the unique (up to sign) irreducible +polynomial in Z[u] which is the defining equation of πA(WA). If codim(πA(WA)) ≥ 2, then +ResA is defined to be the constant polynomial 1. The expression +ResA(f0, . . . , fn) +is the evaluation of ResA at the coefficients of f0, . . . , fn. +Example 2.4. For A0 = {0} , A1 = {0, 1} ⊂ Z, we have that ResA0,A1 = ±u00. +The classical resultant Resd0,...,dn is the special case of the sparse eliminant. Indeed, let +Ai be the set of all integer points in the di-simplex, i.e., Ai = diΣn ∩ Zn and Σn is the +standard unit simplex, that is, +diΣn := {(a0, . . . , an) ∈ Rn+1 : aj ≥ 0, +� +j +aj ≤ di}. +Following [11] and [15], for simplicity let all the sparse polynomials f0, . . . , fn have the +same support Ad = dΣn ∩ Zn for some positive integer d and consider the system +(2.4) + + + +f0 = u01xα1 + . . . + u0dxαn = 0 +... +fn = un1xα1 + . . . + undxαn = 0 +We also let t0, . . . , tn be the homogenous coordinates which are related to x1, . . . , xn by +xi = ti/t0. Then we define the homogenous polynomials +(2.5) +Fi(t0, . . . , tn) = td +0fi(t1/t0, . . . , tn/t0) = td +0fi(x1, . . . , xn), +for 0 ≤ i ≤ n. This method gives n+1 homogenous polynomials of total degree d in +the variables t0, . . . , tn and this procedure is independent of the choice of homogeneous +coordinates. +Proposition 2.5 ([11]). Let Ad = dΣn ∩Zn and consider the systems of polynomials F and +f as above. Then +ResA(f0, . . . , fn) = ±Resd,...,d(F0, . . . , Fn), +where A = (Ad, . . . , Ad). +Using the above proposition, we can give a version of Theorem 2.2 as follows. +5 + +Corollary 2.6. Let f = (f1, . . . , fn) be a system of polynomials with supp(fi) = Ad for +i = 1, . . . , n. Assume that the system F = (F0, . . . , Fn) consists the homogenizations of fi +according to process in (2.5) and denote the set of simultaneous nonzero solutions of F by +Z(F ). Suppose that Z(F ) ∩ H∞(t0) = ∅ where H∞(t0) is the hyperplane at infinity for t0 = +0. Then the system of polynomials f = 0 has no solution if and only if ResAd(f0, . . . , fn) ̸= 0. +Proof. If ResAd(f0, . . . , fn) ̸= 0, then by definition of the sparse resultant the system +f0(x) = . . . = fn(x) = 0 +has no solution. Conversely, letting Fi be the homogenization of fi as in (2.5) with the +corresponding variable t = (t0, . . . , tn), i.e. Fi(t) = td +0fi(x). If the system of polynomials +f = 0 has no solution then Fi(t) = 0 for i = 1, . . . , n if and only if t0 = 0 which contradicts +our assumption. Hence, by Theorem 2.2 we have +±ResAd(f0, . . . , fn) = Resd0,...,dn(F0, . . . , Fn) ̸= 0. +□ +2.2.3. Sparse Resultant. In spite of being a generalization of the multipolynomial resul- +tant and involving considerable large amount of the system of polynomials, the sparse +eliminant does not satisfy some essential properties which is necessary in many applica- +tions, such as additivity property and Poisson formula. In 2014, D’Andrea and Sombra +[12] introduced the following version of the sparse resultant which has the desired fea- +tures. +Definition 2.7. The sparse resultant, denoted by ResA, is defined as any primitive poly- +nomial in Z[u] that is the defining equation of the direct image of WA, (πA)∗(WA) = +deg(πA|WA)πA(WA) if this variety has codimension one, and otherwise we set ResA = 1. +The expression +ResA(f0, . . . , fn) +is the evaluation of ResA at the coefficients of f0, . . . , fn. +According to this definition, the sparse resultant is not irreducible but it is a power of +the irreducible sparse eliminant, i.e., +ResA = ±Res +deg(πA|WA) +A +where deg(πA|WA) is the degree of the projection πA. We also remark that ResA ̸= 1 +whenever ResA ̸= 1. For more details we refer the reader to the manuscripts [12] and +[13]. +Example 2.8. Let A0 = A1 = A2 = {(0, 0), (2, 0), (0, 2)}. Then ResA = det(ui,j) and +ResA = ±[det(ui,j)]4. +6 + +2.2.4. Directional Resultant. For a subset B ⊂ Zn and a polynomial f(x) = � +b∈B βbxb +with support B, we write +Bv := {b ∈ B : ⟨b, v⟩ = sQ(v)} +and +f v(x) = +� +b∈Bv +βbxb +where Q = conv(B) and v ∈ Rn and sconv(B)(v) is defined as equation (2.1). +Definition 2.9. Let A1, . . . , An ⊂ Zn be a family of n non-empty finite subsets, v ∈ Zn\{0}, +and v⊥ ⊂ Rn the orthogonal subspace. Then there exists bi,v ∈ Zn such that +Av +i − bi,v ⊂ Zn ∩ v⊥ +for i = 1, . . . , n. The resultant of A1, . . . , An in the direction of v, denoted ResAv +1 ,...,Avn is +defined as the resultant of the family of the finite subsets Av +i − bi,v. +Let fi ∈ C[x±1 +1 , . . . , x±1 +n ] be Laurent polynomials with support supp(fi) ⊂ Ai i = 1, . . . , n. +For each i = 1, . . . , n, we write f v +i = xbi,vgi,v for a Laurent polynomial gi,v ∈ C[Zn ∩ v⊥] ≃ +C[y±1 +1 , . . . , y±1 +n−1] with supp(gi,v) ⊂ Av +i − bi,v. The expression +ResAv +1 ,...,Avn(f v +1 , . . . , f v +n) +is defined as the evaluation of this resultant at the coefficients of the gi,v. +We remark that the definition of directional resultant is independent of the choice of +the vector bi,v (see [12, Proposition 3.3]). Moreover, the directional resultant ResAv +1 ,...,Avn ̸= +1 only if the direction vector v is an inward pointing normal to a facet of the Minkowski +sum �n +i=1 conv(Ai). Therefore, the nontrivial directional resultants of the family A1, . . . , An +is finitely many. +Example 2.10. Let f(x) = a0 + . . . + anxn ∈ C[x] be a polynomial of degree n. Then the +nontrivial directional resultants are +ResA(f v) = +� +±a0 +if +v = 1, +±an +if +v = −1 +for the polytope conv(A) = [0, n] ⊂ R. +3. EQUIDISTRIBUTION OF ZEROS +3.1. Random Polynomial Systems. First, we recall a theorem of Kozma and Zeitouni +[21] asserts that overdetermined random Bernoulli polynomial systems have no common +zeros with overwhelming probability: +Theorem 3.1. Let f1, . . . , fn+1 ∈ Z[x1, . . . , xn] be n + 1 independent random Bernoulli +polynomials of degree d and +P(d, n) := Probd{∃x ∈ Cn : fi(x) = 0 for i = 1 . . . , n + 1} +denote the probability that the system f1 = . . . = fn+1 = 0 has a common solution. Then +there exists a dimensional constant K = K(n) < ∞ such that +P(d, n) ≤ K/d +7 + +for all d ∈ Z+. +Next, we prove our main result: +Proof of Theorem 1. Let fd,i be a random Bernoulli polynomial of the form +(3.1) +fd,i = +� +|J|≤d +αi,JxJ ∈ Z[x1, . . . , xn], +where {αi,J} is a family of independent Bernoulli random variables for i = 1, . . . , n. +We investigate the directional resultants of the system f for all nonzero primitive di- +rection vectors v ∈ Zn. By [12, Proposition 3.3] it is enough to check the inward normals +to the Minkowski sum of the supports ndΣn which has n + 1 facets with n + 1 inward +normals given by vm = em for m = 1, . . . , n and vn+1 = − �n +m=1 em where {em}n +m=1 is +the standard basis of Rn. +For vm = em the intersection of a support A with the supporting hyperplane in the +direction em is of the form +(3.2) +Avm = +� +(j1, . . . , jn) ∈ A : jm = 0, +n +� +l=1 +jl ≤ d +� +m = 1, . . . , n. Hence, the polynomials f vm +i +can be written as +(3.3) +f vm +i +:= +� +J∈Avm +αi,JxJ +for i = 1, . . . , n. Note that polynomials f vm +i +depend on n − 1 variables. Following the +Definition 2.9, if we choose the vector bi,vm = 0 such that Avm − bi,vm ⊂ Zn ∩ vm⊥, +we see that the functions gi,vm := f vm +i +satisfies the equation f vm +i += xbi,vmgi,vm for each +i = 1, . . . , n. +Recall that for two univariate polynomials h1, h2 ∈ C[x], their resultant Res(h1, h2) is +zero if and only if h1 and h2 have a common solution in C. Therefore, if n = 2 the +necessary and sufficient condition for g1,vm and g2,vm have zero resultant is that they +have a common zero. Theorem 3.1 implies that there exists a constant Km which is +independent of d so that the aforementioned event has probability at most Km/d. +On the other hand, when n > 2, we perform the homogenization process to each (n−1) +variable polynomial gi,vm for i = 1, . . . , n as described in equation (2.5). We obtain the n +variable homogenous polynomials Gi,vm of the form +(3.4) +Gi,vm(t, x) = +� +J∈Avm +αi,Jtd−|J|xJ. +In order to compare the sparse resultant of the polynomials gi,vm and the multipolynomial +resultant of the homogeneous polynomials Gi,vm, we check the conditions of Corollary +2.6. Let Z(G) be the set of nontrivial solutions of the system G = (G1,vm, . . . , Gn,vm) +and suppose that G has a solution ξ = (t, ξ2, . . . , ξn) in the hyperplane at infinity H∞(t). +Evaluating these homogeneous polynomials at t = 0, we obtain the top degree homoge- +neous part of the polynomials gi,vm for i = 1, . . . , n. Since ξ ∈ H∞(t), it has a nonzero +coordinate ξk for some k ∈ {2, . . . , n}. For simplicity, let us assume k = 2 and define the +8 + +new variables zi := ξi+2/ξ2 for i = 1, . . . , n − 2. Applying this change of variables, we +obtained +(3.5) +�Gi,vm(z1, . . . , zn−2) = +� +|J|≤d +αi,Jzϕ(J) +where ϕ : Rn → Rn−2 with ϕ(j1, . . . , jn) = (j3, . . . , jn). This gives n random Bernoulli +polynomials of degree d in n − 2 variables. Hence by Theorem 3.1, there exists a pos- +itive constant Ci, depending only the dimension n such that the probability that the +overdetermined system of Bernoulli polynomials �Gi,vm(z1, . . . , zn−2) have a common so- +lution is less than Ci/d. We infer that the system of homogenized polynomials Gi,vm +has no common zero at hyperplane at infinity H∞(t) except a set that has probability +at most Ci/d. Then by Corollary 2.6, outside of a set of small probability, the system of +polynomials consisting gi,vm has a common solution if and only if the directional resul- +tant ResAvm +1 +,...,Avn(f v +1 , . . . , f v +n ) = 0. Now, since the system of Bernoulli polynomials gi,vm +contains n polynomials in n − 1 variables, by Theorem 3.1, there is a dimensional con- +stant ˜Ci so that the probability that this system has common solution is at most ˜Ci/d. +Hence outside of a set that has probability Ki/d := Ci/d + ˜Ci/d , the directional resultant +ResAvmf vm +d +̸= 0 for all vm for m = 1, . . . , n. +Next, for the inward normal vector vn+1 = − �n +m=1 em, we find the minimal weight in +this direction as Avn+1 = {J ∈ A : |J| = d}. Hence the polynomials in this directions are +of the form +(3.6) +f vn+1 +i += +� +|J|=d +αi,JxJ +In this case Avn+1 is not a subspace of Zn ∩ v⊥ +n+1, hence we need to translate it by sub- +tracting a suitable vector bi,vn+1. For Laurent polynomial systems, the sparse resultant is +invariant under translations of supports (see [12], Proposition 3.3). Since the polynomi- +als fd,i are not Laurent, we need to determine the effects of this translations. Consider +the system of Bernoulli polynomials f d and set of its simultaneous zeros Z(f d). For a +solution x = (x1, . . . , xn) ∈ Z(f d) and assume that x1 = 0. In order to examine the +incidence of this case, we evaluate the system f d at x1 = 0 and we obtain a new system +of n Bernoulli polynomials with n − 1 variables. By Theorem 3.1, there exists a constant +C1 which is independent of d such that this system has a common solution with proba- +bility at most C1/d. Therefore the probability of the event that x1 = 0 is less than C1/d. +Hence there is no harm of translation of supports outside of a set that has probability at +most C/d, where C := �n +i=1 Ci. Now, choosing the vector bi,vn+1 = (d, 0, . . . , 0) so that +Avn+1 − bi,vn+1 ⊂ Zn ∩ v⊥ +n+1, we obtain the polynomials of the form +(3.7) +gi,vn+1 = +� +J∈Avn+1−bi,vn+1 +αi,Jxw(J) +with w : Rn → Rn satisfying (j1, j2, . . . , jn) �→ (−d + j1, j2, . . . , jn). We substitute the new +variables yi := xi+1/x1 into gi,vn+1, i = 1, . . . , n − 1 and obtain +9 + +(3.8) +gi,vn+1(y) = +� +|J|≤d +αi,Jyσ(J) +for y ∈ Cn−1 and σ : Rn → Rn with σ(j1, j2, . . . , jn) = (0, j2, . . . , jn). The system con- +taining the polynomials gi,vn+1(y), i = 1, . . . , n contains n random Bernoulli polynomials +with n − 1 random variable as in the cases vm = em. By applying the same steps, it can +be shown that ResAvn+1f vn+1 +d +̸= 0 outside of a set that has probability at most Ki+1/d. +Now, we define the exceptional set En,d as a subset of Polyn,d which contains the sys- +tems f d that has a zero directional resultant for some nonzero primitive vector v or the +systems f d have a common solution x ∈ Cn with xi = 0 for some i = 1, . . . , n. More +precisely, letting +En,d := {f d ∈ Polyn,d : ∃ v ∈ Zn \ {0} ∋ ResAvf v +d = 0} +(3.9) +� +{f d ∈ Polyn,d : ∃ x ∈ Z(f d) ∋ +� +xi = 0}. +we see that there exists a positive constant K which is independent of d such that +Prob{En,d} ≤ d−1K +where K := �n+1 +i=1 Ki + C. +□ +Next, we recall a deterministic equidistribution results for the solutions of systems of +integer coefficient polynomials [13]. For a polynomial f ∈ C[x1, . . . , xn], the supremum +norm of f on the unit torus is defined as +∥f∥sup := +sup +|w1|=...=|wn|=1 +|f(w1, . . . , wn)| . +Let νHaar be the Haar measure on Cn with support (S1)n and of total mass 1. Assume that +f ∈ Polyn,d be a polynomial mapping such that the set of simultaneous zeros Z(f) is a +discrete set. We denote by denote the discrete probability measure on Cn associated to +the Z(f) by δZ(f). The following result gives the asymptotic distribution of the zeros of +such a system f if the coefficients are integer: +Theorem 3.2. [13] Let f = (f1, . . . , fn) be a polynomial mapping with fi ∈ Z[x1, . . . , xn] +of degree d ≥ 1 for each i = 1, . . . , n. Assume that ResAv +1 ,...,Avn(f v +1 , . . . , f v +n) ̸= 0 for all +v ∈ Zn \ {0} and log ||fi||sup = o(d). Then +lim +d→∞ δZ(f) = νHaar. +As a corollary of Theorem 1.1 and Theorem 3.2, we have the following equidistribution +result for random Bernoulli polynomial mappings: +Proof of Corollary 1.2. Consider the system of Bernoulli polynomials f d = (fd,1, . . . , fd,n). +Since all the coefficients are 1 or −1, by triangle inequality +(3.10) +∥fd,i∥sup = +sup +|w1|=...=|wn|=1 +|fd,i(w1, . . . , wn)| ≤ +�n + d +d +� += O(dn) +10 + +which implies that log ∥fd,i∥sup = o(d). Moreover, by Theorem 1.1 for each sequence +f d ∈ Polyn,d \ En,d we have +ResAv +1 ,...,Avn(f v +1 , . . . , f v +n ) ̸= 0 +for all v ∈ Zn \ {0}. Hence, by Theorem 3.2 +lim +d→∞ δZ(f d) = νHaar +in weak topology. In particular, δZ(f d) → νHaar in probability since Prob{En,d} → 0 as +d → ∞. +□ +4. EXPECTED ZERO DISTRIBUTION +In this section, we introduce radial and angle discrepancies for random Bernoulli poly- +nomial mappings in order to study asymptotics of expected zero measures. We adapt +these concepts from [13] and refer the reader to the manuscript [13] and references +therein for a detailed account of the preliminary results this section. +Let Z be a 0-dimensional effective cycle in Cn that is there is a non-empty finite col- +lection of points ξ = (ξ1, . . . , ξn) ∈ Cn and mξ ∈ N, called the multiplicity of ξ, such +that Z = � +ξ mξ[ξ]. The degree of Z is defined by deg(Z) = � +ξ mξ which is a positive +number. +Definition 4.1. [13] Let Z be a 0-dimensional effective cycle in Cn. For each α = (α1, . . . , αn) +and β = (β1, . . . , βn) ∈ Rn such that −π ≤ αj < βj ≤ π, j = 1, . . . , n we consider the cycle +Zα,β := +� +{ξ∈Z:αj 0. Combining the +equations (4.9) and (4.11), we deduce that lim +d→∞ E[∆ang(Z(f d))] = 0. +The proof of the second assertion is analogous and we omit it. +□ +Proof of Theorem 1.3. We adapt the argument in [13, Theorem 1.8] to our setting. Let us +denote νd := E[ �Z(f d)] +dn +, where E[ �Z(f d)] is the expected zero measure and νHaar be the Haar +probability on (S1)n. We need to show that for each continuous function ϕ with compact +13 + +support in Cn we have +� +ϕdνd → +� +ϕdνHaar as d → ∞. To this end, it is enough to prove +the claim for characteristic functions ϕU of the open sets +(4.12) +U := {(z1, . . . , zn) ∈ Cn : r1,j < |zj| < r2,j and αj < arg(zj) < βj} +where 0 ≤ r1,j < r2,j ≤ ∞, ri,j ̸= 1 for i = 1, 2 and −π < αj < βj ≤ π. +First, we consider the case when U ∩ (S1)n = ∅. Then one can find an 0 < ε < 1 such +that U is disjoint from the set +(4.13) +{(ξ1, . . . , ξn) ∈ Cn : 1 − ε < |ξj| < (1 − ε)−1 for all j}. +Let En,d be the exceptional set as in the proof of Theorem 1.1. If f d ∈ Polyn,d \ En,d then +Z(f d) is discrete and +#{U ∩ Z(f d)} ≤ deg(Z(f d))∆rad(f d, ε) ≤ dn∆rad(f d, ε). +On the other hand, if f d ∈ En,d then by definition deg( �Z(f d)|U) = 0. Hence, +νd(U) ≤ E[∆rad( �Z(f d, ε))] +and by Proposition 4.4, +lim +d→∞ +� +P olyn,d +ϕUdνd = 0 = νHaar(U). +If U ∩ (S1)n ̸= ∅ let +(4.14) +�U = {z : αj ≤ arg(zj) ≤ βj for all j }. +Then we have +νd(U) − +n +� +j=1 +βj − αj +2π += +� +νd(�U) − +n +� +j=1 +βj − αj +2π +� +− νd(�U \ U). +By Theorem 1.1 we have +�����νd(�U) − +n +� +j=1 +βj − αj +2π +����� = +� +P olyn,d\En,d +����� +deg(Z(fd)α,β) +dn +− +n +� +j=1 +βj − αj +2π +����� dProbd(f d) + Kn +d +≤ +� +P olyn,d\En,d +∆ang(Z(f d))dProbd(f d) + Kn +d . +(4.15) +Note that the set �U \U is a union of a finite number of subsets Um of the form (4.12) such +that Um ∩ (S1)n = ∅ for all m, we have limd→∞ νd(Um) = 0 by previous case and hence +limd→∞ νd(U \ U) = 0. Therefore, by Proposition 4.4 and (4.15), +lim +d→∞ νd(U) = lim +d→∞(�U) = +n +� +j=1 +βj − αj +2π += νHaar(U) +which completes the proof. +□ +14 + +REFERENCES +[1] T. Bayraktar, Equidistribution of Zeros of Random Holomorphic Sections, Indiana Univ. Math. J., 5 +(2016), 1759-1793. +[2] T. Bayraktar, Zero distribution of random sparse polynomials, Michigan Math. J. 66 (2017), 389-419 +[3] T. Bayraktar, Global universality of random zeros, Hacet. J. Math. 48 (2019),384-398. +[4] T. Bayraktar, D. Coman, H. Herrmann and G. Marinescu. A survey on zeros of random holomorphic +sections, Dolomit. Res. Notes Approx. 11 (2018), 1-20. +[5] T. Bayraktar, T. Bloom and N. Levenberg. Zeros of Random Polynomial Mappings in Several Complex +Variables, arXiv preprint arXiv:2112.00880. +[6] D.N. Bernstein, The number of roots of a system of equations, Funktsional. Anal. , Prilozhen. 9 (1975), +no.3, 1-4. +[7] T. Bloom, Random polynomials and (pluri)potential theory, Ann. Polon. Math. 91 (2007), 131-141. +[8] T. Bloom and D. Dauvergne, Asymptotic zero distribution of random orthogonal polynomials, The An- +nals of Probability, 47(5) 2019, pp.3202-3230. +[9] T. Bloom and B. Shiffman, Zeros of random polynomials on Cm, Math. Res. Lett.14 (2007), 469-479. +[10] T. Bloom, N. Levenberg, Random Polynomials and Pluripotential Theoretic Extremal Functions, Poten- +tial. Anal. , 42, (2015), 311-334. +[11] D.A. Cox, J. Little, and D. O’Shea, Using Algebraic Geometry, second edition, Grad. Texts in Math., +185, Springer, New York, 2005. +[12] C. D’Andrea and M. Sombra, A Poisson Formula for the Sparse Resultant Proc. Lond. Math. Soc. (3) +110 (2015), no. 4, 932–964. +[13] C. D’Andrea and A. Galligo and M. Sombra, Quantitative equidistribution for the solutions of systems +of sparse polynomial equations, Amer. J. of Math., 136 (2014), 1543-1579. +[14] P. Erd¨os and P. Tur´an, On the distribution of roots of polynomials, Ann. of Math. 2 (1950), 105-119. +[15] I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional +Determinants, Birkh¨ause, 1994. +[16] J.M. Hammersley, The zeros of random polynomials, Proceedings of the third Berkeley symposium on +the mathematical statistics and probability, 1954-1955, vol. II, pp. 89-111. +[17] C. P. Hughes and A. Nikeghbali, The zeros of random polynomials cluster uniformly near the unit circle, +Compos. Math. 144 (2008), no. 212, 1541-1555. +[18] I. Ibragimov and O. Zeitouni, On roosts of random polynomials, Trans.Amer. Soc. 6, (1997), 2427- +2441. +[19] I. Ibragimov and D. Zaporozhets, On Distribution of Random Polynomials in Complex Plane, Prokhorov +and Contemporary Probability Theory, Springer Proc. Math. Stat., 33, Springer, Heidelberg, (2013), +303–323. +[20] M. Kac, On the average number of real roots of a random algebraic equations, Bull. Amer. Math. Soc. +49 (1943), 314-320. +[21] G. Kozma and I. Zeitoni, On Common Roots of Random Bernoulli Polynomials, Int. Math. Res. Not. 18 +(2013), 4334-4347. +[22] J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. III, Rec. +Math. [Mat. Sbornik] N.S. 12(54) (1943), 277–286. +[23] L. A. Shepp and R. J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. +347 (1995), no. 11, 4365–4384. +[24] B. Shiffman, Convergence of random zeros on complex manifolds, Science in China, no.4 Vol 51, (2008), +707-720. +[25] B. Shiffman and S. Zelditch, Equilibrium distribution of zeros of random polynomials, Int. Math. Res. +Not. 1 (2003), 25-49. +[26] B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive +line bundles, Comm. Math. Phys. 200(3):661–683, 1999. +15 + +[27] T. Tao and V. Vu, Local Universality of Random Polynomials, Int. Math. Res. Not. IMRN, (2015), 5053- +5139. +FACULTY OF ENGINEERING AND NATURAL SCIENCES, SABANCI UNIVERSITY, ˙ISTANBUL, TURKEY +Email address: tbayraktar@sabanciuniv.edu +Email address: cigdemcelik@sabanciuniv.edu +16 + diff --git a/5dE2T4oBgHgl3EQf6wiO/content/tmp_files/load_file.txt b/5dE2T4oBgHgl3EQf6wiO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..56e7ac15d0a96bf1f4a09412484b776a807326a3 --- /dev/null +++ b/5dE2T4oBgHgl3EQf6wiO/content/tmp_files/load_file.txt @@ -0,0 +1,1040 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf,len=1039 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='04203v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='CV] 10 Jan 2023 ZERO DISTRIBUTION OF RANDOM BERNOULLI POLYNOMIAL MAPPINGS TURGAY BAYRAKTAR & C¸˙I˘GDEM C¸EL˙IK ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In this note, we study asymptotic zero distribution of multivariable full sys- tem of random polynomials with independent Bernoulli coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We prove that with overwhelming probability their simultaneous zeros sets are discrete and the associated normalized empirical measure of zeros asymptotic to the Haar measure on the unit torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' INTRODUCTION A random Kac polynomial on the complex plane is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1) fd(z) = d � j=0 ajzj where the coefficients aj are independent copies of the (real or complex) standard Gauss- ian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' A classical result due to Kac, Hammersley and Shepp & Vanderbei [20, 16, 23] asserts that almost surely the normalized empirical measure of zeros δZ(fd) := 1 d � fd(ζ)=0 δζ, con- verges to normalized arc length measure on S1 := {|z| = 1} as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Asymptotic zero distribution of Kac polynomials with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' discrete random coefficients have also been studied extensively (see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' [22, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' More recently, Ibragimov and Zaporozhets [19] proved that the empirical measure of zeros δZ(fd) almost surely converges to the the normalized arc length measure if and only if the moment condition E[log(1 + |ai|)] < ∞ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' This property can be considered as a global universality property of the zeros of random polynomials (see also [27] for a local version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Building upon the work of Shiffman and Zelditch [26], equilibrium distribution of random systems of polynomials with Gaussian coefficients was obtained by Bloom & Shiffman [9] and Shiffman [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' More recently, these results were generalized for inde- pendent identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=') random coefficients with bounded density [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We refer the reader to the survey [4] and references therein for the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' On the other hand, asymptotic zero distribution of random polynomial mappings with dis- crete random coefficients remained open (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' [3, 8, 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In this note, we study asymptotic zero distribution of multivariable full system of random polynomials with independent Bernoulli coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Statement of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' A random Bernoulli polynomial is of the form fd,i(x) = � |J|≤d αi,JxJ ∈ C [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , xn] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' and C¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='C¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' are partially supported by T¨UB˙ITAK grant ARDEB-1001/119F184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 1 where xJ = xj1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' xjn n and αi,J are ±1 Bernoulli random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Throughout this work, we consider systems (fd,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fd,n) of random Bernoulli polynomials with independent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We write f d = (fd,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fd,n) for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We denote the collection of all systems of polynomials in n variables and of degree d by Polyn,d that is endowed with the product probability measure Probd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let f d = (fd,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fd,n) be a system of random polynomials with independent ±1 valued Bernoulli coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then there exists a dimensional constant K = K(n) > 0 and an exceptional set En,d ⊂ Polyn,d such that Probd(En,d) ≤ K/d and for all f d ∈ Polyn,d(A)\\En,d the simultaneous solutions of the system f d are isolated with #Z(f d) = dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For a system f d ∈ Polyn,d, if the simultaneous zeros Z(f d) are isolated we denote the corresponding normalized empirical measure by δZ(f d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' That is δZ(fd) is a probabil- ity measure supported on the isolated zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We also let νHaar denote the Haar measure of (S1)n of total mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' As an application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1 together with a determinis- tic equidistribution result [13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='7], we obtain asymptotic zero distribution of random Bernoulli polynomial mappings: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let f d = (fd,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fd,n) be system of random polynomials with independent ±1 valued Bernoulli coefficients and En,d ⊂ Polyn,d be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then for each sequence f d ∈ Polyn,d \\ En,d we have lim d→∞ δZ(fd) = νHaar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' in weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In particular, δZ(f d) → νHaar in probability Probd as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Finally, we consider the measure valued random variables (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2) �Z(f d) = �� ξi∈Z(f d) δ(ξi) for f d ∈ Polyn,d \\ En,d 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' and define the expected zero measure by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='3) � E[ �Z(f d)], ϕ � = � P olyn,d\\En,d � ξi∈Z(f d) ϕ(ξi) dProbd(f d) where ϕ is a continuous function with compact support in Cn and En,d denote the excep- tional set given by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let f d = (fd,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fd,n) be a system of random polynomials with independent ±1 valued Bernoulli coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then lim d→∞ d−nE[Z(f d)] = νHaar in weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The outline of this work as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In §2, we introduce some algebraic background on resultants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In particular, we recall multi-polynomial resultant and sparse resultant for polynomial systems [15, 11] as well as directional resultant [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In §3, we prove the main result Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Finally, in §4 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' PRELIMINARIES In this section, we collect some basic facts and algebraic background related to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' More precisely, we discuss the multi-homogenous (classical) resultant and the sparse eliminant as well as the relation of these two notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For a detailed account of the subject and proofs we refer the reader to [15, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We also discuss the sparse resultant introduced by D’Andrea and Sombra, and corresponding directional sparse resultants [13, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Lattice points, polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For a nonempty subset P ⊂ Rn, we denote its convex hull in Rn by conv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For two nonempty convex sets Q1, Q2, their Minkowski sum is defined as Q1 + Q2 := {q1 + q2 : q1 ∈ Q1, q2 ∈ Q2} and for λ ∈ R, the scaled polytope is of the form λQ := {λq : q ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' It is well known that V oln(d1Q1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' + dnQn) is a homogenous polynomial of degree n in the variables d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , dn ∈ Z+ where V oln denotes the normalized volume of the subsets in Rn with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The coefficient of the monomial d1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' dn is called the mixed volume of Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Qn and denoted by MV (Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' One can use the polarization formula to compute the mixed volume of the convex sets Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Namely, MVn(Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Qn) = n � k=1 � 1≤j1≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='≤jk≤n (−1)n−kV oln(Qj1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' + Qjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In particular, if Q = Q1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' = Qn then MVn(Q) := MVn(Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Q) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='V oln(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For a convex set Q ⊂ Rn its support function sQ : Rn → R is defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1) sQ(v) := inf q∈Q ⟨q, v⟩ where ⟨·, ·⟩ represents the Euclidean inner product of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then the equation ⟨q, v⟩ = sQ(v) defines supporting hyperplane of Q and v is called an inward pointing normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The inter- section of Q with the supporting hyperplane in the direction v ∈ Rn is denoted by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2) Qv := {q ∈ Q : ⟨q, v⟩ = sQ(v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Qv is called the face of Q determined by v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' If Qv has codimension 1, it is called a facet of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Resultant of polynomial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Multipolynomial Resultant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We consider homogenous polynomials of degree di Fi(t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , tn) = � |J|=di ui,JtJ for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n where J is a multi-index (j0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , jn) and tJ indicates the monomial tj0 0 · · · tjn n which is of degree |J| = �n i=0 ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' One can see that the homogenous polynomi- als of degree di form an affine space by identifying � |J|=di ui,JtJ with point (ui,J)|J|=di ∈ CN(di), where N(di) = �n+di−1 n−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Letting N := �n i=0 N(di), recall that the incidence variety is defined by W = � (a, t) ∈ CN × Pn : F0(a0, t) = · · · = Fn(an, t) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We let π : CN × Pn → CN be the projection onto first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' By Projective Extension Theorem (see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' [11]) the image π(W) forms a variety in the affine space CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The multipolynomial resultant Resd0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',dn is defined as the irreducible unique (up to a sign) polynomial in Z[a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , an] which is the defining equation of the variety π(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The resultant of the homogeneous polynomials F0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Fn is the evaluation of Resd0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',dn at the coefficients of F0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Fn and it is denoted by Resd0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',dn(F0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Note that if d0 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' = dn = 1, then the evaluation of multipolynomial resultant Resd0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',dn at the coefficients of F0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Fn is the determinant of the coefficient matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For more general cases, we have the following result: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2 ([15],[11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let F0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Fn ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , xn] be homogenous polynomials of positive total degrees d0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then the system F0 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' = Fn = 0 has a solution in the complex projective space Pn if and only if Resd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',dn(F0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Fn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2 gives a characterization to determine the existence of nontrivial solutions for the systems of homogenous polynomials based on the coefficients of the polynomials in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' However, not all the systems of equations are homogenous, and in the power series expansions not all the monomial terms appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Hence, we need to introduce a more general version of the multi-homogenous resultant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Sparse Eliminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Following [15], we will recall the definition of sparse resultant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let A0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , An be a non-empty finite subsets of Zn, and let ui = {ui,a}a∈Ai be a group of #Ai variables, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n and set u = {u0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , un} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For each i, the general Laurent polynomial fi with support supp(fi) = Ai given by f(x) = � a∈Ai ui,axa ∈ C[u][x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , x±1 n ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We let A = (A0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , An) and consider the incidence variety in this setting defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='3) WA = � (u, x) ∈ n � i=0 P(CAi) × (C∗)n : f0(ui, x) = · · · = fn(un, x) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 4 Consider the canonical projection on the first coordinate πA : n � i=0 P(CAi) × (C∗)n → n � i=0 P(CAi) and let πA(WA) denote the Zariski closure of WA under the projection π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The sparse eliminant, denoted by ResA, is defined as follows: if the variety πA(WA) has codimension 1, then the sparse eliminant is the unique (up to sign) irreducible polynomial in Z[u] which is the defining equation of πA(WA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' If codim(πA(WA)) ≥ 2, then ResA is defined to be the constant polynomial 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The expression ResA(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn) is the evaluation of ResA at the coefficients of f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For A0 = {0} , A1 = {0, 1} ⊂ Z, we have that ResA0,A1 = ±u00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The classical resultant Resd0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',dn is the special case of the sparse eliminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Indeed, let Ai be the set of all integer points in the di-simplex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=', Ai = diΣn ∩ Zn and Σn is the standard unit simplex, that is, diΣn := {(a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , an) ∈ Rn+1 : aj ≥ 0, � j aj ≤ di}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Following [11] and [15], for simplicity let all the sparse polynomials f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn have the same support Ad = dΣn ∩ Zn for some positive integer d and consider the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='4) \uf8f1 \uf8f2 \uf8f3 f0 = u01xα1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' + u0dxαn = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' fn = un1xα1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' + undxαn = 0 We also let t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , tn be the homogenous coordinates which are related to x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , xn by xi = ti/t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then we define the homogenous polynomials (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='5) Fi(t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , tn) = td 0fi(t1/t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , tn/t0) = td 0fi(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , xn), for 0 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' This method gives n+1 homogenous polynomials of total degree d in the variables t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , tn and this procedure is independent of the choice of homogeneous coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='5 ([11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let Ad = dΣn ∩Zn and consider the systems of polynomials F and f as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then ResA(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn) = ±Resd,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',d(F0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Fn), where A = (Ad, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Ad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Using the above proposition, we can give a version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 5 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let f = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn) be a system of polynomials with supp(fi) = Ad for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Assume that the system F = (F0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Fn) consists the homogenizations of fi according to process in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='5) and denote the set of simultaneous nonzero solutions of F by Z(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Suppose that Z(F ) ∩ H∞(t0) = ∅ where H∞(t0) is the hyperplane at infinity for t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then the system of polynomials f = 0 has no solution if and only if ResAd(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' If ResAd(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn) ̸= 0, then by definition of the sparse resultant the system f0(x) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' = fn(x) = 0 has no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Conversely, letting Fi be the homogenization of fi as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='5) with the corresponding variable t = (t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , tn), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Fi(t) = td 0fi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' If the system of polynomials f = 0 has no solution then Fi(t) = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n if and only if t0 = 0 which contradicts our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Hence, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2 we have ±ResAd(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn) = Resd0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',dn(F0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Fn) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Sparse Resultant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In spite of being a generalization of the multipolynomial resul- tant and involving considerable large amount of the system of polynomials, the sparse eliminant does not satisfy some essential properties which is necessary in many applica- tions, such as additivity property and Poisson formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In 2014, D’Andrea and Sombra [12] introduced the following version of the sparse resultant which has the desired fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The sparse resultant, denoted by ResA, is defined as any primitive poly- nomial in Z[u] that is the defining equation of the direct image of WA, (πA)∗(WA) = deg(πA|WA)πA(WA) if this variety has codimension one, and otherwise we set ResA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The expression ResA(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn) is the evaluation of ResA at the coefficients of f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' According to this definition, the sparse resultant is not irreducible but it is a power of the irreducible sparse eliminant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=', ResA = ±Res deg(πA|WA) A where deg(πA|WA) is the degree of the projection πA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We also remark that ResA ̸= 1 whenever ResA ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For more details we refer the reader to the manuscripts [12] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let A0 = A1 = A2 = {(0, 0), (2, 0), (0, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then ResA = det(ui,j) and ResA = ±[det(ui,j)]4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Directional Resultant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For a subset B ⊂ Zn and a polynomial f(x) = � b∈B βbxb with support B, we write Bv := {b ∈ B : ⟨b, v⟩ = sQ(v)} and f v(x) = � b∈Bv βbxb where Q = conv(B) and v ∈ Rn and sconv(B)(v) is defined as equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , An ⊂ Zn be a family of n non-empty finite subsets, v ∈ Zn\\{0}, and v⊥ ⊂ Rn the orthogonal subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then there exists bi,v ∈ Zn such that Av i − bi,v ⊂ Zn ∩ v⊥ for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The resultant of A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , An in the direction of v, denoted ResAv 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',Avn is defined as the resultant of the family of the finite subsets Av i − bi,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let fi ∈ C[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , x±1 n ] be Laurent polynomials with support supp(fi) ⊂ Ai i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n, we write f v i = xbi,vgi,v for a Laurent polynomial gi,v ∈ C[Zn ∩ v⊥] ≃ C[y±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , y±1 n−1] with supp(gi,v) ⊂ Av i − bi,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The expression ResAv 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',Avn(f v 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , f v n) is defined as the evaluation of this resultant at the coefficients of the gi,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We remark that the definition of directional resultant is independent of the choice of the vector bi,v (see [12, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Moreover, the directional resultant ResAv 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',Avn ̸= 1 only if the direction vector v is an inward pointing normal to a facet of the Minkowski sum �n i=1 conv(Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Therefore, the nontrivial directional resultants of the family A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , An is finitely many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let f(x) = a0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' + anxn ∈ C[x] be a polynomial of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then the nontrivial directional resultants are ResA(f v) = � ±a0 if v = 1, ±an if v = −1 for the polytope conv(A) = [0, n] ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' EQUIDISTRIBUTION OF ZEROS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Random Polynomial Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' First, we recall a theorem of Kozma and Zeitouni [21] asserts that overdetermined random Bernoulli polynomial systems have no common zeros with overwhelming probability: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn+1 ∈ Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , xn] be n + 1 independent random Bernoulli polynomials of degree d and P(d, n) := Probd{∃x ∈ Cn : fi(x) = 0 for i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n + 1} denote the probability that the system f1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' = fn+1 = 0 has a common solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then there exists a dimensional constant K = K(n) < ∞ such that P(d, n) ≤ K/d 7 for all d ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Next, we prove our main result: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let fd,i be a random Bernoulli polynomial of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1) fd,i = � |J|≤d αi,JxJ ∈ Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , xn], where {αi,J} is a family of independent Bernoulli random variables for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We investigate the directional resultants of the system f for all nonzero primitive di- rection vectors v ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' By [12, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='3] it is enough to check the inward normals to the Minkowski sum of the supports ndΣn which has n + 1 facets with n + 1 inward normals given by vm = em for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n and vn+1 = − �n m=1 em where {em}n m=1 is the standard basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For vm = em the intersection of a support A with the supporting hyperplane in the direction em is of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2) Avm = � (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , jn) ∈ A : jm = 0, n � l=1 jl ≤ d � m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Hence, the polynomials f vm i can be written as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='3) f vm i := � J∈Avm αi,JxJ for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Note that polynomials f vm i depend on n − 1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Following the Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='9, if we choose the vector bi,vm = 0 such that Avm − bi,vm ⊂ Zn ∩ vm⊥, we see that the functions gi,vm := f vm i satisfies the equation f vm i = xbi,vmgi,vm for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Recall that for two univariate polynomials h1, h2 ∈ C[x], their resultant Res(h1, h2) is zero if and only if h1 and h2 have a common solution in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Therefore, if n = 2 the necessary and sufficient condition for g1,vm and g2,vm have zero resultant is that they have a common zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1 implies that there exists a constant Km which is independent of d so that the aforementioned event has probability at most Km/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' On the other hand, when n > 2, we perform the homogenization process to each (n−1) variable polynomial gi,vm for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n as described in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We obtain the n variable homogenous polynomials Gi,vm of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='4) Gi,vm(t, x) = � J∈Avm αi,Jtd−|J|xJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In order to compare the sparse resultant of the polynomials gi,vm and the multipolynomial resultant of the homogeneous polynomials Gi,vm, we check the conditions of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let Z(G) be the set of nontrivial solutions of the system G = (G1,vm, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , Gn,vm) and suppose that G has a solution ξ = (t, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , ξn) in the hyperplane at infinity H∞(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Evaluating these homogeneous polynomials at t = 0, we obtain the top degree homoge- neous part of the polynomials gi,vm for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Since ξ ∈ H∞(t), it has a nonzero coordinate ξk for some k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For simplicity, let us assume k = 2 and define the 8 new variables zi := ξi+2/ξ2 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Applying this change of variables, we obtained (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='5) �Gi,vm(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , zn−2) = � |J|≤d αi,Jzϕ(J) where ϕ : Rn → Rn−2 with ϕ(j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , jn) = (j3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' This gives n random Bernoulli polynomials of degree d in n − 2 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Hence by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1, there exists a pos- itive constant Ci, depending only the dimension n such that the probability that the overdetermined system of Bernoulli polynomials �Gi,vm(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , zn−2) have a common so- lution is less than Ci/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We infer that the system of homogenized polynomials Gi,vm has no common zero at hyperplane at infinity H∞(t) except a set that has probability at most Ci/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='6, outside of a set of small probability, the system of polynomials consisting gi,vm has a common solution if and only if the directional resul- tant ResAvm 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',Avn(f v 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , f v n ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Now, since the system of Bernoulli polynomials gi,vm contains n polynomials in n − 1 variables, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1, there is a dimensional con- stant ˜Ci so that the probability that this system has common solution is at most ˜Ci/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Hence outside of a set that has probability Ki/d := Ci/d + ˜Ci/d , the directional resultant ResAvmf vm d ̸= 0 for all vm for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Next, for the inward normal vector vn+1 = − �n m=1 em, we find the minimal weight in this direction as Avn+1 = {J ∈ A : |J| = d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Hence the polynomials in this directions are of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='6) f vn+1 i = � |J|=d αi,JxJ In this case Avn+1 is not a subspace of Zn ∩ v⊥ n+1, hence we need to translate it by sub- tracting a suitable vector bi,vn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For Laurent polynomial systems, the sparse resultant is invariant under translations of supports (see [12], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Since the polynomi- als fd,i are not Laurent, we need to determine the effects of this translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Consider the system of Bernoulli polynomials f d and set of its simultaneous zeros Z(f d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For a solution x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , xn) ∈ Z(f d) and assume that x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In order to examine the incidence of this case, we evaluate the system f d at x1 = 0 and we obtain a new system of n Bernoulli polynomials with n − 1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1, there exists a constant C1 which is independent of d such that this system has a common solution with proba- bility at most C1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Therefore the probability of the event that x1 = 0 is less than C1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Hence there is no harm of translation of supports outside of a set that has probability at most C/d, where C := �n i=1 Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Now, choosing the vector bi,vn+1 = (d, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , 0) so that Avn+1 − bi,vn+1 ⊂ Zn ∩ v⊥ n+1, we obtain the polynomials of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='7) gi,vn+1 = � J∈Avn+1−bi,vn+1 αi,Jxw(J) with w : Rn → Rn satisfying (j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , jn) �→ (−d + j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We substitute the new variables yi := xi+1/x1 into gi,vn+1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n − 1 and obtain 9 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='8) gi,vn+1(y) = � |J|≤d αi,Jyσ(J) for y ∈ Cn−1 and σ : Rn → Rn with σ(j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , jn) = (0, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The system con- taining the polynomials gi,vn+1(y), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n contains n random Bernoulli polynomials with n − 1 random variable as in the cases vm = em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' By applying the same steps, it can be shown that ResAvn+1f vn+1 d ̸= 0 outside of a set that has probability at most Ki+1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Now, we define the exceptional set En,d as a subset of Polyn,d which contains the sys- tems f d that has a zero directional resultant for some nonzero primitive vector v or the systems f d have a common solution x ∈ Cn with xi = 0 for some i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' More precisely, letting En,d := {f d ∈ Polyn,d : ∃ v ∈ Zn \\ {0} ∋ ResAvf v d = 0} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='9) � {f d ∈ Polyn,d : ∃ x ∈ Z(f d) ∋ � xi = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' we see that there exists a positive constant K which is independent of d such that Prob{En,d} ≤ d−1K where K := �n+1 i=1 Ki + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' □ Next, we recall a deterministic equidistribution results for the solutions of systems of integer coefficient polynomials [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For a polynomial f ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , xn], the supremum norm of f on the unit torus is defined as ∥f∥sup := sup |w1|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='=|wn|=1 |f(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , wn)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let νHaar be the Haar measure on Cn with support (S1)n and of total mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Assume that f ∈ Polyn,d be a polynomial mapping such that the set of simultaneous zeros Z(f) is a discrete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We denote by denote the discrete probability measure on Cn associated to the Z(f) by δZ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The following result gives the asymptotic distribution of the zeros of such a system f if the coefficients are integer: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' [13] Let f = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fn) be a polynomial mapping with fi ∈ Z[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , xn] of degree d ≥ 1 for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Assume that ResAv 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',Avn(f v 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , f v n) ̸= 0 for all v ∈ Zn \\ {0} and log ||fi||sup = o(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Then lim d→∞ δZ(f) = νHaar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' As a corollary of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2, we have the following equidistribution result for random Bernoulli polynomial mappings: Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Consider the system of Bernoulli polynomials f d = (fd,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , fd,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Since all the coefficients are 1 or −1, by triangle inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='10) ∥fd,i∥sup = sup |w1|=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='=|wn|=1 |fd,i(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , wn)| ≤ �n + d d � = O(dn) 10 which implies that log ∥fd,i∥sup = o(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Moreover, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1 for each sequence f d ∈ Polyn,d \\ En,d we have ResAv 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=',Avn(f v 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , f v n ) ̸= 0 for all v ∈ Zn \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Hence, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='2 lim d→∞ δZ(f d) = νHaar in weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' In particular, δZ(f d) → νHaar in probability since Prob{En,d} → 0 as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' EXPECTED ZERO DISTRIBUTION In this section, we introduce radial and angle discrepancies for random Bernoulli poly- nomial mappings in order to study asymptotics of expected zero measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' We adapt these concepts from [13] and refer the reader to the manuscript [13] and references therein for a detailed account of the preliminary results this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Let Z be a 0-dimensional effective cycle in Cn that is there is a non-empty finite col- lection of points ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , ξn) ∈ Cn and mξ ∈ N, called the multiplicity of ξ, such that Z = � ξ mξ[ξ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' The degree of Z is defined by deg(Z) = � ξ mξ which is a positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' [13] Let Z be a 0-dimensional effective cycle in Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' For each α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , αn) and β = (β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , βn) ∈ Rn such that −π ≤ αj < βj ≤ π, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQf6wiO/content/2301.04203v1.pdf'} +page_content=' , n we consider the cycle Zα,β := � {ξ∈Z:αj 0, +wij ← wij + α(R − R)wijC +� +− e−c(τ−T−) + e−c(T+−τ)� +. +(2.3) +Supervised learning is more commonly formulated in loss functions than rewards. Because a +high reward corresponds to a small loss and vice versa, L := −R is a loss function, L = −R +4 + +is the anticipated loss, and the updating formula becomes +wij ← wij + α(L − L)wijC +� +e−c(τ−T−) − e−c(T+−τ)� +. +(2.4) +A key observation is that these updating formulas are derivative-free in the sense that they +involve the reward (or loss) but not its gradient. +Hebbian learning rules, such as (2.4), model the updating of individual weights, but do not +explain how the brain can learn a task. A brief overview about relevant existing ideas on +learning in BNNs is given in Section 5 +3 +Zero-order optimization +Suppose we want to fit a d-dimensional parameter vector θ to the data and write L(θ) +for the (training) loss incurred by parameter θ. Derivative-free optimization procedures +do not require computation of the gradient of the loss. A simple iterative derivative-free +scheme would be to randomly pick in each round a new candidate parameter and update +the parameter if the loss is decreased. Standard references for derivative-free optimization +include [36, 6, 10, 16, 20]. +Zero-order methods (sometimes also called zero-th order methods) are specific derivative- +free optimization procedures. To explain the concept, recall that standard gradient descent +is an iterative procedure aiming to minimize the loss function θ �→ L(θ) by the iterative +scheme +θk+1 = θk − αk+1∇L(θk), +k = 0, 1, . . . +where the initial values θ0 are chosen in some way, αk+1 > 0 is the learning rate and ∇L(θk) +denotes the gradient of the loss function at θk. In contrast, zero-order methods are only +allowed to access the loss function but not the gradient of the loss. From the loss, one +can build, however, an estimator for the gradient of the loss. 1-point zero-order methods +replace −∇L(θk) by +βL(θk + ξk)ξk +with ξk a d-dimensional random vector and β a constant. To see how this relates to the +gradient, consider the specific case that ξk is multivariate normal with zero mean vector +and covariance matrix σ2Id, where Id denotes the d × d identity matrix. The multivariate +5 + +version of Stein’s lemma [38] states that +E[L(θk + ξk)ξk] = σ2E[∇L(θk + ξk)] +(3.1) +under weak regularity conditions ensuring that all expectations are well-defined. +This +means that σ−2L(θk + ξk)ξk estimates the gradient at θk + ξk, that is, ∇L(θk + ξk) = +∇L(θk)+errork. The hope is that over many iterations the noise contributions cancel out +such that in the long-run, the 1-point zero-order dynamics behaves similarly as gradient +descent. The argument above can be extended to general symmetric distributions of ξk +that are not necessarily Gaussian. +Unfortunately, the variance of the 1-point zero-order gradient estimator (3.1) can be ex- +tremely large and often scales quadratically in the number of parameters d. As an example, +suppose that the data are stored in a d-dimensional vector Y = (Y1, . . . , Yd)⊤ and con- +sider the least squares loss L(θ) = ∥Y − θ∥2 +2. Taking ξk = (ξk1, . . . , ξkd) ∼ N(0, σ2Id) and +β = σ−2, as above, we have for the j-th component of βL(θk + ξk)ξk that +σ−2��Y − θk − ξk +��2 +2ξkj = σ−2� +Yj − θkj − ξkj +�2ξkj + σ−2 � +ℓ:ℓ̸=j +� +Yℓ − θkℓ − ξkℓ +�2ξkj. +The second term on the right hand side has zero mean. It is pure noise and does not help +to estimate the gradient. This sum is over d − 1 summands and its variance scales with +O(d2) in the number of parameters d. +Due to the large variance, there are many scenarios for which 1-point zero-order dynamics +quickly diverges to infinity. Indeed if one iterate θk is already far away from the minimum, +the large loss can result in a parameter update θk+1 which is much further away from the +minimizer than θk, leading to an even larger loss and an exponential growth of the loss as +the number of iterations is further increased. +Regarding theory of zero-order methods, [10] studies a related zero-order methods and +mirror descent. Assuming that the parameter vector lies in an Euclidean ball, they obtain +in their Corollary 1 the rate +� +d/k with k the number of iterations and also provide a +corresponding lower bound proving that this rate is optimal (their Proposition 1). The +large noise causes the factor +√ +d in the rate, suggesting slow convergence in the high- +dimensional regime. [25] also finds a suboptimality of order d if zero-order methods are +compared to gradient descent. Table 1 in [20] shows that the factor +√ +d or d occurs in all +known convergence rates unless second-order information is used. +Due to the large noise, derivative-free methods are in general thought to be inferior com- +pared to gradient descent. +This is for instance remarked in [6], Section 1.3: ”Finally, +6 + +we want to make a strong statement that often councils against the use of derivative-free +methods: if you can obtain clean derivatives (even if it requires considerable effort) and the +functions defining your problem are smooth and free of noise you should not use derivative- +free methods.” +Zero-order methods are also not necessarily much faster to compute than gradient descent +iterates. For the gradient-based backpropagation of ANNs, the number of operations re- +quired for the forward pass is of the same order as the number of operations required for +the backwards pass. Evaluation of the loss is therefore not substantially cheaper than com- +puting the gradient and zero-order methods cannot be computed at a faster order than +backpropagation. +Despite these rather discouraging remarks, there is a rapidly increasing interest in derivative- +free methods and they are successfully applied in practice, for example by Google [13]. +4 +Hebbian learning as zero-order optimization method +The updating formula (2.4) allows to address supervised learning tasks, where we want to +learn the functional relationship between inputs and outputs given observations (or training +data) from input-output pairs (X1, Y1), (X2, Y2), . . . that are all generated from the same, +unknown distribution as the vector (X, Y ). Well-known examples for this framework are +classification and regression. For instance to classify cat and dog images, Xi is the i-th +image containing all the pixel values of the i-th cat image and Yi is the corresponding label +”cat” or ”dog”, coded as 0 or 1. +Consider now a feedforward biological neural network (BNN) with m neurons. This means +that the neurons/nodes form a directed acyclic graph (DAG) with input neurons receiving +information from the data Xi and possibly several output neurons. For the subsequent +analysis, we neither have to specify a layered structure as commonly done for ANNs nor +conversion rules how vector valued inputs are converted into spike trains or output spike +trains are cast into response variables, such as conversion into labels in a classification +problem. +In the k-th instance, we feed the k-th input vector Xk in the BNN, let the BNN run and +receive then as output the predicted response �Yk. The loss at this round is a measure for the +difference between the predicted response �Yk and the real response Yk. It will be denoted +by L(�Yk, Yk) in the following. The anticipated loss that occurs in (2.4) could be modelled +by a (weighted) average over past iterations. Here we use the loss of the previous iterate +L(�Yk−1, Yk−1). +7 + +During each instance, several spikes can be sent between any two connected neurons. We +impose the (strong) assumption that for every run, and any connection, exactly one spike +will be released. +Number the m nodes, that represent the neurons in the graph, by 1, . . . , m and denote the +edge set by T . A pair (i, j) is in T if and only if neuron i is a presynaptic neuron for neuron +j. Equivalently, (i, j) ∈ T iff there is an arrow from i to j in the underlying DAG. We +consider the case that the BNN topology is static, that is, the edge set T does not change +during learning. +If w(k) +ij +is the BNN weight after the k-th round, it is then updated in the (k +1)-st iteration +according to (2.4) +w(k+1) +ij +(4.1) += w(k) +ij + αk+1 +� +L(�Yk, Yk) − L(�Yk−1, Yk−1) +� +w(k) +ij C +� +e−c(τ (k) +ij −T (k) +−,j) − e−c(T (k) ++,j−τ (k) +ij )� +, +for all (i, j) ∈ T and αk+1 > 0 the learning rate. Here T (k) +−,j and T (k) ++,j are the closest spike +times of the j-th neuron before/after the arrival time τ (k) +ij +of the spike that is sent from +neuron i to neuron j. The constant C can be integrated into the loss function and is from +now on set to one. +For the updating, the location of τ (k) +ij +is important within the interval [T (k) +−,j, T (k) ++,j], while the +interval length seems to play a minor role. Therefore, we assume that the interval length +is constant and set A := (T (k) ++,j − T (k) +−,j)/2. We assume moreover that the arrival time of the +spike from neuron i to neuron j has a negligible influence on the spike times of neuron j, +that the spike times τ (k) +ij +are all independent of each other, and follow a uniform distribution +on the interval [T (k) +−,j, T (k) ++,j]. As mentioned before, to trigger a spike, it needs of the order of +20 − 50 presynaptic neurons to fire in a short time interval. The influence of an individual +neuron seems therefore rather minor, justifying the previous assumption. The assumptions +above show that the random variable U (k) +ij +:= τ (k) +ij +− 1 +2(T (k) ++,j + T (k) +−,j) are jointly independent +and uniformly distributed on [−A, A]. Hence, (4.1) becomes +w(k+1) +ij += w(k) +ij + αk+1 +� +L(�Yk, Yk) − L(�Yk−1, Yk−1) +� +w(k) +ij +� +e−c(A+U(k) +i,j ) − e−c(A−U(k) +i,j )� +, +for all (i, j) ∈ T . The factor e−cA can be absorbed into the loss function and the constant c +can be absorbed into the hyperparameter A. By reparametrization, we obtain the updating +formula +w(k+1) +ij += w(k) +ij + αk+1 +� +L(�Yk, Yk) − L(�Yk−1, Yk−1) +� +w(k) +ij +� +e−U(k) +i,j − eU(k) +i,j +� +, +(4.2) +8 + +for all (i, j) ∈ T . +To further analyze this scheme, it is important to understand how the predicted response +�Yk depends on the parameters. We now argue that, under the same assumptions as before, +�Yk is a function of the variables w(k) +ij + eU(k) +i,j . The high-level rationale is that in this neural +model, all the information that is further transmitted in the BNN about the parameter +w(k) +ij +sits in the spike times of neuron j and the interarrival spike times only depend on w(k) +ij +through w(k) +ij + eU(k) +i,j . To see this, fix neuron j. The only information that this node/neuron +releases to its descendants in the DAG are the spike times of this neuron. This means that +from all the incoming information that neuron j receives from presynaptic neurons (parent +nodes) only the part is transmitted that affects the spike times of neuron j. As mentioned in +Section 2, a spike arriving at neuron j from neuron i at time τ (k) +ij +causes the potential t �→ +w(k) +ij eτ (k) +ij −t1(t ≥ τ (k) +ij ) at node j. If every incoming neuron spikes once, the overall potential +of neuron j is � +i:(i,j)∈T w(k) +ij eτ (k) +ij −t1(t ≥ τ (k) +ij ). If S denotes the threshold value for the +potential at which a neuron spikes, then at the spike time T (k) ++,j of the j-th neuron, we have +by the definition of U (k) +ij , S = � +i:(i,j)∈T w(k) +ij eτ (k) +ij −T (k) ++,j = � +i:(i,j)∈T w(k) +ij eU(k) +ij − 1 +2 (T (k) ++,j−T (k) +−,j). +Rearranging this equation shows that the interarrival spike time T (k) ++,j−T (k) +−,j can be expressed +in terms of the variables w(k) +ij eU(k) +ij . Introduce wk := (w(k) +ij )(i,j)∈T , Uk := (U (k) +ij )(i,j)∈T and +write wkeUk for (w(k) +ij eU(k) +i,j )(i,j)∈T . The previous argument indicates that the predictor �Yk +is a function of wkeUk and Xk. Thus, the loss L(�Yk, Yk) can be written as a function of the +form L +� +wkeUk, Xk, Yk +� +and (4.2) becomes +w(k+1) +ij +(4.3) += w(k) +ij + αk+1 +� +L +� +wkeUk, Xk, Yk +� +− L +� +wk−1eUk−1, Xk−1, Yk−1 +�� +w(k) +ij +� +e−U(k) +i,j − eU(k) +i,j +� +. +In a BNN, the parameters w(k) +ij are non-negative. We now introduce the real-valued variables +θ(k) +ij += log(w(k) +ij ) and θk = (θ(k) +ij )(i,j)∈T . This means that w(k) +ij += eθ(k) +ij . A first order Taylor +expansion shows that for real numbers u, v, ∆ such that e−v∆ is small, eu = ev + ∆ gives +u = log(ev + ∆) = v + log(1 + e−v∆) ≈ v + e−v∆. Working with this approximation, we +can rewrite the formula (4.3) in terms of the θ’s as +θ(k+1) +ij +(4.4) += θ(k) +ij + αk+1 +� +L +� +θk + Uk, Xk, Yk +� +− L +� +θk−1 + Uk−1, Xk−1, Yk−1 +��� +e−U(k) +i,j − eU(k) +i,j +� +. +Relating this formula to gradient descent and the weight transportation problem mentioned +in the introduction, we see that the update of one parameter only depends on all the other +9 + +parameters through the value of the loss function. +In vector notation, the previous equality becomes +θk+1 +(4.5) += θk + αk+1 +� +L(θk + Uk, Xk, Yk) − L(θk−1 + Uk−1, Xk−1, Yk−1) +�� +e−Uk − eUk� +, +where eUk and e−Uk should be understood as componentwise applying the functions x �→ ex +and x �→ e−x to the vector Uk. In particular, the loss is always a scalar and eUk, e−Uk are +d-dimensional vectors. +So far, we have not specified any initial conditions. From now on, we assume that the +initial values θ0, θ−1 are given and that all the other parameter updates are determined by +(4.5) for k = 0, 1, . . . with U−1, U0, U1, U2, . . . drawn i.i.d. from the uniform distribution +U([−A, A]d). +As an analogue of (3.1), the next result shows that in average, this dynamic can also be +understood as a gradient descent method with gradient evaluated not exactly at θk but at +a random perturbation θk + Uk. +Theorem 1. Write Uk = (Uk1, . . . , Ukd)⊤ and let (eA −eUk)(eA −e−Uk) be the vector with +components (eA − eUkj)(eA − e−Ukj). Denoting by ⊙ the Hadamard product (componentwise +product) of two matrices/vectors of the same dimension(s), we have +E +� +θk+1 +� += E +� +θk +� +− αk+1e−AE +� +∇θkL(θk + Uk, Xk, Yk) ⊙ +� +eA − eUk�� +eA − e−Uk�� +. (4.6) +Instead of taking the expectation over all randomness, the statement is also true if we only +take the expectation with respect to Uk, which is the same as the conditional expectation +E[·|U−1, U0, U1, . . . , Uk−1, (Xℓ, Yℓ)ℓ≥1]. +Note that (eA − eUkj)(eA − e−Ukj) is non-negative. Thus fA(x) = C(A)−1(eA − ex)(eA − +e−x)1(−A ≤ x ≤ A) defines a probability density function for the positive normalization +constant C(A) = 2A(e2A + 1) + 2 − 2e2A = +� A +−A(eA − ex)(eA − e−x) dx. +Denoting by +∂jL(v, Xk, Yk) the partial derivative of L with respect to the j-th component of v, we can +state the previous result componentwise as +E +� +θk+1,j +� += E +� +θkj +� +− αk+1e−AC(A)E +� +∂jL(θk + U(j) +k , Xk, Yk) +� +, +(4.7) +for a random vector U(j) +k += (Uk1, . . . , Uk,j−1, Vkj, Uk,j+1, . . . , Ukd)⊤, with jointly indepen- +dent random variables Vkj ∼ fA and Ukℓ ∼ U[−A, A], ℓ = 1, . . . , j − 1, j + 1, . . . , d. +10 + +Proof of Theorem 1. Throughout the proof, we omit the dependence of the loss function L +on the data. By conditioning on (U−1, U0, . . . , Uk−1, (Xℓ, Yℓ)ℓ≥1) and the fact that e−Uk +and eUk have the same distribution, it follows that +E +� +L(θk−1 + Uk−1) +� +e−Uk − eUk�� += E +� +L(θk−1 + Uk−1)E +�� +e−Uk − eUk� ��� U−1, U0, . . . , Uk−1, (Xℓ, Yℓ)ℓ≥1 +�� += 0. +(4.8) +With u = (u1, . . . , ud)⊤, the j-th component of e−AE[∇θkL(θk+Uk)⊙(eA−eUk)(eA−e−Uk)] +is +e−A +(2A)d +� +[−A,A]d ∂jL(θk + u) +� +eA − euj�� +eA − e−uj� +du += e−A +(2A)d +� +[−A,A]d−1 +� A +−A +∂jL(θk + u) +� +eA − euj�� +eA − e−uj� +dujdu1 . . . duj−1duj+1 . . . dud, +Observe that (eA − euj)(eA − e−uj) vanishes at the boundaries uj ∈ {−A, A} and ∂uj(eA − +euj)(eA − e−uj) = eA−uj − eA+uj. Thus, applying integration by parts formula to the inner +integral yields +� A +−A +∂jL(θk + u) +� +eA − euj�� +eA − e−uj� +duj = −eA +� A +−A +L(θk + u) +� +e−uj − euj� +duj +and therefore +e−A +(2A)d +� +[−A,A]d ∂jL(θk + u) +� +eA − euj�� +eA − e−uj� +du += − +1 +(2A)d +� +[−A,A]d L(θk + u) +� +e−uj − euj� +du += −E +� +L(θk + Uk) +� +e−Ukj − eUkj�� +. +This holds for all j = 1, . . . , d. The minus on the right hand side cancels out the first minus +in (4.6). Together with (4.8), the claim follows. +Equation (4.8) in the proof shows that the theorem still holds if the term L(θk−1 + +Uk−1, Xk−1, Yk−1) in (4.5) is replaced by zero or any other value that is independent of +Uk. +To obtain a proper zero-order method, a crucial assumption is to choose the amplitude +functions A+, A− in (2.1) to be the same. In the brain, these functions are close, but some +11 + +authors argue that there is a slight difference [34]. Such differences would lead to additional, +small contributions in the iterations that cannot be linked to the gradient. +A statistical analysis of the zero-order method (4.5) is challenging, even for simple models +such as data generated from the linear regression model. +Another open problem is to +determine whether the convergence rate of (4.5) scales in the number of parameters d in +the same way as other zero-order methods. +5 +Literature on learning with BNNs +This literature survey is aimed to give a quick overview. For a more detailed summary of +related literature, see [37, 41]. +To train BNNs on data, a natural idea is to ignore Hebbian learning and to fit BNNs via +gradient descent. Similar as backpropagation efficiently computes the gradient in ANNs, +SpikeProp [3, 4] is an algorithm to compute the gradient for spiking neural networks. +The weight transportation problem is caused by the parameter dependence in the backwards +pass of the backpropagation algorithm. Feedback alignment [18, 26, 17, 1, 19] avoids this +by using the backpropagation algorithm with random weights. In a network, the feedback +could be then transmitted via specific feedback neurons. +If the brain does a version of backpropagation, the difficulty is always the feedback from +the output backwards to the neurons. Contrastive Hebbian learning [27] assumes that there +are two different phases. During the first phase the network does prediction and the second +phase starts after the prediction error is revealed. In one of the phases the learning is +Hebbian and in the other one, the learning is anti-Hebbian. Anti-Hebbian learning means +that if two neurons fire together, the connecting weight parameter is decreased instead of +increased. Equilibrium propagation [29] overcomes the two types of learning in the different +phases but requires again the computation of a gradient. +For a biologically more plausible implementation of the weight transportation problem, +predictive coding [40, 41, 35, 22, 23] uses two types of neurons, named error nodes and value +nodes. These two nodes are associated to each other and process forward and backward +information locally. +[31] proposes the concept of a ”hedonistic synapse” that follows a Hebbian learning rule +and takes the global reward into account. For the learning, a hedonistic synapse has to be +able to store information from previous trials in a so-called eligibility trace. +Closest to our approach is weight perturbation [39]. Weight perturbation adds random +12 + +noise to the parameters or the outputs and compares the loss with and without added +noise to estimate the gradient. Whereas the cause of the noise perturbation is not entirely +clear in the weight perturbation framework, we have shown in this work, how the spike +train structure in BNNs implies a random perturbation of the parameters in the loss with +uniformly distributed noise and how this leads to a specific derivative-free updating formula +for the weights that also involves the difference of the loss function evaluated for different +instance of the noisy parameters. +A more statistical approach is [24] considering unsupervised classification using a small +BNN. This work identifies a closer link between a Hebbian learning rule and the EM- +algorithm for mixtures of multinomial distributions. +Some other ideas on unsupervised +learning in BNNs are moreover provided in [12], Section 19.3. +To summarize, there are various theories that are centered around the idea that the learning +in BNNs should be linked to gradient descent. 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A crit- +ical window for cooperation and competition among developing retinotectal synapses. +Nature 395, 6697 (1998), 37–44. +17 + diff --git a/5tFKT4oBgHgl3EQfSy3E/content/tmp_files/load_file.txt b/5tFKT4oBgHgl3EQfSy3E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0866b31c64b21b7b97c3bc3f64dbc757c3d2a3bb --- /dev/null +++ b/5tFKT4oBgHgl3EQfSy3E/content/tmp_files/load_file.txt @@ -0,0 +1,631 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf,len=630 +page_content='Interpreting learning in biological neural networks as zero-order optimization method Johannes Schmidt-Hieber∗ Abstract Recently, significant progress has been made regarding the statistical understanding of artificial neural networks (ANNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' ANNs are motivated by the functioning of the brain, but differ in several crucial aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In particular, it is biologically implausible that the learning of the brain is based on gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In this work we look at the brain as a statistical method for supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The main contribution is to relate the local updating rule of the connection parameters in biological neural networks (BNNs) to a zero-order optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Keywords: Biological neural networks, zero-order optimization, derivative-free methods, supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' 1 Introduction Compared to artificial neural networks (ANNs), the brain learns faster, generalizes better to new situations and consumes much less energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' A child only requires a few examples to learn to discriminate a dog from a cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' And people only need a few hours to learn how to drive a car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' AI systems, however, need thousands of training samples for image recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' And the self-driving car is still under development, despite the availability of data for millions of kilometers of test drives and billions of kilometers of simulated drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The ∗University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Email: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='schmidt-hieber@utwente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='nl This work has tremendously profited from several discussions with Wouter Koolen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The author is moreover extremely grateful for helpful suggestions and several interesting remarks that were brought up by Matus Telgarsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The research has been supported by the NWO/STAR grant 613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='034b and the NWO Vidi grant VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='Vidi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='11777v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='LG] 27 Jan 2023 Figure 1: Artificial neurons receive and output numbers, biological neurons receive and output spike trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' superhuman performance of AI for some tasks [30, 33, 5] has to be related to the huge databases and the enormous computing power required for the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' When identifying the causes for the differences in statistical behavior, it is important to emphasize that although ANNs are inspired by the functioning of the brain, they are very different from biological neural networks (BNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Each biological neuron emits a so called spike train that can be modelled as a stochastic process or, more precisely, as a point process [8, 9] and all computations in BNNs, including the updating of the network parameters, are local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The signal in ANNs, however, is passed instantaneously through the whole network without a time component such as a spike train structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In conclusion, ANNs generate functions and BNNs point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Another difference between ANNs and BNNs is the learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Fitting the network param- eters in large ANNs is based on variations of stochastic gradient descent (SGD) using the backpropagation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The parameter update at each network weight is global in the sense that every component of the gradient depends, in general, on all the other, possibly millions of network weights in the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This means that SGD methods require knowledge of the state of the whole network to update one parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This is also known as the weight transportation problem [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' As neurons in a biological network do not have the capacity to transport all the information about the state of the other weights, learning in BNNs cannot be driven by gradient descent [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In [7], Francis Crick writes: ”Neverthe- less, as far as the learning process is concerned, it is unlikely that the brain actually uses backpropagation.” In this work, we link the local updating rule for the parameters in a BNN to a derivative-free (or more specifically, a zero-order) optimization method that does not require evaluation of the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Theorem 1 shows that, in expectation, this scheme does approximately gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' 2 ARTIEICIALNEURON BIOLOGICALNEURON OUTPUT a (W,X, +W,X2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' + W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='Xd) OUTPUT W1 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' W1 Wd W2 W2 INPUT INPUT Wd W2 W2 W1 W,2 A brief introduction to biological neural networks (BNNs) Using graph theory terminology, a BNN is a directed Figure 2: Receiving three spike trains, the biological neuron su- perimposes them and releases spikes, whenever the threshold value (in red) is exceeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' graph, with nodes representing neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The nodes/ neurons can receive spikes via incoming edges and emit spikes via outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In the directed graph, parent nodes are also called presynaptic neurons and children nodes are called postsynaptic neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' A simple, first model is to think of a spike that is emit- ted at time τ as a signal or function t �→ eτ−t1(t ≥ τ) with 1(·) the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' If neuron i emits a spike at time τ and is connected to neuron j, then neuron j receives the signal wijeτ−t1(t ≥ τ), where wij is the weight parameter measuring the strength of the connection between the neurons i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Due to the exponential decay, the signal fades out quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' When does neuron j fire/emits a spike?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Suppose neuron j has incoming edges from neurons i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' These neurons will occasionally send spikes to j and the overall received signal/potential at j is the su- perposition of the weighted incoming signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' If the combined signal exceeds a threshold S, j fires and all children nodes (postsynaptic neurons) of j receive a signal from j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The generation of the spike trains is illustrated in Figure 2 for m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The three incoming spike trains were chosen for illustrative purposes as triggering a spike in a BNN requires between 20 to 50 incoming spikes within a short time period [12], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' After a spike, neuron j enters a short rest phase before it gets back to its normal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Although this rest phase might play a role in the learning, it will be ignored in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The parameters in the BNN are the non-negative weights measuring the strength of the connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Plasticity is the neuroscience term to describe the changes of the network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Spike time dependent plasticity (STDP) predicts that the parameter wij mea- suring the signal strength between the neurons i and j is decreased if a spike is sent from i to j and increased if neuron j emits a spike [21, 2, 42, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The increase becomes bigger if the time lag between the arrived spike and the firing of neuron j gets smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This is known as ”fire together, wire together” and is the main principle underlying Hebbian learning [15, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' 3 threshold received signal emitted signalAmong specific forms for the updating formula, as simple but realistic model is to assume that if the spike from neuron i to neuron j arrives at time τ, the weight is decreased by A−(wij)Ce−c(τ−T−) at time τ and increased by A+(wij)Ce−c(T+−τ) at time T+, where c, C are constants and T−, T+ are the last/first spike time of neuron j before/after τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Regarding the amplitude functions, A±(wij), different choices are possible, see Section 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='2 in [12] From now on we will study the case that A±(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' For C ≤ 1, this choice guarantees that the change of the weight is always smaller than the weight itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Thus positive weights remain positive and the network topology does not change during learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Combining both updating steps into one formula, we have wij ← wij + wijC(−e−c(τ−T−) + e−c(T+−τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='1) As a reward for how well the task has been completed compared to earlier trials and also accounting for the total number of trials, a neurotransmitter such as dopamine is released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The higher the reward, the more the parameters are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' If the brain performed poorly in the past and suddenly manages to solve a task well, much more neurotransmitter is released than if the same task has already been completed equally well in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' To take this into account, it has been argued in the neural coding literature that the realized reward is the objective reward, how well the task has been completed, minus the expected reward measuring how well the brain anticipated to do this task [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Denote by R the reward and let R be a measure for the anticipated reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The reward-based synaptic plasticity updating rule becomes then wij ← wij + (R − R)wijC � − e−c(τ−T−) + e−c(T+−τ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='2) The reward is only released after the prediction has been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In the meantime, several spikes could have been sent from neuron i to neuron j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This requires that the system has a short-term memory, [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' If the brain has to complete a similar task more frequently, it becomes less exciting over time, resulting in a smaller reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This can be incorporated into the dynamics by including a learning rate α > 0, wij ← wij + α(R − R)wijC � − e−c(τ−T−) + e−c(T+−τ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='3) Supervised learning is more commonly formulated in loss functions than rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Because a high reward corresponds to a small loss and vice versa, L := −R is a loss function, L = −R 4 is the anticipated loss, and the updating formula becomes wij ← wij + α(L − L)wijC � e−c(τ−T−) − e−c(T+−τ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='4) A key observation is that these updating formulas are derivative-free in the sense that they involve the reward (or loss) but not its gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Hebbian learning rules, such as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='4), model the updating of individual weights, but do not explain how the brain can learn a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' A brief overview about relevant existing ideas on learning in BNNs is given in Section 5 3 Zero-order optimization Suppose we want to fit a d-dimensional parameter vector θ to the data and write L(θ) for the (training) loss incurred by parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Derivative-free optimization procedures do not require computation of the gradient of the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' A simple iterative derivative-free scheme would be to randomly pick in each round a new candidate parameter and update the parameter if the loss is decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Standard references for derivative-free optimization include [36, 6, 10, 16, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Zero-order methods (sometimes also called zero-th order methods) are specific derivative- free optimization procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' To explain the concept, recall that standard gradient descent is an iterative procedure aiming to minimize the loss function θ �→ L(θ) by the iterative scheme θk+1 = θk − αk+1∇L(θk), k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' where the initial values θ0 are chosen in some way, αk+1 > 0 is the learning rate and ∇L(θk) denotes the gradient of the loss function at θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In contrast, zero-order methods are only allowed to access the loss function but not the gradient of the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' From the loss, one can build, however, an estimator for the gradient of the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' 1-point zero-order methods replace −∇L(θk) by βL(θk + ξk)ξk with ξk a d-dimensional random vector and β a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' To see how this relates to the gradient, consider the specific case that ξk is multivariate normal with zero mean vector and covariance matrix σ2Id, where Id denotes the d × d identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The multivariate 5 version of Stein’s lemma [38] states that E[L(θk + ξk)ξk] = σ2E[∇L(θk + ξk)] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='1) under weak regularity conditions ensuring that all expectations are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This means that σ−2L(θk + ξk)ξk estimates the gradient at θk + ξk, that is, ∇L(θk + ξk) = ∇L(θk)+errork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The hope is that over many iterations the noise contributions cancel out such that in the long-run, the 1-point zero-order dynamics behaves similarly as gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The argument above can be extended to general symmetric distributions of ξk that are not necessarily Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Unfortunately, the variance of the 1-point zero-order gradient estimator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='1) can be ex- tremely large and often scales quadratically in the number of parameters d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' As an example, suppose that the data are stored in a d-dimensional vector Y = (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , Yd)⊤ and con- sider the least squares loss L(θ) = ∥Y − θ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Taking ξk = (ξk1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , ξkd) ∼ N(0, σ2Id) and β = σ−2, as above, we have for the j-th component of βL(θk + ξk)ξk that σ−2��Y − θk − ξk ��2 2ξkj = σ−2� Yj − θkj − ξkj �2ξkj + σ−2 � ℓ:ℓ̸=j � Yℓ − θkℓ − ξkℓ �2ξkj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The second term on the right hand side has zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' It is pure noise and does not help to estimate the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This sum is over d − 1 summands and its variance scales with O(d2) in the number of parameters d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Due to the large variance, there are many scenarios for which 1-point zero-order dynamics quickly diverges to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Indeed if one iterate θk is already far away from the minimum, the large loss can result in a parameter update θk+1 which is much further away from the minimizer than θk, leading to an even larger loss and an exponential growth of the loss as the number of iterations is further increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Regarding theory of zero-order methods, [10] studies a related zero-order methods and mirror descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Assuming that the parameter vector lies in an Euclidean ball, they obtain in their Corollary 1 the rate � d/k with k the number of iterations and also provide a corresponding lower bound proving that this rate is optimal (their Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The large noise causes the factor √ d in the rate, suggesting slow convergence in the high- dimensional regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' [25] also finds a suboptimality of order d if zero-order methods are compared to gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Table 1 in [20] shows that the factor √ d or d occurs in all known convergence rates unless second-order information is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Due to the large noise, derivative-free methods are in general thought to be inferior com- pared to gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This is for instance remarked in [6], Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='3: ”Finally, 6 we want to make a strong statement that often councils against the use of derivative-free methods: if you can obtain clean derivatives (even if it requires considerable effort) and the functions defining your problem are smooth and free of noise you should not use derivative- free methods.” Zero-order methods are also not necessarily much faster to compute than gradient descent iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' For the gradient-based backpropagation of ANNs, the number of operations re- quired for the forward pass is of the same order as the number of operations required for the backwards pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Evaluation of the loss is therefore not substantially cheaper than com- puting the gradient and zero-order methods cannot be computed at a faster order than backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Despite these rather discouraging remarks, there is a rapidly increasing interest in derivative- free methods and they are successfully applied in practice, for example by Google [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' 4 Hebbian learning as zero-order optimization method The updating formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='4) allows to address supervised learning tasks, where we want to learn the functional relationship between inputs and outputs given observations (or training data) from input-output pairs (X1, Y1), (X2, Y2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' that are all generated from the same, unknown distribution as the vector (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Well-known examples for this framework are classification and regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' For instance to classify cat and dog images, Xi is the i-th image containing all the pixel values of the i-th cat image and Yi is the corresponding label ”cat” or ”dog”, coded as 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Consider now a feedforward biological neural network (BNN) with m neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This means that the neurons/nodes form a directed acyclic graph (DAG) with input neurons receiving information from the data Xi and possibly several output neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' For the subsequent analysis, we neither have to specify a layered structure as commonly done for ANNs nor conversion rules how vector valued inputs are converted into spike trains or output spike trains are cast into response variables, such as conversion into labels in a classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In the k-th instance, we feed the k-th input vector Xk in the BNN, let the BNN run and receive then as output the predicted response �Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The loss at this round is a measure for the difference between the predicted response �Yk and the real response Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' It will be denoted by L(�Yk, Yk) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The anticipated loss that occurs in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='4) could be modelled by a (weighted) average over past iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Here we use the loss of the previous iterate L(�Yk−1, Yk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' 7 During each instance, several spikes can be sent between any two connected neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' We impose the (strong) assumption that for every run, and any connection, exactly one spike will be released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Number the m nodes, that represent the neurons in the graph, by 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , m and denote the edge set by T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' A pair (i, j) is in T if and only if neuron i is a presynaptic neuron for neuron j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Equivalently, (i, j) ∈ T iff there is an arrow from i to j in the underlying DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' We consider the case that the BNN topology is static, that is, the edge set T does not change during learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' If w(k) ij is the BNN weight after the k-th round, it is then updated in the (k +1)-st iteration according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='4) w(k+1) ij (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='1) = w(k) ij + αk+1 � L(�Yk, Yk) − L(�Yk−1, Yk−1) � w(k) ij C � e−c(τ (k) ij −T (k) −,j) − e−c(T (k) +,j−τ (k) ij )� , for all (i, j) ∈ T and αk+1 > 0 the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Here T (k) −,j and T (k) +,j are the closest spike times of the j-th neuron before/after the arrival time τ (k) ij of the spike that is sent from neuron i to neuron j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The constant C can be integrated into the loss function and is from now on set to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' For the updating, the location of τ (k) ij is important within the interval [T (k) −,j, T (k) +,j], while the interval length seems to play a minor role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Therefore, we assume that the interval length is constant and set A := (T (k) +,j − T (k) −,j)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' We assume moreover that the arrival time of the spike from neuron i to neuron j has a negligible influence on the spike times of neuron j, that the spike times τ (k) ij are all independent of each other, and follow a uniform distribution on the interval [T (k) −,j, T (k) +,j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' As mentioned before, to trigger a spike, it needs of the order of 20 − 50 presynaptic neurons to fire in a short time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The influence of an individual neuron seems therefore rather minor, justifying the previous assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The assumptions above show that the random variable U (k) ij := τ (k) ij − 1 2(T (k) +,j + T (k) −,j) are jointly independent and uniformly distributed on [−A, A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Hence, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='1) becomes w(k+1) ij = w(k) ij + αk+1 � L(�Yk, Yk) − L(�Yk−1, Yk−1) � w(k) ij � e−c(A+U(k) i,j ) − e−c(A−U(k) i,j )� , for all (i, j) ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The factor e−cA can be absorbed into the loss function and the constant c can be absorbed into the hyperparameter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' By reparametrization, we obtain the updating formula w(k+1) ij = w(k) ij + αk+1 � L(�Yk, Yk) − L(�Yk−1, Yk−1) � w(k) ij � e−U(k) i,j − eU(k) i,j � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='2) 8 for all (i, j) ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' To further analyze this scheme, it is important to understand how the predicted response �Yk depends on the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' We now argue that, under the same assumptions as before, �Yk is a function of the variables w(k) ij + eU(k) i,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The high-level rationale is that in this neural model, all the information that is further transmitted in the BNN about the parameter w(k) ij sits in the spike times of neuron j and the interarrival spike times only depend on w(k) ij through w(k) ij + eU(k) i,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' To see this, fix neuron j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The only information that this node/neuron releases to its descendants in the DAG are the spike times of this neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This means that from all the incoming information that neuron j receives from presynaptic neurons (parent nodes) only the part is transmitted that affects the spike times of neuron j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' As mentioned in Section 2, a spike arriving at neuron j from neuron i at time τ (k) ij causes the potential t �→ w(k) ij eτ (k) ij −t1(t ≥ τ (k) ij ) at node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' If every incoming neuron spikes once, the overall potential of neuron j is � i:(i,j)∈T w(k) ij eτ (k) ij −t1(t ≥ τ (k) ij ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' If S denotes the threshold value for the potential at which a neuron spikes, then at the spike time T (k) +,j of the j-th neuron, we have by the definition of U (k) ij , S = � i:(i,j)∈T w(k) ij eτ (k) ij −T (k) +,j = � i:(i,j)∈T w(k) ij eU(k) ij − 1 2 (T (k) +,j−T (k) −,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Rearranging this equation shows that the interarrival spike time T (k) +,j−T (k) −,j can be expressed in terms of the variables w(k) ij eU(k) ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Introduce wk := (w(k) ij )(i,j)∈T , Uk := (U (k) ij )(i,j)∈T and write wkeUk for (w(k) ij eU(k) i,j )(i,j)∈T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The previous argument indicates that the predictor �Yk is a function of wkeUk and Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Thus, the loss L(�Yk, Yk) can be written as a function of the form L � wkeUk, Xk, Yk � and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='2) becomes w(k+1) ij (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='3) = w(k) ij + αk+1 � L � wkeUk, Xk, Yk � − L � wk−1eUk−1, Xk−1, Yk−1 �� w(k) ij � e−U(k) i,j − eU(k) i,j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In a BNN, the parameters w(k) ij are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' We now introduce the real-valued variables θ(k) ij = log(w(k) ij ) and θk = (θ(k) ij )(i,j)∈T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This means that w(k) ij = eθ(k) ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' A first order Taylor expansion shows that for real numbers u, v, ∆ such that e−v∆ is small, eu = ev + ∆ gives u = log(ev + ∆) = v + log(1 + e−v∆) ≈ v + e−v∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Working with this approximation, we can rewrite the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='3) in terms of the θ’s as θ(k+1) ij (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='4) = θ(k) ij + αk+1 � L � θk + Uk, Xk, Yk � − L � θk−1 + Uk−1, Xk−1, Yk−1 ��� e−U(k) i,j − eU(k) i,j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Relating this formula to gradient descent and the weight transportation problem mentioned in the introduction, we see that the update of one parameter only depends on all the other 9 parameters through the value of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In vector notation, the previous equality becomes θk+1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='5) = θk + αk+1 � L(θk + Uk, Xk, Yk) − L(θk−1 + Uk−1, Xk−1, Yk−1) �� e−Uk − eUk� , where eUk and e−Uk should be understood as componentwise applying the functions x �→ ex and x �→ e−x to the vector Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In particular, the loss is always a scalar and eUk, e−Uk are d-dimensional vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' So far, we have not specified any initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' From now on, we assume that the initial values θ0, θ−1 are given and that all the other parameter updates are determined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='5) for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' with U−1, U0, U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' from the uniform distribution U([−A, A]d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' As an analogue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='1), the next result shows that in average, this dynamic can also be understood as a gradient descent method with gradient evaluated not exactly at θk but at a random perturbation θk + Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Write Uk = (Uk1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , Ukd)⊤ and let (eA −eUk)(eA −e−Uk) be the vector with components (eA − eUkj)(eA − e−Ukj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Denoting by ⊙ the Hadamard product (componentwise product) of two matrices/vectors of the same dimension(s), we have E � θk+1 � = E � θk � − αk+1e−AE � ∇θkL(θk + Uk, Xk, Yk) ⊙ � eA − eUk�� eA − e−Uk�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='6) Instead of taking the expectation over all randomness, the statement is also true if we only take the expectation with respect to Uk, which is the same as the conditional expectation E[·|U−1, U0, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , Uk−1, (Xℓ, Yℓ)ℓ≥1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Note that (eA − eUkj)(eA − e−Ukj) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Thus fA(x) = C(A)−1(eA − ex)(eA − e−x)1(−A ≤ x ≤ A) defines a probability density function for the positive normalization constant C(A) = 2A(e2A + 1) + 2 − 2e2A = � A −A(eA − ex)(eA − e−x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Denoting by ∂jL(v, Xk, Yk) the partial derivative of L with respect to the j-th component of v, we can state the previous result componentwise as E � θk+1,j � = E � θkj � − αk+1e−AC(A)E � ∂jL(θk + U(j) k , Xk, Yk) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='7) for a random vector U(j) k = (Uk1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , Uk,j−1, Vkj, Uk,j+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , Ukd)⊤, with jointly indepen- dent random variables Vkj ∼ fA and Ukℓ ∼ U[−A, A], ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , j − 1, j + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' 10 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Throughout the proof, we omit the dependence of the loss function L on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' By conditioning on (U−1, U0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , Uk−1, (Xℓ, Yℓ)ℓ≥1) and the fact that e−Uk and eUk have the same distribution, it follows that E � L(θk−1 + Uk−1) � e−Uk − eUk�� = E � L(θk−1 + Uk−1)E �� e−Uk − eUk� ��� U−1, U0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , Uk−1, (Xℓ, Yℓ)ℓ≥1 �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='8) With u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , ud)⊤, the j-th component of e−AE[∇θkL(θk+Uk)⊙(eA−eUk)(eA−e−Uk)] is e−A (2A)d � [−A,A]d ∂jL(θk + u) � eA − euj�� eA − e−uj� du = e−A (2A)d � [−A,A]d−1 � A −A ∂jL(θk + u) � eA − euj�� eA − e−uj� dujdu1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' duj−1duj+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' dud, Observe that (eA − euj)(eA − e−uj) vanishes at the boundaries uj ∈ {−A, A} and ∂uj(eA − euj)(eA − e−uj) = eA−uj − eA+uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Thus, applying integration by parts formula to the inner integral yields � A −A ∂jL(θk + u) � eA − euj�� eA − e−uj� duj = −eA � A −A L(θk + u) � e−uj − euj� duj and therefore e−A (2A)d � [−A,A]d ∂jL(θk + u) � eA − euj�� eA − e−uj� du = − 1 (2A)d � [−A,A]d L(θk + u) � e−uj − euj� du = −E � L(θk + Uk) � e−Ukj − eUkj�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This holds for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The minus on the right hand side cancels out the first minus in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='8), the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='8) in the proof shows that the theorem still holds if the term L(θk−1 + Uk−1, Xk−1, Yk−1) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='5) is replaced by zero or any other value that is independent of Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' To obtain a proper zero-order method, a crucial assumption is to choose the amplitude functions A+, A− in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='1) to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In the brain, these functions are close, but some 11 authors argue that there is a slight difference [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Such differences would lead to additional, small contributions in the iterations that cannot be linked to the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' A statistical analysis of the zero-order method (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='5) is challenging, even for simple models such as data generated from the linear regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Another open problem is to determine whether the convergence rate of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='5) scales in the number of parameters d in the same way as other zero-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' 5 Literature on learning with BNNs This literature survey is aimed to give a quick overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' For a more detailed summary of related literature, see [37, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' To train BNNs on data, a natural idea is to ignore Hebbian learning and to fit BNNs via gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Similar as backpropagation efficiently computes the gradient in ANNs, SpikeProp [3, 4] is an algorithm to compute the gradient for spiking neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' The weight transportation problem is caused by the parameter dependence in the backwards pass of the backpropagation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Feedback alignment [18, 26, 17, 1, 19] avoids this by using the backpropagation algorithm with random weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In a network, the feedback could be then transmitted via specific feedback neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' If the brain does a version of backpropagation, the difficulty is always the feedback from the output backwards to the neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Contrastive Hebbian learning [27] assumes that there are two different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' During the first phase the network does prediction and the second phase starts after the prediction error is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' In one of the phases the learning is Hebbian and in the other one, the learning is anti-Hebbian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Anti-Hebbian learning means that if two neurons fire together, the connecting weight parameter is decreased instead of increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Equilibrium propagation [29] overcomes the two types of learning in the different phases but requires again the computation of a gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' For a biologically more plausible implementation of the weight transportation problem, predictive coding [40, 41, 35, 22, 23] uses two types of neurons, named error nodes and value nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' These two nodes are associated to each other and process forward and backward information locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' [31] proposes the concept of a ”hedonistic synapse” that follows a Hebbian learning rule and takes the global reward into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' For the learning, a hedonistic synapse has to be able to store information from previous trials in a so-called eligibility trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Closest to our approach is weight perturbation [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Weight perturbation adds random 12 noise to the parameters or the outputs and compares the loss with and without added noise to estimate the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Whereas the cause of the noise perturbation is not entirely clear in the weight perturbation framework, we have shown in this work, how the spike train structure in BNNs implies a random perturbation of the parameters in the loss with uniformly distributed noise and how this leads to a specific derivative-free updating formula for the weights that also involves the difference of the loss function evaluated for different instance of the noisy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' A more statistical approach is [24] considering unsupervised classification using a small BNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' This work identifies a closer link between a Hebbian learning rule and the EM- algorithm for mixtures of multinomial distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' Some other ideas on unsupervised learning in BNNs are moreover provided in [12], Section 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' To summarize, there are various theories that are centered around the idea that the learning in BNNs should be linked to gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' All of these approaches, however, contain still biological implausibilities and lack a theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=' References [1] Bartunov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFKT4oBgHgl3EQfSy3E/content/2301.11777v1.pdf'} +page_content=', Santoro, A.' metadata={'source': 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PREPRINT VERSION. ACCEPTED DEC, 2022 +1 +Learning from Guided Play: +Improving Exploration for Adversarial Imitation +Learning with Simple Auxiliary Tasks +Trevor Ablett1, Bryan Chan2, and Jonathan Kelly1 +Abstract—Adversarial imitation learning (AIL) has become a +popular alternative to supervised imitation learning that reduces +the distribution shift suffered by the latter. However, AIL requires +effective exploration during an online reinforcement learning +phase. In this work, we show that the standard, na¨ıve approach +to exploration can manifest as a suboptimal local maximum +if a policy learned with AIL sufficiently matches the expert +distribution without fully learning the desired task. This can +be particularly catastrophic for manipulation tasks, where the +difference between an expert and a non-expert state-action pair +is often subtle. We present Learning from Guided Play (LfGP), +a framework in which we leverage expert demonstrations of +multiple exploratory, auxiliary tasks in addition to a main task. +The addition of these auxiliary tasks forces the agent to explore +states and actions that standard AIL may learn to ignore. +Additionally, this particular formulation allows for the reusability +of expert data between main tasks. Our experimental results in +a challenging multitask robotic manipulation domain indicate +that LfGP significantly outperforms both AIL and behaviour +cloning, while also being more expert sample efficient than these +baselines. To explain this performance gap, we provide further +analysis of a toy problem that highlights the coupling between +a local maximum and poor exploration, and also visualize the +differences between the learned models from AIL and LfGP.3 +Index Terms—Imitation Learning, Reinforcement Learning, +Transfer Learning +I. INTRODUCTION +E +XPLORATION is a crucial part of effective reinforce- +ment learning (RL). A variety of methods have attempted +to optimize the exploration-exploitation trade-off of RL agents +[1]–[3], but the development of a technique that generalizes +across domains remains an open research problem. A simple, +well-known approach to reduce the need for random explo- +ration is to provide a dense, or “shaped,” reward to learn from, +but this can be very challenging to design appropriately [4]. +Furthermore, the environment may not directly provide the +low-level state information required for such a reward. An +alternative to providing a dense reward is to learn a reward +Manuscript received: Nov. 3, 2022; Accepted: Dec. 18, 2022. +This paper was recommended for publication by Editor Jens Kober upon +evaluation of the Associate Editor and Reviewers’ comments. +1Authors are with the Space & Terrestrial Autonomous Robotic Systems +(STARS) Laboratory at the University of Toronto Institute for Aerospace +Studies (UTIAS), Toronto, Ontario, Canada, M3H 5T6. Email: .@robotics.utias.utoronto.ca +2Author is with the Department of Computing Science at the Uni- +versity +of +Alberta, +Edmonton, +Alberta, +Canada, +T6G +2E8. +Email: +bryan.chan@ualberta.ca +Digital Object Identifier (DOI): see top of this page. +3Code, Blog, Appendix: https://papers.starslab.ca/lfgp +Fig. 1: Learning from Guided Play (LfGP) finds an effective stacking +policy by learning to compose multiple simple auxiliary tasks (only +Reach is shown, for this episode) along with stacking. Discrim- +inator Actor-Critic (DAC) [7], or off-policy AIL, reaches a local +maximum action-value function and policy, failing to solve the task. +Arrow direction indicates mean policy velocity action, red-to-yellow +(background) indicates low-to-high learned value, while arrow colour +indicates probability of closing (green) or opening (blue) the gripper. +function from expert demonstrations of a task, in a process +known as inverse RL (IRL) [5]. Many modern approaches +to IRL are part of the adversarial imitation learning (AIL) +family [6]. In AIL, rather than learning a reward function +directly, the policy and a learned discriminator form a two- +player min-max optimization problem, where the policy aims +to confuse the discriminator by producing expert-like data, +while the discriminator attempts to classify expert and non- +expert data. +Although AIL has been shown to be more expert sample +efficient than supervised imitation learning (also known as be- +havioural cloning, or BC) in continuous-control environments +[6]–[8], its application to long-horizon robotic manipulation +tasks with a wide distribution of possible initial configurations +remains challenging [7], [9]. In this work, we investigate the +use of AIL in a multitask robotic manipulation domain. We +find that a state-of-the-art AIL method, in which off-policy +learning is used to maximize environment sample efficiency [7] +(i.e., reduce the quantity of environment interaction required +from the online RL portion of AIL), is outperformed by BC +arXiv:2301.00051v1 [cs.LG] 30 Dec 2022 + +LfGP +DAC +Reach +Stack +Pre-Grasp +Post-Grasp2 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022 +Multitask Environment +Reach +Lift +Bring +Together +Insert +Stack +Guided Expert Play +Guide +Expert +bring_0 +together +stack_01 +Multitask Environment +Reach( ) +Lift( ) +Bring( ) +Insert( ) +Stack( ) +Multitask Environment +Reach +Lift +Bring +Together +Insert +Stack +Guided Expert Play +Expert +lift( ) +Guide +Guide +stack( ) +Guided Agent Play +Move( ) +RESET +NEXT +Expert +lift( ) +Sched +( ) +stack( ) +Sched +(lift( )) +Agent +Multitask AIL +Update +Fig. 2: The main components of our system for learning from guided play. In a multitask environment, a guide prompts an expert for a mix +of multitask demonstrations, after which we learn a multitask policy through scheduled hierarchical AIL. +with an equivalent amount of expert data, contradicting previ- +ous results [6]–[8]. Through a simplified example, simulated +robotic experiments, and learned model analysis, we show +that this outcome occurs because a model learned with expert +data and a discriminator is susceptible to the deceptive reward +problem [10]. In other words, while AIL, and more generally +IRL, can provide something akin to a dense reward, this reward +is not necessarily optimal for teaching, and AIL alone does +not enforce sufficiently diverse exploration to escape locally +optimal but globally poor models. A locally-optimal policy has +converged to match a subset of the expert data, but in doing +so, avoids crucial states and actions (e.g., in Fig. 1, grasping +the blue block) required to globally match the full expert set. +To overcome this limitation of AIL, we present Learning +from Guided Play (LfGP),4 in which we combine AIL with a +scheduled approach to hierarchical RL (HRL) [12], allowing +an agent to ‘play’ in the environment with an expert guide. +Using expert demonstrations of multiple relevant auxiliary +tasks (e.g., Reach, Lift, Move-Object), along with a main task +(e.g., Stack, Bring, Insert), our scheduled hierarchical agent +is able to learn tasks where AIL alone fails. Crucially, our +formulation also allows auxiliary expert data to be reused +between main tasks, further emphasizing the expert sample +efficiency of our method. +We use the word play to describe an agent that simulta- +neously attempts and learns numerous tasks at once, freely +composing them together, inspired by the playful (as opposed +to goal-directed) phase of learning experienced by children +[12]. In our case, guided represents two separate but related +ideas: first, that the expert guides this play, as opposed to +requiring hand-crafted sparse rewards as in [12] (right side +of Fig. 2), and second, that the expert gathering of multitask, +semi-structured demonstrations is guided by uniform-random +task selection (middle of Fig. 2), rather than requiring the +expert to choose transitions between goals, as in [13], [14]. +Our specific contributions are the following: +1) A novel application of a hierarchical framework [12] to +AIL that learns a reward and policy for a challenging +4Originally presented as a non-archival workshop paper [11]. +main task by simultaneously learning rewards and poli- +cies for auxiliary tasks. +2) Manipulation experiments in which we demonstrate that +AIL fails, while LfGP significantly outperforms both +AIL and BC. +3) A thorough ablation study to examine the effects of +various design choices for LfGP and our baselines. +4) Empirical analysis, including a simplified representative +example and visualization of the learned models of LfGP +and AIL, to better understand why AIL fails and how +LfGP improves upon it. +II. PROBLEM FORMULATION +A Markov decision process (MDP) is defined as M = +⟨S, A, R, P, ρ0, γ⟩, where the sets S and A are respectively +the state and action space, R : S×A → R is a reward function, +P is the state-transition environment dynamics distribution, ρ0 +is the initial state distribution, and γ is the discount factor. +Actions are sampled from a stochastic policy π(a|s). The +policy π interacts with the environment to yield experience +(st, at, rt, st+1) for t = 0, . . . , ∞, where s0 ∼ ρ0(·), at ∼ +π(·|st), st+1 ∼ P(·|st, at), rt = R(st, at). When referring to +finite-horizon tasks, t = T indicates the final timestep of a +trajectory. +For notational convenience, we assume infinite-horizon, +non-terminating environments where t is unbounded, but +the extension to the finite-horizon case is trivial. We aim +to learn a policy π that maximizes the expected return +J(π) += +Eπ [G(τ0:∞)] += +Eπ [�∞ +t=0 γtR(st, at)], where +τt:∞ = {(st, at), . . . } is the trajectory starting with (st, at), +and G(τt:∞) is the return of trajectory τ. +In this work, we focus on imitation learning (IL), where +R is unknown and instead we are given a finite set of expert +demonstration (s, a) pairs BE = +� +(s, a)E, . . . +� +. In AIL, we +attempt to simultaneously learn π and a discriminator D : S × +A → [0, 1] that differentiates between expert samples (s, a)E +and policy samples (s, a)π and subsequently define R using D +[6], [7]. To accommodate hierarchical learning, we augment +M to contain auxiliary tasks, where Taux = {T1, . . . , TK} are +separate MDPs that share S, A, P, ρ0 and γ with the main +task Tmain but have their own reward functions, Rk. With this + +ABLETT et al.: LEARNING FROM GUIDED PLAY +3 +Fig. 3: An MDP, analogous to stacking, with an expert demonstration. +Poor exploration can lead AIL to learn a suboptimal policy. +modification, we refer to entities in our model that are specific +to task T ∈ Tall, Tall = Taux ∪ {Tmain}, as (·)T . We assume +that we have a set of expert data BE +T for each task. +III. LOCAL MAXIMUM WITH OFF-POLICY AIL +In this section, we provide a representative example of how +AIL can fail by reaching a locally maximum policy due to a +learned deceptive reward [10] coupled with poor exploration. +A simple six-state MDP is shown in Fig. 3, with ten state- +conditional actions. We refer to actions as at = anm and states +as st = sn where t, n and m refer to the current timestep, +current state, and next state, respectively. The reward function +is R(s5, a55) = +1, R(s1, a15) = −5 and 0 for all other state- +action pairs. The initial state s1 is always s1, the fixed horizon +length is 5, and no discounting is used. +The MDP is meant to be roughly analogous to a stacking +manipulation task: s2, s3, s4 and s6 represent the first block +being reached, grasped, lifted, and dropped respectively. State +s5 represents the gripper hovering over the second block +(whether the first block has been stacked or not), while s1 is +the reset state, and a15 represents reaching s5 without grasping +the first block. Taking action a15 results in a total return of +-1 (because R(s1, a15) = −5), since the first block has not +actually been grasped. In our case, the agent does not receive +any reward, and instead an expert demonstration of the optimal +trajectory is provided. We will assume access to a learned +(perfect) discriminator, and will use the AIRL [8] reward, so +state-action pairs in the expert set receive +1 reward and all +others receive -1. +We define the action-value Q(st, at) as the expected +value of taking action at in state st, and initialize it to +zero for all (s, a) pairs. We define our update rule as the +standard Q-Learning update [1], Q(st, at) = Q(st, at) + +α (R(st, at) + maxa Q(st+1, a) − Q(st, at)), with α = 0.1. +The agent uses ϵ-greedy exploration, storing each (st, at, st+1) +tuple into a buffer. After each episode, all Q values are updated +to convergence using the whole buffer. +After the first complete episode of {a15, a55, a55, a55, a55}, +Q(s1, a15) += +2.7, and Q(s1, a12) += +0. In the second +({a12, a26, a61, a15, a55}) and third ({a12, a23, a36, a61, a15}) +episodes, the agent initially moves in the correct direction, but +ultimately still fails. The final Q values in s1 are Q(s1, a15) = +0.49 and Q(s1, a12) = 0.13.5 +A policy maximizing Q, having simultaneously learned to +avoid s6 (by avoiding s2 and s3) and exploiting the (s5, a55) +expert pair, will choose a1 = a15, giving a final return of +-1 in the real MDP. This behaviour matches what we see in +Fig. 1: due to the large negative reward from dropping the +block, AIL learns a policy that avoids stacking altogether and +merely reaches the second block, just as AIL here learns to +skip s2 and s3 and exploit a55. In both cases, poor initial +exploration leads to a deceptive reward, which exacerbates +poor exploration. +IV. LEARNING FROM GUIDED PLAY (LFGP) +We now introduce Learning from Guided Play (LfGP). Our +primary goal is to learn a policy πTmain that can solve the main +task Tmain, with a secondary goal of also learning auxiliary task +policies πT1, . . . , πTK that are used for improved exploration. +More specifically, we derive a hierarchical learning objective +that is decomposed into three parts: i) recovering the reward +function of each task with expert demonstrations, ii) training +all policies to achieve their respective goals, and iii) using all +policies for effective exploration in Tmain. For a summary of +the algorithm, see supplementary material link in Footnote 3. +A. Learning the Reward Function +We first describe how to recover the reward functions from +expert demonstrations. For each task T ∈ Tall, we learn a dis- +criminator DT (s, a) that is used to define the reward function +for policy optimization. We construct the joint discriminator +loss following [7] to train each discriminator in an off-policy +manner: +L(D) = − +� +T ∈Tall +EB [log (1 − DT (s, a))] ++EBE +T [log (DT (s, a))] . +(1) +Each resulting discriminator DT attempts to differentiate the +occupancy measure between the distributions induced by BE +T +and B. We can use DT to define various reward functions [7]; +following [8], we define the reward function for each task T +to be RT (st, at) = log (DT (st, at)) − log (1 − DT (st, at)). +B. Learning the Hierarchical Agent +We adapt Scheduled Auxiliary Control (SAC-X) [12] to +learn the hierarchical agent. The agent includes low-level +intention policies (equivalently referred to as intentions), a +high-level scheduler policy, as well as the Q-functions and the +discriminators. The intentions aim to solve their corresponding +tasks (i.e., the intention πT aims to maximize the task return +J(πT )), whereas the scheduler aims to maximize the expected +return for Tmain by selecting a sequence of intentions to interact +with the environment. For the remainder of the paper, when +we refer to a policy, we are referring to an intention policy, +as opposed to the scheduler, unless otherwise specified. +5See six_state_mdp.py from open source code to reproduce. + +Legend +2 +S +-5 +MDP +C +2 +3 +S +S +S +6 +S +a5 +Expert Demo +a4 +a1 +2 +S +S +a2 +a3 +S +a1 +a2-5 +Suboptimal AIL Policy +S4 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022 +1) Learning the Intentions: We learn each intention using +Soft Actor-Critic (SAC) [15], an actor-critic algorithm that +maximizes the entropy-regularized objective, though any off- +policy RL algorithm would suffice. The objective is +J(πT ) = EπT +� ∞ +� +t=0 +γt (RT (st, at) + αH(πT (·|st))) +� +, +(2) +where the learned temperature α determines the importance +of the entropy term and H(πT (·|st)) is the entropy of the +intention πT at state st. The soft Q-function is +QT (st, at) = RT (st, at) ++ EπT +� ∞ +� +t=0 +γt(RT (st+1, at+1) + αH(πT (·|st+1))) +� +. +(3) +The intentions maximize the joint policy objective +L(πint) = +� +T ∈Tall +Es∼Ball,a∼πT (·|s) [QT (s, a) − α log πT (a|s)] , +(4) +where πint refers to the set of intentions {πTmain, πT1, . . . , πTK} +and Ball refers to buffer containing every transition from +interactions and demonstrations, as is done in [16], [17]. +For policy evaluation, the soft Q-functions QT for each πT +minimize the joint soft Bellman residual +L(Q) = +� +T ∈Tall +E(s,a,s′)∼Ball,a′∼πT (·|s′) +� +(QT (s, a) − δT )2� +, +(5) +δT = RT (s, a) + γ (QT (s′, a′) − α log πT (a′|s′)) . +(6) +Crucially, because each task shares the common S, A, P, ρ0, +and γ, and we are using off-policy learning, all tasks can learn +from all data, as in [12]. +2) The Scheduler: SAC-X formulates learning the sched- +uler by maximizing the expected return of the main task +[12]. In particular, let H be the number of possible intention +switches within an episode and let each chosen intention +execute for ξ timesteps. The H intention choices made within +the episode are defined as T 0:H−1 = +� +T (0), . . . , T (H−1)� +, +where T (h) ∈ Tall. The return of the main task, given chosen +intentions, is then defined as +GTmain(T 0:H−1) = +H−1 +� +h=0 +(h+1)ξ−1 +� +t=hξ +γtRTmain(st, at), +(7) +where at ∼ πT (h)(·|st) is the action taken at timestep t, +sampled from the chosen intention T (h) in the hth scheduler +period. The scheduler for the hth period P h +S aims to maxi- +mize the expected main task return: E +� +GTmain(T h:H−1)|P h +S +� +. +Although SAC-X describes a method to learn the scheduler +[12], we find that a combination of two simple task-agnostic +heuristics performs similarly in practice (see Section V-C2). +Specifically, we use a weighted random scheduler (WRS) +combined with handcrafted trajectories (HC). The WRS forms +a prior categorical distribution over the set of tasks, with a +higher probability mass pTmain for the main task and +pTmain +K +for +all other tasks. This approach is comparable to the uniform +scheduler from [12], with a bias towards the main task. The +HC component is a small set of handcrafted trajectories of +tasks that are sampled half of the time, forcing the scheduler +to explore trajectories that would clearly be beneficial for +completing the main task. The chosen handcrafted trajectories +can be found in our code and in our supplementary material. +C. Breaking Out of Local Maxima with LfGP +Returning to the discussion in Section III, resolving the +local maximum problem with LfGP is straightforward. Sup- +pose we include a go-right auxiliary task with BE +go-right = +{(s1, a12), (s2, a23), (s3, a34)}. When the scheduler chooses +the go-right intention, the agent does not exploit the a55 action +because the go-right discriminator learns that R(s5, a55) = +−1. Since the transitions are stored in the shared buffer that +the main intention also samples from, the agent can quickly +obtain the correct, optimal value. +D. Expert Data Collection +We assume that each T ∈ Tall has, for evaluation purposes +only, a binary indicator of success. In single-task imitation +learning where this assumption is valid, expert data is typically +collected by allowing the expert to control the agent until +success conditions are met. At that point, the environment is +reset following ρ0 and collection is repeated for a fixed number +of episodes or (s, a) pairs. We collect our expert data in this +way for each T separately. +V. EXPERIMENTS +In this work, we are interested in answering the following +questions about LfGP: +1) How does the performance of LfGP compare with BC +and AIL in challenging manipulation tasks, in terms of +success rate and expert sample efficiency? +2) What parts of LfGP are necessary for success? +3) How do the policies and action value functions differ +between AIL and LfGP? +A. Experimental Setup +We complete experiments in a simulation environment con- +taining a Franka Emika Panda manipulator, one green and +one blue block in a tray, fixed zones corresponding to the +green and blue blocks, and one slot in each zone with < 1mm +Fig. 4: Example successful runs of our four main tasks. Top to +bottom: Stack, Unstack-Stack, Bring, Insert. + +ABLETT et al.: LEARNING FROM GUIDED PLAY +5 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Stack +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Unstack-Stack +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Bring +1 +2 +3 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Insert +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Updates/steps (millions) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Success Rate +LfGP (multi) +BC (multi) +DAC (single) +BC (single) +Expert +Fig. 5: Performance results for LfGP, multitask BC, single-task BC, and DAC on all four tasks considered in this work. The x-axis corresponds +to both gradient updates and environments steps for LfGP and DAC, and gradient updates only for both versions of BC. The shaded area +corresponds to standard deviation across five seeds. LfGP significantly outperforms the baselines on all tasks, and even in Bring where it is +matched by single-task BC, it is far more expert sample efficient. +tolerance for fitting the blocks (see bottom right of Fig. 4). +The robot is controlled via delta-position commands, and the +blocks and end-effector can both be reset anywhere above the +tray. The environment is designed such that several different +challenging tasks can be completed within a common observa- +tion and action space. The main tasks that we investigate are +Stack, Unstack-Stack, Bring, and Insert (see Fig. 4). For more +details on our environment and definitions of task success, see +supplementary material link in Footnote 3. We also define a set +of auxiliary tasks: Open-Gripper, Close-Gripper, Reach, Lift, +Move-Object, and Bring (Bring is both a main task and an +auxiliary task for Insert), all of which are reusable between +main tasks. +We compare our method to several standard multitask and +single-task baselines. A multitask algorithm simultaneously +learns to complete a main task as well as auxiliary tasks, +while the single-task algorithms only learn to complete the +main task. In general, we consider a multitask algorithm to be +more useful than a single-task algorithm, given the potential +to reuse expert data and trained models for learning new tasks. +To ensure a fair comparison, we provide single-task algorithms +with an equivalent amount of total expert data as our multitask +methods, as shown in Table I. +In our main experiments, we compare LfGP to a mul- +titask variant of behavioural cloning (BC), single-task BC, +and Discriminator-Actor-Critic (DAC) [7], a state-of-the-art +approach to AIL. We train multitask BC with a multitask mean +squared error objective, +L(πint) = +� +T ∈Tall +� +(s,a)∈BE +T +(πT (s) − a)2 , +(8) +while BC is trained with the corresponding single task version. +Following recent trends in improving BC performance, we +train our BC baselines with the same number of gradient +updates as LfGP and DAC, evaluating the policies at the same +frequency. This adjustment has been shown to dramatically +increase the performance of BC [18], [19], particularly com- +pared to the more common practice of using early stopping, +as is done in [6], [7]. We validate that this change signifi- +cantly improves BC performance in our ablation study (see +Section V-C4). +We gather expert data by first training an expert policy using +Scheduled Auxiliary Control (SAC-X) [12]. We then run the +Task +Dataset Sizes +Reuse +Single Total +Multi +Stack +SOCRLM: 1k/task +5k +1k +6k +task +U-Stack +UOCRLM: 1k/task +5k +1k +6k +Bring +BOCRLM: 1k/task +6k +0 +6k +Insert +IBOCRLM: 1k/task +6k +1k +7k +Single +Stack +S: 6k +0 +6k +6k +Task +U-Stack +U: 6k +0 +6k +6k +Bring +B: 6k +0 +6k +6k +Insert +I: 6k +0 +7k +7k +TABLE I: The number of (s, a) pairs used for each main and auxiliary +task. The table illustrates the reusability of the expert data used to +generate the performance results described in Section V-B. Each letter +under “Dataset Sizes” is the first letter of a single (auxiliary) task, +and bolded letters indicate that a dataset was reused for more than +one main task (e.g., Open-Gripper was used for all four main tasks). +Multitask methods (e.g., LfGP) are able to reuse a large portion of the +expert data, while single-task methods (e.g., single-task BC) cannot. +expert policies to collect various amounts of expert data as +described in Section IV-D and Table I. We also collect an extra +200 expert (sT , 0) pairs per auxiliary task, where T refers to +the final timestep of an individual episode and 0 is an action +of all zeros. This is equivalent to adding example data, as is +done in example-based RL [20]. This addition improved final +task performance, likely because it biases the reward towards +completing the final task. It is worth noting that, in the real +world, final states are easier to collect than full demonstrations, +and LfGP does not require any modifications to accommodate +these extra examples. Finally, even without this addition, LfGP +still outperforms the baselines (see Section V-C1). +B. Performance Results +Performance results for all methods and main tasks are +shown in Fig. 5. We freeze the policies every 100k steps +and evaluate those policies for 50 randomized episodes, using +only the mean action outputs for stochastic policies. For all +algorithms, we test across five seeds and report the mean and +standard deviation of all seeds. +In Stack, Unstack-Stack, and Insert, LfGP achieves expert +performance, while the baselines all perform significantly +worse. In Bring, LfGP does not quite achieve expert per- +formance, and is matched by single-task BC. However, we +note that LfGP is much more expert data efficient than single- +task BC because it reuses auxiliary task data (see Table I). +A more direct comparison is multitask BC, which performs + +6 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022 +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Stack (no ablations) +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.5|BE +orig| +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.5|BE +orig| +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Subsampled BE +0.5 +1.0 +1.5 +2.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0No Extra Final Examples +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Updates/steps (millions) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Success Rate +LfGP (multi) +BC (multi) +DAC (single) +BC (single) +Expert +Fig. 6: Various dataset ablations for LfGP and all baselines, including dataset size, subsampling of expert dataset, and replacement of extra +(sT , 0) pairs with an equivalent amount of regular trajectory (s, a) pairs. In all cases, LfGP still significantly outperforms all baselines. +1 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +LfGP Scheduler +0.0 +0.5 +1.0 +Updates/steps (millions) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Success Rate +WRS + HC +WRS only +Learned +No Sched. +Expert +1 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Expert Sampling +0.0 +0.5 +1.0 +Updates/steps (millions) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Success Rate +LfGP +LfGP (BE +for D only) +LfGP (No +(sT , 0) bias) +DAC +DAC (BE +for D only) +DAC (No +(sT , 0) bias) +Expert +1 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +BC/DAC Alternatives +0.0 +0.5 +1.0 +Updates/steps (millions) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Success Rate +BC (multi) +BC (multi, +early stop) +DAC +GAIL +BC +BC (early +stop) +Expert +Fig. 7: Left: Scheduler ablations for training LfGP, WRS is weighted random scheduler, HC is handcraft; Middle: Expert sampling ablations +for training LfGP/DAC; Right: Baseline ablations for training BC/DAC. +much more poorly than LfGP across all tasks, including Bring. +Intriguingly, DAC also performs very poorly on all tasks, a +phenomenon that we further explore in Section VI. +C. Ablation Study +While the fundamental idea of LfGP is relatively straight- +forward, it is worth considering alternatives to some of the +specific choices made for our experiments. In this section, +we complete an ablation study where we vary (a) the expert +dataset, including size, subsampling, and inclusion of extra +(sT , 0) pairs; (b) the type of scheduler used for LfGP (see +Section IV-B2); (c) the sampling strategy used for expert data; +and (d) the method for training our baselines. To reduce the +computational load of completing these experiments, all of +these variations were carried out exclusively for our Stack task. +All ablation results are shown in Fig. 6 and Fig. 7. +1) Dataset Ablations: We tested the following dataset vari- +ations: (a) half and one and a half times the original expert +dataset size; (b) subsampling BE, taking only every 20th +timestep, as is done in [6], [7]; and (c) replacing the 200 extra +(sT , 0) pairs in each buffer with 200 regular trajectory (s, a) +pairs. Notably, even in the challenging regimes of halving +and subsampling the dataset, LfGP still learns an expert-level +policy (albeit more slowly). +2) Scheduler Ablations: We tested the following scheduler +variations: (a) Weighted Random Scheduler (WRS) only, re- +moving the Handcrafted (HC) addition; (b) a learned sched- +uler, as is used in [12]; and (c) no scheduler, in which only the +main task is attempted, akin to the Intentional Unintentional +Agent [12], [21]. Both WRS versions learn slightly faster than +the learned scheduler, but all three methods outperform the No +Scheduler ablation, replicating results from [12] demonstrating +the importance of actually exploring all auxiliary tasks. Per- +haps surprisingly, the HC modification made little difference +compared with WRS only, but it is possible that for even more +complex tasks, this could change. +3) Expert Sampling Ablations: For our main performance +experiments, we modified standard AIL in two ways: (a) we +added expert buffer sampling to π and Q updates, in addition +to the D updates, as is done in [16], [17]; and (b) we biased the +sampling of BE when training D to be 95% final (sT , 0) pairs. +We tested both LfGP and DAC without these additions. For +LfGP, although these modifications improve learning speed, +they are not required to generate an expert policy. For DAC, +performance is quite poor regardless of these adjustments. +4) Baseline Ablations: To verify that we evaluated against +fair baselines, we tested two alternatives to those used for our +main performance experiments: (a) an early stopping variation +of BC, in which each expert buffer is divided into a 70%/30% +train/validation split, taking the policy after validation error has +not improved for 100 epochs; and (b) the on-policy variant +of DAC, also known as Generative Adversarial Imitation +Learning (GAIL) [6]. Notably, the early stopping variants of +BC, commonly used as baselines in other AIL work [6], [7], +[22] perform dramatically more poorly than those used in our +experiments, verifying recent trends [18], [19]. +VI. LEARNED MODEL ANALYSIS +In this section, we further examine the learned Stack models +of LfGP and DAC. We take snapshots of the average per- +forming models from LfGP and DAC at four points during +learning: 200k, 400k, 600k, and 800k model updates and +environment steps. Although the initial gripper and block +positions are randomized between episodes during learning, +for each snapshot, we reset the stacking environment to a +single set of representative initial conditions. We then run the + +ABLETT et al.: LEARNING FROM GUIDED PLAY +7 +LfGP – Open-Gripper +LfGP – Close-Gripper +LfGP – Reach +LfGP – Lift +LfGP – Move-Object +LfGP – Stack +DAC – Stack +Fig. 8: The policy outputs (arrows) and Q values (background) for each LfGP task and for DAC at 200k environment steps. The arrows show +velocity direction/magnitude, blue → green indicates open-gripper → close-gripper. For Q values, red → yellow indicates low → high. The +LfGP policies and Q functions are reasonable for all tasks, while DAC has only learned to reach toward and above the green block. +snapshot policies for a single exploratory trajectory, using the +stochastic outputs of each policy as well as, for LfGP, the +WRS+HC scheduler. Trajectories from these runs are shown +in Fig. 9. +DAC is unable to learn to grasp or even reach the blue +block and ultimately settles on a policy that learns to reach +and hover near the green block. This is understandable—DAC +learns a deceptive reward for hovering above the green block +regardless of the position of the blue block, because it has not +sufficiently explored the alternative of first grasping the blue +block. Even if hovering above the green block does not fully +match the expert data, the DAC policy receives some reward +for doing so, as evidenced by the learned Q value on the right +side of Fig. 8. +In comparison, even after only 200k environment steps, +LfGP learns to reach and push the blue block, and by 600k +steps, grasp, move, and nearly stack it. By enforcing explo- +ration of sub-tasks that are crucial to completing the main task, +LfGP ensures that the distribution of expert stacking data is +fully matched. +VII. RELATED WORK +Imitation learning is often divided into two main categories: +behavioural cloning (BC) [23], [24] and inverse reinforcement +learning (IRL) [5], [25]. BC recovers the expert policy via +supervised learning, but it suffers from compounding errors +due to covariate shift [23], [26]. Alternatively, IRL partially +alleviates the covariate shift problem by estimating the reward +function and then applying RL using the learned reward. +A popular approach to IRL is adversarial imitation learning +(AIL) [6], [7], [27], in which the expert policy is recovered +by matching the occupancy measure between the generated +data and the demonstration data. Our proposed method en- +hances existing AIL algorithms by enabling exploration of +Fig. 9: LfGP and DAC trajectories of the gripper, blue block, and +green block for four stack episodes with consistent initial conditions +throughout the learning process. The LfGP episodes, each including +auxiliary task sub-trajectories, demonstrate significantly more variety +than the DAC trajectories. +key auxiliary tasks via the use of a scheduled multitask model, +simultaneously resolving the susceptibility of AIL to deceptive +rewards. +Agents learned via hierarchical reinforcement learning +(HRL), which act over multiple levels of temporal abstractions +in long planning horizon tasks, are shown to provide more +effective exploration than agents operating over only a single +level of abstraction [12], [28], [29]. Our approach for learning +agents most closely resembles hierarchical AIL methods that +attempt to combine AIL with HRL [27], [30]–[32]. Existing +work [30]–[32] often formulates the hierarchical agent using +the Options framework [28] and learns the reward function +with AIL [6]. Both [30] and [32] leverage task-specific expert +demonstrations to learn options using mixture-of-experts and +expectation-maximization strategies, respectively. In contrast, +our work focuses on expert demonstrations that include multi- +ple reusable auxiliary tasks, each of which has clear semantic +meaning. +In the multitask setting, [27] and [31] leverage unsegmented, +multitask expert demonstrations to learn low-level policies via +a latent variable model. Other work has used a large corpus +of unsegmented but semantically meaningful “play” expert +data to bootstrap policy learning [13], [14]. We define our +expert dataset as being derived from guided play, in that the +expert completes semantically meaningful auxiliary tasks with +provided transitions, reducing the burden on the expert to +generate these data arbitrarily and simultaneously providing +auxiliary task labels. Compared with learning from unseg- +mented demonstrations, the use of segmented demonstrations, +as in [33], ensures that we know which auxiliary tasks our +model will be learning, and opens up the possibility of expert +data reuse and also transfer learning. Finally, we deviate from +the Options framework and build upon Scheduled Auxiliary +Control (SAC-X) to train our hierarchical agent, since SAC- +X has been shown to work well for challenging manipulation +tasks [12]. +VIII. LIMITATIONS +Our approach is not without limitations. While we were +able to use LfGP in six and seven-task settings, the number +of tasks for which this method would become intractable is +unclear. LfGP needs access to segmented expert data as well; +in many cases, this is reasonable, and is also required to +be able to reuse auxiliary task data between main tasks, but +it does necessitate extra care during expert data collection. +Also, LfGP requires pre-defined auxiliary tasks: while this is +a common approach to hierarchical RL (see [34], Section 3.1, +for numerous examples), choosing these tasks may sometimes +present a challenge. Finally, compared with methods that use +offline data exclusively (e.g., BC), for our tasks, LfGP requires + +200k +400k +600k +800k +LfGP +DAC8 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022 +many online environment steps to learn a high-quality policy. +This data gathering could be costly if human supervision was +necessary. It is worth noting that, because LfGP is already a +multitask method, this final point could be partially resolved +through the use of multitask reset-free RL [35]. +IX. CONCLUSION +We have shown how adversarial imitation learning can fail +at challenging manipulation tasks because it learns deceptive +rewards. We demonstrated that this can be resolved with +Learning from Guided Play (LfGP), in which we introduce +auxiliary tasks and the corresponding expert data, guiding the +agent to playfully explore parts of the state and action space +that would have been avoided otherwise. We demonstrated that +our method dramatically outperforms both BC and AIL base- +lines, particularly in the case of AIL. Furthermore, our method +can leverage reusable expert data, making it significantly more +expert sample efficient than the highest-performing baseline, +and its learned auxiliary task models can be applied to transfer +learning. 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Robotics and Automation (ICRA’18), Brisbane, Australia, May +21–25 2018, pp. 4693–4700. +[34] S. Pateria, B. Subagdja, A.-h. Tan, and C. Quek, “Hierarchical Re- +inforcement Learning: A Comprehensive Survey,” ACM Computing +Surveys, vol. 54, no. 5, pp. 109:1–109:35, June 2021. +[35] A. Gupta, et al., “Reset-Free Reinforcement Learning via Multi-Task +Learning: Learning Dexterous Manipulation Behaviors without Human +Intervention,” in Proc. 2021 IEEE Int. Conf. Robotics and Automation +(ICRA’21), Apr. 2021. + +ABLETT et al.: LEARNING FROM GUIDED PLAY +9 +APPENDIX A +LEARNING FROM GUIDED PLAY ALGORITHM +The complete pseudo-code is given in Algorithm 1. Our +implementation builds on RL Sandbox [36], an open-source +PyTorch [37] framework for RL algorithms. For learning +the discriminators, we follow DAC and apply a gradient +penalty for regularization [7], [38]. We optimize the intentions +via the reparameterization trick [40]. As is commonly done +in deep RL, we use the Clipped Double Q-Learning trick +[41] to mitigate overestimation bias [42] and use a target +network to mitigate learning instability [43] when training +the policies and Q-functions. We also learn the temperature +parameter αT separately for each task T (see Section 5 of [44] +for more details on learning α). For Generative Adversarial +Imitation Learning (GAIL), we use a common open-source +PyTorch implementation [45]. The hyperparameters chosen for +all methods are provided in Section G. Please see videos at +papers.starslab.ca/lfgp for examples of what LfGP looks like +in practice. +Algorithm 1 Learning from Guided Play (LfGP) +Input: Expert replay buffers BE +main, BE +1 , . . . , BE +K, scheduler +period ξ, sample batch size N +Parameters: Intentions πT with corresponding Q-functions +QT and discriminators DT , and scheduler πS (e.g. with Q- +table QS) +1: Initialize replay buffer B +2: for t = 1, . . . , do +3: +# Interact with environment +4: +For every ξ steps, select intention πT using πS +5: +Select action at using πT +6: +Execute action at and observe next state s′ +t +7: +Store transition ⟨st, at, s′ +t⟩ in B +8: +9: +# Update discriminator DT ′ for each task T ′ +10: +Sample {(si, ai)}N +i=1 ∼ B +11: +for each task T ′ do +12: +Sample {(s′ +i, a′ +i)}B +i=1 ∼ BE +k +13: +Update DT ′ following Eq. (1) using GAN + Gradient +Penalty +14: +end for +15: +16: +# Update intentions πT ′ and Q-functions QT ′ for each +task T ′ +17: +Sample {(si, ai)}N +i=1 ∼ B +18: +Compute reward DT ′(si, ai) for each task T ′ +19: +Update π and Q following Eq. (4) and Eq. (5) +20: +21: +# Optional Update learned scheduler πS +22: +if at the end of effective horizon then +23: +Compute main task return GTmain using reward esti- +mate from Dmain +24: +Update πS +(e.g. update Q-table QS +following +Eq. (A.3) and recompute Boltzmann distribution) +25: +end if +26: end for +A. Scheduler Details +1) Learning the Scheduler: As stated in our paper, our +main experiments used a simple weighted random scheduler +with handcrafted trajectories. In this section, we provide the +details of our learned scheduler. Following [12], let H be the +total number of possible intention switches within an episode +and let each chosen intention execute for ξ timesteps. The +H intention choices made within the episode are defined as +T 0:H−1 = +� +T (0), . . . , T (H−1)� +, where T (h) ∈ Tall. The main +task’s return given chosen intentions is then defined as +GTmain(T 0:H−1) = +H−1 +� +h=0 +(h+1)ξ−1 +� +t=hξ +γtRTmain(st, at), +(A.1) +where +at +∼ +πT (h)(·|st) +is +the +action +taken +at +timestep +t, +sampled +from +the +chosen +intention +T (h) +in +the +hth +scheduler +period. +We +further +define +the +Q-function +for +the +scheduler +as +QS(T 0:h−1, T (h)) += +ET h:H−1∼P h:H−1 +S +� +GTmain(T h:H−1)|T 0:h−1� +and represent the +scheduler for the hth period as a softmax distribution P h +S over +{QS(T 0:h−1, Tmain), QS(T 0:h−1, T1), . . . , QS(T 0:h−1, TK)}. +The scheduler maximizes the expected return of the main +task following the scheduler: +L(S) = ET (0)∼P 0 +S +� +QS(∅, T (0)) +� +. +(A.2) +We use Monte Carlo returns to estimate QS, estimating the +expected return using the exponential moving average: +QS(T 0:h−1, T (h)) = (1 − φ)QS(T 0:h−1, T (h)) ++φ GTmain(T h:H), +(A.3) +where φ ∈ [0, 1] represents the amount of discounting on older +returns and GTmain(T h:H) is the cumulative discounted return +of the trajectory starting at timestep hξ. +B. Weighted Random Scheduler Plus Handcrafted Trajectories +As stated in our paper, the main experiments were com- +pleted with the described weighted random scheduler (WRS) +combined with some simple handcrafted trajectories (HC) +that we expected to be beneficial for learning each of +the main tasks. In this section, we provide further de- +tails of these handcrafted scheduler trajectories. Given a +chosen proportion hyperparameter (0.5 in our experiments), +we randomly sampled full trajectories from the lists below +at the beginning of training episodes, and otherwise sam- +pled from the regular WRS. For all four tasks Main = +{Stack, Unstack-Stack, Bring, Insert}, we provided the fol- +lowing set of trajectories: +1) Reach, Lift, Main, Open-Gripper, Reach, Lift, Main, +Open-Gripper. +2) Reach, Lift, Move-Object, Main, Open-Gripper, Reach, +Lift, Move-Object. +3) Lift, Main, Open-Gripper, Lift, Main, Open-Gripper, +Lift, Main. +4) Main, Open-Gripper, Main, Open-Gripper, Main, Open- +Gripper, Main, Open-Gripper. + +10 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022 +TABLE II: The components used in our environment observations, +common to all tasks. Grip finger position is a continuous value from +0 (closed) to 1 (open). +Component +Dim +Unit +Privileged? +Extra info +EE pos. +3 +m +No +rel. to base +EE velocity +3 +m/s +No +rel. to base +Grip finger pos. +6 +[0, 1] +No +current, last 2 +Block pos. +6 +m +Yes +both blocks +Block rot. +8 +quat +Yes +both blocks +Block trans vel. +6 +m/s +Yes +rel. to base +Block rot vel. +6 +rad/s +Yes +rel. to base +Block rel to EE +6 +m +Yes +both blocks +Block rel to block +3 +m +Yes +in base frame +Block rel to slot +6 +m +Yes +both blocks +Force-torque +6 +N,Nm +No +at wrist +Total +59 +5) Move-Object, Main, Open-Gripper, Move-Object, Main, +Open-Gripper, Move-Object, Main. +For insert, in addition to the trajectories listed above, we added +two more trajectories to specifically accommodate Bring as an +auxiliary task: +1) Bring, +Insert, +Open-Gripper, +Bring, +Insert, +Open- +Gripper, Bring, Insert. +2) Reach, Lift, Bring, Insert, Open-Gripper, Reach, Lift, +Bring. +APPENDIX B +ENVIRONMENT DETAILS +Fig. 10: An image of our multitask environment immediately after a +reset has been carried out. +A screenshot of our environment, simulated in PyBullet +[47], is shown in Fig. 10. We chose this environment because +we desired tasks that a) have a large distribution of possible +initial states, representative of manipulation in the real world, +b) have a shared observation/action space with several other +tasks, allowing the use of auxiliary tasks and transfer learning, +and c) require a reasonably long horizon and significant use of +contact to solve. The environment contains a tray with sloped +edges (to keep the blocks within the reachable workspace of +the end-effector), as well as a green and a blue block, each +of which is 4 cm × 4 cm × 4 cm and has a mass of 100 g. +The dimensions of the lower part of the tray, before reaching +the sloped edges, are 30 cm × 30 cm. The dimensions of the +‘bring’ boundaries (shaded blue and green regions) are 8 cm +× 8 cm, while the dimensions of the insertion slots, which +are directly in the center of each shaded region, are 4.1 cm × +4.1 cm × 1 cm. The boundaries for end-effector movement, +relative to the tool center point that is directly between the +gripper fingers, are a 30 cm × 30 cm × 14.5 cm box, where +the bottom boundary is low enough to allow the gripper to +interact with objects, but not to collide with the bottom of the +tray. +See Table II for a summary of our environment observations. +In this work, we use privileged state information (e.g., block +poses), but adapting our method to exclusively use image- +based data is straightforward since we do not use hand-crafted +reward functions as in [12]. +The environment movement actions are 3-DOF translational +position changes, where the position change is relative to the +current end-effector position. We leverage PyBullet’s built-in +position-based inverse kinematics function to generate joint +commands. Our actions also contain a fourth dimension that +corresponds to actuating the gripper. To allow for the use +of policy models with exclusively continuous outputs, this +dimension accepts any real number, with any value greater +than 0 commanding the gripper to open, and any number less +than 0 commanding it to close. Actions are supplied at a rate +of 20 Hz, and each training episode is limited to 18 seconds, +corresponding to 360 time steps per episode. For play-based +expert data collection, we also reset the environment manually +every 360 time steps. Between episodes, block positions are +randomized to any pose within the tray, and the end-effector +is randomized to any position between 5 and 14.5 cm above +the tray, within the earlier stated end-effector bounds, with +the gripper fully opened. The only exception to these initial +conditions is during expert data collection and agent training +of the Unstack-Stack task: in this case, the green block is +manually set to be on top of the blue block at the start of the +episode. +APPENDIX C +PERFORMANCE RESULTS FOR AUXILIARY TASKS +The performance results for all multitask methods and +all auxiliary tasks are shown in Fig. 11. Multitask BC has +gradually decreasing performance on many of the auxiliary +tasks as the number of updates increases, which is consistent +with mild overfitting. Intriguingly, however, multitask BC +does achieve quite reasonable performance on many of the +auxiliary tasks (such as Lift) without needing any of the extra +environment interactions required by an online method such +as LfGP or DAC. An interesting direction for future work is to +determine whether pretraining via multitask BC could provide + +ABLETT et al.: LEARNING FROM GUIDED PLAY +11 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Stack +Stack +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Open +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Close +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Lift +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Reach +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Move +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Unstack-Stack +Unstack-Stack +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Open +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Close +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Lift +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Reach +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Move +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Bring +Bring +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Open +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Close +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Lift +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Reach +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +Move +1 +2 +3 +4 +0.0 +0.5 +1.0 +Insert +Insert +1 +2 +3 +4 +0.0 +0.5 +1.0 +Open +1 +2 +3 +4 +0.0 +0.5 +1.0 +Close +1 +2 +3 +4 +0.0 +0.5 +1.0 +Bring +1 +2 +3 +4 +0.0 +0.5 +1.0 +Lift +1 +2 +3 +4 +0.0 +0.5 +1.0 +Reach +1 +2 +3 +4 +0.0 +0.5 +1.0 +Move +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Updates/steps (millions) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Success Rate +LfGP (multi) +BC (multi) +DAC (single) +BC (single) +Fig. 11: Performance for LfGP and the multitask baselines across all tasks, shaded area corresponds to standard deviation. +any improvements in environment sample efficiency. We did +attempt to do this, but found that it resulted in poorer final +performance than training from scratch. +APPENDIX D +PROCEDURE FOR OBTAINING EXPERTS +As stated, we used SAC-X [12] to train models that we +used for generating expert data. We used the same hyperpa- +rameters that we used for LfGP (see Table III), apart from +the discriminator, which, of course, does not exist in SAC-X. +See Section E for details on the hand-crafted rewards that we +used for training these models. For an example of gathering +play-based expert data, please see our attached video. +We made two modifications to regular SAC-X to speed up +learning. First, we pre-trained a Move-Object model before +transferring this model to each of our main tasks, as we did +in Section 5.3 of our main paper, since we found that SAC-X +would plateau when we tried to learn the more challenging +tasks from scratch. The need for this modification demon- +strates another noteworthy benefit of LfGP—when training +LfGP, main tasks could be learned from scratch, and generally +in fewer time steps, than it took to train our experts. Second, +during transfer to the main tasks, we used what we called a +conditional weighted scheduler instead of a Q-Table: we de- +fined weights for every combination of tasks, so that the sched- +uler would pick each task with probability P(T (h)|T (h−1)), +ensuring that ∀T ′ ∈ Tall, � +T ∈Tall P(T |T ′) = 1. The weights +that we used were fairly consistent between main tasks, and +can be found in our packaged code. The conditional weighted +scheduler ensured that every task was still explored throughout +the learning process, so that we would have high-quality +experts for every auxiliary task in addition to the main task. +This scheduler can be considered as a more complex alter- +native to the weighted random scheduler or the addition with +handcrafted trajectories from our main paper, and again shows +the flexibility of using a semantically-meaningful multitask +policy with a common observation and action space. +APPENDIX E +EVALUATION +As stated in our paper, we evaluated all algorithms by +testing the mean output of the main task policy head in +our environment and determining a success rate based on 50 +randomly selected resets. These evaluation episodes were run +for 360 time steps to match our training process, and if a +condition for success was met within that time, they were +recorded as a success. The rest of this section describes in +detail how we evaluated ‘success’ for each of our main and +auxiliary tasks. +As previously stated, we trained experts using a modified +SAC-X [12] that required us to define a set of reward functions +for each task, which we include in this section. The authors +of [12] focused on sparse rewards but also showed a few +experiments in which dense rewards reduced the time to learn +adequate policies, so we chose to use dense rewards. We note +that many of these reward functions are particularly com- +plex and required significant manual shaping effort, further +motivating the use of an imitation learning scheme like the +one presented in our paper. It is possible that we could have +made do with sparse rewards, such as those used in [12], but +our compute resources made this impractical—for example, +in [12], their agent took 5000 episodes × 36 actors × 360 +time steps = 64.8 M time steps to learn their stacking task, +which would have taken over a month of wall clock time on +our fastest machine. To see the specific values used for the +rewards and success conditions described in these sections, +please review our code. +Unless otherwise stated, each of the success conditions in +this section had to be held for 10 time steps, or 0.5 seconds, +before being registered as a success. This choice was made +to prevent registering a success when, for example, the blue +block slipped off the green block during the Stack task. + +12 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022 +A. Common +For each of these functions, we use the following common +labels: +• pb: blue block position, +• vb: blue block velocity, +• ab: blue block acceleration, +• pg: green block position, +• pe: end-effector tool center point position (TCP), +• ps: center of a block pushed into one of the slots, +• g1: (scalar) gripper finger 1 position, +• g2: (scalar) gripper finger 2 position, and +• ag: (scalar) gripper open/close action. +A block is flat on the tray when pb,z = 0 or pg,z = 0. To +further reduce training time for SAC-X experts, all rewards +were set to 0 if ∥pb −pe∥ > 0.1 and ∥pg −pe∥ > 0.1 (i.e., the +TCP must be within 10 cm of either block). During training +while using the Unstack-Stack variation of our environment, +a penalty of -0.1 was added to each reward if ∥pg,z∥ > 0.001 +(i.e., there was a penalty to all rewards if the green block was +not flat on the tray). +B. Stack/Unstack-Stack +The evaluation conditions for Stack and Unstack-Stack are +identical, but in our Unstack-Stack experiments, the environ- +ment is manually set to have the green block start on top of +the blue block. +1) Success: Using internal PyBullet commands, we check +to see whether the blue block is in contact with the green +block and is not in contact with either the tray or the gripper. +2) Reward: We include a term for checking the distance +between the blue block and the spot above the the green block, +a term for rewarding increasing distance between the block and +the TCP once the block is stacked, a term for shaping lifting +behaviour, a term to reward closing the gripper when the block +is within a tight reaching tolerance, and a term for rewarding +the opening the gripper once the block is stacked. +C. Bring/Insert +We use the same success and reward calculations for Bring +and Insert, but for Bring the threshold for success is 3 cm, +and for insert, it is 2.5 mm. +1) Success: We check that the distance between pb and +ps is less than the defined threshold, that the blue block is +touching the tray, and that the end-effector is not touching the +block. For Insert, the block can only be within 2.5 mm of the +insertion target if it is correctly inserted. +2) Reward: We include a term for checking the distance +between the pb and ps and a term for rewarding increas- +ing distance between pb and pe once the blue block is +brought/inserted. +D. Open-Gripper/Close-Gripper +We use the same success and reward calculations for Open- +Gripper and Close-Gripper, apart from inverting the condition. +1) Success: For Open-Gripper and Close-Gripper, we check +to see if ag < 0 or ag > 0 respectively. +2) Reward: We include a term for checking the action, as +we do in the success condition, and also include a shaping term +that discourages high magnitudes of the movement action. +E. Lift +1) Success: We check to see if pb,z > 0.06. +2) Reward: We add a dense reward for checking the height +of the block, but specifically also check that the gripper +positions correspond to being closed around the block, so that +the block does not simply get pushed up the edges of the tray. +We also include a shaping term for encouraging the gripper +to close when the block is reached. +F. Reach +1) Success: We check to see if ∥pe − pb∥ < 0.015. +2) Reward: We have a single dense term to check the +distance between pe and pb. +G. Move-Object +For Move-Object, we changed the required holding time for +success to 1 second, or 20 time steps. +1) Success: We check to see if the vb > 0.05 and ab < 5. +The acceleration condition ensures that the arm has learned to +move the block by following a smooth trajectory, rather than +vigorously shaking it or continuously picking up and dropping +it. +2) Reward: We include a velocity term and an acceleration +penalty, as in the success condition, but also include a dense +bonus for lifting the block. +APPENDIX F +RETURN PLOTS +As previously stated, we generated hand-crafted reward +functions for each of our tasks for the purpose of training +our SAC-X experts. Given that we have these rewards, we +can also generate return plots corresponding to our results +to add extra insight (see Fig. 12 and Fig. 13). The patterns +displayed in these plots are, for the most part, quite similar +to the success rate plots. One notable exception is that there +is an eventual increase in performance when training DAC on +Insert, indicating that, perhaps for certain tasks, DAC alone +can eventually make progress. Nevertheless, it is clear that +LfGP improves learning efficiency, and it is unclear whether +DAC would plateau even if it was trained for a longer period. +APPENDIX G +MODEL ARCHITECTURES AND HYPERPARAMETERS +All the single-task models share the same network architec- +tures and all the multitask models share the same network +architectures. All layers are initialized using the PyTorch +default methods [37]. +For the single-task variant, the policy is a fully-connected +network with two hidden layers followed by ReLU activation. +Each hidden layer consists of 256 hidden units. The output of +the policy for LfGP and DAC is split into two vectors, mean +ˆµ and variance ˆσ2. For both variants of BC, only the mean ˆµ + +ABLETT et al.: LEARNING FROM GUIDED PLAY +13 +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +600 +Stack +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +600 +800 +1000 +Unstack-Stack +0.5 +1.0 +1.5 +2.0 +0 +100 +200 +300 +400 +500 +Bring +0 +1 +2 +3 +4 +100 +200 +300 +400 +500 +Insert +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Updates/steps (millions) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Episode Return +LfGP (multi) +BC (multi) +DAC (single) +BC (single) +Expert +Fig. 12: Episode return for LfGP compared with all baselines. Shaded area corresponds to standard deviation. +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +600 +Stack +Stack +0.5 +1.0 +1.5 +2.0 +200 +250 +300 +Open +0.5 +1.0 +1.5 +2.0 +100 +150 +200 +250 +300 +Close +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +Lift +0.5 +1.0 +1.5 +2.0 +100 +150 +200 +250 +300 +Reach +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +Move +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +600 +800 +Unstack-Stack +Unstack-Stack +0.5 +1.0 +1.5 +2.0 +200 +250 +300 +Open +0.5 +1.0 +1.5 +2.0 +150 +200 +250 +300 +Close +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +Lift +0.5 +1.0 +1.5 +2.0 +0 +100 +200 +Reach +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +Move +0.5 +1.0 +1.5 +2.0 +0 +100 +200 +300 +400 +Bring +Bring +0.5 +1.0 +1.5 +2.0 +200 +250 +300 +Open +0.5 +1.0 +1.5 +2.0 +100 +200 +300 +Close +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +Lift +0.5 +1.0 +1.5 +2.0 +100 +200 +300 +Reach +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +Move +1 +2 +3 +4 +200 +400 +Insert +Insert +1 +2 +3 +4 +250 +275 +300 +325 +Open +1 +2 +3 +4 +100 +200 +300 +Close +1 +2 +3 +4 +100 +200 +300 +400 +500 +Bring +1 +2 +3 +4 +0 +200 +400 +Lift +1 +2 +3 +4 +0 +100 +200 +300 +Reach +1 +2 +3 +4 +0 +200 +400 +Move +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Updates/steps (millions) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Episode Return +LfGP (multi) +BC (multi) +DAC (single) +BC (single) +Fig. 13: Episode return for LfGP compared with multitask baselines on all tasks. Shaded area corresponds to standard deviation. +output is used. The vectors define a Gaussian distribution (i.e. +N(ˆµ, ˆσ2I), where I is the identity matrix). When computing +actions, we squash the samples using the tanh function and +bound the actions to be in range [−1, 1], as done in SAC +[44]. The variance ˆσ2 is computed by applying a softplus +function followed by a sum with an epsilon ϵ = 1e-7 to +prevent underflow: ˆσi = softplus(ˆxi) + ϵ. The Q-functions +are fully-connected networks with two hidden layers followed +by ReLU activations. Each hidden layer consists of 256 units. +The output of the Q-function is a scalar corresponding to the +value estimate given the current state-action pair. Finally, the +discriminator is a fully-connected network with two hidden +layers followed by tanh activations. Each hidden layer consists +of 256 units. The output of the discriminator is a scalar logit +to be used as an input to the sigmoid function. The sigmoid +function output can be viewed as the probability of the current +state-action pair coming from the expert distribution. +For multitask variant, the policies and the Q-functions share +their initial layers. There are two shared, fully-connected +layers followed by ReLU activations. Each layer consists of +256 units. The output of the last shared layer is then fed into +the policies and Q-functions. Each policy head and Q-function +head corresponds to one task and has the same architecture: +a two-layered fully-connected network followed by ReLU +activations. The output of the policy head corresponds to the +parameters of a Gaussian distribution, as described previously. +Similarly, the output of the Q-function head corresponds to the +value estimate. Finally, the discriminator is a fully-connected +network with two hidden layers followed by tanh activations. +Each hidden layer consists of 256 units. The output of the +discriminator is a vector, where the ith entry corresponds to +the logit input to the sigmoid function for task Ti. The ith +sigmoid function output corresponds to the probability of the +current state-action pair coming from the expert distribution +in task Ti. +The hyperparameters for our experiments are listed in +Table III and Table V. In the early-stopping variant of BC, +overfit tolerance refers to the number of full dataset training +epochs without an improvement in validation error before we +stop training. All models are optimized using Adam Optimizer +[48] with PyTorch default values, unless specified otherwise. +APPENDIX H +OPEN-ACTION AND CLOSE-ACTION + +14 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED DEC, 2022 +TABLE III: Hyperparameters for AIL algorithms across all tasks. +Parameters that do not appear in the original version of DAC are +shown in blue. +Algorithm +LfGP +DAC +Total Interactions +2M (4M for Insert) +Buffer Size +2M (4M for Insert) +Buffer Warmup +25k +Initial Exploration +50k +Evaluations per task +50 +Evaluation frequency +100k interactions +Intention +γ +0.99 +Batch Size +256 +Q Update Freq. +1 +Target Q Update Freq. +1 +π Update Freq. +1 +Polyak Averaging +1e-4 +Q Learning Rate +3e-4 +π Learning Rate +1e-5 +α Learning Rate +3e-4 +Initial α +1e-2 +Target Entropy +−dim(a) = −4 +Max. Gradient Norm +10 +π Weight Decay +1e-2 +Q Weight Decay +1e-2 +BE sampling proportion +0.1 +BE sampling decay +0.99999 +Discriminator +Learning Rate +3e-4 +Batch Size +256 +Gradient Penalty λ +10 +Weight Decay +1e-2 +(sT , 0) sampling bias +0.95 +TABLE IV: Hyperparameters for LfGP schedulers. +Scheduler +Learned +WRS +WRS + HC +ξ +45 +N/A +N/A +φ +0.6 +N/A +N/A +Initial Temp. +360 +N/A +N/A +Temp. Decay +0.9995 +N/A +N/A +Min. Temp. +0.1 +N/A +N/A +Main Task Rate +N/A +0.5 +0.5 +Handcraft Rate +N/A +N/A +0.5 +DISTRIBUTION MATCHING +There was one exception to the method we used for col- +lecting our expert data. Specifically, our Open-Gripper and +Close-Gripper tasks required additional considerations. It is +worth reminding the reader that our Open-Gripper and Close- +Gripper tasks were meant to simply open or close the gripper, +respectively, while remaining reasonably close to either block. +If we were to use the approach described above verbatim, +the Open-Gripper and Close-Gripper data would contain no +(s, a) pairs where the gripper actually released or grasped +the block, instead immediately opening or closing the gripper +while simply hovering near the blocks. Perhaps unsurprisingly, +this was detrimental to our algorithm’s performance: as one +example, an agent attempting to learn Stack would, if Open- +Gripper was selected while the blue block was held above +TABLE V: Hyperparameters for BC algorithms (both single-task and +multitask) across all tasks. +Version +Main Results +Early Stopping +Batch Size +256 +Learning Rate +1e-5 +Weight Decay +1e-2 +Total Updates +2M (4M for Insert) +N/A +Overfit Tolerance +N/A +100 +the green block, move the grasped blue block away from the +green block before dropping it on the tray. This behaviour, of +course, is not what we would want, but it better matches an +expert distribution when the environment is reset in between +each task execution. +To mitigate this, our Open-Gripper data actually contain a +mix of each of the other sub-tasks called for the first 45 time +steps, followed by a switch to Open-Gripper, ensuring that +the expert dataset contains some degree of block-releasing, +with the trade-off being that 50% of the Open-Gripper expert +data is specific to whatever the main task happens to be. We +left this additional detail out of our main paper for clarity, +since it corresponds to only a small portion of the expert +data (every other auxiliary task was fully reused). Similarly, +the Close-Gripper data calls Lift for 15 time steps before +switching to Close-Gripper, ensuring that the Close-gripper +dataset will contain a large proportion of data where the block +is actually grasped. For the Closer-gripper data, however, this +modification did still allow data to be reused between main +tasks. +APPENDIX I +ATTEMPTED AND FAILED EXPERIMENTS +In this section, we provide a list of experiments and modi- +fications that did not improve performance, in addition to the +alternatives that did. +1) Pretraining with BC: We attempted to pretrain LfGP +using multitask BC, and then to transition to online +learning with LfGP, but we found that this tended to +produce significantly poorer final performance. Some +existing work [49], [50] has investigated transitioning +from BC to online RL, but achieving this consistently, +especially with off-policy RL, remains an open research +problem. +2) Handcrafted Open-Gripper/Close-Gripper policies: +Given the simplicity of designing a reward function in +these two cases, a natural question is whether Open- +Gripper and Close-Gripper could use hand-crafted re- +ward functions, or even hand-crafted policies, instead of +these specialized datasets. In our experiments, both of +these alternatives proved to be quite detrimental to our +algorithm. +3) Penalizing Q values: In our early experiments, we +found that LfGP training progress was harmed by ex- +ploding Q values. This problem was particularly exac- +erbated when we added BE sampling to our Q and π +updates. It appears that this occurs because, at the begin- +ning of training, the differences between discriminator + +ABLETT et al.: LEARNING FROM GUIDED PLAY +15 +outputs for expert data and non-expert data are so large +that the bootstrap Q updates quickly jump to unrealistic +values. We attempted to use various forms of Q penalties +to resolve this, akin to Conservative Q Learning (CQL) +[51], but found that all of our modifications ultimately +harmed final performance. Some of the things we tried, +in addition to the CQL loss, were reducing γ (.95, .9), +clipping Q losses to -5, +5, smooth L1 loss, huber loss, +increased gradient penalty λ for D (50, 100), decreased +reward scaling (.1), more discriminator updates per π/Q +update (10), and weight decay in D only (as is done +in [9]). We ultimately resolved exploding Q values by +i) decreasing polyak averaging to a significantly lower +value than is used in much other work (1e-4 as opposed +to the SAC default of 5e-3), and ii) adding in weight +decay (with a significantly higher value used than is +used in other work) to π, Q, and D training (which was +required to not overfit with the reduced polyak averaging +value). Without the added weight decay, performance +started to plateau and eventually to decrease. +4) Higher Update-to-Data (UTD) Ratio: Recent work in +RL has started increasing the UTD ratio (i.e., increas- +ing the number of policy/Q updates per environment +interaction), with the goal of improving environment +sample efficiency [53]. We were actually able to increase +this from 1 to 2 and achieve a marginal improvement +in environment sample efficiency, but this also nearly +doubled the running time of our experiments, so we +opted not to include this modification in our final results. +Higher values of the UTD ratio also caused our Q values +to explode. +APPENDIX J +EXPERIMENTAL HARDWARE +For a list of the software we used in this work, see our code +and instructions. We used a number of different computers and +GPUs when completing our experiments: +1) GPU: NVidia Quadro RTX 8000, CPU: AMD - Ryzen +5950x 3.4 GHz 16-core 32-thread, RAM: 64GB, OS: +Ubuntu 20.04. +2) GPU: NVidia V100 SXM2, CPU: Intel Gold 6148 +Skylake @ 2.4 GHz (only used 4 threads), RAM: 32GB, +OS: CentOS 7. +3) GPU: Nvidia GeForce RTX 2070, CPU: RYZEN +Threadripper 2990WX, RAM: 32GB, OS: Ubuntu 20.04. +REFERENCES +[36] B. Chan, “RL sandbox,” https://github.com/chanb/rl sandbox public, +2020. +[37] A. 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Bai, “PyBullet, a Python module for physics +simulation for games, robotics and machine learning,” http://pybullet.org, +2016. +[48] D. P. Kingma and J. Ba, “Adam: A Method for Stochastic Optimization,” +in Proc. Int. Conf. Learning Representations (ICLR’15), San Diego, +USA, May 7–9 2015. +[49] A. Rajeswaran*, et al., “Learning Complex Dexterous Manipulation with +Deep Reinforcement Learning and Demonstrations,” in Proc. Robotics: +Science and Systems (RSS’18), Pittsburgh, USA, Jun. 26–30 2018. +[50] Y. Wu, M. Mozifian, and F. Shkurti, “Shaping Rewards for Rein- +forcement Learning with Imperfect Demonstrations using Generative +Models,” arXiv:2011.01298 [cs], Nov. 2020. +[51] A. Kumar, A. Zhou, G. Tucker, and S. Levine, “Conservative Q-Learning +for Offline Reinforcement Learning,” arXiv:2006.04779 [cs, stat], Aug. +2020. +[52] M. Orsini, et al., “What Matters for Adversarial Imitation Learning?” +in Conf. Neural Inf. Processing Systems, June 2021. +[53] X. Chen, C. Wang, Z. Zhou, and K. Ross, “Randomized Ensembled +Double Q-Learning: Learning Fast Without a Model,” arXiv:2101.05982 +[cs], Mar. 2021. + diff --git a/7tAyT4oBgHgl3EQfQvZV/content/tmp_files/load_file.txt b/7tAyT4oBgHgl3EQfQvZV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ddb73ff4602678d9a2ceb59e4448a2eca7fa4c3 --- /dev/null +++ b/7tAyT4oBgHgl3EQfQvZV/content/tmp_files/load_file.txt @@ -0,0 +1,1448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf,len=1447 +page_content='IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ACCEPTED DEC, 2022 1 Learning from Guided Play: Improving Exploration for Adversarial Imitation Learning with Simple Auxiliary Tasks Trevor Ablett1, Bryan Chan2, and Jonathan Kelly1 Abstract—Adversarial imitation learning (AIL) has become a popular alternative to supervised imitation learning that reduces the distribution shift suffered by the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' However, AIL requires effective exploration during an online reinforcement learning phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In this work, we show that the standard, na¨ıve approach to exploration can manifest as a suboptimal local maximum if a policy learned with AIL sufficiently matches the expert distribution without fully learning the desired task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This can be particularly catastrophic for manipulation tasks, where the difference between an expert and a non-expert state-action pair is often subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We present Learning from Guided Play (LfGP), a framework in which we leverage expert demonstrations of multiple exploratory, auxiliary tasks in addition to a main task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The addition of these auxiliary tasks forces the agent to explore states and actions that standard AIL may learn to ignore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Additionally, this particular formulation allows for the reusability of expert data between main tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Our experimental results in a challenging multitask robotic manipulation domain indicate that LfGP significantly outperforms both AIL and behaviour cloning, while also being more expert sample efficient than these baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' To explain this performance gap, we provide further analysis of a toy problem that highlights the coupling between a local maximum and poor exploration, and also visualize the differences between the learned models from AIL and LfGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='3 Index Terms—Imitation Learning, Reinforcement Learning, Transfer Learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' INTRODUCTION E XPLORATION is a crucial part of effective reinforce- ment learning (RL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A variety of methods have attempted to optimize the exploration-exploitation trade-off of RL agents [1]–[3], but the development of a technique that generalizes across domains remains an open research problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A simple, well-known approach to reduce the need for random explo- ration is to provide a dense, or “shaped,” reward to learn from, but this can be very challenging to design appropriately [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Furthermore, the environment may not directly provide the low-level state information required for such a reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' An alternative to providing a dense reward is to learn a reward Manuscript received: Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Accepted: Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 18, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This paper was recommended for publication by Editor Jens Kober upon evaluation of the Associate Editor and Reviewers’ comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1Authors are with the Space & Terrestrial Autonomous Robotic Systems (STARS) Laboratory at the University of Toronto Institute for Aerospace Studies (UTIAS), Toronto, Ontario, Canada, M3H 5T6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Email: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='@robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='utias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='ca 2Author is with the Department of Computing Science at the Uni- versity of Alberta, Edmonton, Alberta, Canada, T6G 2E8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Email: bryan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='chan@ualberta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='ca Digital Object Identifier (DOI): see top of this page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3Code, Blog, Appendix: https://papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='starslab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='ca/lfgp Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1: Learning from Guided Play (LfGP) finds an effective stacking policy by learning to compose multiple simple auxiliary tasks (only Reach is shown, for this episode) along with stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Discrim- inator Actor-Critic (DAC) [7], or off-policy AIL, reaches a local maximum action-value function and policy, failing to solve the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Arrow direction indicates mean policy velocity action, red-to-yellow (background) indicates low-to-high learned value, while arrow colour indicates probability of closing (green) or opening (blue) the gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' function from expert demonstrations of a task, in a process known as inverse RL (IRL) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Many modern approaches to IRL are part of the adversarial imitation learning (AIL) family [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In AIL, rather than learning a reward function directly, the policy and a learned discriminator form a two- player min-max optimization problem, where the policy aims to confuse the discriminator by producing expert-like data, while the discriminator attempts to classify expert and non- expert data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Although AIL has been shown to be more expert sample efficient than supervised imitation learning (also known as be- havioural cloning, or BC) in continuous-control environments [6]–[8], its application to long-horizon robotic manipulation tasks with a wide distribution of possible initial configurations remains challenging [7], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In this work, we investigate the use of AIL in a multitask robotic manipulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We find that a state-of-the-art AIL method, in which off-policy learning is used to maximize environment sample efficiency [7] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', reduce the quantity of environment interaction required from the online RL portion of AIL), is outperformed by BC arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='00051v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='LG] 30 Dec 2022 LfGP DAC Reach Stack Pre-Grasp Post-Grasp2 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ACCEPTED DEC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2022 Multitask Environment Reach Lift Bring Together Insert Stack Guided Expert Play Guide Expert bring_0 together stack_01 Multitask Environment Reach( ) Lift( ) Bring( ) Insert( ) Stack( ) Multitask Environment Reach Lift Bring Together Insert Stack Guided Expert Play Expert lift( ) Guide Guide stack( ) Guided Agent Play Move( ) RESET NEXT Expert lift( ) Sched ( ) stack( ) Sched (lift( )) Agent Multitask AIL Update Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2: The main components of our system for learning from guided play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In a multitask environment, a guide prompts an expert for a mix of multitask demonstrations, after which we learn a multitask policy through scheduled hierarchical AIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' with an equivalent amount of expert data, contradicting previ- ous results [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Through a simplified example, simulated robotic experiments, and learned model analysis, we show that this outcome occurs because a model learned with expert data and a discriminator is susceptible to the deceptive reward problem [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In other words, while AIL, and more generally IRL, can provide something akin to a dense reward, this reward is not necessarily optimal for teaching, and AIL alone does not enforce sufficiently diverse exploration to escape locally optimal but globally poor models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A locally-optimal policy has converged to match a subset of the expert data, but in doing so, avoids crucial states and actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1, grasping the blue block) required to globally match the full expert set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' To overcome this limitation of AIL, we present Learning from Guided Play (LfGP),4 in which we combine AIL with a scheduled approach to hierarchical RL (HRL) [12], allowing an agent to ‘play’ in the environment with an expert guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Using expert demonstrations of multiple relevant auxiliary tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', Reach, Lift, Move-Object), along with a main task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', Stack, Bring, Insert), our scheduled hierarchical agent is able to learn tasks where AIL alone fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Crucially, our formulation also allows auxiliary expert data to be reused between main tasks, further emphasizing the expert sample efficiency of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We use the word play to describe an agent that simulta- neously attempts and learns numerous tasks at once, freely composing them together, inspired by the playful (as opposed to goal-directed) phase of learning experienced by children [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In our case, guided represents two separate but related ideas: first, that the expert guides this play, as opposed to requiring hand-crafted sparse rewards as in [12] (right side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2), and second, that the expert gathering of multitask, semi-structured demonstrations is guided by uniform-random task selection (middle of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2), rather than requiring the expert to choose transitions between goals, as in [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Our specific contributions are the following: 1) A novel application of a hierarchical framework [12] to AIL that learns a reward and policy for a challenging 4Originally presented as a non-archival workshop paper [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' main task by simultaneously learning rewards and poli- cies for auxiliary tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Manipulation experiments in which we demonstrate that AIL fails, while LfGP significantly outperforms both AIL and BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3) A thorough ablation study to examine the effects of various design choices for LfGP and our baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 4) Empirical analysis, including a simplified representative example and visualization of the learned models of LfGP and AIL, to better understand why AIL fails and how LfGP improves upon it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' PROBLEM FORMULATION A Markov decision process (MDP) is defined as M = ⟨S, A, R, P, ρ0, γ⟩, where the sets S and A are respectively the state and action space, R : S×A → R is a reward function, P is the state-transition environment dynamics distribution, ρ0 is the initial state distribution, and γ is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Actions are sampled from a stochastic policy π(a|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The policy π interacts with the environment to yield experience (st, at, rt, st+1) for t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' , ∞, where s0 ∼ ρ0(·), at ∼ π(·|st), st+1 ∼ P(·|st, at), rt = R(st, at).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' When referring to finite-horizon tasks, t = T indicates the final timestep of a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For notational convenience, we assume infinite-horizon, non-terminating environments where t is unbounded, but the extension to the finite-horizon case is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We aim to learn a policy π that maximizes the expected return J(π) = Eπ [G(τ0:∞)] = Eπ [�∞ t=0 γtR(st, at)], where τt:∞ = {(st, at), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' } is the trajectory starting with (st, at), and G(τt:∞) is the return of trajectory τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In this work, we focus on imitation learning (IL), where R is unknown and instead we are given a finite set of expert demonstration (s, a) pairs BE = � (s, a)E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In AIL, we attempt to simultaneously learn π and a discriminator D : S × A → [0, 1] that differentiates between expert samples (s, a)E and policy samples (s, a)π and subsequently define R using D [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' To accommodate hierarchical learning, we augment M to contain auxiliary tasks, where Taux = {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' , TK} are separate MDPs that share S, A, P, ρ0 and γ with the main task Tmain but have their own reward functions, Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' With this ABLETT et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' : LEARNING FROM GUIDED PLAY 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3: An MDP, analogous to stacking, with an expert demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Poor exploration can lead AIL to learn a suboptimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' modification, we refer to entities in our model that are specific to task T ∈ Tall, Tall = Taux ∪ {Tmain}, as (·)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We assume that we have a set of expert data BE T for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' LOCAL MAXIMUM WITH OFF-POLICY AIL In this section, we provide a representative example of how AIL can fail by reaching a locally maximum policy due to a learned deceptive reward [10] coupled with poor exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A simple six-state MDP is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3, with ten state- conditional actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We refer to actions as at = anm and states as st = sn where t, n and m refer to the current timestep, current state, and next state, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The reward function is R(s5, a55) = +1, R(s1, a15) = −5 and 0 for all other state- action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The initial state s1 is always s1, the fixed horizon length is 5, and no discounting is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The MDP is meant to be roughly analogous to a stacking manipulation task: s2, s3, s4 and s6 represent the first block being reached, grasped, lifted, and dropped respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' State s5 represents the gripper hovering over the second block (whether the first block has been stacked or not), while s1 is the reset state, and a15 represents reaching s5 without grasping the first block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Taking action a15 results in a total return of 1 (because R(s1, a15) = −5), since the first block has not actually been grasped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In our case, the agent does not receive any reward, and instead an expert demonstration of the optimal trajectory is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We will assume access to a learned (perfect) discriminator, and will use the AIRL [8] reward, so state-action pairs in the expert set receive +1 reward and all others receive -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We define the action-value Q(st, at) as the expected value of taking action at in state st, and initialize it to zero for all (s, a) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We define our update rule as the standard Q-Learning update [1], Q(st, at) = Q(st, at) + α (R(st, at) + maxa Q(st+1, a) − Q(st, at)), with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The agent uses ϵ-greedy exploration, storing each (st, at, st+1) tuple into a buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' After each episode, all Q values are updated to convergence using the whole buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' After the first complete episode of {a15, a55, a55, a55, a55}, Q(s1, a15) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='7, and Q(s1, a12) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In the second ({a12, a26, a61, a15, a55}) and third ({a12, a23, a36, a61, a15}) episodes, the agent initially moves in the correct direction, but ultimately still fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The final Q values in s1 are Q(s1, a15) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='49 and Q(s1, a12) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 A policy maximizing Q, having simultaneously learned to avoid s6 (by avoiding s2 and s3) and exploiting the (s5, a55) expert pair, will choose a1 = a15, giving a final return of 1 in the real MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This behaviour matches what we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1: due to the large negative reward from dropping the block, AIL learns a policy that avoids stacking altogether and merely reaches the second block, just as AIL here learns to skip s2 and s3 and exploit a55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In both cases, poor initial exploration leads to a deceptive reward, which exacerbates poor exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' LEARNING FROM GUIDED PLAY (LFGP) We now introduce Learning from Guided Play (LfGP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Our primary goal is to learn a policy πTmain that can solve the main task Tmain, with a secondary goal of also learning auxiliary task policies πT1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' , πTK that are used for improved exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' More specifically, we derive a hierarchical learning objective that is decomposed into three parts: i) recovering the reward function of each task with expert demonstrations, ii) training all policies to achieve their respective goals, and iii) using all policies for effective exploration in Tmain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For a summary of the algorithm, see supplementary material link in Footnote 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Learning the Reward Function We first describe how to recover the reward functions from expert demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For each task T ∈ Tall, we learn a dis- criminator DT (s, a) that is used to define the reward function for policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We construct the joint discriminator loss following [7] to train each discriminator in an off-policy manner: L(D) = − � T ∈Tall EB [log (1 − DT (s, a))] +EBE T [log (DT (s, a))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (1) Each resulting discriminator DT attempts to differentiate the occupancy measure between the distributions induced by BE T and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We can use DT to define various reward functions [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' following [8], we define the reward function for each task T to be RT (st, at) = log (DT (st, at)) − log (1 − DT (st, at)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Learning the Hierarchical Agent We adapt Scheduled Auxiliary Control (SAC-X) [12] to learn the hierarchical agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The agent includes low-level intention policies (equivalently referred to as intentions), a high-level scheduler policy, as well as the Q-functions and the discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The intentions aim to solve their corresponding tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', the intention πT aims to maximize the task return J(πT )), whereas the scheduler aims to maximize the expected return for Tmain by selecting a sequence of intentions to interact with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For the remainder of the paper, when we refer to a policy, we are referring to an intention policy, as opposed to the scheduler, unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 5See six_state_mdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='py from open source code to reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Legend 2 S 5 MDP C 2 3 S S S 6 S a5 Expert Demo a4 a1 2 S S a2 a3 S a1 a2-5 Suboptimal AIL Policy S4 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ACCEPTED DEC, 2022 1) Learning the Intentions: We learn each intention using Soft Actor-Critic (SAC) [15], an actor-critic algorithm that maximizes the entropy-regularized objective, though any off- policy RL algorithm would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The objective is J(πT ) = EπT � ∞ � t=0 γt (RT (st, at) + αH(πT (·|st))) � , (2) where the learned temperature α determines the importance of the entropy term and H(πT (·|st)) is the entropy of the intention πT at state st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The soft Q-function is QT (st, at) = RT (st, at) + EπT � ∞ � t=0 γt(RT (st+1, at+1) + αH(πT (·|st+1))) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (3) The intentions maximize the joint policy objective L(πint) = � T ∈Tall Es∼Ball,a∼πT (·|s) [QT (s, a) − α log πT (a|s)] , (4) where πint refers to the set of intentions {πTmain, πT1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' , πTK} and Ball refers to buffer containing every transition from interactions and demonstrations, as is done in [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For policy evaluation, the soft Q-functions QT for each πT minimize the joint soft Bellman residual L(Q) = � T ∈Tall E(s,a,s′)∼Ball,a′∼πT (·|s′) � (QT (s, a) − δT )2� , (5) δT = RT (s, a) + γ (QT (s′, a′) − α log πT (a′|s′)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (6) Crucially, because each task shares the common S, A, P, ρ0, and γ, and we are using off-policy learning, all tasks can learn from all data, as in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) The Scheduler: SAC-X formulates learning the sched- uler by maximizing the expected return of the main task [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In particular, let H be the number of possible intention switches within an episode and let each chosen intention execute for ξ timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The H intention choices made within the episode are defined as T 0:H−1 = � T (0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' , T (H−1)� , where T (h) ∈ Tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The return of the main task, given chosen intentions, is then defined as GTmain(T 0:H−1) = H−1 � h=0 (h+1)ξ−1 � t=hξ γtRTmain(st, at), (7) where at ∼ πT (h)(·|st) is the action taken at timestep t, sampled from the chosen intention T (h) in the hth scheduler period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The scheduler for the hth period P h S aims to maxi- mize the expected main task return: E � GTmain(T h:H−1)|P h S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Although SAC-X describes a method to learn the scheduler [12], we find that a combination of two simple task-agnostic heuristics performs similarly in practice (see Section V-C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Specifically, we use a weighted random scheduler (WRS) combined with handcrafted trajectories (HC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The WRS forms a prior categorical distribution over the set of tasks, with a higher probability mass pTmain for the main task and pTmain K for all other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This approach is comparable to the uniform scheduler from [12], with a bias towards the main task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The HC component is a small set of handcrafted trajectories of tasks that are sampled half of the time, forcing the scheduler to explore trajectories that would clearly be beneficial for completing the main task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The chosen handcrafted trajectories can be found in our code and in our supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Breaking Out of Local Maxima with LfGP Returning to the discussion in Section III, resolving the local maximum problem with LfGP is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Sup- pose we include a go-right auxiliary task with BE go-right = {(s1, a12), (s2, a23), (s3, a34)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' When the scheduler chooses the go-right intention, the agent does not exploit the a55 action because the go-right discriminator learns that R(s5, a55) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Since the transitions are stored in the shared buffer that the main intention also samples from, the agent can quickly obtain the correct, optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Expert Data Collection We assume that each T ∈ Tall has, for evaluation purposes only, a binary indicator of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In single-task imitation learning where this assumption is valid, expert data is typically collected by allowing the expert to control the agent until success conditions are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' At that point, the environment is reset following ρ0 and collection is repeated for a fixed number of episodes or (s, a) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We collect our expert data in this way for each T separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' EXPERIMENTS In this work, we are interested in answering the following questions about LfGP: 1) How does the performance of LfGP compare with BC and AIL in challenging manipulation tasks, in terms of success rate and expert sample efficiency?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) What parts of LfGP are necessary for success?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3) How do the policies and action value functions differ between AIL and LfGP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Experimental Setup We complete experiments in a simulation environment con- taining a Franka Emika Panda manipulator, one green and one blue block in a tray, fixed zones corresponding to the green and blue blocks, and one slot in each zone with < 1mm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 4: Example successful runs of our four main tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Top to bottom: Stack, Unstack-Stack, Bring, Insert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ABLETT et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' : LEARNING FROM GUIDED PLAY 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Insert 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Updates/steps (millions) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Success Rate LfGP (multi) BC (multi) DAC (single) BC (single) Expert Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 5: Performance results for LfGP, multitask BC, single-task BC, and DAC on all four tasks considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The x-axis corresponds to both gradient updates and environments steps for LfGP and DAC, and gradient updates only for both versions of BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The shaded area corresponds to standard deviation across five seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' LfGP significantly outperforms the baselines on all tasks, and even in Bring where it is matched by single-task BC, it is far more expert sample efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' tolerance for fitting the blocks (see bottom right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The robot is controlled via delta-position commands, and the blocks and end-effector can both be reset anywhere above the tray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The environment is designed such that several different challenging tasks can be completed within a common observa- tion and action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The main tasks that we investigate are Stack, Unstack-Stack, Bring, and Insert (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For more details on our environment and definitions of task success, see supplementary material link in Footnote 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We also define a set of auxiliary tasks: Open-Gripper, Close-Gripper, Reach, Lift, Move-Object, and Bring (Bring is both a main task and an auxiliary task for Insert), all of which are reusable between main tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We compare our method to several standard multitask and single-task baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A multitask algorithm simultaneously learns to complete a main task as well as auxiliary tasks, while the single-task algorithms only learn to complete the main task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In general, we consider a multitask algorithm to be more useful than a single-task algorithm, given the potential to reuse expert data and trained models for learning new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' To ensure a fair comparison, we provide single-task algorithms with an equivalent amount of total expert data as our multitask methods, as shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In our main experiments, we compare LfGP to a mul- titask variant of behavioural cloning (BC), single-task BC, and Discriminator-Actor-Critic (DAC) [7], a state-of-the-art approach to AIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We train multitask BC with a multitask mean squared error objective, L(πint) = � T ∈Tall � (s,a)∈BE T (πT (s) − a)2 , (8) while BC is trained with the corresponding single task version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Following recent trends in improving BC performance, we train our BC baselines with the same number of gradient updates as LfGP and DAC, evaluating the policies at the same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This adjustment has been shown to dramatically increase the performance of BC [18], [19], particularly com- pared to the more common practice of using early stopping, as is done in [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We validate that this change signifi- cantly improves BC performance in our ablation study (see Section V-C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We gather expert data by first training an expert policy using Scheduled Auxiliary Control (SAC-X) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We then run the Task Dataset Sizes Reuse Single Total Multi Stack SOCRLM: 1k/task 5k 1k 6k task U-Stack UOCRLM: 1k/task 5k 1k 6k Bring BOCRLM: 1k/task 6k 0 6k Insert IBOCRLM: 1k/task 6k 1k 7k Single Stack S: 6k 0 6k 6k Task U-Stack U: 6k 0 6k 6k Bring B: 6k 0 6k 6k Insert I: 6k 0 7k 7k TABLE I: The number of (s, a) pairs used for each main and auxiliary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The table illustrates the reusability of the expert data used to generate the performance results described in Section V-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Each letter under “Dataset Sizes” is the first letter of a single (auxiliary) task, and bolded letters indicate that a dataset was reused for more than one main task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', Open-Gripper was used for all four main tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Multitask methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', LfGP) are able to reuse a large portion of the expert data, while single-task methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', single-task BC) cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' expert policies to collect various amounts of expert data as described in Section IV-D and Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We also collect an extra 200 expert (sT , 0) pairs per auxiliary task, where T refers to the final timestep of an individual episode and 0 is an action of all zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This is equivalent to adding example data, as is done in example-based RL [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This addition improved final task performance, likely because it biases the reward towards completing the final task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' It is worth noting that, in the real world, final states are easier to collect than full demonstrations, and LfGP does not require any modifications to accommodate these extra examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Finally, even without this addition, LfGP still outperforms the baselines (see Section V-C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Performance Results Performance results for all methods and main tasks are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We freeze the policies every 100k steps and evaluate those policies for 50 randomized episodes, using only the mean action outputs for stochastic policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For all algorithms, we test across five seeds and report the mean and standard deviation of all seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In Stack, Unstack-Stack, and Insert, LfGP achieves expert performance, while the baselines all perform significantly worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In Bring, LfGP does not quite achieve expert per- formance, and is matched by single-task BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' However, we note that LfGP is much more expert data efficient than single- task BC because it reuses auxiliary task data (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A more direct comparison is multitask BC, which performs 6 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ACCEPTED DEC, 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 6: Various dataset ablations for LfGP and all baselines, including dataset size, subsampling of expert dataset, and replacement of extra (sT , 0) pairs with an equivalent amount of regular trajectory (s, a) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In all cases, LfGP still significantly outperforms all baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='2 0.' metadata={'source': 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+page_content=' Expert 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Expert Sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} 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+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Success Rate LfGP LfGP (BE for D only) LfGP (No (sT , 0) bias) DAC DAC (BE for D only) DAC (No (sT , 0) bias) Expert 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 BC/DAC Alternatives 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Updates/steps (millions) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Success Rate BC (multi) BC (multi, early stop) DAC GAIL BC BC (early stop) Expert Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 7: Left: Scheduler ablations for training LfGP, WRS is weighted random scheduler, HC is handcraft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Middle: Expert sampling ablations for training LfGP/DAC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Right: Baseline ablations for training BC/DAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' much more poorly than LfGP across all tasks, including Bring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Intriguingly, DAC also performs very poorly on all tasks, a phenomenon that we further explore in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Ablation Study While the fundamental idea of LfGP is relatively straight- forward, it is worth considering alternatives to some of the specific choices made for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In this section, we complete an ablation study where we vary (a) the expert dataset, including size, subsampling, and inclusion of extra (sT , 0) pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (b) the type of scheduler used for LfGP (see Section IV-B2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (c) the sampling strategy used for expert data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' and (d) the method for training our baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' To reduce the computational load of completing these experiments, all of these variations were carried out exclusively for our Stack task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' All ablation results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1) Dataset Ablations: We tested the following dataset vari- ations: (a) half and one and a half times the original expert dataset size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (b) subsampling BE, taking only every 20th timestep, as is done in [6], [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' and (c) replacing the 200 extra (sT , 0) pairs in each buffer with 200 regular trajectory (s, a) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Notably, even in the challenging regimes of halving and subsampling the dataset, LfGP still learns an expert-level policy (albeit more slowly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Scheduler Ablations: We tested the following scheduler variations: (a) Weighted Random Scheduler (WRS) only, re- moving the Handcrafted (HC) addition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (b) a learned sched- uler, as is used in [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' and (c) no scheduler, in which only the main task is attempted, akin to the Intentional Unintentional Agent [12], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Both WRS versions learn slightly faster than the learned scheduler, but all three methods outperform the No Scheduler ablation, replicating results from [12] demonstrating the importance of actually exploring all auxiliary tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Per- haps surprisingly, the HC modification made little difference compared with WRS only, but it is possible that for even more complex tasks, this could change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3) Expert Sampling Ablations: For our main performance experiments, we modified standard AIL in two ways: (a) we added expert buffer sampling to π and Q updates, in addition to the D updates, as is done in [16], [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' and (b) we biased the sampling of BE when training D to be 95% final (sT , 0) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We tested both LfGP and DAC without these additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For LfGP, although these modifications improve learning speed, they are not required to generate an expert policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For DAC, performance is quite poor regardless of these adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 4) Baseline Ablations: To verify that we evaluated against fair baselines, we tested two alternatives to those used for our main performance experiments: (a) an early stopping variation of BC, in which each expert buffer is divided into a 70%/30% train/validation split, taking the policy after validation error has not improved for 100 epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' and (b) the on-policy variant of DAC, also known as Generative Adversarial Imitation Learning (GAIL) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Notably, the early stopping variants of BC, commonly used as baselines in other AIL work [6], [7], [22] perform dramatically more poorly than those used in our experiments, verifying recent trends [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' LEARNED MODEL ANALYSIS In this section, we further examine the learned Stack models of LfGP and DAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We take snapshots of the average per- forming models from LfGP and DAC at four points during learning: 200k, 400k, 600k, and 800k model updates and environment steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Although the initial gripper and block positions are randomized between episodes during learning, for each snapshot, we reset the stacking environment to a single set of representative initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We then run the ABLETT et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' : LEARNING FROM GUIDED PLAY 7 LfGP – Open-Gripper LfGP – Close-Gripper LfGP – Reach LfGP – Lift LfGP – Move-Object LfGP – Stack DAC – Stack Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 8: The policy outputs (arrows) and Q values (background) for each LfGP task and for DAC at 200k environment steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The arrows show velocity direction/magnitude, blue → green indicates open-gripper → close-gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For Q values, red → yellow indicates low → high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The LfGP policies and Q functions are reasonable for all tasks, while DAC has only learned to reach toward and above the green block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' snapshot policies for a single exploratory trajectory, using the stochastic outputs of each policy as well as, for LfGP, the WRS+HC scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Trajectories from these runs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' DAC is unable to learn to grasp or even reach the blue block and ultimately settles on a policy that learns to reach and hover near the green block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This is understandable—DAC learns a deceptive reward for hovering above the green block regardless of the position of the blue block, because it has not sufficiently explored the alternative of first grasping the blue block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Even if hovering above the green block does not fully match the expert data, the DAC policy receives some reward for doing so, as evidenced by the learned Q value on the right side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In comparison, even after only 200k environment steps, LfGP learns to reach and push the blue block, and by 600k steps, grasp, move, and nearly stack it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' By enforcing explo- ration of sub-tasks that are crucial to completing the main task, LfGP ensures that the distribution of expert stacking data is fully matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' RELATED WORK Imitation learning is often divided into two main categories: behavioural cloning (BC) [23], [24] and inverse reinforcement learning (IRL) [5], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' BC recovers the expert policy via supervised learning, but it suffers from compounding errors due to covariate shift [23], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Alternatively, IRL partially alleviates the covariate shift problem by estimating the reward function and then applying RL using the learned reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A popular approach to IRL is adversarial imitation learning (AIL) [6], [7], [27], in which the expert policy is recovered by matching the occupancy measure between the generated data and the demonstration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Our proposed method en- hances existing AIL algorithms by enabling exploration of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 9: LfGP and DAC trajectories of the gripper, blue block, and green block for four stack episodes with consistent initial conditions throughout the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The LfGP episodes, each including auxiliary task sub-trajectories, demonstrate significantly more variety than the DAC trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' key auxiliary tasks via the use of a scheduled multitask model, simultaneously resolving the susceptibility of AIL to deceptive rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Agents learned via hierarchical reinforcement learning (HRL), which act over multiple levels of temporal abstractions in long planning horizon tasks, are shown to provide more effective exploration than agents operating over only a single level of abstraction [12], [28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Our approach for learning agents most closely resembles hierarchical AIL methods that attempt to combine AIL with HRL [27], [30]–[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Existing work [30]–[32] often formulates the hierarchical agent using the Options framework [28] and learns the reward function with AIL [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Both [30] and [32] leverage task-specific expert demonstrations to learn options using mixture-of-experts and expectation-maximization strategies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In contrast, our work focuses on expert demonstrations that include multi- ple reusable auxiliary tasks, each of which has clear semantic meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In the multitask setting, [27] and [31] leverage unsegmented, multitask expert demonstrations to learn low-level policies via a latent variable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Other work has used a large corpus of unsegmented but semantically meaningful “play” expert data to bootstrap policy learning [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We define our expert dataset as being derived from guided play, in that the expert completes semantically meaningful auxiliary tasks with provided transitions, reducing the burden on the expert to generate these data arbitrarily and simultaneously providing auxiliary task labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Compared with learning from unseg- mented demonstrations, the use of segmented demonstrations, as in [33], ensures that we know which auxiliary tasks our model will be learning, and opens up the possibility of expert data reuse and also transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Finally, we deviate from the Options framework and build upon Scheduled Auxiliary Control (SAC-X) to train our hierarchical agent, since SAC- X has been shown to work well for challenging manipulation tasks [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' LIMITATIONS Our approach is not without limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' While we were able to use LfGP in six and seven-task settings, the number of tasks for which this method would become intractable is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' LfGP needs access to segmented expert data as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' in many cases, this is reasonable, and is also required to be able to reuse auxiliary task data between main tasks, but it does necessitate extra care during expert data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Also, LfGP requires pre-defined auxiliary tasks: while this is a common approach to hierarchical RL (see [34], Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1, for numerous examples), choosing these tasks may sometimes present a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Finally, compared with methods that use offline data exclusively (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', BC), for our tasks, LfGP requires 200k 400k 600k 800k LfGP DAC8 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ACCEPTED DEC, 2022 many online environment steps to learn a high-quality policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This data gathering could be costly if human supervision was necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' It is worth noting that, because LfGP is already a multitask method, this final point could be partially resolved through the use of multitask reset-free RL [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' CONCLUSION We have shown how adversarial imitation learning can fail at challenging manipulation tasks because it learns deceptive rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We demonstrated that this can be resolved with Learning from Guided Play (LfGP), in which we introduce auxiliary tasks and the corresponding expert data, guiding the agent to playfully explore parts of the state and action space that would have been avoided otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We demonstrated that our method dramatically outperforms both BC and AIL base- lines, particularly in the case of AIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Furthermore, our method can leverage reusable expert data, making it significantly more expert sample efficient than the highest-performing baseline, and its learned auxiliary task models can be applied to transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In future work, we intend to investigate transfer learning to determine if overall policy learning time can be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We gratefully acknowledge the Digital Research Alliance of Canada and NVIDIA Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', who provided the GPUs used in this work through their Resources for Research Groups Program and their Hardware Grant Program, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' REFERENCES [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Sutton and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Barto, Reinforcement Learning: An 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2021 IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Robotics and Automation (ICRA’21), Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ABLETT et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' : LEARNING FROM GUIDED PLAY 9 APPENDIX A LEARNING FROM GUIDED PLAY ALGORITHM The complete pseudo-code is given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Our implementation builds on RL Sandbox [36], an open-source PyTorch [37] framework for RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For learning the discriminators, we follow DAC and apply a gradient penalty for regularization [7], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We optimize the intentions via the reparameterization trick [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' As is commonly done in deep RL, we use the Clipped Double Q-Learning trick [41] to mitigate overestimation bias [42] and use a target network to mitigate learning instability [43] when training the policies and Q-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We also learn the temperature parameter αT separately for each task T (see Section 5 of [44] for more details on learning α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For Generative Adversarial Imitation Learning (GAIL), we use a common open-source PyTorch implementation [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The hyperparameters chosen for all methods are provided in Section G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Please see videos at papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='starslab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='ca/lfgp for examples of what LfGP looks like in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Algorithm 1 Learning from Guided Play (LfGP) Input: Expert replay buffers BE main, BE 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' , BE K, scheduler period ξ, sample batch size N Parameters: Intentions πT with corresponding Q-functions QT and discriminators DT , and scheduler πS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' with Q- table QS) 1: Initialize replay buffer B 2: for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' , do 3: # Interact with environment 4: For every ξ steps, select intention πT using πS 5: Select action at using πT 6: Execute action at and observe next state s′ t 7: Store transition ⟨st, at, s′ t⟩ in B 8: 9: # Update discriminator DT ′ for each task T ′ 10: Sample {(si, ai)}N i=1 ∼ B 11: for each task T ′ do 12: Sample {(s′ i, a′ i)}B i=1 ∼ BE k 13: Update DT ′ following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (1) using GAN + Gradient Penalty 14: end for 15: 16: # Update intentions πT ′ and Q-functions QT ′ for each task T ′ 17: Sample {(si, ai)}N i=1 ∼ B 18: Compute reward DT ′(si, ai) for each task T ′ 19: Update π and Q following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (5) 20: 21: # Optional Update learned scheduler πS 22: if at the end of effective horizon then 23: Compute main task return GTmain using reward esti- mate from Dmain 24: Update πS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' update Q-table QS following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='3) and recompute Boltzmann distribution) 25: end if 26: end for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Scheduler Details 1) Learning the Scheduler: As stated in our paper, our main experiments used a simple weighted random scheduler with handcrafted trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In this section, we provide the details of our learned scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Following [12], let H be the total number of possible intention switches within an episode and let each chosen intention execute for ξ timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The H intention choices made within the episode are defined as T 0:H−1 = � T (0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' , T (H−1)� , where T (h) ∈ Tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The main task’s return given chosen intentions is then defined as GTmain(T 0:H−1) = H−1 � h=0 (h+1)ξ−1 � t=hξ γtRTmain(st, at), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1) where at ∼ πT (h)(·|st) is the action taken at timestep t, sampled from the chosen intention T (h) in the hth scheduler period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We further define the Q-function for the scheduler as QS(T 0:h−1, T (h)) = ET h:H−1∼P h:H−1 S � GTmain(T h:H−1)|T 0:h−1� and represent the scheduler for the hth period as a softmax distribution P h S over {QS(T 0:h−1, Tmain), QS(T 0:h−1, T1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' , QS(T 0:h−1, TK)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The scheduler maximizes the expected return of the main task following the scheduler: L(S) = ET (0)∼P 0 S � QS(∅, T (0)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='2) We use Monte Carlo returns to estimate QS, estimating the expected return using the exponential moving average: QS(T 0:h−1, T (h)) = (1 − φ)QS(T 0:h−1, T (h)) +φ GTmain(T h:H), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='3) where φ ∈ [0, 1] represents the amount of discounting on older returns and GTmain(T h:H) is the cumulative discounted return of the trajectory starting at timestep hξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Weighted Random Scheduler Plus Handcrafted Trajectories As stated in our paper, the main experiments were com- pleted with the described weighted random scheduler (WRS) combined with some simple handcrafted trajectories (HC) that we expected to be beneficial for learning each of the main tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In this section, we provide further de- tails of these handcrafted scheduler trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Given a chosen proportion hyperparameter (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 in our experiments), we randomly sampled full trajectories from the lists below at the beginning of training episodes, and otherwise sam- pled from the regular WRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For all four tasks Main = {Stack, Unstack-Stack, Bring, Insert}, we provided the fol- lowing set of trajectories: 1) Reach, Lift, Main, Open-Gripper, Reach, Lift, Main, Open-Gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Reach, Lift, Move-Object, Main, Open-Gripper, Reach, Lift, Move-Object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3) Lift, Main, Open-Gripper, Lift, Main, Open-Gripper, Lift, Main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 4) Main, Open-Gripper, Main, Open-Gripper, Main, Open- Gripper, Main, Open-Gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 10 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ACCEPTED DEC, 2022 TABLE II: The components used in our environment observations, common to all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Grip finger position is a continuous value from 0 (closed) to 1 (open).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Component Dim Unit Privileged?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Extra info EE pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3 m No rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' to base EE velocity 3 m/s No rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' to base Grip finger pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 6 [0, 1] No current, last 2 Block pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 6 m Yes both blocks Block rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 8 quat Yes both blocks Block trans vel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 6 m/s Yes rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' to base Block rot vel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 6 rad/s Yes rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' to base Block rel to EE 6 m Yes both blocks Block rel to block 3 m Yes in base frame Block rel to slot 6 m Yes both blocks Force-torque 6 N,Nm No at wrist Total 59 5) Move-Object, Main, Open-Gripper, Move-Object, Main, Open-Gripper, Move-Object, Main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For insert, in addition to the trajectories listed above, we added two more trajectories to specifically accommodate Bring as an auxiliary task: 1) Bring, Insert, Open-Gripper, Bring, Insert, Open- Gripper, Bring, Insert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Reach, Lift, Bring, Insert, Open-Gripper, Reach, Lift, Bring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' APPENDIX B ENVIRONMENT DETAILS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 10: An image of our multitask environment immediately after a reset has been carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A screenshot of our environment, simulated in PyBullet [47], is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We chose this environment because we desired tasks that a) have a large distribution of possible initial states, representative of manipulation in the real world, b) have a shared observation/action space with several other tasks, allowing the use of auxiliary tasks and transfer learning, and c) require a reasonably long horizon and significant use of contact to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The environment contains a tray with sloped edges (to keep the blocks within the reachable workspace of the end-effector), as well as a green and a blue block, each of which is 4 cm × 4 cm × 4 cm and has a mass of 100 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The dimensions of the lower part of the tray, before reaching the sloped edges, are 30 cm × 30 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The dimensions of the ‘bring’ boundaries (shaded blue and green regions) are 8 cm × 8 cm, while the dimensions of the insertion slots, which are directly in the center of each shaded region, are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1 cm × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1 cm × 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The boundaries for end-effector movement, relative to the tool center point that is directly between the gripper fingers, are a 30 cm × 30 cm × 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 cm box, where the bottom boundary is low enough to allow the gripper to interact with objects, but not to collide with the bottom of the tray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' See Table II for a summary of our environment observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In this work, we use privileged state information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', block poses), but adapting our method to exclusively use image- based data is straightforward since we do not use hand-crafted reward functions as in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The environment movement actions are 3-DOF translational position changes, where the position change is relative to the current end-effector position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We leverage PyBullet’s built-in position-based inverse kinematics function to generate joint commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Our actions also contain a fourth dimension that corresponds to actuating the gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' To allow for the use of policy models with exclusively continuous outputs, this dimension accepts any real number, with any value greater than 0 commanding the gripper to open, and any number less than 0 commanding it to close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Actions are supplied at a rate of 20 Hz, and each training episode is limited to 18 seconds, corresponding to 360 time steps per episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For play-based expert data collection, we also reset the environment manually every 360 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Between episodes, block positions are randomized to any pose within the tray, and the end-effector is randomized to any position between 5 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 cm above the tray, within the earlier stated end-effector bounds, with the gripper fully opened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The only exception to these initial conditions is during expert data collection and agent training of the Unstack-Stack task: in this case, the green block is manually set to be on top of the blue block at the start of the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' APPENDIX C PERFORMANCE RESULTS FOR AUXILIARY TASKS The performance results for all multitask methods and all auxiliary tasks are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Multitask BC has gradually decreasing performance on many of the auxiliary tasks as the number of updates increases, which is consistent with mild overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Intriguingly, however, multitask BC does achieve quite reasonable performance on many of the auxiliary tasks (such as Lift) without needing any of the extra environment interactions required by an online method such as LfGP or DAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' An interesting direction for future work is to determine whether pretraining via multitask BC could provide ABLETT et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' : LEARNING FROM GUIDED PLAY 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Move 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Updates/steps (millions) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Success Rate LfGP (multi) BC (multi) DAC (single) BC (single) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 11: Performance for LfGP and the multitask baselines across all tasks, shaded area corresponds to standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' any improvements in environment sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We did attempt to do this, but found that it resulted in poorer final performance than training from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' APPENDIX D PROCEDURE FOR OBTAINING EXPERTS As stated, we used SAC-X [12] to train models that we used for generating expert data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We used the same hyperpa- rameters that we used for LfGP (see Table III), apart from the discriminator, which, of course, does not exist in SAC-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' See Section E for details on the hand-crafted rewards that we used for training these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For an example of gathering play-based expert data, please see our attached video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We made two modifications to regular SAC-X to speed up learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' First, we pre-trained a Move-Object model before transferring this model to each of our main tasks, as we did in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='3 of our main paper, since we found that SAC-X would plateau when we tried to learn the more challenging tasks from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The need for this modification demon- strates another noteworthy benefit of LfGP—when training LfGP, main tasks could be learned from scratch, and generally in fewer time steps, than it took to train our experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Second, during transfer to the main tasks, we used what we called a conditional weighted scheduler instead of a Q-Table: we de- fined weights for every combination of tasks, so that the sched- uler would pick each task with probability P(T (h)|T (h−1)), ensuring that ∀T ′ ∈ Tall, � T ∈Tall P(T |T ′) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The weights that we used were fairly consistent between main tasks, and can be found in our packaged code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The conditional weighted scheduler ensured that every task was still explored throughout the learning process, so that we would have high-quality experts for every auxiliary task in addition to the main task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This scheduler can be considered as a more complex alter- native to the weighted random scheduler or the addition with handcrafted trajectories from our main paper, and again shows the flexibility of using a semantically-meaningful multitask policy with a common observation and action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' APPENDIX E EVALUATION As stated in our paper, we evaluated all algorithms by testing the mean output of the main task policy head in our environment and determining a success rate based on 50 randomly selected resets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' These evaluation episodes were run for 360 time steps to match our training process, and if a condition for success was met within that time, they were recorded as a success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The rest of this section describes in detail how we evaluated ‘success’ for each of our main and auxiliary tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' As previously stated, we trained experts using a modified SAC-X [12] that required us to define a set of reward functions for each task, which we include in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The authors of [12] focused on sparse rewards but also showed a few experiments in which dense rewards reduced the time to learn adequate policies, so we chose to use dense rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We note that many of these reward functions are particularly com- plex and required significant manual shaping effort, further motivating the use of an imitation learning scheme like the one presented in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' It is possible that we could have made do with sparse rewards, such as those used in [12], but our compute resources made this impractical—for example, in [12], their agent took 5000 episodes × 36 actors × 360 time steps = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='8 M time steps to learn their stacking task, which would have taken over a month of wall clock time on our fastest machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' To see the specific values used for the rewards and success conditions described in these sections, please review our code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Unless otherwise stated, each of the success conditions in this section had to be held for 10 time steps, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 seconds, before being registered as a success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This choice was made to prevent registering a success when, for example, the blue block slipped off the green block during the Stack task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 12 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ACCEPTED DEC, 2022 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Common For each of these functions, we use the following common labels: pb: blue block position, vb: blue block velocity, ab: blue block acceleration, pg: green block position, pe: end-effector tool center point position (TCP), ps: center of a block pushed into one of the slots, g1: (scalar) gripper finger 1 position, g2: (scalar) gripper finger 2 position, and ag: (scalar) gripper open/close action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' A block is flat on the tray when pb,z = 0 or pg,z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' To further reduce training time for SAC-X experts, all rewards were set to 0 if ∥pb −pe∥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1 and ∥pg −pe∥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', the TCP must be within 10 cm of either block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' During training while using the Unstack-Stack variation of our environment, a penalty of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1 was added to each reward if ∥pg,z∥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='001 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', there was a penalty to all rewards if the green block was not flat on the tray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Stack/Unstack-Stack The evaluation conditions for Stack and Unstack-Stack are identical, but in our Unstack-Stack experiments, the environ- ment is manually set to have the green block start on top of the blue block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1) Success: Using internal PyBullet commands, we check to see whether the blue block is in contact with the green block and is not in contact with either the tray or the gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Reward: We include a term for checking the distance between the blue block and the spot above the the green block, a term for rewarding increasing distance between the block and the TCP once the block is stacked, a term for shaping lifting behaviour, a term to reward closing the gripper when the block is within a tight reaching tolerance, and a term for rewarding the opening the gripper once the block is stacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Bring/Insert We use the same success and reward calculations for Bring and Insert, but for Bring the threshold for success is 3 cm, and for insert, it is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1) Success: We check that the distance between pb and ps is less than the defined threshold, that the blue block is touching the tray, and that the end-effector is not touching the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For Insert, the block can only be within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 mm of the insertion target if it is correctly inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Reward: We include a term for checking the distance between the pb and ps and a term for rewarding increas- ing distance between pb and pe once the blue block is brought/inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Open-Gripper/Close-Gripper We use the same success and reward calculations for Open- Gripper and Close-Gripper, apart from inverting the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1) Success: For Open-Gripper and Close-Gripper, we check to see if ag < 0 or ag > 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Reward: We include a term for checking the action, as we do in the success condition, and also include a shaping term that discourages high magnitudes of the movement action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Lift 1) Success: We check to see if pb,z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Reward: We add a dense reward for checking the height of the block, but specifically also check that the gripper positions correspond to being closed around the block, so that the block does not simply get pushed up the edges of the tray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We also include a shaping term for encouraging the gripper to close when the block is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Reach 1) Success: We check to see if ∥pe − pb∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Reward: We have a single dense term to check the distance between pe and pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Move-Object For Move-Object, we changed the required holding time for success to 1 second, or 20 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1) Success: We check to see if the vb > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='05 and ab < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The acceleration condition ensures that the arm has learned to move the block by following a smooth trajectory, rather than vigorously shaking it or continuously picking up and dropping it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Reward: We include a velocity term and an acceleration penalty, as in the success condition, but also include a dense bonus for lifting the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' APPENDIX F RETURN PLOTS As previously stated, we generated hand-crafted reward functions for each of our tasks for the purpose of training our SAC-X experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Given that we have these rewards, we can also generate return plots corresponding to our results to add extra insight (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 12 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The patterns displayed in these plots are, for the most part, quite similar to the success rate plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' One notable exception is that there is an eventual increase in performance when training DAC on Insert, indicating that, perhaps for certain tasks, DAC alone can eventually make progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Nevertheless, it is clear that LfGP improves learning efficiency, and it is unclear whether DAC would plateau even if it was trained for a longer period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' APPENDIX G MODEL ARCHITECTURES AND HYPERPARAMETERS All the single-task models share the same network architec- tures and all the multitask models share the same network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' All layers are initialized using the PyTorch default methods [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For the single-task variant, the policy is a fully-connected network with two hidden layers followed by ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Each hidden layer consists of 256 hidden units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The output of the policy for LfGP and DAC is split into two vectors, mean ˆµ and variance ˆσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For both variants of BC, only the mean ˆµ ABLETT et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' : LEARNING FROM GUIDED PLAY 13 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='0 Episode Return LfGP (multi) BC (multi) DAC (single) BC (single) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 13: Episode return for LfGP compared with multitask baselines on all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Shaded area corresponds to standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' output is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The vectors define a Gaussian distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' N(ˆµ, ˆσ2I), where I is the identity matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' When computing actions, we squash the samples using the tanh function and bound the actions to be in range [−1, 1], as done in SAC [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The variance ˆσ2 is computed by applying a softplus function followed by a sum with an epsilon ϵ = 1e-7 to prevent underflow: ˆσi = softplus(ˆxi) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The Q-functions are fully-connected networks with two hidden layers followed by ReLU activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Each hidden layer consists of 256 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The output of the Q-function is a scalar corresponding to the value estimate given the current state-action pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Finally, the discriminator is a fully-connected network with two hidden layers followed by tanh activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Each hidden layer consists of 256 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The output of the discriminator is a scalar logit to be used as an input to the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The sigmoid function output can be viewed as the probability of the current state-action pair coming from the expert distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For multitask variant, the policies and the Q-functions share their initial layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' There are two shared, fully-connected layers followed by ReLU activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Each layer consists of 256 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The output of the last shared layer is then fed into the policies and Q-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Each policy head and Q-function head corresponds to one task and has the same architecture: a two-layered fully-connected network followed by ReLU activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The output of the policy head corresponds to the parameters of a Gaussian distribution, as described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Similarly, the output of the Q-function head corresponds to the value estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Finally, the discriminator is a fully-connected network with two hidden layers followed by tanh activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Each hidden layer consists of 256 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The output of the discriminator is a vector, where the ith entry corresponds to the logit input to the sigmoid function for task Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The ith sigmoid function output corresponds to the probability of the current state-action pair coming from the expert distribution in task Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' The hyperparameters for our experiments are listed in Table III and Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In the early-stopping variant of BC, overfit tolerance refers to the number of full dataset training epochs without an improvement in validation error before we stop training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' All models are optimized using Adam Optimizer [48] with PyTorch default values, unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' APPENDIX H OPEN-ACTION AND CLOSE-ACTION 14 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' ACCEPTED DEC, 2022 TABLE III: Hyperparameters for AIL algorithms across all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Parameters that do not appear in the original version of DAC are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Algorithm LfGP DAC Total Interactions 2M (4M for Insert) Buffer Size 2M (4M for Insert) Buffer Warmup 25k Initial Exploration 50k Evaluations per task 50 Evaluation frequency 100k interactions Intention γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='99 Batch Size 256 Q Update Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1 Target Q Update Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1 π Update Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1 Polyak Averaging 1e-4 Q Learning Rate 3e-4 π Learning Rate 1e-5 α Learning Rate 3e-4 Initial α 1e-2 Target Entropy −dim(a) = −4 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Gradient Norm 10 π Weight Decay 1e-2 Q Weight Decay 1e-2 BE sampling proportion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1 BE sampling decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='99999 Discriminator Learning Rate 3e-4 Batch Size 256 Gradient Penalty λ 10 Weight Decay 1e-2 (sT , 0) sampling bias 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='95 TABLE IV: Hyperparameters for LfGP schedulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Scheduler Learned WRS WRS + HC ξ 45 N/A N/A φ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='6 N/A N/A Initial Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 360 N/A N/A Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='9995 N/A N/A Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1 N/A N/A Main Task Rate N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 Handcraft Rate N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='5 DISTRIBUTION MATCHING There was one exception to the method we used for col- lecting our expert data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Specifically, our Open-Gripper and Close-Gripper tasks required additional considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' It is worth reminding the reader that our Open-Gripper and Close- Gripper tasks were meant to simply open or close the gripper, respectively, while remaining reasonably close to either block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' If we were to use the approach described above verbatim, the Open-Gripper and Close-Gripper data would contain no (s, a) pairs where the gripper actually released or grasped the block, instead immediately opening or closing the gripper while simply hovering near the blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Perhaps unsurprisingly, this was detrimental to our algorithm’s performance: as one example, an agent attempting to learn Stack would, if Open- Gripper was selected while the blue block was held above TABLE V: Hyperparameters for BC algorithms (both single-task and multitask) across all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Version Main Results Early Stopping Batch Size 256 Learning Rate 1e-5 Weight Decay 1e-2 Total Updates 2M (4M for Insert) N/A Overfit Tolerance N/A 100 the green block, move the grasped blue block away from the green block before dropping it on the tray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This behaviour, of course, is not what we would want, but it better matches an expert distribution when the environment is reset in between each task execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' To mitigate this, our Open-Gripper data actually contain a mix of each of the other sub-tasks called for the first 45 time steps, followed by a switch to Open-Gripper, ensuring that the expert dataset contains some degree of block-releasing, with the trade-off being that 50% of the Open-Gripper expert data is specific to whatever the main task happens to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We left this additional detail out of our main paper for clarity, since it corresponds to only a small portion of the expert data (every other auxiliary task was fully reused).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Similarly, the Close-Gripper data calls Lift for 15 time steps before switching to Close-Gripper, ensuring that the Close-gripper dataset will contain a large proportion of data where the block is actually grasped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' For the Closer-gripper data, however, this modification did still allow data to be reused between main tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' APPENDIX I ATTEMPTED AND FAILED EXPERIMENTS In this section, we provide a list of experiments and modi- fications that did not improve performance, in addition to the alternatives that did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 1) Pretraining with BC: We attempted to pretrain LfGP using multitask BC, and then to transition to online learning with LfGP, but we found that this tended to produce significantly poorer final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Some existing work [49], [50] has investigated transitioning from BC to online RL, but achieving this consistently, especially with off-policy RL, remains an open research problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) Handcrafted Open-Gripper/Close-Gripper policies: Given the simplicity of designing a reward function in these two cases, a natural question is whether Open- Gripper and Close-Gripper could use hand-crafted re- ward functions, or even hand-crafted policies, instead of these specialized datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' In our experiments, both of these alternatives proved to be quite detrimental to our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3) Penalizing Q values: In our early experiments, we found that LfGP training progress was harmed by ex- ploding Q values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' This problem was particularly exac- erbated when we added BE sampling to our Q and π updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' It appears that this occurs because, at the begin- ning of training, the differences between discriminator ABLETT et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' : LEARNING FROM GUIDED PLAY 15 outputs for expert data and non-expert data are so large that the bootstrap Q updates quickly jump to unrealistic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We attempted to use various forms of Q penalties to resolve this, akin to Conservative Q Learning (CQL) [51], but found that all of our modifications ultimately harmed final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Some of the things we tried, in addition to the CQL loss, were reducing γ (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='95, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='9), clipping Q losses to -5, +5, smooth L1 loss, huber loss, increased gradient penalty λ for D (50, 100), decreased reward scaling (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='1), more discriminator updates per π/Q update (10), and weight decay in D only (as is done in [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We ultimately resolved exploding Q values by i) decreasing polyak averaging to a significantly lower value than is used in much other work (1e-4 as opposed to the SAC default of 5e-3), and ii) adding in weight decay (with a significantly higher value used than is used in other work) to π, Q, and D training (which was required to not overfit with the reduced polyak averaging value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Without the added weight decay, performance started to plateau and eventually to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 4) Higher Update-to-Data (UTD) Ratio: Recent work in RL has started increasing the UTD ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=', increas- ing the number of policy/Q updates per environment interaction), with the goal of improving environment sample efficiency [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We were actually able to increase this from 1 to 2 and achieve a marginal improvement in environment sample efficiency, but this also nearly doubled the running time of our experiments, so we opted not to include this modification in our final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' Higher values of the UTD ratio also caused our Q values to explode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' APPENDIX J EXPERIMENTAL HARDWARE For a list of the software we used in this work, see our code and instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' We used a number of different computers and GPUs when completing our experiments: 1) GPU: NVidia Quadro RTX 8000, CPU: AMD - Ryzen 5950x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 GHz 16-core 32-thread, RAM: 64GB, OS: Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 2) GPU: NVidia V100 SXM2, CPU: Intel Gold 6148 Skylake @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='4 GHz (only used 4 threads), RAM: 32GB, OS: CentOS 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 3) GPU: Nvidia GeForce RTX 2070, CPU: RYZEN Threadripper 2990WX, RAM: 32GB, OS: Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfQvZV/content/2301.00051v1.pdf'} +page_content=' 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b/D9E1T4oBgHgl3EQf-QZ6/content/tmp_files/2301.03566v1.pdf.txt @@ -0,0 +1,3880 @@ +Simple Binary Hypothesis Testing under Local Differential +Privacy and Communication Constraints +Ankit Pensia +University of Wisconsin-Madison +ankitp@cs.wisc.edu +Amir R. Asadi +University of Cambridge +aa2345@cam.ac.uk +Varun Jog +University of Cambridge +vj270@cam.ac.uk +Po-Ling Loh +University of Cambridge +pll28@cam.ac.uk +January 10, 2023 +Abstract +We study simple binary hypothesis testing under both local differential privacy (LDP) and +communication constraints. We qualify our results as either minimax optimal or instance opti- +mal: the former hold for the set of distribution pairs with prescribed Hellinger divergence and +total variation distance, whereas the latter hold for specific distribution pairs. For the sample +complexity of simple hypothesis testing under pure LDP constraints, we establish instance- +optimal bounds for distributions with binary support; minimax-optimal bounds for general +distributions; and (approximately) instance-optimal, computationally efficient algorithms for +general distributions. When both privacy and communication constraints are present, we de- +velop instance-optimal, computationally efficient algorithms that achieve the minimum possible +sample complexity (up to universal constants). Our results on instance-optimal algorithms hinge +on identifying the extreme points of the joint range set A of two distributions p and q, defined +as A := {(Tp, Tq)|T ∈ C}, where C is the set of channels characterizing the constraints. +1 +Introduction +Statistical inference on distributed data is becoming increasingly common, due to the proliferation of +massive datasets which cannot be stored on a single server, and greater awareness of the security and +privacy risks of centralized data. An institution (or statistician) that wishes to infer an aggregate +statistic of such distributed data needs to solicit information, such as the raw data or some relevant +statistic, from data owners (individuals). Individuals may be wary of sharing their data due to +its sensitive nature or their lack of trust in the institution. The local differential privacy (LDP) +paradigm suggests a solution by requiring that individuals’ responses divulge only a limited amount +of information about their data to the institution. +Privacy is typically ensured by deliberately +randomizing individuals’ responses, e.g., by adding noise. See Definition 1.1 below for a formal +definition; we refer the reader to Dwork and Roth [DR13] for more details on differential privacy. +In this paper, we study distributed estimation under LDP constraints, focusing on simple bi- +nary hypothesis testing, a fundamental problem in statistical estimation. We will also consider +LDP constraints in tandem with communication constraints. This is a more realistic setting, since +bandwidth considerations often impose constraints on the size of individuals’ communications. The +1 +arXiv:2301.03566v1 [math.ST] 9 Jan 2023 + +case when only communication constraints are present was addressed previously by Pensia, Jog, +and Loh [PJL22]. +Recall that simple binary hypothesis testing is defined as follows: Let p and q be two distributions +over a finite domain X, and let X1, . . . , Xn ∈ X n be n i.i.d. samples drawn from either p or q. +The goal of the statistician is to identify (with high probability) whether the samples were drawn +from p or q. This problem has been extensively studied in both asymptotic and nonasymptotic +settings [NP33; Wal45; Cam86]. For example, it is known that the optimal test for this problem is +the likelihood ratio test, and its performance can be characterized in terms of divergences between p +and q, such as the total variation distance, Hellinger divergence, or Kullback–Leibler divergence. In +particular, the sample complexity of hypothesis testing, defined as the smallest sample size needed +to achieve an error probability smaller than a small constant, say, 0.01, is Θ +� +1 +d2 +h(p,q) +� +, where d2 +h(p, q) +is the Hellinger divergence between p and q. +In the context of local differential privacy, the statistician no longer has access to the original +samples X1, . . . , Xn, but only their privatized counterparts: Y1, . . . , Yn ∈ Yn, for some set Y.1 Each +Xi is transformed to Yi via a private channel Ti, which is simply a probability kernel specifying +Ti(y, x) = P(Yi = y|Xi = x). With a slight abuse of notation, we shall use Ti to denote the +transition kernel in R|Y|×|X|, as well as the stochastic map Yi = Ti(Xi). A formal definition of +privacy is given below: +Definition 1.1 (ϵ-LDP). Let ϵ ∈ R+, and let X and Y be two domains. A channel T : X → Y +satisfies ϵ-LDP if +sup +x,x′∈X +sup +A⊆Y +P[T(x) ∈ A] − eϵ · P[T(x′) ∈ A] ≤ 0, +where we interpret T as a stochastic map on X. Equivalently, if X and Y are countable domains (as +will be the case for us), a channel T is ϵ-LDP if supx,x′∈X supy∈Y +T(y,x) +T(y,x′) ≤ eϵ, where we interpret +T as the transition kernel. +When ϵ = ∞, we may set Yi equal to Xi with probability 1, and we recover the vanilla version +of the problem with no privacy constraints. +Existing results on simple binary hypothesis testing under LDP constraints have focused on the +high-privacy regime of ϵ ∈ (0, c), for a constant c > 0, and have shown that the sample complexity +is Θ +� +1 +ϵ2d2 +TV(p,q) +� +, where dTV(p, q) is the total variation distance between p and q (cf. Fact 2.7). +Thus, when ϵ is a constant, the sample complexity is Θ +� +1 +d2 +TV(p,q) +� +, and when ϵ = ∞ (no privacy), +the sample complexity is Θ +� +1 +d2 +h(p,q) +� +. Although these two divergences satisfy d2 +TV(p, q) ≲ d2 +h(p, q) ≲ +dTV(p, q), the bounds are tight; i.e., the two sample complexities can be quadratically far apart. +Existing results therefore do not inform sample complexity when 1 ≪ ϵ < ∞. This is not an artifact +of analysis: the optimal tests in the low and high privacy regimes are fundamentally different. +The large-ϵ regime has been increasingly used in practice, due to privacy amplification provided +by shuffling [CSUZZ19; BEMMRLRKTS17; FMT21]. Our paper makes progress on the computa- +tional and statistical fronts in the large-ϵ regime, as will be highlighted in Section 1.3 below. +1As shown in Kairouz, Oh, and Viswanath [KOV16], for simple binary hypothesis testing, we can take Y to be X, +with the same sample complexity (up to constant factors); see Fact 2.7. +2 + +1.1 +Problem Setup +For a natural number k, we use [k] to denote the set {1, 2, . . . , k}. In our paper, we focus on the +private-coin, non-interactive protocol.2 As we will be working with both privacy and communication +constraints in this paper, we first define the generic protocol for distributed inference under an +arbitrary set of channels C below: +Definition 1.2 (Simple binary hypothesis testing under channel constraints). Let X and Y be two +countable sets. Let C be a set of channels from X to Y, and let p and q be two distributions on X. +Let {Ui}n +i=1 denote a set of n users who choose channels {Ti}n +i=1 ∈ Cn according to a deterministic +rule3 R : [n] → C. Each user Ui then observes Xi and generates Yi = Ti(Xi) independently, where +X1, . . . , Xn is a sequence of i.i.d. random variables drawn from an (unknown) r ∈ {p, q}. +The +central server U0 observes (Y1, . . . , Yn) and constructs an estimate �r = φ(Y1, . . . , Yn), for some test +φ : ∪∞ +i=1Yi → {p, q}. We refer to this problem as simple binary hypothesis testing under channel +constraints. +In the non-interactive setup, we can assume that all Ti’s are identical equal to some T, as it +will increase the sample complexity by at most a constant factor [PJL22] (cf. Fact 2.7). We now +specialize the setting of Definition 1.2 to the case of LDP constraints: +Definition 1.3 (Simple binary hypothesis testing under LDP constraints). Consider the problem +in Definition 1.2 with Y = N, where C is the set of all ϵ-LDP channels from X to Y. We denote +this problem by B(p, q, ϵ). For a given test-rule pair (φ, R) with φ : ∪∞ +j=1Yj → {p, q}, we say that +(φ, R) solves B(p, q, ϵ) with sample complexity n if +P(X1,...,Xn)∼p⊗n(φ(Y1, . . . , Yn) ̸= p) + P(X1,...,Xn)∼q⊗n(φ(Y1, . . . , Yn) ̸= q) ≤ 0.1. +(1) +We use n∗(p, q, ϵ) to denote the sample complexity of this task, i.e., the smallest n so that there +exists a (φ, R)-pair that solves B(p, q, ϵ). We use B(p, q) and n∗(p, q) to refer to the setting of non- +private testing, i.e., when ϵ = ∞, which corresponds to the case when C is the set of all possible +channels from X to Y. +For any fixed rule R, the optimal choice of φ corresponds to the likelihood ratio test on {Yi}n +i=1. +Thus, in the rest of this paper, our focus will be optimizing the rule R, with the choice of φ made +implicitly. In fact, we can take Y to be X, at the cost of a constant-factor increase in the sample +complexity [KOV16] (cf. Fact 2.7). +We now define the threshold for free privacy, in terms of a large enough universal constant +Cthresh > 0 which can be explicitly deduced from our proofs: +Definition 1.4 (Threshold for free privacy). We define ϵ∗(p, q) (also denoted by ϵ∗ when the context +is clear) to be the smallest ϵ such that n∗(p, q, ϵ) ≤ Cthresh · n∗(p, q); i.e., for all ϵ ≥ ϵ∗(p, q), we can +obtain ϵ-LDP without any substantial increase in sample complexity compared to the non-private +setting. +Next, we study the problem of simple hypothesis testing under both privacy and communication +constraints. By communication constraints, we mean that the channel T maps from X to [ℓ] for +some ℓ ∈ N, which is potentially much smaller than |X|. +2We refer the reader to Acharya, Canonne, Liu, Sun, and Tyagi [ACLST22] for differences between various +protocols. +3When C is a convex set of channels, as will be the case in this paper, the deterministic rules are equivalent to +randomized rules (with independent randomness). +3 + +Definition 1.5 (Simple binary hypothesis testing under LDP and communication constraints). +Consider the problem in Definition 1.2 and Definition 1.3, with C equal to the set of all channels +that satisfy ϵ-LDP and Y = [ℓ]. We denote this problem by B(p, q, ϵ, ℓ), and use n∗(p, q, ϵ, ℓ) to +denote its sample complexity. +Communication constraints are worth studying not only for their practical relevance in dis- +tributed inference, but also for their potential to simplify algorithms without significantly impacting +performance. Indeed, the sample complexities of simple hypothesis testing with and without com- +munication constraints are almost identical [BNOP21; PJL22] (cf. Fact 2.8), even for a single-bit +(ℓ = 2) communication constraint. As we explain later, a similar statement can be made for privacy +constraints, as well. +1.2 +Existing Results +As noted earlier, the problem of simple hypothesis testing with just communication constraints +was addressed in Pensia, Jog, and Loh [PJL22]. Since communication and privacy constraints are +the most popular information constraints studied in the literature, the LDP-only and LDP-with- +communication-constraints settings considered in this paper are natural next steps. Many of our +results, particularly those on minimax-optimal sample complexity bounds, are in a similar vein as +those in Pensia, Jog, and Loh [PJL22]. Before describing our results, let us briefly mention the most +relevant prior work. We discuss further related work in Section 1.4. +Existing results on sample complexity. +Existing results (cf. Duchi, Jordan, and Wainwright [DJW18, +Theorem 1] and Asoodeh and Zhang [AZ22, Theorem 2]) imply that +n∗(p, q, ϵ) ≳ +� +� +� +� +� +� +� +� +� +1 +ϵ2 · d2 +TV(p,q), +if ϵ ∈ (0, 1], +1 +eϵ · d2 +TV(p,q), +if eϵ ∈ +� +e, d2 +h(p,q) +d2 +TV(p,q) +� +, +1 +d2 +h(p,q), +if eϵ > +d2 +h(p,q) +d2 +TV(p,q). +(2) +An upper bound on the sample complexity can be obtained by choosing a specific private channel +T and analyzing the resulting test. A folklore result (see, for example, Joseph, Mao, Neel, and +Roth [JMNR19, Theorem 5.1]) shows that setting T = TRR × TScheffe, where TScheffe maps X to +{0, 1} using a threshold rule based on p(x) +q(x), and TRR is the binary-input binary-output randomized +response channel, gives n∗(p, q, ϵ) ≲ +1 +min(1,ϵ2) · d2 +TV(p,q). This shows that when ϵ ∈ (0, 1] (or (0, c], for +some constant c), the lower bound is tight up to constants. Observe that for any ℓ ≥ 2, the sample +complexity with privacy and communication constraints n∗(p, q, ϵ, ℓ) also satisfies the same lower +and upper bounds, since the channel T has only two outputs. +However, the following questions remain unanswered: +What is the optimal sample complexity for ϵ ≫ 1? In particular, are the existing +lower bounds (2) tight? What is the threshold for free privacy? +In Section 1.3.1, we establish minimax-optimal bounds on the sample complexity for all values +of ϵ, over sets of distribution pairs with fixed total variation distance and Hellinger divergence. +In particular, we show that the lower bounds (2) are tight for binary distributions, but may be +arbitrarily loose for general distributions. +4 + +Existing results on computationally efficient algorithms. +Recall that each user needs to +select a channel T to optimize the sample complexity. Once T is chosen, the optimal test is simply +a likelihood ratio test between Tp and Tq. Thus, the computational complexity lies in determining +T. As noted earlier, for ϵ ≤ 1, the optimal channel is T = TRR ×TScheffe, and this can be computed +efficiently. However, this channel T may no longer be optimal in the regime of ϵ ≫ 1. +As with statistical rates, prior literature on finding optimal channels for ϵ ≫ 1 is scarce. Existing +algorithms either take time exponential in the domain size [KOV16], or their sample complexity is +suboptimal by polynomial factors (depending on +1 +d2 +TV(p,q), as opposed to +1 +d2 +h(p,q)). This raises the +following natural question: +Is there a polynomial-time algorithm that finds a channel T whose sample +complexity is (nearly) optimal? +We answer this question in the affirmative in Section 1.3.3. +1.3 +Our Results +We are now ready to describe our results in this paper, which we outline in the next three subsections. +In particular, Section 1.3.1 focuses on the sample complexity of simple hypothesis testing under local +privacy, Section 1.3.2 focuses on structural properties of the extreme points of the joint range under +channel constraints, and Section 1.3.3 states our algorithmic guarantees. +1.3.1 +Statistical Rates +We begin by analyzing the sample complexity when both p and q are binary distributions. We prove +the following result in Section 3.1, showing that the existing lower bounds (2) are tight for binary +distributions: +Theorem 1.6 (Sample complexity of binary distributions). Let p and q be two binary distributions. +Then +n∗(p, q, ϵ) ≍ +� +� +� +� +� +� +� +� +� +1 +ϵ2 · d2 +TV(p,q), +if ϵ ≤ 1, +1 +eϵ · d2 +TV(p,q), +if eϵ ∈ +� +e, d2 +h(p,q) +d2 +TV(p,q) +� +, +1 +d2 +h(p,q), +if eϵ > +d2 +h(p,q) +d2 +TV(p,q). +(3) +In particular, the threshold ϵ∗ for free privacy (Definition 1.4) satisfies eϵ∗ ≍ +d2 +h(p,q) +d2 +TV(p,q). Note that +the sample complexity n∗(p, q, ϵ) for all ranges of ϵ is completely characterized by the total variation +distance and Hellinger divergence between p and q. A natural set to consider is all distribution +pairs (not just those with binary support) with a prescribed total variation distance and Hellinger +divergence; we investigate minimax-optimal sample complexity over this set. Our next result shows +that removing the binary support condition radically changes the sample complexity, even if the +total variation distance and Hellinger divergence are the same. Specifically, we show that there are +ternary distribution pairs whose sample complexity (as a function of the total variation distance +and Hellinger divergence) is significantly larger than the corresponding sample complexity for binary +distributions. +Theorem 1.7 (Sample complexity lower bound for general distributions). For any ρ ∈ (0, 0.5) and +ν ∈ (0, 0.5) such that 2ν2 ≤ ρ ≤ ν, there exist ternary distributions p and q such that d2 +h(p, q) = ρ, +5 + +dTV(p, q) = ν, and the sample complexity behaves as +n∗(p, q, ϵ) ≍ +� +� +� +� +� +� +� +1 +ϵ2 · d2 +TV(p,q), +if ϵ ≤ 1, +min +� +1 +d2 +TV(p,q), +1 +eϵ · d4 +h(p,q) +� +, +if eϵ ∈ +� +e, +1 +d2 +h(p,q) +� +, +1 +d2 +h(p,q), +if eϵ > +1 +d2 +h(p,q). +(4) +We prove this result in Section 3.2. +Remark 1.8. We highlight the differences between the sample complexity in the binary setting (cf. +equation (3)) and the worst-case general distributions (cf. equation (4)) below (also see Figure 1): +1. (Relaxing privacy may not lead to significant improvements in accuracy.) In equation (4), +there is an arbitrarily large range of ϵ where the sample complexity remains roughly constant. +In particular, when e ≤ eϵ ≲ d2 +TV(p,q) +d4 +h(p,q) , the sample complexity of hypothesis testing remains +roughly the same (up to constants). That is, we are sacrificing privacy without any significant +gains in statistical efficiency. This is in stark contrast to the binary setting, where increasing +eϵ by a large constant factor leads to a constant-factor improvement in sample complexity. +2. (The threshold for free privacy is larger.) Let ϵ∗ := ϵ(p, q) be the threshold for free privacy (cf. +Definition 1.4). In the binary setting, one has eϵ∗ ≍ +d2 +h(p,q) +d2 +TV(p,q), whereas for general distributions, +one may need eϵ∗ ≳ +1 +d2 +h(p,q). The former ϵ∗ can be arbitrarily smaller than the latter. +To complement the result above, which provides a lower bound on the sample complexity for +worst-case distributions, our next result provides an upper bound on the sample complexity that +nearly matches the rates (up to logarithmic factors) for arbitrary distributions. +Moreover, the +proposed algorithm uses an ϵ-LDP channel with binary outputs. The following result is proved in +Section 3.3: +Theorem 1.9 (Sample complexity upper bounds and an efficient algorithm for hypothesis testing +for general distributions). Let p and q be two distributions on [k]. Let ϵ > 0. Then the sample +complexity behaves as +n∗(p, q, ϵ) ≲ +� +� +� +� +� +� +� +1 +ϵ2 · d2 +TV(p,q), +if ϵ ≤ 1, +min +� +1 +d2 +TV(p,q), +α2 +eϵ · d4 +h(p,q) +� +, +if eϵ ∈ +� +e, +α +d2 +h(p,q) +� +, +α +d2 +h(p,q), +if eϵ > +α +d2 +h(p,q), +(5) +where α ≲ log(1/d2 +h(p, q)) ≍ log (n∗(p, q)). +Moreover, the rates above are achieved by an ϵ-LDP channel T that maps [k] to [2] and can be +found in time polynomial in k, for any choice of p, q, and ϵ. +Theorems 1.7 and 1.9 imply that the above sample complexity is minimax optimal (up to +logarithmic factors) over the class of distributions with total variation distance ν and Hellinger +divergence ρ satisfying the conditions in Theorem 1.7. We summarize this in the following theorem: +Theorem 1.10 (Minimax-optimal bounds). Let ρ ∈ (0, 0.5) and ν ∈ (0, 0.5) be such that 2ν2 ≤ ρ ≤ +ν. Let Sρ,ν be the set of all distribution pairs with discrete supports, with total variation distance +and Hellinger divergence being ν and ρ, respectively: +Sρ,ν := {(p, q) : k ∈ N, p ∈ ∆k, q ∈ ∆k, dTV(p, q) = ν, d2 +h(p, q) = ρ}. +6 + +Let n∗(Sρ,ν, ϵ) be the minimax-optimal sample complexity of hypothesis testing under ϵ-LDP con- +straints, defined as +n∗(Sρ,ν, ϵ) = min +(φ,R) +max +(p,q)∈Sρ,ν +n∗(p, q, ϵ), +for test-rule pairs (φ, R), as defined in Definition 1.3. Then +n∗(Sρ,ν, ϵ) = +� +� +� +� +� +� +� +�Θ +� +1 +ϵ2 · ν2 +� +, +if ϵ ≤ 1, +�Θ +� +min +� +1 +ν2 , +1 +eϵ · ρ2 +�� +, +if eϵ ∈ +� +e, 1 +ρ +� +, +�Θ +� +1 +ρ +� +, +if eϵ > 1 +ρ. +(6) +Here, the �Θ notation hides poly-logarithmic factors in 1/ν and 1/ρ. +Remark 1.11. A version of the above theorem may also be stated for privacy and communication +constraints, by defining +n∗(Sρ,ν, ϵ, ℓ) = min +(φ,R) +max +(p,q)∈Sρ,ν +n∗(p, q, ϵ, ℓ). +In fact, it may seen that the same sample complexity bounds continue to hold for n∗(Sρ,ν, ϵ, ℓ), with +ℓ ≥ 2, since the lower bound in Theorem 1.7 continues to hold with communication constraints, as +does the upper bound in Theorem 1.9, which uses a channel with only binary outputs. +Remark 1.12. The above theorem mirrors a minimax optimality result for communication-constrained +hypothesis testing from Pensia, Jog, and Loh [PJL22]. There, the set under consideration was Sρ, +where ρ is the Hellinger divergence between the distribution pair, and the minimax-optimal sample +complexity was shown to be �Θ(1/ρ) even for a binary communication constraint. +Finally, we consider the threshold for free privacy ϵ∗ for general distributions; see Definition 1.4. +Observe that Theorem 1.9 does not provide any upper bounds on ϵ∗, since the sample complexity +in Theorem 1.9 is bounded away from n∗(p, q), due to the logarithmic multiplier α. Recall that +Theorem 1.7 implies eϵ∗ ≳ +1 +d2 +h(p,q) in the worst case. Our next result, proved in Section 3.3, shows +that this is roughly tight, and eϵ∗ ≲ +1 +d2 +h(p,q) · log +� +1 +d2 +h(p,q) +� +for all distributions: +Theorem 1.13. Let p and q be two distributions on [k], and let eϵ ≳ +1 +d2 +h(p,q) log +� +1 +d2 +h(p,q) +� +. Then +n∗(p, q, ϵ) ≍ n∗(p, q). Moreover, there is a channel T achieving this sample complexity that maps [k] +to a domain of size ⌈log(n∗(p, q))⌉, and which can be computed in poly(k, log(⌈n∗(p, q)⌉)) time. +We thereby settle the question of minimax-optimal sample complexity (up to logarithmic fac- +tors) for simple binary hypothesis testing under LDP-only and LDP-with-communication constraints +(over the class of distributions with a given total variation distance and Hellinger divergence). More- +over, the minimax-optimal upper bounds are achieved by computationally efficient, communication- +efficient algorithms. However, there can be a wide gap between instance-optimal and minimax- +optimal procedures; in the next two subsections, we present structural and computational results +for instance-optimal algorithms. +7 + +100 +101 +102 +103 +104 +105 +106 +107 +108 +109 +eϵ +108 +109 +1010 +1011 +min +T∈Pϵ +1 +d2 +h(Tp, Tp) +1 +d2 +h(p,q) +1 +d2 +TV(p,q) +1 +d2 +h(p,q) +d2 +h(p,q) +d2 +TV(p,q) +d2 +TV(p,q) +d4 +h(p,q) +Contraction of Hellinger Divergence under LDP +Binary p and q +Ternary p and q +Figure 1: In this plot, we show the difference between the behavior of sample complexity under ϵ- +LDP constraints for binary distributions and (worst-case) ternary distributions from Theorem 1.7. +We take two pairs of distributions (p, q)—one pair of binary distributions (shown in blue, with +marker ◦) and one pair of ternary distributions (shown in orange, with marker +)—such that +the two pairs have Hellinger divergence d2 +h(p, q) = 10−8 and total variation distance dTV(p, q) = +10−5. For each value of ϵ, shown on the horizontal axis after being mapped to eϵ, we compute +minT∈Pϵ 1/d2 +h(Tp, Tq), where Pϵ is the set of all ϵ-LDP channels, and plot it on vertical axis. Thus, +the vertical axis characterizes the sample complexity n∗(p, q, ϵ) of simple binary hypothesis testing +between p and q with privacy constraints, up to constant factors (cf. Fact 2.7). Both axes are shown +in log-scale here. Since the total variation distance between the two pairs is identical, we see that +their curves overlap for small ϵ (ϵ ≪ 1, which is consistent with the fact that n∗(p, q, ϵ) ≍ +1 +ϵ2d2 +TV(p,q) +for small ϵ). As predicted by Theorem 1.6, the curve for binary distributions decreases rapidly for +ϵ ≫ 1 until it saturates at 1/d2 +h(p, q). Moreover, for eϵ ≍ d2 +h(p, q)/d2 +TV(p, q), the predicted threshold +for free privacy, the vertical axis is within constant factors of its asymptotic value, as predicted. On +the other hand, the curve for ternary distributions seems to have three different phases, as predicted +by Theorem 1.7: (i) for small ϵ, it behaves as 1/(ϵ2d2 +TV(p, q)); (ii) for moderate values of ϵ, such +that e ≪ eϵ ≪ d2 +TV(p,q) +d4 +h(p,q) , it remains stagnant roughly at +1 +d2 +TV(p,q); and (iii) for eϵ ≫ d2 +TV(p,q) +d4 +h(p,q) , the curve +decreases rapidly until it approaches 1/d2 +h(p, q). The phase (ii) corresponds to the phenomenon that +we are leaking privacy without any gains in statistical efficiency. Finally, eϵ needs to be as large +as 1/d2 +h(p, q) for the vertical axis to be within a factor of 10 of its asymptotic value. We refer the +reader to Remark 1.8 for more details. +8 + +1.3.2 +Structure of Extreme Points under the Joint Range +In this section, we present results for the extreme points of the joint range of an arbitrary pair of +distributions when transformed by a set of channels. Formally, if C is a convex set of channels from +X to Y, and p and q are two distributions on X, we are interested in the extreme points of the +set A := {(Tp, Tq) : T ∈ C}, which is a convex subset of ∆|Y| × ∆|Y|.4 The extreme points of a +convex set are naturally insightful for maximizing quasi-convex functions, and we will present the +consequences of the results in this section in Section 1.3.3. +We consider two choices of C: first, when C is the set of all channels from X to Y = [ℓ], and +second, when C is the set of all ϵ-LDP channels from X to Y = [ℓ]. We use Tℓ,k to denote the set of +all channels that map from [k] to [ℓ]. +The following class of deterministic channels plays a critical role in our theory: +Definition 1.14 (Threshold channels). For some k ∈ N, let p and q be two distributions on [k]. For +any ℓ ∈ N, a deterministic channel T ∈ Tℓ,k is a threshold channel if the following property holds +for every u, v ∈ [k]: If p(u) +q(u) < p(v) +q(v) and T(u) = T(v), then any w ∈ [k] such that p(w) +q(w) ∈ +� +p(u) +q(u), p(v) +q(v) +� +satisfies T(w) = T(u)(= T(v)). (The likelihood ratios are assumed to take values on the extended +real line; i.e., on R ∪ {−∞, +∞}.) +Remark 1.15. Threshold channels are intuitively easy to understand when all the likelihood ratios +are distinct (this may be assumed without loss of generality in our paper, as explained later): +Arrange the inputs in increasing order of their likelihood ratios and partition them into ℓ contiguous +blocks. Thus, there are at most kℓ such threshold channels (up to reordering of output labels). +Our first result proved in Section 4 is for the class of communication-constrained channels, and +shows that all extreme points of the joint range are obtained using deterministic threshold channels: +Theorem 1.16 (Extreme points of the joint range under communication constraints). Let p and q +be two distributions on [k]. Let A be the set of all pairs of distributions that are obtained by passing +p and q through a channel of output size ℓ, i.e., +A = {(Tp, Tq) : T ∈ Tℓ,k}. +If (Tp, Tq) is an extreme point of A, then T is a threshold channel. +We note that the above result is quite surprising: (Tp, Tq) is extreme point of A only if T +is an extreme point of Tℓ,k (i.e., a deterministic channel), but Theorem 1.16 demands that T be +a deterministic threshold channel, meaning it lies in a very small subset of deterministic channels. +Indeed, even for ℓ = 2, the number of deterministic channels from [k] to [2] is 2k, whereas the number +of threshold channels is just 2k. We note that the result above is similar in spirit to Tsitsiklis [Tsi93, +Proposition 2.4]. However, the focus there was on a particular objective, the probability of error in +simple hypothesis testing, with non-identical channels for users. Our result is for identical channels +and is generally applicable to quasi-convex objectives, as mentioned later. +We now consider the case where C is the set of ϵ-LDP channels from [k] to [ℓ]. Since C is a set of +private channels, it does not contain any deterministic channels (thus, does not contain threshold +channels). Somewhat surprisingly, we still show that the threshold channels play a fundamental +role in the extreme points of the joint range under C. The following result shows that any extreme +point of the joint range A can be obtained by a threshold channel mapping into [2ℓ2], followed by +an ϵ-LDP channel from [2ℓ2] to [ℓ]: +4For k ∈ N, we use ∆k to denote the probability simplex on a domain of alphabet size k. +9 + +Theorem 1.17 (Extreme points of the joint range under privacy and communication constraints). +Let p and q be distributions on [k]. Let C be the set of ϵ-LDP channels from [k] to [ℓ]. Let A be the +set of all pairs of distributions that are obtained by applying a channel from C to p and q, i.e., +A = {(Tp, Tq) | T ∈ C}. +(7) +If (Tp, Tq) is an extreme point of A for T ∈ C, then T can be written as T = T2 × T1 for some +threshold channel T1 ∈ T2ℓ2,k and some T2 an extreme point of the set of ϵ-LDP channels from [2ℓ2] +to [ℓ]. +We prove this structural result in Section 5, which leads to polynomial-time algorithms for +constant ℓ for maximizing quasi-convex functions, as mentioned in Section 1.3.3. +1.3.3 +Computationally Efficient Algorithms for Instance Optimality +The results from the previous sections characterized the minimax-optimal sample complexity, but +did not address instance optimality. +Instance-optimal performance may be substantially better +than minimax-optimal performance, as seen by comparing the instance-optimal bounds for binary +distributions to the minimax-optimal bounds for general distributions. In this section, we focus +on identifying an instance-optimal channel T (satisfying the necessary constraints) for a given pair +(p, q) of distributions. +Let p and q be fixed distributions over [k]. Let Pϵ +ℓ,k be the set of all ϵ-LDP channels from [k] +to [ℓ], and let Tℓ,k be the set of all channels from [k] to [ℓ]. Let C ∈ {Pϵ +ℓ,k, Tℓ,k}. As before, define +A = {(Tp, Tq) : T ∈ C}. Let g : A → R be a (jointly) quasi-convex function; i.e., for all t ∈ R, the +sublevel sets {(p′, q′) : g(p′, q′) ≤ t} are convex. In this paper, we are primarily interested in functions +corresponding to divergences between the distribution pair. So, unless otherwise mentioned, we +shall assume the quasi-convex functions g in this paper are permutation-invariant; i.e., g(p′, q′) = +g(Πp, Πq) for all permutation matrices Π. However, our algorithmic results will continue to hold +even without this assumption, with an additional factor of ℓ! in the time complexity. +We will +consider the problem of identifying T that solves +max +T∈C g(Tp, Tq). +The quasi-convexity of g implies that the maximum is attained at some T such that (Tp, Tq) is an +extreme point of A. We can thus leverage the results from Section 1.3.2 to search over the subset +of channels satisfying certain structural properties. +Identifying T that maximizes the Hellinger divergence leads to an instance-optimal test for min- +imizing sample complexity for testing between p and q with channel constraints C: This is because +if each user chooses the channel T, the resulting sample complexity will be Θ +� +1 +d2 +h(Tp,Tq) +� +. Thus, the +instance-optimal sample complexity will be obtained by a channel T that attains maxT∈C d2 +h(Tp, Tq). +Note that the Hellinger divergence is convex (and thus quasi-convex) in its arguments. Apart from +the Hellinger divergence, other functions of interest such as the Kullback–Leibler divergence or +Chernoff information (which are also convex) characterize the asymptotic error rates in hypothe- +sis testing, so finding T for these functions identifies instance-optimal channels in the asymptotic +(large-sample) regime. Other potential functions of interest include Rényi divergences of all orders, +which are quasi-convex, but not necessarily convex [EH14]. +As mentioned earlier, the results of Kairouz, Oh, and Viswanath [KOV16] give a linear program +with 2k variables to find an instance-optimal channel under privacy constraints, which is computa- +tionally prohibitive. It is also unclear if their result extends when the channels are further restricted +10 + +to have communication constraints in addition to privacy constraints. We now show how to im- +prove on the guarantees of Kairouz, Oh, and Viswanath [KOV16] in the presence of communication +constraints, using the structural results from the previous subsection. +Corollary 1.18 (Computationally efficient algorithms for maximizing quasi-convex functions). Let +p and q be fixed distributions over [k], let C ∈ {Tℓ,k, Pϵ +ℓ,k}, and let A = {(Tp, Tq) : T ∈ C}. Let +g : A → R be a jointly quasi-convex function. When C = Tℓ,k, there is an algorithm that solves +maxT∈C g(Tp, Tq) in time polynomial in kℓ. When C = Pϵ +ℓ,k, there is an algorithm that solves +maxT∈C g(Tp, Tq) in time polynomial in kℓ2 and 2ℓ3 log ℓ. +We prove Corollary 1.18 in Section 4 and Section 5.3 for C = Tℓ,k and C = Pϵ +ℓ,k, respectively. +Remark 1.19. When ℓ is constant, we obtain a polynomial-time algorithm for maximizing any +quasi-convex function under Tℓ,k or Pϵ +ℓ,k channel constraints. When C = Tℓ,k and g is the Kullback– +Leibler divergence, this exactly solves (for small ℓ) a problem introduced in Carpi, Garg, and +Erkip [CGE21], which proposed a polynomial-time heuristic. +Applying the above result to the Hellinger divergence d2 +h, we obtain the following result for +simple binary hypothesis testing, proved in Section 5.3: +Corollary 1.20 (Computationally efficient algorithms for instance-optimal results under commu- +nication constraints). Let p and q be two distributions on [k]. For any ϵ and any integer ℓ > 1, there +is an algorithm that runs in time polynomial in kℓ2 and 2ℓ3 log ℓ and outputs an ϵ-LDP channel T +mapping from [k] to [ℓ], such that if N denotes the sample complexity of hypothesis testing between +p and q when each individual uses the channel T, then N ≍ n∗(p, q, ϵ, ℓ). +In particular, the sample complexity with T satisfies +N ≲ n∗(p, q, ϵ) · +� +1 + log (n∗ (p, q, ϵ)) +ℓ +� +. +(8) +The channel T may be decomposed as a deterministic threshold channel to a domain of size [2ℓ2], +followed by an ϵ-LDP channel from [2ℓ2] to [ℓ]. +Thus, by choosing ℓ = 2, we obtain a polynomial-time algorithm with nearly instance-optimal +sample complexity (up to logarithmic factors) under just ϵ-LDP constraints. +1.4 +Related Work +Distributed estimation has been studied extensively under resource constraints such as memory, +privacy, and communication. Typically, this line of research considers problems of interest such as +distribution estimation [RT70; LR86; CKO21; BHÖ20], identity or independence testing [ACT20a; +ACT20b; ACFST21], and parameter estimation [Hel74; DJWZ14; DJW18; BGMNW16; DR19; +BCÖ20; DKPP22], and identifies minimax-optimal bounds on the error or sample complexity. In +what follows, we limit our discussion to related work on hypothesis testing under resource con- +straints. +For memory-constrained hypothesis testing, the earliest works in Cover [Cov69] and Hellman and +Cover [HC73] derived tight bounds on the memory size needed to perform asymptotically error-free +testing. Hellman and Cover [HC71] also highlighted the benefits of randomized algorithms. These +benefits were also noted in recent work by Berg, Ordentlich, and Shayevitz [BOS20], which consid- +ered the error exponent in terms of the memory size. Recently, Braverman, Garg, and Zamir [BGZ22] +showed tight bounds on the memory size needed to test between two Bernoulli distributions. +11 + +Communication-constrained hypothesis testing has two different interpretations. In the informa- +tion theory literature, Berger [Ber79], Ahlswede and Csiszár [AC86], and Amari and Han [AH98] +considered a family of problems where two nodes, one which only observes Xi’s and the other which +only observes Yi’s, try to distinguish between PXY and QXY . Communication between the nodes +occurs over rate-limited channels. The second interpretation, also called “decentralized detection” +in Tsitsiklis [Tsi88], is more relevant to this work. Here, the observed Xi’s are distributed amongst +different nodes (one observation per node) that communicate a finite number of messages (bits) +to a central node, which needs to determine the hypothesis. Tsitsiklis [Tsi88; Tsi93] identified the +optimal decision rules for individual nodes and considered asymptotic error rates in terms of the +number of bits. These results were recently extended to the nonasymptotic regime in Pensia, Jog, +and Loh [PJL22; PLJ22]. +Privacy-constrained hypothesis testing has been studied in the asymptotic and nonasymptotic +regimes under different notions of privacy. The local privacy setting, which is relevant to this paper, +is similar to the decentralized detection model in Tsitsiklis [Tsi93], except that the each node’s +communication to the central server is private. This is achieved by passing observations through +private channels. Liao, Sankar, Calmon, and Tan [LSCT17; LSTC17] considered maximizing the +error exponent under local privacy notions defined via maximal leakage and mutual information. +Sheffet [She18] analyzed the performance of the randomized response method for LDP for hypoth- +esis testing. Gopi, Kamath, Kulkarni, Nikolov, Wu, and Zhang [GKKNWZ20] showed that M-ary +hypothesis testing under pure LDP constraints requires exponentially more samples (Ω(M) instead +of O(log M)). Closely related to the instance-optimal algorithms in our paper, Kairouz, Oh, and +Viswanath [KOV16] presented an algorithm to find LDP channels that maximize the output diver- +gence for two fixed probability distributions at the channel input; the proposed algorithm runs in +time exponential in the domain size of the input distributions.5 Note that divergences are directly +related to error exponents and sample complexities in binary hypothesis testing. The results of +Kairouz, Oh, and Viswanath [KOV16] on extreme points of the polytope of LDP channels were +strengthened in Holohan, Leith, and Mason [HLM17], which characterized the extreme points in +special cases. We were able to find only two other papers that consider instance optimality, but +in rather special settings [GGKMZ21; AFT22]. For simple binary hypothesis testing in the global +differential privacy setting, Canonne, Kamath, McMillan, Smith, and Ullman [CKMSU19] identified +the optimal test and corresponding sample complexity. Bun, Kamath, Steinke, and Wu [BKSW19] +showed that O(log M) samples are enough for M-ary hypothesis testing in the global differential +privacy setting. +1.5 +Organization +This paper is organized as follows: Section 2 records standard results. Section 3 focuses on the +sample complexity of hypothesis testing under privacy constraints. Section 4 considers extreme +points of the joint range under communication constraints. Section 5 characterizes the extreme +points under both privacy and communication constraints. +Section 6 explores other notions of +privacy beyond pure LDP. Finally, we conclude with a discussion in Section 7. We defer proofs of +some intermediate results to the appendices. +5We remark, however, that the algorithm in Kairouz, Oh, and Viswanath [KOV16] is applicable to a wider class +of objective functions, which they term “sublinear.” +12 + +2 +Preliminaries and Facts +Notation: +Throughout this paper, we will focus on discrete distributions. For a natural number +k ∈ N, we use [k] to denote the set {1, . . . , k} and ∆k to denote the set of distributions over +[k]. +We represent a probability distribution p ∈ ∆k as a vector in Rk. +Thus, pi denotes the +probability of element i under p. Given two distributions p and q, let dTV(p, q) := 1 +2 +� +i |pi −qi| and +d2 +h(p, q) := � +i(√pi − √qi)2 denote the total variation distance and Hellinger divergence between p +and q, respectively. +We denote channels with bold letters such as T. As the channels between discrete distributions +can be represented by rectangular column-stochastic matrices (each column is nonnegative and +sums to one), we also use bold capital letters, such as T, to denote the corresponding matrices. In +particular, if a channel T is from [k] to [ℓ], we denote it by an ℓ × k matrix, where each of the k +columns is in ∆ℓ. In the same vein, for a column index c ∈ [k] and a row index r ∈ [ℓ], we use T(r, c) +to refer to the entry at the corresponding location. For a channel T : X → Y and a distribution p +over X, we use Tp to denote the distribution over Y when X ∼ p passes through the channel T. +In the notation above, when p is a distribution over [k], represented as a vector in Rk, and T is a +channel from [k] → [ℓ], represented as a matrix T ∈ Rℓ×k, the output distribution Tp corresponds +to the usual matrix-vector product. We shall also use T to denote the stochastic map transforming +the channel input X to the channel output Y = T(X). Similarly, for two channels T1 and T2 +from [k1] to [k2] and [k2] to [k3], respectively, the channel T3 from [k1] to [k3] that corresponds to +applying T2 to the output of T1 is equal to the matrix product T2 × T1. +Let Tℓ,k be the set of all channels that map from [k] to [ℓ]. +We use T thresh +ℓ,k +to denote the +subset of Tℓ,k that corresponds to threshold channels (cf. Definition 1.14). We use Pϵ +ℓ,k to denote +the set of all ϵ-LDP channels from [k] to [ℓ]. Recall that for two distributions p and q, we use +n∗(p, q, ϵ) (respectively, n∗(p, q, ϵ, ℓ)) to denote the sample complexity of simple binary hypothesis +testing under privacy constraints (respectively, both privacy and communication constraints). +For a set A, we use conv(A) to denote the convex hull of A. For a convex set A, we use ext(A) +to denote the set of extreme points of A. Finally, we use the following notations for simplicity: (i) +≲, ≳, and ≍ to hide positive constants, and (ii) the standard asymptotic notation O(·), Ω(·), and +Θ(·). Finally, we use �O(·), �Ω(·), and �Θ to hide poly-logarithmic factors in their arguments. +2.1 +Convexity +We refer the reader to Bertsimas and Tsitsiklis [BT97] for further details. We will use the following +facts repeatedly in the paper, often without mentioning them explicitly: +Fact 2.1 (Extreme points of linear transformations). Let A be a convex, compact set in a finite- +dimensional space. Let T be a linear function on A, and define the set A′ := {Tx : x ∈ A}. Then +A′ is convex and compact, and ext(A′) ⊆ {Tx : x ∈ ext(A)}. +Fact 2.2. Let A be a convex, compact set. If A = conv(B) for some set B, then ext(A) ⊆ B. +Fact 2.3 (Number of vertices and vertex enumeration). Let A ⊆ Rn be a bounded polytope defined +by m linear inequalities. The number of vertices of A is at most +�m +n +� +. Moreover, there is an algorithm +that takes eO(n log m) time and output all the vertices of A.6 +Fact 2.4 (Extreme points of channels). The set of extreme points of Tℓ,k is the set of all deterministic +channels from [k] to [ℓ]. +6Throughout this paper, we assume the bit-complexity of linear inequalities is bounded. +13 + +2.2 +Local Privacy +We state standard facts from the privacy literature here. +Definition 2.5 (Randomized response). For an integer k ≥ 2, the k-ary randomized response +channel with privacy parameter ϵ is a channel from [k] to [k] defined as follows: for any i ∈ [k], +T(i) = i with probability +eϵ +(k−1)+eϵ and T(i) = j with probability +1 +(k−1)+eϵ , for any j ∈ [k]\{i}. The +standard randomized response [War65] corresponds to k = 2, which we denote by Tϵ +RR. We omit ϵ +in the superscript when it is clear from context. +We will also use the following result on the extreme points for Theorem 1.7. +Fact 2.6 (Extreme points of the LDP polytope in special cases [HLM17]). We mention all the +extreme points of Pϵ +ℓ,k (up to permutation of rows and columns; if a channel is an extreme point, +then any permutation of rows and/or columns is an extreme point) below for some special cases. +1. (Trivial extreme points) A channel with one row of all ones and the rest of the rows with zero +values is always an extreme point of Pϵ +ℓ,k. We call such extreme points trivial. +2. (ℓ = 2 and k ≥ 2) All non-trivial extreme points of Pϵ +2,k are of the form (up to permutation +of rows): +� +a +a +· · · +a +1 − a +1 − a +· · · +1 − a +1 − a +1 − a +· · · +1 − a +a +a +· · · +a +� +, +where a/(1 − a) = eϵ. In other words, the columns are of only two types, containing a and +1 − a. +3. (ℓ = 3 and k = 3) There are two types of non-trivial extreme points of Pϵ +3,3: one with two +nonzero rows and another with three nonzero rows. For the former, the nonzero rows are +exactly the extreme points of Pϵ +2,3. For the latter, two extreme points exist, of the following +form: +� +� +1 − 2a +a +a +a +1 − 2a +a +a +a +1 − 2a +� +� , +one with 1−2a +a += eϵ and one with +a +1−2a = eϵ. The case of 1−2a +a += eϵ corresponds to the usual +randomized response (Definition 2.5). +2.3 +Hypothesis Testing +In this section, we state some standard facts regarding hypothesis testing and divergences that will +be used repeatedly. +Fact 2.7 (Hypothesis testing and divergences; see, for example, Tsybakov [Tsy09]). Let p and q be +two arbitrary distributions. Then: +1. We have d2 +TV(p, q) ≤ d2 +h(p, q) ≤ 2dTV(p, q). +2. (Sample complexity of non-private hypothesis testing) We have n∗(p, q) ≍ +1 +d2 +h(p,q). +14 + +3. (Sample complexity in the high-privacy regime) For every ϵ ≤ 1, we have n∗(p, q, ϵ) ≍ +1 +ϵ2d2 +TV(p,q). See the references [DJW18, Theorem 1], [AZ22, Theorem 2], and [JMNR19, Theo- +rem 5.1]. +4. (Restricting the size of the output domain) Let p and q be distributions over [k]. +Then +n∗(p, q, ϵ) ≍ n∗(p, q, ϵ, k). This follows by applying Theorem 2 in Kairouz, Oh, and Viswanath +[KOV16] to d2 +h(·, ·). +5. (Choice of identical channels in Definition 1.2) Let T be a channel that maximizes d2 +h(Tp, Tq) +among all channels in C. Then the sample complexity of hypothesis testing under the channel +constraints of C is Θ +� +1 +d2 +h(Tp,Tq) +� +. See Lemma 4.2 in Pensia, Jog, and Loh [PJL22]. +Fact 2.8 (Preservation of Hellinger distance under communication constraints (Theorem 1 in Bhatt, +Nazer, Ordentlich, and Polyanskiy [BNOP21] and Corollary 3.4 in Pensia, Jog, and Loh [PJL22])). +Let p and q be two distributions on [k]. Then for any ℓ ∈ N, there exists a channel T from [k] to +[ℓ], which can be computed in time polynomial in k, such that +d2 +h(p, q) ≲ d2 +h(Tp, Tq) · +� +1 + +ℓ +min(k, log(1/d2 +h(p, q))) +� +. +(9) +Moreover, this bound is tight in the following sense: for every choice of ρ ∈ (0, 1), there exist two +distributions p and q such that d2 +h(p, q) ≍ ρ, and for every channel T ∈ Tℓ,k, the right-hand side of +inequality (9) is further upper-bounded by O(ρ). +3 +Locally Private Simple Hypothesis Testing +In this section, we provide upper and lower bounds for locally private simple hypothesis testing. +This section is organized as follows: In Section 3.1, we derive instance-optimal bounds when both +distributions are binary. We then prove minimax-optimal bounds for general distributions (with +support size at least three): Lower bounds on sample complexity are proved in Section 3.2 and upper +bounds in Section 3.3. Proofs of some of the technical arguments are deferred to the appendices. +3.1 +Binary Distributions and Instance-Optimality of Randomized Response +We first consider the special case when p and q are both binary distributions. Our main result +characterizes the instance-optimal sample complexity in this setting: +Theorem 1.6 (Sample complexity of binary distributions). Let p and q be two binary distributions. +Then +n∗(p, q, ϵ) ≍ +� +� +� +� +� +� +� +� +� +1 +ϵ2 · d2 +TV(p,q), +if ϵ ≤ 1, +1 +eϵ · d2 +TV(p,q), +if eϵ ∈ +� +e, d2 +h(p,q) +d2 +TV(p,q) +� +, +1 +d2 +h(p,q), +if eϵ > +d2 +h(p,q) +d2 +TV(p,q). +(3) +By Fact 2.7, the proof of Theorem 1.6 is a consequence of the following bound on the strong +data processing inequality for randomized responses: +15 + +Proposition 3.1 (Strong data processing inequality for Hellinger divergence). Let p and q be two +binary distributions. Then +max +T∈Pϵ +2,2 +d2 +h (Tp, Tq) ≍ +� +ϵ2 · d2 +TV(p, q), +if ϵ ≤ 1 +min +� +eϵ · d2 +TV(p, q), d2 +h(p, q) +� +, +otherwise. +Moreover, the maximum is achieved by the randomized response channel. +Proof. Let A = {(Tp, Tq) : T ∈ Pϵ +2,2} be the joint range of p and q under ϵ-LDP privacy constraints. +Since A is a convex set and d2 +h is a convex function over A, the maximizer of d2 +h in A is an extreme +point of A. Since A is a linear transformation of Pϵ +2,2, Fact 2.1 implies that any extreme point of A +is obtained by using a channel T corresponding to an extreme point of Pϵ +2,2. By Fact 2.6, the only +extreme point of Pϵ +2,2 is the randomized response channel Tϵ +RR. Thus, in the rest of the proof, we +consider T = Tϵ +RR. +By abusing notation, we will also use p and q to denote the probabilities of observing 1 under +the two respective distributions. Without loss of generality, we will assume that 0 ≤ p ≤ q and +p ≤ 1/2. We will repeatedly use the following claim, which is proved in Appendix D: +Claim 3.2 (Approximation for Hellinger divergence of binary distributions). Let p, q ∈ [0, 1]. Let +Ber(p) and Ber(q) be the corresponding Bernoulli distributions with min(p, q) ≤ 1/2. Then +d2 +h (Ber(p), Ber(q)) ≍ d2 +TV(Ber(p), Ber(q)) +max(p, q) +. +Applying Claim 3.2, we obtain +d2 +h(p, q) ≍ d2 +TV(p, q) +q +. +(10) +We know that the transformed distributions p′ := Tϵ +RRp and q′ := Tϵ +RRq are binary distributions; +by abusing notation, let p′ and q′ also be the corresponding real-valued parameters associated with +these binary distributions. By the definition of the randomized response, we have +p′ := p(eϵ − 1) + 1 +1 + eϵ +, +and +q′ := q(eϵ − 1) + 1 +1 + eϵ +. +(11) +Consequently, we have 0 ≤ p′ ≤ q′ and p′ ≤ 1/2. We directly see that +dTV(p′, q′) = q′ − p′ = (q − p)(eϵ − 1) +eϵ + 1 += dTV(p, q) · eϵ − 1 +eϵ + 1. +We now apply Claim 3.2 below to the distributions p′ and q′: +d2 +h(p′, q′) ≍ d2 +TV(p′, q′) +q′ += d2 +TV(p, q) · +�eϵ − 1 +eϵ + 1 +�2 +· +1 + eϵ +q(eϵ − 1) + 1 +(using equation (11)) +≍ d2 +TV(p, q) · (eϵ − 1)2 +eϵ + 1 +· min +� +1, +1 +q(eϵ − 1) +� +� +using +1 +a + b ≍ min +�1 +a, 1 +b +� +for a, b > 0 +� +≍ d2 +TV(p, q) · (eϵ − 1)2 +eϵ + 1 +· min +� +1, +d2 +h(p, q) +d2 +TV(p, q)(eϵ − 1) +� +� +using equation (10) and +16 + +min(1, a) ≍ min(1, b) if a ≍ b +� +≍ min +� +d2 +TV(p, q) · (eϵ − 1)2 +eϵ + 1 , d2 +h(p, q) · eϵ − 1 +eϵ + 1 +� +≍ +� +min +� +d2 +TV(p, q) · ϵ2 , d2 +h(p, q) · ϵ +� +, +if ϵ ≤ 1, +min +� +d2 +TV(p, q) · eϵ , d2 +h(p, q) +� +, +otherwise, +(using eϵ − 1 ≍ ϵ for ϵ ≤ 1 and eϵ otherwise) +≍ +� +ϵ2d2 +TV(p, q), +if ϵ ≤ 1, +min +� +d2 +TV(p, q)eϵ, d2 +h(p, q) +� +, +otherwise, +where the last step uses the inequality d2 +TV(p, q) ≤ d2 +h(p, q) from Fact 2.7. +3.2 +General Distributions: Lower Bounds and Higher Cost of Privacy +In this section, we establish lower bounds for the sample complexity of private hypothesis testing +for general distributions. In the subsequent section, the lower bounds will be shown to be tight up +to logarithmic factors. +We formally state the lower bound in the statement below: +Theorem 1.7 (Sample complexity lower bound for general distributions). For any ρ ∈ (0, 0.5) and +ν ∈ (0, 0.5) such that 2ν2 ≤ ρ ≤ ν, there exist ternary distributions p and q such that d2 +h(p, q) = ρ, +dTV(p, q) = ν, and the sample complexity behaves as +n∗(p, q, ϵ) ≍ +� +� +� +� +� +� +� +1 +ϵ2 · d2 +TV(p,q), +if ϵ ≤ 1, +min +� +1 +d2 +TV(p,q), +1 +eϵ · d4 +h(p,q) +� +, +if eϵ ∈ +� +e, +1 +d2 +h(p,q) +� +, +1 +d2 +h(p,q), +if eϵ > +1 +d2 +h(p,q). +(4) +We provide the proof below. We refer the reader to Remark 1.8 for further discussion on differ- +ences between the worst-case sample complexity of general distributions and the sample complexity +of binary distributions (cf. Theorem 1.6). +We note that a similar construction is mentioned in +Canonne, Kamath, McMillan, Smith, and Ullman [CKMSU19, Section 1.3]; however, their focus is +on the central model of differential privacy. +3.2.1 +Proof of Theorem 1.7 +Proof. The case when ϵ ≤ 1 follows from Fact 2.7. Thus, we set ϵ ≥ 1 in the remainder of this +section. We start with a helpful approximation for computing the Hellinger divergence, proved in +Appendix D: +Claim 3.3 (Additive approximation for √· ). There exist constants 0 < c1 ≤ c2 such that for +0 < y ≤ x, we have c1 · y2 +x ≤ (√x − √x − y)2 ≤ c2 · y2 +x . +For some γ ∈ (0, 0.25) and δ > 0 to be decided later, let p and q be the following ternary +distributions: +p = +� +� +0 +1/2 +1/2 +� +� , +and +q = +� +� +2γ1+δ +1/2 + γ − γ1+δ +1/2 − γ − γ1+δ +� +� . +17 + +Since γ ≤ 0.25 and δ ≥ 0, these two are valid distributions. +Observe that dTV(p, q) = γ + γ1+δ ≍ γ and and d2 +h(p, q) ≍ γ1+δ by Claim 3.3. We choose γ and +δ such that ν = dTV(p, q) and ρ = d2 +h(p, q). Such a choice of γ and δ can be made by the argument +given in Appendix D.2 as long as ν ∈ (0, 0.5) and ρ ∈ [2ν2, ν]. Thus, these two distributions satisfy +the first two conditions of the theorem statement. +In the rest of the proof, we will use the facts that γ1+δ ≍ ρ and γ ≍ ν. In particular, we have +γ2 ≲ γ1+δ ≲ γ. +Since both p and q are supported on [3], we can restrict our attention to ternary output channels +(see Fact 2.7). Recall that Pϵ +3,3 is the set of all ϵ-LDP channels from [3] to [3]. We will establish +the following result: for all ϵ such that e ≤ eϵ ≲ +1 +d2 +h(p,q), we have +max +T∈Pϵ +3,3 +d2 +h(Tp, Tq) ≍ max +� +d2 +TV(p, q), d4 +h(p, q)eϵ� +≍ max(γ2, eϵγ2+2δ). +(12) +By Fact 2.7, equation (12) implies that for e ≤ eϵ ≲ +1 +d2 +h(p,q), we have +n∗(p, q, ϵ) ≍ min +� +1 +d2 +TV(p, q), +1 +eϵ · d4 +h(p, q) +� +. +(13) +Let ϵ0 be the right endpoint of the range for ϵ above, i.e., eϵ0 ≍ +1 +d2 +h(p,q). Then equation (13) shows +that n∗(p, q, ϵ0) ≍ 1/d2 +h(p, q) ≍ n∗(p, q). Since for any ϵ such that ϵ > ϵ0, we have n∗(p, q, ϵ) ∈ +[n∗(p, q, ϵ0), n∗(p, q)], the desired conclusion in equation (4) holds for ϵ > ϵ0, as well. Thus, in the +remainder of this proof, we will focus on establishing equation (12). +Since d2 +h(·, ·) is a convex, bounded function and the set of ϵ-LDP channels is a convex polytope, +it suffices to restrict our attention only to the extreme points of the polytope. As mentioned in +Fact 2.6, these extreme points are of three types: +Case I. (Exactly one nonzero row) Any such extreme point T maps the entire domain to a +single point with probability 1. After transformation under this channel, all distributions become +indistinguishable, giving dh(Tp, Tq) = 0. +Case II. (Exactly two nonzero rows) This corresponds to the case when T = Tϵ +RR × T′, where +T′ is a deterministic threshold channel from [3] to [2].7 There are two non-trivial options for choosing +T′, which we analyze below. +The first choice of T′ maps {1} and {2, 3} to different elements. The transformed distributions p′ +and q′ are [0, 1] and [2γ1+δ, 1−2γ1+δ], respectively. Using Claim 3.3, we obtain d2 +h(p′, q′) ≍ γ1+δ and +dTV(p′, q′) ≍ γ1+δ. Let p′′ and q′′ be the corresponding distributions after applying the randomized +response with parameter ϵ. Since p′ and q′ are binary distributions, we can apply Proposition 3.1 +to obtain +d2 +h(p′′, q′′) ≍ min(d2 +h(p′, q′), eϵd2 +TV(p′, q′)) ≍ min(γ1+δ, eϵγ2+2δ) ≍ γ1+δ · min(1, eϵγ1+δ), +which is equal to eϵ · γ2+2δ in the regime of interest and consistent with the desired expression in +equation (12). +The second choice of T′ maps {1, 2} and {3} to different elements. The transformed distributions +p′ and q′ are [1/2, 1/2] and [1/2 + γ + γ1+δ, 1/2 − γ − γ1+δ], respectively. Applying Claim 3.3, we +7We use Theorem 1.17 to restrict our attention only to threshold channels. +18 + +observe that d2 +h(p′, q′) ≍ γ2 and dTV(p′, q′) ≍ γ. Let p′′ and q′′ be the corresponding distributions +after applying the randomized response with parameter ϵ. Applying Proposition 3.1, we obtain +d2 +h(p′′, q′′) ≍ min(d2 +h(p′, q′), eϵd2 +TV(p′, q′)) ≍ min(γ2, eϵγ2) ≍ γ2 +in the regime of interest. Again, this is consistent with equation (12). +Case III. (All nonzero rows) There are two extreme points of this type (up to a permutation +of the rows), both of the following form: +T1 = +� +� +1 − 2α +α +α +α +1 − 2α +α +α +α +1 − 2α +� +� . +For the first extreme point, α satisfies 1−2α +α += eϵ, while the second extreme point has +α +1−2α = +eϵ. These channels are relatively easy to analyze, since they transform the distributions element- +wise: each entry x of the original distribution is transformed to α + x(1 − 3α). Consequently, the +transformed distributions p′ and q′ are +p′ = +� +� +α +1−α +2 +1−α +2 +� +� , +and +q′ = +� +� +α + 2γ1+δ(1 − 3α) +1−α +2 ++ (γ − γ1+δ)(1 − 3α) +1−α +2 ++ (−γ − γ1+δ)(1 − 3α) +� +� . +(14) +We now compute the Hellinger divergence between these two distributions for both the extreme +points. +Let us first consider the case where 1−2α +α += eϵ. Then α = +1 +2+eϵ ≍ e−ϵ, since ϵ ≥ 0. Thus, in +the desired range of eϵ ≲ γ−(1+δ), the parameter α satisfies α ≳ γ1+δ. We will now calculate the +Hellinger divergence between p′ and q′ in equation (14) by analyzing the contribution from each +of the three terms in the sum �3 +i=1( +� +p′ +i − +� +q′ +i)2. For the first term, we apply Claim 3.3 with +x = α+2γ1+δ(1 − 3α) and y = 2γ1+δ(1 − 3α), to see that its contribution is Θ(γ2+2δ/α) ≍ eϵγ2+2δ +(since 1 − 3α ≥ 0.1). Applying Claim 3.3 again, we see that the contributions from the second and +third elements are Θ(γ2), since α ≪ 1 and γ ≪ 1, respectively. Overall, the Hellinger divergence is +O +� +max(γ2, eϵγ2+2δ) +� +, which satisfies equation (12). +Finally, let us consider the case when +α +1−2α = eϵ. We set β = 1 − 2α. Then β = 1/(1 + 2eϵ), +which is much less than 1 in the desired range and is of the order of e−ϵ. Thus, each entry x of the +distribution is mapped to 1 +2(1 − β + x(3β − 1)). The transformed distributions are +p′ = 1 +2 · +� +� +1 − β +1+β +2 +1+β +2 +� +� , +and +q′ = 1 +2 · +� +� +1 − β + 2γ1+δ(3β − 1) +1+β +2 ++ (γ − γ1+δ)(3β − 1) +1+β +2 ++ (−γ − γ1+δ)(3β − 1) +� +� . +(15) +As β is much less than 1 in the desired range of ϵ, we can apply Claim 3.3 to see that contribution +of the first element is Θ(γ2+2δ), and the contributions of both the second and third elements are +Θ(γ2). Overall, the Hellinger divergence is Θ(γ2), which is again consistent with equation (12). +Combining all the cases above, the maximum Hellinger divergence after applying any ϵ-LDP +channel is Θ(γ2 · max(1, eϵγ2δ)), as desired. +19 + +3.3 +General Distributions: Upper Bounds and Minimax Optimality +We now demonstrate an algorithm that finds a private channel matching the minimax rate in +Theorem 1.7 up to logarithmic factors. Moreover, the proposed algorithm is both computationally +efficient and communication efficient. +Theorem 1.9 (Sample complexity upper bounds and an efficient algorithm for hypothesis testing +for general distributions). Let p and q be two distributions on [k]. Let ϵ > 0. Then the sample +complexity behaves as +n∗(p, q, ϵ) ≲ +� +� +� +� +� +� +� +1 +ϵ2 · d2 +TV(p,q), +if ϵ ≤ 1, +min +� +1 +d2 +TV(p,q), +α2 +eϵ · d4 +h(p,q) +� +, +if eϵ ∈ +� +e, +α +d2 +h(p,q) +� +, +α +d2 +h(p,q), +if eϵ > +α +d2 +h(p,q), +(5) +where α ≲ log(1/d2 +h(p, q)) ≍ log (n∗(p, q)). +Moreover, the rates above are achieved by an ϵ-LDP channel T that maps [k] to [2] and can be +found in time polynomial in k, for any choice of p, q, and ϵ. +In comparison with Theorem 1.7, we see that the test above is minimax optimal up to logarithmic +factors over the class of distributions with fixed Hellinger divergence and total variation distance. +The channel T satisfying this rate is of the following simple form: a deterministic binary channel +T′, followed by the randomized response. In fact, we can take T′ to be either Scheffe’s test (which +preserves the total variation distance) or the binary channel from Fact 2.8 (which preserves the +Hellinger divergence), whichever of the two is better. We provide the complete proof in Section 3.3.1. +One obvious shortcoming of Theorem 1.9 is that even when ϵ → ∞, the test does not recover +the optimal sample complexity of 1/d2 +h(p, q), due to the logarithmic multiplier α. We now consider +the case when eϵ ≳ +1 +d2 +h(p,q) and exhibit a channel that achieves the optimal sample complexity as +soon as eϵ ≳ +1 +d2 +h(p,q) log +� +1 +d2 +h(p,q) +� +. Thus, privacy can be attained essentially for free in this regime. +Theorem 1.13. Let p and q be two distributions on [k], and let eϵ ≳ +1 +d2 +h(p,q) log +� +1 +d2 +h(p,q) +� +. Then +n∗(p, q, ϵ) ≍ n∗(p, q). Moreover, there is a channel T achieving this sample complexity that maps [k] +to a domain of size ⌈log(n∗(p, q))⌉, and which can be computed in poly(k, log(⌈n∗(p, q)⌉)) time. +We note that the size of the output domain of ⌈log(n∗(p, q))⌉ is tight in the sense that any channel +that achieves the sample complexity within constant factors of n∗(p, q) must use an output domain of +size at least Ω(log(n∗(p, q))); this follows by the tightness of Fact 2.8 in the worst case. Consequently, +the channel T achieving the rate above is roughly of the form (1) a communication-efficient channel +from Fact 2.8 that preserves the Hellinger divergence up to constant factors, followed by (2) an +ℓ-ary randomized response channel, for ℓ ≥ 2. +We give the proof of Theorem 1.13 in Section 3.3.2 and defer the proofs of some of the interme- +diate results to Appendix A. +3.3.1 +Proof of Theorem 1.9 +In this section, we provide the proof of Theorem 1.9. We first note that this result can be slightly +strengthened, replacing α by +n∗ +binary +n∗ +, where n∗ +binary is the sample complexity of hypothesis testing un- +der binary communication constraints. This choice of α is smaller, by Pensia, Jog, and Loh [PJL22, +Corollary 3.4 and Theorem 4.1]. +20 + +Proof. The case of ϵ ≤ 1 follows from Fact 2.7; thus, we focus on the setting where ϵ > 1. +We will establish these bounds via Proposition 3.1, by using a binary deterministic channel, +followed by the binary randomized response channel. A sample complexity of 1/d2 +TV(p, q) is direct +by using the channel for ϵ = 1. Thus, our focus will be on the term +1 +eϵd4 +h(p,q). +Let T′ ∈ T2,k be a deterministic binary output channel to be decided later. Consider the channel +T = Tϵ +RR × T′. By Proposition 3.1, we have +d2 +h +� +Tϵ +RR × T′p, Tϵ +RR × T′q +� +≍ min +� +eϵd2 +TV +� +T′p, T′q +� +, d2 +h +� +T′p, T′q +�� +≥ min +� +eϵd4 +h +� +T′p, T′q +� +, d2 +h +� +T′p, T′q +�� += d2 +h +� +T′p, T′q +� +· min +� +eϵd2 +h +� +T′p, T′q +� +, 1 +� +, +(16) +where the first inequality uses Fact 2.7 +If we choose the channel T′ from Fact 2.8, we have d2 +h(T′p, T′q) ≥ 1 +α · d2 +h (p, q). Applying this +to inequality (16), we obtain +d2 +h +� +Tϵ +RR × T′p, Tϵ +RR × T′q +� +≳ 1 +α · d2 +h (p, q) · min +� 1 +α · eϵd2 +h (p, q) , 1 +� +. +By Fact 2.7, the sample complexity of Tϵ +RR×T′, which is ϵ-LDP, is at most +α +d2 +h(p,q) ·max +� +1, +α +eϵd2 +h(p,q) +� +, +which is equivalent to the desired statement. +Finally, the claim on the runtime is immediate, since the channel T′ from Fact 2.8 can be found +efficiently. +3.3.2 +Proof of Theorem 1.13 +We will prove a slightly generalized version of Theorem 1.13 below that works for a wider range of +ϵ: +Proposition 3.4. Let p and q be two distributions on [k] and ϵ > 1. Then there exists an ϵ-LDP +channel T from [k] to [ℓ], for ℓ = min(⌈log(1/d2 +h(p, q))⌉, k), such that +d2 +h(Tp, Tq) ≳ d2 +h(p, q) · min +� +1, +eϵ · d2 +h(p, q) +log(1/d2 +h(p, q)) +� +· min +� +1, +eϵ +log(1/d2 +h(p, q)) +� +. +Furthermore, the channel T can be be computed in poly(k, ℓ) time. +By Fact 2.7, Proposition 3.4 implies the following: +n∗(p, q, ϵ) ≲ n∗ · max +� +1, n∗ log(n∗) +eϵ +� +· max +� +1, log n∗ +eϵ +� +, +where n∗ := n∗(p, q). Setting eϵ equal to n∗ log(n∗) proves Theorem 1.13. Thus, we will focus on +proving Proposition 3.4 in the rest of this section. We establish this result with the help of the +following observations: +• (Lemma 3.5) First, we show that the randomized response preserves the contribution to the +Hellinger divergence by “comparable elements” (elements whose likelihood ratio is in the in- +terval +� 1 +2, 2 +� +) when ϵ is large compared to the support size. In particular, we first define the +following sets: +A = +� +i ∈ [k] : pi +qi +∈ +�1 +2, 1 +�� +and A′ = +� +i ∈ [k] : pi +qi +∈ [1, 2] +� +. +(17) +21 + +Let Tϵ,ℓ +RR denote the randomized response channel from [ℓ] to [ℓ] with privacy parameter ϵ (cf. +Definition 2.5). The following result is proved in Appendix A.1: +Lemma 3.5 (Randomized response preserves contribution of comparable elements). Let p +and q be two distributions on [ℓ]. Suppose � +i∈A � A′(√qi − √pi)2 ≥ τ. Then Tϵ,ℓ +RR, for ℓ ≤ eϵ, +satisfies +d2 +h(Tϵ,ℓ +RRp, Tϵ,ℓ +RRq) ≳ min +� +1, eϵ τ +ℓ +� +· τ . +Thus, when eϵ ≳ ℓ +τ , the randomized response preserves the original contribution of comparable +elements. +• (Lemma 3.6) We then show in Lemma 3.6, proved in Appendix A.2, that either we can +reduce the problem to the previous special case (small support size and main contribution to +Hellinger divergence is from comparable elements) or to the case when the distributions are +binary (where Proposition 3.1 is applicable and is, in a sense, the easy case for privacy). +Lemma 3.6 (Reduction to base case). Let p and q be two distributions on [k]. Then there is +a channel T, which can be computed in time polynomial in k, that maps [k] to [ℓ] (for ℓ to be +decided below) such that for p′ = Tp and q′ = Tq, at least one of the following holds: +1. For any ℓ > 2 and ℓ ≤ min +� +k, 1 + log +� +1/d2 +h(p, q) +�� +, we have +� +i∈B � B′ +�� +q′ +i − +� +p′ +i +�2 +≳ d2 +h(p, q) · +ℓ +min +� +k, 1 + log +� +1/d2 +h(p, q) +��, +where B and B′ are defined analogously to A and A′ in equation (17), but with respect +to distributions p′ and q′. +2. ℓ = 2 and d2 +h(p′, q′) ≳ d2 +h(p, q). +We now provide the proof of Proposition 3.4, with the help of Lemmata 3.5 and 3.6. +Proof. (Proof of Proposition 3.4) The channel T will be of the form T = Tϵ,ℓ +RR × T1, where T1 is a +channel from [k] to [ℓ] and ℓ is to be decided. The privacy of T is clear from the construction. +We begin by applying Lemma 3.6. Let T1 be the channel from Lemma 3.6 that maps from [k] +to [ℓ]. Let p′ = T1p and q′ = T1q, and define ˜p = Tϵ,ℓ +RRp′ and ˜q = Tϵ,ℓ +RRq′. The claim on runtime +thus follows from Lemma 3.6. +Suppose for now that T1 from Lemma 3.6 is a binary channel. Then we know that d2 +h(p′, q′) ≳ +d2 +h(p, q) and dTV(p′, q′) ≳ d2 +h(p′, q′), where the latter holds by Fact 2.7. Applying Proposition 3.1, +we have +d2 +h(˜p, ˜q) ≳ min +� +d2 +h(p′, q′), eϵd2 +TV(p′, q′) +� +≳ min +� +d2 +h(p′, q′), eϵd4 +h(p′, q′) +� +≳ d2 +h(p, q) min +� +1, eϵd2 +h(p, q) +� +, +which concludes the proof in this case. +We now consider the case when ℓ > 2 in the guarantee of Lemma 3.6. Then the compara- +ble elements of p′ and q′ preserve a significant fraction of the Hellinger divergence (depending on +the chosen value of ℓ) between p and q. Let k′ = min +� +k, 1 + log(1/d2 +h(p, q)) +� +, and choose ℓ to be +22 + +min(k′, eϵ). Then Lemma 3.6 implies that the contribution to the Hellinger divergence from com- +parable elements of p′ and q′ is at least τ, for τ ≍ d2 +h(p, q) ℓ +k′ . We will now apply Lemma 3.5 to p′ +and q′ with the above choice of τ. Since ℓ ≤ eϵ by construction, applying Lemma 3.5 to p′ and q′, +we obtain +d2 +h(˜p, ˜q) ≳ τ · min +� +1, eϵτ +ℓ +� +≳ d2 +h(p, q) ℓ +k′ · min +� +1, +eϵ · d2 +h(p, q) +log(1/d2 +h(p, q)) +� +≳ d2 +h(p, q) · min +� +1, +eϵ +log(1/d2 +h(p, q)) +� +· min +� +1, +eϵ · d2 +h(p, q) +log(1/d2 +h(p, q)) +� +, +where the last step uses the facts that ℓ = min(eϵ, k′) and k′ ≳ log(1/d2 +h(p, q)). This completes the +proof. +4 +Extreme Points of Joint Range Under Communication Constraints +In this section, our goal is to understand the extreme points of the set A := {(Tp, Tq) : T ∈ Tℓ,k}. +This will allow us to identify the structure of optimizers of quasi-convex functions over A. The +main result of this section is the following: +Theorem 1.16 (Extreme points of the joint range under communication constraints). Let p and q +be two distributions on [k]. Let A be the set of all pairs of distributions that are obtained by passing +p and q through a channel of output size ℓ, i.e., +A = {(Tp, Tq) : T ∈ Tℓ,k}. +If (Tp, Tq) is an extreme point of A, then T is a threshold channel. +We provide the proof of Theorem 1.16 in Section 4.1. Before proving Theorem 1.16, we dis- +cuss some consequences for optimizing quasi-convex functions over A. The following result proves +Corollary 1.18 for C = Tℓ,k: +Corollary 4.1 (Threshold channels maximize quasi-convex functions). Let p and q be two distri- +butions on [k]. Let A := {(Tp, Tq) : T ∈ Tℓ,k}. Let g be a real-valued quasi-convex function over +A. Then +max +T∈Tℓ,k g(Tp, Tq) = +max +T∈T thresh +ℓ,k +g(Tp, Tq). +Moreover, the above optimization problem can be solved in time poly(kℓ).8 +Proof. Observe that A is a closed polytope. Let X be the set of extreme points of A. Observe that +X ⊆ {(Tp, Tq) : T ∈ Tℓ,k and T is deterministic}, and thus is finite. Since A is a closed polytope, +A is convex hull of X. Furthermore, the maximum of g on X is well-defined and finite, as X is a +finite set. Any y ∈ A can be expressed as a convex combination y = � +xi∈X λixi. Recall that an +8Recall that g is assumed to be permutation invariant. +If not, an extra factor of ℓ! will appear in the time +complexity. +23 + +equivalent definition of quasi-convexity is that g satisfies g(λx + (1 − λ)y) ≤ max(g(x), g(y)) for all +λ ∈ [0, 1]. By repeatedly using this fact, we have +g(λ) = g +� +� � +xi∈X +λixi +� +� ≤ max +x∈X g(x). +By Theorem 1.16, any extreme point x ∈ X is obtained by passing p and q through a threshold +channel. Thus, the maximum of g over X is attained by passing p and q through a threshold channel. +The claimed runtime is obtained by trying all possible threshold channels. +Remark 4.2. (Quasi-)convex functions of interest include all f-divergences, Rényi divergences, +Chernoff information, and Lp norms. We note that the above result also holds for post-processing: +For any fixed channel H ∈ Tℓ′,ℓ, we have +max +T∈Tℓ,k g(HTp, HTq) = +max +T∈T thresh +ℓ,k +g(HTp, HTq). +This is because g(Hp′, Hq′) is a quasi-convex function of (p′, q′) ∈ A. +Remark 4.3. If g is the Hellinger divergence and C = Tℓ,k, we can conclude the following result +for the communication-constrained setting: There exists a T ∈ T thresh +ℓ,k +that attains the instance- +optimal sample complexity (up to universal constants) for hypothesis testing under a communication +constraint of size ℓ. This result is implied in Pensia, Jog, and Loh [PJL22] by Theorem 2.9 (which is a +result from Tsitsiklis [Tsi93]) and Lemma 4.2. The above argument provides a more straightforward +proof. +4.1 +Proof of Theorem 1.16 +We now provide the proof of Theorem 1.16. +Proof. (Proof of Theorem 1.16) We first make the following simplifying assumption about the likeli- +hood ratios: there is at most a single element i∗ with qi∗ = 0, and for all other elements i ∈ [k]\{i∗}, +pi/qi is a unique value. If there are two or more elements with the same likelihood ratio, we can +merge those elements into a single alphabet without loss of generality, as we explain next. Let p′ +and q′ be the distributions after merging these elements, and let k′ ≤ k be the new cardinality. Then +for any channel T ∈ Tℓ,k, there exists another channel T′ ∈ Tℓ,k′ such that (Tp, Tq) = (T′p′, T′q′). +We can then apply the following arguments to p′ and q′. See Appendix D.1 for more details. +In the following, we will consider pi/qi to be ∞ if qi = 0, and we introduce the notation +θi := pi/qi. We will further assume, without loss of generality, that pi/qi is strictly increasing in i. +Since the elements are ordered with respect to the likelihood ratio, a threshold channel corresponds +to a map that partitions the set [k] into contiguous blocks. Formally, we have the following definition: +Definition 4.4 (Partitions and threshold partitions). We say that S = (S1, S2, . . . , Sℓ) forms an +ℓ-partition of [k] if ∪ℓ +i=1Si = [k] and Si ∩ Sj = ∅ for 1 ≤ i ̸= j ≤ ℓ. We say that S forms an +ℓ-threshold partition of [k] if in addition, for all i < j ∈ ℓ, every entry of Si is less than every entry +of Sj. +As mentioned before, channels corresponding to ℓ-threshold partitions are precisely the threshold +channels up to a permutation of output labels. The channels corresponding to ℓ-partitions are the +set of all deterministic channels that map [k] to ℓ, which are the extreme points of Tℓ,k (cf. Fact 2.4). +24 + +Observe that A is a convex, compact set, which is a linear transformation of the convex, compact +set Tℓ,k, and any extreme point of A is of the form (Tp, Tq), where T is an extreme point of Tℓ,k +(cf. Fact 2.1). Now suppose (Tp, Tq) is an extreme point of A, but T is not a threshold channel. +Thus, T corresponds to some ℓ-partition S of [k] that is not an ℓ-threshold partition. We will now +show that (Tp, Tq) is not an extreme point of A, by showing that there exist two distinct channels +T1 ∈ Tℓ,k and T2 ∈ Tℓ,k such that the following holds: +1 +2 · T1p + 1 +2 · T2p = Tp, +and +1 +2 · T1q + 1 +2 · T2q = Tq, +(18) +and T1p ̸= Tp. +Since S is not a ℓ-threshold partition, there exist 1 ≤ a < b < c ≤ k and m ̸= n in [ℓ] such that +a, c ∈ Sm and b ∈ Sn, and pa/qa < pb/qb < pc/qc. Among qa, qb, and qc, only qc is potentially zero. +Suppose for now that qc ̸= 0; we will consider the alternative case shortly. +For some ϵ1 ∈ (0, 1) and ϵ2 ∈ (0, 1) to be determined later, let T1 be the following channel: +1. For x ̸∈ {a, b}, T1 maps x to T(x). +2. For x = a (respectively b), T1 maps x to m (respectively n) with probability 1−ϵ1 (respectively +1 − ϵ2) and to n (respectively m) with probability ϵ1 (respectively ϵ2). +Thus, the channels T and T1 have all columns identical, except for those corresponding to inputs +a and b. Let vi be the ith column of T. Observe that va is a degenerate distribution at m ∈ [ℓ] and +vb is a degenerate distribution at n ∈ [ℓ] (equivalently, T(m, a) = 1 and T(n, b) = 1). Thus, we can +write +T1q = Tq + (ϵ2qb − ϵ1qa)(va − vb), +T1p = Tp + (ϵ2pb − ϵ1pa)(va − vb). +If we choose ϵ1qa = ϵ2qb, we have T1q = Tq and +T1p = Tp + (ϵ2pb − ϵ1pa)(va − vb) += Tp + (ϵ2qbθb − ϵ1qaθa)(va − vb) += Tp + ϵ1qa(θb − θa)(va − vb). +(19) +Recall that θb > θa, as mentioned above. +We now define T2. For some ϵ3 ∈ (0, 1) and ϵ4 ∈ (0, 1) to be decided later, we have: +1. For x ̸∈ {b, c}, T2 maps x to T(x). +2. For x = c (respectively b), T2 maps x to m (respectively n) with probability 1−ϵ3 (respectively +1 − ϵ4) and to n (respectively m) with probability ϵ3 (respectively ϵ4). +With the same arguments as before, we have +T2q = Tq + (ϵ4qb − ϵ3qc)(vc − vb), +T1p = Tp + (ϵ4pb − ϵ3pc)(vc − vb). +If we choose ϵ3qc = ϵ4qb, we have T2q = Tq and +T2p = Tp + (ϵ4qbθb − ϵ3qcθc)(vc − vb) += Tp + ϵ3qc(θb − θc)(vc − vb) +25 + += Tp + ϵ3qc(θb − θc)(va − vb), +(20) +where the last line follows by the fact that va = vc, since T maps both a and c to m almost surely. +Let ϵ1 ∈ (0, 1) and ϵ3 ∈ (0, 1) be such that ϵ1qa(θb − θa) = −ϵ3qc(θb − θc). Such a choice always +exists because θb − θa and −(θb − θc) are both strictly positive and finite. Then equations (19) +and (20) imply that Tp = 1 +2T1p + 1 +2T2p and Tq = 1 +2T1q + 1 +2T2q, and T1p ̸= Tp. Moreover, +T1p ̸= Tp. Thus, (Tp, Tq) is not an extreme point of A. +We now outline how to modify the construction above when qc is zero. By setting ϵ4 to be zero, +we obtain T2q = Tq and T2p = Tp + (−ϵ3pc) (va − vb). The desired conclusion follows by choosing +ϵ1 and ϵ3 small enough such that ϵ1qa(θb − θa) = −ϵ3pc. +5 +Extreme Points of Joint Range under Privacy Constraints +In the previous section, we considered the extreme points of the joint range under communication +constraints. Such communication constraints are routinely applied in practice in the presence of +additional constraints such as local differential privacy. However, the results of the previous section +do not apply directly, as the joint range is now a strict subset of the set in Theorem 1.16, and +the extreme points differ significantly. For example, the threshold channels are not even private. +However, we show in this section that threshold channels still play a fundamental role. Our main +result in this section is the following theorem: +Theorem 1.17 (Extreme points of the joint range under privacy and communication constraints). +Let p and q be distributions on [k]. Let C be the set of ϵ-LDP channels from [k] to [ℓ]. Let A be the +set of all pairs of distributions that are obtained by applying a channel from C to p and q, i.e., +A = {(Tp, Tq) | T ∈ C}. +(7) +If (Tp, Tq) is an extreme point of A for T ∈ C, then T can be written as T = T2 × T1 for some +threshold channel T1 ∈ T2ℓ2,k and some T2 an extreme point of the set of ϵ-LDP channels from [2ℓ2] +to [ℓ]. +Actually, our result applies to a broader family of linear programming (LP) channels that we +describe below: +Definition 5.1 (LP family of channels). For any ℓ ∈ N, let ν = (ν1, ν2, . . . , νℓ) and γ = (γ1, γ2, . . . , γℓ) +be two nonnegative vectors in Rℓ ++. For k ∈ N, define the set of linear programming (LP) channels +J γ,ν +ℓ,k , a subset of Tℓ,k, to be the (convex) set of all channels from [k] to [ℓ] that satisfy the following +constraints: +For each row j ∈ [ℓ], and for each i, i′ ∈ [k], we have T(j, i) ≤ γjT(j, i′) + νj. +(21) +When γj = eϵ and νj = 0 for all j ∈ [ℓ], we recover the set of ϵ-LDP channels from [k] to [ℓ]. +Another example will be mentioned in Section 6 for a relaxed version of approximate LDP. +The rest of this section is organized as follows: In Section 5.1, we show that any T that leads to +an extreme point of A cannot have more than 2ℓ2 unique columns (Lemma 5.2). We use this result to +prove Theorem 1.17 in Section 5.2. In Section 5.3, we apply Theorem 1.17 to prove Corollaries 1.18 +and 1.20. +26 + +5.1 +Bound on the Number of Unique Columns +The following result will be critical in the proof of Theorem 1.17, the main result of this section. +Lemma 5.2. Let p and q be distributions on [k]. Let C be the set of channels from [k] to [ℓ], from +Definition 5.1. Let A be the set of all pairs of distributions that are obtained by applying a channel +from C to p and q, i.e., +A = {(Tp, Tq) | T ∈ C}. +(22) +If T has more than 2ℓ2 unique columns, then (Tp, Tq) is not an extreme point of A. +We prove this result in Section 5.1.2 after proving a quantitatively weaker, but simpler, result +in Section 5.1.1. +5.1.1 +Warm-Up: An Exponential Bound on the Number of Unique Columns +In this section, we first prove a weaker version of Lemma 5.2, where we upper-bound the number of +unique columns in the extreme points of C from Definition 5.1 (not just those that lead to extreme +points of A) by an exponential in ℓ. In fact, this bound will be applicable for a broader class of +channels that satisfy the following property: +Condition 5.3 (Only one free entry per column). Let C be a convex set of channels from [k] to [ℓ]. +Let T be an extreme point of C. Then there exist numbers {m1, . . . , mℓ} and {M1, . . . , Mℓ} such +that for every column c ∈ [k], there exists at most a single row r ∈ [ℓ] such that T(r, c) ̸∈ {mr, Mr}. +We call such entries free. +We show in Appendix B that extreme points of the LP channels from Definition 5.1 satisfy +Condition 5.3. The following claim bounds the number of unique columns in any extreme point of +C, and thus also implies a version of Theorem 1.17 with ℓ · 2ℓ−1 instead of 2ℓ2 (cf. Fact 2.1). +Claim 5.4 (Number of unique columns in an extreme point is at most exponential in ℓ). Let C be +a set of channels satisfying the property of Condition 5.3. Let T be an extreme point of C. Then +the number of unique columns in T is at most ℓ · 2ℓ−1. In particular, T can be written as T2 × T1, +where T1 is a deterministic map from [k] to [ℓ′] and T2 is a map from [ℓ′] to [ℓ], for ℓ′ = ℓ · 2ℓ−1. +Proof. We use the notation from Condition 5.3. For each column, there are ℓ possible locations of +a potential free entry. Let this location be j∗; the value at this location is still flexible. Now let us +consider the number of ways to assign values at the remaining locations. For each j ∈ [ℓ] \ {j∗}, +the entry is either mj or Mj (since j is not a free entry). +Thus, there are 2ℓ−1 such possible +assignments. Since the column entries sum to one, each of those 2ℓ−1 assignments fixes the value at +the j∗ location, as well. Thus, there are at most ℓ · 2ℓ−1 unique columns in T. +5.1.2 +Forbidden Structure in Extreme Points Using the Joint Range +In Claim 5.4, we considered the extreme points of LP channels. However, we are actually interested +in a (potentially much) smaller set: the extreme points that correspond to the extreme points of +the joint range A. In this section, we identify a necessary structural property for extreme points of +LP channels that lead to extreme points of the joint range. We begin by defining the notion of a +“loose” entry in a channel in C: +27 + +Definition 5.5 (Loose and tight entries). Let T be a channel in J γ,ν +ℓ,k +from Definition 5.1 that +maps from [k] to [ℓ]. Let {m1, . . . , mℓ} and {M1, . . . , Mℓ} be the row-wise minimum and maximum +entries, respectively. For c ∈ [k] and r ∈ [ℓ], we say an entry T(r, c) is max-tight if T(r, c) = Mr +and Mr = γrmr + νr. An entry T(r, c) is min-tight if T(r, c) = mr and Mr = γrmr + νr. An entry +that is neither max-tight nor min-tight is called loose. +Remark 5.6. Our results in this section continue to hold for a slightly more general definition, +where we replace the linear functions γjx+νj by arbitrary monotonically increasing functions fj(x). +We focus on linear functions for simplicity and clarity. (Also see Remark 5.11.) +If the rest of the row is kept fixed, a max-tight entry cannot be increased without violating +privacy constraints, but it can be decreased. +Similarly, a min-tight entry cannot be decreased +without violating privacy constraints, but it can be increased. Loose entries can be either increased +or decreased without violating privacy constraints. These perturbations need to be balanced by +adjusting other entries in the same column to satisfy column stochasticity; for example, a max- +tight entry can be decreased while simultaneously increasing a min-tight or loose entry in the same +column. This is formalized below: +Condition 5.7 (Mass can be transferred from entries that are not tight). Let C be a set of channels +from [k] to [ℓ]. +Let T be any extreme point of C. +Suppose there are two rows (r, r′) and two +columns (c, c′) (in the display below, we take r < r′ and c < c′ without loss of generality) with +values (m, m′, M, M′), as shown below: +� +����� +· · · +· · · +· · · +· · · +· · · +· · · +M +· · · +m +· · · +· · · +· · · +· · · +· · · +· · · +· · · +m′ +· · · +M′ +· · · +· · · +· · · +· · · +· · · +· · · +� +����� +, +such that: +• T(r, c) and T(r′, c′) are not min-tight (M and M′ above, respectively). +• T(r, c′) and T(r′, c) are not max-tight (m and m′ above, respectively). +Then there exist ϵ′ > 0 and δ′ > 0 such that for all ϵ ∈ [0, ϵ′) and δ ∈ [0, δ′), the following matrix +T′ also belongs to C: +T′ = +� +����� +· · · +· · · +· · · +· · · +· · · +· · · +M − ϵ +· · · +m + δ +· · · +· · · +· · · +· · · +· · · +· · · +· · · +m′ + ϵ +· · · +M′ − δ +· · · +· · · +· · · +· · · +· · · +· · · +� +����� +, +where the omitted entries of T and T′ are the same. +We show that the channels from Definition 5.1 satisfy Condition 5.7 in Appendix B. Using +Condition 5.7, we show that the following structure is forbidden in the channels that lead to extreme +points of the joint range: +28 + +Lemma 5.8. Let p and q be two distributions on [k]. Let C be the set of LP channels from Def- +inition 5.1 (or, more generally, a convex set of channels satisfying Condition 5.7) from [k] to [ℓ]. +Suppose pi/qi is strictly increasing in i. Let T ∈ C have the following structure: there are two rows +(r, r′) (in the display below, r < r′ is taken without loss of generality) and three columns i1 < i2 < i3 +with values (m, m′, m′′, M, M′, M′′), as shown below: +� +����� +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +M +· · · +m +· · · +M′ +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +m′ +· · · +M′′ +· · · +m′′ +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +� +����� +, +such that: +• T(r, i1), T(r, i3), and T(r′, i2) are not min-tight (M, M′, and M′′ above, respectively). +• T(r, i2), T(r′, i1), and T(r′, i3) are not max-tight (m, m′, and m′′ above, respectively). +Let A := {(Tp, Tq) : T ∈ C}. Then (Tp, Tq) cannot be an extreme point of A. +Proof. Firstly, the set A is convex since C is convex. For some ϵ > 0 and δ > 0 to be decided later, +consider the following perturbed matrices: +T′ = +� +����� +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +M − ϵ +· · · +m + δ +· · · +M′ +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +m′ + ϵ +· · · +M′′ − δ +· · · +m′′ +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +� +����� +, +T′′ = +� +����� +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +M +· · · +m + ϵ′ +· · · +M′ − δ′ +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +m′ +· · · +M′′ − ϵ′ +· · · +m′′ + δ′ +· · · +· · · +· · · +· · · +· · · +· · · +· · · +· · · +� +����� +. +To be specific, the entries of T′, T′′, and T match except in the six locations highlighted here. Since +C satisfies Condition 5.7 (see Claim B.2), both T′ and T′′ belong to the set C if ϵ, ϵ′, δ, and δ are +small enough and positive. We will now show that there exist choices of these parameters such that +(Tp, Tq) is a convex combination of (T′p, T′q) and (T′′p, T′′q), and these three points are distinct +elements of A. Consequently, (Tp, Tq) will not be an extreme point of A. +For any j ∈ ℓ, let vj denote the vector in Rℓ that is 1 at the jth location and 0 otherwise. Define +θi := pi/qi to be the likelihood ratio. If θi < ∞, then pi = θiqi. Since θi is strictly increasing in +i, only θi3 may be infinity. Let us first suppose that θi3 < ∞. We will consider the case when θi3 +might be infinity in the end. +Let us begin by analyzing how T′ transforms p and q. Since T′ differs from T only in the four +locations mentioned above, T′p and Tp, both of which are distributions on [ℓ], differ only in the +elements r and r′ of [ℓ]. On the element r, (T′q)r − (Tq)r is equal to −ϵqi1 + δqi2, and equal to its +negation on the element r′. In particular, they satisfy the relation +T′q = Tq + (−ϵqi1 + δqi2) (vr − vr′) . +29 + +If ϵqi1 = δqi2, we have T′q = Tq. Under the same setting, p is transformed as follows: +T′p = Tp + (−ϵpi1 + δpi2) (vr − vr′) += Tp + (−ϵθi1qi1 + δθi2qi2) (vr − vr′) += Tp + ϵqi1(−θi1 + θi2) (vr − vr′) . +We now analyze the effect of T′′, which satisfies +T′′q = Tq + (ϵ′qi2 − δ′qi3) (vr − vr′) . +If ϵ′qi2 = δ′qi3, we have T′′q = Tq. Under the same setting, p is transformed as follows: +T′′p = Tp + (ϵ′pi2 − δ′pi3) (vr − vr′) += Tp + (−ϵ′θi2qi2 + δ′θi3qi3) (vr − vr′) += Tp + ϵ′qi2(−θi2 + θi3) (vr − vr′) . +Now observe that θi1 < θi2 < θi3. By choosing ϵ > 0 and ϵ′ > 0 small enough such that ϵqi1(−θi1 + +θi2) = ϵ′qi2(−θi2 + θi3), we obtain +(Tp, Tq) = 1 +2 · +� +T′p, T′q +� ++ 1 +2 · +� +T′p, T′q +� +, +and all three points are distinct elements of A. Such a choice of ϵ and ϵ′ always exists, since both +qi1(−θi1 + θi2) and qi2(−θi2 + θi3) are positive and finite. Thus, (Tp, Tq) is not an extreme point +of A. +Let us now consider the case when θi3 = ∞, or equivalently, qi3 = 0. Define ϵ′ to be 0, so that +T′′q = Tq and T′′p = Tp − δ′pi3 (vr − vr′). Then choose δ′ > 0 and ϵ > 0 small enough such that +ϵqi1 (θi2 − θi1) = δ′pi3, which is possible since both sides are positive and finite. Thus, (Tp, Tq) is a +non-trivial convex combination of (T′p, T′q) and (T′′p, T′′q), so is not an extreme point of A. +5.1.3 +Proof of Lemma 5.2 +Proof. Without loss of generality, we assume that the likelihood ratios pi/qi are unique and strictly +increasing in i. We refer the reader to the proof of Theorem 1.16 and Claim D.1 for more details. +Let T ∈ C be a channel from [k] to [ℓ] such that (Tp, Tq) is an extreme point of A. Suppose that +there are ℓ′ unique columns in T with ℓ′ > 2ℓ2. From now on, we assume that ℓ′ = 2ℓ2; otherwise, +we apply the following argument to the first 2ℓ2 distinct columns. +Let c, c′ ∈ [k] be such that the cth and c′th columns of T are distinct. Observe that for every +pair of distinct columns c and c′, there are two rows such that cth column has a strictly bigger value +than the c′th column on one row, and vice versa on the another row. This is because both of the +columns sum up to 1, so if a particular column has a larger entry in a row, its entry must be smaller +in a different row. In particular, there exist two rows g(c, c′) and h(c, c′) such that T(g(c, c′), c) > +T(g(c, c′), c′) and T(h(c, c′), c) < T(h(c, c′), c′). As a result, T(g(c, c′), c) and T(h(c, c′), c′) are not +min-tight, and T(g(c, c′), c′) and T(h(c, c′), c) are not max-tight. +We now order the distinct columns of T in the order of their appearance from left to right. +Let i1, i2, . . . , iℓ′ be the indices of the unique columns. For example, the first distinct column i1 is +the first column of T (corresponding to the element 1). The second distinct column i2 is the first +column of T that is different from the first column. The third distinct column is the first column +of T that is different from the first two columns. Let I be the set of unique column indices of T. +30 + +Now, we divide the distinct columns in T into pairs: H = {(i1, i2), (i3, i4), . . . , (iℓ′−1, iℓ′)}. The +total number of possible choices in H is ℓ′/2, and for every (m, m + 1) in H, the possible number +of choices of (g(im, im+1), h(im, im+1)) is at most ℓ(ℓ − 1), since both of these lie in [ℓ] and are +distinct. Thus, there must exist two pairs in H whose corresponding indices are the same, since +ℓ′ +2 = ℓ2 > ℓ(ℓ − 1). +Without loss of generality, we let these pairs of columns be (i1, i2) and (i3, i4). +Let r := +g(i1, i2) = g(i3, i4) and r′ := h(i1, i2) = h(i3, i4). Then the previous discussion implies that: +• T(r, i1) and T(r, i3) are not min-tight, and T(r′, i1) and T(r′, i3) are not max-tight. +• T(r, i2) and T(r, i4) are not max-tight, and T(r′, i2) and T(r′, i4) are not min-tight. +Thus, the columns i1, i2, and i3 satisfy the conditions of Lemma 5.8, i.e., they exhibit the forbidden +structure. This implies that (Tp, Tq) cannot be an extreme point of A. Therefore, ℓ′ ≤ 2ℓ2. +5.2 +Proof of Theorem 1.17: Unique Columns to Threshold Channels +In this section, we provide the proof of Theorem 1.17 using Lemma 5.2. Noting that our main +structural result is more widely applicable (Condition 5.7), we prove a more general version of +Theorem 1.17 below for Definition 5.1. Before doing so, we require an additional property on the +set of our channels, proved in Appendix B: +Claim 5.9 (Closure under pre-processing). The set J γ,ν +ℓ,k +satisfies the following closure property +under pre-processing: +J γ,ν +ℓ,k = +k� +ℓ′=1 +� +T2 × T1 : T2 ∈ J γ,ν +ℓ,ℓ′ and T1 ∈ Tℓ′,k +� +. +(23) +Informally, if we take an arbitrary channel T1 and compose it with an LP private channel T2, +the composition T2 × T1 is also LP private. +The following result is thus a more general version of Theorem 1.17: +Theorem 5.10 (Structure of optimal channels). Let p and q be distributions on [k]. For any ℓ ∈ N, +let C be the set of channels J γ,ν +ℓ,k from Definition 5.1. Let A be the set of all pairs of distributions +that are obtained by applying a channel from C to p and q, i.e., +A = {(Tp, Tq) | T ∈ C}. +(24) +If (Tp, Tq) is an extreme point of A, then T can be written as T = T2 ×T1, for some T1 ∈ T thresh +ℓ′,k +and T2 an extreme point of the set J γ,ν +ℓ,ℓ′ . +Proof. Since C is convex, the joint range A is convex. By Lemma 5.2, we know that if (Tp, Tq) is +an extreme point of A, then T can be written as T2 × T1, where T1 ∈ Tℓ′,k for ℓ′ := 2ℓ2. Using +Claim 5.9, any such channel in C is of the form T2×T1, where T1 ∈ Tℓ′,k and T2 ∈ J γ,ν +ℓ,ℓ′ . Combining +the last two observations, we obtain the following: +A = conv +�� +(T2 × T1p, T2 × T1q) : T2 ∈ J γ,ν +ℓ,ℓ′ , T1 ∈ Tℓ′,k +�� +. +(25) +We now claim that we can further take T1 to be a threshold channel T1 ∈ T thresh +ℓ′,k +and T2 to be +an extreme point of J γ,ν +ℓ,ℓ′ . This claim follows if we can write an arbitrary point in A as a convex +31 + +combination of elements of the set +� +(T2 × T1p, T2 × T1q) : T2 ∈ ext +� +J γ,ν +ℓ,ℓ′ +� +, T1 ∈ T thresh +ℓ′,k +� +. By +equation (25), it suffices to demonstrate this convex combination for all points of the form (T2 × +T1p, T2 × T1q), for some T2 ∈ J γ,ν +ℓ,ℓ′ and T1 ∈ Tℓ′,k. +Let H1, H2, . . . be extreme points of J γ,ν +ℓ,ℓ′ , and let L1, L2, . . . be an enumeration of the threshold +channels T thresh +ℓ′,k +. By definition, any T1 ∈ J γ,ν +ℓ,ℓ′ can be written as � +i αiHi for some convex combi- +nation α1, α2, . . . . Furthermore, Theorem 1.16 implies that any (T2p, T2q), for T2 ∈ Tℓ′,k, can be +written as � +j βj(Ljp, Ljq) = (� +j βjLjp, � +j βjLjq), for some convex combination β1, β2, . . . . +Thus, any arbitrary point (T2 × T1p, T2 × T1q), for T2 ∈ J γ,ν +ℓ,ℓ′ and T1 ∈ Tℓ′,k, can be written +as +(T2 × T1p, T2 × T1q) = +��� +i +αiHi +� +× T1p, +�� +i +αiHi +� +× T1q +� += +� +i +αi (Hi × T1p, Hi × T1q) += +� +i +αi (Hi (T1p) , Hi (T1q)) += +� +i +αi +� +�Hi +� +�� +j +βjLjp +� +� , Hi +� +�� +j +βjLjq +� +� +� +� += +� +i +αi +� +�� +j +βjHi × Ljp, +� +j +βjHi × Ljq +� +� += +� +i +� +j +αiβj (Hi × Ljp, Hi × Ljq) . +Finally, note that {αiβj} are also valid convex combinations of (Hi × Ljp, Hi × Ljq), since they are +nonnegative and sum to 1. +Remark 5.11 (Extending Theorems 1.17 and 5.10 to a more general set of constraints). We note +that Theorem 5.10 can be extended to an arbitrary convex set of channels C that satisfy (appropri- +ately modified versions of) Condition 5.7 and equation (23). (Also see Remark 5.6.) +5.3 +Application to Hypothesis Testing +In Section 3, we showed that the minimax-optimal sample complexity can be obtained by a communication- +efficient and efficiently computable channel, up to logarithmic factors. However, for a particular +(p, q), these guarantees can be significantly improved. For example, consider the extreme case when +p and q are the following two distributions on [k]: for γ small enough, +p = [α, 1 − α − (k − 2)γ, γ, γ, . . . , γ], +q = [β, 1 − β − (k − 2)γ, γ, γ, . . . , γ]. +Let T′ be a deterministic binary channel that maps the first and second elements to different +elements, while assigning the remaining elements arbitrarily. Now consider the following private +channel T: the channel T′, followed by the randomized response over binary distributions. Then +as γ → 0, the performance of T mirrors equation (3), which is much better than the minimax +bound of equation (4). Thus, there is a wide gap between instance-optimal and minimax-optimal +32 + +performance. We thus consider the computational question of optimizing a quasi-convex function +g(Tp, Tq) over all possible ϵ-private channels that map to a domain of size ℓ. +The following result proves Corollary 1.18 for C equal to Pϵ +ℓ,k: +Corollary 5.12 (Computationally efficient algorithms for maximizing quasi-convex functions under +privacy constraints). Let p and q be fixed distributions over [k], let C be the set of channels J γ,ν +ℓ,k +from Definition 5.1, and let A = {(Tp, Tq) : T ∈ C}. Let g : A → R be a jointly quasi-convex +function. Then there is an algorithm that solves maxT∈C g(Tp, Tq) in time polynomial in kℓ2 and +2O(ℓ3 log ℓ).9 +Proof. The algorithm is as follows: we try all threshold channels in T1 ∈ T thresh +ℓ′,k +and all extreme +points of J γ,ν +ℓ,ℓ′ , and output the channel T = T2 × T1 that attains the maximum value of g(Tp, Tq). +By Theorem 5.10 and quasi-convexity of g, we know the algorithm will output a correct value, +since all extreme points are of this form. Thus, we focus on bounding the runtime. We know that +the cardinality of T thresh +ℓ′,k +is bounded by kℓ′ (up to a rotation of output rows). By Fact 2.3, the +time taken to iterate through all the extreme points of ϵ-LDP channels from [ℓ′] to [ℓ] is at most +polynomial in 2ℓ3 log ℓ, since Jℓ,ℓ′ is a polytope in R2ℓ3 with poly(ℓ) inequalities. This completes the +proof. +The proof of Corollary 1.20 is immediate from Fact 2.7, Corollary 1.18, and Proposition 6.2, +stated later. +6 +Extensions to Other Notions of Privacy +In this section, we explore computational and statistical aspects of hypothesis testing under other +notions of privacy. Section 6.1 is on approximate privacy, in which we first focus on (ϵ, δ)-LDP +and then our proposed definition of approximate privacy. Next, we focus on binary communication +constraints for Rényi differential privacy in Section 6.2. This will be possible since our algorithmic +and structural results were not restricted to the case of pure LDP. +We begin by noting that communication constraints have a benign effect on the sample com- +plexity of hypothesis testing for many notions of privacy: +Condition 6.1 (Closure under post-processing). Let k ∈ N. For each r ∈ N, consider sets Cr ⊆ Tr,k +and define C = ∪r∈NCr. We say C satisfies ℓ-post-processing if for every r ∈ N, if T ∈ Cr and H is a +deterministic channel from [r] to [ℓ], the channel H × T also belongs to Cℓ, and thus to C. +Post-processing is satisfied by various notions of privacy: ϵ-pure privacy, (ϵ, δ)-approximate +privacy (see Dwork and Roth [DR13, Proposition 2.1]), and Rényi privacy [Mir17]. For a set of +channels C, we use the notation n∗(p, q, C) to denote the sample complexity of hypothesis testing +under channel constraints of C in Definition 1.2. The following result shows that even with binary +communication constraints, the sample complexity increases by at most a logarithmic factor: +Proposition 6.2 (Benign effect of communication constraints on sample complexity under closure). +Let p and q be any two distributions on [k]. Let C be a set of channels that satisfy ℓ-post-processing +(Condition 6.1) for some ℓ > 1. Let Cℓ denote the subset of channels in C that map to a domain of +size ℓ. Then +n∗(p, q, Cℓ) ≲ n∗(p, q, C) · +� +1 + log (n∗(p, q, C)) +ℓ +� +. +(26) +9Recall that g is assumed to be permutation invariant. +If not, an extra factor of ℓ! will appear in the time +complexity. +33 + +Proof. Let T be the optimal channel in C that maximizes d2 +h(Tp, Tq). Let k′ be the size of the +range of T. By Fact 2.7, we have n∗(p, q, C) ≍ 1/d2 +h(Tp, Tq). By Fact 2.8, we know that there +exists T′ ∈ Tℓ,k′ such that10 +d2 +h(Tp, Tq) ≲ d2 +h(T′(Tp), T′(Tq)) · +� +1 + log(1/d2 +h(Tp, Tq)) +ℓ +� +. +(27) +By the assumed closure of C under post-processing, the channel T′ × T belongs to C. Thus, the +channel T′×T also belongs to Cℓ, since its output is of size ℓ. This implies that the sample complexity +n∗(p, q, Cℓ) is at most 1/d2 +h(T′ × Tp, T′ × Tq). Using the fact that n∗(p, q, C) ≍ 1/d2 +h(Tp, Tq), we +obtain the desired result. +Thus, in the rest of this section, our main focus will be on the setting of binary channels. +6.1 +Approximate Local Privacy +In this section, we first focus on (ϵ, δ)-approximate LDP (Definition 6.3). We begin by showing +upper bounds on the associated sample complexity. On the computational front, we present efficient +algorithms for the case of binary constraints and then propose a relaxation for the case of larger +output domains. +We first recall the definition of (ϵ, δ)-LDP: +Definition 6.3 ((ϵ, δ)-LDP). We say a channel from X to Y is (ϵ, δ)-LDP if for all S ⊆ Y, we have +sup +x,x′∈X +P[T(x) ∈ S)] − eϵ · P[T(x) ∈ S)] − δ ≤ 0. +(28) +What makes the analysis of (ϵ, δ)-LDP different from ϵ-LDP is that when |Y| > 2, the condition +in inequality (28) should be verified for all sets S ⊆ Y, not just singleton sets (|S| = 1). Only when +|Y| = 2 is it enough to consider singleton sets S. +Let n∗(p, q, (ϵ, δ)) denote the sample complexity for the setting in Definition 1.3, with C equal +to the set of all (ϵ, δ)-LDP channels. We directly obtain the following upper bound on the sample +complexity, proved in Appendix C, which happens to be tight for the case of binary distributions: +Claim 6.4 (Sample complexity of approximate LDP). For all δ ∈ (0, 1), we have +n∗(p, q, (ϵ, δ)) ≲ min +� +n∗(p, q, ϵ) · +1 +1 − δ , n∗(p, q) · 1 +δ +� +. +Moreover, this is tight (up to constant factors) when both p and q are binary distributions. +In the rest of this section, we focus on efficient algorithms in the presence of both privacy and +communication constraints. +Turning to computationally efficient algorithms for the case of privacy and communication +constraints, we present two kinds of results: exact results for the case of binary outputs, and sharp +relaxations for the case of multiple outputs. +Binary channels: +Let C be the set of all (ϵ, δ)-approximate LDP channels from [k] to [2], i.e., +binary channels. Let γ = (eϵ, eϵ) and ν = (δ, δ). Observe that C is then equal to J γ,ν +2,k , defined in +Definition 5.1. Thus, Theorem 5.10 and Corollary 5.12 hold in this case, as well. +10If the supremum is not attained, the proof can be modified by considering a suitable sequence of channels and +applying a similar argument. +34 + +Channels with larger output spaces: +Here, we define a new notion of privacy that relaxes +(ϵ, δ)-LDP. It is enough to verify whether the privacy condition holds for singleton events S: +Definition 6.5 ((ϵ, δ)-SLDP). We say a channel X to Y is (ϵ, δ)-singleton-based-LDP ((ϵ, δ)-SLDP) +if for all S ⊆ Y, we have +sup +x,x′∈X +P[T(x) ∈ S)] − eϵ · P[T(x) ∈ S)] − δ · |S| ≤ 0. +The following result shows that (ϵ, δ)-SLDP is a good approximation to (ϵ, δ)-LDP when the +output space is small: +Claim 6.6 (Relations between LDP and SLDP). Consider a channel T from X to [ℓ]. +1. If T is (ϵ, δ)-SLDP, it is (ϵ, ℓδ)-LDP. +2. If T is (ϵ, δ)-LDP, it is (ϵ, δ)-SLDP. +The proof is immediate from the definitions of (ϵ, δ)-LDP and (ϵ, δ)-SLDP, and we omit it. +We now show that it is easy to optimize over SLDP channels in the presence of communication +constraints. For any ℓ ∈ N, let C be the set of all channels from [k] to [ℓ] that satisfy (ϵ, δ)-SLDP. +Let γ = (eϵ, eϵ, . . . , eϵ) and ν = (δ, δ, . . . , δ). +Observe that C is then equal to J γ,ν +ℓ,k , defined in +Definition 5.1. Thus, Theorem 5.10 and Corollary 5.12 imply that we can efficiently optimize over +SLDP channels. +6.2 +Other Notions of Privacy +We briefly note that our computationally efficient algorithms hold for a wider family of channels +defined in Definition 5.1; see also Remark 5.11. +Finally, we consider the case of Rényi differential privacy introduced in Mironov [Mir17]: +Definition 6.7 ((ϵ, α)-Rényi differential privacy). Let ϵ ∈ R+ and α > 1, and let X and Y be two +domains. A channel T : X → Y satisfies (ϵ, α)-RDP if for all x, x′ ∈ X, we have +Dα(T(x)∥T(x′)) ≤ ϵ, +where Dα(p∥q) is the Rényi divergence of order α between two distributions p and q on the same +probability space, defined as +Dα(p∥q) := +1 +α − 1 log EX∼q +��p(X) +q(X) +�α� +. +Rényi divergence is also defined for α = 1 and α = ∞ by taking limits. When α = 1, the +limit yields the Kullback–Leibler divergence, and when α = ∞, it leads to the supremum of the +log-likelihood ratio between p and q. In fact, (∞, ϵ)-RDP is identical to ϵ-LDP. Similarly, (1, ϵ)-RDP +is closely related to mutual information-based privacy [CY16], since the corresponding channel T +has Shannon capacity at most ϵ. +Proposition 6.8 (Rényi differential privacy and binary constraints). Let ϵ > 0 and α > 1. Let C +be the set of (ϵ, α)-RDP channels from [k] to [2]. Let p and q be two distributions on [k], and define +A := {(Tp, Tq) : T ∈ C}. If (Tp, Tq) is an extreme point of A for T ∈ C, then T can be written as +T1 × T2, where T1 is an extreme point of the set of (ϵ, α)-RDP channels from [2] to [2], and T2 is +a binary threshold channel from [k]. +35 + +Proof. Consider two binary distributions [x, 1 − x] and [y, 1 − y], where 0 ≤ x, y ≤ 1. The α-Rényi +divergence between the distributions is given by +Dα(x∥y) := +1 +α − 1 log +� +xαy1−α + (1 − x)α(1 − y)1−α� +. +Observe that the term inside the logarithm is convex in y for fixed x, and is minimized when y = x. +Hence, the Rényi divergence above, as a function of y, is decreasing for y ∈ [0, x] and increasing for +y ∈ [x, 1]. A similar conclusion holds for fixed y and varying x. +Consider a channel T ∈ C given by +T = +� +x1 +x2 +. . . +xk +1 − x1 +1 − x2 +. . . +1 − xk +� +. +Without loss of generality, assume x1 ≤ x2 ≤ · · · ≤ xk Suppose there is an index j such that +x1 < xj < xk. Observe that xj /∈ {0, 1}. By the monotonicity property of the Rényi divergence +noted above, for any index i, we have +max {Dα(xj∥xi), Dα(xi∥xj)} < max {Dα(x1∥xk), Dα(xk∥x1)} ≤ ϵ. +This means that xj can be perturbed up and down by a small enough δ such that the Rényi +divergence constraints continue to be satisfied. Such perturbations will allow T to be written as a +convex combination of two distinct matrices, so T cannot be an extreme point of the (convex) set +C. Thus, an extreme point must have only two distinct columns; i.e., it must have the form +T = +� +x1 +x1 +. . . +x1 +xk +xk +. . . +xk +1 − x1 +1 − x1 +. . . +1 − x1 +1 − xk +1 − xk +. . . +1 − xk +� +. +Equivalently, any extreme point is a deterministic channel from [k] → [2] followed by an RDP- +channel from [2] → [2]. +Since we are only concerned with extreme points that correspond to +extreme points of the joint range A, an argument identical to the one in the proof of Theorem 5.10 +yields that an extreme point must admit a decomposition T1 ×T2, where T2 is a threshold channel +from [k] → [2] and T1 is an extreme point of the set of RDP channels from [2] → [2]. +The above result implies that given a quasi-convex function g : A → R, if we are interested +in maximizing g(Tp, Tq) over T ∈ C, the optimal T can be written as T1 × T2, where T1 is a +binary-input, binary-output Rényi private channel and T2 is a threshold channel. Since there are +only 2k threshold channels, we can try all those choices of T2, and then try to optimize over T1 for +each of those choices. However, each such problem is over binary inputs and binary outputs, and +thus is amenable to grid search. +Remark 6.9. In addition to the convexity of RDP channels, we also used the closure-under-pre- +processing property (see Claim 5.9) and the unimodality of Dα(x∥y) when one of the variables is +fixed and the other is varied. The above proof technique will therefore work for any set of convex +channels from [k] → [2] that are closed under pre-processing, and are defined in terms of such a +unimodal function. In particular, our results will continue to hold for all f-divergence-based private +channels, defined as all T satisfying +Df(T(x)∥T(x′)) ≤ ϵ. +Our results also hold for zero-concentrated differential privacy (z-CDP) [BS16], which is a notion of +privacy defined using Rényi divergences. +36 + +7 +Conclusion +In this paper, we considered the sample complexity of simple binary hypothesis testing under privacy +and communication constraints. We considered two families of problems: finding minimax-optimal +bounds and algorithms, and finding instance-optimal bounds and algorithms. +For minimax optimality, we considered the set of distributions with fixed Hellinger divergences +and total variation distances. +This is a natural family to consider, because these two metrics +characterize the sample complexity in the low- and high-privacy regimes. Prior work did not resolve +the question of sample complexity in the moderate-privacy regime; our work has addressed this gap +in the literature, by establishing a sample-complexity lower bound via a carefully constructed family +of distribution pairs on the ternary alphabet. Our results highlight a curious separation between +the binary and ternary (and larger alphabet) settings, roughly implying that the binary case is +substantially easier (i.e., has a lower sample complexity) than the general case. +Our focus on instance optimality sets our paper apart from most prior work on information- +constrained estimation, which exclusively considered minimax optimality. When only privacy con- +straints are imposed, we established approximately instance-optimal algorithms; i.e., for any distri- +bution pair, we proposed a protocol whose sample complexity is within logarithmic factors of the +true sample complexity. Importantly, the algorithm we proposed to identify this protocol is compu- +tationally efficient, taking time polynomial in k, the support size of the distributions. When both +privacy and communication constraints are in force, we developed instance-optimal algorithms, i.e., +protocols whose sample complexity is within constant factors of the true sample complexity. As +before, these algorithms take time polynomial in k, for any constant communication constraint of +size ℓ. +Our results highlight the critical role played by threshold channels in both communication- and +privacy-constrained settings. We showed that for any distribution pair, the channel with output size +ℓ that maximizes the output divergence (Hellinger, Kullback–Leibler, or any quasi-convex function +in general) among all channels with fixed output size ℓ must be a threshold channel. Furthermore, +optimal private channels with output size ℓ admit a decomposition into a threshold channel cascaded +with a private channel. These two results underpin our algorithmic contributions. +There are many interesting open problems stemming from our work that would be worth explor- +ing. We did not characterize instance-optimal sample complexity in the moderate-privacy regime; +our work shows that it is not characterized in terms of the Hellinger divergence and total varia- +tion distance, but leaves open the possibility of some other divergence, such as the Eγ divergence, +capturing the sample complexity. We identified a forbidden structure for optimal private channels; +however, the best algorithm from Kairouz, Oh, and Viswanath [KOV16] does not use this infor- +mation at all. It would be interesting to see if that algorithm could be made more efficient by +incorporating the extra structural information. Many open questions remain for the approximate +LDP setting, as well. 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In: Journal of the American Statistical Association +60.309 (1965), pp. 63–69. +A +Randomized Response in Low-Privacy Regime +In this section, we prove Lemmata 3.5 and 3.6, which were used to prove Theorem 1.13 in Section 3. +Lemma 3.5 is proved in Appendix A.1 and Lemma 3.6 is proved in Appendix A.2. +A.1 +Proof of Lemma 3.5 +Recall the definitions of A and A′ from equation (17). +Lemma 3.5 (Randomized response preserves contribution of comparable elements). Let p and q +be two distributions on [ℓ]. Suppose � +i∈A � A′(√qi − √pi)2 ≥ τ. Then Tϵ,ℓ +RR, for ℓ ≤ eϵ, satisfies +d2 +h(Tϵ,ℓ +RRp, Tϵ,ℓ +RRq) ≳ min +� +1, eϵ τ +ℓ +� +· τ . +Thus, when eϵ ≳ +ℓ +τ , the randomized response preserves the original contribution of comparable +elements. +41 + +Proof. Without loss of generality, we will assume that � +i∈A(√qi − √pi)2 ≥ τ +2. Let p′ = Tϵ,ℓ +RRp +and q′ = Tϵ,ℓ +RRq. By the definition of the randomized response, each probability x is mapped to +(1 + x(eϵ − 1))/(k − 1 + eϵ). Thus, p′ and q′ are given by +p′ +i = 1 + pi(eϵ − 1) +(ℓ − 1) + eϵ , +and +q′ +i = 1 + qi(eϵ − 1) +(ℓ − 1) + eϵ , +∀i ∈ ℓ. +(29) +Recall that δi = (pi − qi)/qi ∈ [0, 1]. For each i ∈ ℓ, we now define δ′ +i := (p′ +i − q′ +i)/q′ +i, which has the +following expression in terms of δi and qi: +δ′ +i = p′ +i − q′ +i +q′ +i += (eϵ − 1)(pi − qi) +1 + qi(eϵ − 1) += +(eϵ − 1)qi +1 + qi(eϵ − 1) · δi. +(30) +Let r = 0.01 min +� +e−ϵ, τ +ℓ +� +. We define the following subsets of the domain: +E = {i : δi ∈ (0, 1] and qi ≥ e−ϵ} , +(31) +E′ = {i : δi ∈ (0, 1] and qi ∈ (r, e−ϵ)} . +(32) +Observe that E ∪ E′ ⊆ A. +Since eϵ ≥ ℓ, equation (29) implies that q′ +i ≥ 1 +4(e−ϵ + qi). In particular, on i ∈ E′, we have +q′ +i ≥ 0.25e−ϵ, and on i ∈ E, we have q′ +i ≥ 0.25qi. +We now apply these approximations to equation (30): we lower-bound the numerator by 0.5eϵqiδi +and upper-bound the denominator based on whether i ∈ E or i ∈ E′. On E′, the denominator in +equation (30) is upper-bounded by 2, and on E, the denominator is upper-bounded by 2qieϵ. This +is summarized as follows: for i ∈ E ∪ E′, we have +δ′ +i ≥ +� +0.1δiqieϵ, +i ∈ E′ +0.1δi, +i ∈ E, +q′ +i ≥ +� +0.25e−ϵ, +i ∈ E′ +0.25qi, +i ∈ E . +By definition of δ′, it follows that δ′ +i ∈ (0, 1] on i ∈ E ∪ E′. Thus, the contribution from the ith +element to d2 +h(p′, q′) is at least a constant times q′ +i(δ′ +i)2; see Claim 3.3. Applying this element-wise, +we obtain the following: +d2 +h(p′, q′) ≳ +� +i∈E′ +q′ +i(δ′ +i)2 + +� +i∈E +q′ +i(δ′ +i)2 +≳ +� +i∈E′ +e−ϵ (0.1δiqieϵ)2 + +� +i∈E +qi (0.1δi)2 +≳ eϵr +� +i∈E′ +qiδ2 +i + +� +i∈E +qiδ2 +i . +(33) +Now consider the set A = {i : i ∈ A and qi ≥ r}, which is equal to E ∪ E′. The set A preserves the +contribution to Hellinger divergence from comparable elements, as shown below: +� +i∈A +(√qi − √pi)2 = +� +i∈A +(√qi − √pi)2 − +� +i:i∈A,qi≤r +(√qi − √pi)2 ≥ τ +2 − 2ℓr ≥ τ +4, +since r ≤ +τ +10ℓ. +Since A = E1 ∪ E2, one of the two terms � +i∈E′(√qi − √pi)2 or � +i∈E(√qi − √pi)2 must be at +least τ +8. +Now consider the following two cases: +42 + +Case 1: � +i∈E(√qi − √pi)2 ≳ τ. +In this case, we are done by inequality (33). That is, +d2 +h(p′, q′) ≳ +� +i∈E +( +� +q′ +i − +� +p′ +i)2 ≳ +� +i∈E +qiδ2 +i ≳ +� +i∈E +(√qi − √pi)2 ≳ τ, +where we use Claim 3.3 element-wise. +Case 2: � +i∈E′(√qi − √pi)2 ≳ τ. +By inequality (33), we have +d2 +h(p′, q′) ≳ eϵ · r +� +i∈E′ +qiδ2 +i ≳ eϵ · rτ ≳ min +� +1, eϵ τ +ℓ +� +τ, +where we use the definition of r. +Thus, we obtain the desired lower bound in both of the cases. +A.2 +Proof of Lemma 3.6 +Lemma 3.6 (Reduction to base case). Let p and q be two distributions on [k]. Then there is a +channel T, which can be computed in time polynomial in k, that maps [k] to [ℓ] (for ℓ to be decided +below) such that for p′ = Tp and q′ = Tq, at least one of the following holds: +1. For any ℓ > 2 and ℓ ≤ min +� +k, 1 + log +� +1/d2 +h(p, q) +�� +, we have +� +i∈B � B′ +�� +q′ +i − +� +p′ +i +�2 +≳ d2 +h(p, q) · +ℓ +min +� +k, 1 + log +� +1/d2 +h(p, q) +��, +where B and B′ are defined analogously to A and A′ in equation (17), but with respect to +distributions p′ and q′. +2. ℓ = 2 and d2 +h(p′, q′) ≳ d2 +h(p, q). +Proof. Let us begin by considering the case when � +i∈A � A′ +�√qi − √pi +�2 ≤ +d2 +h(p,q) +2 +. +Following +Pensia, Jog, and Loh [PJL22, Theorem 2 (Case 1 in the proof)], there exists a binary channel that +preserves the Hellinger divergence up to constants. This completes the case for ℓ = 2 above. +Suppose for now that � +i∈A � A′ +�√qi − √pi +�2 ≥ d2 +h(p,q) +2 +, i.e., the comparable elements constitute +at least half the Hellinger divergence. Consider the channel T′ that maps the comparable elements +of p and q to distinct elements, and maps the remaining elements to a single super-element. Let +α be the contribution to the Hellinger divergence from the comparable elements in T′p and T′q +(defined analogously to equation (17)). It can be seen that α ≥ d2 +h(p,q) +2 +. Let ℓ ≥ 3 be as in the +statement. Now consider the channel T′′ that compresses T′p and T′q into ℓ-ary distributions that +preserve the Hellinger divergence, from Pensia, Jog, and Loh [PJL22, Theorem 3.2 (Case 2 in the +proof)]. Let βℓ be the contribution to the Hellinger divergence from the comparable elements in +T′′T′p and T′′T′q. Then the result in Pensia, Jog, and Loh [PJL22, Theorem 3.2] implies that +βl ≳ α +� +ℓ/ min(k, 1 + log(1/d2 +h(p, q))) +� +. This completes the proof in this setting. +B +Properties of Private Channels +Recall the definition of the set of channels J γ,ν +ℓ,k from Definition 5.1 below: +43 + +Definition 5.1 (LP family of channels). For any ℓ ∈ N, let ν = (ν1, ν2, . . . , νℓ) and γ = (γ1, γ2, . . . , γℓ) +be two nonnegative vectors in Rℓ ++. For k ∈ N, define the set of linear programming (LP) channels +J γ,ν +ℓ,k , a subset of Tℓ,k, to be the (convex) set of all channels from [k] to [ℓ] that satisfy the following +constraints: +For each row j ∈ [ℓ], and for each i, i′ ∈ [k], we have T(j, i) ≤ γjT(j, i′) + νj. +(21) +We begin by an equivalent characterization of the constraints above. For a channel T from +[k] to [ℓ], let {m1, . . . , mℓ} and {M1, . . . , Mℓ} be the minimum and maximum entries of each row, +respectively. Then the channel T satisfies the conditions (21) if and only if for each j ∈ [ℓ], we have +Mj ≤ γjmj + νj. +(34) +We first show that J γ,ν +ℓ,k +satisfies Condition 5.3. +For the special case of LDP channels, the +following claim was also proved in Holohan, Leith, and Mason [HLM17]: +Claim B.1. J γ,ν +ℓ,k satisfies Condition 5.3. +Proof. Let T be any extreme point of J γ,ν +ℓ,k . Let {m1, . . . , mℓ} and {M1, . . . , Mℓ} be as defined above. +Suppose that there exists c ∈ [k], such that there exist distinct r, r′ ∈ [ℓ] with T(r, c) ∈ (mr, Mr) +and T(r′, c) ∈ (mr′, Mr′). In particular, both T(r, c) and T(r′, c) are strictly positive and less than +1. +We will now show that T is not an extreme point of J γ,ν +ℓ,k . For an ϵ > 0 to be decided later, +consider the channel T′ that is equal to T on all but two entries: +• On (r, c), T′ assigns probability T(r, c) + ϵ. +• On (r′, c), T′ assigns probability T(r′, c) − ϵ. +Now define T′′ similarly, with the difference being that on (r, c), T′′ assigns probability T(r, c) − ϵ, +and on (r′, c), T′′ assigns probability T(r′, c)+ϵ. Both T′ and T′′ are thus valid channels for ϵ small +enough. Let us show that T′ and T′′ belong to C. If we choose ϵ > 0 small enough, the row-wise +maximum and minimum entries of T′ and T′′ are equal to those of T. Here, we critically use the +fact that the entries that were modified were “free.” By inequality (34), both T′ and T′′ belong to +J γ,ν +ℓ,k . Since T is the average of T′ and T′′, it is not an extreme point of J γ,ν +ℓ,k . +We now show that J γ,ν +ℓ,k satisfies Condition 5.7. +Claim B.2. J γ,ν +ℓ,k satisfies Condition 5.7. +Proof. We follow the notation from Condition 5.7. Let T be an extreme point of J γ,ν +ℓ,k , and let r +and r′ be the corresponding rows. We show that T′ (defined in the condition) belongs to J γ,ν +ℓ,k by +showing that entries of T′ satisfy the constraints of the rth row and the r′th row (since the other +rows are unchanged). In fact, we establish these arguments only for the rth row, and the analogous +arguments hold for the r′th row. +Let mr and Mr be the row-wise minimum and maximum entry of this row in T. Let us first +consider the case when Mr < γrmr + νr. +Then there exist positive ϵ′ and δ′ such that Mr + +δ < γr(mr − ϵ) + νr. By inequality (34), as long as the rth row of a channel contains entries in +[mr − ϵ, Mr + δ], the constraints of this particular row will be satisfied. Since the entries in the rth +row of T′ belong to this interval, the constraints of the rth row are satisfied by T′. +Let us now consider the alternate case where Mr = γrmr+νr. Since m and M do not correspond +to the min-tight and max-tight entries, we have mr < M and m < Mr. Consequently, even after +perturbations by ϵ > 0 and δ > 0 small enough, the entries of T′ lie in [mr, Mr]. Thus, inequality (34) +implies that the constraints of the rth row in T′ are satisfied. +44 + +Claim B.3 (Closure under pre-processing). The set J γ,ν +ℓ,k satisfies the following: +J γ,ν +ℓ,k = +k� +ℓ′=1 +� +T2 × T1 : T2 ∈ J γ,ν +ℓ,ℓ′ and T1 ∈ Tℓ′,k +� +. +(35) +Proof. We first show the simple direction that J γ,ν +ℓ,k ⊆ �k +ℓ′=1 +� +T2 × T1 : T2 ∈ J γ,ν +ℓ,ℓ′ and T1 ∈ Tℓ′,k +� +. +Let Ik correspond to the identity channel on [k]. Then every channel T ∈ J γ,ν +ℓ,k , can be written as +T × I. Thus, J γ,ν +ℓ,k ⊆ +� +T2 × Ik : T2 ∈ J γ,ν +ℓ,ℓ′ +� +, and the desired conclusion follows. +We now show that every channel in the right-hand side belongs to J γ,ν +ℓ,k . For an arbitrary ℓ′ ∈ [k], +let T2 ∈ J γ,ν +ℓ,ℓ′ . Define {m1, . . . , mℓ} and {M1, . . . , Mℓ} to be the minimum and maximum entries +of each row in T2, respectively. By inequality (34), for each j ∈ [ℓ], we have Mj ≤ γjmj + νj. Let +T1 ∈ Tℓ′,k be an arbitrary channel. +Let T = T2 × T1 be in Tℓ,k, and let {m1, . . . , mℓ} and {M′ +1, . . . , M′ +ℓ} be the minimum and +maximum entries of each row in T, respectively. In order to show that T ∈ J γ,ν +ℓ,k , we need to show +that for each j ∈ [ℓ], we have M′ +j ≤ γjm′ +j + νj. Since it already holds that Mj ≤ γjmj + νj for all +j, it suffices to show that M′ +j ≤ Mj and m′ +j ≥ mj for all j. Observe that for any c ∈ [k] and r ∈ [ℓ], +the (r, c)-entry of T is a convex combination of the rth row in T2, where the weights in the convex +combination correspond to the cth column in T1. Since the maximum of a collection of items is +always as large as any convex combination of these items, we have M′ +j ≤ Mj for all j. Similarly, we +have m′ +j ≥ mj. This completes the proof. +C +Other Notions of Privacy +We provide the proof of the following result, omitted from Section 6: +Claim 6.4 (Sample complexity of approximate LDP). For all δ ∈ (0, 1), we have +n∗(p, q, (ϵ, δ)) ≲ min +� +n∗(p, q, ϵ) · +1 +1 − δ , n∗(p, q) · 1 +δ +� +. +Moreover, this is tight (up to constant factors) when both p and q are binary distributions. +Proof. Let T be an ϵ-LDP channel that maximizes d2 +h(Tp, Tq) among all ϵ-LDP channels. Let T′ +be the following channel that maps from [k] to [2k]: for any element i ∈ [k], use the channel T, +and with probability δ, map i to k + i. It can be seen that T′ satisfies (ϵ, δ)-LDP. Let p′ and q′ +be the corresponding distributions after transforming p and q using T′. It can be seen that p′ is a +distribution over [2k] such that the first k elements are equal to (1 − δ)Tp coordinate-wise, and the +bottom k elements are equal to δp coordinate-wise. A similar conclusion holds for q′, as well. Thus, +we have +d2 +h(T′p, T′q) = (1 − δ) · d2 +h(Tp, Tq) + δ · d2 +h(p, q) +≍ max +� +(1 − δ) · d2 +h(Tp, Tq), δ · d2 +h(p, q) +� +≍ max +� +(1 − δ) · +1 +n∗(p, q, ϵ), δ · +1 +n∗(p, q) +� +. +By Fact 2.7, the sample complexity n∗(p, q, (ϵ, δ)) is at most 1/d2 +h(T′p, T′q), which gives the upper +bound on n∗(p, q, (ϵ, δ)). +The tightness follows from the result of Kairouz, Oh, and Viswanath [KOV16, Theorem 18], +which implies that T′ defined above is an optimal channel for binary distributions. +45 + +D +Auxiliary Lemmas +D.1 +Degenerate Conditions for Joint Range +We show in this section that we can safely rule out certain degenerate conditions for p and q for +our results. Let p and q be two distributions on [k]. In particular, we would like to assume the +following: +• Consider the likelihood ratio pi/qi, defined to be ∞ if qi = 0 and pi ̸= 0, and undefined if +both pi and qi are 0. Assume that all the likelihood ratios are well-defined and unique. +If these conditions do not hold, define p′ and q′ to be distributions over [k′] for some k′ ≤ k, +constructed as follows: start by removing elements that have zero probability mass under both p +and q, then merge the elements with the same likelihood ratios into super-elements. Let T∗ ∈ Tk′,k +be the corresponding deterministic map, which satisfies p′ = T∗p and q′ = T∗q. We make the +following claim: +Claim D.1. With the notation above, for any ℓ ∈ N and T ∈ Tℓ,k, there exists T′ ∈ Tℓ,k′ such that +(Tp, Tq) = (T′p′, T′q′). In particular, {(Tp, Tq) : T ∈ C} = {(Tp′, Tq′) : T ∈ C′} for two choices +of C and C′: (i) (C, C′) = (Tℓ,k, Tℓ,k′) and (ii) (C, C′) = (Pϵ +ℓ,k, Pϵ +ℓ,k′). +Claim D.1 ensures that the joint ranges of (p, q) and (p′, q′) are identical, so our structural and +algorithmic results continue to hold when applied to p′ and q′. We will now prove Claim D.1. +Proof of Claim D.1. Let {I0, I1, . . . , Ik′} be the smallest partition of [k] such that I0 contains ele- +ments where both pi and qi are zero, and for each i ∈ [k′], the likelihood ratio of elements in Ii are +identical. Then the channel T∗ mentioned above has the following form: T∗(x) = i if x ∈ Ii and +i > 0, and T∗(x) = 1 if x ∈ I0. Observe that for each i ∈ [k′], we have p′ +i = � +j∈Ii pj, q′ +i = � +j∈Ii qj, +and at most one of them is zero. +Now consider a channel T ∈ Tℓ,k, and let {v1, . . . , vk} be the columns of T. It is easy to see +that columns belonging to indices in I0 do not affect (Tp, Tq). For i ∈ [k′], define θ′ +i := p′ +i/q′ +i to +be the likelihood ratio of the transformed distributions. Define T′ to be the channel with columns +v′ +1, . . . , v′ +k such that +v′ +i = +� +� +� +� +j∈Ii vjpj +p′ +i +if p′ +i > 0, +� +j∈Ii vjqj +q′ +i +otherwise. +First consider the case when for all i ∈ [k′], we have 0 < θ′ +i < ∞. Then for all i ∈ [k′], we have +p′ +i = θ′ +iq′ +i and v′ +i = +� +j∈Ii vjpj +p′ +i += +� +j∈Ii vjqj +q′ +i +. Thus, we have +(Tp, Tq) = +� +� � +i∈[k′] +� +j∈Ii +vjpj, +� +i∈[k′] +� +j∈Ii +vjqj +� +� += +� +� � +i∈[k′] +p′ +i · +�� +j∈Ii vjpj +p′ +i +� +, +� +i∈[k′] +q′ +i · +�� +j∈Ii vjqj +q′ +i +�� +� += +� +� � +i∈[k′] +p′ +iv′ +i, +� +i∈[k′] +q′ +iv′ +i +� +� = +� +T′p′, T′q′� +. +46 + +We now consider the case when there is an index a ∈ [k′] such that p′ +a = 0 and an index b ∈ [k′] such +that q′ +b = 0. Then it must be that θ′ +a = 0 and θ′ +b = ∞. Then v′ +a = +� +j∈Ii vjqj +q′ +i +and v′ +b = +� +j∈Ii vjpj +p′ +i +. +Following the calculations above, we obtain � +j∈Ii vjpj = v′ +ip′ +i for each i ∈ [k] \ {a}. In fact, the +same result is true for i = a, since both sides are 0. The same conclusion holds for q and q′, as well. +This completes the proof of the first claim. +We now turn to the final claim, regarding the joint range under the channel constraints of +C. The case C = Tℓ,k is immediate from the preceding discussion. Let T1 ∈ Tk,k′ be such that +(p, q) = (T1p′, T1q′) and T2 ∈ Tk′,k be such that (p′, q′) = (T2p, T2q). For C = Pϵ +ℓ,k and C′ = Pϵ +ℓ,k′, +we only need to show that (i) if T′ ∈ C′, then T′T2 ∈ C; and (ii) if T ∈ C, then TT1 ∈ C′. Both of +these conditions hold because privacy is closed under pre-processing. +D.2 +Valid Choice of Parameters in Theorem 1.7 +We now give the details that were omitted in the proof of Theorem 1.7 in Section 3.2. +We first reparametrize the problem by setting x = γ and y = γ1+δ. The constraint δ > 0 is +equivalent to y < x. Then dTV(p, q) = x + y, and +d2 +h(p, q) = 2y + +�� +1/2 + x − y − +� +1/2 +�2 ++ +�� +1/2 − x − y − +� +1/2 +�2 +. +We begin by setting ν = x + y, which is possible since 0 ≤ y < x < 0.25 and ν ∈ (0, 0.5). Then +x = ν − y, where y ∈ (0, ν/2) and ν ∈ (0, 0.5). Our goal is now to show that there exists a valid +choice of y such that d2 +h(p, q) = ρ, as long as 2ν2 ≤ ρ ≤ ν. +Define g(y) to be the Hellinger divergence between p and q given y, i.e., +g(y) = 2y + +�� +1/2 + ν − 2y − +� +1/2 +�2 ++ +�� +1/2 − ν − +� +1/2 +�2 +. +Since g is a continuous function, it suffices to show that g(0) < 2ν2 and g(ν/2) > ν, which would +imply that there is a choice of y ∈ (0, ν/2) such that g(y) = ρ. We have +g(0) = +�� +1/2 + ν − +� +1/2 +�2 ++ +�� +1/2 − ν − +� +1/2 +�2 +≤ 3ν2/2, +where we use the fact that +��� +� +1/2 + a − +� +1/2 +��� ≤ a for all a ≥ 0, and is less than |a|/2 for a ≤ 0. +On the other hand, g(ν/2) > ν, since ν < 1/2. Thus, there is a choice of y ∈ (0, ν/2) such that +d2 +h(p, q) = ρ. Given these choices of x and y, we can infer the choice of γ ∈ (0, 0.25) and δ > 0. +D.3 +Taylor Approximation to Hellinger Divergence +Claim 3.3 (Additive approximation for √· ). There exist constants 0 < c1 ≤ c2 such that for +0 < y ≤ x, we have c1 · y2 +x ≤ (√x − √x − y)2 ≤ c2 · y2 +x . +Proof. It suffices to prove that for δ ∈ (0, 1], we have 1 − +√ +1 − δ ≍ δ. We first start with the upper +bound: since 1 − δ ≤ +√ +1 − δ, we have 1 − +√ +1 − δ ≤ δ. We now show the lower bound and claim +that 1 − +√ +1 − δ ≥ 0.5δ for all δ ∈ [0, 1]. This inequality is equivalent to showing 1 − 0.5δ ≥ +√ +1 − δ, +which is equivalent to showing that 1 + 0.25δ2 − δ ≥ 1 − δ, which holds since δ2 ≥ 0. +Claim 3.2 (Approximation for Hellinger divergence of binary distributions). Let p, q ∈ [0, 1]. Let +Ber(p) and Ber(q) be the corresponding Bernoulli distributions with min(p, q) ≤ 1/2. Then +d2 +h (Ber(p), Ber(q)) ≍ d2 +TV(Ber(p), Ber(q)) +max(p, q) +. +47 + +Proof. Let q be the larger of the two quantities, so p satisfies p ≤ 1 +2. The total variation distance is +thus q − p. Let δ = (q − p)/q ∈ (0, 1]. Observe that p = q − qδ and the total variation distance is +δq. +We begin by noting that Claim 3.3 implies that +(√q − √p)2 = +�√q − +� +q − δq +�2 +≍ δ2q2 +q +≍ d2 +TV (Ber(p), Ber(q)) +q +. +(36) +We now split the analysis into two cases: +Case 1: q ≤ 1/2. +Then Pensia, Jog, and Loh [PJL22, Claim F.2] implies that d2 +h(Ber(p), Ber(q)) ≍ +(√q − √p)2. Thus, equation (36) implies the result. +Case 2: q ≥ 1/2. +Applying Claim 3.3 again to the second term, we obtain +�� +1 − p − +� +1 − q +�2 += +�� +1 − p − +� +1 − p − qδ +�2 +≍ q2δ2 +1 − p ≍ q2δ2 +q +≍ d2 +TV(Ber(p), Ber(q)) +q +, (37) +where we use the fact that 1 − p ≍ q, since p, q ∈ [0.5, 1]. The desired conclusion follows from +equations (36) and (37). +48 + diff --git a/D9E1T4oBgHgl3EQf-QZ6/content/tmp_files/load_file.txt b/D9E1T4oBgHgl3EQf-QZ6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac8716ddabe703cc3425694024fd7eae086ed158 --- /dev/null +++ b/D9E1T4oBgHgl3EQf-QZ6/content/tmp_files/load_file.txt @@ -0,0 +1,2285 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf,len=2284 +page_content='Simple Binary Hypothesis Testing under Local Differential Privacy and Communication Constraints Ankit Pensia University of Wisconsin-Madison ankitp@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='edu Amir R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Asadi University of Cambridge aa2345@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='uk Varun Jog University of Cambridge vj270@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='uk Po-Ling Loh University of Cambridge pll28@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='uk January 10, 2023 Abstract We study simple binary hypothesis testing under both local differential privacy (LDP) and communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We qualify our results as either minimax optimal or instance opti- mal: the former hold for the set of distribution pairs with prescribed Hellinger divergence and total variation distance, whereas the latter hold for specific distribution pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For the sample complexity of simple hypothesis testing under pure LDP constraints, we establish instance- optimal bounds for distributions with binary support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' minimax-optimal bounds for general distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' and (approximately) instance-optimal, computationally efficient algorithms for general distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' When both privacy and communication constraints are present, we de- velop instance-optimal, computationally efficient algorithms that achieve the minimum possible sample complexity (up to universal constants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our results on instance-optimal algorithms hinge on identifying the extreme points of the joint range set A of two distributions p and q, defined as A := {(Tp, Tq)|T ∈ C}, where C is the set of channels characterizing the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1 Introduction Statistical inference on distributed data is becoming increasingly common, due to the proliferation of massive datasets which cannot be stored on a single server, and greater awareness of the security and privacy risks of centralized data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' An institution (or statistician) that wishes to infer an aggregate statistic of such distributed data needs to solicit information, such as the raw data or some relevant statistic, from data owners (individuals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Individuals may be wary of sharing their data due to its sensitive nature or their lack of trust in the institution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The local differential privacy (LDP) paradigm suggests a solution by requiring that individuals’ responses divulge only a limited amount of information about their data to the institution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Privacy is typically ensured by deliberately randomizing individuals’ responses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', by adding noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' See Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 below for a formal definition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' we refer the reader to Dwork and Roth [DR13] for more details on differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In this paper, we study distributed estimation under LDP constraints, focusing on simple bi- nary hypothesis testing, a fundamental problem in statistical estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will also consider LDP constraints in tandem with communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This is a more realistic setting, since bandwidth considerations often impose constraints on the size of individuals’ communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='03566v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='ST] 9 Jan 2023 case when only communication constraints are present was addressed previously by Pensia, Jog, and Loh [PJL22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Recall that simple binary hypothesis testing is defined as follows: Let p and q be two distributions over a finite domain X, and let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Xn ∈ X n be n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' samples drawn from either p or q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The goal of the statistician is to identify (with high probability) whether the samples were drawn from p or q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This problem has been extensively studied in both asymptotic and nonasymptotic settings [NP33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Wal45;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Cam86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For example, it is known that the optimal test for this problem is the likelihood ratio test, and its performance can be characterized in terms of divergences between p and q, such as the total variation distance, Hellinger divergence, or Kullback–Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, the sample complexity of hypothesis testing, defined as the smallest sample size needed to achieve an error probability smaller than a small constant, say, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='01, is Θ � 1 d2 h(p,q) � , where d2 h(p, q) is the Hellinger divergence between p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In the context of local differential privacy, the statistician no longer has access to the original samples X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Xn, but only their privatized counterparts: Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Yn ∈ Yn, for some set Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Each Xi is transformed to Yi via a private channel Ti, which is simply a probability kernel specifying Ti(y, x) = P(Yi = y|Xi = x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' With a slight abuse of notation, we shall use Ti to denote the transition kernel in R|Y|×|X|, as well as the stochastic map Yi = Ti(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A formal definition of privacy is given below: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 (ϵ-LDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let ϵ ∈ R+, and let X and Y be two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A channel T : X → Y satisfies ϵ-LDP if sup x,x′∈X sup A⊆Y P[T(x) ∈ A] − eϵ · P[T(x′) ∈ A] ≤ 0, where we interpret T as a stochastic map on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Equivalently, if X and Y are countable domains (as will be the case for us), a channel T is ϵ-LDP if supx,x′∈X supy∈Y T(y,x) T(y,x′) ≤ eϵ, where we interpret T as the transition kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' When ϵ = ∞, we may set Yi equal to Xi with probability 1, and we recover the vanilla version of the problem with no privacy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Existing results on simple binary hypothesis testing under LDP constraints have focused on the high-privacy regime of ϵ ∈ (0, c), for a constant c > 0, and have shown that the sample complexity is Θ � 1 ϵ2d2 TV(p,q) � , where dTV(p, q) is the total variation distance between p and q (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, when ϵ is a constant, the sample complexity is Θ � 1 d2 TV(p,q) � , and when ϵ = ∞ (no privacy), the sample complexity is Θ � 1 d2 h(p,q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Although these two divergences satisfy d2 TV(p, q) ≲ d2 h(p, q) ≲ dTV(p, q), the bounds are tight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', the two sample complexities can be quadratically far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Existing results therefore do not inform sample complexity when 1 ≪ ϵ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This is not an artifact of analysis: the optimal tests in the low and high privacy regimes are fundamentally different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The large-ϵ regime has been increasingly used in practice, due to privacy amplification provided by shuffling [CSUZZ19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' BEMMRLRKTS17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' FMT21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our paper makes progress on the computa- tional and statistical fronts in the large-ϵ regime, as will be highlighted in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1As shown in Kairouz, Oh, and Viswanath [KOV16], for simple binary hypothesis testing, we can take Y to be X, with the same sample complexity (up to constant factors);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' see Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Problem Setup For a natural number k, we use [k] to denote the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In our paper, we focus on the private-coin, non-interactive protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 As we will be working with both privacy and communication constraints in this paper, we first define the generic protocol for distributed inference under an arbitrary set of channels C below: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 (Simple binary hypothesis testing under channel constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let X and Y be two countable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C be a set of channels from X to Y, and let p and q be two distributions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let {Ui}n i=1 denote a set of n users who choose channels {Ti}n i=1 ∈ Cn according to a deterministic rule3 R : [n] → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Each user Ui then observes Xi and generates Yi = Ti(Xi) independently, where X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Xn is a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' random variables drawn from an (unknown) r ∈ {p, q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The central server U0 observes (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Yn) and constructs an estimate �r = φ(Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Yn), for some test φ : ∪∞ i=1Yi → {p, q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We refer to this problem as simple binary hypothesis testing under channel constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In the non-interactive setup, we can assume that all Ti’s are identical equal to some T, as it will increase the sample complexity by at most a constant factor [PJL22] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now specialize the setting of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 to the case of LDP constraints: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 (Simple binary hypothesis testing under LDP constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consider the problem in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 with Y = N, where C is the set of all ϵ-LDP channels from X to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We denote this problem by B(p, q, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For a given test-rule pair (φ, R) with φ : ∪∞ j=1Yj → {p, q}, we say that (φ, R) solves B(p, q, ϵ) with sample complexity n if P(X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=',Xn)∼p⊗n(φ(Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Yn) ̸= p) + P(X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=',Xn)∼q⊗n(φ(Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Yn) ̸= q) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (1) We use n∗(p, q, ϵ) to denote the sample complexity of this task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', the smallest n so that there exists a (φ, R)-pair that solves B(p, q, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We use B(p, q) and n∗(p, q) to refer to the setting of non- private testing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', when ϵ = ∞, which corresponds to the case when C is the set of all possible channels from X to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any fixed rule R, the optimal choice of φ corresponds to the likelihood ratio test on {Yi}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, in the rest of this paper, our focus will be optimizing the rule R, with the choice of φ made implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In fact, we can take Y to be X, at the cost of a constant-factor increase in the sample complexity [KOV16] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now define the threshold for free privacy, in terms of a large enough universal constant Cthresh > 0 which can be explicitly deduced from our proofs: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 (Threshold for free privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We define ϵ∗(p, q) (also denoted by ϵ∗ when the context is clear) to be the smallest ϵ such that n∗(p, q, ϵ) ≤ Cthresh · n∗(p, q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', for all ϵ ≥ ϵ∗(p, q), we can obtain ϵ-LDP without any substantial increase in sample complexity compared to the non-private setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Next, we study the problem of simple hypothesis testing under both privacy and communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By communication constraints, we mean that the channel T maps from X to [ℓ] for some ℓ ∈ N, which is potentially much smaller than |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2We refer the reader to Acharya, Canonne, Liu, Sun, and Tyagi [ACLST22] for differences between various protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 3When C is a convex set of channels, as will be the case in this paper, the deterministic rules are equivalent to randomized rules (with independent randomness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 3 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 (Simple binary hypothesis testing under LDP and communication constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consider the problem in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3, with C equal to the set of all channels that satisfy ϵ-LDP and Y = [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We denote this problem by B(p, q, ϵ, ℓ), and use n∗(p, q, ϵ, ℓ) to denote its sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Communication constraints are worth studying not only for their practical relevance in dis- tributed inference, but also for their potential to simplify algorithms without significantly impacting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Indeed, the sample complexities of simple hypothesis testing with and without com- munication constraints are almost identical [BNOP21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' PJL22] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8), even for a single-bit (ℓ = 2) communication constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As we explain later, a similar statement can be made for privacy constraints, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 Existing Results As noted earlier, the problem of simple hypothesis testing with just communication constraints was addressed in Pensia, Jog, and Loh [PJL22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since communication and privacy constraints are the most popular information constraints studied in the literature, the LDP-only and LDP-with- communication-constraints settings considered in this paper are natural next steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Many of our results, particularly those on minimax-optimal sample complexity bounds, are in a similar vein as those in Pensia, Jog, and Loh [PJL22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Before describing our results, let us briefly mention the most relevant prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We discuss further related work in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Existing results on sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Existing results (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Duchi, Jordan, and Wainwright [DJW18, Theorem 1] and Asoodeh and Zhang [AZ22, Theorem 2]) imply that n∗(p, q, ϵ) ≳ � � � � � � � � � 1 ϵ2 · d2 TV(p,q), if ϵ ∈ (0, 1], 1 eϵ · d2 TV(p,q), if eϵ ∈ � e, d2 h(p,q) d2 TV(p,q) � , 1 d2 h(p,q), if eϵ > d2 h(p,q) d2 TV(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (2) An upper bound on the sample complexity can be obtained by choosing a specific private channel T and analyzing the resulting test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A folklore result (see, for example, Joseph, Mao, Neel, and Roth [JMNR19, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1]) shows that setting T = TRR × TScheffe, where TScheffe maps X to {0, 1} using a threshold rule based on p(x) q(x), and TRR is the binary-input binary-output randomized response channel, gives n∗(p, q, ϵ) ≲ 1 min(1,ϵ2) · d2 TV(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This shows that when ϵ ∈ (0, 1] (or (0, c], for some constant c), the lower bound is tight up to constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that for any ℓ ≥ 2, the sample complexity with privacy and communication constraints n∗(p, q, ϵ, ℓ) also satisfies the same lower and upper bounds, since the channel T has only two outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' However, the following questions remain unanswered: What is the optimal sample complexity for ϵ ≫ 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, are the existing lower bounds (2) tight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' What is the threshold for free privacy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1, we establish minimax-optimal bounds on the sample complexity for all values of ϵ, over sets of distribution pairs with fixed total variation distance and Hellinger divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, we show that the lower bounds (2) are tight for binary distributions, but may be arbitrarily loose for general distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 4 Existing results on computationally efficient algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Recall that each user needs to select a channel T to optimize the sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Once T is chosen, the optimal test is simply a likelihood ratio test between Tp and Tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, the computational complexity lies in determining T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As noted earlier, for ϵ ≤ 1, the optimal channel is T = TRR ×TScheffe, and this can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' However, this channel T may no longer be optimal in the regime of ϵ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As with statistical rates, prior literature on finding optimal channels for ϵ ≫ 1 is scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Existing algorithms either take time exponential in the domain size [KOV16], or their sample complexity is suboptimal by polynomial factors (depending on 1 d2 TV(p,q), as opposed to 1 d2 h(p,q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This raises the following natural question: Is there a polynomial-time algorithm that finds a channel T whose sample complexity is (nearly) optimal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We answer this question in the affirmative in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 Our Results We are now ready to describe our results in this paper, which we outline in the next three subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 focuses on the sample complexity of simple hypothesis testing under local privacy, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 focuses on structural properties of the extreme points of the joint range under channel constraints, and Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 states our algorithmic guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Statistical Rates We begin by analyzing the sample complexity when both p and q are binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We prove the following result in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1, showing that the existing lower bounds (2) are tight for binary distributions: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 (Sample complexity of binary distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then n∗(p, q, ϵ) ≍ � � � � � � � � � 1 ϵ2 · d2 TV(p,q), if ϵ ≤ 1, 1 eϵ · d2 TV(p,q), if eϵ ∈ � e, d2 h(p,q) d2 TV(p,q) � , 1 d2 h(p,q), if eϵ > d2 h(p,q) d2 TV(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (3) In particular, the threshold ϵ∗ for free privacy (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4) satisfies eϵ∗ ≍ d2 h(p,q) d2 TV(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Note that the sample complexity n∗(p, q, ϵ) for all ranges of ϵ is completely characterized by the total variation distance and Hellinger divergence between p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A natural set to consider is all distribution pairs (not just those with binary support) with a prescribed total variation distance and Hellinger divergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' we investigate minimax-optimal sample complexity over this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our next result shows that removing the binary support condition radically changes the sample complexity, even if the total variation distance and Hellinger divergence are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Specifically, we show that there are ternary distribution pairs whose sample complexity (as a function of the total variation distance and Hellinger divergence) is significantly larger than the corresponding sample complexity for binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 (Sample complexity lower bound for general distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any ρ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5) and ν ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5) such that 2ν2 ≤ ρ ≤ ν, there exist ternary distributions p and q such that d2 h(p, q) = ρ, 5 dTV(p, q) = ν, and the sample complexity behaves as n∗(p, q, ϵ) ≍ � � � � � � � 1 ϵ2 · d2 TV(p,q), if ϵ ≤ 1, min � 1 d2 TV(p,q), 1 eϵ · d4 h(p,q) � , if eϵ ∈ � e, 1 d2 h(p,q) � , 1 d2 h(p,q), if eϵ > 1 d2 h(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (4) We prove this result in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We highlight the differences between the sample complexity in the binary setting (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' equation (3)) and the worst-case general distributions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' equation (4)) below (also see Figure 1): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Relaxing privacy may not lead to significant improvements in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=') In equation (4), there is an arbitrarily large range of ϵ where the sample complexity remains roughly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, when e ≤ eϵ ≲ d2 TV(p,q) d4 h(p,q) , the sample complexity of hypothesis testing remains roughly the same (up to constants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' That is, we are sacrificing privacy without any significant gains in statistical efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This is in stark contrast to the binary setting, where increasing eϵ by a large constant factor leads to a constant-factor improvement in sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (The threshold for free privacy is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=') Let ϵ∗ := ϵ(p, q) be the threshold for free privacy (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In the binary setting, one has eϵ∗ ≍ d2 h(p,q) d2 TV(p,q), whereas for general distributions, one may need eϵ∗ ≳ 1 d2 h(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The former ϵ∗ can be arbitrarily smaller than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' To complement the result above, which provides a lower bound on the sample complexity for worst-case distributions, our next result provides an upper bound on the sample complexity that nearly matches the rates (up to logarithmic factors) for arbitrary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, the proposed algorithm uses an ϵ-LDP channel with binary outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The following result is proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9 (Sample complexity upper bounds and an efficient algorithm for hypothesis testing for general distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then the sample complexity behaves as n∗(p, q, ϵ) ≲ � � � � � � � 1 ϵ2 · d2 TV(p,q), if ϵ ≤ 1, min � 1 d2 TV(p,q), α2 eϵ · d4 h(p,q) � , if eϵ ∈ � e, α d2 h(p,q) � , α d2 h(p,q), if eϵ > α d2 h(p,q), (5) where α ≲ log(1/d2 h(p, q)) ≍ log (n∗(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, the rates above are achieved by an ϵ-LDP channel T that maps [k] to [2] and can be found in time polynomial in k, for any choice of p, q, and ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9 imply that the above sample complexity is minimax optimal (up to logarithmic factors) over the class of distributions with total variation distance ν and Hellinger divergence ρ satisfying the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We summarize this in the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='10 (Minimax-optimal bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let ρ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5) and ν ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5) be such that 2ν2 ≤ ρ ≤ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let Sρ,ν be the set of all distribution pairs with discrete supports, with total variation distance and Hellinger divergence being ν and ρ, respectively: Sρ,ν := {(p, q) : k ∈ N, p ∈ ∆k, q ∈ ∆k, dTV(p, q) = ν, d2 h(p, q) = ρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 6 Let n∗(Sρ,ν, ϵ) be the minimax-optimal sample complexity of hypothesis testing under ϵ-LDP con- straints, defined as n∗(Sρ,ν, ϵ) = min (φ,R) max (p,q)∈Sρ,ν n∗(p, q, ϵ), for test-rule pairs (φ, R), as defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then n∗(Sρ,ν, ϵ) = � � � � � � � �Θ � 1 ϵ2 · ν2 � , if ϵ ≤ 1, �Θ � min � 1 ν2 , 1 eϵ · ρ2 �� , if eϵ ∈ � e, 1 ρ � , �Θ � 1 ρ � , if eϵ > 1 ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (6) Here, the �Θ notation hides poly-logarithmic factors in 1/ν and 1/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A version of the above theorem may also be stated for privacy and communication constraints, by defining n∗(Sρ,ν, ϵ, ℓ) = min (φ,R) max (p,q)∈Sρ,ν n∗(p, q, ϵ, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In fact, it may seen that the same sample complexity bounds continue to hold for n∗(Sρ,ν, ϵ, ℓ), with ℓ ≥ 2, since the lower bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 continues to hold with communication constraints, as does the upper bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9, which uses a channel with only binary outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The above theorem mirrors a minimax optimality result for communication-constrained hypothesis testing from Pensia, Jog, and Loh [PJL22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' There, the set under consideration was Sρ, where ρ is the Hellinger divergence between the distribution pair, and the minimax-optimal sample complexity was shown to be �Θ(1/ρ) even for a binary communication constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Finally, we consider the threshold for free privacy ϵ∗ for general distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9 does not provide any upper bounds on ϵ∗, since the sample complexity in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9 is bounded away from n∗(p, q), due to the logarithmic multiplier α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Recall that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 implies eϵ∗ ≳ 1 d2 h(p,q) in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our next result, proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3, shows that this is roughly tight, and eϵ∗ ≲ 1 d2 h(p,q) · log � 1 d2 h(p,q) � for all distributions: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k], and let eϵ ≳ 1 d2 h(p,q) log � 1 d2 h(p,q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then n∗(p, q, ϵ) ≍ n∗(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, there is a channel T achieving this sample complexity that maps [k] to a domain of size ⌈log(n∗(p, q))⌉, and which can be computed in poly(k, log(⌈n∗(p, q)⌉)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We thereby settle the question of minimax-optimal sample complexity (up to logarithmic fac- tors) for simple binary hypothesis testing under LDP-only and LDP-with-communication constraints (over the class of distributions with a given total variation distance and Hellinger divergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' More- over, the minimax-optimal upper bounds are achieved by computationally efficient, communication- efficient algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' However, there can be a wide gap between instance-optimal and minimax- optimal procedures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' in the next two subsections, we present structural and computational results for instance-optimal algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 7 100 101 102 103 104 105 106 107 108 109 eϵ 108 109 1010 1011 min T∈Pϵ 1 d2 h(Tp, Tp) 1 d2 h(p,q) 1 d2 TV(p,q) 1 d2 h(p,q) d2 h(p,q) d2 TV(p,q) d2 TV(p,q) d4 h(p,q) Contraction of Hellinger Divergence under LDP Binary p and q Ternary p and q Figure 1: In this plot, we show the difference between the behavior of sample complexity under ϵ- LDP constraints for binary distributions and (worst-case) ternary distributions from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We take two pairs of distributions (p, q)—one pair of binary distributions (shown in blue, with marker ◦) and one pair of ternary distributions (shown in orange, with marker +)—such that the two pairs have Hellinger divergence d2 h(p, q) = 10−8 and total variation distance dTV(p, q) = 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For each value of ϵ, shown on the horizontal axis after being mapped to eϵ, we compute minT∈Pϵ 1/d2 h(Tp, Tq), where Pϵ is the set of all ϵ-LDP channels, and plot it on vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, the vertical axis characterizes the sample complexity n∗(p, q, ϵ) of simple binary hypothesis testing between p and q with privacy constraints, up to constant factors (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Both axes are shown in log-scale here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since the total variation distance between the two pairs is identical, we see that their curves overlap for small ϵ (ϵ ≪ 1, which is consistent with the fact that n∗(p, q, ϵ) ≍ 1 ϵ2d2 TV(p,q) for small ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As predicted by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6, the curve for binary distributions decreases rapidly for ϵ ≫ 1 until it saturates at 1/d2 h(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, for eϵ ≍ d2 h(p, q)/d2 TV(p, q), the predicted threshold for free privacy, the vertical axis is within constant factors of its asymptotic value, as predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' On the other hand, the curve for ternary distributions seems to have three different phases, as predicted by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7: (i) for small ϵ, it behaves as 1/(ϵ2d2 TV(p, q));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (ii) for moderate values of ϵ, such that e ≪ eϵ ≪ d2 TV(p,q) d4 h(p,q) , it remains stagnant roughly at 1 d2 TV(p,q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' and (iii) for eϵ ≫ d2 TV(p,q) d4 h(p,q) , the curve decreases rapidly until it approaches 1/d2 h(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The phase (ii) corresponds to the phenomenon that we are leaking privacy without any gains in statistical efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Finally, eϵ needs to be as large as 1/d2 h(p, q) for the vertical axis to be within a factor of 10 of its asymptotic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We refer the reader to Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 Structure of Extreme Points under the Joint Range In this section, we present results for the extreme points of the joint range of an arbitrary pair of distributions when transformed by a set of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Formally, if C is a convex set of channels from X to Y, and p and q are two distributions on X, we are interested in the extreme points of the set A := {(Tp, Tq) : T ∈ C}, which is a convex subset of ∆|Y| × ∆|Y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 The extreme points of a convex set are naturally insightful for maximizing quasi-convex functions, and we will present the consequences of the results in this section in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We consider two choices of C: first, when C is the set of all channels from X to Y = [ℓ], and second, when C is the set of all ϵ-LDP channels from X to Y = [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We use Tℓ,k to denote the set of all channels that map from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The following class of deterministic channels plays a critical role in our theory: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='14 (Threshold channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For some k ∈ N, let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any ℓ ∈ N, a deterministic channel T ∈ Tℓ,k is a threshold channel if the following property holds for every u, v ∈ [k]: If p(u) q(u) < p(v) q(v) and T(u) = T(v), then any w ∈ [k] such that p(w) q(w) ∈ � p(u) q(u), p(v) q(v) � satisfies T(w) = T(u)(= T(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (The likelihood ratios are assumed to take values on the extended real line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', on R ∪ {−∞, +∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=') Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Threshold channels are intuitively easy to understand when all the likelihood ratios are distinct (this may be assumed without loss of generality in our paper, as explained later): Arrange the inputs in increasing order of their likelihood ratios and partition them into ℓ contiguous blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, there are at most kℓ such threshold channels (up to reordering of output labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our first result proved in Section 4 is for the class of communication-constrained channels, and shows that all extreme points of the joint range are obtained using deterministic threshold channels: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16 (Extreme points of the joint range under communication constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A be the set of all pairs of distributions that are obtained by passing p and q through a channel of output size ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', A = {(Tp, Tq) : T ∈ Tℓ,k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If (Tp, Tq) is an extreme point of A, then T is a threshold channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We note that the above result is quite surprising: (Tp, Tq) is extreme point of A only if T is an extreme point of Tℓ,k (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', a deterministic channel), but Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16 demands that T be a deterministic threshold channel, meaning it lies in a very small subset of deterministic channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Indeed, even for ℓ = 2, the number of deterministic channels from [k] to [2] is 2k, whereas the number of threshold channels is just 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We note that the result above is similar in spirit to Tsitsiklis [Tsi93, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' However, the focus there was on a particular objective, the probability of error in simple hypothesis testing, with non-identical channels for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our result is for identical channels and is generally applicable to quasi-convex objectives, as mentioned later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now consider the case where C is the set of ϵ-LDP channels from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since C is a set of private channels, it does not contain any deterministic channels (thus, does not contain threshold channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Somewhat surprisingly, we still show that the threshold channels play a fundamental role in the extreme points of the joint range under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The following result shows that any extreme point of the joint range A can be obtained by a threshold channel mapping into [2ℓ2], followed by an ϵ-LDP channel from [2ℓ2] to [ℓ]: 4For k ∈ N, we use ∆k to denote the probability simplex on a domain of alphabet size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 9 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17 (Extreme points of the joint range under privacy and communication constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C be the set of ϵ-LDP channels from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A be the set of all pairs of distributions that are obtained by applying a channel from C to p and q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', A = {(Tp, Tq) | T ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (7) If (Tp, Tq) is an extreme point of A for T ∈ C, then T can be written as T = T2 × T1 for some threshold channel T1 ∈ T2ℓ2,k and some T2 an extreme point of the set of ϵ-LDP channels from [2ℓ2] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We prove this structural result in Section 5, which leads to polynomial-time algorithms for constant ℓ for maximizing quasi-convex functions, as mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 Computationally Efficient Algorithms for Instance Optimality The results from the previous sections characterized the minimax-optimal sample complexity, but did not address instance optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Instance-optimal performance may be substantially better than minimax-optimal performance, as seen by comparing the instance-optimal bounds for binary distributions to the minimax-optimal bounds for general distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In this section, we focus on identifying an instance-optimal channel T (satisfying the necessary constraints) for a given pair (p, q) of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be fixed distributions over [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let Pϵ ℓ,k be the set of all ϵ-LDP channels from [k] to [ℓ], and let Tℓ,k be the set of all channels from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C ∈ {Pϵ ℓ,k, Tℓ,k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As before, define A = {(Tp, Tq) : T ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let g : A → R be a (jointly) quasi-convex function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', for all t ∈ R, the sublevel sets {(p′, q′) : g(p′, q′) ≤ t} are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In this paper, we are primarily interested in functions corresponding to divergences between the distribution pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' So, unless otherwise mentioned, we shall assume the quasi-convex functions g in this paper are permutation-invariant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', g(p′, q′) = g(Πp, Πq) for all permutation matrices Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' However, our algorithmic results will continue to hold even without this assumption, with an additional factor of ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' in the time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will consider the problem of identifying T that solves max T∈C g(Tp, Tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The quasi-convexity of g implies that the maximum is attained at some T such that (Tp, Tq) is an extreme point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We can thus leverage the results from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 to search over the subset of channels satisfying certain structural properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Identifying T that maximizes the Hellinger divergence leads to an instance-optimal test for min- imizing sample complexity for testing between p and q with channel constraints C: This is because if each user chooses the channel T, the resulting sample complexity will be Θ � 1 d2 h(Tp,Tq) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, the instance-optimal sample complexity will be obtained by a channel T that attains maxT∈C d2 h(Tp, Tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Note that the Hellinger divergence is convex (and thus quasi-convex) in its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Apart from the Hellinger divergence, other functions of interest such as the Kullback–Leibler divergence or Chernoff information (which are also convex) characterize the asymptotic error rates in hypothe- sis testing, so finding T for these functions identifies instance-optimal channels in the asymptotic (large-sample) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Other potential functions of interest include Rényi divergences of all orders, which are quasi-convex, but not necessarily convex [EH14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As mentioned earlier, the results of Kairouz, Oh, and Viswanath [KOV16] give a linear program with 2k variables to find an instance-optimal channel under privacy constraints, which is computa- tionally prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' It is also unclear if their result extends when the channels are further restricted 10 to have communication constraints in addition to privacy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now show how to im- prove on the guarantees of Kairouz, Oh, and Viswanath [KOV16] in the presence of communication constraints, using the structural results from the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='18 (Computationally efficient algorithms for maximizing quasi-convex functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be fixed distributions over [k], let C ∈ {Tℓ,k, Pϵ ℓ,k}, and let A = {(Tp, Tq) : T ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let g : A → R be a jointly quasi-convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' When C = Tℓ,k, there is an algorithm that solves maxT∈C g(Tp, Tq) in time polynomial in kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' When C = Pϵ ℓ,k, there is an algorithm that solves maxT∈C g(Tp, Tq) in time polynomial in kℓ2 and 2ℓ3 log ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='18 in Section 4 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 for C = Tℓ,k and C = Pϵ ℓ,k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' When ℓ is constant, we obtain a polynomial-time algorithm for maximizing any quasi-convex function under Tℓ,k or Pϵ ℓ,k channel constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' When C = Tℓ,k and g is the Kullback– Leibler divergence, this exactly solves (for small ℓ) a problem introduced in Carpi, Garg, and Erkip [CGE21], which proposed a polynomial-time heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Applying the above result to the Hellinger divergence d2 h, we obtain the following result for simple binary hypothesis testing, proved in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='20 (Computationally efficient algorithms for instance-optimal results under commu- nication constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any ϵ and any integer ℓ > 1, there is an algorithm that runs in time polynomial in kℓ2 and 2ℓ3 log ℓ and outputs an ϵ-LDP channel T mapping from [k] to [ℓ], such that if N denotes the sample complexity of hypothesis testing between p and q when each individual uses the channel T, then N ≍ n∗(p, q, ϵ, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, the sample complexity with T satisfies N ≲ n∗(p, q, ϵ) · � 1 + log (n∗ (p, q, ϵ)) ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (8) The channel T may be decomposed as a deterministic threshold channel to a domain of size [2ℓ2], followed by an ϵ-LDP channel from [2ℓ2] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, by choosing ℓ = 2, we obtain a polynomial-time algorithm with nearly instance-optimal sample complexity (up to logarithmic factors) under just ϵ-LDP constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 Related Work Distributed estimation has been studied extensively under resource constraints such as memory, privacy, and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Typically, this line of research considers problems of interest such as distribution estimation [RT70;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' LR86;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' CKO21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' BHÖ20], identity or independence testing [ACT20a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' ACT20b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' ACFST21], and parameter estimation [Hel74;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' DJWZ14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' DJW18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' BGMNW16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' DR19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' BCÖ20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' DKPP22], and identifies minimax-optimal bounds on the error or sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In what follows, we limit our discussion to related work on hypothesis testing under resource con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For memory-constrained hypothesis testing, the earliest works in Cover [Cov69] and Hellman and Cover [HC73] derived tight bounds on the memory size needed to perform asymptotically error-free testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Hellman and Cover [HC71] also highlighted the benefits of randomized algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' These benefits were also noted in recent work by Berg, Ordentlich, and Shayevitz [BOS20], which consid- ered the error exponent in terms of the memory size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Recently, Braverman, Garg, and Zamir [BGZ22] showed tight bounds on the memory size needed to test between two Bernoulli distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 11 Communication-constrained hypothesis testing has two different interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In the informa- tion theory literature, Berger [Ber79], Ahlswede and Csiszár [AC86], and Amari and Han [AH98] considered a family of problems where two nodes, one which only observes Xi’s and the other which only observes Yi’s, try to distinguish between PXY and QXY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Communication between the nodes occurs over rate-limited channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The second interpretation, also called “decentralized detection” in Tsitsiklis [Tsi88], is more relevant to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Here, the observed Xi’s are distributed amongst different nodes (one observation per node) that communicate a finite number of messages (bits) to a central node, which needs to determine the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Tsitsiklis [Tsi88;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Tsi93] identified the optimal decision rules for individual nodes and considered asymptotic error rates in terms of the number of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' These results were recently extended to the nonasymptotic regime in Pensia, Jog, and Loh [PJL22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' PLJ22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Privacy-constrained hypothesis testing has been studied in the asymptotic and nonasymptotic regimes under different notions of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The local privacy setting, which is relevant to this paper, is similar to the decentralized detection model in Tsitsiklis [Tsi93], except that the each node’s communication to the central server is private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This is achieved by passing observations through private channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Liao, Sankar, Calmon, and Tan [LSCT17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' LSTC17] considered maximizing the error exponent under local privacy notions defined via maximal leakage and mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Sheffet [She18] analyzed the performance of the randomized response method for LDP for hypoth- esis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Gopi, Kamath, Kulkarni, Nikolov, Wu, and Zhang [GKKNWZ20] showed that M-ary hypothesis testing under pure LDP constraints requires exponentially more samples (Ω(M) instead of O(log M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Closely related to the instance-optimal algorithms in our paper, Kairouz, Oh, and Viswanath [KOV16] presented an algorithm to find LDP channels that maximize the output diver- gence for two fixed probability distributions at the channel input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' the proposed algorithm runs in time exponential in the domain size of the input distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 Note that divergences are directly related to error exponents and sample complexities in binary hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The results of Kairouz, Oh, and Viswanath [KOV16] on extreme points of the polytope of LDP channels were strengthened in Holohan, Leith, and Mason [HLM17], which characterized the extreme points in special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We were able to find only two other papers that consider instance optimality, but in rather special settings [GGKMZ21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' AFT22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For simple binary hypothesis testing in the global differential privacy setting, Canonne, Kamath, McMillan, Smith, and Ullman [CKMSU19] identified the optimal test and corresponding sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Bun, Kamath, Steinke, and Wu [BKSW19] showed that O(log M) samples are enough for M-ary hypothesis testing in the global differential privacy setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 Organization This paper is organized as follows: Section 2 records standard results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Section 3 focuses on the sample complexity of hypothesis testing under privacy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Section 4 considers extreme points of the joint range under communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Section 5 characterizes the extreme points under both privacy and communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Section 6 explores other notions of privacy beyond pure LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Finally, we conclude with a discussion in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We defer proofs of some intermediate results to the appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 5We remark, however, that the algorithm in Kairouz, Oh, and Viswanath [KOV16] is applicable to a wider class of objective functions, which they term “sublinear.” 12 2 Preliminaries and Facts Notation: Throughout this paper, we will focus on discrete distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For a natural number k ∈ N, we use [k] to denote the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , k} and ∆k to denote the set of distributions over [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We represent a probability distribution p ∈ ∆k as a vector in Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, pi denotes the probability of element i under p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Given two distributions p and q, let dTV(p, q) := 1 2 � i |pi −qi| and d2 h(p, q) := � i(√pi − √qi)2 denote the total variation distance and Hellinger divergence between p and q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We denote channels with bold letters such as T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As the channels between discrete distributions can be represented by rectangular column-stochastic matrices (each column is nonnegative and sums to one), we also use bold capital letters, such as T, to denote the corresponding matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, if a channel T is from [k] to [ℓ], we denote it by an ℓ × k matrix, where each of the k columns is in ∆ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In the same vein, for a column index c ∈ [k] and a row index r ∈ [ℓ], we use T(r, c) to refer to the entry at the corresponding location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For a channel T : X → Y and a distribution p over X, we use Tp to denote the distribution over Y when X ∼ p passes through the channel T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In the notation above, when p is a distribution over [k], represented as a vector in Rk, and T is a channel from [k] → [ℓ], represented as a matrix T ∈ Rℓ×k, the output distribution Tp corresponds to the usual matrix-vector product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We shall also use T to denote the stochastic map transforming the channel input X to the channel output Y = T(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Similarly, for two channels T1 and T2 from [k1] to [k2] and [k2] to [k3], respectively, the channel T3 from [k1] to [k3] that corresponds to applying T2 to the output of T1 is equal to the matrix product T2 × T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let Tℓ,k be the set of all channels that map from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We use T thresh ℓ,k to denote the subset of Tℓ,k that corresponds to threshold channels (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We use Pϵ ℓ,k to denote the set of all ϵ-LDP channels from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Recall that for two distributions p and q, we use n∗(p, q, ϵ) (respectively, n∗(p, q, ϵ, ℓ)) to denote the sample complexity of simple binary hypothesis testing under privacy constraints (respectively, both privacy and communication constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For a set A, we use conv(A) to denote the convex hull of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For a convex set A, we use ext(A) to denote the set of extreme points of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Finally, we use the following notations for simplicity: (i) ≲, ≳, and ≍ to hide positive constants, and (ii) the standard asymptotic notation O(·), Ω(·), and Θ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Finally, we use �O(·), �Ω(·), and �Θ to hide poly-logarithmic factors in their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Convexity We refer the reader to Bertsimas and Tsitsiklis [BT97] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will use the following facts repeatedly in the paper, often without mentioning them explicitly: Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 (Extreme points of linear transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A be a convex, compact set in a finite- dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T be a linear function on A, and define the set A′ := {Tx : x ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then A′ is convex and compact, and ext(A′) ⊆ {Tx : x ∈ ext(A)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A be a convex, compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If A = conv(B) for some set B, then ext(A) ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 (Number of vertices and vertex enumeration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A ⊆ Rn be a bounded polytope defined by m linear inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The number of vertices of A is at most �m n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, there is an algorithm that takes eO(n log m) time and output all the vertices of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 (Extreme points of channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The set of extreme points of Tℓ,k is the set of all deterministic channels from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 6Throughout this paper, we assume the bit-complexity of linear inequalities is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 Local Privacy We state standard facts from the privacy literature here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 (Randomized response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For an integer k ≥ 2, the k-ary randomized response channel with privacy parameter ϵ is a channel from [k] to [k] defined as follows: for any i ∈ [k], T(i) = i with probability eϵ (k−1)+eϵ and T(i) = j with probability 1 (k−1)+eϵ , for any j ∈ [k]\\{i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The standard randomized response [War65] corresponds to k = 2, which we denote by Tϵ RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We omit ϵ in the superscript when it is clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will also use the following result on the extreme points for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 (Extreme points of the LDP polytope in special cases [HLM17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We mention all the extreme points of Pϵ ℓ,k (up to permutation of rows and columns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' if a channel is an extreme point, then any permutation of rows and/or columns is an extreme point) below for some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Trivial extreme points) A channel with one row of all ones and the rest of the rows with zero values is always an extreme point of Pϵ ℓ,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We call such extreme points trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (ℓ = 2 and k ≥ 2) All non-trivial extreme points of Pϵ 2,k are of the form (up to permutation of rows): � a a · · a 1 − a 1 − a · · 1 − a 1 − a 1 − a · · 1 − a a a · · a � , where a/(1 − a) = eϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In other words, the columns are of only two types, containing a and 1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (ℓ = 3 and k = 3) There are two types of non-trivial extreme points of Pϵ 3,3: one with two nonzero rows and another with three nonzero rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For the former, the nonzero rows are exactly the extreme points of Pϵ 2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For the latter, two extreme points exist, of the following form: � � 1 − 2a a a a 1 − 2a a a a 1 − 2a � � , one with 1−2a a = eϵ and one with a 1−2a = eϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The case of 1−2a a = eϵ corresponds to the usual randomized response (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 Hypothesis Testing In this section, we state some standard facts regarding hypothesis testing and divergences that will be used repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 (Hypothesis testing and divergences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' see, for example, Tsybakov [Tsy09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two arbitrary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We have d2 TV(p, q) ≤ d2 h(p, q) ≤ 2dTV(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Sample complexity of non-private hypothesis testing) We have n∗(p, q) ≍ 1 d2 h(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Sample complexity in the high-privacy regime) For every ϵ ≤ 1, we have n∗(p, q, ϵ) ≍ 1 ϵ2d2 TV(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' See the references [DJW18, Theorem 1], [AZ22, Theorem 2], and [JMNR19, Theo- rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Restricting the size of the output domain) Let p and q be distributions over [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then n∗(p, q, ϵ) ≍ n∗(p, q, ϵ, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This follows by applying Theorem 2 in Kairouz, Oh, and Viswanath [KOV16] to d2 h(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Choice of identical channels in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2) Let T be a channel that maximizes d2 h(Tp, Tq) among all channels in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then the sample complexity of hypothesis testing under the channel constraints of C is Θ � 1 d2 h(Tp,Tq) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' See Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 in Pensia, Jog, and Loh [PJL22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8 (Preservation of Hellinger distance under communication constraints (Theorem 1 in Bhatt, Nazer, Ordentlich, and Polyanskiy [BNOP21] and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 in Pensia, Jog, and Loh [PJL22])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then for any ℓ ∈ N, there exists a channel T from [k] to [ℓ], which can be computed in time polynomial in k, such that d2 h(p, q) ≲ d2 h(Tp, Tq) · � 1 + ℓ min(k, log(1/d2 h(p, q))) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (9) Moreover, this bound is tight in the following sense: for every choice of ρ ∈ (0, 1), there exist two distributions p and q such that d2 h(p, q) ≍ ρ, and for every channel T ∈ Tℓ,k, the right-hand side of inequality (9) is further upper-bounded by O(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 3 Locally Private Simple Hypothesis Testing In this section, we provide upper and lower bounds for locally private simple hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This section is organized as follows: In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1, we derive instance-optimal bounds when both distributions are binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We then prove minimax-optimal bounds for general distributions (with support size at least three): Lower bounds on sample complexity are proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 and upper bounds in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proofs of some of the technical arguments are deferred to the appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Binary Distributions and Instance-Optimality of Randomized Response We first consider the special case when p and q are both binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our main result characterizes the instance-optimal sample complexity in this setting: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 (Sample complexity of binary distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then n∗(p, q, ϵ) ≍ � � � � � � � � � 1 ϵ2 · d2 TV(p,q), if ϵ ≤ 1, 1 eϵ · d2 TV(p,q), if eϵ ∈ � e, d2 h(p,q) d2 TV(p,q) � , 1 d2 h(p,q), if eϵ > d2 h(p,q) d2 TV(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (3) By Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 is a consequence of the following bound on the strong data processing inequality for randomized responses: 15 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 (Strong data processing inequality for Hellinger divergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then max T∈Pϵ 2,2 d2 h (Tp, Tq) ≍ � ϵ2 · d2 TV(p, q), if ϵ ≤ 1 min � eϵ · d2 TV(p, q), d2 h(p, q) � , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, the maximum is achieved by the randomized response channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A = {(Tp, Tq) : T ∈ Pϵ 2,2} be the joint range of p and q under ϵ-LDP privacy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since A is a convex set and d2 h is a convex function over A, the maximizer of d2 h in A is an extreme point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since A is a linear transformation of Pϵ 2,2, Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 implies that any extreme point of A is obtained by using a channel T corresponding to an extreme point of Pϵ 2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6, the only extreme point of Pϵ 2,2 is the randomized response channel Tϵ RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, in the rest of the proof, we consider T = Tϵ RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By abusing notation, we will also use p and q to denote the probabilities of observing 1 under the two respective distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Without loss of generality, we will assume that 0 ≤ p ≤ q and p ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will repeatedly use the following claim, which is proved in Appendix D: Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 (Approximation for Hellinger divergence of binary distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p, q ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let Ber(p) and Ber(q) be the corresponding Bernoulli distributions with min(p, q) ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then d2 h (Ber(p), Ber(q)) ≍ d2 TV(Ber(p), Ber(q)) max(p, q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Applying Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2, we obtain d2 h(p, q) ≍ d2 TV(p, q) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (10) We know that the transformed distributions p′ := Tϵ RRp and q′ := Tϵ RRq are binary distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' by abusing notation, let p′ and q′ also be the corresponding real-valued parameters associated with these binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By the definition of the randomized response, we have p′ := p(eϵ − 1) + 1 1 + eϵ , and q′ := q(eϵ − 1) + 1 1 + eϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (11) Consequently, we have 0 ≤ p′ ≤ q′ and p′ ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We directly see that dTV(p′, q′) = q′ − p′ = (q − p)(eϵ − 1) eϵ + 1 = dTV(p, q) · eϵ − 1 eϵ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now apply Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 below to the distributions p′ and q′: d2 h(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q′) ≍ d2 TV(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q′) q′ = d2 TV(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) · �eϵ − 1 eϵ + 1 �2 1 + eϵ q(eϵ − 1) + 1 (using equation (11)) ≍ d2 TV(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) · (eϵ − 1)2 eϵ + 1 min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1 q(eϵ − 1) � � using 1 a + b ≍ min �1 a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1 b � for a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' b > 0 � ≍ d2 TV(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) · (eϵ − 1)2 eϵ + 1 min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' d2 h(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) d2 TV(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q)(eϵ − 1) � � using equation (10) and 16 min(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' a) ≍ min(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' b) if a ≍ b � ≍ min � d2 TV(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) · (eϵ − 1)2 eϵ + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' d2 h(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) · eϵ − 1 eϵ + 1 � ≍ � min � d2 TV(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) · ϵ2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' d2 h(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) · ϵ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' if ϵ ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' min � d2 TV(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) · eϵ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' d2 h(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (using eϵ − 1 ≍ ϵ for ϵ ≤ 1 and eϵ otherwise) ≍ � ϵ2d2 TV(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' if ϵ ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' min � d2 TV(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q)eϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' d2 h(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' where the last step uses the inequality d2 TV(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) ≤ d2 h(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' q) from Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 General Distributions: Lower Bounds and Higher Cost of Privacy In this section, we establish lower bounds for the sample complexity of private hypothesis testing for general distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In the subsequent section, the lower bounds will be shown to be tight up to logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We formally state the lower bound in the statement below: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 (Sample complexity lower bound for general distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any ρ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5) and ν ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5) such that 2ν2 ≤ ρ ≤ ν, there exist ternary distributions p and q such that d2 h(p, q) = ρ, dTV(p, q) = ν, and the sample complexity behaves as n∗(p, q, ϵ) ≍ � � � � � � � 1 ϵ2 · d2 TV(p,q), if ϵ ≤ 1, min � 1 d2 TV(p,q), 1 eϵ · d4 h(p,q) � , if eϵ ∈ � e, 1 d2 h(p,q) � , 1 d2 h(p,q), if eϵ > 1 d2 h(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (4) We provide the proof below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We refer the reader to Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8 for further discussion on differ- ences between the worst-case sample complexity of general distributions and the sample complexity of binary distributions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We note that a similar construction is mentioned in Canonne, Kamath, McMillan, Smith, and Ullman [CKMSU19, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' however, their focus is on the central model of differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The case when ϵ ≤ 1 follows from Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, we set ϵ ≥ 1 in the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We start with a helpful approximation for computing the Hellinger divergence, proved in Appendix D: Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 (Additive approximation for √· ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' There exist constants 0 < c1 ≤ c2 such that for 0 < y ≤ x, we have c1 · y2 x ≤ (√x − √x − y)2 ≤ c2 · y2 x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For some γ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='25) and δ > 0 to be decided later, let p and q be the following ternary distributions: p = � � 0 1/2 1/2 � � , and q = � � 2γ1+δ 1/2 + γ − γ1+δ 1/2 − γ − γ1+δ � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 17 Since γ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='25 and δ ≥ 0, these two are valid distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that dTV(p, q) = γ + γ1+δ ≍ γ and and d2 h(p, q) ≍ γ1+δ by Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We choose γ and δ such that ν = dTV(p, q) and ρ = d2 h(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Such a choice of γ and δ can be made by the argument given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 as long as ν ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5) and ρ ∈ [2ν2, ν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, these two distributions satisfy the first two conditions of the theorem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In the rest of the proof, we will use the facts that γ1+δ ≍ ρ and γ ≍ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, we have γ2 ≲ γ1+δ ≲ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since both p and q are supported on [3], we can restrict our attention to ternary output channels (see Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Recall that Pϵ 3,3 is the set of all ϵ-LDP channels from [3] to [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will establish the following result: for all ϵ such that e ≤ eϵ ≲ 1 d2 h(p,q), we have max T∈Pϵ 3,3 d2 h(Tp, Tq) ≍ max � d2 TV(p, q), d4 h(p, q)eϵ� ≍ max(γ2, eϵγ2+2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (12) By Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7, equation (12) implies that for e ≤ eϵ ≲ 1 d2 h(p,q), we have n∗(p, q, ϵ) ≍ min � 1 d2 TV(p, q), 1 eϵ · d4 h(p, q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (13) Let ϵ0 be the right endpoint of the range for ϵ above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', eϵ0 ≍ 1 d2 h(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then equation (13) shows that n∗(p, q, ϵ0) ≍ 1/d2 h(p, q) ≍ n∗(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since for any ϵ such that ϵ > ϵ0, we have n∗(p, q, ϵ) ∈ [n∗(p, q, ϵ0), n∗(p, q)], the desired conclusion in equation (4) holds for ϵ > ϵ0, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, in the remainder of this proof, we will focus on establishing equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since d2 h(·, ·) is a convex, bounded function and the set of ϵ-LDP channels is a convex polytope, it suffices to restrict our attention only to the extreme points of the polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As mentioned in Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6, these extreme points are of three types: Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Exactly one nonzero row) Any such extreme point T maps the entire domain to a single point with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' After transformation under this channel, all distributions become indistinguishable, giving dh(Tp, Tq) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Exactly two nonzero rows) This corresponds to the case when T = Tϵ RR × T′, where T′ is a deterministic threshold channel from [3] to [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 There are two non-trivial options for choosing T′, which we analyze below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The first choice of T′ maps {1} and {2, 3} to different elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The transformed distributions p′ and q′ are [0, 1] and [2γ1+δ, 1−2γ1+δ], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Using Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3, we obtain d2 h(p′, q′) ≍ γ1+δ and dTV(p′, q′) ≍ γ1+δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p′′ and q′′ be the corresponding distributions after applying the randomized response with parameter ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since p′ and q′ are binary distributions, we can apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 to obtain d2 h(p′′, q′′) ≍ min(d2 h(p′, q′), eϵd2 TV(p′, q′)) ≍ min(γ1+δ, eϵγ2+2δ) ≍ γ1+δ · min(1, eϵγ1+δ), which is equal to eϵ · γ2+2δ in the regime of interest and consistent with the desired expression in equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The second choice of T′ maps {1, 2} and {3} to different elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The transformed distributions p′ and q′ are [1/2, 1/2] and [1/2 + γ + γ1+δ, 1/2 − γ − γ1+δ], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Applying Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3, we 7We use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17 to restrict our attention only to threshold channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 18 observe that d2 h(p′, q′) ≍ γ2 and dTV(p′, q′) ≍ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p′′ and q′′ be the corresponding distributions after applying the randomized response with parameter ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1, we obtain d2 h(p′′, q′′) ≍ min(d2 h(p′, q′), eϵd2 TV(p′, q′)) ≍ min(γ2, eϵγ2) ≍ γ2 in the regime of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Again, this is consistent with equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Case III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (All nonzero rows) There are two extreme points of this type (up to a permutation of the rows), both of the following form: T1 = � � 1 − 2α α α α 1 − 2α α α α 1 − 2α � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For the first extreme point, α satisfies 1−2α α = eϵ, while the second extreme point has α 1−2α = eϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' These channels are relatively easy to analyze, since they transform the distributions element- wise: each entry x of the original distribution is transformed to α + x(1 − 3α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consequently, the transformed distributions p′ and q′ are p′ = � � α 1−α 2 1−α 2 � � , and q′ = � � α + 2γ1+δ(1 − 3α) 1−α 2 + (γ − γ1+δ)(1 − 3α) 1−α 2 + (−γ − γ1+δ)(1 − 3α) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (14) We now compute the Hellinger divergence between these two distributions for both the extreme points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let us first consider the case where 1−2α α = eϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then α = 1 2+eϵ ≍ e−ϵ, since ϵ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, in the desired range of eϵ ≲ γ−(1+δ), the parameter α satisfies α ≳ γ1+δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will now calculate the Hellinger divergence between p′ and q′ in equation (14) by analyzing the contribution from each of the three terms in the sum �3 i=1( � p′ i − � q′ i)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For the first term, we apply Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 with x = α+2γ1+δ(1 − 3α) and y = 2γ1+δ(1 − 3α), to see that its contribution is Θ(γ2+2δ/α) ≍ eϵγ2+2δ (since 1 − 3α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Applying Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 again, we see that the contributions from the second and third elements are Θ(γ2), since α ≪ 1 and γ ≪ 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Overall, the Hellinger divergence is O � max(γ2, eϵγ2+2δ) � , which satisfies equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Finally, let us consider the case when α 1−2α = eϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We set β = 1 − 2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then β = 1/(1 + 2eϵ), which is much less than 1 in the desired range and is of the order of e−ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, each entry x of the distribution is mapped to 1 2(1 − β + x(3β − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The transformed distributions are p′ = 1 2 · � � 1 − β 1+β 2 1+β 2 � � , and q′ = 1 2 · � � 1 − β + 2γ1+δ(3β − 1) 1+β 2 + (γ − γ1+δ)(3β − 1) 1+β 2 + (−γ − γ1+δ)(3β − 1) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (15) As β is much less than 1 in the desired range of ϵ, we can apply Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 to see that contribution of the first element is Θ(γ2+2δ), and the contributions of both the second and third elements are Θ(γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Overall, the Hellinger divergence is Θ(γ2), which is again consistent with equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Combining all the cases above, the maximum Hellinger divergence after applying any ϵ-LDP channel is Θ(γ2 · max(1, eϵγ2δ)), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 General Distributions: Upper Bounds and Minimax Optimality We now demonstrate an algorithm that finds a private channel matching the minimax rate in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 up to logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, the proposed algorithm is both computationally efficient and communication efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9 (Sample complexity upper bounds and an efficient algorithm for hypothesis testing for general distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then the sample complexity behaves as n∗(p, q, ϵ) ≲ � � � � � � � 1 ϵ2 · d2 TV(p,q), if ϵ ≤ 1, min � 1 d2 TV(p,q), α2 eϵ · d4 h(p,q) � , if eϵ ∈ � e, α d2 h(p,q) � , α d2 h(p,q), if eϵ > α d2 h(p,q), (5) where α ≲ log(1/d2 h(p, q)) ≍ log (n∗(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, the rates above are achieved by an ϵ-LDP channel T that maps [k] to [2] and can be found in time polynomial in k, for any choice of p, q, and ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In comparison with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7, we see that the test above is minimax optimal up to logarithmic factors over the class of distributions with fixed Hellinger divergence and total variation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The channel T satisfying this rate is of the following simple form: a deterministic binary channel T′, followed by the randomized response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In fact, we can take T′ to be either Scheffe’s test (which preserves the total variation distance) or the binary channel from Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8 (which preserves the Hellinger divergence), whichever of the two is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We provide the complete proof in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' One obvious shortcoming of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9 is that even when ϵ → ∞, the test does not recover the optimal sample complexity of 1/d2 h(p, q), due to the logarithmic multiplier α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now consider the case when eϵ ≳ 1 d2 h(p,q) and exhibit a channel that achieves the optimal sample complexity as soon as eϵ ≳ 1 d2 h(p,q) log � 1 d2 h(p,q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, privacy can be attained essentially for free in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k], and let eϵ ≳ 1 d2 h(p,q) log � 1 d2 h(p,q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then n∗(p, q, ϵ) ≍ n∗(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, there is a channel T achieving this sample complexity that maps [k] to a domain of size ⌈log(n∗(p, q))⌉, and which can be computed in poly(k, log(⌈n∗(p, q)⌉)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We note that the size of the output domain of ⌈log(n∗(p, q))⌉ is tight in the sense that any channel that achieves the sample complexity within constant factors of n∗(p, q) must use an output domain of size at least Ω(log(n∗(p, q)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' this follows by the tightness of Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8 in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consequently, the channel T achieving the rate above is roughly of the form (1) a communication-efficient channel from Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8 that preserves the Hellinger divergence up to constant factors, followed by (2) an ℓ-ary randomized response channel, for ℓ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='13 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 and defer the proofs of some of the interme- diate results to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9 In this section, we provide the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We first note that this result can be slightly strengthened, replacing α by n∗ binary n∗ , where n∗ binary is the sample complexity of hypothesis testing un- der binary communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This choice of α is smaller, by Pensia, Jog, and Loh [PJL22, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 20 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The case of ϵ ≤ 1 follows from Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' thus, we focus on the setting where ϵ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will establish these bounds via Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1, by using a binary deterministic channel, followed by the binary randomized response channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A sample complexity of 1/d2 TV(p, q) is direct by using the channel for ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, our focus will be on the term 1 eϵd4 h(p,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T′ ∈ T2,k be a deterministic binary output channel to be decided later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consider the channel T = Tϵ RR × T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1, we have d2 h � Tϵ RR × T′p, Tϵ RR × T′q � ≍ min � eϵd2 TV � T′p, T′q � , d2 h � T′p, T′q �� ≥ min � eϵd4 h � T′p, T′q � , d2 h � T′p, T′q �� = d2 h � T′p, T′q � min � eϵd2 h � T′p, T′q � , 1 � , (16) where the first inequality uses Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 If we choose the channel T′ from Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8, we have d2 h(T′p, T′q) ≥ 1 α · d2 h (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Applying this to inequality (16), we obtain d2 h � Tϵ RR × T′p, Tϵ RR × T′q � ≳ 1 α · d2 h (p, q) · min � 1 α · eϵd2 h (p, q) , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7, the sample complexity of Tϵ RR×T′, which is ϵ-LDP, is at most α d2 h(p,q) ·max � 1, α eϵd2 h(p,q) � , which is equivalent to the desired statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Finally, the claim on the runtime is immediate, since the channel T′ from Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8 can be found efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='13 We will prove a slightly generalized version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='13 below that works for a wider range of ϵ: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k] and ϵ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then there exists an ϵ-LDP channel T from [k] to [ℓ], for ℓ = min(⌈log(1/d2 h(p, q))⌉, k), such that d2 h(Tp, Tq) ≳ d2 h(p, q) · min � 1, eϵ · d2 h(p, q) log(1/d2 h(p, q)) � min � 1, eϵ log(1/d2 h(p, q)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Furthermore, the channel T can be be computed in poly(k, ℓ) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 implies the following: n∗(p, q, ϵ) ≲ n∗ · max � 1, n∗ log(n∗) eϵ � max � 1, log n∗ eϵ � , where n∗ := n∗(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Setting eϵ equal to n∗ log(n∗) proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, we will focus on proving Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We establish this result with the help of the following observations: (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5) First, we show that the randomized response preserves the contribution to the Hellinger divergence by “comparable elements” (elements whose likelihood ratio is in the in- terval � 1 2, 2 � ) when ϵ is large compared to the support size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, we first define the following sets: A = � i ∈ [k] : pi qi ∈ �1 2, 1 �� and A′ = � i ∈ [k] : pi qi ∈ [1, 2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (17) 21 Let Tϵ,ℓ RR denote the randomized response channel from [ℓ] to [ℓ] with privacy parameter ϵ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The following result is proved in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 (Randomized response preserves contribution of comparable elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Suppose � i∈A � A′(√qi − √pi)2 ≥ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then Tϵ,ℓ RR, for ℓ ≤ eϵ, satisfies d2 h(Tϵ,ℓ RRp, Tϵ,ℓ RRq) ≳ min � 1, eϵ τ ℓ � τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, when eϵ ≳ ℓ τ , the randomized response preserves the original contribution of comparable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6) We then show in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6, proved in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2, that either we can reduce the problem to the previous special case (small support size and main contribution to Hellinger divergence is from comparable elements) or to the case when the distributions are binary (where Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 is applicable and is, in a sense, the easy case for privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 (Reduction to base case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then there is a channel T, which can be computed in time polynomial in k, that maps [k] to [ℓ] (for ℓ to be decided below) such that for p′ = Tp and q′ = Tq, at least one of the following holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any ℓ > 2 and ℓ ≤ min � k, 1 + log � 1/d2 h(p, q) �� , we have � i∈B � B′ �� q′ i − � p′ i �2 ≳ d2 h(p, q) · ℓ min � k, 1 + log � 1/d2 h(p, q) ��, where B and B′ are defined analogously to A and A′ in equation (17), but with respect to distributions p′ and q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' ℓ = 2 and d2 h(p′, q′) ≳ d2 h(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now provide the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4, with the help of Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4) The channel T will be of the form T = Tϵ,ℓ RR × T1, where T1 is a channel from [k] to [ℓ] and ℓ is to be decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The privacy of T is clear from the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We begin by applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T1 be the channel from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 that maps from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p′ = T1p and q′ = T1q, and define ˜p = Tϵ,ℓ RRp′ and ˜q = Tϵ,ℓ RRq′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The claim on runtime thus follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Suppose for now that T1 from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 is a binary channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then we know that d2 h(p′, q′) ≳ d2 h(p, q) and dTV(p′, q′) ≳ d2 h(p′, q′), where the latter holds by Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1, we have d2 h(˜p, ˜q) ≳ min � d2 h(p′, q′), eϵd2 TV(p′, q′) � ≳ min � d2 h(p′, q′), eϵd4 h(p′, q′) � ≳ d2 h(p, q) min � 1, eϵd2 h(p, q) � , which concludes the proof in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now consider the case when ℓ > 2 in the guarantee of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then the compara- ble elements of p′ and q′ preserve a significant fraction of the Hellinger divergence (depending on the chosen value of ℓ) between p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let k′ = min � k, 1 + log(1/d2 h(p, q)) � , and choose ℓ to be 22 min(k′, eϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 implies that the contribution to the Hellinger divergence from com- parable elements of p′ and q′ is at least τ, for τ ≍ d2 h(p, q) ℓ k′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will now apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 to p′ and q′ with the above choice of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since ℓ ≤ eϵ by construction, applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 to p′ and q′, we obtain d2 h(˜p, ˜q) ≳ τ · min � 1, eϵτ ℓ � ≳ d2 h(p, q) ℓ k′ · min � 1, eϵ · d2 h(p, q) log(1/d2 h(p, q)) � ≳ d2 h(p, q) · min � 1, eϵ log(1/d2 h(p, q)) � min � 1, eϵ · d2 h(p, q) log(1/d2 h(p, q)) � , where the last step uses the facts that ℓ = min(eϵ, k′) and k′ ≳ log(1/d2 h(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 4 Extreme Points of Joint Range Under Communication Constraints In this section, our goal is to understand the extreme points of the set A := {(Tp, Tq) : T ∈ Tℓ,k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This will allow us to identify the structure of optimizers of quasi-convex functions over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The main result of this section is the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16 (Extreme points of the joint range under communication constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A be the set of all pairs of distributions that are obtained by passing p and q through a channel of output size ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', A = {(Tp, Tq) : T ∈ Tℓ,k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If (Tp, Tq) is an extreme point of A, then T is a threshold channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We provide the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Before proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16, we dis- cuss some consequences for optimizing quasi-convex functions over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The following result proves Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='18 for C = Tℓ,k: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 (Threshold channels maximize quasi-convex functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distri- butions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A := {(Tp, Tq) : T ∈ Tℓ,k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let g be a real-valued quasi-convex function over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then max T∈Tℓ,k g(Tp, Tq) = max T∈T thresh ℓ,k g(Tp, Tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, the above optimization problem can be solved in time poly(kℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that A is a closed polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let X be the set of extreme points of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that X ⊆ {(Tp, Tq) : T ∈ Tℓ,k and T is deterministic}, and thus is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since A is a closed polytope, A is convex hull of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Furthermore, the maximum of g on X is well-defined and finite, as X is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Any y ∈ A can be expressed as a convex combination y = � xi∈X λixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Recall that an 8Recall that g is assumed to be permutation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If not, an extra factor of ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' will appear in the time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 23 equivalent definition of quasi-convexity is that g satisfies g(λx + (1 − λ)y) ≤ max(g(x), g(y)) for all λ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By repeatedly using this fact, we have g(λ) = g � � � xi∈X λixi � � ≤ max x∈X g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16, any extreme point x ∈ X is obtained by passing p and q through a threshold channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, the maximum of g over X is attained by passing p and q through a threshold channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The claimed runtime is obtained by trying all possible threshold channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Quasi-)convex functions of interest include all f-divergences, Rényi divergences, Chernoff information, and Lp norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We note that the above result also holds for post-processing: For any fixed channel H ∈ Tℓ′,ℓ, we have max T∈Tℓ,k g(HTp, HTq) = max T∈T thresh ℓ,k g(HTp, HTq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This is because g(Hp′, Hq′) is a quasi-convex function of (p′, q′) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If g is the Hellinger divergence and C = Tℓ,k, we can conclude the following result for the communication-constrained setting: There exists a T ∈ T thresh ℓ,k that attains the instance- optimal sample complexity (up to universal constants) for hypothesis testing under a communication constraint of size ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This result is implied in Pensia, Jog, and Loh [PJL22] by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9 (which is a result from Tsitsiklis [Tsi93]) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The above argument provides a more straightforward proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16 We now provide the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16) We first make the following simplifying assumption about the likeli- hood ratios: there is at most a single element i∗ with qi∗ = 0, and for all other elements i ∈ [k]\\{i∗}, pi/qi is a unique value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If there are two or more elements with the same likelihood ratio, we can merge those elements into a single alphabet without loss of generality, as we explain next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p′ and q′ be the distributions after merging these elements, and let k′ ≤ k be the new cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then for any channel T ∈ Tℓ,k, there exists another channel T′ ∈ Tℓ,k′ such that (Tp, Tq) = (T′p′, T′q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We can then apply the following arguments to p′ and q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In the following, we will consider pi/qi to be ∞ if qi = 0, and we introduce the notation θi := pi/qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will further assume, without loss of generality, that pi/qi is strictly increasing in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since the elements are ordered with respect to the likelihood ratio, a threshold channel corresponds to a map that partitions the set [k] into contiguous blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Formally, we have the following definition: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 (Partitions and threshold partitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We say that S = (S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Sℓ) forms an ℓ-partition of [k] if ∪ℓ i=1Si = [k] and Si ∩ Sj = ∅ for 1 ≤ i ̸= j ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We say that S forms an ℓ-threshold partition of [k] if in addition, for all i < j ∈ ℓ, every entry of Si is less than every entry of Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As mentioned before, channels corresponding to ℓ-threshold partitions are precisely the threshold channels up to a permutation of output labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The channels corresponding to ℓ-partitions are the set of all deterministic channels that map [k] to ℓ, which are the extreme points of Tℓ,k (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 24 Observe that A is a convex, compact set, which is a linear transformation of the convex, compact set Tℓ,k, and any extreme point of A is of the form (Tp, Tq), where T is an extreme point of Tℓ,k (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Now suppose (Tp, Tq) is an extreme point of A, but T is not a threshold channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, T corresponds to some ℓ-partition S of [k] that is not an ℓ-threshold partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will now show that (Tp, Tq) is not an extreme point of A, by showing that there exist two distinct channels T1 ∈ Tℓ,k and T2 ∈ Tℓ,k such that the following holds: 1 2 · T1p + 1 2 · T2p = Tp, and 1 2 · T1q + 1 2 · T2q = Tq, (18) and T1p ̸= Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since S is not a ℓ-threshold partition, there exist 1 ≤ a < b < c ≤ k and m ̸= n in [ℓ] such that a, c ∈ Sm and b ∈ Sn, and pa/qa < pb/qb < pc/qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Among qa, qb, and qc, only qc is potentially zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Suppose for now that qc ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' we will consider the alternative case shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For some ϵ1 ∈ (0, 1) and ϵ2 ∈ (0, 1) to be determined later, let T1 be the following channel: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For x ̸∈ {a, b}, T1 maps x to T(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For x = a (respectively b), T1 maps x to m (respectively n) with probability 1−ϵ1 (respectively 1 − ϵ2) and to n (respectively m) with probability ϵ1 (respectively ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, the channels T and T1 have all columns identical, except for those corresponding to inputs a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let vi be the ith column of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that va is a degenerate distribution at m ∈ [ℓ] and vb is a degenerate distribution at n ∈ [ℓ] (equivalently, T(m, a) = 1 and T(n, b) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, we can write T1q = Tq + (ϵ2qb − ϵ1qa)(va − vb), T1p = Tp + (ϵ2pb − ϵ1pa)(va − vb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If we choose ϵ1qa = ϵ2qb, we have T1q = Tq and T1p = Tp + (ϵ2pb − ϵ1pa)(va − vb) = Tp + (ϵ2qbθb − ϵ1qaθa)(va − vb) = Tp + ϵ1qa(θb − θa)(va − vb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (19) Recall that θb > θa, as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now define T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For some ϵ3 ∈ (0, 1) and ϵ4 ∈ (0, 1) to be decided later, we have: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For x ̸∈ {b, c}, T2 maps x to T(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For x = c (respectively b), T2 maps x to m (respectively n) with probability 1−ϵ3 (respectively 1 − ϵ4) and to n (respectively m) with probability ϵ3 (respectively ϵ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' With the same arguments as before, we have T2q = Tq + (ϵ4qb − ϵ3qc)(vc − vb), T1p = Tp + (ϵ4pb − ϵ3pc)(vc − vb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If we choose ϵ3qc = ϵ4qb, we have T2q = Tq and T2p = Tp + (ϵ4qbθb − ϵ3qcθc)(vc − vb) = Tp + ϵ3qc(θb − θc)(vc − vb) 25 = Tp + ϵ3qc(θb − θc)(va − vb), (20) where the last line follows by the fact that va = vc, since T maps both a and c to m almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let ϵ1 ∈ (0, 1) and ϵ3 ∈ (0, 1) be such that ϵ1qa(θb − θa) = −ϵ3qc(θb − θc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Such a choice always exists because θb − θa and −(θb − θc) are both strictly positive and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then equations (19) and (20) imply that Tp = 1 2T1p + 1 2T2p and Tq = 1 2T1q + 1 2T2q, and T1p ̸= Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, T1p ̸= Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, (Tp, Tq) is not an extreme point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now outline how to modify the construction above when qc is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By setting ϵ4 to be zero, we obtain T2q = Tq and T2p = Tp + (−ϵ3pc) (va − vb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The desired conclusion follows by choosing ϵ1 and ϵ3 small enough such that ϵ1qa(θb − θa) = −ϵ3pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 5 Extreme Points of Joint Range under Privacy Constraints In the previous section, we considered the extreme points of the joint range under communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Such communication constraints are routinely applied in practice in the presence of additional constraints such as local differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' However, the results of the previous section do not apply directly, as the joint range is now a strict subset of the set in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16, and the extreme points differ significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For example, the threshold channels are not even private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' However, we show in this section that threshold channels still play a fundamental role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our main result in this section is the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17 (Extreme points of the joint range under privacy and communication constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C be the set of ϵ-LDP channels from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A be the set of all pairs of distributions that are obtained by applying a channel from C to p and q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', A = {(Tp, Tq) | T ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (7) If (Tp, Tq) is an extreme point of A for T ∈ C, then T can be written as T = T2 × T1 for some threshold channel T1 ∈ T2ℓ2,k and some T2 an extreme point of the set of ϵ-LDP channels from [2ℓ2] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Actually, our result applies to a broader family of linear programming (LP) channels that we describe below: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 (LP family of channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any ℓ ∈ N, let ν = (ν1, ν2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , νℓ) and γ = (γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , γℓ) be two nonnegative vectors in Rℓ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For k ∈ N, define the set of linear programming (LP) channels J γ,ν ℓ,k , a subset of Tℓ,k, to be the (convex) set of all channels from [k] to [ℓ] that satisfy the following constraints: For each row j ∈ [ℓ], and for each i, i′ ∈ [k], we have T(j, i) ≤ γjT(j, i′) + νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (21) When γj = eϵ and νj = 0 for all j ∈ [ℓ], we recover the set of ϵ-LDP channels from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Another example will be mentioned in Section 6 for a relaxed version of approximate LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The rest of this section is organized as follows: In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1, we show that any T that leads to an extreme point of A cannot have more than 2ℓ2 unique columns (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We use this result to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3, we apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17 to prove Corollaries 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='18 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Bound on the Number of Unique Columns The following result will be critical in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17, the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C be the set of channels from [k] to [ℓ], from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A be the set of all pairs of distributions that are obtained by applying a channel from C to p and q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', A = {(Tp, Tq) | T ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (22) If T has more than 2ℓ2 unique columns, then (Tp, Tq) is not an extreme point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We prove this result in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 after proving a quantitatively weaker, but simpler, result in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Warm-Up: An Exponential Bound on the Number of Unique Columns In this section, we first prove a weaker version of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2, where we upper-bound the number of unique columns in the extreme points of C from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 (not just those that lead to extreme points of A) by an exponential in ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In fact, this bound will be applicable for a broader class of channels that satisfy the following property: Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 (Only one free entry per column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C be a convex set of channels from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T be an extreme point of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then there exist numbers {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , mℓ} and {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Mℓ} such that for every column c ∈ [k], there exists at most a single row r ∈ [ℓ] such that T(r, c) ̸∈ {mr, Mr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We call such entries free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We show in Appendix B that extreme points of the LP channels from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 satisfy Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The following claim bounds the number of unique columns in any extreme point of C, and thus also implies a version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17 with ℓ · 2ℓ−1 instead of 2ℓ2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 (Number of unique columns in an extreme point is at most exponential in ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C be a set of channels satisfying the property of Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T be an extreme point of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then the number of unique columns in T is at most ℓ · 2ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, T can be written as T2 × T1, where T1 is a deterministic map from [k] to [ℓ′] and T2 is a map from [ℓ′] to [ℓ], for ℓ′ = ℓ · 2ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We use the notation from Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For each column, there are ℓ possible locations of a potential free entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let this location be j∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' the value at this location is still flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Now let us consider the number of ways to assign values at the remaining locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For each j ∈ [ℓ] \\ {j∗}, the entry is either mj or Mj (since j is not a free entry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, there are 2ℓ−1 such possible assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since the column entries sum to one, each of those 2ℓ−1 assignments fixes the value at the j∗ location, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, there are at most ℓ · 2ℓ−1 unique columns in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 Forbidden Structure in Extreme Points Using the Joint Range In Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4, we considered the extreme points of LP channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' However, we are actually interested in a (potentially much) smaller set: the extreme points that correspond to the extreme points of the joint range A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In this section, we identify a necessary structural property for extreme points of LP channels that lead to extreme points of the joint range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We begin by defining the notion of a “loose” entry in a channel in C: 27 Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 (Loose and tight entries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T be a channel in J γ,ν ℓ,k from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 that maps from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , mℓ} and {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Mℓ} be the row-wise minimum and maximum entries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For c ∈ [k] and r ∈ [ℓ], we say an entry T(r, c) is max-tight if T(r, c) = Mr and Mr = γrmr + νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' An entry T(r, c) is min-tight if T(r, c) = mr and Mr = γrmr + νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' An entry that is neither max-tight nor min-tight is called loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our results in this section continue to hold for a slightly more general definition, where we replace the linear functions γjx+νj by arbitrary monotonically increasing functions fj(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We focus on linear functions for simplicity and clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Also see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=') If the rest of the row is kept fixed, a max-tight entry cannot be increased without violating privacy constraints, but it can be decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Similarly, a min-tight entry cannot be decreased without violating privacy constraints, but it can be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Loose entries can be either increased or decreased without violating privacy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' These perturbations need to be balanced by adjusting other entries in the same column to satisfy column stochasticity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' for example, a max- tight entry can be decreased while simultaneously increasing a min-tight or loose entry in the same column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This is formalized below: Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 (Mass can be transferred from entries that are not tight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C be a set of channels from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T be any extreme point of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Suppose there are two rows (r, r′) and two columns (c, c′) (in the display below, we take r < r′ and c < c′ without loss of generality) with values (m, m′, M, M′), as shown below: � ����� · · · · · · · · · · · · M · · m · · · · · · · · · · · · · · m′ · · M′ · · · · · · · · · · · · � ����� , such that: T(r, c) and T(r′, c′) are not min-tight (M and M′ above, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' T(r, c′) and T(r′, c) are not max-tight (m and m′ above, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then there exist ϵ′ > 0 and δ′ > 0 such that for all ϵ ∈ [0, ϵ′) and δ ∈ [0, δ′), the following matrix T′ also belongs to C: T′ = � ����� · · · · · · · · · · · · M − ϵ · · m + δ · · · · · · · · · · · · · · m′ + ϵ · · M′ − δ · · · · · · · · · · · · � ����� , where the omitted entries of T and T′ are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We show that the channels from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 satisfy Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Using Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7, we show that the following structure is forbidden in the channels that lead to extreme points of the joint range: 28 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C be the set of LP channels from Def- inition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 (or, more generally, a convex set of channels satisfying Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7) from [k] to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Suppose pi/qi is strictly increasing in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T ∈ C have the following structure: there are two rows (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' r′) (in the display below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' r < r′ is taken without loss of generality) and three columns i1 < i2 < i3 with values (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' m′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' m′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' M′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' M′′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' as shown below: � ����� · · · · · · · · · · · · · · · · M · · m · · M′ · · · · · · · · · · · · · · · · · · m′ · · M′′ · · m′′ · · · · · · · · · · · · · · · · � ����� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' such that: T(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' i1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' T(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' i3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' and T(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' i2) are not min-tight (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' M′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' and M′′ above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' T(r, i2), T(r′, i1), and T(r′, i3) are not max-tight (m, m′, and m′′ above, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A := {(Tp, Tq) : T ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then (Tp, Tq) cannot be an extreme point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Firstly, the set A is convex since C is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For some ϵ > 0 and δ > 0 to be decided later,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' consider the following perturbed matrices: T′ = � ����� · · · · · · · · · · · · · · · · M − ϵ · · m + δ · · M′ · · · · · · · · · · · · · · · · · · m′ + ϵ · · M′′ − δ · · m′′ · · · · · · · · · · · · · · · · � ����� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' T′′ = � ����� · · · · · · · · · · · · · · · · M · · m + ϵ′ · · M′ − δ′ · · · · · · · · · · · · · · · · · · m′ · · M′′ − ϵ′ · · m′′ + δ′ · · · · · · · · · · · · · · · · � ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' To be specific, the entries of T′, T′′, and T match except in the six locations highlighted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since C satisfies Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 (see Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2), both T′ and T′′ belong to the set C if ϵ, ϵ′, δ, and δ are small enough and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will now show that there exist choices of these parameters such that (Tp, Tq) is a convex combination of (T′p, T′q) and (T′′p, T′′q), and these three points are distinct elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consequently, (Tp, Tq) will not be an extreme point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any j ∈ ℓ, let vj denote the vector in Rℓ that is 1 at the jth location and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Define θi := pi/qi to be the likelihood ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If θi < ∞, then pi = θiqi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since θi is strictly increasing in i, only θi3 may be infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let us first suppose that θi3 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will consider the case when θi3 might be infinity in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let us begin by analyzing how T′ transforms p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since T′ differs from T only in the four locations mentioned above, T′p and Tp, both of which are distributions on [ℓ], differ only in the elements r and r′ of [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' On the element r, (T′q)r − (Tq)r is equal to −ϵqi1 + δqi2, and equal to its negation on the element r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, they satisfy the relation T′q = Tq + (−ϵqi1 + δqi2) (vr − vr′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 29 If ϵqi1 = δqi2, we have T′q = Tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Under the same setting, p is transformed as follows: T′p = Tp + (−ϵpi1 + δpi2) (vr − vr′) = Tp + (−ϵθi1qi1 + δθi2qi2) (vr − vr′) = Tp + ϵqi1(−θi1 + θi2) (vr − vr′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now analyze the effect of T′′, which satisfies T′′q = Tq + (ϵ′qi2 − δ′qi3) (vr − vr′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If ϵ′qi2 = δ′qi3, we have T′′q = Tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Under the same setting, p is transformed as follows: T′′p = Tp + (ϵ′pi2 − δ′pi3) (vr − vr′) = Tp + (−ϵ′θi2qi2 + δ′θi3qi3) (vr − vr′) = Tp + ϵ′qi2(−θi2 + θi3) (vr − vr′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Now observe that θi1 < θi2 < θi3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By choosing ϵ > 0 and ϵ′ > 0 small enough such that ϵqi1(−θi1 + θi2) = ϵ′qi2(−θi2 + θi3), we obtain (Tp, Tq) = 1 2 · � T′p, T′q � + 1 2 · � T′p, T′q � , and all three points are distinct elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Such a choice of ϵ and ϵ′ always exists, since both qi1(−θi1 + θi2) and qi2(−θi2 + θi3) are positive and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, (Tp, Tq) is not an extreme point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let us now consider the case when θi3 = ∞, or equivalently, qi3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Define ϵ′ to be 0, so that T′′q = Tq and T′′p = Tp − δ′pi3 (vr − vr′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then choose δ′ > 0 and ϵ > 0 small enough such that ϵqi1 (θi2 − θi1) = δ′pi3, which is possible since both sides are positive and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, (Tp, Tq) is a non-trivial convex combination of (T′p, T′q) and (T′′p, T′′q), so is not an extreme point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Without loss of generality, we assume that the likelihood ratios pi/qi are unique and strictly increasing in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We refer the reader to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16 and Claim D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T ∈ C be a channel from [k] to [ℓ] such that (Tp, Tq) is an extreme point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Suppose that there are ℓ′ unique columns in T with ℓ′ > 2ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' From now on, we assume that ℓ′ = 2ℓ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' otherwise, we apply the following argument to the first 2ℓ2 distinct columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let c, c′ ∈ [k] be such that the cth and c′th columns of T are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that for every pair of distinct columns c and c′, there are two rows such that cth column has a strictly bigger value than the c′th column on one row, and vice versa on the another row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This is because both of the columns sum up to 1, so if a particular column has a larger entry in a row, its entry must be smaller in a different row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, there exist two rows g(c, c′) and h(c, c′) such that T(g(c, c′), c) > T(g(c, c′), c′) and T(h(c, c′), c) < T(h(c, c′), c′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As a result, T(g(c, c′), c) and T(h(c, c′), c′) are not min-tight, and T(g(c, c′), c′) and T(h(c, c′), c) are not max-tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now order the distinct columns of T in the order of their appearance from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , iℓ′ be the indices of the unique columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For example, the first distinct column i1 is the first column of T (corresponding to the element 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The second distinct column i2 is the first column of T that is different from the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The third distinct column is the first column of T that is different from the first two columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let I be the set of unique column indices of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 30 Now, we divide the distinct columns in T into pairs: H = {(i1, i2), (i3, i4), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , (iℓ′−1, iℓ′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The total number of possible choices in H is ℓ′/2, and for every (m, m + 1) in H, the possible number of choices of (g(im, im+1), h(im, im+1)) is at most ℓ(ℓ − 1), since both of these lie in [ℓ] and are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, there must exist two pairs in H whose corresponding indices are the same, since ℓ′ 2 = ℓ2 > ℓ(ℓ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Without loss of generality, we let these pairs of columns be (i1, i2) and (i3, i4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let r := g(i1, i2) = g(i3, i4) and r′ := h(i1, i2) = h(i3, i4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then the previous discussion implies that: T(r, i1) and T(r, i3) are not min-tight, and T(r′, i1) and T(r′, i3) are not max-tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' T(r, i2) and T(r, i4) are not max-tight, and T(r′, i2) and T(r′, i4) are not min-tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, the columns i1, i2, and i3 satisfy the conditions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', they exhibit the forbidden structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This implies that (Tp, Tq) cannot be an extreme point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Therefore, ℓ′ ≤ 2ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17: Unique Columns to Threshold Channels In this section, we provide the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17 using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Noting that our main structural result is more widely applicable (Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7), we prove a more general version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17 below for Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Before doing so, we require an additional property on the set of our channels, proved in Appendix B: Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9 (Closure under pre-processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The set J γ,ν ℓ,k satisfies the following closure property under pre-processing: J γ,ν ℓ,k = k� ℓ′=1 � T2 × T1 : T2 ∈ J γ,ν ℓ,ℓ′ and T1 ∈ Tℓ′,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (23) Informally, if we take an arbitrary channel T1 and compose it with an LP private channel T2, the composition T2 × T1 is also LP private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The following result is thus a more general version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='10 (Structure of optimal channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any ℓ ∈ N, let C be the set of channels J γ,ν ℓ,k from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let A be the set of all pairs of distributions that are obtained by applying a channel from C to p and q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', A = {(Tp, Tq) | T ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (24) If (Tp, Tq) is an extreme point of A, then T can be written as T = T2 ×T1, for some T1 ∈ T thresh ℓ′,k and T2 an extreme point of the set J γ,ν ℓ,ℓ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since C is convex, the joint range A is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2, we know that if (Tp, Tq) is an extreme point of A, then T can be written as T2 × T1, where T1 ∈ Tℓ′,k for ℓ′ := 2ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Using Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9, any such channel in C is of the form T2×T1, where T1 ∈ Tℓ′,k and T2 ∈ J γ,ν ℓ,ℓ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Combining the last two observations, we obtain the following: A = conv �� (T2 × T1p, T2 × T1q) : T2 ∈ J γ,ν ℓ,ℓ′ , T1 ∈ Tℓ′,k �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (25) We now claim that we can further take T1 to be a threshold channel T1 ∈ T thresh ℓ′,k and T2 to be an extreme point of J γ,ν ℓ,ℓ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This claim follows if we can write an arbitrary point in A as a convex 31 combination of elements of the set � (T2 × T1p, T2 × T1q) : T2 ∈ ext � J γ,ν ℓ,ℓ′ � , T1 ∈ T thresh ℓ′,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By equation (25), it suffices to demonstrate this convex combination for all points of the form (T2 × T1p, T2 × T1q), for some T2 ∈ J γ,ν ℓ,ℓ′ and T1 ∈ Tℓ′,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let H1, H2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' be extreme points of J γ,ν ℓ,ℓ′ , and let L1, L2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' be an enumeration of the threshold channels T thresh ℓ′,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By definition, any T1 ∈ J γ,ν ℓ,ℓ′ can be written as � i αiHi for some convex combi- nation α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Furthermore, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='16 implies that any (T2p, T2q), for T2 ∈ Tℓ′,k, can be written as � j βj(Ljp, Ljq) = (� j βjLjp, � j βjLjq), for some convex combination β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, any arbitrary point (T2 × T1p, T2 × T1q), for T2 ∈ J γ,ν ℓ,ℓ′ and T1 ∈ Tℓ′,k, can be written as (T2 × T1p, T2 × T1q) = ��� i αiHi � × T1p, �� i αiHi � × T1q � = � i αi (Hi × T1p, Hi × T1q) = � i αi (Hi (T1p) , Hi (T1q)) = � i αi � �Hi � �� j βjLjp � � , Hi � �� j βjLjq � � � � = � i αi � �� j βjHi × Ljp, � j βjHi × Ljq � � = � i � j αiβj (Hi × Ljp, Hi × Ljq) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Finally, note that {αiβj} are also valid convex combinations of (Hi × Ljp, Hi × Ljq), since they are nonnegative and sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='11 (Extending Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='17 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='10 to a more general set of constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We note that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='10 can be extended to an arbitrary convex set of channels C that satisfy (appropri- ately modified versions of) Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 and equation (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (Also see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=') 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 Application to Hypothesis Testing In Section 3, we showed that the minimax-optimal sample complexity can be obtained by a communication- efficient and efficiently computable channel, up to logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' However, for a particular (p, q), these guarantees can be significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For example, consider the extreme case when p and q are the following two distributions on [k]: for γ small enough, p = [α, 1 − α − (k − 2)γ, γ, γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , γ], q = [β, 1 − β − (k − 2)γ, γ, γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T′ be a deterministic binary channel that maps the first and second elements to different elements, while assigning the remaining elements arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Now consider the following private channel T: the channel T′, followed by the randomized response over binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then as γ → 0, the performance of T mirrors equation (3), which is much better than the minimax bound of equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, there is a wide gap between instance-optimal and minimax-optimal 32 performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We thus consider the computational question of optimizing a quasi-convex function g(Tp, Tq) over all possible ϵ-private channels that map to a domain of size ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The following result proves Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='18 for C equal to Pϵ ℓ,k: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='12 (Computationally efficient algorithms for maximizing quasi-convex functions under privacy constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be fixed distributions over [k], let C be the set of channels J γ,ν ℓ,k from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1, and let A = {(Tp, Tq) : T ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let g : A → R be a jointly quasi-convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then there is an algorithm that solves maxT∈C g(Tp, Tq) in time polynomial in kℓ2 and 2O(ℓ3 log ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The algorithm is as follows: we try all threshold channels in T1 ∈ T thresh ℓ′,k and all extreme points of J γ,ν ℓ,ℓ′ , and output the channel T = T2 × T1 that attains the maximum value of g(Tp, Tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='10 and quasi-convexity of g, we know the algorithm will output a correct value, since all extreme points are of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, we focus on bounding the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We know that the cardinality of T thresh ℓ′,k is bounded by kℓ′ (up to a rotation of output rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3, the time taken to iterate through all the extreme points of ϵ-LDP channels from [ℓ′] to [ℓ] is at most polynomial in 2ℓ3 log ℓ, since Jℓ,ℓ′ is a polytope in R2ℓ3 with poly(ℓ) inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='20 is immediate from Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='18, and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2, stated later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 6 Extensions to Other Notions of Privacy In this section, we explore computational and statistical aspects of hypothesis testing under other notions of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 is on approximate privacy, in which we first focus on (ϵ, δ)-LDP and then our proposed definition of approximate privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Next, we focus on binary communication constraints for Rényi differential privacy in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This will be possible since our algorithmic and structural results were not restricted to the case of pure LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We begin by noting that communication constraints have a benign effect on the sample com- plexity of hypothesis testing for many notions of privacy: Condition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 (Closure under post-processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For each r ∈ N, consider sets Cr ⊆ Tr,k and define C = ∪r∈NCr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We say C satisfies ℓ-post-processing if for every r ∈ N, if T ∈ Cr and H is a deterministic channel from [r] to [ℓ], the channel H × T also belongs to Cℓ, and thus to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Post-processing is satisfied by various notions of privacy: ϵ-pure privacy, (ϵ, δ)-approximate privacy (see Dwork and Roth [DR13, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1]), and Rényi privacy [Mir17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For a set of channels C, we use the notation n∗(p, q, C) to denote the sample complexity of hypothesis testing under channel constraints of C in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The following result shows that even with binary communication constraints, the sample complexity increases by at most a logarithmic factor: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 (Benign effect of communication constraints on sample complexity under closure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be any two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C be a set of channels that satisfy ℓ-post-processing (Condition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1) for some ℓ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let Cℓ denote the subset of channels in C that map to a domain of size ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then n∗(p, q, Cℓ) ≲ n∗(p, q, C) · � 1 + log (n∗(p, q, C)) ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (26) 9Recall that g is assumed to be permutation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If not, an extra factor of ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' will appear in the time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 33 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T be the optimal channel in C that maximizes d2 h(Tp, Tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let k′ be the size of the range of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7, we have n∗(p, q, C) ≍ 1/d2 h(Tp, Tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8, we know that there exists T′ ∈ Tℓ,k′ such that10 d2 h(Tp, Tq) ≲ d2 h(T′(Tp), T′(Tq)) · � 1 + log(1/d2 h(Tp, Tq)) ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (27) By the assumed closure of C under post-processing, the channel T′ × T belongs to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, the channel T′×T also belongs to Cℓ, since its output is of size ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This implies that the sample complexity n∗(p, q, Cℓ) is at most 1/d2 h(T′ × Tp, T′ × Tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Using the fact that n∗(p, q, C) ≍ 1/d2 h(Tp, Tq), we obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, in the rest of this section, our main focus will be on the setting of binary channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Approximate Local Privacy In this section, we first focus on (ϵ, δ)-approximate LDP (Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We begin by showing upper bounds on the associated sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' On the computational front, we present efficient algorithms for the case of binary constraints and then propose a relaxation for the case of larger output domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We first recall the definition of (ϵ, δ)-LDP: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 ((ϵ, δ)-LDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We say a channel from X to Y is (ϵ, δ)-LDP if for all S ⊆ Y, we have sup x,x′∈X P[T(x) ∈ S)] − eϵ · P[T(x) ∈ S)] − δ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (28) What makes the analysis of (ϵ, δ)-LDP different from ϵ-LDP is that when |Y| > 2, the condition in inequality (28) should be verified for all sets S ⊆ Y, not just singleton sets (|S| = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Only when |Y| = 2 is it enough to consider singleton sets S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let n∗(p, q, (ϵ, δ)) denote the sample complexity for the setting in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3, with C equal to the set of all (ϵ, δ)-LDP channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We directly obtain the following upper bound on the sample complexity, proved in Appendix C, which happens to be tight for the case of binary distributions: Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 (Sample complexity of approximate LDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For all δ ∈ (0, 1), we have n∗(p, q, (ϵ, δ)) ≲ min � n∗(p, q, ϵ) · 1 1 − δ , n∗(p, q) · 1 δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, this is tight (up to constant factors) when both p and q are binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In the rest of this section, we focus on efficient algorithms in the presence of both privacy and communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Turning to computationally efficient algorithms for the case of privacy and communication constraints, we present two kinds of results: exact results for the case of binary outputs, and sharp relaxations for the case of multiple outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Binary channels: Let C be the set of all (ϵ, δ)-approximate LDP channels from [k] to [2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', binary channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let γ = (eϵ, eϵ) and ν = (δ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that C is then equal to J γ,ν 2,k , defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='10 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='12 hold in this case, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 10If the supremum is not attained, the proof can be modified by considering a suitable sequence of channels and applying a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 34 Channels with larger output spaces: Here, we define a new notion of privacy that relaxes (ϵ, δ)-LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' It is enough to verify whether the privacy condition holds for singleton events S: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 ((ϵ, δ)-SLDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We say a channel X to Y is (ϵ, δ)-singleton-based-LDP ((ϵ, δ)-SLDP) if for all S ⊆ Y, we have sup x,x′∈X P[T(x) ∈ S)] − eϵ · P[T(x) ∈ S)] − δ · |S| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The following result shows that (ϵ, δ)-SLDP is a good approximation to (ϵ, δ)-LDP when the output space is small: Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 (Relations between LDP and SLDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consider a channel T from X to [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If T is (ϵ, δ)-SLDP, it is (ϵ, ℓδ)-LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If T is (ϵ, δ)-LDP, it is (ϵ, δ)-SLDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The proof is immediate from the definitions of (ϵ, δ)-LDP and (ϵ, δ)-SLDP, and we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now show that it is easy to optimize over SLDP channels in the presence of communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any ℓ ∈ N, let C be the set of all channels from [k] to [ℓ] that satisfy (ϵ, δ)-SLDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let γ = (eϵ, eϵ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , eϵ) and ν = (δ, δ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that C is then equal to J γ,ν ℓ,k , defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='10 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='12 imply that we can efficiently optimize over SLDP channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 Other Notions of Privacy We briefly note that our computationally efficient algorithms hold for a wider family of channels defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' see also Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Finally, we consider the case of Rényi differential privacy introduced in Mironov [Mir17]: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 ((ϵ, α)-Rényi differential privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let ϵ ∈ R+ and α > 1, and let X and Y be two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A channel T : X → Y satisfies (ϵ, α)-RDP if for all x, x′ ∈ X, we have Dα(T(x)∥T(x′)) ≤ ϵ, where Dα(p∥q) is the Rényi divergence of order α between two distributions p and q on the same probability space, defined as Dα(p∥q) := 1 α − 1 log EX∼q ��p(X) q(X) �α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Rényi divergence is also defined for α = 1 and α = ∞ by taking limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' When α = 1, the limit yields the Kullback–Leibler divergence, and when α = ∞, it leads to the supremum of the log-likelihood ratio between p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In fact, (∞, ϵ)-RDP is identical to ϵ-LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Similarly, (1, ϵ)-RDP is closely related to mutual information-based privacy [CY16], since the corresponding channel T has Shannon capacity at most ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='8 (Rényi differential privacy and binary constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let ϵ > 0 and α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let C be the set of (ϵ, α)-RDP channels from [k] to [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k], and define A := {(Tp, Tq) : T ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If (Tp, Tq) is an extreme point of A for T ∈ C, then T can be written as T1 × T2, where T1 is an extreme point of the set of (ϵ, α)-RDP channels from [2] to [2], and T2 is a binary threshold channel from [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 35 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consider two binary distributions [x, 1 − x] and [y, 1 − y], where 0 ≤ x, y ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The α-Rényi divergence between the distributions is given by Dα(x∥y) := 1 α − 1 log � xαy1−α + (1 − x)α(1 − y)1−α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that the term inside the logarithm is convex in y for fixed x, and is minimized when y = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Hence, the Rényi divergence above, as a function of y, is decreasing for y ∈ [0, x] and increasing for y ∈ [x, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A similar conclusion holds for fixed y and varying x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consider a channel T ∈ C given by T = � x1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' xk 1 − x1 1 − x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1 − xk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Without loss of generality, assume x1 ≤ x2 ≤ · · · ≤ xk Suppose there is an index j such that x1 < xj < xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that xj /∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By the monotonicity property of the Rényi divergence noted above, for any index i, we have max {Dα(xj∥xi), Dα(xi∥xj)} < max {Dα(x1∥xk), Dα(xk∥x1)} ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This means that xj can be perturbed up and down by a small enough δ such that the Rényi divergence constraints continue to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Such perturbations will allow T to be written as a convex combination of two distinct matrices, so T cannot be an extreme point of the (convex) set C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, an extreme point must have only two distinct columns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', it must have the form T = � x1 x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' x1 xk xk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' xk 1 − x1 1 − x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1 − x1 1 − xk 1 − xk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 1 − xk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Equivalently, any extreme point is a deterministic channel from [k] → [2] followed by an RDP- channel from [2] → [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since we are only concerned with extreme points that correspond to extreme points of the joint range A, an argument identical to the one in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='10 yields that an extreme point must admit a decomposition T1 ×T2, where T2 is a threshold channel from [k] → [2] and T1 is an extreme point of the set of RDP channels from [2] → [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The above result implies that given a quasi-convex function g : A → R, if we are interested in maximizing g(Tp, Tq) over T ∈ C, the optimal T can be written as T1 × T2, where T1 is a binary-input, binary-output Rényi private channel and T2 is a threshold channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since there are only 2k threshold channels, we can try all those choices of T2, and then try to optimize over T1 for each of those choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' However, each such problem is over binary inputs and binary outputs, and thus is amenable to grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In addition to the convexity of RDP channels, we also used the closure-under-pre- processing property (see Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='9) and the unimodality of Dα(x∥y) when one of the variables is fixed and the other is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The above proof technique will therefore work for any set of convex channels from [k] → [2] that are closed under pre-processing, and are defined in terms of such a unimodal function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, our results will continue to hold for all f-divergence-based private channels, defined as all T satisfying Df(T(x)∥T(x′)) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our results also hold for zero-concentrated differential privacy (z-CDP) [BS16], which is a notion of privacy defined using Rényi divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 36 7 Conclusion In this paper, we considered the sample complexity of simple binary hypothesis testing under privacy and communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We considered two families of problems: finding minimax-optimal bounds and algorithms, and finding instance-optimal bounds and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For minimax optimality, we considered the set of distributions with fixed Hellinger divergences and total variation distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This is a natural family to consider, because these two metrics characterize the sample complexity in the low- and high-privacy regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Prior work did not resolve the question of sample complexity in the moderate-privacy regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' our work has addressed this gap in the literature, by establishing a sample-complexity lower bound via a carefully constructed family of distribution pairs on the ternary alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our results highlight a curious separation between the binary and ternary (and larger alphabet) settings, roughly implying that the binary case is substantially easier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', has a lower sample complexity) than the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our focus on instance optimality sets our paper apart from most prior work on information- constrained estimation, which exclusively considered minimax optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' When only privacy con- straints are imposed, we established approximately instance-optimal algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', for any distri- bution pair, we proposed a protocol whose sample complexity is within logarithmic factors of the true sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Importantly, the algorithm we proposed to identify this protocol is compu- tationally efficient, taking time polynomial in k, the support size of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' When both privacy and communication constraints are in force, we developed instance-optimal algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', protocols whose sample complexity is within constant factors of the true sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' As before, these algorithms take time polynomial in k, for any constant communication constraint of size ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our results highlight the critical role played by threshold channels in both communication- and privacy-constrained settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We showed that for any distribution pair, the channel with output size ℓ that maximizes the output divergence (Hellinger, Kullback–Leibler, or any quasi-convex function in general) among all channels with fixed output size ℓ must be a threshold channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Furthermore, optimal private channels with output size ℓ admit a decomposition into a threshold channel cascaded with a private channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' These two results underpin our algorithmic contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' There are many interesting open problems stemming from our work that would be worth explor- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We did not characterize instance-optimal sample complexity in the moderate-privacy regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' our work shows that it is not characterized in terms of the Hellinger divergence and total varia- tion distance, but leaves open the possibility of some other divergence, such as the Eγ divergence, capturing the sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We identified a forbidden structure for optimal private channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' however, the best algorithm from Kairouz, Oh, and Viswanath [KOV16] does not use this infor- mation at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' It would be interesting to see if that algorithm could be made more efficient by incorporating the extra structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Many open questions remain for the approximate LDP setting, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' There is no known upper bound on the number of outputs that suffice for op- timal approximate LDP channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' It is plausible, but unknown, if instance-optimal private channels with ℓ > 2 outputs admit decompositions into threshold channels cascaded with private channels, similar to the pure LDP setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' It would be interesting to see if optimal SLDP channels, which are efficient to find, are nearly instance optimal for approximate LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 37 References [AC86] R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Warner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' “Randomized Response: A Survey Technique for Eliminating Evasive Answer Bias”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In: Journal of the American Statistical Association 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='309 (1965), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 63–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A Randomized Response in Low-Privacy Regime In this section, we prove Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6, which were used to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='13 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 is proved in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 is proved in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 Recall the definitions of A and A′ from equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5 (Randomized response preserves contribution of comparable elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Suppose � i∈A � A′(√qi − √pi)2 ≥ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then Tϵ,ℓ RR, for ℓ ≤ eϵ, satisfies d2 h(Tϵ,ℓ RRp, Tϵ,ℓ RRq) ≳ min � 1, eϵ τ ℓ � τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, when eϵ ≳ ℓ τ , the randomized response preserves the original contribution of comparable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 41 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Without loss of generality, we will assume that � i∈A(√qi − √pi)2 ≥ τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p′ = Tϵ,ℓ RRp and q′ = Tϵ,ℓ RRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By the definition of the randomized response, each probability x is mapped to (1 + x(eϵ − 1))/(k − 1 + eϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, p′ and q′ are given by p′ i = 1 + pi(eϵ − 1) (ℓ − 1) + eϵ , and q′ i = 1 + qi(eϵ − 1) (ℓ − 1) + eϵ , ∀i ∈ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (29) Recall that δi = (pi − qi)/qi ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For each i ∈ ℓ, we now define δ′ i := (p′ i − q′ i)/q′ i, which has the following expression in terms of δi and qi: δ′ i = p′ i − q′ i q′ i = (eϵ − 1)(pi − qi) 1 + qi(eϵ − 1) = (eϵ − 1)qi 1 + qi(eϵ − 1) · δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (30) Let r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='01 min � e−ϵ, τ ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We define the following subsets of the domain: E = {i : δi ∈ (0, 1] and qi ≥ e−ϵ} , (31) E′ = {i : δi ∈ (0, 1] and qi ∈ (r, e−ϵ)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (32) Observe that E ∪ E′ ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since eϵ ≥ ℓ, equation (29) implies that q′ i ≥ 1 4(e−ϵ + qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, on i ∈ E′, we have q′ i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='25e−ϵ, and on i ∈ E, we have q′ i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='25qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now apply these approximations to equation (30): we lower-bound the numerator by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5eϵqiδi and upper-bound the denominator based on whether i ∈ E or i ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' On E′, the denominator in equation (30) is upper-bounded by 2, and on E, the denominator is upper-bounded by 2qieϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This is summarized as follows: for i ∈ E ∪ E′, we have δ′ i ≥ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1δiqieϵ, i ∈ E′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1δi, i ∈ E, q′ i ≥ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='25e−ϵ, i ∈ E′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='25qi, i ∈ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By definition of δ′, it follows that δ′ i ∈ (0, 1] on i ∈ E ∪ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, the contribution from the ith element to d2 h(p′, q′) is at least a constant times q′ i(δ′ i)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' see Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Applying this element-wise, we obtain the following: d2 h(p′, q′) ≳ � i∈E′ q′ i(δ′ i)2 + � i∈E q′ i(δ′ i)2 ≳ � i∈E′ e−ϵ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1δiqieϵ)2 + � i∈E qi (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1δi)2 ≳ eϵr � i∈E′ qiδ2 i + � i∈E qiδ2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (33) Now consider the set A = {i : i ∈ A and qi ≥ r}, which is equal to E ∪ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The set A preserves the contribution to Hellinger divergence from comparable elements, as shown below: � i∈A (√qi − √pi)2 = � i∈A (√qi − √pi)2 − � i:i∈A,qi≤r (√qi − √pi)2 ≥ τ 2 − 2ℓr ≥ τ 4, since r ≤ τ 10ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since A = E1 ∪ E2, one of the two terms � i∈E′(√qi − √pi)2 or � i∈E(√qi − √pi)2 must be at least τ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Now consider the following two cases: 42 Case 1: � i∈E(√qi − √pi)2 ≳ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In this case, we are done by inequality (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' That is, d2 h(p′, q′) ≳ � i∈E ( � q′ i − � p′ i)2 ≳ � i∈E qiδ2 i ≳ � i∈E (√qi − √pi)2 ≳ τ, where we use Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 element-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Case 2: � i∈E′(√qi − √pi)2 ≳ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By inequality (33), we have d2 h(p′, q′) ≳ eϵ · r � i∈E′ qiδ2 i ≳ eϵ · rτ ≳ min � 1, eϵ τ ℓ � τ, where we use the definition of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, we obtain the desired lower bound in both of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='6 (Reduction to base case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then there is a channel T, which can be computed in time polynomial in k, that maps [k] to [ℓ] (for ℓ to be decided below) such that for p′ = Tp and q′ = Tq, at least one of the following holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any ℓ > 2 and ℓ ≤ min � k, 1 + log � 1/d2 h(p, q) �� , we have � i∈B � B′ �� q′ i − � p′ i �2 ≳ d2 h(p, q) · ℓ min � k, 1 + log � 1/d2 h(p, q) ��, where B and B′ are defined analogously to A and A′ in equation (17), but with respect to distributions p′ and q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' ℓ = 2 and d2 h(p′, q′) ≳ d2 h(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let us begin by considering the case when � i∈A � A′ �√qi − √pi �2 ≤ d2 h(p,q) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Following Pensia, Jog, and Loh [PJL22, Theorem 2 (Case 1 in the proof)], there exists a binary channel that preserves the Hellinger divergence up to constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This completes the case for ℓ = 2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Suppose for now that � i∈A � A′ �√qi − √pi �2 ≥ d2 h(p,q) 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', the comparable elements constitute at least half the Hellinger divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consider the channel T′ that maps the comparable elements of p and q to distinct elements, and maps the remaining elements to a single super-element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let α be the contribution to the Hellinger divergence from the comparable elements in T′p and T′q (defined analogously to equation (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' It can be seen that α ≥ d2 h(p,q) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let ℓ ≥ 3 be as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Now consider the channel T′′ that compresses T′p and T′q into ℓ-ary distributions that preserve the Hellinger divergence, from Pensia, Jog, and Loh [PJL22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 (Case 2 in the proof)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let βℓ be the contribution to the Hellinger divergence from the comparable elements in T′′T′p and T′′T′q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then the result in Pensia, Jog, and Loh [PJL22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2] implies that βl ≳ α � ℓ/ min(k, 1 + log(1/d2 h(p, q))) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This completes the proof in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' B Properties of Private Channels Recall the definition of the set of channels J γ,ν ℓ,k from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 below: 43 Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 (LP family of channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For any ℓ ∈ N, let ν = (ν1, ν2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , νℓ) and γ = (γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , γℓ) be two nonnegative vectors in Rℓ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For k ∈ N, define the set of linear programming (LP) channels J γ,ν ℓ,k , a subset of Tℓ,k, to be the (convex) set of all channels from [k] to [ℓ] that satisfy the following constraints: For each row j ∈ [ℓ], and for each i, i′ ∈ [k], we have T(j, i) ≤ γjT(j, i′) + νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (21) We begin by an equivalent characterization of the constraints above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For a channel T from [k] to [ℓ], let {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , mℓ} and {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Mℓ} be the minimum and maximum entries of each row, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then the channel T satisfies the conditions (21) if and only if for each j ∈ [ℓ], we have Mj ≤ γjmj + νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (34) We first show that J γ,ν ℓ,k satisfies Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For the special case of LDP channels, the following claim was also proved in Holohan, Leith, and Mason [HLM17]: Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' J γ,ν ℓ,k satisfies Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T be any extreme point of J γ,ν ℓ,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , mℓ} and {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Mℓ} be as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Suppose that there exists c ∈ [k], such that there exist distinct r, r′ ∈ [ℓ] with T(r, c) ∈ (mr, Mr) and T(r′, c) ∈ (mr′, Mr′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, both T(r, c) and T(r′, c) are strictly positive and less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will now show that T is not an extreme point of J γ,ν ℓ,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For an ϵ > 0 to be decided later, consider the channel T′ that is equal to T on all but two entries: On (r, c), T′ assigns probability T(r, c) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' On (r′, c), T′ assigns probability T(r′, c) − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Now define T′′ similarly, with the difference being that on (r, c), T′′ assigns probability T(r, c) − ϵ, and on (r′, c), T′′ assigns probability T(r′, c)+ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Both T′ and T′′ are thus valid channels for ϵ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let us show that T′ and T′′ belong to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If we choose ϵ > 0 small enough, the row-wise maximum and minimum entries of T′ and T′′ are equal to those of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Here, we critically use the fact that the entries that were modified were “free.” By inequality (34), both T′ and T′′ belong to J γ,ν ℓ,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since T is the average of T′ and T′′, it is not an extreme point of J γ,ν ℓ,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now show that J γ,ν ℓ,k satisfies Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' J γ,ν ℓ,k satisfies Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We follow the notation from Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T be an extreme point of J γ,ν ℓ,k , and let r and r′ be the corresponding rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We show that T′ (defined in the condition) belongs to J γ,ν ℓ,k by showing that entries of T′ satisfy the constraints of the rth row and the r′th row (since the other rows are unchanged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In fact, we establish these arguments only for the rth row, and the analogous arguments hold for the r′th row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let mr and Mr be the row-wise minimum and maximum entry of this row in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let us first consider the case when Mr < γrmr + νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then there exist positive ϵ′ and δ′ such that Mr + δ < γr(mr − ϵ) + νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By inequality (34), as long as the rth row of a channel contains entries in [mr − ϵ, Mr + δ], the constraints of this particular row will be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since the entries in the rth row of T′ belong to this interval, the constraints of the rth row are satisfied by T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let us now consider the alternate case where Mr = γrmr+νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since m and M do not correspond to the min-tight and max-tight entries, we have mr < M and m < Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Consequently, even after perturbations by ϵ > 0 and δ > 0 small enough, the entries of T′ lie in [mr, Mr].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, inequality (34) implies that the constraints of the rth row in T′ are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 44 Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 (Closure under pre-processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The set J γ,ν ℓ,k satisfies the following: J γ,ν ℓ,k = k� ℓ′=1 � T2 × T1 : T2 ∈ J γ,ν ℓ,ℓ′ and T1 ∈ Tℓ′,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (35) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We first show the simple direction that J γ,ν ℓ,k ⊆ �k ℓ′=1 � T2 × T1 : T2 ∈ J γ,ν ℓ,ℓ′ and T1 ∈ Tℓ′,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let Ik correspond to the identity channel on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then every channel T ∈ J γ,ν ℓ,k , can be written as T × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, J γ,ν ℓ,k ⊆ � T2 × Ik : T2 ∈ J γ,ν ℓ,ℓ′ � , and the desired conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now show that every channel in the right-hand side belongs to J γ,ν ℓ,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For an arbitrary ℓ′ ∈ [k], let T2 ∈ J γ,ν ℓ,ℓ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Define {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , mℓ} and {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Mℓ} to be the minimum and maximum entries of each row in T2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By inequality (34), for each j ∈ [ℓ], we have Mj ≤ γjmj + νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T1 ∈ Tℓ′,k be an arbitrary channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T = T2 × T1 be in Tℓ,k, and let {m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , mℓ} and {M′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , M′ ℓ} be the minimum and maximum entries of each row in T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In order to show that T ∈ J γ,ν ℓ,k , we need to show that for each j ∈ [ℓ], we have M′ j ≤ γjm′ j + νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since it already holds that Mj ≤ γjmj + νj for all j, it suffices to show that M′ j ≤ Mj and m′ j ≥ mj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that for any c ∈ [k] and r ∈ [ℓ], the (r, c)-entry of T is a convex combination of the rth row in T2, where the weights in the convex combination correspond to the cth column in T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since the maximum of a collection of items is always as large as any convex combination of these items, we have M′ j ≤ Mj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Similarly, we have m′ j ≥ mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' C Other Notions of Privacy We provide the proof of the following result, omitted from Section 6: Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='4 (Sample complexity of approximate LDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For all δ ∈ (0, 1), we have n∗(p, q, (ϵ, δ)) ≲ min � n∗(p, q, ϵ) · 1 1 − δ , n∗(p, q) · 1 δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Moreover, this is tight (up to constant factors) when both p and q are binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T be an ϵ-LDP channel that maximizes d2 h(Tp, Tq) among all ϵ-LDP channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T′ be the following channel that maps from [k] to [2k]: for any element i ∈ [k], use the channel T, and with probability δ, map i to k + i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' It can be seen that T′ satisfies (ϵ, δ)-LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p′ and q′ be the corresponding distributions after transforming p and q using T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' It can be seen that p′ is a distribution over [2k] such that the first k elements are equal to (1 − δ)Tp coordinate-wise, and the bottom k elements are equal to δp coordinate-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' A similar conclusion holds for q′, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, we have d2 h(T′p, T′q) = (1 − δ) · d2 h(Tp, Tq) + δ · d2 h(p, q) ≍ max � (1 − δ) · d2 h(Tp, Tq), δ · d2 h(p, q) � ≍ max � (1 − δ) · 1 n∗(p, q, ϵ), δ · 1 n∗(p, q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' By Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7, the sample complexity n∗(p, q, (ϵ, δ)) is at most 1/d2 h(T′p, T′q), which gives the upper bound on n∗(p, q, (ϵ, δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The tightness follows from the result of Kairouz, Oh, and Viswanath [KOV16, Theorem 18], which implies that T′ defined above is an optimal channel for binary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 45 D Auxiliary Lemmas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 Degenerate Conditions for Joint Range We show in this section that we can safely rule out certain degenerate conditions for p and q for our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p and q be two distributions on [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, we would like to assume the following: Consider the likelihood ratio pi/qi, defined to be ∞ if qi = 0 and pi ̸= 0, and undefined if both pi and qi are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Assume that all the likelihood ratios are well-defined and unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' If these conditions do not hold, define p′ and q′ to be distributions over [k′] for some k′ ≤ k, constructed as follows: start by removing elements that have zero probability mass under both p and q, then merge the elements with the same likelihood ratios into super-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T∗ ∈ Tk′,k be the corresponding deterministic map, which satisfies p′ = T∗p and q′ = T∗q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We make the following claim: Claim D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' With the notation above, for any ℓ ∈ N and T ∈ Tℓ,k, there exists T′ ∈ Tℓ,k′ such that (Tp, Tq) = (T′p′, T′q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In particular, {(Tp, Tq) : T ∈ C} = {(Tp′, Tq′) : T ∈ C′} for two choices of C and C′: (i) (C, C′) = (Tℓ,k, Tℓ,k′) and (ii) (C, C′) = (Pϵ ℓ,k, Pϵ ℓ,k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Claim D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1 ensures that the joint ranges of (p, q) and (p′, q′) are identical, so our structural and algorithmic results continue to hold when applied to p′ and q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We will now prove Claim D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof of Claim D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let {I0, I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , Ik′} be the smallest partition of [k] such that I0 contains ele- ments where both pi and qi are zero, and for each i ∈ [k′], the likelihood ratio of elements in Ii are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then the channel T∗ mentioned above has the following form: T∗(x) = i if x ∈ Ii and i > 0, and T∗(x) = 1 if x ∈ I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that for each i ∈ [k′], we have p′ i = � j∈Ii pj, q′ i = � j∈Ii qj, and at most one of them is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Now consider a channel T ∈ Tℓ,k, and let {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , vk} be the columns of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' It is easy to see that columns belonging to indices in I0 do not affect (Tp, Tq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For i ∈ [k′], define θ′ i := p′ i/q′ i to be the likelihood ratio of the transformed distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Define T′ to be the channel with columns v′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' , v′ k such that v′ i = � � � � j∈Ii vjpj p′ i if p′ i > 0, � j∈Ii vjqj q′ i otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' First consider the case when for all i ∈ [k′], we have 0 < θ′ i < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then for all i ∈ [k′], we have p′ i = θ′ iq′ i and v′ i = � j∈Ii vjpj p′ i = � j∈Ii vjqj q′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, we have (Tp, Tq) = � � � i∈[k′] � j∈Ii vjpj, � i∈[k′] � j∈Ii vjqj � � = � � � i∈[k′] p′ i · �� j∈Ii vjpj p′ i � , � i∈[k′] q′ i · �� j∈Ii vjqj q′ i �� � = � � � i∈[k′] p′ iv′ i, � i∈[k′] q′ iv′ i � � = � T′p′, T′q′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 46 We now consider the case when there is an index a ∈ [k′] such that p′ a = 0 and an index b ∈ [k′] such that q′ b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then it must be that θ′ a = 0 and θ′ b = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then v′ a = � j∈Ii vjqj q′ i and v′ b = � j∈Ii vjpj p′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Following the calculations above, we obtain � j∈Ii vjpj = v′ ip′ i for each i ∈ [k] \\ {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' In fact, the same result is true for i = a, since both sides are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The same conclusion holds for q and q′, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This completes the proof of the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now turn to the final claim, regarding the joint range under the channel constraints of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The case C = Tℓ,k is immediate from the preceding discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let T1 ∈ Tk,k′ be such that (p, q) = (T1p′, T1q′) and T2 ∈ Tk′,k be such that (p′, q′) = (T2p, T2q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' For C = Pϵ ℓ,k and C′ = Pϵ ℓ,k′, we only need to show that (i) if T′ ∈ C′, then T′T2 ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' and (ii) if T ∈ C, then TT1 ∈ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Both of these conditions hold because privacy is closed under pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 Valid Choice of Parameters in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 We now give the details that were omitted in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='7 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We first reparametrize the problem by setting x = γ and y = γ1+δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The constraint δ > 0 is equivalent to y < x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then dTV(p, q) = x + y, and d2 h(p, q) = 2y + �� 1/2 + x − y − � 1/2 �2 + �� 1/2 − x − y − � 1/2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We begin by setting ν = x + y, which is possible since 0 ≤ y < x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='25 and ν ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then x = ν − y, where y ∈ (0, ν/2) and ν ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Our goal is now to show that there exists a valid choice of y such that d2 h(p, q) = ρ, as long as 2ν2 ≤ ρ ≤ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Define g(y) to be the Hellinger divergence between p and q given y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=', g(y) = 2y + �� 1/2 + ν − 2y − � 1/2 �2 + �� 1/2 − ν − � 1/2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Since g is a continuous function, it suffices to show that g(0) < 2ν2 and g(ν/2) > ν, which would imply that there is a choice of y ∈ (0, ν/2) such that g(y) = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We have g(0) = �� 1/2 + ν − � 1/2 �2 + �� 1/2 − ν − � 1/2 �2 ≤ 3ν2/2, where we use the fact that ��� � 1/2 + a − � 1/2 ��� ≤ a for all a ≥ 0, and is less than |a|/2 for a ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' On the other hand, g(ν/2) > ν, since ν < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, there is a choice of y ∈ (0, ν/2) such that d2 h(p, q) = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Given these choices of x and y, we can infer the choice of γ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='25) and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 Taylor Approximation to Hellinger Divergence Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 (Additive approximation for √· ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' There exist constants 0 < c1 ≤ c2 such that for 0 < y ≤ x, we have c1 · y2 x ≤ (√x − √x − y)2 ≤ c2 · y2 x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' It suffices to prove that for δ ∈ (0, 1], we have 1 − √ 1 − δ ≍ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We first start with the upper bound: since 1 − δ ≤ √ 1 − δ, we have 1 − √ 1 − δ ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We now show the lower bound and claim that 1 − √ 1 − δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5δ for all δ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' This inequality is equivalent to showing 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5δ ≥ √ 1 − δ, which is equivalent to showing that 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='25δ2 − δ ≥ 1 − δ, which holds since δ2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2 (Approximation for Hellinger divergence of binary distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let p, q ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let Ber(p) and Ber(q) be the corresponding Bernoulli distributions with min(p, q) ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then d2 h (Ber(p), Ber(q)) ≍ d2 TV(Ber(p), Ber(q)) max(p, q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 47 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let q be the larger of the two quantities, so p satisfies p ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The total variation distance is thus q − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Let δ = (q − p)/q ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Observe that p = q − qδ and the total variation distance is δq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' We begin by noting that Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 implies that (√q − √p)2 = �√q − � q − δq �2 ≍ δ2q2 q ≍ d2 TV (Ber(p), Ber(q)) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' (36) We now split the analysis into two cases: Case 1: q ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Then Pensia, Jog, and Loh [PJL22, Claim F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='2] implies that d2 h(Ber(p), Ber(q)) ≍ (√q − √p)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Thus, equation (36) implies the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Case 2: q ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' Applying Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='3 again to the second term, we obtain �� 1 − p − � 1 − q �2 = �� 1 − p − � 1 − p − qδ �2 ≍ q2δ2 1 − p ≍ q2δ2 q ≍ d2 TV(Ber(p), Ber(q)) q , (37) where we use the fact that 1 − p ≍ q, since p, q ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content='5, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' The desired conclusion follows from equations (36) and (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} +page_content=' 48' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf'} diff --git a/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf b/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..539ced061022c65eb2ca2043a9f7548077abd94e --- /dev/null +++ b/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cab6e19960ac8db9ef37e97e28a0e896acb647c77696cafe26fb030422cc5dfe +size 4396506 diff --git a/F9E5T4oBgHgl3EQfVg_E/vector_store/index.faiss b/F9E5T4oBgHgl3EQfVg_E/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..0f1a7729705c269a59b7e15d9e5442e802c1774f --- /dev/null +++ b/F9E5T4oBgHgl3EQfVg_E/vector_store/index.faiss @@ -0,0 +1,3 @@ +version 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opportunities and risks that technology holds in connection to our safe and sus- +tainable future, it is becoming increasingly important to involve a larger portion of our society in +becoming active co-creators of our digitalized future—moving from the passenger seat to the +driver seat. Yet, despite extensive efforts around the world, little progress has been made in +growing the representation of certain communities and groups in software engineering. This +chapter shares one successful project, called Czechitas, triggering a major social change in +Czechia, involving 1 000+ volunteers to support 50 000+ women on their way towards software +engineering education and career. +arXiv:2301.12000v1 [cs.SE] 27 Jan 2023 + +CHAPTER 1 +Introduction +The past decade has witnessed the emergence of hundreds of initiatives around the world +supporting various underrepresented groups on their pathway towards software engineering, +whether connected to universities [13], companies [15], or run as independent non-profit or- +ganizations [14]. Although the initiatives often start with a great vision and high volunteering +commitment, after a few years into the activities, it becomes challenging to sustain the volun- +teering energy and commitment in face of the very slow progress towards the better. In those +moments, the success cases by others can be what helps us keep going. +The initiative featured in this chapter, called Czechitas [6], started in 2014 in Czechia, with +a simple idea to bring tech closer to girls and girls closer to tech, in reaction to the strong +under-representation of women in tech in the country (see Figure 1.1). The prompt snowball +effect helped us to build a community around the joint vision to empower and encourage girls +and women to engage in computing education and career transition, and to show them that +software engineering is an interesting career direction that is not necessarily difficult nor limited +to one gender. Initially established to provide women in Czechia with an opportunity to put their +hands on programming, it now contributes to a major social change in the country. +Over time, Czechitas has become a movement that has attracted a strong community of +tech-professional volunteers (over 1 000) and companies (over 100), and given rise to a portfo- +lio of women-tailored courses in various areas of software engineering, such as programming, +Figure 1.1: Women ICT Professional (Eurostat, 2019 data) [8]. +2 + +30% +25% +20% +EU = 17.9% +15% +10% +5% +0%CHAPTER 1 +web development, mobile app development, data science, cybersecurity or testing (over 1 300 +courses delivered so far). We have influenced over 50 000 women (over 30 000 via live events +and over 20 000 via online tutorials) who graduated from our courses to use their new tech +skills to change their education path or advance their careers. +Czechitas Mission: We inspire, train and guide new talents towards stronger +diversity and competitiveness in tech. +Thanks to the success of our education activities with hundreds of events a year (each +receiving more registrations than its capacity), we have become recognized as the leading +platform in Czechia actively addressing gender diversity in tech. In this chapter, we share +the lessons we learned about the low representation of women in tech, effective strategies in +supporting women on their way to software engineering, discuss the ingredients that helped us +succeed, the obstacles and challenges we faced, and the progress yet to be made. +Why are There so Few Women in Tech? +Across Europe, only 19.1% of tech professionals are women (according to 2021 data) [8], with +Czechia being the last on the list. The major reasons behind the trend in our region according +to our recent study (with 70% of participants from Czechia and Germany) [9] are: +1. Access. The first hole in the leaky pipeline on girls’ pathway towards software engi- +neering is linked to the missing access to encouragement and support, together with the +missing access to suitable education that would be able to build on the interests of girls +that often span across multiple disciplines. +2. Stereotypes. The ability to see herself as a software engineer is then challenged by the +perception of the software engineering as a field not leading to a purpose the girl would +like to dedicate her future to. Often, the close family and friends step-in in this moment +to direct girls away from software engineering with the intention to protect them from +a future where they cannot really imagine the girls becoming successful. Interestingly, +3 + +CHAPTER 1 +the intentions are meant well, to protect the girls, which shows how crucial it is to help +parents (and mainly mothers) to understand that software engineering can be a great +career choice for their daughters. +3. Confidence. The next hole on the leaky pipeline comes when girls find themselves in +the classroom, often surrounded by more-experienced learners (typically boys). For the +little girls who often excel in other subjects, it can be hard to fall in the category of a slow +novice learner. The girls often mention frustrations of low self-efficacy, inadequacy and +missing experience of success in presence of a classroom dynamic being monopolized +by the earlier technology adopters. +4. Sense of Belonging. The girls who resist through the earlier three challenges and find +themselves on the education pathway towards software engineering, find themselves in +classrooms surrounded predominantly by boys. While this is a comfortable environment +for some, many in the study reported not feeling comfortable to express themselves, fac- +ing sexism or unwanted attention and missing relatable role models and mentors, which +led them to reconsider whether this is the environment they would be willing to spend the +rest of their lives in. +5. Feeling Valued. The last hole in the leaky pipeline challenges the women who entered +software engineering careers, as some of them emphasize the struggle of not feeling +valued at workplace. The reasons are different for the women with stereotypical talent +spectrum (that matches the talent spectrum typical among their men colleagues, typically +being very technical) and non-stereotypical talent spectrum (bringing not-that-common +talents to the table, typically more multidisciplinary and human-oriented). While the first +group feels ”tired of proving them wrong”, the second group feel frustrated from their +strengths viewed as second class and from missing appreciation. +Supporting Women on their Way to Tech +In Czechitas, we understand that plumbing the leaky pipeline can hardly be done by isolated +and uncoordinated efforts. This section discusses the interlinked pillars of our activities (see +Figure 1.2), listing examples of the activities and events we delivered in 2022. +4 + +CHAPTER 1 +Czechitas Pillar I – Awareness +One of the crucial success factors for a change towards improving gender balance in soft- +ware engineering is the actual understanding that we are in a disbalanced state that further +reinforces itself due to the factors discussed earlier. The efforts towards encouraging women +to join software engineering cannot make a difference unless the society, education system +and corporate environment welcomes and supports the change (understanding it as a push +towards the real equilibrium, not a push out of it). +In Czechitas, we are investing substantial effort in awareness around the topic. In 2022 +alone, we participated in over 20 conferences and panel discussions, gave numerous inter- +views in TV, radio and other media, organized talks to students and teachers at high schools, +and to tech professionals in our partner companies. We were visible with a booth at 15 festivals +and family days across Czechia. Over 2022, Czechitas was mentioned in 508 articles, reach- +ing major part of Czech population. In 2021, we also launched a Czechitas podcast, which in +2022 reached over 14 676 listens. Furthermore, our website was in 2022 visited by 123 785 +unique visitors, and our newsletter was followed by 25 983 subscribers. +The next step in raising awareness among the general public is to make it as easy as +possible to get the first exposure to coding in a fun, enjoyable and community way. To this end, +we for instance organize an Advent Christmas Coding campaign (following the tradition of an +advent calendar, in which instead of a sweet treat, each day holds a coding assignment along a +story of bringing Mr. Gingerbread home for Christmas), which is being followed by hundreds of +Figure 1.2: The Pillars of Czechitas Activities. +5 + +CAREER +AWARENESS +TRAINING +TRANSITION +COMMUNITYCHAPTER 1 +people. Furthermore, in collaboration with the Ministry of Education, Youth and Sports, we e.g. +co-organized the #DIGIEDUHACK hackathon. And in collaboration with Czech universities, we +run the Czechitas Thesis Award to give visibility to exceptional bachelor theses authored by +girls. All these activities typically repeat every year. +Czechitas Pillar II – Training +Since the start of our activities in 2014, we are improving the education design of our courses +to reflect the needs of our audience—women and girls who are very often later technology +adopters or career changers—with an emphasis on providing suitable first contact with soft- +ware engineering, creating safe and supportive environment for novice learners, accommodat- +ing differences in the learning speed of each student, building self-confidence, and supporting +sustaining long-term interest, which we also publish [2, 10]. In 2022, we delivered 242 live +software-engineering courses with 15 316 participants, with the courses around web develop- +ment and data science scoring as the most popular ones. +Although most of the training is targeted to women and girls, we are also investing in training +Figure 1.3: Czechitas-participation (Data as of June 2022). +6 + +21,767 +1.482+ +57.683+ +8,686 +educational +participants +events +428 +10 +11,763 +299 +3,752 +8,268 +238 +418 +13,081 +5,821 +3,969 +145 +141 +4,271 +295 +102 +6 +8,011 +84 +2,654 +45 +135 +1,873 +1,266 +4,299 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +2022* +2015 +2016 +2017 +2018 +2019 +2020 +2021 +2022* +Number of educational events +Number of video tutorials +Educational events participants +Educational video tutorials participantsCHAPTER 1 +elementary-school and high-school teachers (irrespective of gender). And some mixed-gender +activities were organized also for children (7 week-long summer camps in the summer of 2022, +besides others) and high-school kids, although in case of high schools, it is already important +to offer also girl-only courses (3 week-long summer schools for high-school girls were given +in 2022). Besides, training courses for mixed audience are also provided on events such as +Family Days (we were present at over 20 such events in 2022). +Czechitas Pillar III – Career Transition +As many women in our community intend to enter software engineering as their future profes- +sion, some of our activities are intentionally designed to facilitate this journey, whether software +engineering is to become their first job or they intend to change their career [3]. +In cooperation with our partner companies, we have identified three career pathways that +appear to be the most suitable entry points to software engineering in Czechia. These are +(1) web development (including courses on JavaScript, React, HTML/CSS, Bootstrap, Git, +UX design, and others), (2) data analytics (including courses on Python, databases, SQL, +statistics, Power BI, and others), (3) testing (including courses on requirements engineering, +agile processes, manual testing, issue tracking, regression testing, smoke testing, basics of +automated testing, browsers, API, databases, version control, and others). +For the three directions, we have developed a complex career-transition support within so- +called Digital Academies. A Digital Academy is a four-month program for a group of 30 women +(and involving around 5-15 partner companies), which besides individual courses covering the +topics outlined above and taking place 3-4 times a week (evenings on working days, full days +on weekends) includes also pairing of the students with mentors from the companies to support +them in developing their own projects, a hackathon, career support, and further events offered +by the partner companies. In 2022, we have run 10 Digital Academies across four major cities +in Czechia, with over 60% of the graduates receiving a job offer within three months from +graduating from the academy. +To facilitate the career transition also for the women who opt to customize their training +journey (not attending a Digital Academy), our career consultants provide hundreds of career +consultations each year (327 in 2022), and we twice a year organize a Czechitas Job Fair, +7 + +CHAPTER 1 +where our graduates can meet the representatives of our partner companies (each Job Fair +attended by about 350 graduates and 30 companies). +Czechitas Foundation – Community +The foundation that supports all our activities is the community, which involves the participants +and graduates of our courses, tech professionals who teach with us, mentors, course facili- +tators, and our partner companies. The fact that many members in our community are men +helps us not only engage more tech-professional allies in our vision, but also influence a more +supportive environment in tech companies where our graduates land. To support the blending +of the community and increasing the sense of belonging of our graduates also in the mixed- +gender environment, we regularly engage in organization of Tech Meet-ups and Hackathons, +as well as informal CzechiPubs that regularly take place in 10 different cities across Czechia. +Making a Difference +The positive influence of Czechitas activities in Czechia is already visible in the shifted percep- +tion of software engineering as an education pathway and career choice to be considered by +any gender. That not only motivates many girls to consider software engineering in their choice +of a university study field (with the representation of women among ICT students changing from +12% in 2016 to 17% in 2021 in Czechia [7, 5], moving the country closer to the European av- +erage, see Figure 1.4) but is likely also having secondary influence on all who so far hesitated +to join software engineering. +What Helped us Succeed +Building Czechitas was only possible thanks to a coordinated effort of hundreds of people (90 +employees and over 1,000 volunteers). Over the past eight years of our existence, we came to +understand the ingredients without which this would not be possible: +• Great leadership and love for what we do is giving us the sense of purpose, energy and +direction, holding us together and keeping us focused. Mentors from partner companies +8 + +CHAPTER 1 +Figure 1.4: Women ICT Students (Czech Statistical Office, 2021 data) [5]. +and beyond have been of great help to guide us through the design of our leadership and +expansion strategy. +• Visual and playful communication is giving us the fresh flavour of fun and joy that we +all (students as well as trainers and volunteers) enjoy joining even after a tiring day at +school or work. The informal and visually attractive communication helps us to share the +love for our brand. +• Community and sense of belonging is crucial for connecting those who strive to learn +with those who strive to share and teach, and those who want to support the connection. +It helps our student to feel home and make it easier for them to keep going even when +learning gets hard. +• Inclusive environment and encouragement makes it safe for our students to make mis- +takes and experience success, have the opportunity to exchange knowledge, collaborate, +and get personalized feedback and guidance. Specific strategies and interventions we +have developed to support novice learners and their self-efficacy have been key in this +direction [2]. +• Knowledge and understanding is crucial for us to design our activities with insight into +the frustrations steering women away from software engineering [9] and effective strate- +gies to support girls and women in tech education [10] and career transition [3]. We +9 + +30% +25% +EU = 20% +20% +15% +10% +5% +0%CHAPTER 1 +invest our time in sharing the lessons we have learned [2, 9, 3], and learning from other +initiatives from across the world (e.g., within the EUGAIN network, see https://eugain.eu/). +• Creating and sharing stories helps us to inspire our students, bring them closer to +relatable role models, and to give them hope and confidence that with some work and +dedication, a transition into software engineering is possible. The stories (each featuring +an inspiring woman who changed her career towards tech) are published in our blog, +communicated via social networks, and used in media articles. These women inspire +others as speakers and panelists in our events, and as guests in Czechitas Podcast. +• Sustainable financial model helps us to sustain a team employed to run the organi- +zation. The model stands on financial participation of the students, partner companies, +foundations and individual donors, with an intention to reach out also to the government +level in the future. The most crucial pillar of our financial sustainability is the partner com- +panies, which are beside their yearly partnership contributions (depending on the level of +partnership) helping us to cover certain costs (e.g., offering their office spaces for events, +motivating their employees to volunteer as mentors), and opening doors towards further +funding opportunities (e.g. with global foundations connected to their company). +Obstacles and Challenges we Faced +As any organization that has substantially outgrown its own plans and expectations, Czechitas +has undergone numerous changes and readjustments over its course of existence. And al- +though we are trying to publish the effective setup that works for us now [2, 3, 4], our first steps +were highly organic and experimental, which was key to learning what works for the context +we were in. With our enthusiasm and ”always yes” spirit, we walked many paths that we failed +and rolled back, but we also faced numerous obstacles and challenges that we withstood. +• Scaling the organization. Turning a non-profit start-up into a scale-up is a challenge on +its own, as the means for achieving stability are different from traditional companies – be- +sides the discussed financial stability, also in terms of sustained volunteering involvement +and brand building. We needed to learn to manage the mix of the innovative and largely +self-sacrificing founding community with the necessary systematic and organized spirit of +10 + +CHAPTER 1 +new employees. We needed to learn to prioritize and say no to some activities that the +team felt strongly for. +• Being misunderstood. As a large organization, we needed to learn to communicate +our mission well so that it is not misunderstood, knowing that anything that damages +the brand may sink the whole boat. Namely, we needed to help our partner companies +understand what level of expertise is realistic to achieve in our students, help our students +understand what time investment and commitment it takes to change direction towards +tech, and help our society understand why our focus on women is key to the success of +our society as a whole. +• Quantifying the impact of our activities. One of the important challenges that we are +still facing is our ability to quantify the impact of our individual interventions and activities, +as it is difficult to isolate the effects of each one of them. More so that the impact is often +very subtle and propagates over long periods of time (e.g., a woman making a few steps +towards tech education inspiring her friend to make a major shift towards tech, who then +inspires her daughter to study CS at university). So although we have a Data & Impact +team at Czechitas, with substantial data available, the numbers we have (e.g., the number +of women who change their career to tech each year) are still only the tip of the iceberg +of the real impact we strive for, which is the shift in the collective mindset of the entire +society, leading to a sustained change. +Progress yet to be Made +With the increasing number of Czechitas graduates who are joining software engineering in- +dustry, often as very junior (in terms of their software-engineering expertise) and diverse (in +terms of their talents and competencies) members, we find it crucial to assist the companies +to improve the inclusiveness of their environment to integrate and leverage the new diverse +talent. In 2020, we made the first step towards that goal via designing a Diversity Awareness +Training, which was since then delivered to over 300 managers (mostly from Central and East- +ern Europe) across some of our partner companies. The concepts that have shown to be the +most crucial to discuss and understand during these trainings are outlined below: +11 + +CHAPTER 1 +Figure 1.5: Tuckman’s Model of Team Dynamics with an illustration of different dynamics +observed in homogeneous and heterogeneous teams. +• Diversity does not come easy, but it pays off. Avoiding diversity is natural to human in- +dividuals, but dangerous to humankind1. The same is true for corporate environment. We +need to acknowledge that diverse teams might have a harder time at start (as illustrated +with the Tuckman’s Model of Team Dynamics in Figure 1.5), but in long-term, diversity is +firmly correlated with higher performance [11, 12]. +• We too often lose talented people by missing the talent in them. We are all talented, +in many diverse ways. It is the task of the manager to recognize and direct the talent to- +wards team success. The fact that a person uses a different talent spectrum (approaches +problems and situations differently) does not make them more/less suitable for software +engineering as such. There is no such thing as a second-class citizen when it comes to +the talents we need in software engineering. +• Biases evolved to help us navigate complexity, but they are not serving us well +when making assumptions about the potential in people. The dark side of biases is +that we tend to judge people’s potential based on how their talent spectrum matches the +talent of already-successful ones. Without realizing that the successful ones embody the +skills and conditions that worked when they joined the field (in the past) while we are now +choosing the software engineers for the future. +• Connection is built through communication. There are many unhealthy communica- +1Our quote inspired by the statement ”Diversity is the new Darwinism” by the Great British Diversity Experiment [1]. +12 + +7 +Forming +Strorming +Norming +Performing +Adjourning +个 +Effectiveness +Homogenous +team +Heterogenous +team +TimeCHAPTER 1 +tion patterns around diversity, which often go against the purpose of making us all feel +the sense of belonging. It is important to create safe space, in which we can learn to +communicate our differences but also ask about the differences of others. Mistakes are +part of that learning, and forgiveness of the mistakes shall be encouraged if the mistakes +were done in the process of learning and not repeated blindly. It is important to create a +safe space to acknowledge our biases and stop shaming one another for them. +• Avoid the quick fixes, remove the barriers instead. Encourage curiosity about why cer- +tain communities are under-represented in software engineering. What are the barriers +they face and what can we do to remove them or make their journey lighter in presence of +the barriers (e.g. the care-taking on the side of most women)? Avoiding the conversation +and looking away from the differences in our experiences might lead the community to as- +sume that the under-representation is the lower-fit problem, which is dangerous because +it leads to push-back on any diversity support one might try to introduce. +• Change takes time. Promoting I&D is more complex than it might seem at first. It is +crucial to know how to start to see the first positive effects soon and be able to use them +to get more people on board towards promoting I&D further. Choose your first steps well +and invest in them. The investment will pay off. +Conclusion +Making a difference in improving gender balance in software engineering on the scale of the +whole country is not easy, but is possible. And it is very rewarding to be part of such a move- +ment. In 2021, the social impact of Czechitas activities was recognized at the European Union +level via winning the EU Social Economy Award (over 180 organizations nominated) in the +Digitalisation and Skills category, and in 2022 winning the global Equals in Tech Award (155 +organizations nominated) in the Skills category. We hope our example can inspire others, +which is also why we are eager to share the lessons learned from our journey. +13 + +Bibliography +[1] Amanda Bennett. Case study: The great British diversity experiment, 2016. FairPlay Ltd. +[2] Barbora Buhnova and Lucia Happe. Girl-friendly computer science classroom: Czechitas +experience report. In European Conference on Software Architecture, pages 125–137. +Springer, 2020. +[3] Barbora Buhnova, Lucie Jurystova, and Dita Prikrylova. +Assisting women in career +change towards software engineering: experience from czechitas ngo. In Proceedings of +the 13th European Conference on Software Architecture-Volume 2, pages 88–93, 2019. +[4] Barbora Buhnova and Dita Prikrylova. +Women want to learn tech: Lessons from the +czechitas education project. In 2019 IEEE/ACM 2nd International Workshop on Gender +Equality in Software Engineering (GE), pages 25–28. IEEE, 2019. +[5] Czech Statistical Office. Human resources in information technology, 2021. Available on- +line at URL https://www.czso.cz/documents/10180/165376696/063015-21.pdf/c7e96151- +b285-4388-9384-532e55f4a318?version=1.2. +[6] Czechitas. +Czechitas +annual +report +2021, +2022. +Available +online +at +URL +https://is.muni.cz/go/u6ji13. +[7] Eurostat. +Female students under-represented in ICT, 2016. +Available online at URL +https://ec.europa.eu/eurostat/web/products-eurostat-news/-/edn-20190425-1. +[8] Eurostat. +ICT +specialists +in +employment, +2022. +Available +online +at +URL +https://ec.europa.eu/eurostat/statistics-explained/index.php?title=ICT specialists in em- +ployment. + +CHAPTER 1 +[9] Lucia Happe and Barbora Buhnova. Frustrations steering women away from software +engineering. IEEE Software, 39(4):63–69, 2022. +[10] Lucia Happe, Barbora Buhnova, Anne Koziolek, and Ingo Wagner. Effective measures to +foster girls’ interest in secondary computer science education. Education and Information +Technologies, 26(3):2811–2829, 2021. +[11] Dame Vivian Hunt, Dennis Layton, and Sara Prince. +Why diversity matters, 2015. +McKinsey. Available online at URL https://www.mckinsey.com/capabilities/people-and- +organizational-performance/our-insights/why-diversity-matters. +[12] Rocio Lorenzo and Martin Reeves. How and where diversity drives financial performance. +Business Harward Review, 2018. Available online at URL https://hbr.org/2018/01/how- +and-where-diversity-drives-financial-performance. +[13] Minerva Informatics Equality Award. Best practices in supporting women, 2022. Available +online at URL https://www.informatics-europe.org/society/minerva-informatics-equality- +award/best-practices-in-supporting-women.html. +[14] Sarah K. White. 19 organizations advancing women in tech, 2022. Available online at +URL https://www.cio.com/article/215709/16-organizations-for-women-in-tech.html. +[15] Hannah Williams. +Best initiatives for women in tech, 2017. +Available online at URL +https://techmonitor.ai/technology/hardware/best-initiatives-women-tech. +Acknowledgement +This chapter was made possible thanks to the great dedication and support of the entire +Czechitas team. Besides, it has been supported by the COST Action CA19122 – European +Network for Gender Balance in Informatics (EUGAIN). +15 + diff --git a/H9FLT4oBgHgl3EQfIi9F/content/tmp_files/load_file.txt b/H9FLT4oBgHgl3EQfIi9F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b1ea96d2587c37033513922f0fee673b12b7ef2 --- /dev/null +++ b/H9FLT4oBgHgl3EQfIi9F/content/tmp_files/load_file.txt @@ -0,0 +1,267 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf,len=266 +page_content='Chapter 1 Beyond Classroom: Making a Difference in Diversity in Tech Barbora Buhnova With all the opportunities and risks that technology holds in connection to our safe and sus- tainable future, it is becoming increasingly important to involve a larger portion of our society in becoming active co-creators of our digitalized future—moving from the passenger seat to the driver seat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Yet, despite extensive efforts around the world, little progress has been made in growing the representation of certain communities and groups in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' This chapter shares one successful project, called Czechitas, triggering a major social change in Czechia, involving 1 000+ volunteers to support 50 000+ women on their way towards software engineering education and career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='12000v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='SE] 27 Jan 2023 CHAPTER 1 Introduction The past decade has witnessed the emergence of hundreds of initiatives around the world supporting various underrepresented groups on their pathway towards software engineering, whether connected to universities [13], companies [15], or run as independent non-profit or- ganizations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Although the initiatives often start with a great vision and high volunteering commitment, after a few years into the activities, it becomes challenging to sustain the volun- teering energy and commitment in face of the very slow progress towards the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In those moments, the success cases by others can be what helps us keep going.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The initiative featured in this chapter, called Czechitas [6], started in 2014 in Czechia, with a simple idea to bring tech closer to girls and girls closer to tech, in reaction to the strong under-representation of women in tech in the country (see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The prompt snowball effect helped us to build a community around the joint vision to empower and encourage girls and women to engage in computing education and career transition, and to show them that software engineering is an interesting career direction that is not necessarily difficult nor limited to one gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Initially established to provide women in Czechia with an opportunity to put their hands on programming, it now contributes to a major social change in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Over time, Czechitas has become a movement that has attracted a strong community of tech-professional volunteers (over 1 000) and companies (over 100), and given rise to a portfo- lio of women-tailored courses in various areas of software engineering, such as programming, Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='1: Women ICT Professional (Eurostat, 2019 data) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 2 30% 25% 20% EU = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='9% 15% 10% 5% 0%CHAPTER 1 web development, mobile app development, data science, cybersecurity or testing (over 1 300 courses delivered so far).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' We have influenced over 50 000 women (over 30 000 via live events and over 20 000 via online tutorials) who graduated from our courses to use their new tech skills to change their education path or advance their careers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Czechitas Mission: We inspire, train and guide new talents towards stronger diversity and competitiveness in tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Thanks to the success of our education activities with hundreds of events a year (each receiving more registrations than its capacity), we have become recognized as the leading platform in Czechia actively addressing gender diversity in tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In this chapter, we share the lessons we learned about the low representation of women in tech, effective strategies in supporting women on their way to software engineering, discuss the ingredients that helped us succeed, the obstacles and challenges we faced, and the progress yet to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Why are There so Few Women in Tech?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Across Europe, only 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='1% of tech professionals are women (according to 2021 data) [8], with Czechia being the last on the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The major reasons behind the trend in our region according to our recent study (with 70% of participants from Czechia and Germany) [9] are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The first hole in the leaky pipeline on girls’ pathway towards software engi- neering is linked to the missing access to encouragement and support, together with the missing access to suitable education that would be able to build on the interests of girls that often span across multiple disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Stereotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The ability to see herself as a software engineer is then challenged by the perception of the software engineering as a field not leading to a purpose the girl would like to dedicate her future to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Often, the close family and friends step-in in this moment to direct girls away from software engineering with the intention to protect them from a future where they cannot really imagine the girls becoming successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Interestingly, 3 CHAPTER 1 the intentions are meant well, to protect the girls, which shows how crucial it is to help parents (and mainly mothers) to understand that software engineering can be a great career choice for their daughters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The next hole on the leaky pipeline comes when girls find themselves in the classroom, often surrounded by more-experienced learners (typically boys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' For the little girls who often excel in other subjects, it can be hard to fall in the category of a slow novice learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The girls often mention frustrations of low self-efficacy, inadequacy and missing experience of success in presence of a classroom dynamic being monopolized by the earlier technology adopters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Sense of Belonging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The girls who resist through the earlier three challenges and find themselves on the education pathway towards software engineering, find themselves in classrooms surrounded predominantly by boys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' While this is a comfortable environment for some, many in the study reported not feeling comfortable to express themselves, fac- ing sexism or unwanted attention and missing relatable role models and mentors, which led them to reconsider whether this is the environment they would be willing to spend the rest of their lives in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Feeling Valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The last hole in the leaky pipeline challenges the women who entered software engineering careers, as some of them emphasize the struggle of not feeling valued at workplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The reasons are different for the women with stereotypical talent spectrum (that matches the talent spectrum typical among their men colleagues, typically being very technical) and non-stereotypical talent spectrum (bringing not-that-common talents to the table, typically more multidisciplinary and human-oriented).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' While the first group feels ”tired of proving them wrong”, the second group feel frustrated from their strengths viewed as second class and from missing appreciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Supporting Women on their Way to Tech In Czechitas, we understand that plumbing the leaky pipeline can hardly be done by isolated and uncoordinated efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' This section discusses the interlinked pillars of our activities (see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='2), listing examples of the activities and events we delivered in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 4 CHAPTER 1 Czechitas Pillar I – Awareness One of the crucial success factors for a change towards improving gender balance in soft- ware engineering is the actual understanding that we are in a disbalanced state that further reinforces itself due to the factors discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The efforts towards encouraging women to join software engineering cannot make a difference unless the society, education system and corporate environment welcomes and supports the change (understanding it as a push towards the real equilibrium, not a push out of it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In Czechitas, we are investing substantial effort in awareness around the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In 2022 alone, we participated in over 20 conferences and panel discussions, gave numerous inter- views in TV, radio and other media, organized talks to students and teachers at high schools, and to tech professionals in our partner companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' We were visible with a booth at 15 festivals and family days across Czechia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Over 2022, Czechitas was mentioned in 508 articles, reach- ing major part of Czech population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In 2021, we also launched a Czechitas podcast, which in 2022 reached over 14 676 listens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Furthermore, our website was in 2022 visited by 123 785 unique visitors, and our newsletter was followed by 25 983 subscribers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The next step in raising awareness among the general public is to make it as easy as possible to get the first exposure to coding in a fun, enjoyable and community way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' To this end, we for instance organize an Advent Christmas Coding campaign (following the tradition of an advent calendar, in which instead of a sweet treat, each day holds a coding assignment along a story of bringing Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Gingerbread home for Christmas), which is being followed by hundreds of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='2: The Pillars of Czechitas Activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 5 CAREER AWARENESS TRAINING TRANSITION COMMUNITYCHAPTER 1 people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Furthermore, in collaboration with the Ministry of Education, Youth and Sports, we e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' co-organized the #DIGIEDUHACK hackathon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' And in collaboration with Czech universities, we run the Czechitas Thesis Award to give visibility to exceptional bachelor theses authored by girls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' All these activities typically repeat every year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Czechitas Pillar II – Training Since the start of our activities in 2014,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' we are improving the education design of our courses to reflect the needs of our audience—women and girls who are very often later technology adopters or career changers—with an emphasis on providing suitable first contact with soft- ware engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' creating safe and supportive environment for novice learners,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' accommodat- ing differences in the learning speed of each student,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' building self-confidence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' and supporting sustaining long-term interest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' which we also publish [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In 2022, we delivered 242 live software-engineering courses with 15 316 participants, with the courses around web develop- ment and data science scoring as the most popular ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Although most of the training is targeted to women and girls, we are also investing in training Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='3: Czechitas-participation (Data as of June 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 6 21,767 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='482+ 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='683+ 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='686 educational participants events 428 10 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='763 299 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='752 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='268 238 418 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='081 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='821 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='969 145 141 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='271 295 102 6 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='011 84 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='654 45 135 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='873 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='266 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='299 2015 2016 2017 2018 2019 2020 2021 2022* 2015 2016 2017 2018 2019 2020 2021 2022* Number of educational events Number of video tutorials Educational events participants Educational video tutorials participantsCHAPTER 1 elementary-school and high-school teachers (irrespective of gender).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' And some mixed-gender activities were organized also for children (7 week-long summer camps in the summer of 2022, besides others) and high-school kids, although in case of high schools, it is already important to offer also girl-only courses (3 week-long summer schools for high-school girls were given in 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Besides, training courses for mixed audience are also provided on events such as Family Days (we were present at over 20 such events in 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Czechitas Pillar III – Career Transition As many women in our community intend to enter software engineering as their future profes- sion, some of our activities are intentionally designed to facilitate this journey, whether software engineering is to become their first job or they intend to change their career [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In cooperation with our partner companies, we have identified three career pathways that appear to be the most suitable entry points to software engineering in Czechia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' These are (1) web development (including courses on JavaScript, React, HTML/CSS, Bootstrap, Git, UX design, and others), (2) data analytics (including courses on Python, databases, SQL, statistics, Power BI, and others), (3) testing (including courses on requirements engineering, agile processes, manual testing, issue tracking, regression testing, smoke testing, basics of automated testing, browsers, API, databases, version control, and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' For the three directions, we have developed a complex career-transition support within so- called Digital Academies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' A Digital Academy is a four-month program for a group of 30 women (and involving around 5-15 partner companies), which besides individual courses covering the topics outlined above and taking place 3-4 times a week (evenings on working days, full days on weekends) includes also pairing of the students with mentors from the companies to support them in developing their own projects, a hackathon, career support, and further events offered by the partner companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In 2022, we have run 10 Digital Academies across four major cities in Czechia, with over 60% of the graduates receiving a job offer within three months from graduating from the academy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' To facilitate the career transition also for the women who opt to customize their training journey (not attending a Digital Academy), our career consultants provide hundreds of career consultations each year (327 in 2022), and we twice a year organize a Czechitas Job Fair, 7 CHAPTER 1 where our graduates can meet the representatives of our partner companies (each Job Fair attended by about 350 graduates and 30 companies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Czechitas Foundation – Community The foundation that supports all our activities is the community, which involves the participants and graduates of our courses, tech professionals who teach with us, mentors, course facili- tators, and our partner companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The fact that many members in our community are men helps us not only engage more tech-professional allies in our vision, but also influence a more supportive environment in tech companies where our graduates land.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' To support the blending of the community and increasing the sense of belonging of our graduates also in the mixed- gender environment, we regularly engage in organization of Tech Meet-ups and Hackathons, as well as informal CzechiPubs that regularly take place in 10 different cities across Czechia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Making a Difference The positive influence of Czechitas activities in Czechia is already visible in the shifted percep- tion of software engineering as an education pathway and career choice to be considered by any gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' That not only motivates many girls to consider software engineering in their choice of a university study field (with the representation of women among ICT students changing from 12% in 2016 to 17% in 2021 in Czechia [7, 5], moving the country closer to the European av- erage, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='4) but is likely also having secondary influence on all who so far hesitated to join software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' What Helped us Succeed Building Czechitas was only possible thanks to a coordinated effort of hundreds of people (90 employees and over 1,000 volunteers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Over the past eight years of our existence, we came to understand the ingredients without which this would not be possible: Great leadership and love for what we do is giving us the sense of purpose, energy and direction, holding us together and keeping us focused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Mentors from partner companies 8 CHAPTER 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='4: Women ICT Students (Czech Statistical Office, 2021 data) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' and beyond have been of great help to guide us through the design of our leadership and expansion strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Visual and playful communication is giving us the fresh flavour of fun and joy that we all (students as well as trainers and volunteers) enjoy joining even after a tiring day at school or work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The informal and visually attractive communication helps us to share the love for our brand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Community and sense of belonging is crucial for connecting those who strive to learn with those who strive to share and teach, and those who want to support the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' It helps our student to feel home and make it easier for them to keep going even when learning gets hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Inclusive environment and encouragement makes it safe for our students to make mis- takes and experience success, have the opportunity to exchange knowledge, collaborate, and get personalized feedback and guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Specific strategies and interventions we have developed to support novice learners and their self-efficacy have been key in this direction [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Knowledge and understanding is crucial for us to design our activities with insight into the frustrations steering women away from software engineering [9] and effective strate- gies to support girls and women in tech education [10] and career transition [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' We 9 30% 25% EU = 20% 20% 15% 10% 5% 0%CHAPTER 1 invest our time in sharing the lessons we have learned [2, 9, 3], and learning from other initiatives from across the world (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=', within the EUGAIN network, see https://eugain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='eu/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Creating and sharing stories helps us to inspire our students, bring them closer to relatable role models, and to give them hope and confidence that with some work and dedication, a transition into software engineering is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The stories (each featuring an inspiring woman who changed her career towards tech) are published in our blog, communicated via social networks, and used in media articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' These women inspire others as speakers and panelists in our events, and as guests in Czechitas Podcast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Sustainable financial model helps us to sustain a team employed to run the organi- zation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The model stands on financial participation of the students, partner companies, foundations and individual donors, with an intention to reach out also to the government level in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The most crucial pillar of our financial sustainability is the partner com- panies, which are beside their yearly partnership contributions (depending on the level of partnership) helping us to cover certain costs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=', offering their office spaces for events, motivating their employees to volunteer as mentors), and opening doors towards further funding opportunities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' with global foundations connected to their company).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Obstacles and Challenges we Faced As any organization that has substantially outgrown its own plans and expectations, Czechitas has undergone numerous changes and readjustments over its course of existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' And al- though we are trying to publish the effective setup that works for us now [2, 3, 4], our first steps were highly organic and experimental, which was key to learning what works for the context we were in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' With our enthusiasm and ”always yes” spirit, we walked many paths that we failed and rolled back, but we also faced numerous obstacles and challenges that we withstood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Scaling the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Turning a non-profit start-up into a scale-up is a challenge on its own, as the means for achieving stability are different from traditional companies – be- sides the discussed financial stability, also in terms of sustained volunteering involvement and brand building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' We needed to learn to manage the mix of the innovative and largely self-sacrificing founding community with the necessary systematic and organized spirit of 10 CHAPTER 1 new employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' We needed to learn to prioritize and say no to some activities that the team felt strongly for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Being misunderstood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' As a large organization, we needed to learn to communicate our mission well so that it is not misunderstood, knowing that anything that damages the brand may sink the whole boat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Namely, we needed to help our partner companies understand what level of expertise is realistic to achieve in our students, help our students understand what time investment and commitment it takes to change direction towards tech, and help our society understand why our focus on women is key to the success of our society as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Quantifying the impact of our activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' One of the important challenges that we are still facing is our ability to quantify the impact of our individual interventions and activities, as it is difficult to isolate the effects of each one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' More so that the impact is often very subtle and propagates over long periods of time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=', a woman making a few steps towards tech education inspiring her friend to make a major shift towards tech, who then inspires her daughter to study CS at university).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' So although we have a Data & Impact team at Czechitas, with substantial data available, the numbers we have (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=', the number of women who change their career to tech each year) are still only the tip of the iceberg of the real impact we strive for, which is the shift in the collective mindset of the entire society, leading to a sustained change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Progress yet to be Made With the increasing number of Czechitas graduates who are joining software engineering in- dustry, often as very junior (in terms of their software-engineering expertise) and diverse (in terms of their talents and competencies) members, we find it crucial to assist the companies to improve the inclusiveness of their environment to integrate and leverage the new diverse talent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In 2020, we made the first step towards that goal via designing a Diversity Awareness Training, which was since then delivered to over 300 managers (mostly from Central and East- ern Europe) across some of our partner companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The concepts that have shown to be the most crucial to discuss and understand during these trainings are outlined below: 11 CHAPTER 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='5: Tuckman’s Model of Team Dynamics with an illustration of different dynamics observed in homogeneous and heterogeneous teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Diversity does not come easy, but it pays off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Avoiding diversity is natural to human in- dividuals, but dangerous to humankind1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The same is true for corporate environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' We need to acknowledge that diverse teams might have a harder time at start (as illustrated with the Tuckman’s Model of Team Dynamics in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='5), but in long-term, diversity is firmly correlated with higher performance [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' We too often lose talented people by missing the talent in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' We are all talented, in many diverse ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' It is the task of the manager to recognize and direct the talent to- wards team success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The fact that a person uses a different talent spectrum (approaches problems and situations differently) does not make them more/less suitable for software engineering as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' There is no such thing as a second-class citizen when it comes to the talents we need in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Biases evolved to help us navigate complexity, but they are not serving us well when making assumptions about the potential in people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The dark side of biases is that we tend to judge people’s potential based on how their talent spectrum matches the talent of already-successful ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Without realizing that the successful ones embody the skills and conditions that worked when they joined the field (in the past) while we are now choosing the software engineers for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Connection is built through communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' There are many unhealthy communica- 1Our quote inspired by the statement ”Diversity is the new Darwinism” by the Great British Diversity Experiment [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 12 7 Forming Strorming Norming Performing Adjourning 个 Effectiveness Homogenous team Heterogenous team TimeCHAPTER 1 tion patterns around diversity, which often go against the purpose of making us all feel the sense of belonging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' It is important to create safe space, in which we can learn to communicate our differences but also ask about the differences of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Mistakes are part of that learning, and forgiveness of the mistakes shall be encouraged if the mistakes were done in the process of learning and not repeated blindly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' It is important to create a safe space to acknowledge our biases and stop shaming one another for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Avoid the quick fixes, remove the barriers instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Encourage curiosity about why cer- tain communities are under-represented in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' What are the barriers they face and what can we do to remove them or make their journey lighter in presence of the barriers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' the care-taking on the side of most women)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Avoiding the conversation and looking away from the differences in our experiences might lead the community to as- sume that the under-representation is the lower-fit problem, which is dangerous because it leads to push-back on any diversity support one might try to introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Change takes time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Promoting I&D is more complex than it might seem at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' It is crucial to know how to start to see the first positive effects soon and be able to use them to get more people on board towards promoting I&D further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Choose your first steps well and invest in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' The investment will pay off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Conclusion Making a difference in improving gender balance in software engineering on the scale of the whole country is not easy, but is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' And it is very rewarding to be part of such a move- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In 2021, the social impact of Czechitas activities was recognized at the European Union level via winning the EU Social Economy Award (over 180 organizations nominated) in the Digitalisation and Skills category, and in 2022 winning the global Equals in Tech Award (155 organizations nominated) in the Skills category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' We hope our example can inspire others, which is also why we are eager to share the lessons learned from our journey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 13 Bibliography [1] Amanda Bennett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Case study: The great British diversity experiment, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' FairPlay Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [2] Barbora Buhnova and Lucia Happe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Girl-friendly computer science classroom: Czechitas experience report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In European Conference on Software Architecture, pages 125–137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [3] Barbora Buhnova, Lucie Jurystova, and Dita Prikrylova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Assisting women in career change towards software engineering: experience from czechitas ngo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In Proceedings of the 13th European Conference on Software Architecture-Volume 2, pages 88–93, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [4] Barbora Buhnova and Dita Prikrylova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Women want to learn tech: Lessons from the czechitas education project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' In 2019 IEEE/ACM 2nd International Workshop on Gender Equality in Software Engineering (GE), pages 25–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [5] Czech Statistical Office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Human resources in information technology, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Available on- line at URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='czso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='cz/documents/10180/165376696/063015-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='pdf/c7e96151- b285-4388-9384-532e55f4a318?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='version=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [6] Czechitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Czechitas annual report 2021, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Available online at URL https://is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='muni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='cz/go/u6ji13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [7] Eurostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Female students under-represented in ICT, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Available online at URL https://ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='eu/eurostat/web/products-eurostat-news/-/edn-20190425-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [8] Eurostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' ICT specialists in employment, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Available online at URL https://ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='eu/eurostat/statistics-explained/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='title=ICT specialists in em- ployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' CHAPTER 1 [9] Lucia Happe and Barbora Buhnova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Frustrations steering women away from software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' IEEE Software, 39(4):63–69, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [10] Lucia Happe, Barbora Buhnova, Anne Koziolek, and Ingo Wagner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Effective measures to foster girls’ interest in secondary computer science education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Education and Information Technologies, 26(3):2811–2829, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [11] Dame Vivian Hunt, Dennis Layton, and Sara Prince.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Why diversity matters, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' McKinsey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Available online at URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='mckinsey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='com/capabilities/people-and- organizational-performance/our-insights/why-diversity-matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [12] Rocio Lorenzo and Martin Reeves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' How and where diversity drives financial performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Business Harward Review, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Available online at URL https://hbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='org/2018/01/how- and-where-diversity-drives-financial-performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [13] Minerva Informatics Equality Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Best practices in supporting women, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Available online at URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='informatics-europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='org/society/minerva-informatics-equality- award/best-practices-in-supporting-women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [14] Sarah K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 19 organizations advancing women in tech, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Available online at URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='cio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='com/article/215709/16-organizations-for-women-in-tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' [15] Hannah Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Best initiatives for women in tech, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Available online at URL https://techmonitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content='ai/technology/hardware/best-initiatives-women-tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Acknowledgement This chapter was made possible thanks to the great dedication and support of the entire Czechitas team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' Besides, it has been supported by the COST Action CA19122 – European Network for Gender Balance in Informatics (EUGAIN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FLT4oBgHgl3EQfIi9F/content/2301.12000v1.pdf'} diff --git a/HdE2T4oBgHgl3EQf_Alo/content/tmp_files/2301.04244v1.pdf.txt b/HdE2T4oBgHgl3EQf_Alo/content/tmp_files/2301.04244v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..68817443354c70b07d39a8b8480a17b0121177c2 --- /dev/null +++ b/HdE2T4oBgHgl3EQf_Alo/content/tmp_files/2301.04244v1.pdf.txt @@ -0,0 +1,470 @@ +Elastic Cash +Anup Rao +University of Washington +anuprao@cs.washington.edu +January 12, 2023 +Abstract +Elastic Cash is a new decentralized mechanism for regulating the money supply. The mech- +anism operates by modifying the supply so that an interest rate determined by a public market +is kept approximately fixed. It can be incorporated into the conventional monetary system to +improve the elasticity of the US Dollar, and it can be used to design new elastic cryptocurrencies +that remain decentralized. +1 +Introduction +Money is as old as recorded history, and yet it continues to evolve. Even the mighty US Dollar +has been repeatedly updated over the last 200 years. The recent emergence of Bitcoin and other +cryptocurrencies is another step in that evolution, and it prompts us to revisit the mechanisms +that ensure the desirable properties of money. An important property of money is its elasticity: +money is elastic if the money supply increases in response to demand. In this work, I present a +new decentralized mechanism to ensure the elasticity of money. +The US Dollar is elastic, but it uses a convoluted system to achieve its elasticity. The Dollar +system was not designed from first principles; it was iteratively amended in response to financial +crises. Elasticity is currently achieved by the combined actions of many institutions: banks and +non-banks, public and private, domestic and international. No single entity is entirely in charge +of the money supply, and a relatively small number of investor-owned institutions have undue +influence. I discuss the mechanics of the US Dollar system and its limitations in Section 2. +In contrast, Bitcoin was designed to be inelastic. Bitcoin caps the total possible supply at 21M, +and the available supply, currently about 19.2M, will slowly increase until it hits this limit. Satoshi +Nakamoto, the creator of Bitcoin, criticized conventional mechanisms for achieving elasticity in an +early forum post: +The root problem with conventional currency is all the trust that’s required to make +it work. The central bank must be trusted not to debase the currency, but the history +of fiat currencies is full of breaches of that trust. Banks must be trusted to hold our +money and transfer it electronically, but they lend it out in waves of credit bubbles with +barely a fraction in reserve. +Nakamoto eliminated the need for the kind of infrastructure used to ensure Dollar elasticity by +simply choosing to make Bitcoin inelastic. Additional technological innovations in Bitcoin further +eliminated the need for any trusted central authority to carry out transactions. +1 +arXiv:2301.04244v1 [q-fin.GN] 10 Jan 2023 + +To illustrate the negative consequences of inelasticity, consider the trajectory of home prices +denominated in Bitcoin over a long period. As the population grows, the demand for houses is +sure to grow. Even if the supply of houses keeps pace, home prices must fall, because the supply of +Bitcoin cannot keep pace. Certainly, if the number of concurrent transactions involving home sales +grows and the prices of houses remain stable, the amount of Bitcoin involved in these transactions +would have to grow, yet it cannot grow beyond 21M. So, a fixed money supply leads to falling +prices in a growing economy, even if the underlying supply and demand of items keep pace with +each other. +At the other extreme, if the money supply is increased excessively, the currency is debased; the +excess supply leads to rising prices and inflation. So, a mechanism for correctly setting the money +supply is essential, because this is the foundation upon which stable prices are built. As much as +possible, prices should reflect the tension between the supply and demand for items, and nothing +else. But how much money is too much? It is not just a matter of trust, the problem is that the +appropriate supply is difficult for anyone to calculate! Is there a principled method to compute +the supply, and a fair way to create new money? This work gives my answers to these important +questions. +Ideally, a mechanism for achieving elasticity should be transparent. +It should not rely on +the judgment or integrity of small groups of people, or a few institutions. I will describe a new +decentralized mechanism called Elastic Cash that enjoys these features. Elastic Cash can replace +the role played by private institutions in providing elasticity for the US Dollar, and it can be +combined with the concept of blockchains to make new cryptocurrencies that are elastic, yet do +not require any trusted central authority. +Here I will give a high-level description of Elastic Cash; a full description is in Section 3. I will +refer to the central bank and the algorithm as cash authorities. At the heart of the mechanism is +a new financial contract issued by the cash authority called a cashbond, and a public market that +allows anyone to buy and sell cashbonds using public auctions in the market. Each cashbond can be +redeemed for $1 on a specific date, so it executes a risk-free loan from the holder of the cashbond to +the cash authority. The loans are risk-free because the cash authority can always create new money +to repay the loans. Perhaps it is counterintuitive, but the purpose of the market in cashbonds is +not to give the cash authority a way to borrow money; instead, the function of the market is to use +the trading activity of participants to compute the risk-free rate of interest, which can be computed +from the prices of cashbonds in the market. The mechanism requires that cashbonds can only be +transfered by selling and buying them in the market. For example, no entity should be permitted +to use cashbonds as collateral to borrow money. This forces participants to liquidate cashbonds +when they need money, and keeps the mechanism informed about the demands for money. +It is market participants that determine the money supply in Elastic Cash. +The supply is +increased or decreased according to transparent rules ensuring that the risk-free rate of interest +remains approximately fixed. Trade in cashbonds leads to fluctuations in the rate of interest, and +the mechanism responds by creating corresponding fluctuations in the money supply. Intuitively, +the risk-free rate of interest encodes the cost of renting money. By regulating the supply of money to +keep the cost fixed, the mechanism ensures that the supply stays in equilibrium with the demand for +liquidity. When the supply is increased, market participants acquire any newly-generated money. +The supply is decreased by incentivizing market participants to exchange their money for cashbonds. +Anyone can participate in this market, and money is distributed to or taken from participants +according to transparent rules, so no single entity can control the flow of money. +2 + +Source: Board of Governors of the Federal Reserve System (US) +fred.stlouisfed.org +Billions of Dollars +1960 +1970 +1980 +1990 +2000 +2010 +2020 +0 +4,000 +8,000 +12,000 +16,000 +20,000 +24,000 +M2 +Figure 1: M2, a measure of the supply of US Dollars +Elastic Cash can be implemented within the framework of conventional currencies like the US +Dollar by creating regulations that require the central bank to implement the market for cashbonds +and generate money according to the rules of the mechanism. +It can be implemented in the +framework of cryptocurrencies by setting up a distributed algorithm to implement the market for +cashbonds using the blockchain. Money is generated by the algorithm according to the rules of the +mechanism. So, one can obtain the positive features of traditional currencies and cryptocurrencies +in addition to the elasticity of Elastic Cash. +Outline of this paper +In Section 2, I give more details about how the US Dollar achieves its +current elasticity, before turning to describe the new mechanism in Section 3. I discuss how the new +mechanism can be incorporated into the Dollar system in Section 4 and how it can be implemented +on the blockchain to give new cryptocurrencies in Section 7. +2 +US Dollar elasticity: the Fed, banks, and shadow banks +Figure 1 shows how the supply of US Dollars held as deposits has changed over time. In this section +I briefly review the history and mechanics of the US Dollar system. I recommend the following +3 + +二Dexcellent sources for additional background on the history of the Dollar system [1, 4], and the book +[2] for a longer history of the technology of money. +The supply of US Dollars has consistently been deemed too important to be left entirely in +the control of a government agency. Instead, we have developed a system where money is created +by investor-owned private entities. These include banks and so-called shadow banks — non-bank +financial institutions that are able to create Dollars. This privately created money is then back- +stopped by the central bank, which is the Federal Reserve, or Fed in the US. The Fed is governed +by laws written by Congress, but these laws are somewhat ambiguous about the Fed’s powers, and +Congress has repeatedly made adjustments. The Fed has responded to various crises by incentiviz- +ing or directly making big changes to the money supply, and by sometimes deciding that a new +category of privately-issued financial instruments can be exchanged for US Dollars. +At this point, there are at least four kinds of financial institutions that are able to generate +financial instruments that are de facto US Dollars. It is helpful to view the evolution of the US +Dollar system according to the events that elevated these instruments to the stature of US Dollars: +Commercial bank deposits By the early 1900s, deposits of US Dollars were being issued by +a number of investor-owned private banks. +These deposits were treated by depositors as +equivalent to US Dollars, even though the banks were issuing loans by creating deposits that +did not correspond to cash reserves. This led to a run on the banks in 1907, and the Fed was +created in 1913 to solve the problem. The Fed was given the power to backstop the money +created by commercial banks by lending to the banks. This meant that deposits could always +be exchanged for money created by the Fed. This solved the problem of bank runs, but also +elevated bank deposits to the stature of cash issued by the Fed, and effectively meant that +commercial banks were given the authority to create legitimate US Dollars in the process of +issuing loans. Currently, about $17T is held in commercial bank deposits. +Reverse purchase agreements (repos) In the 1950s, broker dealers began to enter the banking +industry using a financial instrument called reverse purchase agreements, or repo. Repos allow +dealers to borrow money from cash providers using government securities as collateral. Cash +providers began to treat the repos they had purchased from dealers as equivalent to cash. +By itself, such an arrangement does not create any new money, because if repos crash in +value, then they cannot actually be exchanged for cash. However, the properties of repos +were substantially altered when the Fed decided to backstop dealers by giving them access to +overnight loans. This meant that the private holders of repos were guaranteed that repos could +always be exchanged for US Dollars via the Fed’s repo facility, and so repos were elevated to +the same stature as cash and bank deposits. In 1991, Congress reduced restrictions on the +Fed to make it even easier to backstop the repo market. Effectively, broker dealers were given +the power to create new US Dollars. Currently, the size of the repo market is about $4T. +Eurodollars Foreign companies including both banks and non-banks (e.g. insurance companies) +have been issuing financial instruments called eurodollars that can be redeemed for US Dollars. +These eurodollars were not initially backed by actual Dollars. The oil shock of 1973-1974 led +to problems in the eurodollar market that eventually brought down a domestic US bank. The +Fed responded by promising to backstop eurodollars by providing actual US Dollars in the +form of loans to the corresponding foreign central banks. Effectively, the Fed permitted the +banking systems of other countries to create deposits that could be exchanged for US Dollars. +The size of the eurodollar market was estimated at about $13T in 2016 [3]. +4 + +Money market mutual funds These funds emerged in the 1970s as investments whose share +price was pegged at $1. In reality, these funds held assets whose value could drop below the +peg, and so it was not possible to guarantee the peg during a financial crisis. In response +to the great financial crisis of 2008, the Fed began to backstop these funds using its Money +Market Mutual Fund Liquidity Facility, and so elevated deposits in these funds to the stature +of US Dollars. Total assets in these funds is about $4.8T. +In addition to recognizing new forms of the US Dollar, the Fed has resorted to buying assets +in order to inject liquidity under the Quantitative Easing program. During the Great Financial +Crisis of 2008, the Fed kicked off this program by buying mortgage backed securities and treasuries, +which are loans to the US Treasury. Throughout the last decade, the Fed has continued to expand +its balance sheet, mostly with treasuries. In 2020 the Fed once again bought significant quantities +of these assets. Currently the Fed holds about $8.5T on its balance sheet. +The history of the US Dollar is full of ad hoc amendments to maintain stability in the face of +financial crises. At its heart, the problem is that there is currently no principled way to regulate +the supply of money. This becomes apparent during times of financial crises, but the imbalance in +supply is probably always brewing, even in normal times. By now, the pattern of Fed actions is +familiar, and it is inevitable that it will repeat. During times of crisis, the Fed must act to protect +money or face a significant crash in the entire system. Because private financial institutions know +that the Fed will protect their financial instruments from the most negative consequences of their +choices, they do not have the correct incentives. +Elastic Cash is meant to provide a clean, transparent, and principled mechanism to achieve +robust elasticity. +We do not need to cede control of the money supply to private institutions +or foreign banks. We do not need to elevate invented forms of money to the stature of the US +Dollar. Once Elastic Cash is adopted, I believe we can safely bar all private creation of Dollars. +The mechanism will generate new US Dollars when required, and financial institutions can obtain +liquidity by participating in the mechanism, just like everyone else. +Extricating ourselves from the current system and its vested interests is likely to be challenging, +to say the least. Nevertheless, I describe a path to incorporating the new mechanism in Section 4. +3 +Elastic Cash: the details +Elastic Cash uses trade in cashbonds to determine a risk-free rate of interest. The money supply +is regulated to ensure that this interest rate remains approximately fixed. Cashbonds are issued +by the central bank (in the case of conventional currencies) or by the distributed algorithm (in the +case of cryptocurrencies). I refer to these entities as cash authorities. +The contract cashbond(d) promises that the cash authority will pay the holder of the contract $1 +on the date d. Let us reserve d0 to denote the current date. On date d0, the cash authority pays each +holder of cashbond(d0) $1, and these contracts expire. Elastic Cash requires that the cash authority +implement a public market in cashbonds. On the date d0, contracts of the type cashbond(d) for +d > d0 will be available in the market for cashbonds maintained by the cash authority. +Cashbonds are a special class of asset, and they should not be treated like other securities. +Elastic Cash requires that cashbonds can be generated and traded only in the public market that +is administered by the cash authority. Cashbonds are not transferable, meaning they cannot be +exchanged outside of the public market, and they cannot be used as collateral for loans. Because of +5 + +these restrictions, cashbonds cannot themselves play the same role as money. The purpose of these +rules is to ensure that holders of cashbonds that desire liquidity will sell their cashbonds in the +market and so keep the mechanism informed about the demand for liquidity. For the same reasons, +trade in cashbonds should not be taxed. The transactions of buying and selling cashbonds should +be viewed as similar to transactions that move money between savings accounts paying varying +rates of interest, and treated similarly under the law. +3.1 +Risk-free rate of interest +The price at which cashbonds trade implies interest rates for risk-free loans of varying durations. +Let rate(t) denote the interest rate for duration t. Let us write price(d) to denote the price at which +cashbond(d) last traded in the market. Then, if the current date is d0, the prices of cashbonds can +be used to compute implied interest rates according to the formula: +price(t + d0) · (1 + rate(t))t = 1, +which implies that the interest rate can be expressed as +rate(t) = price(t + d0)− 1 +t − 1. +Because the loans executed by cashbonds are risk-free, the values rate(t) capture something about +the market’s belief about the opportunity cost of making risk-free loans for duration t. Generally, +one would expect rate(t) to be a monotone function of t, meaning that rate(t) > rate(t′) if t > t′, +because loans of longer duration usually command higher interest rates. Moreover, if t is much +larger than t′, then we might expect rate(t) to have higher variance than rate(t′), because predictions +about the distant future can diverge much more than predictions about the immediate future. +These rates encode important information about the demand for liquidity. The goal of the +mechanism is to regulate the money supply so that one of these rates is held approximately fixed. +It makes the most sense to pick a rate for a relatively short duration, because these rates are likely +to have the least variance. With that in mind, let τ denote a short time period, say 1 week. The +goal of the mechanism will be to keep +rate(τ) ≈ 0.02. +There is nothing special about 0.02, except that it is convention for central banks around the world +to use 2% as the target rate of longterm inflation. +Let us set +p− = (1 + 0.021)−τ, +and +p+ = (1 + 0.019)−τ. +The goal of the mechanism will be to regulate the money supply so that +p− ≤ price(τ + d0) ≤ p+, +where again d0 is the current date. This corresponds to keeping +0.019 ≤ rate(τ) ≤ 0.021. +6 + +3.2 +Using the market to regulate the money supply +Participants in the cashbond market can put in orders to sell a specific number of cashbonds that +they hold at a specific price, and can also put in orders to buy a specific number of cashbond(d) +at a specific price. The cash authority acts as a market maker to match buy orders to sell orders +and so conduct transactions at a specific price between market participants. Ideally, the market +for cashbonds will support auctions1 for sellers to sell their cashbonds when needed. +The cash authority will itself participate in this public market by buying and selling cash bonds +in prescribed ways. The goal of the mechanism is to maintain rate(τ) approximately fixed, and to +keep the market in cashbonds sufficiently liquid, so that the money supply can be quickly adjusted +based on changes to rate(τ). Here is the proposed scheme for buying and selling cashbonds: +1. The cash authority will buy and sell cashbonds to keep rate(τ) ≈ 0.02. The cash authority +will place a standing order to buy an infinite number of contracts cashbond(τ + d0) at price +p−, and a separate standing order to sell an infinite number of cashbond(τ + d0) contracts at +price p+. +Because the cash authority is able to generate arbitrary amounts of both money and cash- +bonds, it will always be able to satisfy any of the resulting transactions. This ensures that +p− ≤ price(τ + d0) ≤ p+, +as discussed above. +2. When cashbonds are redeemed for money, the cash authority will need to sell new cashbonds +to restore the balance between money and cashbonds. +It makes sense to pick a particular target distribution on outstanding cashbonds that is +maintained during normal times. If the current date is d0, we say that cashbond(δ + d0) has +duration δ. For example, the cash authority might aim to maintain the invariant that at any +point in time, 1/4 of the outstanding cashbonds have duration between 0 and 1 month, 1/4 +have duration between 1 month and 1 year, 1/4 have duration between 1 year and 4 years, +and 1/4 have duration between 4 years and 10 years.2. +Given such a target distribution, the redeemed cashbonds should be replaced by selling new +cashbonds at auction, picking the dates of the new cashbonds so that the overall distribution +on duration is maintained as much as possible. +3. When the demand for money is high, we are likely to reach the point where all of the available +bonds cashbond(τ + d0) have been purchased by the cash authority. +In such times, the +mechanism has run out of the means to inject money into the financial system at a fast enough +pace according to rule 1. This can be resolved by selling large quantities of cashbond(2τ +d0) +contracts at auction in the market. Market participants will be incentivized to buy these +cashbonds and then sell them back after time τ; at that time the cash authority itself will +be willing to buy the cashbonds at price p−. The net effect will be to inject money into the +system, while preserving the number of outstanding cashbonds. +1I will not commit to a specific style of auction here, though any implementation must carefully specifying how +the cash authority behaves as a market maker and what the rules of the auctions are. +2There are many considerations for how to choose the target distribution, but here I will not dwell on the choices +too much. +7 + +The number of cashbonds sold in this process is a design choice. The goal should be inject +significant liquidity, so I would favor an exponentially escalating volume of sales. For exam- +ple, the cash authority might first sell a quantity that corresponds to 1% of all outstanding +cashbonds, and a week later escalate it to 2%, then 4%, and so on until the cashbond market +returns to the state where market participants are no longer willing to sell back the cash- +bonds of duration τ to the cash authority at p−. These actions may temporarily distort the +distribution on the durations of outstanding cashbonds, but the distribution will be quickly +restored when the new cashbonds are redeemed and rule 2 is applied. +An actual implementation of Elastic Cash would need to resolve many smaller technical details. +Let me now make a few comments and observations about the Elastic Cash mechanism as I have +defined it. +3.3 +Discussion +Elastic Cash is quite different from a system where the central bank simply allows deposits for +all with interest rate 2%—such a scheme does not give the central bank a method to inject large +amounts of money when the liquidity is needed. History has shown that the Fed needs a tool like +Elastic Cash to inject liquidity into the financial system, since interest rates have proven too weak +as a tool to inject large quantities of liquidity. As we discussed in Section 2, this has led to the +Fed buying assets or propping up assets that were liable to crash in value. In doing so, the Fed +is forced to pick and choose between market participants that get first access to the new liquidity +that it provides. +Central bankers should not be attempting to directly reason about the demands for liquidity; +they do not have enough data to make those decisions. But if they must take such dramatic actions, +the scheme of Elastic Cash at the very least gives a fair way to do it by trading cashbonds along +the lines I have suggested above. This removes the ability of the finance sector to control the flow +of the new money. It is also preferable to having the Fed buy treasuries, because it disentangles the +actions of the Fed from the needs of the Treasury. There is no need to tie increases in the money +supply to increases in government spending. +Cashbonds should not be confused with conventional government securities like US treasuries. +These instruments are significantly different from each other, and one cannot make inferences about +the cashbond market, which does not yet exist, based on the behavior of the US treasury market. Let +me highlight some key differences. The issuance of cashbonds is controlled by strict and transparent +rules, and is not tied to the spending of the US government. There is no analogue of debt ceilings, +or any chance that the central bank will default. Cashbonds cannot be used as collateral for loans, +cannot be transferred outside of the Elastic Cash market, and trade in cashbonds is not taxed. +It is important for the functioning of Elastic Cash to maintain a large volume of outstanding +cashbonds of varying durations. Ideally, we would like there to be broad participation in the cash- +bond market from all kinds of financial entities: banks, companies, pension funds, and individuals. +Because these participants will be willing to trade at different durations, participation will be in- +creased if a wide range of durations are available, and the market is liquid at all durations. Even +though the cash authority only regulates the interest rate for duration τ, this action will affect the +rates for all durations. One would expect that banks and other sophisticated players will trade +cashbonds of shorter duration, and perform the arbitrage necessary for information about demands +for liquidity of all durations to propagate to the shorter durations. I suspect that there is a prin- +8 + +cipled way to choose the ideal distribution on durations of outstanding cashbonds, but I have not +yet been able to convince myself about what it ought to be. +4 +Adopting Elastic Cash in the US Dollar system +As discussed in Section 2, the Dollar system involves many different kinds of institutions that are +currently creating instruments that can be exchanged for US Dollars. Changing the system is not +going to be straightforward. +However, I do believe that there is a path to making the change +somewhat gradually, so that all the parties involved have time to adapt to the new system. Here +is a proposed sequence of steps to adopting Elastic Cash for the US Dollar: +1. The Fed begins to populate the cashbond market by gradually selling cashbonds of varying +duration. Cashbonds are held at accounts maintained by the Fed, which allows the Fed to +enforce that cashbonds cannot be transferred outside the cashbond market. At this point, +cashbonds that expire are replaced according to the rules of Elastic Cash, but the risk-free +rate of interest is allowed to float freely. I would expect this floating rate to converge close to +the current Fed funds rate. +2. Once the market for cashbonds is running at significant scale, regulations should be enacted to +curtail the private creation of US Dollars. This can be done gradually by raising the interest +rate at which the Fed lends to private entities through its discount window. At the same +time, the Fed should begin to put bounds on the risk-free rate determined by cashbond, by +trading in the cashbond market. Eventually, we should end up with a high rate for borrowing +from the Fed via the discount window, while the risk-free rate in the cashbond market should +be close to 2%. This will incentivize private entities to participate in the cashbond market +and raise money there. The current creators of US Dollars can be handled as follows: +(a) Commercial banks should be barred from creating new deposits that are not backed +by cash reserves. Banks should fund new lending activity by selling corporate bonds +instead. +(b) The Fed’s repo facility and money market fund facility should be closed. +(c) The eurodollar market is, perhaps, a bigger problem, both because of its size and the fact +that the institutions cannot be regulated by US law. Still, the Fed can wind down its +swap lines with foreign central banks gradually, until eurodollars lose their Fed backing. +Foreign central banks and governments should be allowed and encouraged to participate +in the cashbond market to obtain liquidity. +3. The inevitable tantrums in the financial sector should be treated with stoicism. +It is an understatement that moving from our current system of private money creation to +Elastic Cash would be a dramatic change. There are likely to be many challenges that need to +be overcome to implement it, not least the resistance of the finance industry, whose raison d’ˆetre +is to control the flow of money. Elastic Cash represents a significant loss of control for financial +firms, and a democratization of the flow of money. For these reasons, it is perhaps more easily +implemented in a cryptocurrency, as I discuss next. +9 + +5 +Elastic Cash in cryptocurrencies +A major advantage of Elastic Cash over conventional mechanisms for elasticity is that it can be +implemented in a truly decentralized way, without any trusted central authority. Bitcoin made +a technological leap when it introduced the concept of a blockchain. +Since then, a number of +cryptocurrencies have emerged, with different ways to implement the blockchain. Any of these +systems can be used to implement Elastic Cash, so here I will keep the discussion at a high level, only +talking about how the blockchain can be utilized. Because Elastic Cash involves making significant +changes to the money supply, I do believe that implementing it requires new cryptocurrencies. I +do not think it can be implemented using a layer built on top of Bitcoin, for example. +Here is how one can implement Elastic Cash on a blockchain at a high level: +1. At any point in time, each user of the cryptocurrency is known to hold some amount of money, +as well as various cashbonds. +2. Users of the currency can announce transactions of money, as well as orders placed in the +cashbond market. The orders can be placed with a specific expiry date. +3. Miners will add both money transactions and orders in the cashbond market to the next block +of the blockchain. To implement the market in cashbonds: +(a) Miners will act as market makers to map buy orders to sell orders and so execute the +trade in cashbonds. There are some subtle issues that need to be addressed here. For +example, a miner may be incentivized to pick some orders over others to include on +the blockchain, and choose to ignore some orders when acting as a market maker. In +particular, miners should themselves be paid the spread between buy and sell orders as +a transaction fee to carry out their market making function. This removes the incentives +to manipulate the orders that are added to the most recent block. +(b) Miners will also execute the algorithm to simulate the activities of the cash authority in +the cashbond market. New money and cashbonds will be created according to the rules +of the mechanism, and these will be traded with users based on the orders that have +been added to the blockchain. +6 +Conclusions and Questions +It is an exciting time to think about the technology of money. The US Dollar is experiencing a +once-in-a-lifetime contraction (see Figure 1), and the demands for a stable global currency have +never been larger. Elastic Cash is a broad scheme to enable elastic money. I have purposefully +left the mechanism underspecified, because I believe that more work is required to understand the +details and trade-offs involved in the particulars of the mechanism. +Here are some important questions that I feel remain unanswered: +1. How should the market maker behave in the cashbond market? In the context of conventional +currencies, can private entities function as market makers? In the context of cryptocurrencies, +how should the algorithm be set up so that miners do not have an incentive to behave +dishonestly when they are carrying out the role of market maker? +10 + +2. What style of auction would give the best results for the cashbond market? +3. What is the ideal target distribution on cashbonds? If the cashbonds are concentrated on +very short durations, this gives the most power for the mechanism to inject large quantities +of money, but it also means that the market loses information about the demand for liquidity +over long durations. +So, there is a trade-off between various choices for distributions on +durations. +4. How can we gradually transition the current US Dollar system to such a mechanism? The +steps I discussed in Section 4 are likely to be difficult to execute. Perhaps there is a way +to use cashbonds and incentivize the large players in the financial system to adopt Elastic +Cash without being forced to do it. What is needed is a mechanism to transition to the new +mechanism! +5. How should we expect the free floating rate curve rate(t) to behave as a function of t during +normal times? I would expect this function to be monotone, but I am not sure how to reason +about it beyond that. +7 +Acknowledgements +Thanks to Paul Beame, Siddharth Iyer, Travis Kriplean, James Lee, Noam Nisan, Darcy Rao, +Eli Ben-Sasson, Oscar Sprumont, Michael Whitmeyer and Amir Yehudayoff for many helpful and +entertaining conversations about money. +References +[1] Fed history overview. https://www.federalreservehistory.org/time-period. +[2] Christine Desan. +Making Money: Coin, Currency, and the Coming of Capitalism. +Oxford +University Press, 2014. +[3] Neels Heyneke and Mehul Daya. +The rise and fall of the eurodollar system. +https: +//www.nedbank.co.za/content/dam/nedbank-crp/reports/Strategy/NeelsAndMehul/ +2016/September/TheRiseAndFallOfTheEurodollarSystem_160907.pdf, 2016. +[4] Lev Menand. The Fed-Unbound: Central Banking in a Time of Crisis. 2022. +11 + diff --git a/HdE2T4oBgHgl3EQf_Alo/content/tmp_files/load_file.txt b/HdE2T4oBgHgl3EQf_Alo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb0483f055afee64cb01a724fc79c24124691d20 --- /dev/null +++ b/HdE2T4oBgHgl3EQf_Alo/content/tmp_files/load_file.txt @@ -0,0 +1,312 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf,len=311 +page_content='Elastic Cash Anup Rao University of Washington anuprao@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='edu January 12, 2023 Abstract Elastic Cash is a new decentralized mechanism for regulating the money supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The mech- anism operates by modifying the supply so that an interest rate determined by a public market is kept approximately fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It can be incorporated into the conventional monetary system to improve the elasticity of the US Dollar, and it can be used to design new elastic cryptocurrencies that remain decentralized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 1 Introduction Money is as old as recorded history, and yet it continues to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Even the mighty US Dollar has been repeatedly updated over the last 200 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The recent emergence of Bitcoin and other cryptocurrencies is another step in that evolution, and it prompts us to revisit the mechanisms that ensure the desirable properties of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' An important property of money is its elasticity: money is elastic if the money supply increases in response to demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In this work, I present a new decentralized mechanism to ensure the elasticity of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The US Dollar is elastic, but it uses a convoluted system to achieve its elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The Dollar system was not designed from first principles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' it was iteratively amended in response to financial crises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Elasticity is currently achieved by the combined actions of many institutions: banks and non-banks, public and private, domestic and international.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' No single entity is entirely in charge of the money supply, and a relatively small number of investor-owned institutions have undue influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I discuss the mechanics of the US Dollar system and its limitations in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In contrast, Bitcoin was designed to be inelastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Bitcoin caps the total possible supply at 21M, and the available supply, currently about 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='2M, will slowly increase until it hits this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Satoshi Nakamoto, the creator of Bitcoin, criticized conventional mechanisms for achieving elasticity in an early forum post: The root problem with conventional currency is all the trust that’s required to make it work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The central bank must be trusted not to debase the currency, but the history of fiat currencies is full of breaches of that trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Banks must be trusted to hold our money and transfer it electronically, but they lend it out in waves of credit bubbles with barely a fraction in reserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Nakamoto eliminated the need for the kind of infrastructure used to ensure Dollar elasticity by simply choosing to make Bitcoin inelastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Additional technological innovations in Bitcoin further eliminated the need for any trusted central authority to carry out transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='04244v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='GN] 10 Jan 2023 To illustrate the negative consequences of inelasticity, consider the trajectory of home prices denominated in Bitcoin over a long period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' As the population grows, the demand for houses is sure to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Even if the supply of houses keeps pace, home prices must fall, because the supply of Bitcoin cannot keep pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Certainly, if the number of concurrent transactions involving home sales grows and the prices of houses remain stable, the amount of Bitcoin involved in these transactions would have to grow, yet it cannot grow beyond 21M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' So, a fixed money supply leads to falling prices in a growing economy, even if the underlying supply and demand of items keep pace with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' At the other extreme, if the money supply is increased excessively, the currency is debased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' the excess supply leads to rising prices and inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' So, a mechanism for correctly setting the money supply is essential, because this is the foundation upon which stable prices are built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' As much as possible, prices should reflect the tension between the supply and demand for items, and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' But how much money is too much?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It is not just a matter of trust, the problem is that the appropriate supply is difficult for anyone to calculate!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Is there a principled method to compute the supply, and a fair way to create new money?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This work gives my answers to these important questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Ideally, a mechanism for achieving elasticity should be transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It should not rely on the judgment or integrity of small groups of people, or a few institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I will describe a new decentralized mechanism called Elastic Cash that enjoys these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Elastic Cash can replace the role played by private institutions in providing elasticity for the US Dollar, and it can be combined with the concept of blockchains to make new cryptocurrencies that are elastic, yet do not require any trusted central authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Here I will give a high-level description of Elastic Cash;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' a full description is in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I will refer to the central bank and the algorithm as cash authorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' At the heart of the mechanism is a new financial contract issued by the cash authority called a cashbond, and a public market that allows anyone to buy and sell cashbonds using public auctions in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Each cashbond can be redeemed for $1 on a specific date, so it executes a risk-free loan from the holder of the cashbond to the cash authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The loans are risk-free because the cash authority can always create new money to repay the loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Perhaps it is counterintuitive, but the purpose of the market in cashbonds is not to give the cash authority a way to borrow money;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' instead, the function of the market is to use the trading activity of participants to compute the risk-free rate of interest, which can be computed from the prices of cashbonds in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The mechanism requires that cashbonds can only be transfered by selling and buying them in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' For example, no entity should be permitted to use cashbonds as collateral to borrow money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This forces participants to liquidate cashbonds when they need money, and keeps the mechanism informed about the demands for money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It is market participants that determine the money supply in Elastic Cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The supply is increased or decreased according to transparent rules ensuring that the risk-free rate of interest remains approximately fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Trade in cashbonds leads to fluctuations in the rate of interest, and the mechanism responds by creating corresponding fluctuations in the money supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Intuitively, the risk-free rate of interest encodes the cost of renting money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' By regulating the supply of money to keep the cost fixed, the mechanism ensures that the supply stays in equilibrium with the demand for liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' When the supply is increased, market participants acquire any newly-generated money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The supply is decreased by incentivizing market participants to exchange their money for cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Anyone can participate in this market, and money is distributed to or taken from participants according to transparent rules, so no single entity can control the flow of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 2 Source: Board of Governors of the Federal Reserve System (US) fred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='stlouisfed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='org Billions of Dollars 1960 1970 1980 1990 2000 2010 2020 0 4,000 8,000 12,000 16,000 20,000 24,000 M2 Figure 1: M2, a measure of the supply of US Dollars Elastic Cash can be implemented within the framework of conventional currencies like the US Dollar by creating regulations that require the central bank to implement the market for cashbonds and generate money according to the rules of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It can be implemented in the framework of cryptocurrencies by setting up a distributed algorithm to implement the market for cashbonds using the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Money is generated by the algorithm according to the rules of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' So, one can obtain the positive features of traditional currencies and cryptocurrencies in addition to the elasticity of Elastic Cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Outline of this paper In Section 2, I give more details about how the US Dollar achieves its current elasticity, before turning to describe the new mechanism in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I discuss how the new mechanism can be incorporated into the Dollar system in Section 4 and how it can be implemented on the blockchain to give new cryptocurrencies in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 2 US Dollar elasticity: the Fed, banks, and shadow banks Figure 1 shows how the supply of US Dollars held as deposits has changed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In this section I briefly review the history and mechanics of the US Dollar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I recommend the following 3 二Dexcellent sources for additional background on the history of the Dollar system [1, 4], and the book [2] for a longer history of the technology of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The supply of US Dollars has consistently been deemed too important to be left entirely in the control of a government agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Instead, we have developed a system where money is created by investor-owned private entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' These include banks and so-called shadow banks — non-bank financial institutions that are able to create Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This privately created money is then back- stopped by the central bank, which is the Federal Reserve, or Fed in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The Fed is governed by laws written by Congress, but these laws are somewhat ambiguous about the Fed’s powers, and Congress has repeatedly made adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The Fed has responded to various crises by incentiviz- ing or directly making big changes to the money supply, and by sometimes deciding that a new category of privately-issued financial instruments can be exchanged for US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' At this point, there are at least four kinds of financial institutions that are able to generate financial instruments that are de facto US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It is helpful to view the evolution of the US Dollar system according to the events that elevated these instruments to the stature of US Dollars: Commercial bank deposits By the early 1900s, deposits of US Dollars were being issued by a number of investor-owned private banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' These deposits were treated by depositors as equivalent to US Dollars, even though the banks were issuing loans by creating deposits that did not correspond to cash reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This led to a run on the banks in 1907, and the Fed was created in 1913 to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The Fed was given the power to backstop the money created by commercial banks by lending to the banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This meant that deposits could always be exchanged for money created by the Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This solved the problem of bank runs, but also elevated bank deposits to the stature of cash issued by the Fed, and effectively meant that commercial banks were given the authority to create legitimate US Dollars in the process of issuing loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Currently, about $17T is held in commercial bank deposits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Reverse purchase agreements (repos) In the 1950s, broker dealers began to enter the banking industry using a financial instrument called reverse purchase agreements, or repo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Repos allow dealers to borrow money from cash providers using government securities as collateral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Cash providers began to treat the repos they had purchased from dealers as equivalent to cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' By itself, such an arrangement does not create any new money, because if repos crash in value, then they cannot actually be exchanged for cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' However, the properties of repos were substantially altered when the Fed decided to backstop dealers by giving them access to overnight loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This meant that the private holders of repos were guaranteed that repos could always be exchanged for US Dollars via the Fed’s repo facility, and so repos were elevated to the same stature as cash and bank deposits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In 1991, Congress reduced restrictions on the Fed to make it even easier to backstop the repo market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Effectively, broker dealers were given the power to create new US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Currently, the size of the repo market is about $4T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Eurodollars Foreign companies including both banks and non-banks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' insurance companies) have been issuing financial instruments called eurodollars that can be redeemed for US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' These eurodollars were not initially backed by actual Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The oil shock of 1973-1974 led to problems in the eurodollar market that eventually brought down a domestic US bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The Fed responded by promising to backstop eurodollars by providing actual US Dollars in the form of loans to the corresponding foreign central banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Effectively, the Fed permitted the banking systems of other countries to create deposits that could be exchanged for US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The size of the eurodollar market was estimated at about $13T in 2016 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 4 Money market mutual funds These funds emerged in the 1970s as investments whose share price was pegged at $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In reality, these funds held assets whose value could drop below the peg, and so it was not possible to guarantee the peg during a financial crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In response to the great financial crisis of 2008, the Fed began to backstop these funds using its Money Market Mutual Fund Liquidity Facility, and so elevated deposits in these funds to the stature of US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Total assets in these funds is about $4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='8T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In addition to recognizing new forms of the US Dollar, the Fed has resorted to buying assets in order to inject liquidity under the Quantitative Easing program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' During the Great Financial Crisis of 2008, the Fed kicked off this program by buying mortgage backed securities and treasuries, which are loans to the US Treasury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Throughout the last decade, the Fed has continued to expand its balance sheet, mostly with treasuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In 2020 the Fed once again bought significant quantities of these assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Currently the Fed holds about $8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='5T on its balance sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The history of the US Dollar is full of ad hoc amendments to maintain stability in the face of financial crises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' At its heart, the problem is that there is currently no principled way to regulate the supply of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This becomes apparent during times of financial crises, but the imbalance in supply is probably always brewing, even in normal times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' By now, the pattern of Fed actions is familiar, and it is inevitable that it will repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' During times of crisis, the Fed must act to protect money or face a significant crash in the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Because private financial institutions know that the Fed will protect their financial instruments from the most negative consequences of their choices, they do not have the correct incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Elastic Cash is meant to provide a clean, transparent, and principled mechanism to achieve robust elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' We do not need to cede control of the money supply to private institutions or foreign banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' We do not need to elevate invented forms of money to the stature of the US Dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Once Elastic Cash is adopted, I believe we can safely bar all private creation of Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The mechanism will generate new US Dollars when required, and financial institutions can obtain liquidity by participating in the mechanism, just like everyone else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Extricating ourselves from the current system and its vested interests is likely to be challenging, to say the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Nevertheless, I describe a path to incorporating the new mechanism in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 3 Elastic Cash: the details Elastic Cash uses trade in cashbonds to determine a risk-free rate of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The money supply is regulated to ensure that this interest rate remains approximately fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Cashbonds are issued by the central bank (in the case of conventional currencies) or by the distributed algorithm (in the case of cryptocurrencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I refer to these entities as cash authorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The contract cashbond(d) promises that the cash authority will pay the holder of the contract $1 on the date d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Let us reserve d0 to denote the current date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' On date d0, the cash authority pays each holder of cashbond(d0) $1, and these contracts expire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Elastic Cash requires that the cash authority implement a public market in cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' On the date d0, contracts of the type cashbond(d) for d > d0 will be available in the market for cashbonds maintained by the cash authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Cashbonds are a special class of asset, and they should not be treated like other securities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Elastic Cash requires that cashbonds can be generated and traded only in the public market that is administered by the cash authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Cashbonds are not transferable, meaning they cannot be exchanged outside of the public market, and they cannot be used as collateral for loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Because of 5 these restrictions, cashbonds cannot themselves play the same role as money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The purpose of these rules is to ensure that holders of cashbonds that desire liquidity will sell their cashbonds in the market and so keep the mechanism informed about the demand for liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' For the same reasons, trade in cashbonds should not be taxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The transactions of buying and selling cashbonds should be viewed as similar to transactions that move money between savings accounts paying varying rates of interest, and treated similarly under the law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='1 Risk-free rate of interest The price at which cashbonds trade implies interest rates for risk-free loans of varying durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Let rate(t) denote the interest rate for duration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Let us write price(d) to denote the price at which cashbond(d) last traded in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Then, if the current date is d0, the prices of cashbonds can be used to compute implied interest rates according to the formula: price(t + d0) · (1 + rate(t))t = 1, which implies that the interest rate can be expressed as rate(t) = price(t + d0)− 1 t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Because the loans executed by cashbonds are risk-free, the values rate(t) capture something about the market’s belief about the opportunity cost of making risk-free loans for duration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Generally, one would expect rate(t) to be a monotone function of t, meaning that rate(t) > rate(t′) if t > t′, because loans of longer duration usually command higher interest rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Moreover, if t is much larger than t′, then we might expect rate(t) to have higher variance than rate(t′), because predictions about the distant future can diverge much more than predictions about the immediate future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' These rates encode important information about the demand for liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The goal of the mechanism is to regulate the money supply so that one of these rates is held approximately fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It makes the most sense to pick a rate for a relatively short duration, because these rates are likely to have the least variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' With that in mind, let τ denote a short time period, say 1 week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The goal of the mechanism will be to keep rate(τ) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' There is nothing special about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='02, except that it is convention for central banks around the world to use 2% as the target rate of longterm inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Let us set p− = (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='021)−τ, and p+ = (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='019)−τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The goal of the mechanism will be to regulate the money supply so that p− ≤ price(τ + d0) ≤ p+, where again d0 is the current date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This corresponds to keeping 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='019 ≤ rate(τ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='2 Using the market to regulate the money supply Participants in the cashbond market can put in orders to sell a specific number of cashbonds that they hold at a specific price, and can also put in orders to buy a specific number of cashbond(d) at a specific price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The cash authority acts as a market maker to match buy orders to sell orders and so conduct transactions at a specific price between market participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Ideally, the market for cashbonds will support auctions1 for sellers to sell their cashbonds when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The cash authority will itself participate in this public market by buying and selling cash bonds in prescribed ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The goal of the mechanism is to maintain rate(τ) approximately fixed, and to keep the market in cashbonds sufficiently liquid, so that the money supply can be quickly adjusted based on changes to rate(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Here is the proposed scheme for buying and selling cashbonds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The cash authority will buy and sell cashbonds to keep rate(τ) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The cash authority will place a standing order to buy an infinite number of contracts cashbond(τ + d0) at price p−, and a separate standing order to sell an infinite number of cashbond(τ + d0) contracts at price p+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Because the cash authority is able to generate arbitrary amounts of both money and cash- bonds, it will always be able to satisfy any of the resulting transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This ensures that p− ≤ price(τ + d0) ≤ p+, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' When cashbonds are redeemed for money, the cash authority will need to sell new cashbonds to restore the balance between money and cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It makes sense to pick a particular target distribution on outstanding cashbonds that is maintained during normal times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' If the current date is d0, we say that cashbond(δ + d0) has duration δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' For example, the cash authority might aim to maintain the invariant that at any point in time, 1/4 of the outstanding cashbonds have duration between 0 and 1 month, 1/4 have duration between 1 month and 1 year, 1/4 have duration between 1 year and 4 years, and 1/4 have duration between 4 years and 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Given such a target distribution, the redeemed cashbonds should be replaced by selling new cashbonds at auction, picking the dates of the new cashbonds so that the overall distribution on duration is maintained as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' When the demand for money is high, we are likely to reach the point where all of the available bonds cashbond(τ + d0) have been purchased by the cash authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In such times, the mechanism has run out of the means to inject money into the financial system at a fast enough pace according to rule 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This can be resolved by selling large quantities of cashbond(2τ +d0) contracts at auction in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Market participants will be incentivized to buy these cashbonds and then sell them back after time τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' at that time the cash authority itself will be willing to buy the cashbonds at price p−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The net effect will be to inject money into the system, while preserving the number of outstanding cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 1I will not commit to a specific style of auction here, though any implementation must carefully specifying how the cash authority behaves as a market maker and what the rules of the auctions are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 2There are many considerations for how to choose the target distribution, but here I will not dwell on the choices too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 7 The number of cashbonds sold in this process is a design choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The goal should be inject significant liquidity, so I would favor an exponentially escalating volume of sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' For exam- ple, the cash authority might first sell a quantity that corresponds to 1% of all outstanding cashbonds, and a week later escalate it to 2%, then 4%, and so on until the cashbond market returns to the state where market participants are no longer willing to sell back the cash- bonds of duration τ to the cash authority at p−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' These actions may temporarily distort the distribution on the durations of outstanding cashbonds, but the distribution will be quickly restored when the new cashbonds are redeemed and rule 2 is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' An actual implementation of Elastic Cash would need to resolve many smaller technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Let me now make a few comments and observations about the Elastic Cash mechanism as I have defined it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='3 Discussion Elastic Cash is quite different from a system where the central bank simply allows deposits for all with interest rate 2%—such a scheme does not give the central bank a method to inject large amounts of money when the liquidity is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' History has shown that the Fed needs a tool like Elastic Cash to inject liquidity into the financial system, since interest rates have proven too weak as a tool to inject large quantities of liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' As we discussed in Section 2, this has led to the Fed buying assets or propping up assets that were liable to crash in value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In doing so, the Fed is forced to pick and choose between market participants that get first access to the new liquidity that it provides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Central bankers should not be attempting to directly reason about the demands for liquidity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' they do not have enough data to make those decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' But if they must take such dramatic actions, the scheme of Elastic Cash at the very least gives a fair way to do it by trading cashbonds along the lines I have suggested above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This removes the ability of the finance sector to control the flow of the new money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It is also preferable to having the Fed buy treasuries, because it disentangles the actions of the Fed from the needs of the Treasury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' There is no need to tie increases in the money supply to increases in government spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Cashbonds should not be confused with conventional government securities like US treasuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' These instruments are significantly different from each other, and one cannot make inferences about the cashbond market, which does not yet exist, based on the behavior of the US treasury market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Let me highlight some key differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The issuance of cashbonds is controlled by strict and transparent rules, and is not tied to the spending of the US government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' There is no analogue of debt ceilings, or any chance that the central bank will default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Cashbonds cannot be used as collateral for loans, cannot be transferred outside of the Elastic Cash market, and trade in cashbonds is not taxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It is important for the functioning of Elastic Cash to maintain a large volume of outstanding cashbonds of varying durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Ideally, we would like there to be broad participation in the cash- bond market from all kinds of financial entities: banks, companies, pension funds, and individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Because these participants will be willing to trade at different durations, participation will be in- creased if a wide range of durations are available, and the market is liquid at all durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Even though the cash authority only regulates the interest rate for duration τ, this action will affect the rates for all durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' One would expect that banks and other sophisticated players will trade cashbonds of shorter duration, and perform the arbitrage necessary for information about demands for liquidity of all durations to propagate to the shorter durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I suspect that there is a prin- 8 cipled way to choose the ideal distribution on durations of outstanding cashbonds, but I have not yet been able to convince myself about what it ought to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 4 Adopting Elastic Cash in the US Dollar system As discussed in Section 2, the Dollar system involves many different kinds of institutions that are currently creating instruments that can be exchanged for US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Changing the system is not going to be straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' However, I do believe that there is a path to making the change somewhat gradually, so that all the parties involved have time to adapt to the new system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Here is a proposed sequence of steps to adopting Elastic Cash for the US Dollar: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The Fed begins to populate the cashbond market by gradually selling cashbonds of varying duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Cashbonds are held at accounts maintained by the Fed, which allows the Fed to enforce that cashbonds cannot be transferred outside the cashbond market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' At this point, cashbonds that expire are replaced according to the rules of Elastic Cash, but the risk-free rate of interest is allowed to float freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I would expect this floating rate to converge close to the current Fed funds rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Once the market for cashbonds is running at significant scale, regulations should be enacted to curtail the private creation of US Dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This can be done gradually by raising the interest rate at which the Fed lends to private entities through its discount window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' At the same time, the Fed should begin to put bounds on the risk-free rate determined by cashbond, by trading in the cashbond market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Eventually, we should end up with a high rate for borrowing from the Fed via the discount window, while the risk-free rate in the cashbond market should be close to 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This will incentivize private entities to participate in the cashbond market and raise money there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The current creators of US Dollars can be handled as follows: (a) Commercial banks should be barred from creating new deposits that are not backed by cash reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Banks should fund new lending activity by selling corporate bonds instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' (b) The Fed’s repo facility and money market fund facility should be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' (c) The eurodollar market is, perhaps, a bigger problem, both because of its size and the fact that the institutions cannot be regulated by US law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Still, the Fed can wind down its swap lines with foreign central banks gradually, until eurodollars lose their Fed backing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Foreign central banks and governments should be allowed and encouraged to participate in the cashbond market to obtain liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The inevitable tantrums in the financial sector should be treated with stoicism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' It is an understatement that moving from our current system of private money creation to Elastic Cash would be a dramatic change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' There are likely to be many challenges that need to be overcome to implement it, not least the resistance of the finance industry, whose raison d’ˆetre is to control the flow of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Elastic Cash represents a significant loss of control for financial firms, and a democratization of the flow of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' For these reasons, it is perhaps more easily implemented in a cryptocurrency, as I discuss next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 9 5 Elastic Cash in cryptocurrencies A major advantage of Elastic Cash over conventional mechanisms for elasticity is that it can be implemented in a truly decentralized way, without any trusted central authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Bitcoin made a technological leap when it introduced the concept of a blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Since then, a number of cryptocurrencies have emerged, with different ways to implement the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Any of these systems can be used to implement Elastic Cash, so here I will keep the discussion at a high level, only talking about how the blockchain can be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Because Elastic Cash involves making significant changes to the money supply, I do believe that implementing it requires new cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I do not think it can be implemented using a layer built on top of Bitcoin, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Here is how one can implement Elastic Cash on a blockchain at a high level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' At any point in time, each user of the cryptocurrency is known to hold some amount of money, as well as various cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Users of the currency can announce transactions of money, as well as orders placed in the cashbond market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The orders can be placed with a specific expiry date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Miners will add both money transactions and orders in the cashbond market to the next block of the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' To implement the market in cashbonds: (a) Miners will act as market makers to map buy orders to sell orders and so execute the trade in cashbonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' There are some subtle issues that need to be addressed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' For example, a miner may be incentivized to pick some orders over others to include on the blockchain, and choose to ignore some orders when acting as a market maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In particular, miners should themselves be paid the spread between buy and sell orders as a transaction fee to carry out their market making function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' This removes the incentives to manipulate the orders that are added to the most recent block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' (b) Miners will also execute the algorithm to simulate the activities of the cash authority in the cashbond market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' New money and cashbonds will be created according to the rules of the mechanism, and these will be traded with users based on the orders that have been added to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 6 Conclusions and Questions It is an exciting time to think about the technology of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The US Dollar is experiencing a once-in-a-lifetime contraction (see Figure 1), and the demands for a stable global currency have never been larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Elastic Cash is a broad scheme to enable elastic money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I have purposefully left the mechanism underspecified, because I believe that more work is required to understand the details and trade-offs involved in the particulars of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Here are some important questions that I feel remain unanswered: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' How should the market maker behave in the cashbond market?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In the context of conventional currencies, can private entities function as market makers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' In the context of cryptocurrencies, how should the algorithm be set up so that miners do not have an incentive to behave dishonestly when they are carrying out the role of market maker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' What style of auction would give the best results for the cashbond market?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' What is the ideal target distribution on cashbonds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' If the cashbonds are concentrated on very short durations, this gives the most power for the mechanism to inject large quantities of money, but it also means that the market loses information about the demand for liquidity over long durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' So, there is a trade-off between various choices for distributions on durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' How can we gradually transition the current US Dollar system to such a mechanism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The steps I discussed in Section 4 are likely to be difficult to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Perhaps there is a way to use cashbonds and incentivize the large players in the financial system to adopt Elastic Cash without being forced to do it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' What is needed is a mechanism to transition to the new mechanism!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' How should we expect the free floating rate curve rate(t) to behave as a function of t during normal times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' I would expect this function to be monotone, but I am not sure how to reason about it beyond that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 7 Acknowledgements Thanks to Paul Beame, Siddharth Iyer, Travis Kriplean, James Lee, Noam Nisan, Darcy Rao, Eli Ben-Sasson, Oscar Sprumont, Michael Whitmeyer and Amir Yehudayoff for many helpful and entertaining conversations about money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' References [1] Fed history overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='federalreservehistory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='org/time-period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' [2] Christine Desan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Making Money: Coin, Currency, and the Coming of Capitalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' Oxford University Press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' [3] Neels Heyneke and Mehul Daya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The rise and fall of the eurodollar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='nedbank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='za/content/dam/nedbank-crp/reports/Strategy/NeelsAndMehul/ 2016/September/TheRiseAndFallOfTheEurodollarSystem_160907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content='pdf, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' [4] Lev Menand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' The Fed-Unbound: Central Banking in a Time of Crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf'} diff --git a/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf b/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..34644a55b1e9a1cd13acb51e24147052dfb6bbd5 --- /dev/null +++ b/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7e5c6b4c2082d0abbfa6f7c349a9e0b5187521da25771c9a88e595268edc15e4 +size 960592 diff --git a/HdFJT4oBgHgl3EQfFCyj/vector_store/index.faiss b/HdFJT4oBgHgl3EQfFCyj/vector_store/index.faiss new file mode 100644 index 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b/HdFJT4oBgHgl3EQfuS10/content/tmp_files/2301.11621v1.pdf.txt @@ -0,0 +1,1639 @@ +Event Causality Extraction with Event Argument Correlations +Shiyao Cui1,2 Jiawei Sheng1,2 Xin Cong1,2 +QuanGang Li1,2∗Tingwen Liu1,2 Jinqiao Shi3,1 +1Institute of Information Engineering, Chinese Academy of Sciences. Beijing, China +2School of Cyber Security, University of Chinese Academy of Sciences. Beijing, China +3Beijing University of Posts and Telecommunications. Beijing, China +{cuishiyao, shengjiawei, congxin, liquangang}@iie.ac.cn +liutingwen@iie.ac.cn shijinqiao@bupt.edu.cn +Abstract +Event Causality Identification (ECI), which +aims to detect whether a causality relation +exists between two given textual events, is +an important task for event causality under- +standing. However, the ECI task ignores cru- +cial event structure and cause-effect causality +component information, making it struggle for +downstream applications. +In this paper, we +explore a novel task, namely Event Causality +Extraction (ECE), aiming to extract the cause- +effect event causality pairs with their struc- +tured event information from plain texts. The +ECE task is more challenging since each event +can contain multiple event arguments, posing +fine-grained correlations between events to de- +cide the cause-effect event pair. +Hence, we +propose a method with a dual grid tagging +scheme to capture the intra- and inter-event ar- +gument correlations for ECE. Further, we de- +vise a event type-enhanced model architecture +to realize the dual grid tagging scheme. Ex- +periments demonstrate the effectiveness of our +method, and extensive analyses point out sev- +eral future directions for ECE. +1 +Introduction +Event causality (Liu et al., 2020; Cao et al., 2021) +denotes an explicit causal relation between two +events, constituting a specific cause-effect event +pair. As shown in Figure 1, a causal relation exists +between a Price Rise event (The worldwide +rise of oil prices) and a Cost Rise event (in- +creases the cost of international shipping industry). +Understanding such event causality could facilitate +various downstream applications including event +forecasting (Hashimoto et al., 2014), intelligent +search (Rudnik et al., 2019) and question answer- +ing (Costa et al., 2020), which is important for +natural language understanding. +In recent years, it has aroused the research in- +terest for Event Causality Identification (ECI) (Liu +∗Corresponding Author +The worldwide rise of oil prices increases the cost of international shipping +industry and stimulates the demand for new energy such as Ammonia fuel. +No. +Event Type +Event Roles +Product +Region +Industry +(1) +Cause +Price Rise +oil +worldwide +None +Effect +Cost Rise +None +international +shipping industry +(2) +Cause +Price Rise +oil +worldwide +None +Effect +Demand Rise +new energy +None +Ammonia fuel +Causality +Component +Figure 1: Illustration for ECE which takes the raw text +as input, and outputs the structured event causality pair. +et al., 2020; Cao et al., 2021; Zuo et al., 2021a,b, +2020; Tran Phu and Nguyen, 2021), which aims +to detect whether the causality exists between two +given events. Despite of its success, there exist two +issues that the ECI task fails to address. 1) Event +Structure Missing, where each event in ECI is +only expressed using a word or phrase which re- +flects its occurrence, but ignores the explicit event +type and event arguments (i.e., entities which par- +ticipate in the event). The absence of such event +structure would lose valuable clues for understand- +ing event causality. As shown in Figure 1, “oil” +plays a Product role in a Price Rise-typed +cause event, implying a consequent Cost Rise- +typed effect event towards “shipping industry”. 2) +Causality Component Missing, where ECI only +predicts the existence of causality between the +given event pairs, ignoring to discriminate the spe- +cific cause/effect event causality component. Lim- +ited by these issues, ECI insufficiently explores the +causality between events, which demands further +promotion to the understanding of event causality. +Motivated by discussion about event causality +in CCKS (2021), we therefore formulate a task +termed as Event Causality Extraction (ECE). As +Figure 1 shows, ECE aims to end-to-end extract +the cause-effect event pairs with structured event +information from plain texts. Comparing with ECI, +ECE illustrates the event causality including the +event structure, namely event types and arguments, +arXiv:2301.11621v1 [cs.CL] 27 Jan 2023 + +and the specific cause-effect causality component, +making it more informative to support the various +downstream applications (Wang et al., 2021a). +Intuitively, ECE could be achieved by succes- +sively extracting the structured event and then clas- +sifying their causality relation. Unfortunately, such +a paradigm would easily suffer from the redundant +event-pair problem, where the causality-unrelated +events would be inevitably extracted, confusing the +causality decision. Another promising direction +is to borrow ideas from relational triple extraction +(RTE), which shares the similar task formulation. +However, comparing with the entity-centric RTE +task, the event-centric ECE raises new challenges: +1) Intra-event Argument Correlations. Specif- +ically, ECE focuses on event, which is a struc- +ture maintaining interactive correlations among +its arguments. For example in Figure 1, the ar- +gument “new energy” and “Ammonia fuel” in the +Demand Rise event have strong semantic cor- +relation. While RTE focuses on individual entity, +thus simply adapting RTE models cannot capture +such correlations to derive the event structure. 2) +Inter-event Argument Correlations. Concretely, +the event arguments involved in a cause/effect +event pair usually show semantic correlations for +causality deduction. As shown in Figure 1, event +Pricing_Rise which occurs in “worldwide” +Region could imply event Cost_Rise in “inter- +national” Region. It demonstrates that the inter- +event argument correlations not only provide im- +portant clues to decide causality, but also benefit +reliable cause/effect event extraction with mutual +confirmation between the cause-effect pair. +In this paper, we propose an effective approach +named DualCor, which explores both the intra- +event and inter-event argument Correlations with +a dual grid tagging scheme for ECE. Specifically, +DualCor contains two grid tagging tables regarding +event types and the input sentence, to respectively +derive the event structures for cause and effect +events. In each table, DualCor extracts structured +event arguments according to different event types, +naturally considering intra-event argument correla- +tions. Further, when predicting the event arguments +in the cause/effect table, DualCor also predicts their +corresponding effect/cause event arguments, serv- +ing as auxiliary arguments to promote inter-event +argument correlations. By confirming the auxiliary +arguments in the other table, DualCor matches re- +liable cause-effect event pairs as predictions. To +realize the above dual grid tagging scheme, we +further devise a type-aware encoder, which refines +textual representations with essential event type +information to enhance argument prediction. We +conduct the dual grid tagging on the type-aware tex- +tual representations to derive the final cause-effect +event pair. Overall, our main contributions include: +(1) To promote the understanding to event causal- +ity, we formulate a new task named Event Causality +Extraction (ECE), which succeeds ECI to push for- +ward the research of event causality understanding. +(2) We propose a novel approach, DualCor, to +exploit the intra-event and inter-event argument +correlations for ECE, and present it as a baseline +to inspire the following research. +(3) Experiments1 on the ECE dataset reflect the +effectiveness of DualCor, and extensive analyses +show potential research directions for future works. +2 +Related Works +This paper explores a novel ECE task, which aims +to extract the cause-effect event pairs with struc- +tured event information from plain texts. Existing +event-causality-related researches mostly focus on +event causality identification, which predicts the +causality for the previously given event pairs. They +can be roughly categorized into three groups: (1) +Early works exploit the linguistic features (Riaz +and Girju, 2013; Gao et al., 2019), causal pat- +terns (Hu et al., 2017; Do et al., 2011) and sta- +tistical causal associations (Riaz and Girju, 2014) +to explore the causality between events. (2) Recent +researchers (Liu et al., 2020; Cao et al., 2021; Zuo +et al., 2021a,b, 2020) pay the major focus on in- +corporating external knowledge for causality iden- +tification with limited training data. (3) Different +from works above which conduct ECI within a sin- +gle sentence, Tran Phu and Nguyen (2021) focus on +document-level ECI, where the given events scat- +ter in multiple sentences. Despite of their success, +they all suffer from the issues of event structure +missing and causality component missing. To +our best knowledge, ECE is the first to simultane- +ously derive the structured event information and +explicit causality component, which could better +support the downstream applications. +Other than ECE, Relational Triple Extraction +(RTE) (Yu et al., 2020; Cong et al., 2022) has +the similar task formulation, the ideas of which +1Dataset and source code for implementation are available +here: https://github.com/cuishiyao96/ECE + +could actually be adapted for ECE. Concretely, +RTE detects entity pairs in a sentence and predicts +pre-defined relation types between them. Exist- +ing approaches for RTE could be roughly catego- +rized into two lines. (1) Traditional joint methods, +which solve RTE through sequential interrelated +steps via task decomposition (Wei et al., 2020; Yu +et al., 2020; Cong et al., 2020) or sequence gen- +eration (Zeng et al., 2018; Nayak and Ng, 2020). +Unfortunately, these methods all suffer from the +exposure bias (Wang et al., 2020) problem due to +the gap from training to inference between multiple +steps. (2) Unified joint methods, which simultane- +ously derive the triplet entities and relations in one- +stage without cascading between steps, and is thus +free from the exposure bias. These methods solve +RTE in either a sequence-labeling manner (Zheng +et al., 2017) or grid-filling manner (Wang et al., +2021b, 2020). However, the entity-centric RTE +methods seem to struggle for the event-centric task, +since events present more complicated argument +correlations either intra- and inter events. +3 +Task Formulation +Event causality extraction (ECE) aims to derive the +cause-effect event pairs from plain texts. Here, a +cause-effect event pair contains a Cause component +and an Effect component, where each component +denotes an event with a specific event type and its +event arguments with their event roles. Given a +piece of text, an event causality extraction system +is required to predict all the cause-effect event pairs +from it as Figure 1 shows. +4 +Dual Grid Tagging Scheme +This section introduces our proposed dual grid tag- +ging scheme for the ECE, including the tagging +scheme and its decoding strategy. The specific +model implementation is introduced in Section. 5. +4.1 +Tagging Scheme +In general, we construct two grid tagging tables +respectively for the cause/effect events, where each +table extracts all the possible events occurring in +the sentence. Formally, given an n-token sentence +and m predefined event types, we construct two +m×n grid tables for the cause and effect events re- +spectively. As shown in Figure 2, each row denotes +arguments within the same event type, while each +column denotes tags assigned to the token in the +sentence based on the event type. For each entry +3 +4 +1 2 +3 +4 +11 +12 +7 +8 +Prices of +agri. products ... nationwide ... corn seeds ... corn planting across the country +Frost +... +Price +Rise +... +Flood +(a) Grid tagging for in the cause table. +7 8 +9 +10 +5 +6 +1 +2 +Prices +of +agri. products ... nationwide ... corn seeds ... corn planting across the country +Frost +... +Profit +Dec +... +Flood +(b) Grid tagging in the effect table. +1 +Intra-Region-S +2 +Intra-Region-E +3 +Intra-Product-S +4 +Intra-Product-E +5 +Intra-Industry-S +6 +Intra-Industry-E +7 +Inter-Region-S +8 +Inter-Region-E +9 +Inter-Product-S +10 +Inter-Product-E +11 +Inter-Industry-S +12 +Inter-Industry-E +Tags Map +(c) Tags map. +Figure 2: Tagging scheme illustration with the sentence +“Prices of agricultural products rose, but the nationwide +soaring prices of corn seeds decreased the profit in corn +planting across the country.”, where the boundary-field +is short as “S” and “E”. +in the tables, we fill it with a tag in the form of +{Cor-Rol-Bdy} consisting of three fields, namely +correlation-field, role-field and boundary-field: +(1) For the boundary-field: Bdy ∈ {Sta, End}, +we devise it to denote start and end position of +argument spans. For example in Figure 2(a), we +match the argument “corn seeds” by matching the +Cor-Rol-Sta and Cor-Rol-End tags. +(2) For the role-field: Rol∈{Roli}i (i for role in- +dex), we devise it to denote the event role for each +argument in an event, thus constituting an event +structure. For example in Figure 2(a), we decide the +argument “corn seeds” as a Product-role argu- +ment in Price_Rising-type event based on its +Cor-Product-Bdy tag in Price_Rising row. +(3) For correlation-field: Cor∈{Intra, Inter}, +we devise it to denote event argument correlations +in the cause-effect event pair. Specifically, Intra +denotes the arguments belonging to the same event +in the one causality component, while Inter de- +notes the arguments in the other causality com- +ponent. For example, when predicting the cause +event in the cause table, we predict not only the +cause arguments (marked with Intra) with cause +event type, but also the potential effect arguments +(marked with Inter) as auxiliary arguments for +mutual confirmation in causality pair matching. As + +Figure 2(a) shows, we not only predict argument +“corn seeds” with Intra for the Price_Rise- +type cause event, but also predict “corn planting” +with Inter tag as effect event argument. By match- +ing argument “corn planting” with Intra tag in the +effect table, we can derive a Price_Rise-typed +and Profit_Declination-typed event pair. +Building upon the tagging scheme, the model +can naturally extract causality event pairs with their +arguments. Besides, the scheme learns event argu- +ments for each type within separate type row, allow- +ing the model to consider intra-event argument cor- +relations with type-specific information. Moreover, +the tagging scheme enforces the model to extract +arguments in one causality component perceiving +arguments in the other causality component, thus +capturing inter-event argument correlations. +4.2 +Decoding Strategy +Based upon the tagging scheme, we introduce the +decoding strategy for the tagging results. Specifi- +cally, we decompose the process into three steps, +including argument span decoding, event structure +decoding and causality pair decoding. Appendix A +also provides figure illustration to these three steps. +Step 1. Argument span decoding. To derive ar- +gument spans for cause/effect events, we adopt +the nearest start-end match principle (Wei et al., +2020). Specifically, for entry tags having the same +correlation-field and role-field in the same row, we +match the start position to the nearest end posi- +tion according to the position-field to obtain candi- +date argument spans. For example in Figure 2(a), +this step ought to predict “agriculture products”, +“nationwide”, “corn seeds”, “corn planting” and +“across the country” as candidate argument spans. +Step 2. Event structure decoding. To derive +event structure for cause/effect events, we collect +candidate argument spans attached to the same +event type. Specifically, we merge the event ar- +guments with correlation-field Intra belonging +to the same row, resulting in structured candi- +date events. For example in Figure 2(a), given +the candidate argument spans in Step 1, this step +ought to select “agriculture products”, “nationwide” +and “corn seeds” with Intra tags as the candidate +Price_Rising-type cause event arguments. +Step 3. Causality pair decoding. To derive +causality pairs, we match inter-event correlated ar- +guments between candidate cause and effect events. +Specifically, we search the arguments co-occurring +[CLS] e1 [M1] ... [SEP] Prices... rose,but the ... country [SEP] +Encoding Layer +Grid Representation Layer +Grid Representation Layer +Decoding +Decoding +Causality +Event Type +Event Roles +Product +Region +Industry +Cause +Price Rise +corn seeds +nationwide +None +Effect +Profit Declination +None +across the country +corn planting +Figure 3: A toy illustration to our model architecture. +in both event tables simultaneously associating +with correlation-field Intra and Inter, and then +confirm cause-effect event arguments. For exam- +ple in Figure 2(a), given the candidate event ar- +guments in Step 2, this step ought to select “na- +tionwide” and “corn seeds” as the true cause event +arguments, since there also exist “nationwide” and +“corn seeds” with Inter tags in the effect tables +(Figure 2(b)). Similarly, this step also selects “corn +planting”, “across the country” as the arguments in +the Profit_Declination-type effect events. +Accordingly, it predicts the Price_Rise-type +cause and Profit_Declination-type effect +event pair as Figure 3 shows. Note that though +“agriculture product” is also an event argument can- +didate of a Price_Rise-type event in Step 2, it is +not included in the causality pair due to the absence +of Inter correlation in the effect table. +5 +Model +In this section, we introduce the model architecture +to implement DualCor as Figure 3 shows. +5.1 +Encoding Layer +This layer derives the contextualized representa- +tions of tokens in the sentence and event types. To +facilitate the following event argument prediction, +we intend to conduct event type-aware encoding +which refines textual representations with event +type information. Specifically, we concatenate the +event types ahead of the sentence, and employ +BERT (Devlin et al., 2019) for encoding thanks +to its deep self-attention architectures (Vaswani +et al., 2017). Supposing that a text consisting of +n tokens {t1, t2, ..., tn} and m predefined event +types {e1, e2, ..., em} are given, the input sequence + +is organized in the form as follows: +[CLS] e1 [M1] e2 [M2]... em [Mm] [SEP]t1 ... tn [SEP] +(1) +where [Mj] is the marker for the jth event types ej. +We feed the input sequence into the encoder and +use the output representations H = h1, h2, ..., hn +corresponding to the sentence as token represen- +tations. Then, we gather representations of event +type markers as event type representations, which +is denoted as E = e1, e2, ..., em. +5.2 +Grid Representation Layer +This section first details the function for producing +entry representations, and then introduces how to +apply it in both grid tables. +5.2.1 +Semantic Fusion Function +Each entry in the grid respectively models the re- +lation between one token and an event type for +event argument deduction. For an entry connecting +the jth event type ej and ith token in the sentence, +its representation gj,i could be obtained via a fu- +sion function φ by integrating the semantics of ti +and ej as gj,i = φ(ej, hi). Intuitively, φ could be +achieved in various semantic fusion ways includ- +ing concatenation or addition. Considering that +the same event argument span could play different +role in different event types (Sheng et al., 2021), +the decision of event arguments are conditioned on +the event type. Hence, φ should imply the condi- +tional dependency between event types and tokens. +Accordingly, we adopt Conditional Layer Normal- +ization (CLN) (Su, 2019) to implement φ. CLN +is mostly based on the Layer Normalization (Ba +et al., 2016), but it dynamically computes the gain +γ and bias β based on the prior condition instead +of directly deploying them as learnable parameters +in neural networks. Given the event type represen- +tation ej as condition and a token representation +hi, the fusion function φ is achieved via CLN as: +φ(ej, hi) = CLN(ej, hi) = γj ⊙ (hi − µi +σi +) + βj, +γj = Wγej + bγ, βj = Wβej + bβ, +(2) +where µi ∈ R and σi ∈ R are the mean and stan- +dard variance taken across the elements of hi, and +γj ∈ Rd and βj ∈ Rd are respectively the condi- +tional gain and bias. In this way, the event type in- +formation is expressed as conditional information, +and is thus integrated with token representations. +5.2.2 +Grid Representation +We employ two semantic fusion functions, φc, φr, +to respectively derive entry representations for the +cause and effect grid table . Each semantic fusion +function is implemented by a layer of CLN, and +thus the entry representation is obtained as: +gc +j,i = φc(ej, hi) = CLNc(ej, hi), +gr +j,i = φr(ej, hi) = CLNr(ej, hi), +(3) +where gc +j,i, gr +j,i are respectively the entry represen- +tation in the cause and effect table for grid tagging. +5.3 +Training and Inference +Since multiple tags could be simultaneously as- +signed towards (ej, ti) in each table, we conduct +multi-label classification upon entry representa- +tions. Specifically, a fully-connected network pre- +dicts the probability of each tag for (ej, ti) as: +pI +j,i = sigmoid(gI +j,iWI + bI) +(4) +where I ∈ {c, r} is the symbol of grid field de- +noting the cause and effect grid table respectively, +and each dimension of pI +j,i denotes the probability +for a tag between (ej, ti). Consequently, we adapt +Cross-Entropy loss as the loss function: +LI = − +m +� +j=1 +n +� +i=1 +� +k∈C +I(yI +ji = k)log(pI +j,i[k]), +(5) +where C is the set of predefined tags, pI +j,i[k] ∈ +[0, 1] is the predicted probability of tag k between +(ej, ti) and yI +ji is the ground truth tag between +(ej, ti). I is a switching function which equals +to 1 when yI +ji = k, otherwise 0. Following equa- +tion 5, we obtain losses from both grid tables, and +aggregate them for the final training objective: +J (θ) = Lc + Lr. +(6) +For inference, pI +j,i is converted into tags whose +probability overweights the scalar threshold τ I ∈ +[0, 1], which is a manually tuned hyper-parameter. +6 +Experiments +6.1 +Dataset and Evaluation +Dataset We conduct experiments on the cor- +pus (Tianchi, 2021) released by China Conference +on Knowledge Graph and Semantic Computing +2021 (CCKS2021). The corpus comes from the + +public news and reports, containing 7,000 sen- +tences with an average length of 104 tokens. It an- +notates 15,816 events containing 7908 event causal- +ity pairs, covering 39 types of events and 3 types +of event roles, namely Product, Region and +Industry. To adapt this corpus into ECE task, +we divide the corpus into training/validation/test +set based on Cause-Effect event types. Specifically, +CCKS2021 is divided into training/validation/test +set with the proportion of 8 : 1 : 1. We rename the +split dataset as ECE-CCKS. +Evaluation We evaluate our model using Precision +(P), Recall (R) and Micro-F1 (F1) of three metrics. +(1) Event Argument Extraction (EAE) Metric +evaluates the model’s ability to extract event ar- +guments of interests. Like prior works +(Yang +et al., 2019), an argument is correctly predicted +when its event type, span and event role simulta- +neously match the gold label. (2) Cause-Effect +Type (CET) Metric measures whether both the +predicted cause and effect event type match the +golden answer. (3) Event Causality Extraction +(ECE) Metric synthesizes the above two metrics, +where an argument in ECE is correctly extracted +when its predicted cause-effect event type, span +and event role simultaneously meet the gold label. +6.2 +Implementation Details +We employ BERTbase (Devlin et al., 2019) as the +encoder for our model and baselines. For DualCor, +we manually tune all the hyper-parameters on the +validation set. AdamW with learning rate of 3e- +5 is adopted for model optimization. The model +is trained 10 epoches with batch size of 8. The +max length of sentence is 150 by padding shorter +sentences and cutting longer ones. The threshold +τ c, τ r are both set as 0.5. +6.3 +Baselines +We employ a variety of baselines which could be +classified into two streams. +Event-then-Causality methods. These methods +first extract events from texts and then classify the +causal relation. For event extraction, we choose +three typical models. (1)BERT-Softmax (Devlin +et al., 2019) adopts BERT to learn textual represen- +tations, and conducts sequence labeling for event +extraction; (2) BERT-CRF utilizes conditional ran- +dom field (CRF) to capture label dependencies +upon the textual representations (Du and Cardie, +2020). (3) DMBERT (Wang et al., 2019) adopts +dynamic multi-pooling (Chen et al., 2015) upon +BERT to aggregate features for event extraction. +(4) PLMEE (Yang et al., 2019) further adopts role- +specific argument tagger upon BERT to solve the +argument overlapping issue. After obtaining the +events, we enumerate all possible cause-effect pairs +and follow Zuo et al. (2021b) to build a Multilayer +Perceptron classifier to decide the causality. +Event-with-Causality methods. Instead of sepa- +rately deriving events and causality, these meth- +ods conduct event extraction with the causality +taken into consideration and thus simultaneously +derive the events and causality pair. To do this, we +adapt three typical RTE methods as follows. (1) +Novel-tagging designs a unified label space com- +bining causality component (cause/effect), event +types, event roles and argument boundary, and con- +ducts ECE via sequence-labeling following Zheng +et al. (2017). (2) CasECE, which is inspired by +CasRel (Wei et al., 2020), first extracts the cause +event, conditioned on which to derive the effect +event. (3) Pair-linking works in a grid tagging +manner following Wang et al. (2020). It first con- +ducts event-type-level pair linking to derive the +cause-effect event-type, which is then used as con- +ditional information for token-pair linking to de- +rive event arguments. Appendix B provides details +about how we adapt these methods for ECE. +6.4 +Main Results +We report the overall results in Table 1, and have +observations as follows. +(1) The event-then-causality baselines generally +produce weak performances, especially on the Pre- +cision indicator. The reason lies in that these meth- +ods extract events without considering the interest +of causality. As a result, many causality-unrelated +events are wrongly extracted, which would confuse +the causality decision. +(2) Performances of the event-with-causality +baselines are superior to the event-then-causality +models, since the events are extracted with causal- +ity modeling, thus reducing the number of redun- +dant events. However, their performances are still +barely satisfactory, since the entity-oriented rela- +tion modeling strategy could not sufficiently to ex- +plore intra- and inter- correlations between events. +(3) DualCor achieves the best results among all +baselines, we attribute this to that our designed +dual grid tagging schema effectively explore the +intra- and inter-event argument correlations. De- +spite of this, the overall ECE performance is far + +EAE(%) +CET(%) +ECE(%) +P +R +F1 +P +R +F1 +P +R +F1 +BERT-softmax+Causality +32.55 +35.11 +33.78 +49.61 +64.20 +55.97 +30.47 +31.52 +30.99 +BERT-CRF+Causality +35.52 +34.10 +34.79 +53.22 +60.95 +56.82 +31.02 +31.28 +31.15 +DMBERT+Causality +34.27 +38.18 +36.12 +52.87 +63.20 +57.58 +30.08 +34.93 +32.33 +PLMEE+Causality +34.22 +40.70 +37.18 +58.11 +60.20 +59.13 +29.98 +41.14 +34.69 +Novel-tagging +59.40 +28.47 +38.49 +49.79 +61.70 +55.11 +51.52 +26.75 +35.22 +CasECE +36.88 +36.72 +36.80 +58.26 +59.70 +58.97 +31.30 +41.81 +35.80 +Pair-tagging +47.08 +46.49 +46.79 +55.78 +62.95 +59.14 +39.24 +47.69 +43.05 +DualCor +58.05 +47.60 +52.31 +61.75 +58.19 +59.92 +48.56 +44.85 +46.63 +Table 1: Overall results. The Wilcoxons test shows significant difference (p<0.05) between DualCor and baselines. +Overall +Single subset +Multi subset +20 +30 +40 +50 +ECE F1 score (%) +DualCor +Pair-linking +PLMEE+Cau +Figure 4: ECE performances on overall test set, Single +and Multi subset. Appendix D shows detailed values. +from satisfactory. This reflects that ECE requires +investigations from future works to improve it. +6.5 +Single pair vs. Multi pairs +We notice that nearly 10% sentences in our dataset +express multiple event causality pairs, and thus +probe how the number of causality pairs influ- +ences the ECE performance. Specifically, we di- +vide the test set into a Single subset where each +sentence contains only one event causality pair, +otherwise, Multi subset. +Apart from DualCor, +PLMEE+Causality (PLMEE+Cau in short) and +Pair-linking are chosen as representatives for com- +pared baselines, and we present their performances +in Figure 4. Reading from the figure, we could +see that (1) all models present a decreasing per- +formance from Single to Multi subset, reflecting +that ECE towards multiple causality pairs is much +tricky. (2) Reasons for the weak performance on +the Multi subset may be that the increasing number +of causality pairs come from the increase of men- +tioned events, which demands more complicated +inter-event correlations modeling (Sheng et al., +2022). (3) Since the performance on the Multi sub- +set is obviously inferior to the overall and Single +subset performances, we argue that Multi-pairs +could be one great challenge which deserves inves- +tigation from future ECE works. +Method +EAE +CET +ECE +DualCor +52.50 +61.60 +47.58 +w/o Intra Cor +20.47 +14.38 +10.37 +w/o Inter Cor +48.57 +56.82 +43.36 +w/o type-aware encoding +47.69 +56.00 +43.16 +φ → Concatenation +51.39 +59.56 +45.88 +φ → Addition +51.96 +61.08 +46.83 +Table 2: Ablation Study (F1%) on the validation set . +Appendix E illustrates ablation on the test set. +7 +Analysis and Discussion +7.1 +Ablation Study +To study how each module contributes to the per- +formance, we ablate to DualCor on the validation +set as Table 2 shows. +We probe the argument correlations via ablation +to the tagging scheme. (1) w/o Intra-event argu- +ment correlations (Intra-Cor): To explore the neces- +sity of Intra-Cor, we remove tags whose correlation- +field are Intra in the tagging scheme. This leads to +the sharp performance drops since Intra-Cor is the +key to derive individual event from each grid. (2) +w/o Inter-event argument correlations (Inter-Cor): +To certify the effectiveness of Inter-Cor, we remove +tags whose correlation-field are Inter. Without +Inter-cor, the causality pairs are obtained by ex- +haustive enumeration between the cause and effect +event which are individually derived from two ta- +bles. The ECE performance declines 4.42%, reflect- +ing the importance of Inter-Cor. (3) We observe +that the removing of either type of tags would hurt +performances, verifying that these two correlations +are both beneficial and functional for ECE. +We explore the influence of the model architec- +ture via ablation to the encoding and grid represen- +tation layer. (1) w/o type-aware encoding: Instead +of the collaborative encoding as Equation 1 shows, + +Category +Example +Wrong Cause- +Effect Type +Instance#1: The falling of stainless steel prices was caused by the drop in the cost of pure nickel. +Gold: {Event typecause: Cost Declination, event typeeffect: Price Declination } +Predicted: {Event typecause: Price Declination, event typeeffect: Price Declination } +Redundant +Arguments +Instance#2: Feed prices rise across the country, reducing the profits in poultry industry. +Gold: {Regioncause: across the country, Regioneffect: None } +Predicted: {Regioncause: across the country, Regioneffect: across the country } +Missing +Arguments +Instance#3: 50% of coke enterprises in Shanxi, Ningxia and 30% of those in Inner Mongolia have +restricted their production, for which the coke output decreased. +Gold: {Regionreason: Shanxi, Ningxia, Inner Mongolia } +Predicted: {Regionreason: Shanxi, Ningxia } +Table 3: Error analysis, where we only present the associated event types and event arguments due to the space +limitation. Appendix F provides the complete event causality pair for these three instances. +#Para. +Training +Inference +DualCor +107.10M +18.6sents/s +38.9sents/s +Pair-linking +107.63M +6.4sents/s +19.4sents/s +Table 4: Efficiency comparison. +when we encode sentence using BERT while ob- +tain event type embeddings by random initializa- +tion, the final performance declines by 4.42%. This +manifests the importance of capturing semantic de- +pendency between event types and each sentence. +(2) φ → Concatenation or Addition: To explore the +impact of the semantic fusion function φ in Sec- +tion 5.2.1, we respectively replace CLN as concate- +nation and addition. The performance degradation +upon two variants signifies that CLN could bet- +ter enhance token representations with event types, +producing more expressive entry representations. +7.2 +Efficiency Discussion +Since Pair-Linking also works in a grid tagging +manner and achieves the comparable performance +with DualCor, we discuss the efficiency of these +two architectures from two aspects: parameter +amount and running speed. For the sake of fairness, +we run them on the same GPU server. Reading +from Table 4, we notice that the amount of pa- +rameters of our model and Pair-Linking is roughly +equal. We attribute this to that they both exploits +the same basic encoding and grid representations +learning strategy. However, we observe that the +training and inference speeds of our model are re- +spectively about 2.91 and 2.01 times faster than +Pair-Linking. This is mainly because that the rep- +resentation learning for two grids are carried in +parallel in our model, while those of Pair-Linking +are sequentially conducted. Considering analysis +above, we could conclude that our model also main- +tains efficiency advantage over Pair-linking. +7.3 +Error Analysis +To probe the drawbacks of DualCor and promote +future works, we conduct error analysis towards +100 randomly selected failure instances. Here, we +discuss three typical error types as Table 3 shows. +(1) Wrong Cause-Effect Type refers to predicting +the wrong combination of cause-effect event types +as Instance#1. This error can severely hurt the fi- +nal performances, since event arguments under the +wrong cause-effect type would be regarded as false +positive in ECE. We notice that almost 40% error +cases of DualCor belong to this type, while that +of Pair-linking is 32%. We attribute this to that +our method mainly focus on correlations between +event arguments, which lacks exact cause-effect +modeling between event types, while the event- +type-level pair linking in Pair-linking accounts for +this. (2) Redundant Arguments denotes that the +model predicts an argument which actually does +not exist, as the redundant region for effect event +in Instance#2. This kind of errors usually appear +between the cause and effect event upon the same +event role, which demonstrates the difficulty of de- +ducing causality-specific event arguments. Though +redundant arguments accounts for nearly 30% er- +ror cases of DualCor, it is almost 10% lower than +that of Pair-linking. This reveals the importance of +exploring intra- and inter- event argument correla- +tions to discriminate the cause / effect event argu- +ments. (3) Missing Arguments refers to that the +model fails to predict the existed event argument, +as the missed “Inner Mongolia” in Instance#3. We +observe that it usually occurs for event roles which +contains multiple event arguments, where more +sophisticated modeling of intra-event arguments + +correlations are required. +8 +Conclusion +In this paper, we formulate a new task, Event +Causality Extraction (ECE), which aims to extract +the cause-effect event pairs with structured event +information from plain texts. We propose a method +based on an elaborately devised dual grid tagging +scheme, which explores the intra- and inter-event +argument correlations for the task. Experiment re- +sults prove the effectiveness of our method, and +extensive analyses are conducted to point out sev- +eral promising directions to inspire future works. +Acknowledgements +We would like to thank Bowen Yu and Jiangxia Cao +for helpful discussions, support, and feedback on +earlier versions of this work. We would also like to +thank the anonymous reviewers for their insightful +comments and suggestions. This work is supported +by the National Key Research and Development +Program of China (grant No.2021YFB3100600), +the Strategic Priority Research Program of Chinese +Academy of Sciences (grant No.XDC02040400) +, the Youth Innovation Promotion Association of +CAS (Grant No. 2021153). +References +Lei Jimmy Ba, Jamie Ryan Kiros, and Geoffrey E. Hin- +ton. 2016. Layer normalization. CoRR. +Pengfei Cao, Xinyu Zuo, Yubo Chen, Kang Liu, Jun +Zhao, Yuguang Chen, and Weihua Peng. 2021. +Knowledge-enriched event causality identification +via latent structure induction networks. In Proceed- +ings of the 59th Annual Meeting of the Association +for Computational Linguistics and the 11th Interna- +tional Joint Conference on Natural Language Pro- +cessing (Volume 1: Long Papers), pages 4862–4872, +Online. Association for Computational Linguistics. +CCKS. 2021. Joint extraction of financial events and +causal relations. +Yubo Chen, Liheng Xu, Kang Liu, Daojian Zeng, and +Jun Zhao. 2015. 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Association for Computa- +tional Linguistics. +Xinyu Zuo, Pengfei Cao, Yubo Chen, Kang Liu, Jun +Zhao, Weihua Peng, and Yuguang Chen. 2021b. +LearnDA: Learnable knowledge-guided data aug- +mentation for event causality identification. In Pro- +ceedings of the 59th Annual Meeting of the Associa- +tion for Computational Linguistics and the 11th In- +ternational Joint Conference on Natural Language +Processing (Volume 1: Long Papers), pages 3558– +3571, Online. Association for Computational Lin- +guistics. +Xinyu Zuo, Yubo Chen, Kang Liu, and Jun Zhao. 2020. +KnowDis: Knowledge enhanced data augmentation +for event causality detection via distant supervision. +In Proceedings of the 28th International Conference +on Computational Linguistics, pages 1544–1550, +Barcelona, Spain (Online). International Committee +on Computational Linguistics. + +A +Decoding strategy +Step 1. Argument span decoding. In this stage, +we derive argument spans for cause/effect events +using the nearest start-end match principle (Wei +et al., 2020). Specifically, for those entry tags hav- +ing the same correlation-field and role-field in the +same row, we match the start position to the near- +est end position according to the position-field to +obtain candidate argument spans Figure 5.(a). +Step 2. +Event structure decoding. +In this +stage, we collect candidate argument spans at- +tached to the same event types. Specifically, we +merge the event arguments with correlation-field +Intra belonging to the same row, resulting in struc- +tured candidate events in Figure 5.(b). +Step 3. Causality pair decoding. To derive +causality pairs, we match inter-event correlated ar- +guments between candidate cause and effect events. +Specifically, we first obtain event argument with +correlation-field Inter in each table as Figure 5.(c). +Then, we search the arguments co-occurring in +both event tables simultaneously associating with +correlation-field Intra and Inter, and merge +them to form the cause-effect pair in Figure 5.(d). +Table +Event Type +Event Roles +Product +Region +Industry +Cause(Intra) +Price Rise +agriculture products +corn seeds +nationwide +None +Effect(Intra) +Profit Declination +None +across the country +corn planting +Table +Event Type +Event Roles +Product +Region +Industry +Cause(Inter) +Price Rise +None +across the country +corn planting +Effect(Inter) +Profit Declination +corn seeds +nationwide +None +Causality +Event Type +Event Roles +Product +Region +Industry +Cause +Price Rise +corn seeds +nationwide +None +Effect +Prifit Declination +None +across the country +corn planting +Table +Argument Spans +Cause +agriculture products, corn seeds, nationwide, corn planting, across the country +Effect +corn seeds, nationwide, corn planting, across the country +(a) Argument spans derived from Step 1. +(b) Event candidates derived from Step 2 via Intra field. +(c) Inter-arguments derived via Inter field. +(d) The cause-effect Event pair, which is derived by merging arguments which co- +appear in both event tables with correlation-field Intra and Inter. +Figure 5: Detailed illustration to decoding strategy. +B +Details about adapted baseliens +We detail the adaption of RTE methods to ECE. +(1) Novel-tagging is adapted from +Zheng +et al. (2017). It performs RTE through sequence +labeling with a novel tagging scheme, which com- +bines the label spaces of relation types and relation +roles (subject and object), Similarly, we adopt a uni- +fied label space combining cause/effect, event type, +event roles and argument boundary tag, namely +{Causality-EventType-EventRole-Bdy}. +Given the 2 types of causality components, 39 +predefined event types, 3 predefined event roles +and the Start/End boundary indicator, the capacity +of the unified label space is 2 × 39 × 3 × 2 = 468. +We employ the unified labels to tag the tokens in a +sequence labeling manner with BERT+Softmax. +Note that we only deploy BERT+Softmax for +sequence labeling here, since the joint label space +is too large for BERT-CRF to implement on our +experiment devices. +(2) CasECE is adapted from CasRel (Wei et al., +2020), which conducts RTE by modeling the rela- +tions as functions mapping subject entity to object +entity. Similarly, we regard the causality relation +as the function which maps the cause event to the +effect event. Following CasRel, we first extract the +cause event, and then conditioned on it to derive the +effect event. During this process, PLMEE (Yang +et al., 2019) is utilized as the event extractor. +(3) +Pair-linking +is +adapted +from +TPLinker (Wang et al., 2020), +where RTE +is formulated as a token pair linking problem +which aligns the boundary tokens of entity pairs +under each relation type. Similarly, we intend to +respectively extract event arguments under specific +cause-effect event types. +Specifically, we first +conduct event-type-level pair linking to derive +the cause-effect event types. +Then, we utilize +CLN to refine textual representations enhanced +with cause-effect event type pair information, and +conduct token-pair-linking to extract the event +arguments for the specific cause and effect event. +C +Other encoders +We report performances of DualCor using different +basic encoders in Table 5. +D +Single pair vs. Multi pairs +We detail ECE performances on the overall test, +Single and Multi subset in Table 6. +E +Ablation on the test +We provide ablation study on the test set in Table 7. +F +Error analysis +This section provides the complete instances for +error analysis in Table 8. + +EAE(%) +CET(%) +ECE(%) +P +R +F1 +P +R +F1 +P +R +F1 +DualCorBERTbase +58.05 +47.60 +52.31 +61.75 +58.19 +59.92 +48.56 +44.85 +46.63 +DualCorRobertabase +61.46 +46.29 +52.80 +66.14 +58.19 +61.91 +52.801 +44.89 +48.53 +DualCorRobertalarge +63.44 +50.48 +56.22 +67.52 +62.70 +65.02 +54.67 +49.80 +52.12 +DualCorMacBERT +67.64 +49.19 +56.96 +70.68 +60.95 +65.45 +58.29 +48.02 +52.66 +Table 5: Overall results on the test set. +Overall(%) +Single Subset(%) +Multi Subset(%) +P +R +F1 +P +R +F1 +P +R +F1 +PLMEE+Causality +29.98 +41.14 +34.69 +29.89 +46.42 +36.37 +30.58 +23.76 +26.74 +Pair-linking +39.24 +47.69 +43.05 +40.31 +54.32 +46.28 +32.15 +24.82 +28.03 +DualCor +48.64 +44.85 +46.67 +49.39 +51.56 +50.46 +43.65 +22.72 +29.89 +Table 6: ECE performances on the overall test set, Single subset and Multi subset. +Method +EAE +CET +ECE +DualCor +52.36 +59.96 +46.67 +w/o Intra Cor +19.11 +12.51 +9.32 +w/o Inter Cor +49.29 +55.04 +42.88 +w/o type-aware encoding +47.21 +54.72 +41.79 +φ → Concatenation +50.91 +57.99 +44.04 +φ → Addition +51.52 +58.90 +45.05 +Table 7: Ablation Study: F1% upon the three metrics on the test set. +Category +Example +Wrong Cause- +Effect Type +Instance#1: The falling of stainless steel prices was caused by the drop in the cost of pure nickel. +Gold: {Event typecause: Cost Declination, Event typeeffect: Price Declination, +Productcause: pure nickel, +Producteffect: stainless steel, +Regioncause: None, +Industryeffect: None +Industrycause: None, +Industryeffect: None } +Predicted: {Event typecause: Price Declination, event typeeffect: Price Declination, +Productcause: pure nickel, +Producteffect: stainless steel, +Regioncause: None, +Industryeffect: None +Industrycause: None, +Industryeffect: None } +Redundant +Arguments +Instance#2: Feed prices rise across the country, reducing the profits in poultry industry. +Gold: { Event typecause: Price Rise, +Event typeeffect: Profit Declination, +Productcause: feed, +Producteffect: None, +Regioncause: across the country, +Regioneffect: None, +Industrycause: None, +Industryeffect: poultry industry } +Predicted: { Event typecause: Price Rising, +Event typeeffect: Profit Declination, +Productcause: feed, +Producteffect: None, +Regioncause: across the country, +Regioneffect: across the country, +Industrycause: None, +Industryeffect: poultry industry } +Missing +Arguments +Instance#3: 50% of coke enterprises in Shanxi, Ningxia and 30% of those in Inner Mongolia have +restricted their production, for which the supply of coke output. +Gold: {Event typecause: Production Restriction, +Event typeeffect: Supply Reduction, +Productcause: coke, +Producteffect: coke, +Regioncause: Shanxi, Ningxia, Inner Mongolia, +Regioneffect: None, +Industrycause: None, +Industryeffect: None } +Predicted: {Event typecause: Production Restriction, +Event typeeffect: Supply Reduction, +Productcause: coke, +Producteffect: coke, +Regioncause: Shanxi, Ningxia, +Regioneffect: None, +Industrycause: None, +Industryeffect: None } +Table 8: The complete results of error analysis. + diff --git a/HdFJT4oBgHgl3EQfuS10/content/tmp_files/load_file.txt b/HdFJT4oBgHgl3EQfuS10/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f295174e86a60cb161fd4d3726f6eed97d10c13 --- /dev/null +++ b/HdFJT4oBgHgl3EQfuS10/content/tmp_files/load_file.txt @@ -0,0 +1,876 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf,len=875 +page_content='Event Causality Extraction with Event Argument Correlations Shiyao Cui1,2 Jiawei Sheng1,2 Xin Cong1,2 QuanGang Li1,2∗Tingwen Liu1,2 Jinqiao Shi3,1 1Institute of Information Engineering, Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Beijing, China 2School of Cyber Security, University of Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Beijing, China 3Beijing University of Posts and Telecommunications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Beijing, China {cuishiyao, shengjiawei, congxin, liquangang}@iie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='cn liutingwen@iie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='cn shijinqiao@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='cn Abstract Event Causality Identification (ECI), which aims to detect whether a causality relation exists between two given textual events, is an important task for event causality under- standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' However, the ECI task ignores cru- cial event structure and cause-effect causality component information, making it struggle for downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In this paper, we explore a novel task, namely Event Causality Extraction (ECE), aiming to extract the cause- effect event causality pairs with their struc- tured event information from plain texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The ECE task is more challenging since each event can contain multiple event arguments, posing fine-grained correlations between events to de- cide the cause-effect event pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Hence, we propose a method with a dual grid tagging scheme to capture the intra- and inter-event ar- gument correlations for ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Further, we de- vise a event type-enhanced model architecture to realize the dual grid tagging scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Ex- periments demonstrate the effectiveness of our method, and extensive analyses point out sev- eral future directions for ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 1 Introduction Event causality (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2021) denotes an explicit causal relation between two events, constituting a specific cause-effect event pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' As shown in Figure 1, a causal relation exists between a Price Rise event (The worldwide rise of oil prices) and a Cost Rise event (in- creases the cost of international shipping industry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Understanding such event causality could facilitate various downstream applications including event forecasting (Hashimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2014), intelligent search (Rudnik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2019) and question answer- ing (Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020), which is important for natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In recent years, it has aroused the research in- terest for Event Causality Identification (ECI) (Liu ∗Corresponding Author The worldwide rise of oil prices increases the cost of international shipping industry and stimulates the demand for new energy such as Ammonia fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Event Type Event Roles Product Region Industry (1) Cause Price Rise oil worldwide None Effect Cost Rise None international shipping industry (2) Cause Price Rise oil worldwide None Effect Demand Rise new energy None Ammonia fuel Causality Component Figure 1: Illustration for ECE which takes the raw text as input, and outputs the structured event causality pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2021a,b, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Tran Phu and Nguyen, 2021), which aims to detect whether the causality exists between two given events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Despite of its success, there exist two issues that the ECI task fails to address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 1) Event Structure Missing, where each event in ECI is only expressed using a word or phrase which re- flects its occurrence, but ignores the explicit event type and event arguments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', entities which par- ticipate in the event).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The absence of such event structure would lose valuable clues for understand- ing event causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' As shown in Figure 1, “oil” plays a Product role in a Price Rise-typed cause event, implying a consequent Cost Rise- typed effect event towards “shipping industry”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 2) Causality Component Missing, where ECI only predicts the existence of causality between the given event pairs, ignoring to discriminate the spe- cific cause/effect event causality component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Lim- ited by these issues, ECI insufficiently explores the causality between events, which demands further promotion to the understanding of event causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Motivated by discussion about event causality in CCKS (2021), we therefore formulate a task termed as Event Causality Extraction (ECE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' As Figure 1 shows, ECE aims to end-to-end extract the cause-effect event pairs with structured event information from plain texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Comparing with ECI, ECE illustrates the event causality including the event structure, namely event types and arguments, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='11621v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='CL] 27 Jan 2023 and the specific cause-effect causality component, making it more informative to support the various downstream applications (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Intuitively, ECE could be achieved by succes- sively extracting the structured event and then clas- sifying their causality relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Unfortunately, such a paradigm would easily suffer from the redundant event-pair problem, where the causality-unrelated events would be inevitably extracted, confusing the causality decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Another promising direction is to borrow ideas from relational triple extraction (RTE), which shares the similar task formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' However, comparing with the entity-centric RTE task, the event-centric ECE raises new challenges: 1) Intra-event Argument Correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specif- ically, ECE focuses on event, which is a struc- ture maintaining interactive correlations among its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For example in Figure 1, the ar- gument “new energy” and “Ammonia fuel” in the Demand Rise event have strong semantic cor- relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' While RTE focuses on individual entity, thus simply adapting RTE models cannot capture such correlations to derive the event structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 2) Inter-event Argument Correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Concretely, the event arguments involved in a cause/effect event pair usually show semantic correlations for causality deduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' As shown in Figure 1, event Pricing_Rise which occurs in “worldwide” Region could imply event Cost_Rise in “inter- national” Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' It demonstrates that the inter- event argument correlations not only provide im- portant clues to decide causality, but also benefit reliable cause/effect event extraction with mutual confirmation between the cause-effect pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In this paper, we propose an effective approach named DualCor, which explores both the intra- event and inter-event argument Correlations with a dual grid tagging scheme for ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, DualCor contains two grid tagging tables regarding event types and the input sentence, to respectively derive the event structures for cause and effect events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In each table, DualCor extracts structured event arguments according to different event types, naturally considering intra-event argument correla- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Further, when predicting the event arguments in the cause/effect table, DualCor also predicts their corresponding effect/cause event arguments, serv- ing as auxiliary arguments to promote inter-event argument correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' By confirming the auxiliary arguments in the other table, DualCor matches re- liable cause-effect event pairs as predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' To realize the above dual grid tagging scheme, we further devise a type-aware encoder, which refines textual representations with essential event type information to enhance argument prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We conduct the dual grid tagging on the type-aware tex- tual representations to derive the final cause-effect event pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Overall, our main contributions include: (1) To promote the understanding to event causal- ity, we formulate a new task named Event Causality Extraction (ECE), which succeeds ECI to push for- ward the research of event causality understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) We propose a novel approach, DualCor, to exploit the intra-event and inter-event argument correlations for ECE, and present it as a baseline to inspire the following research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) Experiments1 on the ECE dataset reflect the effectiveness of DualCor, and extensive analyses show potential research directions for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 2 Related Works This paper explores a novel ECE task, which aims to extract the cause-effect event pairs with struc- tured event information from plain texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Existing event-causality-related researches mostly focus on event causality identification, which predicts the causality for the previously given event pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' They can be roughly categorized into three groups: (1) Early works exploit the linguistic features (Riaz and Girju, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2019), causal pat- terns (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Do et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2011) and sta- tistical causal associations (Riaz and Girju, 2014) to explore the causality between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) Recent researchers (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2021a,b, 2020) pay the major focus on in- corporating external knowledge for causality iden- tification with limited training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) Different from works above which conduct ECI within a sin- gle sentence, Tran Phu and Nguyen (2021) focus on document-level ECI, where the given events scat- ter in multiple sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Despite of their success, they all suffer from the issues of event structure missing and causality component missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' To our best knowledge, ECE is the first to simultane- ously derive the structured event information and explicit causality component, which could better support the downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Other than ECE, Relational Triple Extraction (RTE) (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Cong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2022) has the similar task formulation, the ideas of which 1Dataset and source code for implementation are available here: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='com/cuishiyao96/ECE could actually be adapted for ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Concretely, RTE detects entity pairs in a sentence and predicts pre-defined relation types between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Exist- ing approaches for RTE could be roughly catego- rized into two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (1) Traditional joint methods, which solve RTE through sequential interrelated steps via task decomposition (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Cong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020) or sequence gen- eration (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Nayak and Ng, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Unfortunately, these methods all suffer from the exposure bias (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020) problem due to the gap from training to inference between multiple steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) Unified joint methods, which simultane- ously derive the triplet entities and relations in one- stage without cascading between steps, and is thus free from the exposure bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' These methods solve RTE in either a sequence-labeling manner (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2017) or grid-filling manner (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2021b, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' However, the entity-centric RTE methods seem to struggle for the event-centric task, since events present more complicated argument correlations either intra- and inter events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 3 Task Formulation Event causality extraction (ECE) aims to derive the cause-effect event pairs from plain texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Here, a cause-effect event pair contains a Cause component and an Effect component, where each component denotes an event with a specific event type and its event arguments with their event roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Given a piece of text, an event causality extraction system is required to predict all the cause-effect event pairs from it as Figure 1 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 4 Dual Grid Tagging Scheme This section introduces our proposed dual grid tag- ging scheme for the ECE, including the tagging scheme and its decoding strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The specific model implementation is introduced in Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='1 Tagging Scheme In general, we construct two grid tagging tables respectively for the cause/effect events, where each table extracts all the possible events occurring in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Formally, given an n-token sentence and m predefined event types, we construct two m×n grid tables for the cause and effect events re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' As shown in Figure 2, each row denotes arguments within the same event type, while each column denotes tags assigned to the token in the sentence based on the event type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For each entry 3 4 1 2 3 4 11 12 7 8 Prices of agri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' products .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' nationwide .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' corn seeds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' corn planting across the country Frost .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Price Rise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Flood (a) Grid tagging for in the cause table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 7 8 9 10 5 6 1 2 Prices of agri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' products .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' nationwide .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' corn seeds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' corn planting across the country Frost .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Profit Dec .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Flood (b) Grid tagging in the effect table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 1 Intra-Region-S 2 Intra-Region-E 3 Intra-Product-S 4 Intra-Product-E 5 Intra-Industry-S 6 Intra-Industry-E 7 Inter-Region-S 8 Inter-Region-E 9 Inter-Product-S 10 Inter-Product-E 11 Inter-Industry-S 12 Inter-Industry-E Tags Map (c) Tags map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Figure 2: Tagging scheme illustration with the sentence “Prices of agricultural products rose, but the nationwide soaring prices of corn seeds decreased the profit in corn planting across the country.”, where the boundary-field is short as “S” and “E”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' in the tables, we fill it with a tag in the form of {Cor-Rol-Bdy} consisting of three fields, namely correlation-field, role-field and boundary-field: (1) For the boundary-field: Bdy ∈ {Sta, End}, we devise it to denote start and end position of argument spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For example in Figure 2(a), we match the argument “corn seeds” by matching the Cor-Rol-Sta and Cor-Rol-End tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) For the role-field: Rol∈{Roli}i (i for role in- dex), we devise it to denote the event role for each argument in an event, thus constituting an event structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For example in Figure 2(a), we decide the argument “corn seeds” as a Product-role argu- ment in Price_Rising-type event based on its Cor-Product-Bdy tag in Price_Rising row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) For correlation-field: Cor∈{Intra, Inter}, we devise it to denote event argument correlations in the cause-effect event pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, Intra denotes the arguments belonging to the same event in the one causality component, while Inter de- notes the arguments in the other causality com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For example, when predicting the cause event in the cause table, we predict not only the cause arguments (marked with Intra) with cause event type, but also the potential effect arguments (marked with Inter) as auxiliary arguments for mutual confirmation in causality pair matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' As Figure 2(a) shows, we not only predict argument “corn seeds” with Intra for the Price_Rise- type cause event, but also predict “corn planting” with Inter tag as effect event argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' By match- ing argument “corn planting” with Intra tag in the effect table, we can derive a Price_Rise-typed and Profit_Declination-typed event pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Building upon the tagging scheme, the model can naturally extract causality event pairs with their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Besides, the scheme learns event argu- ments for each type within separate type row, allow- ing the model to consider intra-event argument cor- relations with type-specific information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Moreover, the tagging scheme enforces the model to extract arguments in one causality component perceiving arguments in the other causality component, thus capturing inter-event argument correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='2 Decoding Strategy Based upon the tagging scheme, we introduce the decoding strategy for the tagging results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifi- cally, we decompose the process into three steps, including argument span decoding, event structure decoding and causality pair decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Appendix A also provides figure illustration to these three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Argument span decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' To derive ar- gument spans for cause/effect events, we adopt the nearest start-end match principle (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, for entry tags having the same correlation-field and role-field in the same row, we match the start position to the nearest end posi- tion according to the position-field to obtain candi- date argument spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For example in Figure 2(a), this step ought to predict “agriculture products”, “nationwide”, “corn seeds”, “corn planting” and “across the country” as candidate argument spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Event structure decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' To derive event structure for cause/effect events, we collect candidate argument spans attached to the same event type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, we merge the event ar- guments with correlation-field Intra belonging to the same row, resulting in structured candi- date events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For example in Figure 2(a), given the candidate argument spans in Step 1, this step ought to select “agriculture products”, “nationwide” and “corn seeds” with Intra tags as the candidate Price_Rising-type cause event arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Causality pair decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' To derive causality pairs, we match inter-event correlated ar- guments between candidate cause and effect events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, we search the arguments co-occurring [CLS] e1 [M1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' [SEP] Prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' rose,but the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' country [SEP] Encoding Layer Grid Representation Layer Grid Representation Layer Decoding Decoding Causality Event Type Event Roles Product Region Industry Cause Price Rise corn seeds nationwide None Effect Profit Declination None across the country corn planting Figure 3: A toy illustration to our model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' in both event tables simultaneously associating with correlation-field Intra and Inter, and then confirm cause-effect event arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For exam- ple in Figure 2(a), given the candidate event ar- guments in Step 2, this step ought to select “na- tionwide” and “corn seeds” as the true cause event arguments, since there also exist “nationwide” and “corn seeds” with Inter tags in the effect tables (Figure 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Similarly, this step also selects “corn planting”, “across the country” as the arguments in the Profit_Declination-type effect events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Accordingly, it predicts the Price_Rise-type cause and Profit_Declination-type effect event pair as Figure 3 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Note that though “agriculture product” is also an event argument can- didate of a Price_Rise-type event in Step 2, it is not included in the causality pair due to the absence of Inter correlation in the effect table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 5 Model In this section, we introduce the model architecture to implement DualCor as Figure 3 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='1 Encoding Layer This layer derives the contextualized representa- tions of tokens in the sentence and event types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' To facilitate the following event argument prediction, we intend to conduct event type-aware encoding which refines textual representations with event type information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, we concatenate the event types ahead of the sentence, and employ BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2019) for encoding thanks to its deep self-attention architectures (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Supposing that a text consisting of n tokens {t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', tn} and m predefined event types {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', em} are given, the input sequence is organized in the form as follows: [CLS] e1 [M1] e2 [M2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' em [Mm] [SEP]t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' tn [SEP] (1) where [Mj] is the marker for the jth event types ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We feed the input sequence into the encoder and use the output representations H = h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', hn corresponding to the sentence as token represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Then, we gather representations of event type markers as event type representations, which is denoted as E = e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='2 Grid Representation Layer This section first details the function for producing entry representations, and then introduces how to apply it in both grid tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='1 Semantic Fusion Function Each entry in the grid respectively models the re- lation between one token and an event type for event argument deduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For an entry connecting the jth event type ej and ith token in the sentence, its representation gj,i could be obtained via a fu- sion function φ by integrating the semantics of ti and ej as gj,i = φ(ej, hi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Intuitively, φ could be achieved in various semantic fusion ways includ- ing concatenation or addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Considering that the same event argument span could play different role in different event types (Sheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2021), the decision of event arguments are conditioned on the event type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Hence, φ should imply the condi- tional dependency between event types and tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Accordingly, we adopt Conditional Layer Normal- ization (CLN) (Su, 2019) to implement φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' CLN is mostly based on the Layer Normalization (Ba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2016), but it dynamically computes the gain γ and bias β based on the prior condition instead of directly deploying them as learnable parameters in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Given the event type represen- tation ej as condition and a token representation hi, the fusion function φ is achieved via CLN as: φ(ej, hi) = CLN(ej, hi) = γj ⊙ (hi − µi σi ) + βj, γj = Wγej + bγ, βj = Wβej + bβ, (2) where µi ∈ R and σi ∈ R are the mean and stan- dard variance taken across the elements of hi, and γj ∈ Rd and βj ∈ Rd are respectively the condi- tional gain and bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In this way, the event type in- formation is expressed as conditional information, and is thus integrated with token representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='2 Grid Representation We employ two semantic fusion functions, φc, φr, to respectively derive entry representations for the cause and effect grid table .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Each semantic fusion function is implemented by a layer of CLN, and thus the entry representation is obtained as: gc j,i = φc(ej, hi) = CLNc(ej, hi), gr j,i = φr(ej, hi) = CLNr(ej, hi), (3) where gc j,i, gr j,i are respectively the entry represen- tation in the cause and effect table for grid tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='3 Training and Inference Since multiple tags could be simultaneously as- signed towards (ej, ti) in each table, we conduct multi-label classification upon entry representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, a fully-connected network pre- dicts the probability of each tag for (ej, ti) as: pI j,i = sigmoid(gI j,iWI + bI) (4) where I ∈ {c, r} is the symbol of grid field de- noting the cause and effect grid table respectively, and each dimension of pI j,i denotes the probability for a tag between (ej, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Consequently, we adapt Cross-Entropy loss as the loss function: LI = − m � j=1 n � i=1 � k∈C I(yI ji = k)log(pI j,i[k]), (5) where C is the set of predefined tags, pI j,i[k] ∈ [0, 1] is the predicted probability of tag k between (ej, ti) and yI ji is the ground truth tag between (ej, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' I is a switching function which equals to 1 when yI ji = k, otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Following equa- tion 5, we obtain losses from both grid tables, and aggregate them for the final training objective: J (θ) = Lc + Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (6) For inference, pI j,i is converted into tags whose probability overweights the scalar threshold τ I ∈ [0, 1], which is a manually tuned hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 6 Experiments 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='1 Dataset and Evaluation Dataset We conduct experiments on the cor- pus (Tianchi, 2021) released by China Conference on Knowledge Graph and Semantic Computing 2021 (CCKS2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The corpus comes from the public news and reports, containing 7,000 sen- tences with an average length of 104 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' It an- notates 15,816 events containing 7908 event causal- ity pairs, covering 39 types of events and 3 types of event roles, namely Product, Region and Industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' To adapt this corpus into ECE task, we divide the corpus into training/validation/test set based on Cause-Effect event types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, CCKS2021 is divided into training/validation/test set with the proportion of 8 : 1 : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We rename the split dataset as ECE-CCKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Evaluation We evaluate our model using Precision (P), Recall (R) and Micro-F1 (F1) of three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (1) Event Argument Extraction (EAE) Metric evaluates the model’s ability to extract event ar- guments of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Like prior works (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2019), an argument is correctly predicted when its event type, span and event role simulta- neously match the gold label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) Cause-Effect Type (CET) Metric measures whether both the predicted cause and effect event type match the golden answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) Event Causality Extraction (ECE) Metric synthesizes the above two metrics, where an argument in ECE is correctly extracted when its predicted cause-effect event type, span and event role simultaneously meet the gold label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='2 Implementation Details We employ BERTbase (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2019) as the encoder for our model and baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For DualCor, we manually tune all the hyper-parameters on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' AdamW with learning rate of 3e- 5 is adopted for model optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The model is trained 10 epoches with batch size of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The max length of sentence is 150 by padding shorter sentences and cutting longer ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The threshold τ c, τ r are both set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='3 Baselines We employ a variety of baselines which could be classified into two streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Event-then-Causality methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' These methods first extract events from texts and then classify the causal relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For event extraction, we choose three typical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (1)BERT-Softmax (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2019) adopts BERT to learn textual represen- tations, and conducts sequence labeling for event extraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) BERT-CRF utilizes conditional ran- dom field (CRF) to capture label dependencies upon the textual representations (Du and Cardie, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) DMBERT (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2019) adopts dynamic multi-pooling (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2015) upon BERT to aggregate features for event extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (4) PLMEE (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2019) further adopts role- specific argument tagger upon BERT to solve the argument overlapping issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' After obtaining the events, we enumerate all possible cause-effect pairs and follow Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2021b) to build a Multilayer Perceptron classifier to decide the causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Event-with-Causality methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Instead of sepa- rately deriving events and causality, these meth- ods conduct event extraction with the causality taken into consideration and thus simultaneously derive the events and causality pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' To do this, we adapt three typical RTE methods as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (1) Novel-tagging designs a unified label space com- bining causality component (cause/effect), event types, event roles and argument boundary, and con- ducts ECE via sequence-labeling following Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) CasECE, which is inspired by CasRel (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020), first extracts the cause event, conditioned on which to derive the effect event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) Pair-linking works in a grid tagging manner following Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' It first con- ducts event-type-level pair linking to derive the cause-effect event-type, which is then used as con- ditional information for token-pair linking to de- rive event arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Appendix B provides details about how we adapt these methods for ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='4 Main Results We report the overall results in Table 1, and have observations as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (1) The event-then-causality baselines generally produce weak performances, especially on the Pre- cision indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The reason lies in that these meth- ods extract events without considering the interest of causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' As a result, many causality-unrelated events are wrongly extracted, which would confuse the causality decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) Performances of the event-with-causality baselines are superior to the event-then-causality models, since the events are extracted with causal- ity modeling, thus reducing the number of redun- dant events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' However, their performances are still barely satisfactory, since the entity-oriented rela- tion modeling strategy could not sufficiently to ex- plore intra- and inter- correlations between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) DualCor achieves the best results among all baselines, we attribute this to that our designed dual grid tagging schema effectively explore the intra- and inter-event argument correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' De- spite of this, the overall ECE performance is far EAE(%) CET(%) ECE(%) P R F1 P R F1 P R F1 BERT-softmax+Causality 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='55 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='11 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='78 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='61 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='20 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='97 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='47 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='52 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='99 BERT-CRF+Causality 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='52 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='10 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='79 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='22 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='95 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='82 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='02 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='28 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='15 DMBERT+Causality 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='27 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='18 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='12 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='87 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='20 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='58 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='08 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='93 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='33 PLMEE+Causality 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='22 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='70 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='18 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='11 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='20 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='13 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='98 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='14 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='69 Novel-tagging 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='40 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='47 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='49 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='79 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='70 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='11 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='52 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='75 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='22 CasECE 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='88 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='72 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='80 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='26 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='70 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='97 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='30 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='81 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='80 Pair-tagging 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='08 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='49 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='79 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='78 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='95 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='14 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='24 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='69 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='05 DualCor 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='05 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='60 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='31 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='75 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='19 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='92 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='56 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='85 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='63 Table 1: Overall results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The Wilcoxons test shows significant difference (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='05) between DualCor and baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Overall Single subset Multi subset 20 30 40 50 ECE F1 score (%) DualCor Pair-linking PLMEE+Cau Figure 4: ECE performances on overall test set, Single and Multi subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Appendix D shows detailed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' from satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' This reflects that ECE requires investigations from future works to improve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='5 Single pair vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Multi pairs We notice that nearly 10% sentences in our dataset express multiple event causality pairs, and thus probe how the number of causality pairs influ- ences the ECE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, we di- vide the test set into a Single subset where each sentence contains only one event causality pair, otherwise, Multi subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Apart from DualCor, PLMEE+Causality (PLMEE+Cau in short) and Pair-linking are chosen as representatives for com- pared baselines, and we present their performances in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Reading from the figure, we could see that (1) all models present a decreasing per- formance from Single to Multi subset, reflecting that ECE towards multiple causality pairs is much tricky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) Reasons for the weak performance on the Multi subset may be that the increasing number of causality pairs come from the increase of men- tioned events, which demands more complicated inter-event correlations modeling (Sheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) Since the performance on the Multi sub- set is obviously inferior to the overall and Single subset performances, we argue that Multi-pairs could be one great challenge which deserves inves- tigation from future ECE works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Method EAE CET ECE DualCor 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='50 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='60 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='58 w/o Intra Cor 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='47 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='38 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='37 w/o Inter Cor 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='57 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='82 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='36 w/o type-aware encoding 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='69 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='00 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='16 φ → Concatenation 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='39 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='56 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='88 φ → Addition 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='96 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='08 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='83 Table 2: Ablation Study (F1%) on the validation set .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Appendix E illustrates ablation on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 7 Analysis and Discussion 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='1 Ablation Study To study how each module contributes to the per- formance, we ablate to DualCor on the validation set as Table 2 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We probe the argument correlations via ablation to the tagging scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (1) w/o Intra-event argu- ment correlations (Intra-Cor): To explore the neces- sity of Intra-Cor, we remove tags whose correlation- field are Intra in the tagging scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' This leads to the sharp performance drops since Intra-Cor is the key to derive individual event from each grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) w/o Inter-event argument correlations (Inter-Cor): To certify the effectiveness of Inter-Cor, we remove tags whose correlation-field are Inter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Without Inter-cor, the causality pairs are obtained by ex- haustive enumeration between the cause and effect event which are individually derived from two ta- bles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The ECE performance declines 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='42%, reflect- ing the importance of Inter-Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) We observe that the removing of either type of tags would hurt performances, verifying that these two correlations are both beneficial and functional for ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We explore the influence of the model architec- ture via ablation to the encoding and grid represen- tation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (1) w/o type-aware encoding: Instead of the collaborative encoding as Equation 1 shows, Category Example Wrong Cause- Effect Type Instance#1: The falling of stainless steel prices was caused by the drop in the cost of pure nickel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Gold: {Event typecause: Cost Declination, event typeeffect: Price Declination } Predicted: {Event typecause: Price Declination, event typeeffect: Price Declination } Redundant Arguments Instance#2: Feed prices rise across the country, reducing the profits in poultry industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Gold: {Regioncause: across the country, Regioneffect: None } Predicted: {Regioncause: across the country, Regioneffect: across the country } Missing Arguments Instance#3: 50% of coke enterprises in Shanxi, Ningxia and 30% of those in Inner Mongolia have restricted their production, for which the coke output decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Gold: {Regionreason: Shanxi, Ningxia, Inner Mongolia } Predicted: {Regionreason: Shanxi, Ningxia } Table 3: Error analysis, where we only present the associated event types and event arguments due to the space limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Appendix F provides the complete event causality pair for these three instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' #Para.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Training Inference DualCor 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='10M 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='6sents/s 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='9sents/s Pair-linking 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='63M 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='4sents/s 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='4sents/s Table 4: Efficiency comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' when we encode sentence using BERT while ob- tain event type embeddings by random initializa- tion, the final performance declines by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='42%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' This manifests the importance of capturing semantic de- pendency between event types and each sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) φ → Concatenation or Addition: To explore the impact of the semantic fusion function φ in Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='1, we respectively replace CLN as concate- nation and addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' The performance degradation upon two variants signifies that CLN could bet- ter enhance token representations with event types, producing more expressive entry representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='2 Efficiency Discussion Since Pair-Linking also works in a grid tagging manner and achieves the comparable performance with DualCor, we discuss the efficiency of these two architectures from two aspects: parameter amount and running speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' For the sake of fairness, we run them on the same GPU server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Reading from Table 4, we notice that the amount of pa- rameters of our model and Pair-Linking is roughly equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We attribute this to that they both exploits the same basic encoding and grid representations learning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' However, we observe that the training and inference speeds of our model are re- spectively about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='91 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='01 times faster than Pair-Linking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' This is mainly because that the rep- resentation learning for two grids are carried in parallel in our model, while those of Pair-Linking are sequentially conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Considering analysis above, we could conclude that our model also main- tains efficiency advantage over Pair-linking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='3 Error Analysis To probe the drawbacks of DualCor and promote future works, we conduct error analysis towards 100 randomly selected failure instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Here, we discuss three typical error types as Table 3 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (1) Wrong Cause-Effect Type refers to predicting the wrong combination of cause-effect event types as Instance#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' This error can severely hurt the fi- nal performances, since event arguments under the wrong cause-effect type would be regarded as false positive in ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We notice that almost 40% error cases of DualCor belong to this type, while that of Pair-linking is 32%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We attribute this to that our method mainly focus on correlations between event arguments, which lacks exact cause-effect modeling between event types, while the event- type-level pair linking in Pair-linking accounts for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) Redundant Arguments denotes that the model predicts an argument which actually does not exist, as the redundant region for effect event in Instance#2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' This kind of errors usually appear between the cause and effect event upon the same event role, which demonstrates the difficulty of de- ducing causality-specific event arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Though redundant arguments accounts for nearly 30% er- ror cases of DualCor, it is almost 10% lower than that of Pair-linking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' This reveals the importance of exploring intra- and inter- event argument correla- tions to discriminate the cause / effect event argu- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) Missing Arguments refers to that the model fails to predict the existed event argument, as the missed “Inner Mongolia” in Instance#3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We observe that it usually occurs for event roles which contains multiple event arguments, where more sophisticated modeling of intra-event arguments correlations are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 8 Conclusion In this paper, we formulate a new task, Event Causality Extraction (ECE), which aims to extract the cause-effect event pairs with structured event information from plain texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We propose a method based on an elaborately devised dual grid tagging scheme, which explores the intra- and inter-event argument correlations for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Experiment re- sults prove the effectiveness of our method, and extensive analyses are conducted to point out sev- eral promising directions to inspire future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Acknowledgements We would like to thank Bowen Yu and Jiangxia Cao for helpful discussions, support, and feedback on earlier versions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We would also like to thank the anonymous reviewers for their insightful comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' This work is supported by the National Key Research and Development Program of China (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='2021YFB3100600), the Strategic Priority Research Program of Chinese Academy of Sciences (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='XDC02040400) , the Youth Innovation Promotion Association of CAS (Grant No.' metadata={'source': 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In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 5284–5294, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Association for Compu- tational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Bowen Yu, Zhenyu Zhang, Xiaobo Shu, Tingwen Liu, Yubin Wang, Bin Wang, and Sujian Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Joint extraction of entities and relations based on a novel decomposition strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In ECAI 2020 - 24th European Conference on Artificial Intelligence, 29 August-8 September 2020, Santiago de Compostela, Spain, August 29 - September 8, 2020 - Includ- ing 10th Conference on Prestigious Applications of Artificial Intelligence (PAIS 2020), volume 325 of Frontiers in Artificial Intelligence and Applications, pages 2282–2289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' IOS Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Xiangrong Zeng, Daojian Zeng, Shizhu He, Kang Liu, and Jun Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Extracting relational facts by an end-to-end neural model with copy mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 506–514, Melbourne, Aus- tralia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Suncong Zheng, Feng Wang, Hongyun Bao, Yuexing Hao, Peng Zhou, and Bo Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Joint extrac- tion of entities and relations based on a novel tag- ging scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In Proceedings of the 55th Annual Meeting of the Association for Computational Lin- guistics (Volume 1: Long Papers), pages 1227–1236, Vancouver, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Xinyu Zuo, Pengfei Cao, Yubo Chen, Kang Liu, Jun Zhao, Weihua Peng, and Yuguang Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Improving event causality identification via self-supervised representation learning on external causal statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, pages 2162–2172, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Xinyu Zuo, Pengfei Cao, Yubo Chen, Kang Liu, Jun Zhao, Weihua Peng, and Yuguang Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' LearnDA: Learnable knowledge-guided data aug- mentation for event causality identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In Pro- ceedings of the 59th Annual Meeting of the Associa- tion for Computational Linguistics and the 11th In- ternational Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 3558– 3571, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Association for Computational Lin- guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Xinyu Zuo, Yubo Chen, Kang Liu, and Jun Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' KnowDis: Knowledge enhanced data augmentation for event causality detection via distant supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In Proceedings of the 28th International Conference on Computational Linguistics, pages 1544–1550, Barcelona, Spain (Online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' International Committee on Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' A Decoding strategy Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Argument span decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In this stage, we derive argument spans for cause/effect events using the nearest start-end match principle (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, for those entry tags hav- ing the same correlation-field and role-field in the same row, we match the start position to the near- est end position according to the position-field to obtain candidate argument spans Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Event structure decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' In this stage, we collect candidate argument spans at- tached to the same event types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, we merge the event arguments with correlation-field Intra belonging to the same row, resulting in struc- tured candidate events in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Causality pair decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' To derive causality pairs, we match inter-event correlated ar- guments between candidate cause and effect events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, we first obtain event argument with correlation-field Inter in each table as Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Then, we search the arguments co-occurring in both event tables simultaneously associating with correlation-field Intra and Inter, and merge them to form the cause-effect pair in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Event Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Event Roles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Product ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Industry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Cause(Intra) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Price Rise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='agriculture products ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='corn seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='nationwide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Effect(Intra) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Profit Declination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='across the country ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='corn planting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Event Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Event Roles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Product ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Industry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Cause(Inter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Price Rise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='across the country ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='corn planting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Effect(Inter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Profit Declination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='corn seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='nationwide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Causality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Event Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Event Roles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Product ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Industry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Cause ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Price Rise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='corn seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='nationwide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Effect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Prifit Declination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='across the country ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='corn planting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Argument Spans ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='Cause ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='agriculture products,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' corn seeds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' nationwide,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' corn planting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' across the country Effect corn seeds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' nationwide,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' corn planting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' across the country (a) Argument spans derived from Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (b) Event candidates derived from Step 2 via Intra field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (c) Inter-arguments derived via Inter field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (d) The cause-effect Event pair, which is derived by merging arguments which co- appear in both event tables with correlation-field Intra and Inter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Figure 5: Detailed illustration to decoding strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' B Details about adapted baseliens We detail the adaption of RTE methods to ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (1) Novel-tagging is adapted from Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' It performs RTE through sequence labeling with a novel tagging scheme, which com- bines the label spaces of relation types and relation roles (subject and object), Similarly, we adopt a uni- fied label space combining cause/effect, event type, event roles and argument boundary tag, namely {Causality-EventType-EventRole-Bdy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Given the 2 types of causality components, 39 predefined event types, 3 predefined event roles and the Start/End boundary indicator, the capacity of the unified label space is 2 × 39 × 3 × 2 = 468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' We employ the unified labels to tag the tokens in a sequence labeling manner with BERT+Softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Note that we only deploy BERT+Softmax for sequence labeling here, since the joint label space is too large for BERT-CRF to implement on our experiment devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (2) CasECE is adapted from CasRel (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020), which conducts RTE by modeling the rela- tions as functions mapping subject entity to object entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Similarly, we regard the causality relation as the function which maps the cause event to the effect event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Following CasRel, we first extract the cause event, and then conditioned on it to derive the effect event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' During this process, PLMEE (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2019) is utilized as the event extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' (3) Pair-linking is adapted from TPLinker (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=', 2020), where RTE is formulated as a token pair linking problem which aligns the boundary tokens of entity pairs under each relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Similarly, we intend to respectively extract event arguments under specific cause-effect event types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Specifically, we first conduct event-type-level pair linking to derive the cause-effect event types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Then, we utilize CLN to refine textual representations enhanced with cause-effect event type pair information, and conduct token-pair-linking to extract the event arguments for the specific cause and effect event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' C Other encoders We report performances of DualCor using different basic encoders in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' D Single pair vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Multi pairs We detail ECE performances on the overall test, Single and Multi subset in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' E Ablation on the test We provide ablation study on the test set in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' F Error analysis This section provides the complete instances for error analysis in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' EAE(%) CET(%) ECE(%) P R F1 P R F1 P R F1 DualCorBERTbase 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='05 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='60 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='31 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='75 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='19 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='92 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='56 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='85 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='63 DualCorRobertabase 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='46 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='29 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='80 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='14 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='19 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='91 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='801 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='89 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='53 DualCorRobertalarge 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='44 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='48 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='22 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='52 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='70 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='02 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='67 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='80 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='12 DualCorMacBERT 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='64 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='19 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='96 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='68 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='95 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='45 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='29 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='02 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='66 Table 5: Overall results on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Overall(%) Single Subset(%) Multi Subset(%) P R F1 P R F1 P R F1 PLMEE+Causality 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='98 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='14 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='69 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='89 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='42 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='37 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='58 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='76 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='74 Pair-linking 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='24 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='69 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='05 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='31 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='32 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='28 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='15 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='82 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='03 DualCor 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='64 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='85 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='67 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='39 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='56 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='46 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='65 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='72 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='89 Table 6: ECE performances on the overall test set, Single subset and Multi subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Method EAE CET ECE DualCor 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='36 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='96 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='67 w/o Intra Cor 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='11 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='51 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='32 w/o Inter Cor 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='29 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='04 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='88 w/o type-aware encoding 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='21 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='72 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='79 φ → Concatenation 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='91 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='99 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='04 φ → Addition 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='52 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='90 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content='05 Table 7: Ablation Study: F1% upon the three metrics on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Category Example Wrong Cause- Effect Type Instance#1: The falling of stainless steel prices was caused by the drop in the cost of pure nickel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Gold: {Event typecause: Cost Declination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Event typeeffect: Price Declination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Productcause: pure nickel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Producteffect: stainless steel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Regioncause: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industryeffect: None Industrycause: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industryeffect: None } Predicted: {Event typecause: Price Declination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' event typeeffect: Price Declination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Productcause: pure nickel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Producteffect: stainless steel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Regioncause: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industryeffect: None Industrycause: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industryeffect: None } Redundant Arguments Instance#2: Feed prices rise across the country,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' reducing the profits in poultry industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Gold: { Event typecause: Price Rise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Event typeeffect: Profit Declination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Productcause: feed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Producteffect: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Regioncause: across the country,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Regioneffect: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industrycause: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industryeffect: poultry industry } Predicted: { Event typecause: Price Rising,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Event typeeffect: Profit Declination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Productcause: feed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Producteffect: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Regioncause: across the country,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Regioneffect: across the country,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industrycause: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industryeffect: poultry industry } Missing Arguments Instance#3: 50% of coke enterprises in Shanxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Ningxia and 30% of those in Inner Mongolia have restricted their production,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' for which the supply of coke output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Gold: {Event typecause: Production Restriction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Event typeeffect: Supply Reduction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Productcause: coke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Producteffect: coke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Regioncause: Shanxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Ningxia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Inner Mongolia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Regioneffect: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industrycause: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industryeffect: None } Predicted: {Event typecause: Production Restriction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Event typeeffect: Supply Reduction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Productcause: coke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Producteffect: coke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Regioncause: Shanxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Ningxia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Regioneffect: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industrycause: None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} +page_content=' Industryeffect: None } Table 8: The complete results of error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfuS10/content/2301.11621v1.pdf'} diff --git a/LNAzT4oBgHgl3EQfkP3w/content/tmp_files/2301.01530v1.pdf.txt b/LNAzT4oBgHgl3EQfkP3w/content/tmp_files/2301.01530v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..faec90ffe24d67c5f67763aa5d4dc428012d58dc --- /dev/null +++ b/LNAzT4oBgHgl3EQfkP3w/content/tmp_files/2301.01530v1.pdf.txt @@ -0,0 +1,2671 @@ +arXiv:2301.01530v1 [math.OC] 4 Jan 2023 +Nonlinear conjugate gradient methods: worst-case convergence +rates via computer-assisted analyses∗ +Shuvomoy Das Gupta†, Robert M. Freund‡, Xu Andy Sun§, Adrien Taylor¶ +Abstract +In this paper, we propose a computer-assisted approach to the analysis of the worst-case convergence +of nonlinear conjugate gradient methods (NCGMs). Those methods are known for their generally good +empirical performances for large-scale optimization, while having relatively incomplete analyses. Using +this approach, we establish novel complexity bounds for the Polak-Ribière-Polyak (PRP) and the Fletcher- +Reeves (FR) NCGMs for smooth strongly convex minimization. Conversely, we provide examples showing +that those methods might behave worse than the regular steepest descent on the same class of problems. +1 +Introduction +We consider the standard unconstrained convex minimization problem +f⋆ ≜ min +x∈Rn f(x), +(1) +where f is L-smooth (i.e., it has an L-Lipschitz gradient) and µ-strongly convex. We study the worst-case +performances of a few famous variants of nonlinear conjugate gradient methods (NCGMs) for solving (1). +More specifically, we study Polak-Ribière-Polyak (PRP) [1, 2] and Fletcher-Reeves (FR) [3] schemes with +exact line search. With exact line search, many other NCGMs such as the Hestenes and Stiefel method [4], +the conjugate descent method due to Fletcher [5], and the Dai and Yuan method [6] reduce to either PRP +or FR. Under exact line search, PRP and FR can be presented in the following compact form: +γk ∈ argmin +γ +f(xk − γ dk), +xk+1 = xk − γkdk, +βk = ∥∇f(xk+1)∥2 − η ⟨∇f(xk+1); ∇f(xk)⟩ +∥∇f(xk)∥2 +, +dk+1 = ∇f(xk+1) + βkdk, +(M) +where PRP and FR are respectively obtained by setting η = 1 and η = 0. NCGMs have a long history (see, +e.g., the nice survey [7]), but are much less studied compared to their many first-order competitors. For +instance, even though FR is generally considered the first NCGM [7, §1], we are not aware of non-asymptotic +convergence results for it. Still, some variants are known for their generally good empirical behaviors (which +∗R. M. Freund acknowledges support by AFOSR Grant No. FA9550-22-1-0356. +A. Taylor acknowledges support from +the European Research Council (grant SEQUOIA 724063). +This work was partly funded by the French government under +management of Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference ANR-19-P3IA- +0001 (PRAIRIE 3IA Institute). +†Operations Research Center, Massachusetts Institute of Technology. Email: sdgupta@mit.edu. +‡Sloan School of Management, Massachusetts Institute of Technology. Email: rfreund@mit.edu. +§Sloan School of Management, Massachusetts Institute of Technology. Email: sunx@mit.edu. +¶INRIA, École Normale Supérieure, CNRS, PSL Research University, Paris. adrien.taylor@inria.fr. +1 + +we illustrate on Figure 1) with little of them being backed-up by classical complexity analyses. In this work, +we apply the performance estimation approach [8, 9] to (M) for filling this gap by explicitly computing some +worst-case convergence properties of PRP and FR. +0 +200 +400 +600 +800 +1,000 +10−4 +10−3 +10−2 +10−1 +100 +iterations +f(x) − f∗ +0 +200 +400 +600 +800 +1,000 +10−9 +10−6 +10−3 +100 +iterations +Gradient +Nesterov +Nesterov (SC version) +Optimized gradient +FR +PRP +Figure 1: Convergence of a few first-order methods on a logistic regression problem on the small-sized +Sonar dataset [10]. Experiments with normalized features (zero mean and unit variance). Left: without +regularization. Right: with an ℓ2 regularization of parameter 10−4. All methods were featured with an +exact line search: (i) gradient descent, (ii) Nesterov’s accelerated gradient [11] (exact line search instead of +backtracking), (iii) Nesterov’s accelerated method for strongly convex problems, version [12, Algorithm 28] +with exact line search instead of the gradient step, (iv) optimized gradient descent [13, Algorithm (OGM- +LS)], (v) FR, and (vi) PRP. No method was tuned, the results correspond to the first run for each method +and are only meant for illustrative purposes. +1.1 +Contributions +The contribution of this paper is twofold. First, we compute worst-case convergence bounds and counter- +examples for PRP and FR. Those bounds are obtained by formulating the problems of computing worst-case +scenarios as nonconvex quadratically constrained quadratic optimization problems (QCQPs) and then by +solving them to global optimality. +Second, these computations also allow us to construct mathematical +proofs that establish an improved non-asymptotic convergence bound for PRP, and, to the best of our +knowledge, the first non-asymptotic convergence bound for FR. Furthermore, the worst-case bounds for +PRP and FR obtained numerically show that there are simple adversarial examples on which those methods +do not behave better than gradient descent with an exact line search (GDEL), thus leaving very few room +for improvements on this class of problems. +From a methodological point of view, the approach of computing worst-case scenarios and bounds through +optimization is often referred to as performance estimation. In many situations, those problems are amenable +to convex semidefinite programs [8, 9, 14], but it is generally not the case for adaptive first-order methods +such as PRP and FR. For studying those methods, we evaluate the worst-case performances of (M) by +solving nonconvex QCQPs, extending the more standard SDP-based approach from [8, 9, 14] developed +for non-adaptive methods. This contribution is similar in spirit with that in [15] which was developed for +devising optimal (but non-adaptive) first-order methods. +Organization. +The paper is organized as follows. In Section 2, we establish non-asymptotic convergence +rates for PRP and FR by viewing the search direction dk in (M) as an approximate gradient direction. In +Section 3, we compute the exact numerical values of the worst-case f(xN)−f⋆/f(x0)−f⋆ and f(xk+N)−f⋆/f(xk)−f⋆ +for PRP and FR by formulating the problems as nonconvex QCQPs and then solving them to certifiable +global optimality using a custom spatial branch-and-bound algorithm. +2 + +1.2 +Related works +Conjugate gradient (CG) methods are particularly popular choices for solving systems of linear equations +and quadratic minimization problems; in this context, they are known to be information-optimal in the class +of first-order methods [16, Chapter 12 & Chapter 13] or [17, Chapter 5]. There are many extensions beyond +quadratics, commonly referred to as nonlinear conjugate gradient methods (NCGMs). They are discussed +at length in the textbooks [18, Chapter 5 & Chapter 7] and [19, Chapter 5] and in the nice survey [7]. In +particular, when exact line searches are used, many variants become equivalent and can be seen as instances +of quasi-Newton methods, see [18, Chapter 7, §“Relationship with conjugate gradient methods”] or [19, +Chapter 5, §5.5]. +For instance, it is well known that standard variants such as Hestenes-Stiefel [4] and +Dai-Yuan [6] are equivalent to (M) when exact line searches are used, while being different in the presence +of more popular line search procedures (such as Wolfe’s [18, Chapter 3]). Beyond quadratics, obtaining +convergence guarantees is often reduced to the problem of ensuring the search direction to be a descent +direction, see for instance [17, §5.5 “Extensions to non-quadratic problems”] or [20, 21]. Without exact line +searches, even when f is strongly convex, there are counter-examples showing that even popular variants +may not generate descent directions [22]. Note that NCGMs are often used together with restart strategies, +which we do not consider here; see, e.g., [23] and the references therein. +In this work, we use the performance estimation framework [8, 9, 14]. This methodology is essentially +mature for analyzing “fixed-step” (i.e., non-adaptive) first-order methods (and for methods whose analyses +are amenable to those of fixed-step methods), whose stepsizes are essentially chosen in advance. This type of +methods include many common first-order methods and operator splitting schemes, including the heavy-ball +method [24] and Nesterov’s accelerated gradient [11, 25]. Only very few adaptive methods were studied using +the PEP methodology, namely gradient descent with exact line searches [26], greedy first-order methods [13], +and Polyak stepsizes [27]. A premise to the study of NCGMs using PEPs was done in [28, §4.5.2]. This work +is also closely related in spirit with the technique developed in [15] for optimizing coefficients of fixed-step +first-order methods using nonconvex optimization. +1.3 +Preliminaries +In this short section, we recall the definition and a result on smooth strongly convex functions, as well as a +base result on steepest descent with an exact line search. +We use the standard notation ⟨ · ; · ⟩ : Rn × Rn → R to denote the Euclidean inner product, and the +corresponding induced Euclidean norm ∥ · ∥. The class of L-smooth µ-strongly convex functions is standard +and can be defined as follows. +Definition 1.1. Let f : Rn → R be a proper, closed, and convex function, and consider two constants +0 ⩽ µ < L < ∞. The function f is L-smooth and µ-strongly convex (notation f ∈ Fµ,L(Rn)), if +• (L-smooth) for all x, y ∈ Rn, it holds that f(x) ⩽ f(y) + ⟨∇f(y); x − y⟩ + L +2 ∥x − y∥2, +• (µ-strongly convex) for all x, y ∈ Rn, it holds that f(x) ⩾ f(y) + ⟨∇f(y); x − y⟩ + µ +2 ∥x − y∥2. +We simply denote f ∈ Fµ,L when the dimension is either clear from the context or unspecified. We also +denote by q ≜ µ +L the inverse condition number. For readability, we do not explicitly treat the (trivial) case +L = µ. +Smooth strongly convex functions satisfy many inequalities, see e.g., [29, Theorem 2.1.5]. For the devel- +opments below, we need only one specific inequality characterizing functions in Fµ,L. The following result +can be found in [9, Theorem 4] and is key in our analysis. +Theorem 1.1. [9, Theorem 4, Fµ,L-interpolation] Let I be an index set and S = {(xi, gi, fi)}i∈I ⊆ Rn × +Rn × R be a set of triplets. There exists f ∈ Fµ,L satisfying f(xi) = fi and ∇f(xi) = gi for all i ∈ I if and +only if +fi ⩾ fj+⟨gj; xi − xj⟩ + 1 +2L∥gi − gj∥2 + +µ +2(1 − µ/L)∥xi − xj − 1 +L(gi − gj)∥2 +(2) +holds for all i, j ∈ I. +3 + +Another related result from [30, §2.1] that we record next involves constructing a strongly-convex smooth +function from a given set of triplets. +Theorem 1.2. [30, §2.1, strongly convex and smooth extension] Suppose I is a set of indices and S = +{(xi, gi, fi)}i∈I ⊆ Rn ×Rn ×R is a set of triplets such that (2) holds for all i, j ∈ I for some 0 ⩽ µ < L < ∞. +Then the function f : Rn → R defined by +f(y) = max +α∈∆ +�L +2 ∥y∥2 − L − µ +2 +∥y − +1 +L − µ +� +i∈I +αi(gi − µxi)∥2 ++ +� +i∈I +αi +� +fi + +1 +2(L − µ)∥gi − Lxi∥2 − L +2 ∥xi∥2�� +(3) +where ∆ = {α ∈ Rn | α ⩾ 0, �n +i=1 αi = 1}, satisfies f ∈ Fµ,L(Rn), f(xi) = fi and ∇f(xi) = gi for all i ∈ I. +Finally, consider a function f ∈ Fµ,L and the approximate steepest descent method: +γk = argmin +γ +f(xk − γdk) +xk+1 = xk − γkdk, +(4) +where the search direction dk satisfies a relative accuracy criterion: +∥∇f(xk) − dk∥ ⩽ ǫ∥∇f(xk)∥. +(5) +In particular, (5) holds when | sin θ| ⩽ ǫ with θ being the angle between ∇f(xk) and dk. With line searches, +this amounts to checking that dk is a descent direction. We will use the following result in Section 2. +Theorem 1.3. [26, Theorem 5.1] Let f ∈ Fµ,L(Rn), x⋆ ≜ argminx∈Rnf(x) be a minimizer of f, and +f⋆ ≜ f(x⋆). For any xk ∈ Rn and search direction dk satisfying (5), we have: +f(xk+1) − f⋆ ⩽ +�1 − qǫ +1 + qǫ +�2 +(f(xk) − f⋆) , +(6) +where xk+1 is computed as (4) and qǫ ≜ µ(1−ǫ)/L(1+ǫ). +Note that similar results (without line searches) to that of Theorem 1.3 can be found in [31], which might +help in future analyses of NCGMs without line searches. +2 +Base descent properties of NCGMs +In this section, we analyze NCGMs as approximate steepest descent methods through a computer-assisted +approach. Because the NCGMs make use of exact line searches, only the generated search directions matter, +and not their magnitudes. This renders the analysis somehow simpler, and we argue that this is a reasonable +setting for improving the analysis and understanding of NCGMs. +This section builds on the idea that when | sin θk| (where θk is the angle between minus the gradient and +the search direction at iteration k) is upper bounded in an appropriate fashion, one can use Theorem 1.3 for +obtaining convergence guarantees. In particular, we get nontrivial convergence guarantees as soon as θk can +be bounded away from ± π +2 , i.e., sin θk should be bounded away from 1 for ensuring that dk’s are descent +directions. Of course, viewing NCGMs as approximate gradient methods is very adversarial by nature, as it +misses the point that the directions of NCGMs are meant to be better than those of vanilla gradient descent, +while such analyses can only provide worse rates. +Albeit being pessimistic by construction, the analyses of this section are, to the best of our knowledge, +already better than the state-of-the-art bounds for NCGMs. Further, we show in the next sections that there +is actually nearly no room for improving those analyses. +4 + +Properties of NCGMs with exact line search. +Before going into the detailed approach, let us review a +few properties of the iterates of (M). First, note that the exact line search condition γk = argminγf(xk−γdk) +in (M) implies the following equalities: +⟨∇f(xk+1); dk⟩ = 0, +⟨∇f(xk+1); xk − xk+1⟩ = 0, +⟨∇f(xk); dk⟩ = ∥∇f(xk)∥2, +(7) +which we can show as follows. The exact line search condition is equivalent to +0 = [∇γf(xk − γdk)]γ=γk += − ⟨∇f(xk − γkdk); dk⟩ += − ⟨∇f(xk+1); dk⟩ +(8) +thereby obtaining the first line of (7). Then, the definition of xk+1 implies the second equality. The last line +follows from applying the first line to +⟨∇f(xk); dk⟩ = ⟨∇f(xk); ∇f(xk) + βk−1dk−1⟩ = ∥∇f(xk)∥2. +(9) +Combining (9) with ⟨∇f(xk); dk⟩ = ∥∇f(xk)∥∥dk∥ cosθk, we obtain that ∥∇f(xk)∥/∥dk∥ = cos θk, thereby +reaching sin2 θk = 1 − ∥∇f(xk)∥2/∥dk∥2. Thus, any upper bound on the ratio ∥dk∥/∥∇f(xk)∥ can be converted to +a worst-case convergence rate using Theorem 1.3. +Section organization. +For obtaining the desired bounds measuring the quality of the angle θk, Section 2.1 +first frames the problems of computing the worst-case ∥dk∥/∥∇f(xk)∥ for PRP and FR as optimization prob- +lems, referred to as performance estimation problems (PEPs). These PEPs are nonconvex but practically +tractable QCQPs and can be solved numerically to certifiable global optimality using spatial branch-and- +bound algorithms (detailed in Appendix D), which allows (i) to construct “bad” functions on which the +worst-case ∥dk∥/∥∇f(xk)∥ for PRP and FR is achieved, and (ii) to identify closed-form solutions to the PEPs +leading to proofs that can be verified in a standard and mathematically rigorous way. The convergence rates +for PRP and FR are provided and proved in Section 2.2. +2.1 +Computing worst-case search directions +In this section, we formulate the problems of computing the worst-case ratios of ∥dk∥/∥∇f(xk)∥ as optimization +problems. Following a classical steps introduced in [9, 14], we show that it can be cast as a nonconvex QCQP. +For doing that, we assume that at iteration k−1 the NCGM has not reached optimality, so ∇f(xk−1) ̸= 0. +Because ∥∇f(xk−1)∥2 ⩽ ∥dk−1∥2 (follows from applying Cauchy–Schwarz inequality to (9)), without loss of +generality we define the ratio ck−1 ≜ ∥dk−1∥2/∥∇f(xk−1)∥2 where ck−1 ⩾ 1. Then, denoting by ck the worst- +case ratio ∥dk∥2/∥∇f(xk)∥2 arising when applying (M) to the minimization of an L-smooth µ-strongly convex +function, we will compute ck as a function of L, µ, and ck−1. In other words, we use a Lyapunov-type point +of view and take the stand of somewhat forgetting about how dk−1 was generated (except through the fact +that it satisfies (7)). Then, we compute the worst possible next search direction dk that the algorithm could +generate given that dk−1 satisfies a certain quality. Thereby, we obtain an upper bound on the evolution of +the quality of the search directions (quantified by ck) obtained throughout the iterative procedure. Formally, +we compute +ck(µ, L, ck−1) ≜ + + + + + + + + +maximize +f,n,xk−1,dk−1 +xk,dk,βk−1 +∥dk∥2 +∥∇f(xk)∥2 +subject to +n ∈ N, f ∈ Fµ,L(Rn), dk−1, xk−1 ∈ Rn, +xk, dk and βk−1 generated by (M) from xk−1 and dk−1, +⟨∇f(xk−1); dk−1⟩ = ∥∇f(xk−1)∥2, +∥dk−1∥2 = ck−1∥∇f(xk−1)∥2. + + + + + + + + +(10) +5 + +For computing ck(µ, L, ck−1), we reformulate (10) as follows. Denote I ≜ {k − 1, k}. An appropriate +sampling of the variable f (which is inconveniently infinite-dimensional) allows us to cast (10) as: +ck(µ, L, ck−1) = + + + + + + + + + + + + + + + + + + + + + + +maximize +n,{di}i∈I,γk−1,βk−1 +{(xi,gi,fi)}i∈I +∥dk∥2 +∥gk∥2 +subject to +n ∈ N, βk−1 ∈ R, dk−1, dk ∈ Rn, +{(xi, gi, fi)}i∈I ⊂ Rn × Rn × R, +∃f ∈ Fµ,L : +� f(xi) = fi +∇f(xi) = gi +∀i ∈ I, +γk−1 = argmin +γ +f(xk−1 − γ dk−1), +xk = xk−1 − γk−1dk−1, +βk−1 = ∥gk∥2−η⟨gk; gk−1⟩ +∥gk−1∥2 +, +dk = gk + βk−1dk−1, +⟨∇f(xk−1); dk−1⟩ = ∥gk−1∥2, +∥dk−1∥2 = ck−1∥gk−1∥2. + + + + + + + + + + + + + + + + + + + + + + +(11) +Using Theorem 1.1, the existence constraint can be replaced by a set of linear/quadratic inequalities (2) +for all pairs of triplets in {(xi, gi, fi)}i∈I without changing the objective value. Furthermore, if βk−1 and +γk were pre-defined parameters (instead of variables), the problem would be amenable to a convex semidef- +inite program [9, 14]. So, applying Theorem 1.1 to (11) followed by an homogeneity argument and a few +substitutions based on (7), we arrive at: +ck(µ, L, ck−1) = + + + + + + + + + + + + + + + + + + + + + + + + + +maximize +n,{di}i∈I,γk−1,βk−1 +{(xi,gi,fi)}i∈I +∥dk∥2 +subject to +n ∈ N, dk−1, xk−1 ∈ Rn, +fi ⩾ fj + ⟨gj; xi − xj⟩ + +1 +2(1− µ +L ) +� +1 +L∥gi − gj∥2 ++µ∥xi − xj∥2 − 2 µ +L ⟨gi − gj; xi − xj⟩ +� +, +i, j ∈ I, +⟨gk−1; dk−1⟩ = ∥gk−1∥2, +⟨gk; dk−1⟩ = 0, +⟨gk; xk−1 − xk⟩ = 0, +xk = xk−1 − γk−1dk−1, +βk−1 = ∥gk∥2−η⟨gk; gk−1⟩ +∥gk−1∥2 +, +dk = gk + βk−1dk−1 +∥dk−1∥2 = ck−1∥gk−1∥2, +∥gk∥2 = 1. + + + + + + + + + + + + + + + + + + + + + + + + + +(D) +Note that without the variable n this problem is amenable to a nonconvex QCQP (see Appendix B). Fortu- +nately standard arguments (e.g., [9, Theorem 5], or Appendix B) allows setting n = 4 without changing the +optimal value of this problem, thereby discarding this dimension issue. We can then solve (D) to certifiable +global optimality using a custom branch-and-bound algorithm. Reformulation details are provided in Ap- +pendix B, whereas a description of the custom spatial branch-and-bound algorithm is given in Appendix D. +Finally, we recall that numerical solutions to (D) correspond to worst-case functions that can be obtained +through the reconstruction procedure from Theorem 1.2. +In addition, numerical solutions can serve as +inspirations for devising rigorous mathematical proofs, as presented next. +2.2 +Worst-case bounds for PRP and FR +In this section, we provide explicit solutions to (D) for PRP and FR. Those results are then used for deducing +simple convergence bounds through a straightforward application of Theorem 1.3. +6 + +2.2.1 +A worst-case bound for Polak-Ribière-Polyak (PRP) +Solving (D) with η = 1 to global optimality allows obtaining the following worst-case bound for PRP +quantifying the quality of the search direction with respect to the gradient direction. +Lemma 2.1 (Worst-case search direction for PRP). Let f ∈ Fµ,L, and let xk−1, dk−1 ∈ Rn and xk, dk be +generated by the PRP method (i.e., (M) with η = 1). It holds that: +∥dk∥2 +∥∇f(xk)∥2 ⩽ (1 + q)2 +4q +, +(12) +with q ≜ µ/L. Equivalently, ∥dk − ∇f(xk)∥ ⩽ ǫ∥∇f(xk)∥ holds with ǫ = 1−q/1+q. +Proof. Recall that xk = xk−1 − γk−1 dk−1 and dk = ∇f(xk) + βk−1dk−1. The proof consists of the following +weighted sum of inequalities: +• optimality condition of the line search, with weight λ1 = −β2 +k−1 +1+q +Lγk−1q : +⟨∇f(xk); dk−1⟩ = 0, +• smoothness and strong convexity of f between xk−1 and xk, with weight λ2 = +β2 +k−1(1+q)2 +Lγ2 +k−1(1−q)q : +f(xk−1) ⩾f(xk) + ⟨∇f(xk); xk−1 − xk⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥xk−1 − xk − 1 +L(∇f(xk−1) − ∇f(xk))∥2 +=f(xk) + γk−1⟨∇f(xk); dk−1⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥γk−1dk−1 − 1 +L(∇f(xk−1) − ∇f(xk))∥2 +• smoothness and strong convexity of f between xk and xk−1, with weight λ3 = λ2: +f(xk) ⩾f(xk−1) + ⟨∇f(xk−1); xk − xk−1⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥xk−1 − xk − 1 +L(∇f(xk−1) − ∇f(xk))∥2 +=f(xk−1) − γk−1⟨∇f(xk−1), dk−1⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥γk−1dk−1 − 1 +L(∇f(xk−1) − ∇f(xk))∥2 +• definition of βk−1 with weight λ4 = βk−1(1+q) +Lγk−1q : +0 = ⟨∇f(xk−1); ∇f(xk)⟩ − ∥∇f(xk)∥2 + βk−1∥∇f(xk−1)∥2 += ⟨∇f(xk−1); ∇f(xk)⟩ − ∥∇f(xk)∥2 + βk−1⟨∇f(xk−1); dk−1⟩. +We arrive at the following weighted sum: +0 ⩾λ1⟨∇f(xk); dk−1⟩ ++ λ2 +� +f(xk) − f(xk−1) + γk−1⟨∇f(xk); dk−1⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥γk−1dk−1 − 1 +L(∇f(xk−1) − ∇f(xk))∥2 +� ++ λ3 +� +f(xk−1) − f(xk) − γk−1⟨∇f(xk−1); dk−1⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥γk−1dk−1 − 1 +L(∇f(xk−1) − ∇f(xk))∥2 +� ++ λ4 +� +⟨∇f(xk−1); ∇f(xk)⟩ − ∥∇f(xk)∥2 + βk−1⟨∇f(xk−1); dk−1⟩ +� +7 + +which can be reformulated exactly as (expand both expressions and observe that all terms match) +0 ⩾∥dk∥2 − (1 + q)2 +4q +∥∇f(xk)∥2 ++ 4β2 +k−1q +(1 − q)2 +���dk−1 − +1+q +2Lγk−1q∇f(xk−1) + 2βk−1(1+q)−Lγk−1(1−q)2 +4βk−1Lγk−1q +∇f(xk) +��� +2 +, +⩾∥dk∥2 − (1 + q)2 +4q +∥∇f(xk)∥2, +thereby arriving to the desired conclusion. +In Appendix A we numerically showcase the tightness of the worst-case bounds (12) for PRP. By tightness, +we mean that we verified numerically that there exist n ∈ N, functions f ∈ Fµ,L and xk−1, dk−1 ∈ Rn such +that ∥dk∥2 = (1+q)2/4q∥∇f(xk)∥2. This is done by exhibiting feasible points to (D) (obtained by solving (D) +numerically for η = 1) for different values of the inverse condition number q and ck−1. Those feasible points +were verified through other (independent) software [32, 33]. +The following rate is a direct consequence of Lemma 2.1 and Theorem 1.3. Perhaps surprisingly, the +following guaranteed convergence rate for PRP corresponds to that of gradient descent with an exact line +search (Theorem 1.3 with ǫ = 0) when the condition number is squared. +Theorem 2.1 (Worst-case bound for PRP). Let f ∈ Fµ,L, and xk, dk ∈ Rn and xk+1, dk+1 ∈ Rn be +generated by respectively k ⩾ 0 and k + 1 iterations of the PRP method (i.e., (M) with η = 1). It holds that +f(xk+1) − f⋆ ⩽ +�1 − q2 +1 + q2 +�2 +(f(xk) − f⋆) , +with q ≜ µ/L. +Proof. The desired claim is a direct consequence of Theorem 1.3 with ǫ = 1−q +1+q . That is, the PRP scheme +can be seen as a descent method with direction dk satisfying ∥dk − ∇f(xk)∥ ⩽ ǫ∥∇f(xk)∥. +As a take-away from this theorem, we obtained an improved bound on the convergence rate of PRP, +but possibly not in the most satisfying way: this analysis strategy does not allow beating steepest de- +scent. Furthermore, this bound is tight for one iteration assuming that the current search direction satisfies +∥dk∥2/∥∇f(xk)∥2 = (1+q)2/4q. However, it does not specify whether such an angle can be achieved on the same +worst-case instances as those where Theorem 1.3 is achieved. In other words, there might be no worst-case +instances where the bounds (6) and (12) are tight simultaneously, possibly leaving room for improvement in +the analysis of PRP. We show in the sequel that we could indeed slightly improve this bound by taking into +account the history of the method in a more appropriate way. +Remark. The only worst-case complexity result that we are aware of in the context of PRP for smooth +strongly convex problems was provided by Polyak in [1, Theorem 2]: +f(xk+1) − f⋆ ⩽ +q +1 + 1 +q2 +(f(xk) − f⋆) . +This bound is about two times worse compared to the rate achieved by gradient descent (1−q/1+q) when the +condition number is put to the cube. From what we can tell, this is due to two main weaknesses in the proof +of Polyak [1, Theorem 2]: a weaker analysis of gradient descent, and a weaker analysis of the direction (and +in particular that ∥dk∥2/∥∇f(xk)∥2 ⩽ 1 + 1/q2). That is, whereas gradient descent with exact line searches is +guaranteed to achieve an accuracy f(xk) − f⋆ ⩽ ε in O(1/q log 1/ε), our analysis provides an O(1/q2 log 1/ε) +guarantee for PRP, where Polyak’s guarantee for PRP is O(1/q3 log 1/ε). As a reference, note that the lower +complexity bound (achieved by a few methods, including many variations of Nesterov’s accelerated gradients) +is of order O( +� +1/q log 1/ε). +8 + +2.2.2 +A worst-case bound for Fletcher-Reeves (FR) +Similar to the obtaining of the bound for PRP, our bound for FR follows from solving (D) (for η = 0) in +closed-form. We start by quantifying the quality of the search direction with respect to the steepest descent +direction. For doing that, we first establish the following bound on the FR update parameter βk−1. +Lemma 2.2 (Bound on βk−1 for FR). Let f ∈ Fµ,L, and let xk−1, dk−1 ∈ Rn and xk, dk be generated by the +FR method (i.e., (M) with η = 0). For any ck−1 ∈ R such that ∥dk−1∥2/∥∇f(xk−1)∥2 = ck−1, where ck−1 > 1, +it holds that: +0 ⩽ βk−1 ⩽ +1 +ck−1 +� +1 − q + 2 +� +(ck−1 − 1)q +�2 +4q +, +(13) +where q ≜ µ/L. +Proof. First, note that βk−1 ⩾ 0 by definition. The other part of the proof consists of the following weighted +sum of inequalities: +• relation between ∇f(xk−1) and dk−1 with weight λ1 = γk−1(L + µ) − +2√ +βk−1 +√ +(ck−1−1)ck−1 : +0 = ⟨∇f(xk−1); dk−1⟩ − ∥∇f(xk−1)∥2, +• optimality condition of the line search with weight λ2 = +2 +ck−1 − γk−1(L + µ): +0 = ⟨∇f(xk); dk−1⟩ , +• definition of βk−1 with weight λ3 = +√ +ck−1−1 +√ +βk−1ck−1 : +0 = ∥∇f(xk)∥2 − βk−1∥∇f(xk−1)∥2, +• initial condition on the ratio +∥dk−1∥2 +∥∇f(xk−1)∥2 with weight λ4 = −γ2 +k−1Lµ + +√ +βk−1 +ck−1√ +(ck−1−1)ck−1 : +0 = ∥dk−1∥2 − ck−1∥gk−1∥2 +• smoothness and strong convexity of f between xk−1 and xk, with weight λ5 = L − µ: +0 ⩾ − f(xk−1) + f(xk) + ⟨∇f(xk); xk−1 − xk⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥xk−1 − xk − 1 +L(∇f(xk−1) − ∇f(xk))∥2 +=f(xk) + γk−1⟨∇f(xk); dk−1⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥γk−1dk−1 − 1 +L(∇f(xk−1) − ∇f(xk))∥2 +• smoothness and strong convexity of f between xk and xk−1, with weight λ6 = λ5: +0 ⩾ − f(xk) + f(xk−1) + ⟨∇f(xk−1); xk − xk−1⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥xk−1 − xk − 1 +L(∇f(xk−1) − ∇f(xk))∥2 +=f(xk−1) − γk−1⟨∇f(xk−1); dk−1⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥γk−1dk−1 − 1 +L(∇f(xk−1) − ∇f(xk))∥2 +9 + +The weighted sum can be written as: +0 ⩾ λ1 +� +⟨∇f(xk−1); dk−1⟩ − ∥∇f(xk−1)∥2� ++ λ2 [⟨∇f(xk); dk−1⟩] ++ λ3 +� +∥∇f(xk)∥2 − βk−1∥∇f(xk−1)∥2� ++ λ4 +� +∥dk−1∥2 − ck−1∥gk−1∥2� ++ λ5 +� +f(xk) + γk−1⟨∇f(xk); dk−1⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥γk−1dk−1 − 1 +L(∇f(xk−1) − ∇f(xk))∥2� ++ λ6 +� +f(xk−1) − γk−1⟨∇f(xk−1); dk−1⟩ + +1 +2L∥∇f(xk−1) − ∇f(xk)∥2 ++ +µ +2(1−µ/L)∥γk−1dk−1 − 1 +L(∇f(xk−1) − ∇f(xk))∥2� +, +which can be reformulated exactly as (expand the expressions and observe that all terms match): +0 ⩾∥∇f(xk)∥2 − ν(βk−1, γk−1, ck−1, µ, L)∥∇f(xk−1)∥2 ++ +����� +4 +� +βk−1 +(ck−1 − 1)c3 +k−1 +dk−1 − +4 +� +βk−1ck−1 +ck−1 − 1 ∇f(xk−1) + +4 +� +ck−1 − 1 +βk−1ck−1 +∇f(xk) +����� +2 +⩾∥∇f(xk)∥2 − ν(βk−1, γk−1, ck−1, µ, L)∥∇f(xk−1)∥2, +where +ν(βk−1, γk−1, ck−1, µ, L) = 2 +� +1 − +1 +ck−1 +� +βk−1 − ck−1γ2 +k−1Lµ + γk−1(L + µ) − 1. +So, we have: +βk−1 ⩽ ν(βk−1, γk−1, ck−1, µ, L) +⇔ βk−1 − 2 +� +1 − +1 +ck−1 +� +βk−1 ⩽ −ck−1γ2 +k−1Lµ + γk−1(L + µ) − 1 +⇒ βk−1 − 2 +� +1 − +1 +ck−1 +� +βk−1 ⩽ max +γ +� +−ck−1γ2 +k−1Lµ + γk−1(L + µ) − 1 +� +. +Because, −ck−1γ2 +k−1Lµ + γk−1(L + µ) − 1 is a concave function in γk−1, its maximum can be achieved by +differentiating the term with respect to γk−1, equating it to 0, and then solving for γk−1. The corresponding +maximum value is equal to (L+µ)2/4ck−1Lµ−1 and achieved at γk−1 = L+µ/2ck−1Lµ. Hence, the last inequality +becomes: +βk−1−2 +� +1 − +1 +ck−1 +� +βk−1 − (L + µ)2 +4ck−1Lµ + 1 ⩽ 0 +⇔ +�� +βk−1 +�2 +− 2 +� +1 − +1 +ck−1 +� +βk−1 + +�� +1 − +1 +ck−1 +�2 +− (L + µ)2 +4ck−1Lµ − +�� +1 − +1 +ck−1 +�2 ++ 1 ⩽ 0 +⇔ +� +� +βk−1 − +� +1 − +1 +ck−1 +�2 +⩽ (L + µ)2 +4ck−1Lµ + ✁1 − +1 +ck−1 +− ✁1 = +1 +ck−1 +�(L + µ)2 +4Lµ +− 1 +� +⇔ +� +βk−1 ⩽ +� +1 − +1 +ck−1 ++ +� +(L + µ)2 +4ck−1Lµ − +1 +ck−1 +. +10 + +Thereby, squaring both sides (which are nonnegative) of the last inequality and then through some algebra, +we reach +βk−1 ⩽ 1 + (L − µ) +ck−1 +� +(ck−1 − 1) +µL ++ µ2 − 6µL + L2 +4ck−1µL += +1 +ck−1 +� +1 − q + 2 +� +(ck−1 − 1)q +�2 +4q +. +As βk−1 ⩾ 0 by definition, we have thus proven the desired statement. +Next, we prove a bound quantifying the quality of the search directions of FR. +Lemma 2.3 (Worst-case search direction for FR). Let f ∈ Fµ,L, and let xk−1, dk−1 ∈ Rn and xk, dk be +generated by the FR method (i.e., (M) with η = 0). For any ck−1 ∈ R such that ∥dk−1∥2/∥∇f(xk−1)∥2 = ck−1, +where ck−1 > 1, it holds that: +∥dk∥2 +∥∇f(xk)∥2 ⩽ ck ≜ 1 + +� +1 − q + 2 +� +(ck−1 − 1)q +�2 +4q +, +(14) +with q ≜ µ/L. Equivalently, ∥dk − ∇f(xk)∥ ⩽ ǫ∥∇f(xk)∥ holds with ǫ = +� +1 − 1/ck. +Proof. The proof consists of the following weighted sum of inequalities: +• optimality condition of the line search with weight λ1 = 2βk−1: +0 = ⟨∇f(xk); dk−1⟩, +• the quality of the search direction with weight λ2 = β2 +k−1: +0 = ∥dk−1∥2 − ck−1∥∇f(xk−1)∥2, +• definition of βk−1 with weight λ3 = −ck−1βk−1: +0 = ∥∇f(xk)∥2 − βk−1∥∇f(xk−1)∥2. +The weighted sum can be written as +0 ⩾λ1 [⟨∇f(xk); dk−1⟩] + λ2 +� +∥dk−1∥2 − ck−1∥∇f(xk−1)∥2� ++ λ3 +� +−∥∇f(xk)∥2 + βk−1∥∇f(xk−1)∥2� +, +and can be reformulated exactly as +0 ⩾ ∥dk∥2 − (1 + ck−1βk−1)∥∇f(xk)∥2 ⇔ ∥dk∥2 ⩽ (1 + ck−1βk−1)∥∇f(xk)∥2 +⩽ + + +1 + +� +1 − q + 2 +� +(ck−1 − 1)q +�2 +4q + + + ∥∇f(xk)∥2, +where in the last line we have used the upper bound on βk−1 from (13). +Similar to PRP, we compare this last bound with the worst example that we were able to find numerically +(i.e., worst feasible points to (D)) in Appendix A. Thereby, we conclude tightness of the bound on the quality +of the search direction (14). That is, we claim that for all values of q and ck−1, there exist n ∈ N, functions +f ∈ Fµ,L and xk−1, dk−1 ∈ Rn such that the bound from Lemma 2.3 is achieved with equality. +That being said, this bound only allows obtaining unsatisfactory convergence results for FR, although +not letting much room for improvements, as showed in the next sections. +11 + +Theorem 2.2 (Worst-case bound). Let f ∈ Fµ,L, and xk, dk ∈ Rn and xk+1, dk+1 ∈ Rn be generated by +respectively k ⩾ 0 and k + 1 iterations of the FR method (i.e., (M) with η = 0). It holds that +f(xk+1) − f⋆ ⩽ +� +1 − q 1−ǫk +1+ǫk +1 + q 1−ǫk +1+ǫk +�2 +(f(xk) − f⋆) , +with ǫk = +� +(1−q)2(k−1)2/4q+(1−q)2(k−1)2. +Proof. The desired claim is a direct consequence of Theorem 1.3 with Lemma 2.3. Indeed, it follows from +ck ⩽ 1 + +� +1 − µ +L + 2 +� +(ck−1 − 1) µ +L +�2 +4µ +L +(the guarantee from Lemma 2.3 for the quality of the search direction) which we can rewrite as +� +ck+1 − 1 ⩽ 1 − q + 2 +� +(ck − 1)q +2√q +with c0 −1 = 0, thereby arriving to ck ⩽ 1+k2(1−q)2/4q by recursion. For applying Theorem 1.3, we compute +ǫk = +� +1 − 1/ck ⩽ +� +(1−q)2k2/4q+(1−q)2k2 and reach the desired statement. +It is clear that the statement of Theorem 2.2 is rather very disappointing, as the convergence rate of +the FR variation can become arbitrarily close to 1. While this guarantee clearly does not give a total and +fair picture of the true behavior of FR in practice, it seems in line with the practical necessity to effectively +restart the method as it runs [7]. +The next section is devoted to studying the possibilities for obtaining tighter guarantees for PRP and FR +beyond the simple single-iteration worst-case analyses of this section (which are tight for one iteration, but +not beyond), showing that we cannot hope to improve the convergence rates for those methods without +further assumptions on the problems at hand. +3 +Obtaining better worst-case bounds for NCGMs +In the previous section, we established closed-form bounds on ratios between consecutive function values for +NCGMs by characterizing worst-case search directions. Albeit being tight for the analysis of NCGMs for one +iteration, the bounds that we obtained are disappointingly inferior to those of the vanilla gradient descent. +In this section, we investigate the possibility of obtaining better worst-case guarantees for NCGMs. For +doing this using our framework, one natural possibility for us is to go beyond the study of a single iteration +(since our results appear to be tight for this situation). Therefore, in contrast with the previous section, we +now proceed only numerically and provide worst-case bounds without closed-forms. +More precisely, we solve the corresponding PEPs in two regimes. In short, the difference between the +two regimes resides in the type of bounds under consideration. +1. The first type of bounds can be thought to as a “Lyapunov” approach which studies N iterations +of (M) starting at some iterate (xk, dk) (for which we “neglect” how it was generated). In this first +setup, we numerically compute worst-case bounds on f(xk+N)−f⋆/f(xk)−f⋆ for different values of N +(namely N ∈ {1, 2, 3, 4}). As for the results of Section 2, we quantify the quality of the couple (xk, dk) +by requiring that ∥dk∥2 ⩽ ck∥∇f(xk)∥2. When N = 1, this setup corresponds to that of Section 2. +Stemming from the fact the worst-case behaviors observed for N = 1 might not be compatible between +consecutive iterations, we expect the quality of the bounds to improve with N. Of course, the main +weakness of this approach stands in the fact that we neglect how (xk, dk) was generated. +2. As a natural complementary alternative, the second type of bounds studies N iterations of (M) initi- +ated at x0 (with d0 = ∇f(x0)). Whereas the first type of bounds is by construction more conservative, +12 + +it has the advantage of being recursive: it is valid for all k ⩾ 0. On the other side, the second type of +bounds is only valid for the first N iterations (the bound cannot be used recursively), but it cannot be +improved at all. That is, we study exact worst-case ratio f(xN)−f⋆/f(x0)−f⋆ for a few different values +of N (namely N ∈ {1, 2, 3, 4}). In this setup, we obtain worst-case bounds that are only valid close +to initialization. However, it has the advantage of being unimprovable, as we do not neglect how the +search direction is generated. +Section organization. +This section is organized as follows. First, Section 3.1 presents the performance es- +timation problems for (M) specifically for computing the worst-case ratios f(xk+N)−f⋆/f(xk)−f⋆ and f(xN)−f⋆/f(x0)−f⋆. +Then, Section 3.2 and Section 3.3 presents our findings for respectively PRP and FR. Details on how we +managed to solve the resulting nonconvex QCQPs numerically are provided in Appendix C. +3.1 +Computing numerical worst-case scenarios +Similar to (10), the problem of computing the worst-case ratio f(xk+N)−f⋆/f(xk)−f⋆ is framed as the following +nonconvex maximization problem (for c ⩾ 1 and q ≜ µ/L): +ρN(q, c) ≜ + + + + + + + + + + + + +maximize +f,n,{xk+i},{dk+i}i, +{γk+i}i,{βk+i}i +f(xk+N)−f⋆ +f(xk)−f⋆ +subject to +n ∈ N, f ∈ Fq,1(Rn), dk, xk ∈ Rn, +⟨∇f(xk); dk⟩ = ∥∇f(xk)∥2, +∥dk∥2 ⩽ c∥∇f(xk)∥2, + + +xk+1 +dk+1 +βk + + , . . . , + + +xk+N +dk+N +βk+N−1 + + generated by (M) from xk and dk. + + + + + + + + + + + + +(BLyapunov) +We proceed similarly for f(xN)−f⋆/f(x0)−f⋆: +ρN,0(q) ≜ + + + + + + + + + + +maximize +f,n,{xk+i},{dk+i}i, +{γk+i}i,{βk+i}i +f(xN)−f⋆ +f(x0)−f⋆ +subject to +n ∈ N, f ∈ Fq,1(Rn), x0 ∈ Rn, +d0 = ∇f(x0), + + +x1 +d1 +β0 + + , . . . , + + +xN +dN +βN−1 + + generated by (M) from xk and dk. + + + + + + + + + + +(Bexact) +Obviously, ρN(q, c) ⩾ ρN,0(q) for any c ⩾ 1. We solve (BLyapunov) and (Bexact) numerically to high precision +(details in Appendix C) for N ∈ {1, 2, 3, 4} and report the corresponding results in what follows. In the +numerical experiments, we fix the values of c using Lemma 2.1 for PRP in (BLyapunov), thereby computing +ρN +� +q, (1+q)2/4q +� +whose results are provided in Figure 2. +For FR, c can become arbitrarily bad and we +therefore only compute ρN,0(q) via (Bexact). The numerical values for ρN,0(q) respectively PRP and FR are +provided in Figure 3 and Figure 4. The next sections discuss and draw a few conclusions from the numerical +worst-case convergence results for PRP and FR. +3.2 +Improved worst-case bounds for PRP +Figure 2 reports the worst-case values of the “Lyapunov” ratio f(xk+N)−f⋆/f(xk)−f⋆ as a function of the inverse +condition number q ≜ µ/L and for c = (1+q)2/4q and N = 1, 2, 3, 4. This worst-case ratio seem to improve +as N grows, but does not outperform gradient descent with exact line search (GDEL). The diminishing +improvements with N also suggests the worst-case performance of PRP in this regime might not outperform +GDEL even for larger values of N ⩾ 4, albeit probably getting close to the same asymptotic worst-case +convergence rate. +13 + +As a complement, Figure 3 shows how PRP’s worst-case ratio fN −f⋆/f0−f⋆ evolves as a function of q +for N = 1, 2, 3, 4. The worst-case performance of PRP in this setup seems to be similar to that of GDEL. +Further, for small q (which is typically the only regime of interest for large-scale optimization), PRP’s worst- +case performance seems to be slightly better than than of GDEL. On the other hand, for larger q, PRP +performs slightly worse than GDEL. +As a conclusion, we believe there is no hope to prove uniformly better worst-case bounds for PRP +than those for GDEL for base smooth strongly convex minimization. However, we might be able to prove +improvements for small values of q at the cost of possibly very technical proofs. +As for the Lyapunov +approach, the numerical results from this section could be improved by further increasing N, but we believe +that the transient does not suggest this direction to be promising. We recall that we computed the bounds +by solving an optimization problem whose feasible points correspond to worst-case examples. Therefore, the +numerical results provided in this section are backed-up by numerically constructed examples on which PRP +behaves “badly” (more details in Appendix C). +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1 +q +N +� +ρN +� +q, (1+q)2 +4q +� +GDEL : fk+1−f⋆/fk−f⋆ +PRP:N = 1 +PRP:N = 2 +PRP:N = 3 +PRP:N = 4 +(1 − √q)2 +Figure 2: This figure reports the worst-case values for the “Lyapunov” ratio +N� +f(xk+N)−f⋆/f(xk)−f⋆ vs. the +(inverse) condition ratio q ≜ µ +L for PRP. We compute ρN(q, c) with c = (1+q)2/4q for N = 1, 2, 3, 4. As N +increases, the worst-case +N� +fk+N−f⋆/fk−f⋆ improves, but remains worse than that of gradient descent with +exact line search (GDEL). The curve (1 − √q)2 (orange) corresponds to the rate of the lower complexity +bounds for this class of problems [30]. +14 + +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1 +q +N� +ρN,0(q) +GDEL : f1−f⋆ +f0−f⋆ +PRP:N = 1 +PRP:N = 2 +PRP:N = 3 +PRP:N = 4 +(1 − √q)2 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1 +q +N� +ρN,0(q) +GDEL : f1−f⋆ +f0−f⋆ +PRP:N = 1 +PRP:N = 2 +PRP:N = 3 +PRP:N = 4 +(1 − √q)2 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1 +q +N� +ρN,0(q) +GDEL : f1−f⋆ +f0−f⋆ +PRP:N = 1 +PRP:N = 2 +PRP:N = 3 +PRP:N = 4 +(1 − √q)2 +Figure 3: This figure reports the worst-case values for the ratio +N� +fN −f⋆/f0−f⋆ vs. q for PRP for N = 1, 2, 3, 4. +For N = 1, PRP and GDEL perform the same iteration. For N = 2, 3, 4, the worst-case ratio of PRP is +better than that of GDEL for q ⩽ 0.1. The curve (1 − √q)2 (orange) corresponds to the rate of the lower +complexity bounds for this class of problems [30]. +15 + +3.3 +Improved worst-case bounds for FR +Figure 4 reports the worst-case values for the ratio fN −f⋆/f0−f⋆ as a function of q, for N ∈ {1, 2, 3, 4}. +The convergence bounds appears to be marginally better than GDEL for some sufficiently small inverse +condition numbers. +Further, the range of values of q for which there is an improvement appears to be +decreasing with N ⩾ 2. Beyond this range, the worst-case values become significantly worse than that of +GDEL. Though apparently not as dramatic as the worst-case bound from Theorem 2.2, the quality of the +bound appears to be decreasing with N, which stands in line with the practical need to restart the method [7]. +As in the previous section, we recall that those curves were obtained by numerically constructing “bad” +worst-case examples satisfying our assumptions. In other words, there is no hope to obtain better results +without adding assumptions or changing the types of bounds under consideration. +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1 +q +N� +ρN,0(q) +GDEL : f1−f⋆ +f0−f⋆ +FR: N = 1 +FR: N = 2 +FR: N = 3 +FR: N = 4 +(1 − √q)2 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1 +q +N� +ρN,0(q) +GDEL : f1−f⋆ +f0−f⋆ +FR: N = 1 +FR: N = 2 +FR: N = 3 +FR: N = 4 +(1 − √q)2 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1 +q +N� +ρN,0(q) +GDEL : f1−f⋆ +f0−f⋆ +FR: N = 1 +FR: N = 2 +FR: N = 3 +FR: N = 4 +(1 − √q)2 +Figure 4: This figure reports the worst-case values for the ratio +N� +fN −f⋆/f0−f⋆ vs. q for FR for N = 1, 2, 3, 4. +For N = 1, FR and GDEL perform the same iteration. For N = 2, 3, 4, the worst-case bound for FR is +slightly better than that of GDEL for small enough values of q, and gets larger than GDEL for larger values +of q. The range of q for which FR is better than GDEL gets smaller as N ⩾ 2 increases. The curve (1−√q)2 +(orange) corresponds to the rate of the lower complexity bounds for this class of problems [30]. +4 +Conclusion +This works studies the iteration complexity of two variants of nonlinear conjugate gradients, namely the +Polak-Ribière-Polyak (PRP) and the Fletcher-Reeves (FR) methods. We provide new improved complexity +bounds for both those methods, and show that albeit unsatisfying, not much can a priori be gained from a +worst-case perspective, as both method appear to behave similar or worse to regular steepest descent in the +worst-case. Further, those results suggest that explaining the good practical performances of NCGMs might +be out of reach for traditional worst-case complexity analyses on classical classes of problems. +A limitation of this work stands in the fact that only somewhat “ideal” variants of nonlinear conjugate +gradients were considered, as we make explicit use of exact line search procedures. However, there is a +priori no reason to believe that different line search procedures would help avoiding the possibly bad worst- +case behaviors. Further, the performance estimation methodology allows tackling such alternate line search +procedures into account, so the same methodology could be applied for tackling those questions. We let such +investigations for future work. +16 + +Code. +All the numerical results in this paper were obtained on MIT Supercloud Computing Cluster with +Intel-Xeon-Platinum-8260 processor with 48 cores and 128 GB of RAM running Ubuntu 18.04.6 LTS with +Linux 4.14.250-llgrid-10ms kernel [34]. We used JuMP—a domain specific modeling language for mathematical +optimization embedded in the open-source programming language Julia [35]—to model the optimization +problems. To solve the optimization problems, we use the following solvers: Mosek 9.3 [36], KNITRO 13.0.0 +[37], and Gurobi 10.0.0, which are free for academic use. The relative feasibility tolerance and relative +optimality tolerance of all the solvers are set at 1e-6. We validated the “bad” worst-case scenarios produced +by our methodology using the PEPit package [32], which is an open-source Python library allowing to use +the PEP framework. +The codes used to generate and validate the results in this paper are available at: +https://github.com/Shuvomoy/NCG-PEP-code. +References +[1] Boris T. Polyak. The conjugate gradient method in extremal problems. USSR Computational Mathe- +matics and Mathematical Physics, 9(4):94–112, 1969. +[2] Elijah Polak and Gerard Ribiere. Note sur la convergence de méthodes de directions conjuguées. Revue +française d’informatique et de recherche opérationnelle. Série rouge, 3(16):35–43, 1969. +[3] Reeves Fletcher and Colin M. Reeves. Function minimization by conjugate gradients. The computer +journal, 7(2):149–154, 1964. +[4] Magnus R. Hestenes and Eduard Stiefel. Methods of conjugate gradients for solving. Journal of research +of the National Bureau of Standards, 49(6):409, 1952. +[5] Roger Fletcher. 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Table 1 details the organization of this additional material. +Section +Content +Appendix A +Numerical illustration of tightness of the worst-case search direction (12) for PRP +and (14) for FR. +Appendix B +Nonconvex QCQP reformulation of (D). +Appendix C +Nonconvex QCQP reformulation of (BLyapunov) (Appendix C.1). +Nonconvex QCQP reformulation of (Bexact) (Appendix C.2). +The relative gap between the lower bounds and upper bounds (Appendix C.3). +Appendix D +Description of the custom spatial branch-and-bound algorithm that is used to solve +the nonconvex QCQP formulations of the performance estimation problems. +Table 1: Organization of the appendix. +Notation. +We denote by (· ⊙ ·) : Rn ×Rn → Rn×n the symmetric outer product, that is, for any x, y ∈ Rn: +x ⊙ y = 1 +2 +� +xy⊤ + yx⊤� +. +A +Tightness of the worst-case search directions +Figure 5 and Figure 6 illustrate the tightness of the bounds (12) and (14) for PRP and FR respectively. That +is, we compare the numerical bounds (discrete points) with closed-forms (continuous lines) for a few different +values of q and ck−1. Numerical bounds are obtained by solving (D) with η = 1 for PRP and η = 0 for +FR. These numerical examples strongly suggest that our bounds cannot be improved in general. Absolute +relative differences between closed-form expressions and numerical ratios is less than 1e − 6 in all cases. +20 + +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +1 +1.5 +2 +2.5 +3 +q +ck(µ, L, ck−1) +(1+q)2/4q +PRP: ck−1 = 1.01 +PRP: ck−1 = 2 +PRP: ck−1 = 10 +PRP: ck−1 = 50 +Figure 5: Worst-case bound (12) (continuous line) and numerical bounds (discrete points) from (D) with +η = 1 (for PRP) for different values of q and ck−1. The bound appear to match to numerical precision. +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0 +20 +40 +60 +q +ck(µ, L, ck−1) +Analytical: ck−1 = 1.01 +Analytical: ck−1 = 2 +Analytical: ck−1 = 10 +Analytical: ck−1 = 50 +Numerical: ck−1 = 1.01 +Numerical: ck−1 = 2 +Numerical: ck−1 = 10 +Numerical: ck−1 = 50 +Figure 6: Worst-case bound (14) (continuous line) and numerical bounds (discrete points) from (D) with +η = 0 (for FR) for different values of q and ck−1. The bound appear to match to numerical precision. +21 + +B +Nonconvex QCQP reformulation of (D) +To reformulate (D) as a nonconvex QCQP, we introduce the following Grammian matrices that is a common +step in performance estimation literature [9, 14]: +H = [xk−1 | gk−1 | gk | dk−1] ∈ Rn×4, +G = H⊤H ∈ S4 ++, +rank G ⩽ n +F = [fk−1 | fk] ∈ R1×2. +(15) +Because we maximize over n, we can ignore rank G ⩽ n and also confine H ∈ R4×4 without loss of +generality [9, Theorem 5], [14, Remark 2.8]. We next define the following notation for selecting columns and +elements of H and F: +xk−1 = e1, gk−1 = e2, gk = e3, dk−1 = e4, (all in R4) +fk−1 = e1, fk = e2, (all in R2), +xk = xk−1 − γk−1dk−1, (all in R4), +dk = gk + βk−1dk−1, (all in R4). +(16) +This ensures that xi = Hxi, gi = Hgi, di = Hdi, fi = Ffi, for all i, j ∈ I. Next, for appropriate choices of +matrices Ai,j, Bi,j, Ci,j, �Ci,j, Di,j, �Di,j, Ei,j, and vector ai,j, we can ensure that the following reformulations +hold for all i, j ∈ I: +⟨gj; xi − xj⟩ = tr GAi,j, +∥xi − xj∥2 = tr GBi,j, +∥gi − gj∥2 = tr GCi,j, ∥gi∥2 = tr GCi,⋆, +∥di − dj∥2 = tr G �Ci,j, ∥di∥2 = tr G �Ci,⋆, +⟨gi; gj⟩ = tr GDi,j, +⟨gi; dj⟩ = tr G �Di,j, +⟨gi − gj; xi − xj⟩ = tr GEi,j, +fj − fi = Fai,j, +(17) +where, using (16), we define +Ai,j = gj ⊙ (xi − xj) +Bi,j = (xi − xj) ⊙ (xi − xj) +Ci,j = (gi − gj) ⊙ (gi − gj), Ci,⋆ = gi ⊙ gi, +�Ci,j = (di − dj) ⊙ (di − dj), �Ci,⋆ = di ⊙ di, +Di,j = gi ⊙ gj, +�Di,j = gi ⊙ dj, +Ei,j = (gi − gj) ⊙ (xi − xj), +ai,j = fj − fi. +(18) +Using (18) and using the definition of G = H⊤H, where H ∈ R4×4, we can write (D) as the following +22 + +nonconvex QCQP: +ck(µ, L, ck−1) = + + + + + + + + + + + + + + + + + + + + + +maximize +tr G �Ck,⋆ +subject to +tr G �Dk−1,k−1 = tr GCk−1,⋆, +tr G �Dk,k−1 = 0, +tr GAk−1,k = 0, +βk−1 × tr GCk−1,⋆ = tr G (Ck,⋆ − ηDk,k−1) , +tr G �Ck−1,⋆ ⩽ ck−1 tr GCk−1,⋆, +Fai,j + tr G +� +Ai,j ++ +1 +2(1− µ +L ) +� 1 +LCi,j + µΘi,j − 2 µ +LEi,j +� � +⩽ 0, i, j ∈ I, +Θi,j = Bi,j, i, j ∈ I, +G = H⊤H, +tr GCk,⋆ = 1, + + + + + + + + + + + + + + + + + + + + + +(19) +where G, F, H, Θ, γk−1, βk−1 are the decision variables. This nonconvex QCQP can be solved to certifiable +global optimality using a custom spatial branch-and-bound algorithm described in Appendix D. +C +Nonconvex QCQP reformulations of (BLyapunov) and (Bexact) +Similar to the reformulations from Appendix D, (BLyapunov) and (Bexact) can be cast as nonconvex QCQPs, +where the number of nonconvex constraints grow quadratically with N. Thereby, solving them to global +optimality in reasonable time for N = 3, 4 is already challenging. +Therefore, rather than solving the nonconvex QCQP reformulations of (BLyapunov) and (Bexact) directly, +we compute upper bounds and lower bounds by solving more tractable nonconvex QCQP formulations. We +then show that the relative gap between the upper and lower bounds is less than 10% which thereby indicates +that there is essentially no room for further improvement. +C.1 +Nonconvex QCQP reformulation of (BLyapunov) +This section presents our upper bound ρN(q, c) and lower bound ρN(q, c) on ρN(q, c). +C.1.1 +Computing ρN(q, c) +Using (7), we have the following relaxation of (BLyapunov), which provides upper bounds on ρN(q, c): + + + + + + + + + + + + + + + + +maximize +f,n,{xk+i}i∈[0:N],{dk+i}i∈[0:N] +f(xk+N)−f⋆ +f(xk)−f⋆ +subject to +n ∈ N, f ∈ Fµ,L(Rn), +xk+i, dk+i ∈ Rn, +i ∈ [0 : N] +∥dk∥2 ⩽ c∥∇f(xk)∥2, +⟨∇f(xk+i+1); dk+i⟩ = 0, +i ∈ [0 : N − 1], +⟨∇f(xk+i+1); xk+i − xk+i+1⟩ = 0, +i ∈ [0 : N − 1], +⟨∇f(xk+i); dk+i⟩ = ∥∇f(xk+i)∥2, +i ∈ [0 : N − 1], +dk+i+1 = gk+i+1 + βk+idk+i, +i ∈ [0 : N − 2], +βk+i = ∥gk+i+1∥2−η⟨gk+i+1; gk+i⟩ +∥gk+i∥2 +, +i ∈ [0 : N − 2]. + + + + + + + + + + + + + + + + +(20) +23 + +Using the notation gi ≜ ∇f(xi) and fi ≜ f(xi) again, and then applying an homogeneity argument, we write +(20) as: +ρN(q, c) = + + + + + + + + + + + + + + + + + +maximize +fk+N − f⋆ +subject to +n ∈ N, f ∈ Fµ,L(Rn), +xk+i, dk+i ∈ Rn, +i ∈ [0 : N] +∥dk∥2 ⩽ c∥gk∥2, +⟨gk+i+1; dk+i⟩ = 0, +i ∈ [0 : N − 1], +⟨gk+i+1; xk+i − xk+i+1⟩ = 0, +i ∈ [0 : N − 1], +⟨gk+i; dk+i⟩ = ∥gk+i∥2, +i ∈ [0 : N − 1], +dk+i+1 = gk+i+1 + βk+idk+i, +i ∈ [0 : N − 2], +βk+i−1 = ∥gk+i∥2−η⟨gk+i; gk+i−1⟩ +∥gi−1∥2 +, +i ∈ [1 : N − 1], +fk − f⋆ = 1, + + + + + + + + + + + + + + + + + +(21) +where f, n, {xk+i}i∈[0:N], {dk+i}i∈[0:N] are the decision variables. Define I⋆ +N = {⋆, k, k + 1, . . . , k + N}. Next, +note that the equation dk+i+1 = gk+i+1 + βk+idk+i for i ∈ [0 : N − 2], can be written equivalently as the +following set of equations: +χj,i = χj,i−1βk+i−1, +i ∈ [1 : N − 1], j ∈ [0 : i − 2], +χi−1,i = βk+i−1, +i ∈ [1 : N − 1], +dk+i = gk+i + +i−1 +� +j=1 +χj,igk+j + χ0,idk, +i ∈ [1 : N − 1], +(22) +where we have introduced the intermediate variables χj,i, which will aid us in formulating (21) as a nonconvex +QCQP down the line. Next, using (22) and Theorem 1.1, we can equivalently write (21) as: +ρN(q, c) = + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +maximize +fk+N − f⋆ +subject to +n ∈ N, +fi ⩾ fj + ⟨gj; xi − xj⟩ + +1 +2(1− µ +L ) +� +1 +L∥gi − gj∥2 ++µ∥xi − xj∥2 − 2 µ +L ⟨gi − gj; xi − xj⟩ +� +, +i, j ∈ I⋆ +N, +∥dk∥2 ⩽ c∥gk∥2, +⟨gk+i+1; dk+i⟩ = 0, +i ∈ [0 : N − 1], +⟨gk+i+1; xk+i − xk+i+1⟩ = 0, +i ∈ [0 : N − 1], +⟨gk+i; dk+i⟩ = ∥gk+i∥2, +i ∈ [0 : N − 1], +βk+i−1 = ∥gk+i∥2−η⟨gk+i; gk+i−1⟩ +∥gk+i−1∥2 +, +i ∈ [1 : N − 1], +χj,i = χj,i−1βk+i−1, +i ∈ [1 : N − 1], j ∈ [0 : i − 2], +χi−1,i = βk+i−1, +i ∈ [1 : N − 1], +dk+i = gk+i + �i−1 +j=1 χj,igk+j + χ0,idk, +i ∈ [1 : N − 1], +fk − f⋆ = 1, +g⋆ = 0, x⋆ = 0, f⋆ = 0, +{xi, gi, fi}i∈I⋆ +N ⊂ Rn × Rn × R, {di}i∈I⋆ +N\{k+N} ⊂ Rn, +{βk+i}i∈[0:N−2] ⊂ R, {χj,i}j∈[0:N−2],i∈[0:N−1] ⊂ R, + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +(23) +where {xk+i, gk+i, fk+i}i, n, {dk+i}i∈, {βk+i}i, {χj,i}j,i are the decision variables. +Note that we have set +g⋆ = 0, x⋆ = 0, and f⋆ = 0 without loss of generality, because both the objective and the function class are +closed and invariant under shifting variables and function values. We introduce Grammian matrices again: +H = [dk | gk | gk+1 | gk+2 | · · · | gk+N | xk | xk+1 | xk+2 | · · · | xk+N] ∈ Rn×(2N+3), +G = H⊤H ∈ S(2N+3) ++ +, rank G ⩽ n, +F = [fk | fk+1 | . . . | fk+N] ∈ R1×(N+1). +(24) +24 + +As we maximize over n, we can ignore the constraint rank G ⩽ n, and confine H to be in R(2N+3)×(2N+3) +without loss of generality [9, Theorem 5], [14, Remark 2.8]. Next, define the following notation for selecting +columns and elements of H and F: +x⋆ = 0 ∈ R2N+3, dk = e1 ∈ R2N+3, gk+i = ei+2 ∈ R2N+3, +xk+i = e(N+2)+(i+1) ∈ R2N+3, +f⋆ = 0, fk+i = ei+1 ∈ R(N+1), +dk+i = gk+i + +i−1 +� +j=1 +χj,igk+j + χ0,idk ∈ R2N+3, +(25) +where i ∈ [0 : N]. This ensures that we have xi = Hxi, gi = Hgi, di = Hdi,, fi = Ffi for all i ∈ I⋆ +N. For +appropriate choices of matrices Ai,j,Bi,j, Ci,j, �Ci,j, Di,j, �Di,j, Ei,j, and vector ai,j as defined in (17), where +xi, gi, fi, di are taken from (25) now, we can ensure that the identities in (18) hold for all i, j ∈ I⋆ +N. Using +those identities and using the definition of G = H⊤H, where H ∈ R(2N+3)×(2N+3), we can write (23) as the +following nonconvex QCQP: +ρN(q, c) = + + + + + + + + + + + + + + + + + + + + + + + + +maximize +Fa⋆,k+N +subject to +Fai,j + tr G +� +Ai,j + +1 +2(1− µ +L ) +� 1 +LCi,j + µBi,j − 2 µ +LEi,j +�� +⩽ 0, +i, j ∈ I⋆ +N, +tr G �Ck,⋆ ⩽ c tr GCk,⋆, +tr G �Dk+i+1,k+i = 0, +i ∈ [0 : N − 1], +tr GAk+i,k+i+1 = 0, +i ∈ [0 : N − 1], +tr G �Dk+i,k+i = tr GCk+i,⋆ +i ∈ [0 : N − 1], +βk+i−1 × tr GCk+i−1,⋆ = tr G (Ck+i,⋆ − ηDk+i,k+i−1) , +i ∈ [1 : N − 1], +χj,i = χj,i−1βk+i−1, +i ∈ [1 : N − 1], j ∈ [0 : i − 2], +χi−1,i = βk+i−1, +i ∈ [1 : N − 1], +Fa⋆,k = 1, +G = H⊤H, +F ∈ RN+1, G ∈ S2N+3, H ∈ R(2N+3)×(2N+3), +{βk+i}i∈[0:N−2] ⊂ R, {χj,i}j∈[0:N−2],i∈[0:N−1] ⊂ R, + + + + + + + + + + + + + + + + + + + + + + + + +(26) +where F, G, H, {χj,i}j,i, {βk+i}i are the decision variables. +C.1.2 +Computing ρN(q, c) +We now discuss how to compute ρN(q, c). Once we have solved (26), it provides us with the corresponding +CG update parameters, which we denote by βi. If we can solve (BLyapunov) with the CG update parameters +fixed to the βi found from (26), then it will provide us with the lower bound ρN(µ, L, c)s along with a +bad function, which we show now. Using the notation gi ≜ ∇f(xi) and fi ≜ f(xi), then applying the +homogeneity argument, we can compute ρN(q, c) by finding a feasible solution to the following optimization +problem: + + + + + + + + + + + + + + + +maximize +fk+N − f⋆ +subject to +n ∈ N, f ∈ Fµ,L(Rn), +xk+i, dk+i ∈ Rn, +i ∈ [0 : N] +∥dk∥2 ⩽ c∥gk∥2, +γk+i = argminγf(xk+i − γdk+i), +i ∈ [0 : N − 1], +xk+i+1 = xk+i − γk+idk+i, +i ∈ [0 : N − 1], +dk+i+1 = gk+i+1 + βk+idk+i, +i ∈ [0 : N − 2], +βk+i−1 = ∥gk+i∥2−η⟨gk+i; gk+i−1⟩ +∥gk+i−1∥2 +, +i ∈ [1 : N − 1], +fk − f⋆ = 1, + + + + + + + + + + + + + + + +(27) +25 + +where f, n, {xk+i}, {dk+i}i, {γk+i}i are the decision variables. Next, note that the NCGM iteration scheme +in (27) can be equivalently written as: +χj,i = χj,i−1βk+i−1, +i ∈ [1 : N − 1], j ∈ [0 : i − 2] +χi−1,i = βk+i−1, +i ∈ [1 : N − 1] +αi,i−1 = γk+i−1, +i ∈ [1 : N], +αi,j = γk+j + +i−1 +� +ℓ=j+1 +γk+ℓχj,ℓ, +i ∈ [1 : N], j ∈ [0 : i − 2], +xk+i = xk − +i−1 +� +j=1 +αi,jgk+j − αi,0dk, +i ∈ [1 : N], +dk+i = gk+i + +i−1 +� +j=1 +χj,igk+j + χ0,idk, +i ∈ [1 : N − 1]. +(28) +where we have introduced intermediate variables χj,i and αi,j which will aid us in formulating (27) as a +nonconvex QCQP. Define I⋆ +N = {⋆, k, k + 1, . . . , k + N}. Now using (28), Theorem 1.1, and (7), we can +equivalently write (21) as: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +maximize +fk+N − f⋆ +subject to +n ∈ N, +fi ⩾ fj + ⟨gj; xi − xj⟩ + +1 +2(1− µ +L ) +� +1 +L∥gi − gj∥2 ++µ∥xi − xj∥2 − 2 µ +L ⟨gi − gj; xi − xj⟩ +� +, +i, j ∈ I⋆ +N, +∥dk∥2 ⩽ c∥gk∥2, +⟨gk+i+1; dk+i⟩ = 0, +i ∈ [0 : N − 1], +⟨gk+i+1; xk+i − xk+i+1⟩ = 0, +i ∈ [0 : N − 1], +⟨gk+i; dk+i⟩ = ∥gk+i∥2, +i ∈ [0 : N − 1], +χj,i = χj,i−1βk+i−1, +i ∈ [1 : N − 1], j ∈ [0 : i − 2] +χi−1,i = βk+i−1, +i ∈ [1 : N − 1] +αi,i−1 = γk+i−1, +i ∈ [1 : N], +αi,j = γk+j + �i−1 +ℓ=j+1 γk+ℓχj,ℓ, +i ∈ [1 : N], j ∈ [0 : i − 2], +xk+i = xk − �i−1 +j=1 αi,jgk+j − αi,0dk, +i ∈ [1 : N], +dk+i = gk+i + �i−1 +j=1 χj,igk+j + χ0,idk, +i ∈ [1 : N − 1]. +βk+i−1 = ∥gk+i∥2−η⟨gk+i; gk+i−1⟩ +∥gk+i−1∥2 +, +i ∈ [1 : N − 1], +fk − f⋆ = 1, +g⋆ = 0, x⋆ = 0, f⋆ = 0, +{xi, gi, fi}i∈I⋆ +N ⊂ Rn × Rn × R, {di}i∈I⋆ +N\{k+N} ⊂ Rn, +{χj,i}j∈[0:N−2],i∈[0:N−1] ⊂ R, +{γk+i}i∈[0:N] ⊂ R, {αi,j}i∈[1:N],j∈[0:N−1] ⊂ R, + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +(29) +where {xk+i, gk+i, fk+i}i, n, {γk+i}i, {χj,i}j,i, {αi,j}i,j are the decision variables. We introduce the Gram- +mian transformation: +H = [xk | gk | gk+1 | . . . | gk+N | dk] ∈ Rn×(N+3), +G = H⊤H ∈ SN+3 ++ +, rank G ⩽ n, +F = [fk | fk+1 | . . . | fk+N] ∈ R1×(N+1). +(30) +As we maximize over n, we again ignore the constraint rank G ⩽ n and can let H ∈ R(N+3)×(N+3) +without loss of generality [9, Theorem 5], [14, Remark 2.8]. +We next define the following notation for +26 + +selecting columns and elements of H and F: +g⋆ = 0 ∈ RN+3, gk+i = ei+2 ∈ RN+3, +i ∈ [0 : N], +dk = eN+3 ∈ RN+3, +xk = e1 ∈ RN+2, x⋆ = 0 ∈ RN+2, +xk+i(α) = xk − +i−1 +� +j=1 +αi,jgk+j − αi,0dk ∈ RN+3, +i ∈ [1 : N], +dk+i(χ) = gk+i + +i−1 +� +j=1 +χj,igk+j + χ0,idk, +i ∈ [1 : N − 1], +f⋆ = 0 ∈ RN+1, fk+i = ei+1 ∈ RN+1, +i ∈ [0 : N], +(31) +which ensure xi = Hxi, gi = Hgi, fi = Ffi, di = Hdi for i ∈ I⋆ +N. For appropriate choices of matrices +Ai,j,Bi,j, Ci,j, �Ci,j, Di,j, �Di,j, Ei,j, and vector ai,j as defined in (17), where xi, gi, fi, di are from (31), we +can ensure that the identities in (18) hold for all i, j ∈ I⋆ +N. Using those identities and using the definition of +G = H⊤H, where H ∈ R(N+3)×(N+3), we can write (29) as the following nonconvex QCQP: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +maximize +Fa⋆,N +subject to +Fai,j + tr G +� +Ai,j + +1 +2(1− µ +L ) +� 1 +LCi,j + µΘi,j − 2 µ +LEi,j +�� +⩽ 0, +i, j ∈ I⋆ +N, +Θi,j = Bi,j, +i, j ∈ I⋆ +N, +tr G �Ck,⋆ ⩽ c tr GCk,⋆, +tr G �Dk+i+1,k+i = 0, +i ∈ [0 : N − 1], +tr GAk+i,k+i+1 = 0, +i ∈ [0 : N − 1], +tr G �Dk+i,k+i = tr GCk+i,⋆ +i ∈ [0 : N − 1], +χj,i = χj,i−1βk+i−1, +i ∈ [1 : N − 1], j ∈ [0 : i − 2] +χi−1,i = βk+i−1, +i ∈ [1 : N − 1] +αi,i−1 = γk+i−1, +i ∈ [1 : N], +αi,j = γk+j + �i−1 +ℓ=j+1 γk+ℓχj,ℓ, +i ∈ [1 : N], j ∈ [0 : i − 2], +βk+i−1 × tr GCk+i−1,⋆ = tr G (Ck+i,⋆ − ηDk+i,k+i−1) , +i ∈ [1 : N − 1], +Fa⋆,k = 1, +G = H⊤H, +F ∈ RN+1, G ∈ SN+3, H ∈ R(N+3)×(N+3), +{χj,i}j∈[0:N−2],i∈[0:N−1] ⊂ R, +{γk+i}i∈[0:N] ⊂ R, {αi,j}i∈[1:N],j∈[0:N−1] ⊂ R, + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +(32) +where G, F, Θ, H, γ, α, χ are the decision variables. Note that {Θi,j}i,j∈I⋆ +N is introduced as a separate decision +variable to formulate the cubic constraints arising from Bi,j as quadratic constraints. Note that to compute +ρN(q, c), it suffices to find just a feasible solution to (32), in Appendix D we will discuss how to do so using +our custom spatial branch-and-bound algorithm. From the solution to (32) we construct the associated +triplets {xi, gi, fi}i∈I⋆ +N and then apply Theorem 1.2 construct the corresponding bad function. +C.2 +Nonconvex QCQP reformulation of (Bexact) +Now we discuss how we compute the upper bound ρN,0(q) and lower bound ρN,0(q) to ρN,0(q) defined in +(Bexact). The bound computation process is very similar to that of (BLyapunov). Observe that, in (BLyapunov), +if we remove the constraint ∥dk∥2 ⩽ c∥∇f(xk)∥2, set k ≜ 0 , and then add the constraint d0 = ∇f(x0), then +it is identical to (Bexact) (the constraint ⟨∇f(x0); d0⟩ = ∥∇f(x0)∥2 in (BLyapunov) is a valid but redundant +constraint for (Bexact)). +27 + +So, to compute the upper bound ρN,0(q), we can follow a transformation process very similar to Ap- +pendix C.1.1 but with a few changes. In (21) and (23), we remove the constraint ∥dk∥2 ⩽ c∥gk∥2, and +then add the constraint gk = dk. Second, the Grammian matrices defined in (24) stays the same, and in +(25) the vectors {xi, gi, fi}i∈I⋆ +N stays the same except we set dk = gk = e2 ∈ R2N+3, which ensures that +dk = Fdk = gk. We then remove the constraint tr G �Ck,⋆ ⩽ c tr GCk,⋆ from (26) and finally set k ≜ 0 in the +resultant QCQP. The solution to the nonconvex QCQP will provide us the upper bound ρN,0(q) in (Bexact). +To compute the lower bound ρN,0(q), we follow the same set of changes described in the last paragraph +but to (27) in Appendix C.1.2. +C.3 +The relative gap between the lower bounds and upper bounds +Tables 2, 3, 4 record the relative gap between lower bounds and upper bounds for a few representative values +of q obtained by solving the aforementioned nonconvex QCQPs associated with (BLyapunov) and (Bexact) using +our custom spatial branch-and-bound algorithm described in Appendix D. Note that the tables contain a +few negative entries close to zero which are due to the absolute gap being of the same order as the accuracy +of the solver (1e − 6). For the full list for all values, we refer to our open-source code in Section 4, which +also allows for computing these bounds for a user-specified value of q as well. In all cases, the relative gap +is less than 10%. In most cases, it is significantly better. +q = +0.001 +0.005 +0.02 +0.04 +0.06 +0.08 +0.1 +0.3 +0.5 +N = 1 +3e−8 +−1e−6 +3e−9 +6e−8 +9e−8 +2e−7 +2e−7 +1e−6 +3e−7 +N = 2 +2e−6 +6e−7 +−3e−8 +9e−8 +1e−7 +8e−8 +3e−7 +8e−3 +4e−4 +N = 3 +5e−6 +5e−4 +7e−3 +2e−2 +3e−2 +4e−2 +2e−2 +5e−2 +−3e−7 +N = 4 +2e−4 +3e−3 +2e−2 +7e−2 +1e−1 +3e−2 +4e−2 +4e−2 +4e−2 +Table 2: Relative gaps ρN(q,c)−ρN (q,c)/ρN(q,c) for PRP with c = (1+q)2/4q. +q = +0.001 +0.005 +0.02 +0.04 +0.06 +0.08 +0.1 +0.3 +0.5 +N = 2 +7e−6 +2e−4 +2e−3 +7e−3 +1e−2 +1e−2 +2e−2 +1e−2 +1e−6 +N = 3 +5e−5 +9e−4 +1e−2 +3e−2 +5e−2 +6e−2 +6e−2 +5e−3 +−7e−6 +N = 4 +3e−4 +4e−3 +3e−2 +4e−2 +9e−2 +9e−2 +7e−2 +3e−2 +7e−2 +Table 3: Relative gap ρN,0(q)−ρN,0(q)/ρN,0(q) for PRP where N = 2, 3, 4. The case N = 1 is omitted, as PRP +is equivalent to GDEL in this case. +q = +0.001 +0.005 +0.02 +0.04 +0.06 +0.08 +0.1 +0.3 +0.5 +N = 2 +9e−6 +2e−4 +1e−3 +7e−3 +1e−2 +1e−2 +2e−2 +1e−2 +8e−7 +N = 3 +7e−5 +1e−3 +1e−2 +2e−2 +3e−2 +3e−2 +3e−2 +3e−7 +−1e−7 +N = 4 +2e−4 +3e−3 +2e−2 +3e−2 +3e−2 +2e−2 +1e−2 +1e−6 +4e−2 +Table 4: The relative gap ρN,0(q)−ρN,0(q)/ρN,0(q) for FR where N = 2, 3, 4. The case N = 1 is omitted again, +as in this case FR is equivalent to GDEL. +28 + +D +Custom spatial branch-and-bound algorithm +This section discusses implementation details for solving the nonconvex QCQPs of this paper (namely (19), (26), +or (32)) using a custom spatial branch-and-bound method. This strategy proceeds in three stages, as follows. +• Stage 1: Compute a feasible solution. First, we construct a feasible solution to the nonconvex +QCQP. We do that by generating a random µ-strongly convex and L-smooth quadratic function, +and by applying the corresponding nonlinear conjugate gradient method on it. The corresponding +iterates, gradient and function values correspond to a feasible point for the nonconvex QCQPs under +consideration. +• Stage 2: Compute a locally optimal solution by warm-starting at Stage 1 solution. Stage 2 +computes a locally optimal solution to the nonconvex QCQPs using an interior-point algorithm, warm- +starting at the feasible solution produced by Stage 1. When a good warm-starting point is provided, +interior-point algorithms can quickly converge to a locally optimal solution under suitable regularity +conditions [38, 39], [40, §3.3]. In the situation where the interior-point algorithm fails to converge, we +go back to the feasible solution from Stage 1. We have empirically observed that Stage 2 consistently +provides a locally optimal solution. +• Stage 3: Compute a globally optimal solution by warm-starting at Stage 2 solution. Stage +3 computes a globally optimal solution to the nonconvex QCQP using a spatial branch-and-bound +algorithm [41, 42], warm-starting at the locally-optimal solution produced by Stage 2. For details +about how spatial branch-and-bound algorithm works, we refer the reader to [15, §4.1]. +Remark. In stage 3, the most numerically challenging nonconvex quadratic constraint in (19), (26) or (32) +is G = PP ⊤. To solve those problems in reasonable times, we use the lazy constraints approach, [15, §4.2.5]. +In short, we replace the constraint G = PP ⊤ by the infinite set of linear constraints tr +� +Gyy⊤� +⩾ 0 for +all y, which we then sample to obtain a finite set of linear constraints (we recursively add additional linear +constraints afterwards if need be). More precisely, we use +tr +� +Gyy⊤� +⩾ 0, +y ∈ Y, +(33) +where the initial Y is generated randomly as a set of unit vectors following the methodology described in [43, +§5.1]. By replacing G = PP ⊤ by (33) we obtain a simpler (but relaxed) QCQP. Then, we update the solution +G lazily by repeating the following steps until G ≽ 0 is satisfied subject to a termination criterion. Practically +speaking, our termination criterion is that the minimal eigenvalue of G is larger than ǫ ≈ −1e − 6; until +then, we repeat the following procedure: +1. Solve the relaxation of the nonconvex QCQPs, where (33) is used instead of G = PP ⊤, which provides +us an upper bound on the original nonconvex QCQP. +2. Compute the minimal eigenvalue eigmin(G) and the corresponding eigenvector u of G. If eigmin(G) ≥ 0, +we reached an optimal solution to the nonconvex QCQP and we terminate. +3. If eigmin(G) < 0, we add a constraint tr(Guu⊤) ⩾ 0 lazily, which makes the current G infeasible for the +new relaxation. We use the lazy constraint callback interface of JuMP to add constraints lazily, which +means that after adding one additional linear constraint, updating the solution in step 1 is efficient +since Gurobi and all modern solvers based on the simplex algorithm can quickly update a solution when +only one linear constraint is added [44, pp. 205-207]. +29 + diff --git a/LNAzT4oBgHgl3EQfkP3w/content/tmp_files/load_file.txt b/LNAzT4oBgHgl3EQfkP3w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea7de832adafc948478f46366a00ba8cac3c404c --- /dev/null +++ b/LNAzT4oBgHgl3EQfkP3w/content/tmp_files/load_file.txt @@ -0,0 +1,1153 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf,len=1152 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='01530v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='OC] 4 Jan 2023 Nonlinear conjugate gradient methods: worst-case convergence rates via computer-assisted analyses∗ Shuvomoy Das Gupta†, Robert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Freund‡, Xu Andy Sun§, Adrien Taylor¶ Abstract In this paper, we propose a computer-assisted approach to the analysis of the worst-case convergence of nonlinear conjugate gradient methods (NCGMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Those methods are known for their generally good empirical performances for large-scale optimization, while having relatively incomplete analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Using this approach, we establish novel complexity bounds for the Polak-Ribière-Polyak (PRP) and the Fletcher- Reeves (FR) NCGMs for smooth strongly convex minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Conversely, we provide examples showing that those methods might behave worse than the regular steepest descent on the same class of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 1 Introduction We consider the standard unconstrained convex minimization problem f⋆ ≜ min x∈Rn f(x), (1) where f is L-smooth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', it has an L-Lipschitz gradient) and µ-strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We study the worst-case performances of a few famous variants of nonlinear conjugate gradient methods (NCGMs) for solving (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' More specifically, we study Polak-Ribière-Polyak (PRP) [1, 2] and Fletcher-Reeves (FR) [3] schemes with exact line search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' With exact line search, many other NCGMs such as the Hestenes and Stiefel method [4], the conjugate descent method due to Fletcher [5], and the Dai and Yuan method [6] reduce to either PRP or FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Under exact line search, PRP and FR can be presented in the following compact form: γk ∈ argmin γ f(xk − γ dk), xk+1 = xk − γkdk, βk = ∥∇f(xk+1)∥2 − η ⟨∇f(xk+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ∇f(xk)⟩ ∥∇f(xk)∥2 , dk+1 = ∇f(xk+1) + βkdk, (M) where PRP and FR are respectively obtained by setting η = 1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' NCGMs have a long history (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', the nice survey [7]), but are much less studied compared to their many first-order competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For instance, even though FR is generally considered the first NCGM [7, §1], we are not aware of non-asymptotic convergence results for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Still, some variants are known for their generally good empirical behaviors (which ∗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Freund acknowledges support by AFOSR Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' FA9550-22-1-0356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Taylor acknowledges support from the European Research Council (grant SEQUOIA 724063).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This work was partly funded by the French government under management of Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference ANR-19-P3IA- 0001 (PRAIRIE 3IA Institute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' †Operations Research Center, Massachusetts Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Email: sdgupta@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ‡Sloan School of Management, Massachusetts Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Email: rfreund@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' §Sloan School of Management, Massachusetts Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Email: sunx@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ¶INRIA, École Normale Supérieure, CNRS, PSL Research University, Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' adrien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='taylor@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 1 we illustrate on Figure 1) with little of them being backed-up by classical complexity analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In this work, we apply the performance estimation approach [8, 9] to (M) for filling this gap by explicitly computing some worst-case convergence properties of PRP and FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 0 200 400 600 800 1,000 10−4 10−3 10−2 10−1 100 iterations f(x) − f∗ 0 200 400 600 800 1,000 10−9 10−6 10−3 100 iterations Gradient Nesterov Nesterov (SC version) Optimized gradient FR PRP Figure 1: Convergence of a few first-order methods on a logistic regression problem on the small-sized Sonar dataset [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Experiments with normalized features (zero mean and unit variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Left: without regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Right: with an ℓ2 regularization of parameter 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' All methods were featured with an exact line search: (i) gradient descent, (ii) Nesterov’s accelerated gradient [11] (exact line search instead of backtracking), (iii) Nesterov’s accelerated method for strongly convex problems, version [12, Algorithm 28] with exact line search instead of the gradient step, (iv) optimized gradient descent [13, Algorithm (OGM- LS)], (v) FR, and (vi) PRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' No method was tuned, the results correspond to the first run for each method and are only meant for illustrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 Contributions The contribution of this paper is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' First, we compute worst-case convergence bounds and counter- examples for PRP and FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Those bounds are obtained by formulating the problems of computing worst-case scenarios as nonconvex quadratically constrained quadratic optimization problems (QCQPs) and then by solving them to global optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Second, these computations also allow us to construct mathematical proofs that establish an improved non-asymptotic convergence bound for PRP, and, to the best of our knowledge, the first non-asymptotic convergence bound for FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Furthermore, the worst-case bounds for PRP and FR obtained numerically show that there are simple adversarial examples on which those methods do not behave better than gradient descent with an exact line search (GDEL), thus leaving very few room for improvements on this class of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' From a methodological point of view, the approach of computing worst-case scenarios and bounds through optimization is often referred to as performance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In many situations, those problems are amenable to convex semidefinite programs [8, 9, 14], but it is generally not the case for adaptive first-order methods such as PRP and FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For studying those methods, we evaluate the worst-case performances of (M) by solving nonconvex QCQPs, extending the more standard SDP-based approach from [8, 9, 14] developed for non-adaptive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This contribution is similar in spirit with that in [15] which was developed for devising optimal (but non-adaptive) first-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In Section 2, we establish non-asymptotic convergence rates for PRP and FR by viewing the search direction dk in (M) as an approximate gradient direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In Section 3, we compute the exact numerical values of the worst-case f(xN)−f⋆/f(x0)−f⋆ and f(xk+N)−f⋆/f(xk)−f⋆ for PRP and FR by formulating the problems as nonconvex QCQPs and then solving them to certifiable global optimality using a custom spatial branch-and-bound algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 Related works Conjugate gradient (CG) methods are particularly popular choices for solving systems of linear equations and quadratic minimization problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' in this context, they are known to be information-optimal in the class of first-order methods [16, Chapter 12 & Chapter 13] or [17, Chapter 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' There are many extensions beyond quadratics, commonly referred to as nonlinear conjugate gradient methods (NCGMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' They are discussed at length in the textbooks [18, Chapter 5 & Chapter 7] and [19, Chapter 5] and in the nice survey [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In particular, when exact line searches are used, many variants become equivalent and can be seen as instances of quasi-Newton methods, see [18, Chapter 7, §“Relationship with conjugate gradient methods”] or [19, Chapter 5, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For instance, it is well known that standard variants such as Hestenes-Stiefel [4] and Dai-Yuan [6] are equivalent to (M) when exact line searches are used, while being different in the presence of more popular line search procedures (such as Wolfe’s [18, Chapter 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Beyond quadratics, obtaining convergence guarantees is often reduced to the problem of ensuring the search direction to be a descent direction, see for instance [17, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 “Extensions to non-quadratic problems”] or [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Without exact line searches, even when f is strongly convex, there are counter-examples showing that even popular variants may not generate descent directions [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Note that NCGMs are often used together with restart strategies, which we do not consider here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', [23] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In this work, we use the performance estimation framework [8, 9, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This methodology is essentially mature for analyzing “fixed-step” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', non-adaptive) first-order methods (and for methods whose analyses are amenable to those of fixed-step methods), whose stepsizes are essentially chosen in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This type of methods include many common first-order methods and operator splitting schemes, including the heavy-ball method [24] and Nesterov’s accelerated gradient [11, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Only very few adaptive methods were studied using the PEP methodology, namely gradient descent with exact line searches [26], greedy first-order methods [13], and Polyak stepsizes [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' A premise to the study of NCGMs using PEPs was done in [28, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This work is also closely related in spirit with the technique developed in [15] for optimizing coefficients of fixed-step first-order methods using nonconvex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 Preliminaries In this short section, we recall the definition and a result on smooth strongly convex functions, as well as a base result on steepest descent with an exact line search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We use the standard notation ⟨ · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' · ⟩ : Rn × Rn → R to denote the Euclidean inner product, and the corresponding induced Euclidean norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The class of L-smooth µ-strongly convex functions is standard and can be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Let f : Rn → R be a proper, closed, and convex function, and consider two constants 0 ⩽ µ < L < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The function f is L-smooth and µ-strongly convex (notation f ∈ Fµ,L(Rn)), if (L-smooth) for all x, y ∈ Rn, it holds that f(x) ⩽ f(y) + ⟨∇f(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' x − y⟩ + L 2 ∥x − y∥2, (µ-strongly convex) for all x, y ∈ Rn, it holds that f(x) ⩾ f(y) + ⟨∇f(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' x − y⟩ + µ 2 ∥x − y∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We simply denote f ∈ Fµ,L when the dimension is either clear from the context or unspecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We also denote by q ≜ µ L the inverse condition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For readability, we do not explicitly treat the (trivial) case L = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Smooth strongly convex functions satisfy many inequalities, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', [29, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For the devel- opments below, we need only one specific inequality characterizing functions in Fµ,L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The following result can be found in [9, Theorem 4] and is key in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' [9, Theorem 4, Fµ,L-interpolation] Let I be an index set and S = {(xi, gi, fi)}i∈I ⊆ Rn × Rn × R be a set of triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' There exists f ∈ Fµ,L satisfying f(xi) = fi and ∇f(xi) = gi for all i ∈ I if and only if fi ⩾ fj+⟨gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xi − xj⟩ + 1 2L∥gi − gj∥2 + µ 2(1 − µ/L)∥xi − xj − 1 L(gi − gj)∥2 (2) holds for all i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 3 Another related result from [30, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1] that we record next involves constructing a strongly-convex smooth function from a given set of triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' [30, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1, strongly convex and smooth extension] Suppose I is a set of indices and S = {(xi, gi, fi)}i∈I ⊆ Rn ×Rn ×R is a set of triplets such that (2) holds for all i, j ∈ I for some 0 ⩽ µ < L < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Then the function f : Rn → R defined by f(y) = max α∈∆ �L 2 ∥y∥2 − L − µ 2 ∥y − 1 L − µ � i∈I αi(gi − µxi)∥2 + � i∈I αi � fi + 1 2(L − µ)∥gi − Lxi∥2 − L 2 ∥xi∥2�� (3) where ∆ = {α ∈ Rn | α ⩾ 0, �n i=1 αi = 1}, satisfies f ∈ Fµ,L(Rn), f(xi) = fi and ∇f(xi) = gi for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Finally, consider a function f ∈ Fµ,L and the approximate steepest descent method: γk = argmin γ f(xk − γdk) xk+1 = xk − γkdk, (4) where the search direction dk satisfies a relative accuracy criterion: ∥∇f(xk) − dk∥ ⩽ ǫ∥∇f(xk)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' (5) In particular, (5) holds when | sin θ| ⩽ ǫ with θ being the angle between ∇f(xk) and dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' With line searches, this amounts to checking that dk is a descent direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We will use the following result in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' [26, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1] Let f ∈ Fµ,L(Rn), x⋆ ≜ argminx∈Rnf(x) be a minimizer of f, and f⋆ ≜ f(x⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For any xk ∈ Rn and search direction dk satisfying (5), we have: f(xk+1) − f⋆ ⩽ �1 − qǫ 1 + qǫ �2 (f(xk) − f⋆) , (6) where xk+1 is computed as (4) and qǫ ≜ µ(1−ǫ)/L(1+ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Note that similar results (without line searches) to that of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 can be found in [31], which might help in future analyses of NCGMs without line searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 2 Base descent properties of NCGMs In this section, we analyze NCGMs as approximate steepest descent methods through a computer-assisted approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Because the NCGMs make use of exact line searches, only the generated search directions matter, and not their magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This renders the analysis somehow simpler, and we argue that this is a reasonable setting for improving the analysis and understanding of NCGMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This section builds on the idea that when | sin θk| (where θk is the angle between minus the gradient and the search direction at iteration k) is upper bounded in an appropriate fashion, one can use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 for obtaining convergence guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In particular, we get nontrivial convergence guarantees as soon as θk can be bounded away from ± π 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', sin θk should be bounded away from 1 for ensuring that dk’s are descent directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Of course, viewing NCGMs as approximate gradient methods is very adversarial by nature, as it misses the point that the directions of NCGMs are meant to be better than those of vanilla gradient descent, while such analyses can only provide worse rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Albeit being pessimistic by construction, the analyses of this section are, to the best of our knowledge, already better than the state-of-the-art bounds for NCGMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Further, we show in the next sections that there is actually nearly no room for improving those analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 4 Properties of NCGMs with exact line search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Before going into the detailed approach, let us review a few properties of the iterates of (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' First, note that the exact line search condition γk = argminγf(xk−γdk) in (M) implies the following equalities: ⟨∇f(xk+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk⟩ = 0, ⟨∇f(xk+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xk − xk+1⟩ = 0, ⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk⟩ = ∥∇f(xk)∥2, (7) which we can show as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The exact line search condition is equivalent to 0 = [∇γf(xk − γdk)]γ=γk = − ⟨∇f(xk − γkdk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk⟩ = − ⟨∇f(xk+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk⟩ (8) thereby obtaining the first line of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Then, the definition of xk+1 implies the second equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The last line follows from applying the first line to ⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk⟩ = ⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ∇f(xk) + βk−1dk−1⟩ = ∥∇f(xk)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' (9) Combining (9) with ⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk⟩ = ∥∇f(xk)∥∥dk∥ cosθk, we obtain that ∥∇f(xk)∥/∥dk∥ = cos θk, thereby reaching sin2 θk = 1 − ∥∇f(xk)∥2/∥dk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Thus, any upper bound on the ratio ∥dk∥/∥∇f(xk)∥ can be converted to a worst-case convergence rate using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Section organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For obtaining the desired bounds measuring the quality of the angle θk, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 first frames the problems of computing the worst-case ∥dk∥/∥∇f(xk)∥ for PRP and FR as optimization prob- lems, referred to as performance estimation problems (PEPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' These PEPs are nonconvex but practically tractable QCQPs and can be solved numerically to certifiable global optimality using spatial branch-and- bound algorithms (detailed in Appendix D), which allows (i) to construct “bad” functions on which the worst-case ∥dk∥/∥∇f(xk)∥ for PRP and FR is achieved, and (ii) to identify closed-form solutions to the PEPs leading to proofs that can be verified in a standard and mathematically rigorous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The convergence rates for PRP and FR are provided and proved in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 Computing worst-case search directions In this section, we formulate the problems of computing the worst-case ratios of ∥dk∥/∥∇f(xk)∥ as optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Following a classical steps introduced in [9, 14], we show that it can be cast as a nonconvex QCQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For doing that, we assume that at iteration k−1 the NCGM has not reached optimality, so ∇f(xk−1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Because ∥∇f(xk−1)∥2 ⩽ ∥dk−1∥2 (follows from applying Cauchy–Schwarz inequality to (9)), without loss of generality we define the ratio ck−1 ≜ ∥dk−1∥2/∥∇f(xk−1)∥2 where ck−1 ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Then, denoting by ck the worst- case ratio ∥dk∥2/∥∇f(xk)∥2 arising when applying (M) to the minimization of an L-smooth µ-strongly convex function, we will compute ck as a function of L, µ, and ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In other words, we use a Lyapunov-type point of view and take the stand of somewhat forgetting about how dk−1 was generated (except through the fact that it satisfies (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Then, we compute the worst possible next search direction dk that the algorithm could generate given that dk−1 satisfies a certain quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Thereby, we obtain an upper bound on the evolution of the quality of the search directions (quantified by ck) obtained throughout the iterative procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Formally, we compute ck(µ, L, ck−1) ≜ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize f,n,xk−1,dk−1 xk,dk,βk−1 ∥dk∥2 ∥∇f(xk)∥2 subject to n ∈ N, f ∈ Fµ,L(Rn), dk−1, xk−1 ∈ Rn, xk, dk and βk−1 generated by (M) from xk−1 and dk−1, ⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ = ∥∇f(xk−1)∥2, ∥dk−1∥2 = ck−1∥∇f(xk−1)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (10) 5 For computing ck(µ, L, ck−1), we reformulate (10) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Denote I ≜ {k − 1, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' An appropriate sampling of the variable f (which is inconveniently infinite-dimensional) allows us to cast (10) as: ck(µ, L, ck−1) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize n,{di}i∈I,γk−1,βk−1 {(xi,gi,fi)}i∈I ∥dk∥2 ∥gk∥2 subject to n ∈ N, βk−1 ∈ R, dk−1, dk ∈ Rn, {(xi, gi, fi)}i∈I ⊂ Rn × Rn × R, ∃f ∈ Fµ,L : � f(xi) = fi ∇f(xi) = gi ∀i ∈ I, γk−1 = argmin γ f(xk−1 − γ dk−1), xk = xk−1 − γk−1dk−1, βk−1 = ∥gk∥2−η⟨gk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' gk−1⟩ ∥gk−1∥2 , dk = gk + βk−1dk−1, ⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ = ∥gk−1∥2, ∥dk−1∥2 = ck−1∥gk−1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (11) Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1, the existence constraint can be replaced by a set of linear/quadratic inequalities (2) for all pairs of triplets in {(xi, gi, fi)}i∈I without changing the objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Furthermore, if βk−1 and γk were pre-defined parameters (instead of variables), the problem would be amenable to a convex semidef- inite program [9, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' So, applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 to (11) followed by an homogeneity argument and a few substitutions based on (7), we arrive at: ck(µ, L, ck−1) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize n,{di}i∈I,γk−1,βk−1 {(xi,gi,fi)}i∈I ∥dk∥2 subject to n ∈ N, dk−1, xk−1 ∈ Rn, fi ⩾ fj + ⟨gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xi − xj⟩ + 1 2(1− µ L ) � 1 L∥gi − gj∥2 +µ∥xi − xj∥2 − 2 µ L ⟨gi − gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xi − xj⟩ � , i, j ∈ I, ⟨gk−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ = ∥gk−1∥2, ⟨gk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ = 0, ⟨gk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xk−1 − xk⟩ = 0, xk = xk−1 − γk−1dk−1, βk−1 = ∥gk∥2−η⟨gk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' gk−1⟩ ∥gk−1∥2 , dk = gk + βk−1dk−1 ∥dk−1∥2 = ck−1∥gk−1∥2, ∥gk∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (D) Note that without the variable n this problem is amenable to a nonconvex QCQP (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Fortu- nately standard arguments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', [9, Theorem 5], or Appendix B) allows setting n = 4 without changing the optimal value of this problem, thereby discarding this dimension issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We can then solve (D) to certifiable global optimality using a custom branch-and-bound algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Reformulation details are provided in Ap- pendix B, whereas a description of the custom spatial branch-and-bound algorithm is given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Finally, we recall that numerical solutions to (D) correspond to worst-case functions that can be obtained through the reconstruction procedure from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In addition, numerical solutions can serve as inspirations for devising rigorous mathematical proofs, as presented next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 Worst-case bounds for PRP and FR In this section, we provide explicit solutions to (D) for PRP and FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Those results are then used for deducing simple convergence bounds through a straightforward application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 A worst-case bound for Polak-Ribière-Polyak (PRP) Solving (D) with η = 1 to global optimality allows obtaining the following worst-case bound for PRP quantifying the quality of the search direction with respect to the gradient direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 (Worst-case search direction for PRP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Let f ∈ Fµ,L, and let xk−1, dk−1 ∈ Rn and xk, dk be generated by the PRP method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', (M) with η = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' It holds that: ∥dk∥2 ∥∇f(xk)∥2 ⩽ (1 + q)2 4q , (12) with q ≜ µ/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Equivalently, ∥dk − ∇f(xk)∥ ⩽ ǫ∥∇f(xk)∥ holds with ǫ = 1−q/1+q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Recall that xk = xk−1 − γk−1 dk−1 and dk = ∇f(xk) + βk−1dk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The proof consists of the following weighted sum of inequalities: optimality condition of the line search, with weight λ1 = −β2 k−1 1+q Lγk−1q : ⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ = 0, smoothness and strong convexity of f between xk−1 and xk, with weight λ2 = β2 k−1(1+q)2 Lγ2 k−1(1−q)q : f(xk−1) ⩾f(xk) + ⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xk−1 − xk⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥xk−1 − xk − 1 L(∇f(xk−1) − ∇f(xk))∥2 =f(xk) + γk−1⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥γk−1dk−1 − 1 L(∇f(xk−1) − ∇f(xk))∥2 smoothness and strong convexity of f between xk and xk−1, with weight λ3 = λ2: f(xk) ⩾f(xk−1) + ⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xk − xk−1⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥xk−1 − xk − 1 L(∇f(xk−1) − ∇f(xk))∥2 =f(xk−1) − γk−1⟨∇f(xk−1), dk−1⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥γk−1dk−1 − 1 L(∇f(xk−1) − ∇f(xk))∥2 definition of βk−1 with weight λ4 = βk−1(1+q) Lγk−1q : 0 = ⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ∇f(xk)⟩ − ∥∇f(xk)∥2 + βk−1∥∇f(xk−1)∥2 = ⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ∇f(xk)⟩ − ∥∇f(xk)∥2 + βk−1⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We arrive at the following weighted sum: 0 ⩾λ1⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ + λ2 � f(xk) − f(xk−1) + γk−1⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥γk−1dk−1 − 1 L(∇f(xk−1) − ∇f(xk))∥2 � + λ3 � f(xk−1) − f(xk) − γk−1⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥γk−1dk−1 − 1 L(∇f(xk−1) − ∇f(xk))∥2 � + λ4 � ⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ∇f(xk)⟩ − ∥∇f(xk)∥2 + βk−1⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ � 7 which can be reformulated exactly as (expand both expressions and observe that all terms match) 0 ⩾∥dk∥2 − (1 + q)2 4q ∥∇f(xk)∥2 + 4β2 k−1q (1 − q)2 ���dk−1 − 1+q 2Lγk−1q∇f(xk−1) + 2βk−1(1+q)−Lγk−1(1−q)2 4βk−1Lγk−1q ∇f(xk) ��� 2 , ⩾∥dk∥2 − (1 + q)2 4q ∥∇f(xk)∥2, thereby arriving to the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In Appendix A we numerically showcase the tightness of the worst-case bounds (12) for PRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' By tightness, we mean that we verified numerically that there exist n ∈ N, functions f ∈ Fµ,L and xk−1, dk−1 ∈ Rn such that ∥dk∥2 = (1+q)2/4q∥∇f(xk)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This is done by exhibiting feasible points to (D) (obtained by solving (D) numerically for η = 1) for different values of the inverse condition number q and ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Those feasible points were verified through other (independent) software [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The following rate is a direct consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Perhaps surprisingly, the following guaranteed convergence rate for PRP corresponds to that of gradient descent with an exact line search (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 with ǫ = 0) when the condition number is squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 (Worst-case bound for PRP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Let f ∈ Fµ,L, and xk, dk ∈ Rn and xk+1, dk+1 ∈ Rn be generated by respectively k ⩾ 0 and k + 1 iterations of the PRP method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', (M) with η = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' It holds that f(xk+1) − f⋆ ⩽ �1 − q2 1 + q2 �2 (f(xk) − f⋆) , with q ≜ µ/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The desired claim is a direct consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 with ǫ = 1−q 1+q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' That is, the PRP scheme can be seen as a descent method with direction dk satisfying ∥dk − ∇f(xk)∥ ⩽ ǫ∥∇f(xk)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' As a take-away from this theorem, we obtained an improved bound on the convergence rate of PRP, but possibly not in the most satisfying way: this analysis strategy does not allow beating steepest de- scent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Furthermore, this bound is tight for one iteration assuming that the current search direction satisfies ∥dk∥2/∥∇f(xk)∥2 = (1+q)2/4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' However, it does not specify whether such an angle can be achieved on the same worst-case instances as those where Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In other words, there might be no worst-case instances where the bounds (6) and (12) are tight simultaneously, possibly leaving room for improvement in the analysis of PRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We show in the sequel that we could indeed slightly improve this bound by taking into account the history of the method in a more appropriate way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The only worst-case complexity result that we are aware of in the context of PRP for smooth strongly convex problems was provided by Polyak in [1, Theorem 2]: f(xk+1) − f⋆ ⩽ q 1 + 1 q2 (f(xk) − f⋆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This bound is about two times worse compared to the rate achieved by gradient descent (1−q/1+q) when the condition number is put to the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' From what we can tell, this is due to two main weaknesses in the proof of Polyak [1, Theorem 2]: a weaker analysis of gradient descent, and a weaker analysis of the direction (and in particular that ∥dk∥2/∥∇f(xk)∥2 ⩽ 1 + 1/q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' That is, whereas gradient descent with exact line searches is guaranteed to achieve an accuracy f(xk) − f⋆ ⩽ ε in O(1/q log 1/ε), our analysis provides an O(1/q2 log 1/ε) guarantee for PRP, where Polyak’s guarantee for PRP is O(1/q3 log 1/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' As a reference, note that the lower complexity bound (achieved by a few methods, including many variations of Nesterov’s accelerated gradients) is of order O( � 1/q log 1/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 A worst-case bound for Fletcher-Reeves (FR) Similar to the obtaining of the bound for PRP, our bound for FR follows from solving (D) (for η = 0) in closed-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We start by quantifying the quality of the search direction with respect to the steepest descent direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For doing that, we first establish the following bound on the FR update parameter βk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 (Bound on βk−1 for FR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Let f ∈ Fµ,L, and let xk−1, dk−1 ∈ Rn and xk, dk be generated by the FR method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', (M) with η = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For any ck−1 ∈ R such that ∥dk−1∥2/∥∇f(xk−1)∥2 = ck−1, where ck−1 > 1, it holds that: 0 ⩽ βk−1 ⩽ 1 ck−1 � 1 − q + 2 � (ck−1 − 1)q �2 4q , (13) where q ≜ µ/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' First, note that βk−1 ⩾ 0 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The other part of the proof consists of the following weighted sum of inequalities: relation between ∇f(xk−1) and dk−1 with weight λ1 = γk−1(L + µ) − 2√ βk−1 √ (ck−1−1)ck−1 : 0 = ⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ − ∥∇f(xk−1)∥2, optimality condition of the line search with weight λ2 = 2 ck−1 − γk−1(L + µ): 0 = ⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ , definition of βk−1 with weight λ3 = √ ck−1−1 √ βk−1ck−1 : 0 = ∥∇f(xk)∥2 − βk−1∥∇f(xk−1)∥2, initial condition on the ratio ∥dk−1∥2 ∥∇f(xk−1)∥2 with weight λ4 = −γ2 k−1Lµ + √ βk−1 ck−1√ (ck−1−1)ck−1 : 0 = ∥dk−1∥2 − ck−1∥gk−1∥2 smoothness and strong convexity of f between xk−1 and xk, with weight λ5 = L − µ: 0 ⩾ − f(xk−1) + f(xk) + ⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xk−1 − xk⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥xk−1 − xk − 1 L(∇f(xk−1) − ∇f(xk))∥2 =f(xk) + γk−1⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥γk−1dk−1 − 1 L(∇f(xk−1) − ∇f(xk))∥2 smoothness and strong convexity of f between xk and xk−1, with weight λ6 = λ5: 0 ⩾ − f(xk) + f(xk−1) + ⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xk − xk−1⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥xk−1 − xk − 1 L(∇f(xk−1) − ∇f(xk))∥2 =f(xk−1) − γk−1⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥γk−1dk−1 − 1 L(∇f(xk−1) − ∇f(xk))∥2 9 The weighted sum can be written as: 0 ⩾ λ1 � ⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ − ∥∇f(xk−1)∥2� + λ2 [⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩] + λ3 � ∥∇f(xk)∥2 − βk−1∥∇f(xk−1)∥2� + λ4 � ∥dk−1∥2 − ck−1∥gk−1∥2� + λ5 � f(xk) + γk−1⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥γk−1dk−1 − 1 L(∇f(xk−1) − ∇f(xk))∥2� + λ6 � f(xk−1) − γk−1⟨∇f(xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩ + 1 2L∥∇f(xk−1) − ∇f(xk)∥2 + µ 2(1−µ/L)∥γk−1dk−1 − 1 L(∇f(xk−1) − ∇f(xk))∥2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' which can be reformulated exactly as (expand the expressions and observe that all terms match): 0 ⩾∥∇f(xk)∥2 − ν(βk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' γk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ck−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' L)∥∇f(xk−1)∥2 + ����� 4 � βk−1 (ck−1 − 1)c3 k−1 dk−1 − 4 � βk−1ck−1 ck−1 − 1 ∇f(xk−1) + 4 � ck−1 − 1 βk−1ck−1 ∇f(xk) ����� 2 ⩾∥∇f(xk)∥2 − ν(βk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' γk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ck−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' L)∥∇f(xk−1)∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' where ν(βk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' γk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ck−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' L) = 2 � 1 − 1 ck−1 � βk−1 − ck−1γ2 k−1Lµ + γk−1(L + µ) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' So, we have: βk−1 ⩽ ν(βk−1, γk−1, ck−1, µ, L) ⇔ βk−1 − 2 � 1 − 1 ck−1 � βk−1 ⩽ −ck−1γ2 k−1Lµ + γk−1(L + µ) − 1 ⇒ βk−1 − 2 � 1 − 1 ck−1 � βk−1 ⩽ max γ � −ck−1γ2 k−1Lµ + γk−1(L + µ) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Because, −ck−1γ2 k−1Lµ + γk−1(L + µ) − 1 is a concave function in γk−1, its maximum can be achieved by differentiating the term with respect to γk−1, equating it to 0, and then solving for γk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The corresponding maximum value is equal to (L+µ)2/4ck−1Lµ−1 and achieved at γk−1 = L+µ/2ck−1Lµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Hence, the last inequality becomes: βk−1−2 � 1 − 1 ck−1 � βk−1 − (L + µ)2 4ck−1Lµ + 1 ⩽ 0 ⇔ �� βk−1 �2 − 2 � 1 − 1 ck−1 � βk−1 + �� 1 − 1 ck−1 �2 − (L + µ)2 4ck−1Lµ − �� 1 − 1 ck−1 �2 + 1 ⩽ 0 ⇔ � � βk−1 − � 1 − 1 ck−1 �2 ⩽ (L + µ)2 4ck−1Lµ + ✁1 − 1 ck−1 − ✁1 = 1 ck−1 �(L + µ)2 4Lµ − 1 � ⇔ � βk−1 ⩽ � 1 − 1 ck−1 + � (L + µ)2 4ck−1Lµ − 1 ck−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 10 Thereby, squaring both sides (which are nonnegative) of the last inequality and then through some algebra, we reach βk−1 ⩽ 1 + (L − µ) ck−1 � (ck−1 − 1) µL + µ2 − 6µL + L2 4ck−1µL = 1 ck−1 � 1 − q + 2 � (ck−1 − 1)q �2 4q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' As βk−1 ⩾ 0 by definition, we have thus proven the desired statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Next, we prove a bound quantifying the quality of the search directions of FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 (Worst-case search direction for FR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Let f ∈ Fµ,L, and let xk−1, dk−1 ∈ Rn and xk, dk be generated by the FR method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', (M) with η = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For any ck−1 ∈ R such that ∥dk−1∥2/∥∇f(xk−1)∥2 = ck−1, where ck−1 > 1, it holds that: ∥dk∥2 ∥∇f(xk)∥2 ⩽ ck ≜ 1 + � 1 − q + 2 � (ck−1 − 1)q �2 4q , (14) with q ≜ µ/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Equivalently, ∥dk − ∇f(xk)∥ ⩽ ǫ∥∇f(xk)∥ holds with ǫ = � 1 − 1/ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The proof consists of the following weighted sum of inequalities: optimality condition of the line search with weight λ1 = 2βk−1: 0 = ⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩, the quality of the search direction with weight λ2 = β2 k−1: 0 = ∥dk−1∥2 − ck−1∥∇f(xk−1)∥2, definition of βk−1 with weight λ3 = −ck−1βk−1: 0 = ∥∇f(xk)∥2 − βk−1∥∇f(xk−1)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The weighted sum can be written as 0 ⩾λ1 [⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk−1⟩] + λ2 � ∥dk−1∥2 − ck−1∥∇f(xk−1)∥2� + λ3 � −∥∇f(xk)∥2 + βk−1∥∇f(xk−1)∥2� , and can be reformulated exactly as 0 ⩾ ∥dk∥2 − (1 + ck−1βk−1)∥∇f(xk)∥2 ⇔ ∥dk∥2 ⩽ (1 + ck−1βk−1)∥∇f(xk)∥2 ⩽ \uf8eb \uf8ec \uf8ed1 + � 1 − q + 2 � (ck−1 − 1)q �2 4q \uf8f6 \uf8f7 \uf8f8 ∥∇f(xk)∥2, where in the last line we have used the upper bound on βk−1 from (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Similar to PRP, we compare this last bound with the worst example that we were able to find numerically (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', worst feasible points to (D)) in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Thereby, we conclude tightness of the bound on the quality of the search direction (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' That is, we claim that for all values of q and ck−1, there exist n ∈ N, functions f ∈ Fµ,L and xk−1, dk−1 ∈ Rn such that the bound from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 is achieved with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' That being said, this bound only allows obtaining unsatisfactory convergence results for FR, although not letting much room for improvements, as showed in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 11 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 (Worst-case bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Let f ∈ Fµ,L, and xk, dk ∈ Rn and xk+1, dk+1 ∈ Rn be generated by respectively k ⩾ 0 and k + 1 iterations of the FR method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=', (M) with η = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' It holds that f(xk+1) − f⋆ ⩽ � 1 − q 1−ǫk 1+ǫk 1 + q 1−ǫk 1+ǫk �2 (f(xk) − f⋆) , with ǫk = � (1−q)2(k−1)2/4q+(1−q)2(k−1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The desired claim is a direct consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Indeed, it follows from ck ⩽ 1 + � 1 − µ L + 2 � (ck−1 − 1) µ L �2 4µ L (the guarantee from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 for the quality of the search direction) which we can rewrite as � ck+1 − 1 ⩽ 1 − q + 2 � (ck − 1)q 2√q with c0 −1 = 0, thereby arriving to ck ⩽ 1+k2(1−q)2/4q by recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3, we compute ǫk = � 1 − 1/ck ⩽ � (1−q)2k2/4q+(1−q)2k2 and reach the desired statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' It is clear that the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 is rather very disappointing, as the convergence rate of the FR variation can become arbitrarily close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' While this guarantee clearly does not give a total and fair picture of the true behavior of FR in practice, it seems in line with the practical necessity to effectively restart the method as it runs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The next section is devoted to studying the possibilities for obtaining tighter guarantees for PRP and FR beyond the simple single-iteration worst-case analyses of this section (which are tight for one iteration, but not beyond), showing that we cannot hope to improve the convergence rates for those methods without further assumptions on the problems at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 3 Obtaining better worst-case bounds for NCGMs In the previous section, we established closed-form bounds on ratios between consecutive function values for NCGMs by characterizing worst-case search directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Albeit being tight for the analysis of NCGMs for one iteration, the bounds that we obtained are disappointingly inferior to those of the vanilla gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In this section, we investigate the possibility of obtaining better worst-case guarantees for NCGMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For doing this using our framework, one natural possibility for us is to go beyond the study of a single iteration (since our results appear to be tight for this situation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Therefore, in contrast with the previous section, we now proceed only numerically and provide worst-case bounds without closed-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' More precisely, we solve the corresponding PEPs in two regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In short, the difference between the two regimes resides in the type of bounds under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The first type of bounds can be thought to as a “Lyapunov” approach which studies N iterations of (M) starting at some iterate (xk, dk) (for which we “neglect” how it was generated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In this first setup, we numerically compute worst-case bounds on f(xk+N)−f⋆/f(xk)−f⋆ for different values of N (namely N ∈ {1, 2, 3, 4}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' As for the results of Section 2, we quantify the quality of the couple (xk, dk) by requiring that ∥dk∥2 ⩽ ck∥∇f(xk)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' When N = 1, this setup corresponds to that of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Stemming from the fact the worst-case behaviors observed for N = 1 might not be compatible between consecutive iterations, we expect the quality of the bounds to improve with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Of course, the main weakness of this approach stands in the fact that we neglect how (xk, dk) was generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' As a natural complementary alternative, the second type of bounds studies N iterations of (M) initi- ated at x0 (with d0 = ∇f(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Whereas the first type of bounds is by construction more conservative, 12 it has the advantage of being recursive: it is valid for all k ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' On the other side, the second type of bounds is only valid for the first N iterations (the bound cannot be used recursively), but it cannot be improved at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' That is, we study exact worst-case ratio f(xN)−f⋆/f(x0)−f⋆ for a few different values of N (namely N ∈ {1, 2, 3, 4}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In this setup, we obtain worst-case bounds that are only valid close to initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' However, it has the advantage of being unimprovable, as we do not neglect how the search direction is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Section organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This section is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' First, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 presents the performance es- timation problems for (M) specifically for computing the worst-case ratios f(xk+N)−f⋆/f(xk)−f⋆ and f(xN)−f⋆/f(x0)−f⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Then, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 presents our findings for respectively PRP and FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Details on how we managed to solve the resulting nonconvex QCQPs numerically are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 Computing numerical worst-case scenarios Similar to (10), the problem of computing the worst-case ratio f(xk+N)−f⋆/f(xk)−f⋆ is framed as the following nonconvex maximization problem (for c ⩾ 1 and q ≜ µ/L): ρN(q, c) ≜ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize f,n,{xk+i},{dk+i}i, {γk+i}i,{βk+i}i f(xk+N)−f⋆ f(xk)−f⋆ subject to n ∈ N, f ∈ Fq,1(Rn), dk, xk ∈ Rn, ⟨∇f(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk⟩ = ∥∇f(xk)∥2, ∥dk∥2 ⩽ c∥∇f(xk)∥2, \uf8eb \uf8ed xk+1 dk+1 βk \uf8f6 \uf8f8 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' , \uf8eb \uf8ed xk+N dk+N βk+N−1 \uf8f6 \uf8f8 generated by (M) from xk and dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (BLyapunov) We proceed similarly for f(xN)−f⋆/f(x0)−f⋆: ρN,0(q) ≜ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize f,n,{xk+i},{dk+i}i, {γk+i}i,{βk+i}i f(xN)−f⋆ f(x0)−f⋆ subject to n ∈ N, f ∈ Fq,1(Rn), x0 ∈ Rn, d0 = ∇f(x0), \uf8eb \uf8ed x1 d1 β0 \uf8f6 \uf8f8 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' , \uf8eb \uf8ed xN dN βN−1 \uf8f6 \uf8f8 generated by (M) from xk and dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (Bexact) Obviously, ρN(q, c) ⩾ ρN,0(q) for any c ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We solve (BLyapunov) and (Bexact) numerically to high precision (details in Appendix C) for N ∈ {1, 2, 3, 4} and report the corresponding results in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In the numerical experiments, we fix the values of c using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 for PRP in (BLyapunov), thereby computing ρN � q, (1+q)2/4q � whose results are provided in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For FR, c can become arbitrarily bad and we therefore only compute ρN,0(q) via (Bexact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The numerical values for ρN,0(q) respectively PRP and FR are provided in Figure 3 and Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The next sections discuss and draw a few conclusions from the numerical worst-case convergence results for PRP and FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 Improved worst-case bounds for PRP Figure 2 reports the worst-case values of the “Lyapunov” ratio f(xk+N)−f⋆/f(xk)−f⋆ as a function of the inverse condition number q ≜ µ/L and for c = (1+q)2/4q and N = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This worst-case ratio seem to improve as N grows, but does not outperform gradient descent with exact line search (GDEL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The diminishing improvements with N also suggests the worst-case performance of PRP in this regime might not outperform GDEL even for larger values of N ⩾ 4, albeit probably getting close to the same asymptotic worst-case convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 13 As a complement, Figure 3 shows how PRP’s worst-case ratio fN −f⋆/f0−f⋆ evolves as a function of q for N = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The worst-case performance of PRP in this setup seems to be similar to that of GDEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Further, for small q (which is typically the only regime of interest for large-scale optimization), PRP’s worst- case performance seems to be slightly better than than of GDEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' On the other hand, for larger q, PRP performs slightly worse than GDEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' As a conclusion, we believe there is no hope to prove uniformly better worst-case bounds for PRP than those for GDEL for base smooth strongly convex minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' However, we might be able to prove improvements for small values of q at the cost of possibly very technical proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' As for the Lyapunov approach, the numerical results from this section could be improved by further increasing N, but we believe that the transient does not suggest this direction to be promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We recall that we computed the bounds by solving an optimization problem whose feasible points correspond to worst-case examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Therefore, the numerical results provided in this section are backed-up by numerically constructed examples on which PRP behaves “badly” (more details in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 1 q N � ρN � q, (1+q)2 4q � GDEL : fk+1−f⋆/fk−f⋆ PRP:N = 1 PRP:N = 2 PRP:N = 3 PRP:N = 4 (1 − √q)2 Figure 2: This figure reports the worst-case values for the “Lyapunov” ratio N� f(xk+N)−f⋆/f(xk)−f⋆ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' the (inverse) condition ratio q ≜ µ L for PRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We compute ρN(q, c) with c = (1+q)2/4q for N = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' As N increases, the worst-case N� fk+N−f⋆/fk−f⋆ improves, but remains worse than that of gradient descent with exact line search (GDEL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The curve (1 − √q)2 (orange) corresponds to the rate of the lower complexity bounds for this class of problems [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 14 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 1 q N� ρN,0(q) GDEL : f1−f⋆ f0−f⋆ PRP:N = 1 PRP:N = 2 PRP:N = 3 PRP:N = 4 (1 − √q)2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 1 q N� ρN,0(q) GDEL : f1−f⋆ f0−f⋆ PRP:N = 1 PRP:N = 2 PRP:N = 3 PRP:N = 4 (1 − √q)2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 1 q N� ρN,0(q) GDEL : f1−f⋆ f0−f⋆ PRP:N = 1 PRP:N = 2 PRP:N = 3 PRP:N = 4 (1 − √q)2 Figure 3: This figure reports the worst-case values for the ratio N� fN −f⋆/f0−f⋆ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' q for PRP for N = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For N = 1, PRP and GDEL perform the same iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For N = 2, 3, 4, the worst-case ratio of PRP is better than that of GDEL for q ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The curve (1 − √q)2 (orange) corresponds to the rate of the lower complexity bounds for this class of problems [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 Improved worst-case bounds for FR Figure 4 reports the worst-case values for the ratio fN −f⋆/f0−f⋆ as a function of q, for N ∈ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The convergence bounds appears to be marginally better than GDEL for some sufficiently small inverse condition numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Further, the range of values of q for which there is an improvement appears to be decreasing with N ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Beyond this range, the worst-case values become significantly worse than that of GDEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Though apparently not as dramatic as the worst-case bound from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2, the quality of the bound appears to be decreasing with N, which stands in line with the practical need to restart the method [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' As in the previous section, we recall that those curves were obtained by numerically constructing “bad” worst-case examples satisfying our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In other words, there is no hope to obtain better results without adding assumptions or changing the types of bounds under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 1 q N� ρN,0(q) GDEL : f1−f⋆ f0−f⋆ FR: N = 1 FR: N = 2 FR: N = 3 FR: N = 4 (1 − √q)2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 1 q N� ρN,0(q) GDEL : f1−f⋆ f0−f⋆ FR: N = 1 FR: N = 2 FR: N = 3 FR: N = 4 (1 − √q)2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 1 q N� ρN,0(q) GDEL : f1−f⋆ f0−f⋆ FR: N = 1 FR: N = 2 FR: N = 3 FR: N = 4 (1 − √q)2 Figure 4: This figure reports the worst-case values for the ratio N� fN −f⋆/f0−f⋆ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' q for FR for N = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For N = 1, FR and GDEL perform the same iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For N = 2, 3, 4, the worst-case bound for FR is slightly better than that of GDEL for small enough values of q, and gets larger than GDEL for larger values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The range of q for which FR is better than GDEL gets smaller as N ⩾ 2 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The curve (1−√q)2 (orange) corresponds to the rate of the lower complexity bounds for this class of problems [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 4 Conclusion This works studies the iteration complexity of two variants of nonlinear conjugate gradients, namely the Polak-Ribière-Polyak (PRP) and the Fletcher-Reeves (FR) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We provide new improved complexity bounds for both those methods, and show that albeit unsatisfying, not much can a priori be gained from a worst-case perspective, as both method appear to behave similar or worse to regular steepest descent in the worst-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Further, those results suggest that explaining the good practical performances of NCGMs might be out of reach for traditional worst-case complexity analyses on classical classes of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' A limitation of this work stands in the fact that only somewhat “ideal” variants of nonlinear conjugate gradients were considered, as we make explicit use of exact line search procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' However, there is a priori no reason to believe that different line search procedures would help avoiding the possibly bad worst- case behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Further, the performance estimation methodology allows tackling such alternate line search procedures into account, so the same methodology could be applied for tackling those questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We let such investigations for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 16 Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' All the numerical results in this paper were obtained on MIT Supercloud Computing Cluster with Intel-Xeon-Platinum-8260 processor with 48 cores and 128 GB of RAM running Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 LTS with Linux 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='250-llgrid-10ms kernel [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We used JuMP—a domain specific modeling language for mathematical optimization embedded in the open-source programming language Julia [35]—to model the optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' To solve the optimization problems, we use the following solvers: Mosek 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 [36], KNITRO 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='0 [37], and Gurobi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='0, which are free for academic use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The relative feasibility tolerance and relative optimality tolerance of all the solvers are set at 1e-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We validated the “bad” worst-case scenarios produced by our methodology using the PEPit package [32], which is an open-source Python library allowing to use the PEP framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The codes used to generate and validate the results in this paper are available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='com/Shuvomoy/NCG-PEP-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' References [1] Boris T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Polyak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The conjugate gradient method in extremal problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' USSR Computational Mathe- matics and Mathematical Physics, 9(4):94–112, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' [2] Elijah Polak and Gerard Ribiere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Note sur la convergence de méthodes de directions conjuguées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Revue française d’informatique et de recherche opérationnelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Série rouge, 3(16):35–43, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' [3] Reeves Fletcher and Colin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Reeves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Function minimization by conjugate gradients.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Athena Scientific Belmont, MA, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 19 Organization of the appendix In what follows, we report detailed numerical results and computations that are not presented in the core of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Table 1 details the organization of this additional material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Section Content Appendix A Numerical illustration of tightness of the worst-case search direction (12) for PRP and (14) for FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Appendix B Nonconvex QCQP reformulation of (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Appendix C Nonconvex QCQP reformulation of (BLyapunov) (Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Nonconvex QCQP reformulation of (Bexact) (Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The relative gap between the lower bounds and upper bounds (Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Appendix D Description of the custom spatial branch-and-bound algorithm that is used to solve the nonconvex QCQP formulations of the performance estimation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Table 1: Organization of the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We denote by (· ⊙ ·) : Rn ×Rn → Rn×n the symmetric outer product, that is, for any x, y ∈ Rn: x ⊙ y = 1 2 � xy⊤ + yx⊤� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' A Tightness of the worst-case search directions Figure 5 and Figure 6 illustrate the tightness of the bounds (12) and (14) for PRP and FR respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' That is, we compare the numerical bounds (discrete points) with closed-forms (continuous lines) for a few different values of q and ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Numerical bounds are obtained by solving (D) with η = 1 for PRP and η = 0 for FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' These numerical examples strongly suggest that our bounds cannot be improved in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Absolute relative differences between closed-form expressions and numerical ratios is less than 1e − 6 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 3 q ck(µ, L, ck−1) (1+q)2/4q PRP: ck−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='01 PRP: ck−1 = 2 PRP: ck−1 = 10 PRP: ck−1 = 50 Figure 5: Worst-case bound (12) (continuous line) and numerical bounds (discrete points) from (D) with η = 1 (for PRP) for different values of q and ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The bound appear to match to numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8 0 20 40 60 q ck(µ, L, ck−1) Analytical: ck−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='01 Analytical: ck−1 = 2 Analytical: ck−1 = 10 Analytical: ck−1 = 50 Numerical: ck−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='01 Numerical: ck−1 = 2 Numerical: ck−1 = 10 Numerical: ck−1 = 50 Figure 6: Worst-case bound (14) (continuous line) and numerical bounds (discrete points) from (D) with η = 0 (for FR) for different values of q and ck−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The bound appear to match to numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 21 B Nonconvex QCQP reformulation of (D) To reformulate (D) as a nonconvex QCQP, we introduce the following Grammian matrices that is a common step in performance estimation literature [9, 14]: H = [xk−1 | gk−1 | gk | dk−1] ∈ Rn×4, G = H⊤H ∈ S4 +, rank G ⩽ n F = [fk−1 | fk] ∈ R1×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' (15) Because we maximize over n, we can ignore rank G ⩽ n and also confine H ∈ R4×4 without loss of generality [9, Theorem 5], [14, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We next define the following notation for selecting columns and elements of H and F: xk−1 = e1, gk−1 = e2, gk = e3, dk−1 = e4, (all in R4) fk−1 = e1, fk = e2, (all in R2), xk = xk−1 − γk−1dk−1, (all in R4), dk = gk + βk−1dk−1, (all in R4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' (16) This ensures that xi = Hxi, gi = Hgi, di = Hdi, fi = Ffi, for all i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Next, for appropriate choices of matrices Ai,j, Bi,j, Ci,j, �Ci,j, Di,j, �Di,j, Ei,j, and vector ai,j, we can ensure that the following reformulations hold for all i, j ∈ I: ⟨gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xi − xj⟩ = tr GAi,j, ∥xi − xj∥2 = tr GBi,j, ∥gi − gj∥2 = tr GCi,j, ∥gi∥2 = tr GCi,⋆, ∥di − dj∥2 = tr G �Ci,j, ∥di∥2 = tr G �Ci,⋆, ⟨gi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' gj⟩ = tr GDi,j, ⟨gi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dj⟩ = tr G �Di,j, ⟨gi − gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xi − xj⟩ = tr GEi,j, fj − fi = Fai,j, (17) where, using (16), we define Ai,j = gj ⊙ (xi − xj) Bi,j = (xi − xj) ⊙ (xi − xj) Ci,j = (gi − gj) ⊙ (gi − gj), Ci,⋆ = gi ⊙ gi, �Ci,j = (di − dj) ⊙ (di − dj), �Ci,⋆ = di ⊙ di, Di,j = gi ⊙ gj, �Di,j = gi ⊙ dj, Ei,j = (gi − gj) ⊙ (xi − xj), ai,j = fj − fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' (18) Using (18) and using the definition of G = H⊤H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' where H ∈ R4×4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' we can write (D) as the following 22 nonconvex QCQP: ck(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' ck−1) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize tr G �Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ subject to tr G �Dk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k−1 = tr GCk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr G �Dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k−1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr GAk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' βk−1 × tr GCk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ = tr G (Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ − ηDk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr G �Ck−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ ⩽ ck−1 tr GCk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Fai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j + tr G � Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j + 1 2(1− µ L ) � 1 LCi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j + µΘi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j − 2 µ LEi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j � � ⩽ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' j ∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j = Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' j ∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' G = H⊤H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr GCk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (19) where G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' γk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' βk−1 are the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This nonconvex QCQP can be solved to certifiable global optimality using a custom spatial branch-and-bound algorithm described in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' C Nonconvex QCQP reformulations of (BLyapunov) and (Bexact) Similar to the reformulations from Appendix D, (BLyapunov) and (Bexact) can be cast as nonconvex QCQPs, where the number of nonconvex constraints grow quadratically with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Thereby, solving them to global optimality in reasonable time for N = 3, 4 is already challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Therefore, rather than solving the nonconvex QCQP reformulations of (BLyapunov) and (Bexact) directly, we compute upper bounds and lower bounds by solving more tractable nonconvex QCQP formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We then show that the relative gap between the upper and lower bounds is less than 10% which thereby indicates that there is essentially no room for further improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 Nonconvex QCQP reformulation of (BLyapunov) This section presents our upper bound ρN(q, c) and lower bound ρN(q, c) on ρN(q, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 Computing ρN(q, c) Using (7), we have the following relaxation of (BLyapunov), which provides upper bounds on ρN(q, c): \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize f,n,{xk+i}i∈[0:N],{dk+i}i∈[0:N] f(xk+N)−f⋆ f(xk)−f⋆ subject to n ∈ N, f ∈ Fµ,L(Rn), xk+i, dk+i ∈ Rn, i ∈ [0 : N] ∥dk∥2 ⩽ c∥∇f(xk)∥2, ⟨∇f(xk+i+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk+i⟩ = 0, i ∈ [0 : N − 1], ⟨∇f(xk+i+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xk+i − xk+i+1⟩ = 0, i ∈ [0 : N − 1], ⟨∇f(xk+i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk+i⟩ = ∥∇f(xk+i)∥2, i ∈ [0 : N − 1], dk+i+1 = gk+i+1 + βk+idk+i, i ∈ [0 : N − 2], βk+i = ∥gk+i+1∥2−η⟨gk+i+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' gk+i⟩ ∥gk+i∥2 , i ∈ [0 : N − 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (20) 23 Using the notation gi ≜ ∇f(xi) and fi ≜ f(xi) again, and then applying an homogeneity argument, we write (20) as: ρN(q, c) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize fk+N − f⋆ subject to n ∈ N, f ∈ Fµ,L(Rn), xk+i, dk+i ∈ Rn, i ∈ [0 : N] ∥dk∥2 ⩽ c∥gk∥2, ⟨gk+i+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk+i⟩ = 0, i ∈ [0 : N − 1], ⟨gk+i+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xk+i − xk+i+1⟩ = 0, i ∈ [0 : N − 1], ⟨gk+i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk+i⟩ = ∥gk+i∥2, i ∈ [0 : N − 1], dk+i+1 = gk+i+1 + βk+idk+i, i ∈ [0 : N − 2], βk+i−1 = ∥gk+i∥2−η⟨gk+i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' gk+i−1⟩ ∥gi−1∥2 , i ∈ [1 : N − 1], fk − f⋆ = 1, \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (21) where f, n, {xk+i}i∈[0:N], {dk+i}i∈[0:N] are the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Define I⋆ N = {⋆, k, k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' , k + N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Next, note that the equation dk+i+1 = gk+i+1 + βk+idk+i for i ∈ [0 : N − 2], can be written equivalently as the following set of equations: χj,i = χj,i−1βk+i−1, i ∈ [1 : N − 1], j ∈ [0 : i − 2], χi−1,i = βk+i−1, i ∈ [1 : N − 1], dk+i = gk+i + i−1 � j=1 χj,igk+j + χ0,idk, i ∈ [1 : N − 1], (22) where we have introduced the intermediate variables χj,i, which will aid us in formulating (21) as a nonconvex QCQP down the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Next, using (22) and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1, we can equivalently write (21) as: ρN(q, c) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize fk+N − f⋆ subject to n ∈ N, fi ⩾ fj + ⟨gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xi − xj⟩ + 1 2(1− µ L ) � 1 L∥gi − gj∥2 +µ∥xi − xj∥2 − 2 µ L ⟨gi − gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xi − xj⟩ � , i, j ∈ I⋆ N, ∥dk∥2 ⩽ c∥gk∥2, ⟨gk+i+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk+i⟩ = 0, i ∈ [0 : N − 1], ⟨gk+i+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xk+i − xk+i+1⟩ = 0, i ∈ [0 : N − 1], ⟨gk+i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk+i⟩ = ∥gk+i∥2, i ∈ [0 : N − 1], βk+i−1 = ∥gk+i∥2−η⟨gk+i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' gk+i−1⟩ ∥gk+i−1∥2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i = χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i−1βk+i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' j ∈ [0 : i − 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' χi−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i = βk+i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk+i = gk+i + �i−1 j=1 χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='igk+j + χ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='idk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' fk − f⋆ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' g⋆ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' x⋆ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' f⋆ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' fi}i∈I⋆ N ⊂ Rn × Rn × R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {di}i∈I⋆ N\\{k+N} ⊂ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {βk+i}i∈[0:N−2] ⊂ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i}j∈[0:N−2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i∈[0:N−1] ⊂ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (23) where {xk+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' gk+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' fk+i}i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {dk+i}i∈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {βk+i}i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i}j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i are the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Note that we have set g⋆ = 0, x⋆ = 0, and f⋆ = 0 without loss of generality, because both the objective and the function class are closed and invariant under shifting variables and function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We introduce Grammian matrices again: H = [dk | gk | gk+1 | gk+2 | · · · | gk+N | xk | xk+1 | xk+2 | · · · | xk+N] ∈ Rn×(2N+3), G = H⊤H ∈ S(2N+3) + , rank G ⩽ n, F = [fk | fk+1 | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' | fk+N] ∈ R1×(N+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' (24) 24 As we maximize over n, we can ignore the constraint rank G ⩽ n, and confine H to be in R(2N+3)×(2N+3) without loss of generality [9, Theorem 5], [14, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Next, define the following notation for selecting columns and elements of H and F: x⋆ = 0 ∈ R2N+3, dk = e1 ∈ R2N+3, gk+i = ei+2 ∈ R2N+3, xk+i = e(N+2)+(i+1) ∈ R2N+3, f⋆ = 0, fk+i = ei+1 ∈ R(N+1), dk+i = gk+i + i−1 � j=1 χj,igk+j + χ0,idk ∈ R2N+3, (25) where i ∈ [0 : N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This ensures that we have xi = Hxi, gi = Hgi, di = Hdi,, fi = Ffi for all i ∈ I⋆ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For appropriate choices of matrices Ai,j,Bi,j, Ci,j, �Ci,j, Di,j, �Di,j, Ei,j, and vector ai,j as defined in (17), where xi, gi, fi, di are taken from (25) now, we can ensure that the identities in (18) hold for all i, j ∈ I⋆ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Using those identities and using the definition of G = H⊤H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' where H ∈ R(2N+3)×(2N+3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' we can write (23) as the following nonconvex QCQP: ρN(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' c) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize Fa⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k+N subject to Fai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j + tr G � Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j + 1 2(1− µ L ) � 1 LCi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j + µBi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j − 2 µ LEi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j �� ⩽ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' j ∈ I⋆ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr G �Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ ⩽ c tr GCk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr G �Dk+i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k+i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [0 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr GAk+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k+i+1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [0 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr G �Dk+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k+i = tr GCk+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ i ∈ [0 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' βk+i−1 × tr GCk+i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ = tr G (Ck+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ − ηDk+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k+i−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i = χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i−1βk+i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' j ∈ [0 : i − 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' χi−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i = βk+i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Fa⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' G = H⊤H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' F ∈ RN+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' G ∈ S2N+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' H ∈ R(2N+3)×(2N+3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {βk+i}i∈[0:N−2] ⊂ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i}j∈[0:N−2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i∈[0:N−1] ⊂ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (26) where F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i}j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {βk+i}i are the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 Computing ρN(q, c) We now discuss how to compute ρN(q, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Once we have solved (26), it provides us with the corresponding CG update parameters, which we denote by βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' If we can solve (BLyapunov) with the CG update parameters fixed to the βi found from (26), then it will provide us with the lower bound ρN(µ, L, c)s along with a bad function, which we show now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Using the notation gi ≜ ∇f(xi) and fi ≜ f(xi), then applying the homogeneity argument, we can compute ρN(q, c) by finding a feasible solution to the following optimization problem: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize fk+N − f⋆ subject to n ∈ N, f ∈ Fµ,L(Rn), xk+i, dk+i ∈ Rn, i ∈ [0 : N] ∥dk∥2 ⩽ c∥gk∥2, γk+i = argminγf(xk+i − γdk+i), i ∈ [0 : N − 1], xk+i+1 = xk+i − γk+idk+i, i ∈ [0 : N − 1], dk+i+1 = gk+i+1 + βk+idk+i, i ∈ [0 : N − 2], βk+i−1 = ∥gk+i∥2−η⟨gk+i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' gk+i−1⟩ ∥gk+i−1∥2 , i ∈ [1 : N − 1], fk − f⋆ = 1, \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (27) 25 where f, n, {xk+i}, {dk+i}i, {γk+i}i are the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Next, note that the NCGM iteration scheme in (27) can be equivalently written as: χj,i = χj,i−1βk+i−1, i ∈ [1 : N − 1], j ∈ [0 : i − 2] χi−1,i = βk+i−1, i ∈ [1 : N − 1] αi,i−1 = γk+i−1, i ∈ [1 : N], αi,j = γk+j + i−1 � ℓ=j+1 γk+ℓχj,ℓ, i ∈ [1 : N], j ∈ [0 : i − 2], xk+i = xk − i−1 � j=1 αi,jgk+j − αi,0dk, i ∈ [1 : N], dk+i = gk+i + i−1 � j=1 χj,igk+j + χ0,idk, i ∈ [1 : N − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' (28) where we have introduced intermediate variables χj,i and αi,j which will aid us in formulating (27) as a nonconvex QCQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Define I⋆ N = {⋆, k, k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' , k + N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Now using (28), Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1, and (7), we can equivalently write (21) as: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize fk+N − f⋆ subject to n ∈ N, fi ⩾ fj + ⟨gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xi − xj⟩ + 1 2(1− µ L ) � 1 L∥gi − gj∥2 +µ∥xi − xj∥2 − 2 µ L ⟨gi − gj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xi − xj⟩ � , i, j ∈ I⋆ N, ∥dk∥2 ⩽ c∥gk∥2, ⟨gk+i+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk+i⟩ = 0, i ∈ [0 : N − 1], ⟨gk+i+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' xk+i − xk+i+1⟩ = 0, i ∈ [0 : N − 1], ⟨gk+i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' dk+i⟩ = ∥gk+i∥2, i ∈ [0 : N − 1], χj,i = χj,i−1βk+i−1, i ∈ [1 : N − 1], j ∈ [0 : i − 2] χi−1,i = βk+i−1, i ∈ [1 : N − 1] αi,i−1 = γk+i−1, i ∈ [1 : N], αi,j = γk+j + �i−1 ℓ=j+1 γk+ℓχj,ℓ, i ∈ [1 : N], j ∈ [0 : i − 2], xk+i = xk − �i−1 j=1 αi,jgk+j − αi,0dk, i ∈ [1 : N], dk+i = gk+i + �i−1 j=1 χj,igk+j + χ0,idk, i ∈ [1 : N − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' βk+i−1 = ∥gk+i∥2−η⟨gk+i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' gk+i−1⟩ ∥gk+i−1∥2 , i ∈ [1 : N − 1], fk − f⋆ = 1, g⋆ = 0, x⋆ = 0, f⋆ = 0, {xi, gi, fi}i∈I⋆ N ⊂ Rn × Rn × R, {di}i∈I⋆ N\\{k+N} ⊂ Rn, {χj,i}j∈[0:N−2],i∈[0:N−1] ⊂ R, {γk+i}i∈[0:N] ⊂ R, {αi,j}i∈[1:N],j∈[0:N−1] ⊂ R, \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (29) where {xk+i, gk+i, fk+i}i, n, {γk+i}i, {χj,i}j,i, {αi,j}i,j are the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We introduce the Gram- mian transformation: H = [xk | gk | gk+1 | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' | gk+N | dk] ∈ Rn×(N+3), G = H⊤H ∈ SN+3 + , rank G ⩽ n, F = [fk | fk+1 | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' | fk+N] ∈ R1×(N+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' (30) As we maximize over n, we again ignore the constraint rank G ⩽ n and can let H ∈ R(N+3)×(N+3) without loss of generality [9, Theorem 5], [14, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We next define the following notation for 26 selecting columns and elements of H and F: g⋆ = 0 ∈ RN+3, gk+i = ei+2 ∈ RN+3, i ∈ [0 : N], dk = eN+3 ∈ RN+3, xk = e1 ∈ RN+2, x⋆ = 0 ∈ RN+2, xk+i(α) = xk − i−1 � j=1 αi,jgk+j − αi,0dk ∈ RN+3, i ∈ [1 : N], dk+i(χ) = gk+i + i−1 � j=1 χj,igk+j + χ0,idk, i ∈ [1 : N − 1], f⋆ = 0 ∈ RN+1, fk+i = ei+1 ∈ RN+1, i ∈ [0 : N], (31) which ensure xi = Hxi, gi = Hgi, fi = Ffi, di = Hdi for i ∈ I⋆ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For appropriate choices of matrices Ai,j,Bi,j, Ci,j, �Ci,j, Di,j, �Di,j, Ei,j, and vector ai,j as defined in (17), where xi, gi, fi, di are from (31), we can ensure that the identities in (18) hold for all i, j ∈ I⋆ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Using those identities and using the definition of G = H⊤H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' where H ∈ R(N+3)×(N+3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' we can write (29) as the following nonconvex QCQP: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed maximize Fa⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='N subject to Fai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j + tr G � Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j + 1 2(1− µ L ) � 1 LCi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j + µΘi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j − 2 µ LEi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j �� ⩽ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' j ∈ I⋆ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j = Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' j ∈ I⋆ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr G �Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ ⩽ c tr GCk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr G �Dk+i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k+i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [0 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr GAk+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k+i+1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [0 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' tr G �Dk+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k+i = tr GCk+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ i ∈ [0 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i = χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i−1βk+i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' j ∈ [0 : i − 2] χi−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i = βk+i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N − 1] αi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i−1 = γk+i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' αi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j = γk+j + �i−1 ℓ=j+1 γk+ℓχj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' j ∈ [0 : i − 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' βk+i−1 × tr GCk+i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ = tr G (Ck+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='⋆ − ηDk+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k+i−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' i ∈ [1 : N − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Fa⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' G = H⊤H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' F ∈ RN+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' G ∈ SN+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' H ∈ R(N+3)×(N+3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {χj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i}j∈[0:N−2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='i∈[0:N−1] ⊂ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {γk+i}i∈[0:N] ⊂ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' {αi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j}i∈[1:N],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='j∈[0:N−1] ⊂ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (32) where G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' χ are the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Note that {Θi,j}i,j∈I⋆ N is introduced as a separate decision variable to formulate the cubic constraints arising from Bi,j as quadratic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Note that to compute ρN(q, c), it suffices to find just a feasible solution to (32), in Appendix D we will discuss how to do so using our custom spatial branch-and-bound algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' From the solution to (32) we construct the associated triplets {xi, gi, fi}i∈I⋆ N and then apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 construct the corresponding bad function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2 Nonconvex QCQP reformulation of (Bexact) Now we discuss how we compute the upper bound ρN,0(q) and lower bound ρN,0(q) to ρN,0(q) defined in (Bexact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The bound computation process is very similar to that of (BLyapunov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Observe that, in (BLyapunov), if we remove the constraint ∥dk∥2 ⩽ c∥∇f(xk)∥2, set k ≜ 0 , and then add the constraint d0 = ∇f(x0), then it is identical to (Bexact) (the constraint ⟨∇f(x0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' d0⟩ = ∥∇f(x0)∥2 in (BLyapunov) is a valid but redundant constraint for (Bexact)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 27 So, to compute the upper bound ρN,0(q), we can follow a transformation process very similar to Ap- pendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 but with a few changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In (21) and (23), we remove the constraint ∥dk∥2 ⩽ c∥gk∥2, and then add the constraint gk = dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Second, the Grammian matrices defined in (24) stays the same, and in (25) the vectors {xi, gi, fi}i∈I⋆ N stays the same except we set dk = gk = e2 ∈ R2N+3, which ensures that dk = Fdk = gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We then remove the constraint tr G �Ck,⋆ ⩽ c tr GCk,⋆ from (26) and finally set k ≜ 0 in the resultant QCQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The solution to the nonconvex QCQP will provide us the upper bound ρN,0(q) in (Bexact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' To compute the lower bound ρN,0(q), we follow the same set of changes described in the last paragraph but to (27) in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 The relative gap between the lower bounds and upper bounds Tables 2, 3, 4 record the relative gap between lower bounds and upper bounds for a few representative values of q obtained by solving the aforementioned nonconvex QCQPs associated with (BLyapunov) and (Bexact) using our custom spatial branch-and-bound algorithm described in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Note that the tables contain a few negative entries close to zero which are due to the absolute gap being of the same order as the accuracy of the solver (1e − 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For the full list for all values, we refer to our open-source code in Section 4, which also allows for computing these bounds for a user-specified value of q as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In all cases, the relative gap is less than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In most cases, it is significantly better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 N = 1 3e−8 −1e−6 3e−9 6e−8 9e−8 2e−7 2e−7 1e−6 3e−7 N = 2 2e−6 6e−7 −3e−8 9e−8 1e−7 8e−8 3e−7 8e−3 4e−4 N = 3 5e−6 5e−4 7e−3 2e−2 3e−2 4e−2 2e−2 5e−2 −3e−7 N = 4 2e−4 3e−3 2e−2 7e−2 1e−1 3e−2 4e−2 4e−2 4e−2 Table 2: Relative gaps ρN(q,c)−ρN (q,c)/ρN(q,c) for PRP with c = (1+q)2/4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 N = 2 7e−6 2e−4 2e−3 7e−3 1e−2 1e−2 2e−2 1e−2 1e−6 N = 3 5e−5 9e−4 1e−2 3e−2 5e−2 6e−2 6e−2 5e−3 −7e−6 N = 4 3e−4 4e−3 3e−2 4e−2 9e−2 9e−2 7e−2 3e−2 7e−2 Table 3: Relative gap ρN,0(q)−ρN,0(q)/ρN,0(q) for PRP where N = 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The case N = 1 is omitted, as PRP is equivalent to GDEL in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5 N = 2 9e−6 2e−4 1e−3 7e−3 1e−2 1e−2 2e−2 1e−2 8e−7 N = 3 7e−5 1e−3 1e−2 2e−2 3e−2 3e−2 3e−2 3e−7 −1e−7 N = 4 2e−4 3e−3 2e−2 3e−2 3e−2 2e−2 1e−2 1e−6 4e−2 Table 4: The relative gap ρN,0(q)−ρN,0(q)/ρN,0(q) for FR where N = 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The case N = 1 is omitted again, as in this case FR is equivalent to GDEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 28 D Custom spatial branch-and-bound algorithm This section discusses implementation details for solving the nonconvex QCQPs of this paper (namely (19), (26), or (32)) using a custom spatial branch-and-bound method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' This strategy proceeds in three stages, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Stage 1: Compute a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' First, we construct a feasible solution to the nonconvex QCQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We do that by generating a random µ-strongly convex and L-smooth quadratic function, and by applying the corresponding nonlinear conjugate gradient method on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' The corresponding iterates, gradient and function values correspond to a feasible point for the nonconvex QCQPs under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Stage 2: Compute a locally optimal solution by warm-starting at Stage 1 solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Stage 2 computes a locally optimal solution to the nonconvex QCQPs using an interior-point algorithm, warm- starting at the feasible solution produced by Stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' When a good warm-starting point is provided, interior-point algorithms can quickly converge to a locally optimal solution under suitable regularity conditions [38, 39], [40, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In the situation where the interior-point algorithm fails to converge, we go back to the feasible solution from Stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We have empirically observed that Stage 2 consistently provides a locally optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Stage 3: Compute a globally optimal solution by warm-starting at Stage 2 solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Stage 3 computes a globally optimal solution to the nonconvex QCQP using a spatial branch-and-bound algorithm [41, 42], warm-starting at the locally-optimal solution produced by Stage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' For details about how spatial branch-and-bound algorithm works, we refer the reader to [15, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In stage 3, the most numerically challenging nonconvex quadratic constraint in (19), (26) or (32) is G = PP ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' To solve those problems in reasonable times, we use the lazy constraints approach, [15, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' In short, we replace the constraint G = PP ⊤ by the infinite set of linear constraints tr � Gyy⊤� ⩾ 0 for all y, which we then sample to obtain a finite set of linear constraints (we recursively add additional linear constraints afterwards if need be).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' More precisely, we use tr � Gyy⊤� ⩾ 0, y ∈ Y, (33) where the initial Y is generated randomly as a set of unit vectors following the methodology described in [43, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' By replacing G = PP ⊤ by (33) we obtain a simpler (but relaxed) QCQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Then, we update the solution G lazily by repeating the following steps until G ≽ 0 is satisfied subject to a termination criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Practically speaking, our termination criterion is that the minimal eigenvalue of G is larger than ǫ ≈ −1e − 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' until then, we repeat the following procedure: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Solve the relaxation of the nonconvex QCQPs, where (33) is used instead of G = PP ⊤, which provides us an upper bound on the original nonconvex QCQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' Compute the minimal eigenvalue eigmin(G) and the corresponding eigenvector u of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' If eigmin(G) ≥ 0, we reached an optimal solution to the nonconvex QCQP and we terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' If eigmin(G) < 0, we add a constraint tr(Guu⊤) ⩾ 0 lazily, which makes the current G infeasible for the new relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' We use the lazy constraint callback interface of JuMP to add constraints lazily, which means that after adding one additional linear constraint, updating the solution in step 1 is efficient since Gurobi and all modern solvers based on the simplex algorithm can quickly update a solution when only one linear constraint is added [44, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 205-207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAzT4oBgHgl3EQfkP3w/content/2301.01530v1.pdf'} diff --git a/M9E3T4oBgHgl3EQfBQnj/content/tmp_files/2301.04266v1.pdf.txt b/M9E3T4oBgHgl3EQfBQnj/content/tmp_files/2301.04266v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e4920ced8cb06b03d1e71b564b7b42b655ca34f --- /dev/null +++ b/M9E3T4oBgHgl3EQfBQnj/content/tmp_files/2301.04266v1.pdf.txt @@ -0,0 +1,949 @@ +arXiv:2301.04266v1 [cs.IT] 11 Jan 2023 +1 +Illegal Intelligent Reflecting Surface Based Active Channel +Aging: When Jammer Can Attack Without Power and CSI +Huan Huang, Student Member, IEEE, Ying Zhang, Hongliang Zhang, Member, IEEE, +Chongfu Zhang, Senior Member, IEEE, and Zhu Han, Fellow, IEEE +Abstract—Illegal intelligent reflecting surfaces (I-IRSs), i.e., the +illegal deployment and utilization of IRSs, impose serious harmful +impacts on wireless networks. The existing I-IRS-based illegal +jammer (IJ) requires channel state information (CSI) or extra +power or both, and therefore, the I-IRS-based IJ seems to be +difficult to implement in practical wireless networks. To raise +concerns about significant potential threats posed by I-IRSs, we +propose an alternative method to jam legitimate users (LUs) +without relying on the CSI. By using an I-IRS to actively change +wireless channels, the orthogonality of multi-user beamforming +vectors and the co-user channels is destroyed, and significant +inter-user interference is then caused, which is referred to as +active channel aging. Such a fully-passive jammer (FPJ) can +launch jamming attacks on multi-user multiple-input single- +output (MU-MISO) systems via inter-user interference caused +by active channel aging, where the IJ requires no additional +transmit power and instantaneous CSI. The simulation results +show the effectiveness of the proposed FPJ scheme. Moreover, +we also investigate how the transmit power and the number of +quantization phase shift bits influence the jamming performance. +Index Terms—Intelligent reflecting surface, jamming attacks, +multi-user MISO, low-power wireless networks. +I. INTRODUCTION +D +UE to the intrinsic characteristics of wireless channels, +i.e., broadcast and superposition, wireless networks are +vulnerable to jamming attacks (also referred as to interfer- +ence attacks), and it is difficult to protect transmitted signals +from unauthorized recipients [1]. Intelligent reflecting surfaces +(IRSs) has been an emerging wireless technology for 5G, 6G +and beyond [2], [3]. Legitimate IRSs can be used to provide an +important approach for enhancing the physical layer security +(PLS) in wireless networks [4], [5]. +Therefore, many previous studies have investigated the use +of legitimate IRSs to improve PLS [6], [7]. In [6], IRSs +combined with artificial noise (AN) or friendly jamming at +the access point (AP) are used for security enhancement in the +presence of illegal eavesdroppers. In [7], the authors proposed +an IRS-assisted anti-jamming scheme against jamming attacks, +where a friendly IRS is used to prevent the illegal jammer +(IJ) from jamming legitimate users (LUs). Note that the +This work was supported by the National Key R&D Program of China +(2018YFB1801302). (Corresponding author: Chongfu Zhang) +H. Huang, Y. Zhang, and C. Zhang are with the School of Information +and Communication Engineering, University of Electronic Science and Tech- +nology of China, Chengdu 611731, China (e-mail: hhuang@std.uestc.edu.cn; +yzhang1@std.uestc.edu.cn; cfzhang@uestc.edu.cn). +H. Zhang is with the School of Electronics, Peking University, Beijing +100871, China (email: hongliang.zhang92@gmail.com). +Z. Han is with the University of Houston, Houston, TX 77004, USA (email: +zhan2@uh.edu). +legitimate AP in the legitimate IRS aided scenario knows the +legitimate IRS’s information, like its location, and can control +the reflecting phase shifts of the legitimate IRS. +In contrast, illegal IRSs (I-IRSs) represent the illegal de- +ployment and utilization of IRSs [8], where the legitimate +AP does not know the I-IRSs’ information and also can not +control the I-IRSs. Due to the passive nature, the I-IRSs are +hard to be detect. Consequently, the I-IRSs impose a more +serious harmful impact on PLS. For example, an I-IRS has +been employed to deteriorate signals at LUs in the presence +of jamming attacks [8], where the I-IRS aggravates the AN +generated by the IJ to reduce the received signal-to-noise +ratio (SNR) or the signal-to-interference-noise ratio (SINR). +However, there are two requirements in existing methods to +achieve the I-IRS-based IJ. +1) I-IRSs need to know the channel state information (CSI) +of all channels involved. Yet, the uplink channel estimation +for IRS-aided channels remains difficult due to the passive +nature of IRSs [9]. Acquiring the I-IRS-aided channels’ CSI +at IJ is too idealistic to implement in practice. Although illegal +jamming can be achieved without the CSI by broadcasting the +AN [10], the performance gain obtained by implementing an +I-IRS, in this case, is limited as reflecting phase shifts of the +I-IRS are hard to optimize without the CSI. +2) A large amount of power is needed to transmit jamming +signals continuously. Even a few papers attempt to realize an I- +IRS-based passive jammer (PJ) without the transmit power for +single-user systems [11], which minimizes the received power +at the LU by destructively adding the signal from the AP-IRS- +User channel. However, this I-IRS-based PJ still requires the +CSI of IRS-aided channels to optimize the I-IRS’s reflecting +phase shifts. +Limited by these two requirements above, especially the +CSI acquisition, the I-IRS-based IJ seems to be difficult to +implement in practical wireless networks. So in this paper, we +try to answer the following research question: Can IJs jam +LUs without both the transmit power and the CSI? +To draw attention to the impact of I-IRSs on multi-user +multiple-input single-output (MU-MISO) systems, we propose +an I-IRS-based fully-passive jammer (FPJ) that can launch +jamming attacks without relying on the transmit power and +the CSI. To the best of our knowledge, it is the first time that +an IJ can jam LUs without the CSI. +• An I-IRS is exploited to actively change wireless chan- +nels, and therefore, the orthogonality of the multi-user +active beamforming vectors and the co-user channels is + +2 +AP +LU1 +LU2 +LUK +G +Hd +HI +I-IRS +Φ +Legitimate users +Independent +controller +One-bit controllable +reflecting element +RPT +DT +Random +Fig. 1. Illustration of a MU-MISO system jammed by the I-IRS-based FPJ, +where phase shifts of the I-IRS are randomly generated by the independent +I-IRS controller. RPT: reverse pilot transmission; DT: data transmission. +destroyed, which is referred to as active channel aging1. +• During the reverse pilot transmission (RPT) phase, we +randomly generate reflecting phase shifts for the I-IRS. +During the data transmission (DT) phase, we randomly +generate other reflecting phase shifts. The I-IRS acts like +a “disco ball” without optimizing its phase shifts based +on the CSI. The resulting serious inter-user interference +due to active channel aging jams the LUs effectively. +Notation: We use bold capital type for a matrix, e.g., Φ, +small bold type for a vector, e.g., ϕ, and italic type for a scalar, +e.g., K. Moreover, the superscripts (·)H and (·)T denote the +Hermitian transpose and the transpose. Moreover, the symbols +|·| and ∥·∥ denote the absolute value and the Frobenius norm. +II. SYSTEM STATEMENT +In this section, first, we describe the general mode of an +MU-MISO system jammed by the I-IRS-based FPJ. Then, we +give the optimization metric and state the two communications +phases: the RPT phase and the DT phase. +A. System Model and Channel Model +Figure 1 schematically illustrates a MU-MISO system +jammed by the I-IRS-based FPJ, where the legitimate +AP is equipped with an NA-element uniform linear array +(ULA) and communicates with K single-antenna LUs termed +LU1, LU2, · · · , LUK. An I-IRS comprised of NI one-bit con- +trollable reflecting elements is deployed near the AP2 to jam +LUs. When the data signal sk ∈ C for LUk (1 ≤ k ≤ K) +is normalized to unit power, the signal received at LUk is +expressed as, +yk = hH +com,k +K +� +u=1 +wusu + nk, +(1) +1Channel aging is CSI inaccuracy due to time variation of wireless channels +and delays in the computation [12]. In this work, we actively introduce CSI +inaccuracy by using an I-IRS. To differentiate, we call it active channel aging. +2Based on the existing literature on the IRS’s deployment location [13], the +IRS should be deployed as close to users or as close to the AP as possible +to increase its effect. Yet, in the jamming scenario, we make the more robust +assumption that the IJ does not know any information about LUs, for instance, +LUs’ locations and CSI. Therefore, we deploy the I-IRS near the AP. +where hH +com,k = +� +hH +I,kΦG+hH +d,k +� +∈ C1×NA denotes the +combined channel between the legitimate AP and LUk, hI,k ∈ +CNI×1 denotes the channel between the I-IRS and LUk, +G ∈ CNI×NA denotes the channel between the legitimate AP +and the I-IRS, and hd,k ∈ CNA×1 denotes the direct channel +between the legitimate AP and LUk. +In (1), Φ = diag(ϕ) ∈ CNI×NI represents the reflecting +matrix of the I-IRS, where the one-bit reflecting vector ϕ +is expressed as ϕ = +� +ejϕ1, · · · , ejϕNI �H, and ϕn ∈ Ω = +{0, π} (1 ≤ n ≤ NI) denotes reflecting phase shift of the n-th +reflecting element. The independent I-IRS controller generates +ϕ and then controls the I-IRS to implement the corresponding +phase shifts. Besides, wk denotes the active beamforming at +the AP for LUk, and nk denotes the additive white Gaussian +noise with 0 mean and σ2 variance, i.e., nk ∼ CN +� +0, σ2� +. +For ease of representation, we further define the multi-user +direct channel between the AP and the LUs, the multi-user +channel between the I-IRS and the LUs, as well as the multi- +user combined channel between the AP and all LUs as HH +d = +[hd,1, hd,2, · · · , hd,K]H, HH +I = [hI,1, hI,2, · · · , hI,K]H, and +HH +com = [hcom,1, hcom,2, · · · , hcom,K]H, respectively. Fur- +thermore, the multi-user active beamforming at the AP is +denoted as W=[w1, w2, . . . , wK]. +The multi-user direct channel Hd follows Rayleigh fading, +while the IRS-aided channels G and hI,k follow Rician +fading [14]. Specifically, G and hI,k are modeled as +G=LG +�� +κG +1+κG +GLOS+ +� +1 +1 + κG +GNLOS +� +, +hI,k =LI,k +�� +κI +1+κI +hLOS +I,k + +� +1 +1+κI +hNLOS +I,k +� +, +(2) +where LG and LI,k represent the large-scale path loss be- +tween the AP and the I-IRS and that between the I-IRS and +LUk, and κG and κI are the Rician factors of G and hI,k. +In (2), GLOS and hLOS +I,k +are the line-of-sight (LOS) com- +ponents of G and hI,k, and GNLOS and hNLOS +I,k +are non-line- +of-sight (NLOS) components. The NLOS components follow +Rayleigh fading, while the LOS components are [14], +GLOS = +� +NINAαI (ϑ, θ) αH +A (φ) , +hLoS +I,k = +� +NIαI (ϑI,k, θI,k) , +(3) +where αA and αI are the array responses [14]. +B. Wireless Communications: The RPT and DT Phases +In practice, the main aim of a MU-MISO system is to +maximize a certain performance metric that generally is a +strictly-increasing utility function of SINR [15]. Specifically, +a widely-used performance metric is the sum rate, which +is expressed as Rsum = �K +k=1 Rk = �K +k=1 log2 (1 + γk). +According to (1), the received SINR γk at LUk is stated as, +γk = +���hH +com,kwk +��� +2 +� +u̸=k +���hH +com,kwu +��� +2 ++ σ2 +. +(4) + +3 +1) Acquiring CSI During The RPT Phase: From (4), it can +be seen that the optimization of multi-user active beamforming +W = [w1, w2, . . . , wK] at the AP aims to maximize the +signal term +���hH +com,kwk +��� while minimizing the inter-user inter- +ference term � +u̸=k +���hH +com,kwu +���. In order to optimize W, the +CSI of Hcom must be obtained at the AP3. Generally, the CSI +can be acquired during the RPT phase according to the pilot +estimation, as shown in Fig. 1. More specifically, to acquire the +CSI of hcom,k, the LUk sends pilot signals to the legitimate +AP, and the AP then estimates hcom,k by certain traditional +solutions, for instance, the least square (LS) algorithm [9]. +2) Precoding During The DT Phase: Based on the obtained +CSI in the RPT phase, the multi-user active beamforming used +during the DT phase can be designed. Generally, the multi-user +active beamforming optimization problem is a nondeterminis- +tic polynomial-time (NP)-hard problem, and therefore, com- +puting the optimal multi-user active beamforming is difficult. +To this end, some heuristic beamforming designs, which can +achieve near-optimal performance, have been investigated. +A widely known beamforming solution is the zero-forcing +beamforming (ZFBF) algorithm [15], which causes zero inter- +user interference. Specifically, the multi-user active beamform- +ing WZF computed via the ZFBF algorithm is written as +WZF = Hcom +� +HH +comHcom +�−1P +1 +2 +���Hcom(HH +comHcom)−1��� +2 , +(5) +where P +1 +2 = diag +�√p1, √p2, · · · , √pK +� +, and pk represents +the transmit power allocated to LUk. The power allocation +must satisfy the constraint that �K +k=1 pk ≤ P0, where P0 is the +total transmit power at the AP. The optimal power allocation +can be calculated by the water-filling algorithm [15]. +3) Orthogonal Interference Subspace: According to (4), the +ratio of inter-user interference to noise (I/N) I is equal to +I = +K +� +k=1 +� +u̸=k +���hH +com,kwu +��� +2 +σ2 +. +(6) +Incorporating (5) into (6), it is clear that I += +0 +due to the presence of the pseudoinverse +� +HH +comHcom +�−1. +In +other +words, +ZFBF +causes +zero +inter-user +interfer- +ence +by +projecting the +user +channel hcom,k +onto +the +subspace +that +is +orthogonal +to +the +co-user +channels +hcom,1, · · · , hcom,k−1, hcom,k+1, · · · , hcom,K, i.e., the or- +thogonal interference subspace. +III. I-IRS-BASED FULLY-PASSIVE JAMMER VIA ACTIVE +CHANNEL AGING +To raise concerns about the potential threat that an I-IRS +could launch jamming attacks without the transmit power +3In the MU-MISO system under I-IRS-based jamming attacks, it is im- +practical to acquire the CSI of IRS-aided channels and the direct channel, +respectively. The legitimate AP cannot know any information about the I-IRS, +like its location, much less jointly train the IRS-based channels with the I-IRS. +Namely, the legitimate AP can only obtain the CSI of Hcom. Note that the CSI +of Hcom is easily obtained at the legitimate AP when Φ is determined, which +is the traditional MISO channel estimation. The phase shifts of the I-IRS are +generated at random by the independent I-IRS controller, and therefore, Φ is +always determined for the legitimate AP, as shown in Fig. 1. +or even the CSI, we introduce a CSI-based PJ without the +transmit power in Section III-A, i.e., the extension of [11]. +Furthermore, the results from the CSI-based PJ are used as +benchmarks. In Section III-B, we propose an I-IRS-based FPJ +via active channel aging. By destroying the orthogonality of +the multi-user active beamforming vectors and the co-user +channels, the proposed I-IRS-based FPJ can jam LUs without +the transmit power and the CSI. +A. CSI-Based Jamming Attacks Without Power +To implement the extension of [11], it is necessary to con- +sider the most ideal case for jamming attacks: the legitimate +AP only knows the CSI of Hd and then calculates the multi- +user active beamforming Wd via the ZFBF algorithm, while +the independent I-IRS controller knows the CSI of Hd, HI, +and G as well as Wd. The CSI-based PJ can launch jamming +attacks without the transmit power, where the reflecting vector +for the I-IRS is optimized by minimizing a certain performance +metric. Taking the example of minimizing the sum rate Rsum +received at LUs, the optimization of the one-bit reflecting +vector is mathematically represented as +min +ϕ Rsum = min +ϕ +K +� +k=1 +log2 + + + +1 + +���hH +com,kwd,k +��� +2 +� +u̸=k +���hH +com,kwd,u +��� +2 ++ σ2 + + + + +(7) +s.t. ϕn ∈ Ω, n = 1, 2, · · · , NI. +(8) +The phase shift optimization problem in (7) can be solved +by enumerating all possible {ϕn}NI +n=1 combinations. However, +there are 2NI different combinations, and thus the computa- +tional complexity is large. +To this end, we first relax the discrete phase shift constraint +in (8) to a continuous constraint. Mathematically, the reflecting +vector optimization is relaxed to +max +¯ϕ +K +� +k=1 +−log2 + + + +1 + +��� +� +¯ϕdiag(hH +I,k)G+hH +d,k +� +wd,k +��� +2 +� +u̸=k +��� +� +¯ϕdiag(hH +I,k)G+hH +d,k +� +wd,u +��� +2 ++σ2 + + + + +(9) +s.t. ¯ϕn ∈ [0, 2π] , n = 1, 2, · · · , NI. +(10) +The objective function in (9) is then a continuous and +differentiable function of ¯ϕ, and the constraint in (10) creates a +complex circle manifold. Therefore, the optimization problem +in (9) can be computed by the Riemannian conjugate gradient +(RCG) algorithm [16]. After computing the continuous reflect- +ing vector ¯ϕ, the discrete reflecting vector is obtained by +min +ϕ ∥ϕ − ¯ϕ∥2 +(11) +s.t. (8). +The complexity of the benchmarking CSI-based PJ is +O +� +IRK2N 2 +I +� +, where IR represents the iteration times of +the RCG algorithm. In each iteration, the complexity comes +mainly from calculating the Euclidean gradient [16]. Specifi- +cally, the complexity of the Euclidean gradient calculation is + +4 +Subspace of co-user channels +in the RPT phase +the RPT phase +Subspace of co-user channels +in the DT phase + p +... +LU1 +LU2 +LUK +I-IRS +RPT +DT +Fig. 2. I-IRS-based FPJ via active channel aging, where the I-IRS acts like +a “disco ball” and ZFBF cannot project the user channel to the orthogonal +interference subspace. +O +� +K2N 2 +I +� +. Moreover, the complexity of the discreteization +of ¯ϕ expressed by (11) is O(2NI). When the number of +reflecting elements packed on the I-IRS is large (NI ≫ 1), the +complexity of the discreteization, i.e., O(2NI), can be ignored. +B. I-IRS-Based Jamming Attacks Without Power and CSI +Although the CSI-based PJ proposed in Section III-A can +jam without the transmit power, the CSI of all channels needs +to be obtained at the independent I-IRS controller, which is +difficult to satisfy in practice. In wireless communications, the +AP needs to obtain the CSI during the RPT phase before the +DT phase, as stated in Section II-B. +1) The RPT Phase: During the RPT phase, the one-bit +reflecting vector for the I-IRS is generated by tuning the n- +th reflecting element to a random phase shift belonging to +Ω, i.e., ϕ1 +n ∼ U (Ω). More particularly, the reflecting vector +ϕ1 follows the uniform distribution denoted ϕ1 ∼ U +� +ΩNI� +. +It is worth noting that the independent I-IRS controller in +the proposed scheme does not need to optimize the reflecting +phase shifts of the I-IRS. +Consequently, the multi-user combined channel estimated +by the AP is written as (H1 +com)H = HH +I diag +� +ϕ1� +G+HH +d = +� +h1 +com,1, h1 +com,2, · · · , h1 +com,K +�H. Based on H1 +com, the AP can +compute the multi-user active beamforming used in the DT +phase that is expressed as +W1 +ZF =H1 +com +� +(H1 +com)HH1 +com +�−1P +1 +2 +���H1com((H1com)HH1com)−1��� +2 = +� +w1 +ZF,1,w1 +ZF,2,· · ·,w1 +ZF,K +� +, +(12) +where w1 +ZF,k is orthogonal to the subspace of co-user channels +h1 +com,1, · · · , h1 +com,k−1, h1 +com,k+1, · · · , h1 +com,K. +2) The DT Phase: Then, during the DT phase, the one-bit +reflecting vector of the I-IRS is formed according to another +reflecting vector ϕ2 that also follows the uniform distribution +in Ω, i.e., ϕ2 ∼ U +� +ΩNI� +. Therefore, during the DT phase, the +multi-user combined channel is changed to +(H2 +com)H=HH +I diag +� +ϕ2� +G+HH +d = +� +h2 +com,1,h2 +com,2,· · ·,h2 +com,K +�H . +(13) +Including (12) and (13) into (4), the actual received SINR +¯γk at LUk during the DT phase is +¯γk = +���(h2 +com,k)Hw1 +ZF,k +��� +2 +� +u̸=k +���(h2 +com,k)Hw1 +ZF,u +��� +2 ++ σ2 +. +(14) +The complexity of our proposed scheme comes from ran- +domly generating the two reflecting vectors used in the RPT +phase and the DT phase, which is only O(2NI). Compared +with the benchmarking CSI-based PJ, the I-IRS’s controller in +the proposed I-IRS-based FPJ not only does not require the +CSI of all channels involved, but also the complexity of the +proposed I-IRS-based FPJ is much lower. +3) Active Channel Aging: Based on (12) and (13), the +reflecting vectors for the I-IRS are different and random +during the RPT phase and the DT phase (like a “disco +ball” shown in Fig. 2), which destroys the orthogonality +generated from ZFBF due to active channel aging. The +w1 +ZF,k is only orthogonal to the subspace of co-user channels +h1 +com,1, · · · , h1 +com,k−1, h1 +com,k+1, · · · , h1 +com,K, and thus I in +(6) is then equal to �K +k=1 +� +u̸=k +|(h2 +com,k)Hw1 +ZF,u| +2 +σ2 +, which is +no longer zero due to active channel aging. +As a result, the actual received SINR ¯γk in (14) achieved +under the proposed I-IRS-based FPJ is dramatically reduced +compared to that without attacks. We stated that the reflecting +vector for the I-IRS is different during the RPT phase and the +DT phase. In fact, there is no need for precise synchronization +in practical implementation. Assuming that the periods of the +RPT phase and the DT phase are Tr and Td (Tr ≤ Td), the +reflecting vector changes randomly with a period of no more +than Tr, and active channel aging then occurs. +IV. SIMULATION RESULTS AND DISCUSSION +Consider a MU-MISO system with four single-antenna +LUs, where the legitimate AP is equipped with a 12-element +ULA [15] and an I-IRS contains 1,024 reflecting elements +(NI,y = NI,z = 32). Moreover, the AP is located at (0m, 0m, +0m) and the four LUs are randomly distributed in a circle +centered at (200m, 0m, 0m) with a radius of 10m, while the +I-IRS is deployed at (5m, 5m, 2m). +Most of the existing performance-enhancing IRS-aided sys- +tems make the assumption that Hd has significant large-scale +path loss or is blocked, while the large-scale path losses +of G and HI are much smaller [14], [16]. However, this +assumption is too idealistic for jamming attacks. According to +the 3GPP propagation environment [17], the large-scale path +losses Lk, LG and LI,k are set as Lk =32.6+22log10(dk), +LG = 35.6 + 20log10(dG) and LI,k = 35.6 + 22log10(dI,k), +where dk is the distance between the AP and LUk, dG is the +distance between the AP and and the I-IRS, and dI,k is the +distance between the I-IRS and LUk (1 ≤ k ≤ 4). Moreover, +σ2 = −170 + 10 log10(BW) dBm, where BW denotes the +transmission bandwidth and BW =180 kHz [16]. We compare +the proposed I-IRS-based FPJ with three benchmarks. +1) Benchmark 1: The average sum rates without IJ (w/o IJ) +are computed based on the multi-user direct channel, where +the received SINR γk at LUk is γk = +|hH +d,kwd,k| +2 +� +u̸=k |hH +d,kwd,u| +2+σ2 . +2) Benchmark 2: The average sum rates under the active +jammer (w/ AJ/N) with different ratios of the jamming power +to the noise power (AJ/N) at each LU. More specifically, the +received SINR γk at LUk under active jamming is expressed + +2 +W +ZF ,k1 +W +ZF,k62 +com.1 +com.k-1 +com.k-1> +com,K5 +-15 +-10 +-5 +0 +5 +10 +15 +Total Transmit Power [dBm] +10 +20 +30 +40 +50 +60 +70 +Average Sum Rate [bits/symbol use] +-60 +-40 +-20 +0 +20 +40 +Inter-User Interference/Noise [dB] +Benchmark 1 +Benchmark 2 w/ 5 dB +Benchmark 2 w/ 10 dB +Proposed FPJ +Benchmark 3 +I/N of Benchmark 3 +I/N of Proposed FPJ +I/N of Benchmark 1 +Fig. 3. Average sum rates (left, solid lines) and I/N (right, dash-dot lines) of +different schemes vs total transmit power. +1 +2 +3 +4 +5 +6 +7 +Quantization bits of reflecting phase shifts +40 +45 +50 +55 +60 +65 +Average Sum Rate [bits/symbol use] +-60 +-40 +-20 +0 +20 +40 +Inter-User Interference/Noise [dB] +Benchmark 1 +Benchmark 2 w/ 5 dB +Benchmark 2 w/ 10 dB +Proposed FPJ +Benchmark 3 +I/N of Benchmark 3 +I/N of Proposed FPJ +I/N of Benchmark 1 +Fig. 4. Influence of quantization reflecting phase shift bits. +as γk = +|hH +d,kwd,k| +2 +� +u̸=k |hH +d,kwd,u| +2+PJ +σ2 , where AJ/N = PJ/σ2=5 dB +and 10 dB, respectively. +2) Benchmark 3: The CSI-based PJ in Section III-A, i.e., +the extension of [11]. +Fig. 3 illustrates the average sum rates via the proposed +FPJ and the above three benchmarks, where I generated +from them is also given. By destroying the orthogonality of +the multi-user active beamforming vectors and the co-user +channels, the inter-user interference becomes significant due to +active channel aging. The reflecting vector in the proposed FPJ +affects both the multi-user combined channel and the multi- +user active beamforming, while the reflecting vector in the +CSI-based PJ just impacts the multi-user combined channel. +As shown in Fig. 3, I from the proposed FPJ is more serious +than that from the CSI-aided PJ (Benchmark 3). Therefore, the +proposed FPJ can jam LUs without the transmit power and the +CSI, even more effectively than the CSI-aided PJ. +From Fig. 3, one can see that the sum rate of Proposed FPJ +is smaller than that of Benchmark 2 with 5 dB AJ/N when the +total transmit power is greater than 0 dBm. In contrast to the +active jamming, the jamming launched by the proposed FPJ +cannot be mitigated by increasing the total transmit power. +To show the influence of the number of quantization re- +flecting phase shift bits, the relationships between the average +20 +22 +24 +26 +28 +30 +32 +Number of Reflecting Elements +40 +45 +50 +55 +60 +65 +Average Sum Rate [bits/symbol use] +-60 +-40 +-20 +0 +20 +40 +Inter-User Interference/Noise [dB] +Benchmark 1 +Benchmark 2 w/ 5 dB +Benchmark 2 w/ 10 dB +Proposed FPJ +Benchmark 3 +I/N of Benchmark 3 +I/N of Proposed FPJ +I/N of Benchmark 1 +Fig. 5. Influence of the number of reflecting elements. +sum rates and quantization bits are illustrated in Fig. 4. One +can see that the proposed FPJ is robust to the quantization +bits since the reflecting vector is randomly generated. Based +on the proposed FPJ, the one-bit I-IRS is enough to launch +effective jamming attacks on LUs. The greater the number +of quantization bits, the smaller the difference ∥ϕ − ¯ϕ∥2 +in (11) is. Although the sum rate achieved by Benchmark 3 +decreases with the number of quantization bits, the high-bit +I-IRS requires high physical implementation costs. +Moreover, Fig. 5 shows the relationship between the sum +rates and the number of reflecting elements as well as that +between I/N and the number of reflecting elements. The +difference between the sum rates achieved by Benchmark 3 +and Proposed FPJ increases with the number of reflecting +elements. On the one hand, active channel aging becomes +more significant with the number of reflecting elements, and +thus the corresponding jamming attacks are more effective. On +the other hand, the minimum value of ∥ϕ − ¯ϕ∥2 in (11) gets +bigger with the number of reflecting elements. In practice, an +IRS generally consists of massive reflecting elements, which +is beneficial to the proposed I-IRS-based FPJ. +V. CONCLUSIONS +In this letter, we investigated the impact of I-IRSs on MU- +MISO systems, where an I-IRS-based FPJ was proposed. By +exploiting an I-IRS to cause active channel aging, we have +demonstrated that the proposed FPJ can jam without relying +on the transmit power and the CSI. Due to the impacts on +both the multi-user combined channel and the multi-user active +beamforming, the jamming launched by the proposed FPJ +is even more effective than that launched by the CSI-aided +PJ. Meanwhile, the proposed FPJ is robust to the number +of quantization reflecting phase shift bits. Different from the +active jamming attacks, the jamming attacks launched by the +proposed FPJ cannot be mitigated by increasing the total +transmit power at the legitimate AP. When the legitimate AP +has large transmit power, the proposed FPJ can jam LUs +more effectively. Moreover, the proposed FPJ can be perfectly +hidden in wireless environments because it does not require +additional transmit power and instantaneous CSI. + +6 +REFERENCES +[1] A. Mukherjee, S. A. A. Fakoorian, J. Huang, and A. L. Swindlehurst, +“Principles of physical layer security in multiuser wireless networks: A +survey,” IEEE Commun. Surv. Tut., vol. 16, no. 3, pp. 1550–1573, 3rd +Quarter 2014. +[2] H. Zhang, S. Zeng, B. Di, Y. Tan, M. D. Renzo, M. Debbah, Z. Han, +H. V. Poor, and L. Song, “Intelligent omni-surfaces for full-dimensional +wireless communications: Principles, technology, and implementation,” +IEEE Commun. Mag., vol. 60, no. 2, pp. 39–45, Feb. 2022. +[3] M. A. ElMossallamy, H. Zhang, L. Song, K. G. Seddik, Z. Han, G. Y. Li, +“Reconfigurable intelligent surfaces for wireless communications: Prin- +ciples, challenges, and opportunities,” IEEE Trans. Cogn. Commun., vol. +6, no. 3, pp. 990–1002, Sept. 2020. +[4] W. Tan, C. Zhang, J. Peng, L. Dai, S. Fu, and K. 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Nallanathan, “Wireless com- +munication in the presence of illegal reconfigurable intelligent surface: +Signal leakage and interference attack,” IEEE Wireless Commun., vol. +29, no. 3, pp. 131–138, Jun. 2022. +[9] X. Wei, D. Shen, and L. Dai, “Channel estimation for RIS assisted +wireless communications: Part I-fundamentals, solutions, and future op- +portunities,” IEEE Commun. Lett., vol. 25, no. 5, pp. 1398–1402, May +2021. +[10] N. Romero-Zurita, D. McLernon, M. Ghogho, and A. Swami, “PHY +layer security based on protected zone and artificial noise,” IEEE Signal +Process. Lett., vol. 20, no. 5, pp. 487–490, May 2012. +[11] B. Lyu, D. T. Hoang, S. Gong, D. Niyato, and D. I. Kim, “IRS-based +wireless jamming attacks: When jammers can attack without power,” +IEEE Wireless Commun. Lett., vol. 9, no. 10, pp. 1663–1667, Oct. 2020. +[12] K. T. Truong and R. W. Heath Jr., “Effects of channel aging in massive +MIMO systems,” J. Commun. Netw-S. Kor., vol. 15, no. 4, pp. 338–351, +Aug. 2013. +[13] S. Zhang and R. Zhang, “Intelligent reflecting surface aided multi-user +communication: Capacity region and deployment strategy,” IEEE Trans. +Commun., vol. 69, no. 9, pp. 5790–5806, Spet. 2021. +[14] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless +network via joint active and passive beamforming,” IEEE Trans. Wireless +Commun., vol. 18, no. 11, pp. 5394–5409, Nov. 2019. +[15] E. Bj¨ornson, M. Bengtsson, and B. Ottersten, “Optimal multiuser trans- +mit beamforming: A difficult problem with a simple solution structure,” +IEEE Signal Process. Mag., vol. 31, no. 4, pp. 142–148, Jun. 2014. +[16] H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum- +rate maximization for reconfigurable intelligent surface aided wireless +networks,” IEEE Trans. Wireless Commun., vol. 19, no. 5, pp. 3064– +30769, May 2020. +[17] Further Advancements for E-UTRA Physical Layer Aspects (Release +9), document 3GPP TS 36.814, Mar. 2010. + diff --git a/M9E3T4oBgHgl3EQfBQnj/content/tmp_files/load_file.txt b/M9E3T4oBgHgl3EQfBQnj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb3fd554a336acc094e993c0f03e507ba1f3edf8 --- /dev/null +++ b/M9E3T4oBgHgl3EQfBQnj/content/tmp_files/load_file.txt @@ -0,0 +1,456 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf,len=455 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='04266v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='IT] 11 Jan 2023 1 Illegal Intelligent Reflecting Surface Based Active Channel Aging: When Jammer Can Attack Without Power and CSI Huan Huang, Student Member, IEEE, Ying Zhang, Hongliang Zhang, Member, IEEE, Chongfu Zhang, Senior Member, IEEE, and Zhu Han, Fellow, IEEE Abstract—Illegal intelligent reflecting surfaces (I-IRSs), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', the illegal deployment and utilization of IRSs, impose serious harmful impacts on wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The existing I-IRS-based illegal jammer (IJ) requires channel state information (CSI) or extra power or both, and therefore, the I-IRS-based IJ seems to be difficult to implement in practical wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' To raise concerns about significant potential threats posed by I-IRSs, we propose an alternative method to jam legitimate users (LUs) without relying on the CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' By using an I-IRS to actively change wireless channels, the orthogonality of multi-user beamforming vectors and the co-user channels is destroyed, and significant inter-user interference is then caused, which is referred to as active channel aging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Such a fully-passive jammer (FPJ) can launch jamming attacks on multi-user multiple-input single- output (MU-MISO) systems via inter-user interference caused by active channel aging, where the IJ requires no additional transmit power and instantaneous CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The simulation results show the effectiveness of the proposed FPJ scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Moreover, we also investigate how the transmit power and the number of quantization phase shift bits influence the jamming performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Index Terms—Intelligent reflecting surface, jamming attacks, multi-user MISO, low-power wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' INTRODUCTION D UE to the intrinsic characteristics of wireless channels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', broadcast and superposition, wireless networks are vulnerable to jamming attacks (also referred as to interfer- ence attacks), and it is difficult to protect transmitted signals from unauthorized recipients [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Intelligent reflecting surfaces (IRSs) has been an emerging wireless technology for 5G, 6G and beyond [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Legitimate IRSs can be used to provide an important approach for enhancing the physical layer security (PLS) in wireless networks [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Therefore, many previous studies have investigated the use of legitimate IRSs to improve PLS [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In [6], IRSs combined with artificial noise (AN) or friendly jamming at the access point (AP) are used for security enhancement in the presence of illegal eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In [7], the authors proposed an IRS-assisted anti-jamming scheme against jamming attacks, where a friendly IRS is used to prevent the illegal jammer (IJ) from jamming legitimate users (LUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Note that the This work was supported by the National Key R&D Program of China (2018YFB1801302).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' (Corresponding author: Chongfu Zhang) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Zhang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Zhang are with the School of Information and Communication Engineering, University of Electronic Science and Tech- nology of China, Chengdu 611731, China (e-mail: hhuang@std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' yzhang1@std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' cfzhang@uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Zhang is with the School of Electronics, Peking University, Beijing 100871, China (email: hongliang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='zhang92@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Han is with the University of Houston, Houston, TX 77004, USA (email: zhan2@uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' legitimate AP in the legitimate IRS aided scenario knows the legitimate IRS’s information, like its location, and can control the reflecting phase shifts of the legitimate IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In contrast, illegal IRSs (I-IRSs) represent the illegal de- ployment and utilization of IRSs [8], where the legitimate AP does not know the I-IRSs’ information and also can not control the I-IRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Due to the passive nature, the I-IRSs are hard to be detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Consequently, the I-IRSs impose a more serious harmful impact on PLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' For example, an I-IRS has been employed to deteriorate signals at LUs in the presence of jamming attacks [8], where the I-IRS aggravates the AN generated by the IJ to reduce the received signal-to-noise ratio (SNR) or the signal-to-interference-noise ratio (SINR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' However, there are two requirements in existing methods to achieve the I-IRS-based IJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 1) I-IRSs need to know the channel state information (CSI) of all channels involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Yet, the uplink channel estimation for IRS-aided channels remains difficult due to the passive nature of IRSs [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Acquiring the I-IRS-aided channels’ CSI at IJ is too idealistic to implement in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Although illegal jamming can be achieved without the CSI by broadcasting the AN [10], the performance gain obtained by implementing an I-IRS, in this case, is limited as reflecting phase shifts of the I-IRS are hard to optimize without the CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 2) A large amount of power is needed to transmit jamming signals continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Even a few papers attempt to realize an I- IRS-based passive jammer (PJ) without the transmit power for single-user systems [11], which minimizes the received power at the LU by destructively adding the signal from the AP-IRS- User channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' However, this I-IRS-based PJ still requires the CSI of IRS-aided channels to optimize the I-IRS’s reflecting phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Limited by these two requirements above, especially the CSI acquisition, the I-IRS-based IJ seems to be difficult to implement in practical wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' So in this paper, we try to answer the following research question: Can IJs jam LUs without both the transmit power and the CSI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' To draw attention to the impact of I-IRSs on multi-user multiple-input single-output (MU-MISO) systems, we propose an I-IRS-based fully-passive jammer (FPJ) that can launch jamming attacks without relying on the transmit power and the CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' To the best of our knowledge, it is the first time that an IJ can jam LUs without the CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' An I-IRS is exploited to actively change wireless chan- nels, and therefore, the orthogonality of the multi-user active beamforming vectors and the co-user channels is 2 AP LU1 LU2 LUK G Hd HI I-IRS Φ Legitimate users Independent controller One-bit controllable reflecting element RPT DT Random Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Illustration of a MU-MISO system jammed by the I-IRS-based FPJ, where phase shifts of the I-IRS are randomly generated by the independent I-IRS controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' RPT: reverse pilot transmission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' DT: data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' destroyed, which is referred to as active channel aging1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' During the reverse pilot transmission (RPT) phase, we randomly generate reflecting phase shifts for the I-IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' During the data transmission (DT) phase, we randomly generate other reflecting phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The I-IRS acts like a “disco ball” without optimizing its phase shifts based on the CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The resulting serious inter-user interference due to active channel aging jams the LUs effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Notation: We use bold capital type for a matrix, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', Φ, small bold type for a vector, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', ϕ, and italic type for a scalar, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Moreover, the superscripts (·)H and (·)T denote the Hermitian transpose and the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Moreover, the symbols |·| and ∥·∥ denote the absolute value and the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' SYSTEM STATEMENT In this section, first, we describe the general mode of an MU-MISO system jammed by the I-IRS-based FPJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Then, we give the optimization metric and state the two communications phases: the RPT phase and the DT phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' System Model and Channel Model Figure 1 schematically illustrates a MU-MISO system jammed by the I-IRS-based FPJ, where the legitimate AP is equipped with an NA-element uniform linear array (ULA) and communicates with K single-antenna LUs termed LU1, LU2, · · · , LUK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' An I-IRS comprised of NI one-bit con- trollable reflecting elements is deployed near the AP2 to jam LUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' When the data signal sk ∈ C for LUk (1 ≤ k ≤ K) is normalized to unit power, the signal received at LUk is expressed as, yk = hH com,k K � u=1 wusu + nk, (1) 1Channel aging is CSI inaccuracy due to time variation of wireless channels and delays in the computation [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In this work, we actively introduce CSI inaccuracy by using an I-IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' To differentiate, we call it active channel aging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 2Based on the existing literature on the IRS’s deployment location [13], the IRS should be deployed as close to users or as close to the AP as possible to increase its effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Yet, in the jamming scenario, we make the more robust assumption that the IJ does not know any information about LUs, for instance, LUs’ locations and CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Therefore, we deploy the I-IRS near the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' where hH com,k = � hH I,kΦG+hH d,k � ∈ C1×NA denotes the combined channel between the legitimate AP and LUk, hI,k ∈ CNI×1 denotes the channel between the I-IRS and LUk, G ∈ CNI×NA denotes the channel between the legitimate AP and the I-IRS, and hd,k ∈ CNA×1 denotes the direct channel between the legitimate AP and LUk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In (1), Φ = diag(ϕ) ∈ CNI×NI represents the reflecting matrix of the I-IRS, where the one-bit reflecting vector ϕ is expressed as ϕ = � ejϕ1, · · · , ejϕNI �H, and ϕn ∈ Ω = {0, π} (1 ≤ n ≤ NI) denotes reflecting phase shift of the n-th reflecting element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The independent I-IRS controller generates ϕ and then controls the I-IRS to implement the corresponding phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Besides, wk denotes the active beamforming at the AP for LUk, and nk denotes the additive white Gaussian noise with 0 mean and σ2 variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', nk ∼ CN � 0, σ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' For ease of representation, we further define the multi-user direct channel between the AP and the LUs, the multi-user channel between the I-IRS and the LUs, as well as the multi- user combined channel between the AP and all LUs as HH d = [hd,1, hd,2, · · · , hd,K]H, HH I = [hI,1, hI,2, · · · , hI,K]H, and HH com = [hcom,1, hcom,2, · · · , hcom,K]H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Fur- thermore, the multi-user active beamforming at the AP is denoted as W=[w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' , wK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The multi-user direct channel Hd follows Rayleigh fading, while the IRS-aided channels G and hI,k follow Rician fading [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Specifically, G and hI,k are modeled as G=LG �� κG 1+κG GLOS+ � 1 1 + κG GNLOS � , hI,k =LI,k �� κI 1+κI hLOS I,k + � 1 1+κI hNLOS I,k � , (2) where LG and LI,k represent the large-scale path loss be- tween the AP and the I-IRS and that between the I-IRS and LUk, and κG and κI are the Rician factors of G and hI,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In (2), GLOS and hLOS I,k are the line-of-sight (LOS) com- ponents of G and hI,k, and GNLOS and hNLOS I,k are non-line- of-sight (NLOS) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The NLOS components follow Rayleigh fading, while the LOS components are [14], GLOS = � NINAαI (ϑ, θ) αH A (φ) , hLoS I,k = � NIαI (ϑI,k, θI,k) , (3) where αA and αI are the array responses [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Wireless Communications: The RPT and DT Phases In practice, the main aim of a MU-MISO system is to maximize a certain performance metric that generally is a strictly-increasing utility function of SINR [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Specifically, a widely-used performance metric is the sum rate, which is expressed as Rsum = �K k=1 Rk = �K k=1 log2 (1 + γk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' According to (1), the received SINR γk at LUk is stated as, γk = ���hH com,kwk ��� 2 � u̸=k ���hH com,kwu ��� 2 + σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' (4) 3 1) Acquiring CSI During The RPT Phase: From (4), it can be seen that the optimization of multi-user active beamforming W = [w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' , wK] at the AP aims to maximize the signal term ���hH com,kwk ��� while minimizing the inter-user inter- ference term � u̸=k ���hH com,kwu ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In order to optimize W, the CSI of Hcom must be obtained at the AP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Generally, the CSI can be acquired during the RPT phase according to the pilot estimation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' More specifically, to acquire the CSI of hcom,k, the LUk sends pilot signals to the legitimate AP, and the AP then estimates hcom,k by certain traditional solutions, for instance, the least square (LS) algorithm [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 2) Precoding During The DT Phase: Based on the obtained CSI in the RPT phase, the multi-user active beamforming used during the DT phase can be designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Generally, the multi-user active beamforming optimization problem is a nondeterminis- tic polynomial-time (NP)-hard problem, and therefore, com- puting the optimal multi-user active beamforming is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' To this end, some heuristic beamforming designs, which can achieve near-optimal performance, have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' A widely known beamforming solution is the zero-forcing beamforming (ZFBF) algorithm [15], which causes zero inter- user interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Specifically, the multi-user active beamform- ing WZF computed via the ZFBF algorithm is written as WZF = Hcom � HH comHcom �−1P 1 2 ���Hcom(HH comHcom)−1��� 2 , (5) where P 1 2 = diag �√p1, √p2, · · · , √pK � , and pk represents the transmit power allocated to LUk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The power allocation must satisfy the constraint that �K k=1 pk ≤ P0, where P0 is the total transmit power at the AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The optimal power allocation can be calculated by the water-filling algorithm [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 3) Orthogonal Interference Subspace: According to (4), the ratio of inter-user interference to noise (I/N) I is equal to I = K � k=1 � u̸=k ���hH com,kwu ��� 2 σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' (6) Incorporating (5) into (6), it is clear that I = 0 due to the presence of the pseudoinverse � HH comHcom �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In other words, ZFBF causes zero inter-user interfer- ence by projecting the user channel hcom,k onto the subspace that is orthogonal to the co-user channels hcom,1, · · · , hcom,k−1, hcom,k+1, · · · , hcom,K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', the or- thogonal interference subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' I-IRS-BASED FULLY-PASSIVE JAMMER VIA ACTIVE CHANNEL AGING To raise concerns about the potential threat that an I-IRS could launch jamming attacks without the transmit power 3In the MU-MISO system under I-IRS-based jamming attacks, it is im- practical to acquire the CSI of IRS-aided channels and the direct channel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The legitimate AP cannot know any information about the I-IRS, like its location, much less jointly train the IRS-based channels with the I-IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Namely, the legitimate AP can only obtain the CSI of Hcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Note that the CSI of Hcom is easily obtained at the legitimate AP when Φ is determined, which is the traditional MISO channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The phase shifts of the I-IRS are generated at random by the independent I-IRS controller, and therefore, Φ is always determined for the legitimate AP, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' or even the CSI, we introduce a CSI-based PJ without the transmit power in Section III-A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', the extension of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Furthermore, the results from the CSI-based PJ are used as benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In Section III-B, we propose an I-IRS-based FPJ via active channel aging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' By destroying the orthogonality of the multi-user active beamforming vectors and the co-user channels, the proposed I-IRS-based FPJ can jam LUs without the transmit power and the CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' CSI-Based Jamming Attacks Without Power To implement the extension of [11], it is necessary to con- sider the most ideal case for jamming attacks: the legitimate AP only knows the CSI of Hd and then calculates the multi- user active beamforming Wd via the ZFBF algorithm, while the independent I-IRS controller knows the CSI of Hd, HI, and G as well as Wd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The CSI-based PJ can launch jamming attacks without the transmit power, where the reflecting vector for the I-IRS is optimized by minimizing a certain performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Taking the example of minimizing the sum rate Rsum received at LUs, the optimization of the one-bit reflecting vector is mathematically represented as min ϕ Rsum = min ϕ K � k=1 log2 \uf8eb \uf8ec \uf8ec \uf8ed1 + ���hH com,kwd,k ��� 2 � u̸=k ���hH com,kwd,u ��� 2 + σ2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (7) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' ϕn ∈ Ω, n = 1, 2, · · · , NI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' (8) The phase shift optimization problem in (7) can be solved by enumerating all possible {ϕn}NI n=1 combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' However, there are 2NI different combinations, and thus the computa- tional complexity is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' To this end, we first relax the discrete phase shift constraint in (8) to a continuous constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Mathematically, the reflecting vector optimization is relaxed to max ¯ϕ K � k=1 −log2 \uf8eb \uf8ec \uf8ec \uf8ed1 + ��� � ¯ϕdiag(hH I,k)G+hH d,k � wd,k ��� 2 � u̸=k ��� � ¯ϕdiag(hH I,k)G+hH d,k � wd,u ��� 2 +σ2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (9) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' ¯ϕn ∈ [0, 2π] , n = 1, 2, · · · , NI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' (10) The objective function in (9) is then a continuous and differentiable function of ¯ϕ, and the constraint in (10) creates a complex circle manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Therefore, the optimization problem in (9) can be computed by the Riemannian conjugate gradient (RCG) algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' After computing the continuous reflect- ing vector ¯ϕ, the discrete reflecting vector is obtained by min ϕ ∥ϕ − ¯ϕ∥2 (11) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The complexity of the benchmarking CSI-based PJ is O � IRK2N 2 I � , where IR represents the iteration times of the RCG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In each iteration, the complexity comes mainly from calculating the Euclidean gradient [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Specifi- cally, the complexity of the Euclidean gradient calculation is 4 Subspace of co-user channels in the RPT phase the RPT phase Subspace of co-user channels in the DT phase p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' LU1 LU2 LUK I-IRS RPT DT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' I-IRS-based FPJ via active channel aging, where the I-IRS acts like a “disco ball” and ZFBF cannot project the user channel to the orthogonal interference subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' O � K2N 2 I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Moreover, the complexity of the discreteization of ¯ϕ expressed by (11) is O(2NI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' When the number of reflecting elements packed on the I-IRS is large (NI ≫ 1), the complexity of the discreteization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', O(2NI), can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' I-IRS-Based Jamming Attacks Without Power and CSI Although the CSI-based PJ proposed in Section III-A can jam without the transmit power, the CSI of all channels needs to be obtained at the independent I-IRS controller, which is difficult to satisfy in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In wireless communications, the AP needs to obtain the CSI during the RPT phase before the DT phase, as stated in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 1) The RPT Phase: During the RPT phase, the one-bit reflecting vector for the I-IRS is generated by tuning the n- th reflecting element to a random phase shift belonging to Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', ϕ1 n ∼ U (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' More particularly, the reflecting vector ϕ1 follows the uniform distribution denoted ϕ1 ∼ U � ΩNI� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' It is worth noting that the independent I-IRS controller in the proposed scheme does not need to optimize the reflecting phase shifts of the I-IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Consequently, the multi-user combined channel estimated by the AP is written as (H1 com)H = HH I diag � ϕ1� G+HH d = � h1 com,1, h1 com,2, · · · , h1 com,K �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Based on H1 com, the AP can compute the multi-user active beamforming used in the DT phase that is expressed as W1 ZF =H1 com � (H1 com)HH1 com �−1P 1 2 ���H1com((H1com)HH1com)−1��� 2 = � w1 ZF,1,w1 ZF,2,· · ·,w1 ZF,K � , (12) where w1 ZF,k is orthogonal to the subspace of co-user channels h1 com,1, · · · , h1 com,k−1, h1 com,k+1, · · · , h1 com,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 2) The DT Phase: Then, during the DT phase, the one-bit reflecting vector of the I-IRS is formed according to another reflecting vector ϕ2 that also follows the uniform distribution in Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', ϕ2 ∼ U � ΩNI� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Therefore, during the DT phase, the multi-user combined channel is changed to (H2 com)H=HH I diag � ϕ2� G+HH d = � h2 com,1,h2 com,2,· · ·,h2 com,K �H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' (13) Including (12) and (13) into (4), the actual received SINR ¯γk at LUk during the DT phase is ¯γk = ���(h2 com,k)Hw1 ZF,k ��� 2 � u̸=k ���(h2 com,k)Hw1 ZF,u ��� 2 + σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' (14) The complexity of our proposed scheme comes from ran- domly generating the two reflecting vectors used in the RPT phase and the DT phase, which is only O(2NI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Compared with the benchmarking CSI-based PJ, the I-IRS’s controller in the proposed I-IRS-based FPJ not only does not require the CSI of all channels involved, but also the complexity of the proposed I-IRS-based FPJ is much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 3) Active Channel Aging: Based on (12) and (13), the reflecting vectors for the I-IRS are different and random during the RPT phase and the DT phase (like a “disco ball” shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 2), which destroys the orthogonality generated from ZFBF due to active channel aging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The w1 ZF,k is only orthogonal to the subspace of co-user channels h1 com,1, · · · , h1 com,k−1, h1 com,k+1, · · · , h1 com,K, and thus I in (6) is then equal to �K k=1 � u̸=k |(h2 com,k)Hw1 ZF,u| 2 σ2 , which is no longer zero due to active channel aging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' As a result, the actual received SINR ¯γk in (14) achieved under the proposed I-IRS-based FPJ is dramatically reduced compared to that without attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' We stated that the reflecting vector for the I-IRS is different during the RPT phase and the DT phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In fact, there is no need for precise synchronization in practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Assuming that the periods of the RPT phase and the DT phase are Tr and Td (Tr ≤ Td), the reflecting vector changes randomly with a period of no more than Tr, and active channel aging then occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' SIMULATION RESULTS AND DISCUSSION Consider a MU-MISO system with four single-antenna LUs, where the legitimate AP is equipped with a 12-element ULA [15] and an I-IRS contains 1,024 reflecting elements (NI,y = NI,z = 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Moreover, the AP is located at (0m, 0m, 0m) and the four LUs are randomly distributed in a circle centered at (200m, 0m, 0m) with a radius of 10m, while the I-IRS is deployed at (5m, 5m, 2m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Most of the existing performance-enhancing IRS-aided sys- tems make the assumption that Hd has significant large-scale path loss or is blocked, while the large-scale path losses of G and HI are much smaller [14], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' However, this assumption is too idealistic for jamming attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' According to the 3GPP propagation environment [17], the large-scale path losses Lk, LG and LI,k are set as Lk =32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='6+22log10(dk), LG = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='6 + 20log10(dG) and LI,k = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='6 + 22log10(dI,k), where dk is the distance between the AP and LUk, dG is the distance between the AP and and the I-IRS, and dI,k is the distance between the I-IRS and LUk (1 ≤ k ≤ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Moreover, σ2 = −170 + 10 log10(BW) dBm, where BW denotes the transmission bandwidth and BW =180 kHz [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' We compare the proposed I-IRS-based FPJ with three benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 1) Benchmark 1: The average sum rates without IJ (w/o IJ) are computed based on the multi-user direct channel, where the received SINR γk at LUk is γk = |hH d,kwd,k| 2 � u̸=k |hH d,kwd,u| 2+σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 2) Benchmark 2: The average sum rates under the active jammer (w/ AJ/N) with different ratios of the jamming power to the noise power (AJ/N) at each LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' More specifically, the received SINR γk at LUk under active jamming is expressed 2 W ZF ,k1 W ZF,k62 com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='1 com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='k-1 com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='k-1> com,K5 15 10 5 0 5 10 15 Total Transmit Power [dBm] 10 20 30 40 50 60 70 Average Sum Rate [bits/symbol use] 60 40 20 0 20 40 Inter-User Interference/Noise [dB] Benchmark 1 Benchmark 2 w/ 5 dB Benchmark 2 w/ 10 dB Proposed FPJ Benchmark 3 I/N of Benchmark 3 I/N of Proposed FPJ I/N of Benchmark 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Average sum rates (left, solid lines) and I/N (right, dash-dot lines) of different schemes vs total transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 1 2 3 4 5 6 7 Quantization bits of reflecting phase shifts 40 45 50 55 60 65 Average Sum Rate [bits/symbol use] 60 40 20 0 20 40 Inter-User Interference/Noise [dB] Benchmark 1 Benchmark 2 w/ 5 dB Benchmark 2 w/ 10 dB Proposed FPJ Benchmark 3 I/N of Benchmark 3 I/N of Proposed FPJ I/N of Benchmark 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Influence of quantization reflecting phase shift bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' as γk = |hH d,kwd,k| 2 � u̸=k |hH d,kwd,u| 2+PJ +σ2 , where AJ/N = PJ/σ2=5 dB and 10 dB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 2) Benchmark 3: The CSI-based PJ in Section III-A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=', the extension of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 3 illustrates the average sum rates via the proposed FPJ and the above three benchmarks, where I generated from them is also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' By destroying the orthogonality of the multi-user active beamforming vectors and the co-user channels, the inter-user interference becomes significant due to active channel aging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The reflecting vector in the proposed FPJ affects both the multi-user combined channel and the multi- user active beamforming, while the reflecting vector in the CSI-based PJ just impacts the multi-user combined channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 3, I from the proposed FPJ is more serious than that from the CSI-aided PJ (Benchmark 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Therefore, the proposed FPJ can jam LUs without the transmit power and the CSI, even more effectively than the CSI-aided PJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 3, one can see that the sum rate of Proposed FPJ is smaller than that of Benchmark 2 with 5 dB AJ/N when the total transmit power is greater than 0 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In contrast to the active jamming, the jamming launched by the proposed FPJ cannot be mitigated by increasing the total transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' To show the influence of the number of quantization re- flecting phase shift bits, the relationships between the average 20 22 24 26 28 30 32 Number of Reflecting Elements 40 45 50 55 60 65 Average Sum Rate [bits/symbol use] 60 40 20 0 20 40 Inter-User Interference/Noise [dB] Benchmark 1 Benchmark 2 w/ 5 dB Benchmark 2 w/ 10 dB Proposed FPJ Benchmark 3 I/N of Benchmark 3 I/N of Proposed FPJ I/N of Benchmark 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Influence of the number of reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' sum rates and quantization bits are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' One can see that the proposed FPJ is robust to the quantization bits since the reflecting vector is randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Based on the proposed FPJ, the one-bit I-IRS is enough to launch effective jamming attacks on LUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The greater the number of quantization bits, the smaller the difference ∥ϕ − ¯ϕ∥2 in (11) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Although the sum rate achieved by Benchmark 3 decreases with the number of quantization bits, the high-bit I-IRS requires high physical implementation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' 5 shows the relationship between the sum rates and the number of reflecting elements as well as that between I/N and the number of reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' The difference between the sum rates achieved by Benchmark 3 and Proposed FPJ increases with the number of reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' On the one hand, active channel aging becomes more significant with the number of reflecting elements, and thus the corresponding jamming attacks are more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' On the other hand, the minimum value of ∥ϕ − ¯ϕ∥2 in (11) gets bigger with the number of reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' In practice, an IRS generally consists of massive reflecting elements, which is beneficial to the proposed I-IRS-based FPJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' CONCLUSIONS In this letter, we investigated the impact of I-IRSs on MU- MISO systems, where an I-IRS-based FPJ was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' By exploiting an I-IRS to cause active channel aging, we have demonstrated that the proposed FPJ can jam without relying on the transmit power and the CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Due to the impacts on both the multi-user combined channel and the multi-user active beamforming, the jamming launched by the proposed FPJ is even more effective than that launched by the CSI-aided PJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Meanwhile, the proposed FPJ is robust to the number of quantization reflecting phase shift bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Different from the active jamming attacks, the jamming attacks launched by the proposed FPJ cannot be mitigated by increasing the total transmit power at the legitimate AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' When the legitimate AP has large transmit power, the proposed FPJ can jam LUs more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} +page_content=' Moreover, the proposed FPJ can be perfectly hidden in wireless environments because it does not require additional transmit power and instantaneous CSI.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E3T4oBgHgl3EQfBQnj/content/2301.04266v1.pdf'} diff --git a/N9AzT4oBgHgl3EQfzP7F/content/tmp_files/2301.01767v1.pdf.txt b/N9AzT4oBgHgl3EQfzP7F/content/tmp_files/2301.01767v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f913824919576808f951e7005f54abdb2b84da9c --- /dev/null +++ b/N9AzT4oBgHgl3EQfzP7F/content/tmp_files/2301.01767v1.pdf.txt @@ -0,0 +1,3487 @@ +Self-Supervised Video Forensics by Audio-Visual Anomaly Detection +Chao Feng +Ziyang Chen +Andrew Owens +University of Michigan +Abstract +Manipulated videos often contain subtle inconsistencies +between their visual and audio signals. +We propose a +video forensics method, based on anomaly detection, that +can identify these inconsistencies, and that can be trained +solely using real, unlabeled data. We train an autoregres- +sive model to generate sequences of audio-visual features, +using feature sets that capture the temporal synchronization +between video frames and sound. At test time, we then flag +videos that the model assigns low probability. Despite be- +ing trained entirely on real videos, our model obtains strong +performance on the task of detecting manipulated speech +videos. Project site: https://cfeng16.github.io/ +audio-visual-forensics. +1. Introduction +Supervised learning underlies today’s most successful +methods for image and video forensics. However, the diffi- +culty of collecting large, labeled datasets that fully capture +all of the possible manipulations that one might encounter +in the wild places significant limitations on this approach. +A longstanding goal of the forensics community has been +to design methods that, instead, learn to detect manipula- +tions using cues discovered by analyzing large amounts of +real data through self-supervision [28,48]. +We propose a method that identifies manipulated video +through anomaly detection. Our model learns how audio +and visual data temporally co-occur by training on large +amounts of real, unlabeled video. At test time, we can then +flag videos that our model assigns low probability, such as +those whose video and audio streams are inconsistent. +One might expect that this problem could be posed as +simply detecting out-of-sync examples, such as by finding +cases in which a speaker’s mouth does not open precisely +at the onset of a spoken word. Unfortunately, videos in the +wild are often “naturally” misaligned due to errors in en- +coding or recording, such as by having a single, consistent +shift by a few frames [2,24]. +Instead, we pose the problem as detecting anomalies in +what we call synchronization features: audio-visual features +Input video +Time (frame) +Input audio +Ziyang’s +version +(remove +this text) +Time delay +Figure 1. +Audio-visual anomaly detection. We identify fake +videos by finding anomalies in their audio-visual features, using +generative models trained entirely on real videos. In one variation +of our model (shown here), we use the time delay between the two +modalities as our feature set, i.e., temporal misalignment between +each video frame and the audio stream. We learn the distribution +of these sequences, then flag sequences with low probability. +that are designed to convey the temporal alignment between +vision and sound. We evaluate several feature sets, each ex- +tracted from a model that has been trained to temporally +align audio and visual streams of a video [18, 24, 79]. In +Figure 1, we show one such feature set: the amount of time +that each video frame appears to be temporally offset from +its corresponding sound. To detect anomalies, we fit an au- +toregressive generative model [85,100] to sequences of syn- +chronization features extracted from real videos, and iden- +tify low probability examples. +A key advantage of formulation is that it does not require +any manipulated examples for training. It also does not re- +quire the speakers in the test set to already be present in the +training set. This is contrast to previous audio-visual foren- +sics approaches, which either require finetuning on datasets +of manipulated video [41], or which are based on verifying +that the speaker’s voice matches previously observed exam- +1 +arXiv:2301.01767v1 [cs.CV] 4 Jan 2023 + ++15 +-15ples [26]. +We evaluate our model on videos that have manipulated +a person’s speech and face, using datasets of lip-synced +and audio-driven face reenactment videos, some of which +are also manipulated by faceswap techniques. Our model +obtains strong performance on the FakeAVCeleb [54] and +KoDF [62] datasets, despite the fact that it is trained en- +tirely on real examples obtained from other video datasets. +Our model generalizes to other spoken languages without +retraining and obtains robustness to a variety of postpro- +cessing operations, such as compression and blurring. We +show through our experiments that: +• Video forensics can be posed as an audio-visual anomaly +detection problem. +• Synchronization features convey information about video +manipulations. +• Our model can successfully detect fake videos, while +training solely on real videos. +• Our model generalizes to many types of image postpro- +cessing operations and to speech videos from spoken lan- +guages not observed during training. +2. Related Work +Audio-visual +forensics. +In +early +work, +Malik +and +Farid [72] detected audio manipulations by finding incon- +sistencies in reverberation. +Recent work has focused on +detecting manipulated speech videos using audio-visual in- +consistencies. +Several approaches have directly trained +audio-visual networks through supervised learning, using +labels indicating whether a video is manipulated [21, 74]. +A variety of methods have recently used audio-visual self- +supervision for pretraining supervised models, which are +finetuned with “real or fake” labels. Zeng et al. [106] used +local and global contrastive learning methods to learn video +features. +Haliassos et al. [41] jointly solved a negative- +less contrastive learning problem [39] and a forensics task. +Zhou and Lim [114] used audio-visual synchronization sig- +nal implicitly, and proposed a dataset for audio-visual deep- +fake detection1. Other work [42] pretrains using lip-reading +data. In contrast to these methods, our approach is trained +entirely using real data and does not require any labels or +examples of fake videos. Other work has used speaker ver- +ification [26] and phoneme-viseme mismatches [7] to de- +tect fake videos and it also detects face swap manipula- +tions, which preserve the synchronization between modal- +ities. In contrast, our approach detects misaligned images +and sounds and does not require that examples from the +speaker be present in the training set. +Audio-visual representation learning. +A variety of +methods have been proposed to learn audio-visual repre- +sentation from videos via self-supervision. +Researchers +1Their dataset is not publicly available. +have leveraged the natural semantic correspondence in the +videos between frames and audio tracks [10, 76, 106] to +learn multi-modal features and applied them to downstream +tasks such as sound localization [9,46,75]. Other work stud- +ies temporal synchronization between audio and visual sig- +nals to learn audio-visual features [24, 60, 79], which can +be used for active speaker detection [8,59,96], source sep- +aration [37, 71, 112], lip reading [5, 70, 73] and so on. Our +method uses the off-the-shelf audio-visual synchronization +model to perform anomaly detection. +Visual face forensics. +A major focus of the forensics field +has been on the problem of detecting manipulated videos +of human faces. In recent years, a variety of visual face +manipulation datasets are proposed, such as FaceForen- +sics++ [88], VideoForensicsHQ [35] and FFIW10K [113]. +Meanwhile, many methods are proposed to detect syn- +thetic contents to fight against their potential threats. Some +work [12,40,64] has proposed to use hand-crafted features +to capture inconsistent visual or JPEG artifacts. Other work +has proposed to use deep learning to inspect specific arti- +facts, such as blending [63], frequency domain [32,36,84], +or texture [68]. A variety of methods have studied the gen- +eralization between detection classifiers [16,101]. +Anomaly detection. +A variety of methods have learned +a distribution, then flagged unusual examples. These ex- +amples are often considered anomalies [67,92,105,115] or +outliers [82, 90], and are used as part of open-set recog- +nition [58, 107]. +We formulate video forensics as the +task of detecting anomalies, using a feature set that con- +veys information that would be hard for a forger to cre- +ate. There have been a variety of methods proposed for +learning this distribution, such as GAN discriminators [38, +58, 67, 82, 90, 92, 105, 115], flow-based models [107], +and autoregressive models [94]. +Similarly, our model is +based on an autoregressive generative model [11,85], since +they have achieved strong performance at modeling com- +plex distributions. Other work addresses goals similar to +anomaly detection by creating methods that model uncer- +tainty [65] or that perform outlier exposure [31, 45, 89]. +Some work [13, 27, 29, 48, 51, 55, 80] has used special- +purpose anomaly detection methods for image/video foren- +sics. Other work [34,47,102,110] uses supervised learning +to find anomalous patterns. In contrast, our method builds +the likelihood function entirely on real videos and views +low-probability examples as fake. +3. Method +We formulate the problem of detecting manipulated +videos as an anomaly detection problem. We model the dis- +tribution of audio-visual examples, then flag examples that +have low probability. If we were to fit a model on the raw +data, then this would be a very challenging learning task. +2 + +Audio-visual Synchronization Model +A +V +Autoregressive Prediction +Time +Likelihood +Target +features +··· +ℒ +Predicted +features +··· +-1 ++1 +··· +-2 +Discrete +time delays +Prob. +Distribution +over delays +V +A +Feature +activations +(a) Synchronization feature extraction +(b) Anomaly detection +Figure 2. Audio-visual anomaly detection model. (a) We extract a feature set from an audio-visual synchronization network: the number +of frames of delay between video frames and sound, the distribution over delays at each frame, and feature activations from the audio and +visual subnetworks. (b) We train an autoregressive Transformer model to assign probabilities to synchronization features. At test time, we +flag low probability examples. +Instead, we learn the distribution over a feature set that con- +veys subtle properties that are unlikely to be accurately cap- +tured in manipulated video. +3.1. Estimating audio-visual synchronization +We obtain our feautres from a network that performs +audio-visual synchronization [18, 24, 25, 79]. We use the +model of Chen et al. [18]. We learn a function φ(Vi, Aj) +that indicates how likely video clip Vi temporally co-occurs +with audio clip Aj. We estimate the synchronization score +S(i, j) of all audio-visual pairs in a temporal window: +S(i, j) = +exp (φ(Vi, Aj)) +�i+τ +k=i−τ exp (φ(Vi, Ak)) +, +(1) +where τ is maximum time difference between two streams, +and φ(Vi, Aj) = h (gv(Vi), ga(Aj)) is calculated using late +fusion by a visual encoder gv, audio encoder ga and the +fusion module h. We also interpret S(i, j) as synchroniza- +tion probability. We maximize the synchronization of true +audio-visual pairs (Vi, Ai) using the InfoNCE loss [78]: +Lsync = − 1 +T +T +� +i=1 +log S(i, i), +(2) +for a video of length T. We provide details about the archi- +tecture and training procedure in Appendix E. +After training, we can use the learned model to obtain a +feature set for anomaly detection. For example, we can use +the rows of S, which provide a probability distribution over +possible alignments between video clips and audio clips. +3.2. Audio-visual anomaly detection +We use our learned model to obtain a feature set for +anomaly detection. We learn the distribution of these fea- +tures on a training set of real videos. Then at test time, +videos with low probability will be flagged as potential +fakes. We now explore two key design decisions that go +into such a system: what feature set to use, and how the +distribution is learned. +Given features for each frame, we learn a distribution +pθ(x1, x2, . . . , xN). We generally use autoregressive mod- +els to learn this distribution, given their success in model- +ing complex distributions [14,104]. These models take the +form: +pθ(x1, x2, · · · , xN) = +N−1 +� +i=0 +pθ(xi+1|x1, · · · , xi). +(3) +We train a model ˆxi+1 = fθ(x1, x2, . . . , xi) that estimates +the features of the next frame, given all of the features from +the previous frames. Maximizing the log probability can be +posed as minimizing a per-frame loss, L: +L = +N +� +i=1 +L(ˆxi, xi). +(4) +We now describe different formulations of the loss func- +tion L, the feature representation xi. In each case, we im- +plement fθ as a Transformer [100]. +Discrete time delays. +We first consider a simple model +that uses discrete time delay estimates as our feature rep- +resentation, following the success of autoregressive mod- +3 + +els for fitting discrete data [33, 87, 98]. +Taking inspira- +tion from work on time delay estimation [19, 57], for ev- +ery video frame, we estimate how far ahead (or behind) it +appears to be from the audio signal. For each frame, we +set xi to be the time delay with the highest probability, i.e., +xi = arg maxj(S(i, j)). We then set L to be cross en- +tropy loss between the ground truth and predicted time de- +lay. This amounts to solving a categorization problem with +2τ + 1 possible labels for each frame. +Distributions over delays. +While discrete time delays are +straightforward to represent in the model, they discard im- +portant information, such as when there is ambiguity in the +delay. We therefore propose a model that directly predicts +the entries of the time delay distribution. We set the features +xi to be the rows of S, i.e. the probability of each possible +delay, and use cross entropy loss: +L(ˆxi, xi) = − +2τ+1 +� +j +xi,j log(ˆxi,j). +(5) +We constrain the predictions made by our model fθ to +sum to 1 by applying a softmax. +Audio-visual network activations. +The feature activa- +tions within the audio-visual synchronization network con- +vey information about the time delay. We, therefore, ask +whether these activations can be directly used as features +for anomaly detection. We concatenate the representations +of the visual and audio subnetworks, gv and ga. To pro- +vide a straightforward comparison with the time delay dis- +tribution model, we reduce the dimensionality of the fea- +tures by projecting them onto the top 2τ + 1 principal +components, following other work in autoregressive mod- +els of features [86]. We use squared distance as our loss: +L (ˆxi, xi) = ∥xi − ˆxi∥2. +4. Results +We evaluate the different variations of our model on a +variety of video forensics tasks. +4.1. Implementation details +Synchronization model. +Following Chen et al. [18], we +use ResNet-18 2D+3D [43, 44] as the visual encoder, us- +ing 5 frames frames (25 fps) as input. +The audio en- +coder uses VGG-M [17] and extracts features from 0.2s +audio clips (16kHz). We fuse audio and visual data us- +ing a Transformer that has 3 standard Transformer encoder +blocks [100], 4 attention heads, and 512 channels. We train +using the cropped faces provided by each dataset. Please +see Appendix E for details. +Anomaly detection model. +We use a decoder-only au- +toregressive Transformer [33, 66, 85] to learn the distri- +bution over synchronization features. +We use 2 decoder +blocks [100], each with 16 attention heads with 256 chan- +nels. For models that use time delay, we set the maximum +delay to be τ = 15 frames, resulting in the distribution +Si ∈ R31 for each video frame. We use sequences of length +N = 50 from 2.0s video. +Hyperparameters. +All videos are resampled to 25 fps +and 16kHz mono audio. We represent audio segments as +the mel spectrogram of size 21×80 via Short-Term Fourier +Transform (STFT) with 80 mel filter banks, the hop length +of 160, and the window size of 320. Please see more details +in Appendix E. +4.2. Dataset +We train our model on real, unlabeled speech video, and +evaluate it on forensics datasets. +Training datasets. +We train our models on Lip Reading +Sentences 2 (LRS2, 97k videos) [3] and Lip Reading Sen- +tences 3 (LRS3, 120k videos) [4]. The videos in each con- +tain tightly cropped face tracks. We divide each dataset into +3 splits and train the audio-visual synchronization model +and the autoregressive model on different splits. +Evaluation datasets. +We evaluate on two video foren- +sics datasets, spanning several different types of manip- +ulations that change the speech and face of a human +speaker. FakeAVCeleb [54], which is derived from Vox- +Celeb2 [22]. +This dataset contains 500 real videos and +19,500 fake videos manipulated by Faceswap [61], FS- +GAN [77], and Wav2Lip [83], and fake sounds that are +generated by SV2TTS [49]. The examples in the dataset +contain different combinations of these manipulations. We +use the dataset’s provided face crops. +We sample 2400 +videos (400 real videos and 2000 fake videos) as train/val +splits and 600 videos (100 real videos and 500 fake videos) +as test split. We note that our method does not use any +videos from train/val splits, since it is trained from an- +other dataset (LRS2 or LRS3). +Second, we evaluate on +KoDF [62], a large-scale Korean-language deepfake detec- +tion dataset. It contains 62,166 real videos and 175,776 fake +videos, where fake videos are generated by 6 synthesized +methods: FaceSwap [1], FSGAN [77], DeepFaceLab [81], +FOMM [93], ATFHP [103] and Wav2Lip [83]. We extract +faces by using face detection [108] and alignment [15]. +4.3. Evaluation methods +Following common practice [16,41,42,54,62,63,84,88, +101, 111], we evaluate using average precision (AP) and +AUC. These evaluation metrics are widely used for cross- +dataset generalization and unsupervised models since they +avoid the need to threshold the predictions. We compare our +approach to both supervised and self-supervised methods. +Unless otherwise stated, we use time delay distributions as +our feature set (Sec. 3.2). +4 + +Pretrained +dataset +Category +Method +Modality +RVFA +FVRA-WL +FVFA-FS +FVFA-GAN +FVFA-WL +AVG-FV +AP +AUC +AP +AUC +AP +AUC +AP +AUC +AP +AUC +AP +AUC +Supervised +Xception [88] +V +ImageNet [30] +– +– +88.2 +88.3 +92.3 +93.5 +67.6 +68.5 +91.0 +91.0 +84.8 +85.3 +LipForensics [42] +V +LRW [23] +– +– +97.8 +97.7 +99.9 +99.9 +61.5 +68.1 +98.6 +98.7 +89.4 +91.1 +AD DFD [114] +AV +Kinetics [53] +74.9 +73.3 +97.0 +97.4 +99.6 +99.7 +58.4 +55.4 +100. +100. +88.8 +88.1 +FTCN [111] +V +– +– +– +96.2 +97.4 +100. +100. +77.4 +78.3 +95.6 +96.5 +92.3 +93.1 +RealForensics [41] +V +LRW [23] +– +– +88.8 +93.0 +99.3 +99.1 +99.8 +99.8 +93.4 +96.7 +95.3 +97.1 +Unsupervised +AVBYOL [39,41] +AV +LRW [23] +50.0 +50.0 +73.4 +61.3 +88.7 +80.8 +60.2 +33.8 +73.2 +61.0 +73.9 +59.2 +VQ-GAN [33] +V +FFHQ [52] +– +– +52.7 +53.8 +49.0 +51.2 +63.9 +61.1 +52.0 +54.4 +54.4 +55.1 +Ours +AV +LRS2 [3] +62.4 +71.6 +93.6 +93.7 +95.3 +95.8 +94.1 +94.3 +93.8 +94.1 +94.2 +94.5 +Ours +AV +LRS3 [4] +70.7 +80.5 +91.1 +93.0 +91.0 +92.3 +91.6 +92.7 +91.4 +93.1 +91.3 +92.8 +Table 1. Manipulation detection on FakeAVCeleb. We report AP scores (%) and AUC scores (%), following the evaluation protocol +of Haliassos et al. [42], in which supervised methods are evaluated on unseen manipulation types (unsupervised methods are not trained +with labels). We report results with combinations of real/fake video/audio, using different generation algorithms. We report the average +performance over four fake video (FV) categories in AVG-FV. We retrained all models, since there is no standard dataset split. +Supervised methods. +For supervised methods, we train +several state-of-the-art detectors on the two datasets: +1) Xception [88]: a popular baseline for forensics detec- +tion; 2) LipForensics [42]: a detector is built on high- +level semantic embeddings of mouth and targets irregu- +larities in mouth movements; 3) AD DFD [114]: a mul- +timodal detector with audio and video branches, utilizes +audio-visual synchronization signal implicitly for detection; +4) FTCN [111]: a video forensics detector leverages tem- +poral incoherence to boost generalization capability; 5) Re- +alForensics [41]: it first pretrains the network by audio- +visual BYOL [39] framework and then finetunes the pre- +trained model and forensics datasets by multi-task learning +to obtain robust and general face forgery detection. Since +there is no standard split for our forensics tasks, we retrain +each model. +Self-supervised methods. +Since we are not aware of any +existing methods that consider self-supervised speech video +forensics, we adapt two existing methods to the task. First, +we consider an audio-visual contrastive learning model, +which we call AVBYOL, that learns to determine whether +the visual and audio streams of a video do (or do not) match, +an approach that has been used as a part of other audio- +visual forensics models [26, 41]. We adapt the model of +Haliassos et al. [41], which uses BYOL [39] to learn a joint +audio-visual embedding for pretraining. +Instead of pre- +training, we directly use the model’s audio-visual similarity +score to flag fake examples. Second, we use an off-the- +shelf generative model, VQGAN [33], for anomaly detec- +tion. VQGAN converts an image into a sequence of discrete +codes, then uses an autoregressive Transformer to learn the +distribution of codes. We use the code’s log likelihood, av- +eraged over each video frame, for anomaly detection. +4.4. Evaluation +In real-world scenarios, the deployed detectors are ex- +pected to recognize fake videos manipulated by unseen +techniques. Thus, following the standard procedure used +in [41,42,111,114], we conduct the experiment to evaluate +the cross-manipulation generalization ability of our model +on the FakeAVCeleb dataset [54] which videos are manip- +ulated in various ways. Since our approach and other self- +supervised baselines learn from real, unmanipulated videos +and perform zero-shot fake video detection, all the fake +videos during evaluation are considered as manipulated by +unseen methods. +We split FakeAVCeleb dataset [54] into five cate- +gories based on the manipulation methods and manipu- +lated modalities: 1) RVFA: real video with fake audio +by SV2TTS [49]; 2) FVRA-WL: real audio with fake +video by Wav2Lip [83]; 3) FVFA-WL: fake video by +Wav2Lip [83], and fake audio by SV2TTS [49]; 4) FVFA- +FS: fake video by Faceswap [61] and Wav2Lip [83], and +fake audio by SV2TTS [49]; 5) FVFA-GAN: fake video by +FaceswapGAN [77] and Wav2Lip [83], and fake audio by +SV2TTS [49]. For the supervised methods, we hold out the +evaluated category and train the models on the four remain- +ing categories. Note that some approaches are only able to +detect the manipulation on a certain modality, we do not +report their performance on the categories with the manip- +ulation only on the other modalities. +We show our results in Tab. 1. Our method substantially +outperforms both self-supervised methods AVBYOL [39, +41] and VQGAN [33] on each category by a large mar- +gin. +More importantly, our method works on par with +or outperforms some supervised methods on certain cat- +egories, especially FVFA-GAN, even though our method +does not use any labeled supervision or fake examples. +5 + +Method +Modality +KoDF [62] +AP +AUC +Supervised +(transfer) +Xception [88] +V +76.9 +77.7 +LipForensics [42] +V +89.5 +86.6 +AD DFD [114] +AV +79.6 +82.1 +FTCN [111] +V +66.8 +68.1 +RealForensics [41] +V +95.7 +93.6 +Unsupervised +AVBYOL [39,41] +AV +74.9 +78.9 +VQ-GAN [33] +V +49.0 +50.0 +Ours +AV +87.6 +86.9 +Table 2. Generalization to Korean speech. AP scores (%) and +AUC scores (%) are reported on KoDF dataset [62]. Supervised +methods are trained on FakeAVCeleb dataset [54] and transferred. +Ours is trained on LRS2 [3]. Best results are in bold. +Moreover, our method has quite consistent performances +and it can achieve AP over 90% on the most of categories. +While Xception [88], LipForensics [42], AD DFD [114] +and FTCN [111] work well on 75% of the settings, there are +settings where performance collapses to near-chance (e.g., +AD DFD [114] on FVFA-GAN). Interestingly, the two self- +supervised baselines struggle to detect fake videos, perhaps +because both models do not necessarily capture the sub- +tle information that would be needed to detect manipula- +tions. In addition, VQGAN [33] compresses the visual sig- +nal using a codebook, which might drop the artifact clues +and harm the detection performance. Moreover, our model +trained on LRS2 [3] works on par with the one trained on +LRS3 [4], indicating that our method’s performance is not +tied to a single training set. +Cross-dataset generalization. +We also evaluate the gen- +eralization capability of our model by evaluating it on the +KoDF dataset [62], following [41,42,111,114]. We focus on +the audio-driven synthesis examples in the dataset, where +videos are manipulated by ATFHP [103] or Wav2Lip [83]. +We train the supervised models on FakeAVCeleb [54] to +evaluate their generalization ability. Many of these training +videos share the same technique used in KoDF for synthe- +sis [83]. As the results are shown in Tab. 2, our approach ob- +tains a comparable performance to many supervised meth- +ods. Although our system is trained on the English speech +datasets, it still generalizes to KoDF [62] dataset of Korean +speech, perhaps because it learns low-level lip motion cues +that are broadly useful. We provide more results in Ap- +pendix B. +Qualitative results. +We visualize the ground truth and +predicted time delay distributions generated by our au- +toregressive continuous time delay model (Sec. 3.2) in +Fig. 3. We use the four main categories from FakeAVCeleb +dataset [54]. For each one, we display a heat map indicat- +Figure 3. Time delay predictions for real vs. fake examples. +We visualize the time delay distributions from the synchronization +model and predicted results generated by the autoregressive model +for four random samples from different categories. Synchroniza- +tion probabilities are in a range from 0 +1. We show the +predictions of the autoregressive model when feeding it ground +truth observations of the previous timesteps. We show cumulative +prediction error (indicating the probability of being fake) for each +sample over time steps in the last row. +ing the predicted time delay, using a model that obtains the +ground truth delays of the previous frames as input. We also +plot the cumulative prediction loss (Eq. 4) over time. From +Fig. 3, we can see that our autoregressive model accurately +predicts the ground truth for real video, which results in a +lower score. For fake videos, we can find clear differences +between ground truth and predicted time delay distribution, +leading to higher prediction loss. +4.5. Robustness to unseen perturbations +When the fake video is redistributed, it may undergo +many types of postprocessing that result in corruption, mak- +ing detection more difficult. +Thus, it is important for +forensics models to be robust to the types of postprocess- +ing operations they may encounter in the wild. +Follow- +ing [41, 42, 111], we use the set of visual perturbations +proposed in [50]: 1) Color saturation change; 2) Block- +wise distortion; 3) Color contrast change; 4) Gaussian blur; +6 + +50 +60 +70 +80 +90 +100 +AUC (%) +Saturation +JPEG +Block-wise +Gaussian Noise +Gaussian Blur +0 +2 +4 +Intensity +50 +60 +70 +80 +90 +100 +AUC (%) +Compression +0 +2 +4 +Intensity +Contrast +0 +2 +4 +Intensity +Average +0 +2 +4 +Intensity +Adversarial +Chance +FTCN +Xcepetion +AD DFD +RealForensics +Ours +Figure 4. Robustness to unseen perturbations. AUC scores (%) of different detectors as a function of perturbation intensities. There are +6 intensity levels in total from [50]. “Average” represents the average over 7 perturbations under each intensity. “Adversarial” means we +pick the worst performance across 7 perturbations under each intensity. +5) Gaussian noise; 6) JPEG compression; 7) Video com- +pression rate change. We set the intensity levels from 0 to 5 +for each perturbation. +We compare our model with four supervised meth- +ods XceptionNet [88] FTCN [114], AD DFD [114] and +RealForensics [41]. +The results are reported in Fig. 4. +Our proposed self-supervised model is overall more robust +to unseen visual perturbations on average compared with +these supervised methods, with the exception of RealForen- +sics [41]. This is also true when we consider “worst case” +performance, by taking the minimum performance over all +types of augmentation of a given intensity level. Interest- +ingly, we obtain this performance even though our model is +trained in a very different way from other works, suggesting +that the feature set continues to convey useful information +to the anomaly detection model, even in the presence of sig- +nificant corruption. +4.6. Feature set analysis +We evaluate the effectiveness of different feature sets +used by our anomaly detection model. +As described in +Sec. 3.2, we start with discrete time delays as our feature +representation and optimize the model with cross entropy +loss. Then, we use continuous time delay distributions +for representations instead, where we optimize models with +different objective functions: 1) Soft CE: we use the time +delay distribution as target (akin to a “soft” label) and use +cross entropy loss (Sec. 3.2); 2) CE: we map each distri- +bution into one-hot encoding as target by using arg max +and employ cross entropy loss; 3) BCE: we use the dis- +tribution as the target while treating each synchronization +score S(i, j) (Sec. 3.1) within the same time step indepen- +Model +Feature set +L +AVG-ALL +AVG-FV +AP +AUC +AP +AUC +Bayes +- +- +73.1 +85.1 +72.4 +86.0 +Ours +discrete delay +CE +80.8 +86.5 +80.0 +86.6 +distribution +CE +84.8 +87.9 +90.3 +92.2 +distribution +BCE +78.6 +83.4 +80.5 +84.8 +distribution +Soft CE +87.8 +90.0 +94.2 +94.5 +activation-AV +MSE +86.5 +87.1 +91.5 +91.9 +Act.-AV+dist. +MSE +85.5 +87.0 +90.0 +91.3 +activation-V +MSE +– +– +77.6 +85.9 +discrete prob. +– +83.4 +86.9 +88.6 +91.1 +Table 3. Feature set analysis. AP (%) and AUC (%) are reported +on FakeAVCeleb [54] when using different feature sets. Best re- +sults are in bold. AVG-ALL means the average over all categories. +AVG-FV represents the average over four fake video categories. +dently. We use the sigmoid function and binary cross en- +tropy loss to train the model. We also use our network’s fea- +ture activations as in Sec. 3.2: 1) audio-visual feature ac- +tivations (activation-AV); 2) visual-only feature activations +(activation-V). Besides, we consider using a combination +of different feature sets where we concatenate continuous +time delay distributions and audio-visual feature activa- +tions (Act.-AV+dist.) as a new feature. Similar to audio- +visual feature activations as in Sec. 3.2, we use squared dis- +tance as the loss for the concatenation of these two types of +feature sets. +We also compare with a simple model based on Naive +Bayes and discrete time delays. This model assumes that +each frame’s time delay is independent, and obtains a prob- +ability for the entire sequence by multiplying the probability +7 + +Synchronization +dataset +Auto-regression +dataset +AVG +AP +AUC +LRS2 [3] +LRS2 [3] +87.8 +90.0 +LRS3 [4] +85.0 +89.6 +LRS2 [3]+LRS3 [4] +85.1 +89.9 +LRS3 [4] +LRS2 [3] +86.6 +89.0 +LRS3 [4] +87.2 +90.3 +LRS2 [3]+LRS3 [4] +87.2 +90.6 +Table 4. Dataset ablation. AP scores (%) and AUC scores (%) are +reported on FakeAVCeleb [54] dataset by using different datasets +to train synchronization model or atuoregressive model. Best re- +sults are in bold. +of each frame’s time delay. This amounts to simply detect- +ing large misalignments, since in practice the Naive Bayes +model will assign probability solely based on the magnitude +of each delay. +Finally, we consider a version of the model that autore- +gressively predicts the entire distribution of time delays, in- +spired by autoregresive models, such as PixelCNN [97] that +generate images in a raster scan order. We autoregressively +predict each element of the 2D matrix ˆS(i, j), where ˆS(i, j) +is created by vector quantizing the entries of the synchro- +nization probability S(i, j) using k-means (see Appendix D +for details). +Analysis. +We evaluate each variant on FakeAVCeleb [54] +and report results in Tab. 3. +These results suggest that +all formulations achieve performance significantly better +than chance, indicating that these feature sets are useful for +anomaly detection. As in Tab. 3, the time delay distribu- +tion model outperforms the discrete time delay model, sug- +gesting that there is important information conveyed in the +probability of unlikely delays. The autoregressive model +that uses distribution as input and soft labels (soft CE) per- +forms best, since it forces the output prediction to match the +distribution from the synchronization model. Interestingly, +the model that uses audio-visual feature activations obtains +performance close to that of the soft CE model, indicating +that the networks’ audio-visual features convey useful infor- +mation. Finally, the multimodal activation-AV model sig- +nificantly outperforms the visual-only activation-V model, +suggesting that having access to both modalities is useful +for our anomaly detection model. +4.7. Ablation study +Different training dataset. +We ask how the choice of +dataset affects the quality of the model. To test this, we +train our synchronization and autoregressive models on dif- +ferent datasets to analyze the generalization abilities of +each component, i.e., training the synchronization model +on LRS2/LRS3 and training the autoregressive model on +10 +20 +30 +40 +50 +60 +Sequence length +74 +77 +80 +83 +86 +89 +92 +Average +AP +AUC +11 +21 +31 +41 +51 +Distribution length +80 +82 +84 +86 +88 +90 +92 +Figure 5. +Hyperparameter ablation. +We evaluate with dif- +ferent input sequence lengths for our autoregressive model on +FakeAVCeleb (left), and study the effect of the time delay dis- +tribution’s maximum temporal offset (right). +LRS3/LRS2 or LRS2+LRS3 with the same hyperparame- +ters. As shown in Tab. 4, there is no significant performance +change when we train these two components on different +combinations of datasets, including when they are trained +on the same dataset. This suggests that the distribution of +time delay predictions may be stable between these speech +video datasets. +Influence of sequence length. +To explore the influence +of input sequence length for autoregressive model, we sam- +ple the same amount of training videos for sequence length +N of 10, 20, 30, 40, 50, and 60, and keep other hyperpa- +rameters the same. We test these models on FakeAVCeleb +dataset [54]. Fig. 5 shows that as the sequence length in- +creases, the performance increases with it. +Effect of time delay distribution maximum offset. +We +also study how the length of time delay distribution would +affect the performance of the autoregressive model with dis- +tribution over delays. We experiment with maximum offset +τ ∈ {5, 10, 15, 20, 25} resulting in the delay distribution +length of {11, 21, 31, 41, 51}. We test these models on the +FakeAVCeleb dataset [54]. Fig. 5 shows that as distribu- +tion length increases, the performance first increases, after +which point results plateau or slightly decrease. This may +be due to the fact that when considering larger ranges of +offsets, the distribution spreads over a large number of un- +likely possibilities, making important information less ap- +parent after normalization. +5. Conclusion +We have proposed a method for detecting video manipu- +lation by self-supervised anomaly detection. To do this, we +create novel feature sets that convey audio-visual synchro- +nization. We then show that fake videos can be detected +by flagging examples with unlikely sequences of these fea- +tures, according to a learned distribution. Our model ob- +tains strong performance on the FakeAVCeleb and KoDF +datasets, despite the fact that it was trained only on real +8 + +video. It also obtains robustness to visual postprocessing +operations and to videos containing other spoken languages. +We see our work as opening in two directions. The first is +in posing forensics as an anomaly detection problem with +a self-supervised feature set. While we have proposed one +such model, based on autoregressive sequence models, the +field of anomaly detection offers many possible future ap- +proaches. The second direction is in developing new feature +sets that are well-suited to forensics problems, beyond the +synchronization features used in this work. We will release +code, models, and dataset splits upon acceptance. +Limitations and Broader Impacts. +Our work provides +methods that can potentially be applied to detecting mali- +cious video manipulations and disinformation. While we +have shown that our model is capable of detecting several +types of fake video, there may be other techniques that our +model fails to detect. In particular, due to the design of +our use of synchronization-based features, our model is not +well suited to detecting manipulations that leave the syn- +chronization between motion and sound relatively consis- +tent, such as those that change a speaker’s appearance with- +out significantly changing the motion of their mouth. +Acknowledgements. +We thank David Fouhey, Richard +Higgins, Sarah Jabbour, Yuexi Du, Mandela Patrick, Deva +Ramanan, Haochen Wang, and Aayush Bansal for helpful +discussions. This work was supported in part by DARPA +Semafor and Cisco Systems. The views, opinions and/or +findings expressed are those of the authors and should not +be interpreted as representing the official views or policies +of the Department of Defense or the U.S. Government. +References +[1] Faceswap. https://github.com/deepfakes/faceswap, 2022. 4 +[2] Triantafyllos Afouras, Yuki M. Asano, Francois Fagan, An- +drea Vedaldi, and Florian Metze. 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We show “ground truth” outputs from +the synchronization model and autoregressive predictions +over time. We also show a score indicating the probabil- +ity that the example is fake (Eq. (4), main paper). We use +the time delay distribution as our feature set, and display the +predictions as a heat map. Each step xt of the predicted out- +puts were obtained by providing the ground truth features +from times x1, x2, ..., xt−1 into the autoregressive model. +B. Cross-dataset Generalization +We also experiment with the another state-of-the-art +audio-driven video editing method VideoReTalking [20] +which takes audio-visual synchronization into considera- +tion. +We test on a small dataset consisting of 37 real +videos and 37 fake videos, which are based on the HDTF +dataset [109]. +We show results on Tab. 5. +Our method +leverages AV feature activations as in Sec. 4.6 (number of +principal components is set to 512) and obtains a compa- +rable performance to many supervised methods although it +is only trained on real videos, indicating that our method +possesses certain generalization capability to different ma- +nipulation methods. +Method +Modality +VideoReTalking [20] +AP +AUC +Supervised +(transfer) +Xception [88] +V +61.1 +61.4 +LipForensics [42] +V +98.0 +97.7 +AD DFD [114] +AV +72.7 +73.1 +FTCN [111] +V +70.2 +67.6 +RealForensics [41] +V +100. +100. +Unsupervised +AVBYOL [39,41] +AV +51.3 +57.8 +VQ-GAN [33] +V +51.2 +46.0 +Ours (activation-AV) +AV +71.2 +74.8 +Table 5. +Generalization to VideoReTalking method [20]. +AP scores (%) and AUC scores (%) are reported on HDTF +dataset [109]. +Fake videos are manipulated by VideoReTalk- +ing [20]. +Supervised methods are trained on FakeAVCeleb +dataset [54] and transferred. Ours is trained on LRS2 [3] and uses +the feature set of AV activations. Best results are in bold. +C. Ablation Study +Feature activation. +We study the influence of the num- +ber of principal components D for the variation of the +anomaly detection model that projects the audio-visual fea- +ture activations into a lower dimensional space. We set D to +11, 31, 32, 64, 128, 256, 512 and keep other hyperparame- +ters the same. Fig. 6 shows that as the D increases, the accu- +racy of the anomaly detection model decreases. The reason +11 +31 +32 +64 +128 +256 +512 +Number of principal components +80 +81 +82 +83 +84 +85 +86 +87 +88 +Average +AP +AUC +Figure 6. Number of PCA projections. We evaluate with differ- +ent number of principal components for the model that obtains a +feature set by projecting the audio-visual feature activations to a +lower dimensional space using PCA. We report average results for +the metrics in the FakeAVCeleb dataset [54]. +might be that the higher dimensional prediction problem is +also more challenging. +D. Feature Set Variations +We describe more details about building the autoregres- +sive model on some of our feature sets in this section. +Binary cross entropy (BCE) model. +We assume that the +2τ + 1 possible time delays of each time step are indepen- +dent, and use and BCE loss for probability S(i, j) of each +time delay. Thus, we can decompose pθ(x1, x2, · · · , xN) +as in Eq. (6): +pθ(x1, x2, · · · , xN) = +N−1 +� +i=0 +2τ+1 +� +q=1 +pθ (xi+1,q|x1, · · · , xi) , +(6) +We then maximize pθ (xi+1,q|x1, · · · , xi) using BCE loss: +L(ˆxi, xi) = − +2τ+1 +� +j=1 +xi,j log(ˆxi,j)+(1−xi,j) log(1− ˆxi,j). +(7) +We constrain the prediction ˆxi,j to the range of [0, 1] via +a sigmoid function. +Discrete 2D probability model. +The discrete prob. +model generates the entire 2D frame/time delay matrix au- +toregressively in “raster scan” order, similar to models such +as PixelCNN [91, 99]. We unroll the time delay distribu- +tion sequence into 2D grid like a grayscale image as shown +in Fig. 7. We use K-means (K = 8) clustering to quan- +tize eacn entry and assume each probability ˆS(i, j) of time +delay not only depends on the previous time delay distri- +butions but also the previous probabilities within the same +14 + +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +15 +12 +9 +6 +3 +0 +-3 +-6 +-9 +-12 +-15 +4 +4 +4 +4 +5 +5 +5 +5 +6 +5 +5 +4 +5 +4 +4 +5 +5 +6 +6 +6 +5 +5 +6 +6 +5 +4 +3 +3 +4 +6 +6 +6 +6 +6 +6 +6 +5 +6 +5 +5 +5 +5 +5 +5 +5 +5 +5 +4 +4 +5 +4 +4 +4 +4 +5 +5 +5 +5 +6 +5 +5 +4 +5 +4 +6 +6 +6 +6 +6 +6 +5 +5 +6 +6 +6 +6 +5 +5 +6 +6 +6 +6 +6 +6 +5 +5 +6 +6 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +4 +4 +4 +4 +5 +5 +5 +5 +6 +5 +5 +4 +5 +5 +6 +6 +6 +6 +6 +5 +5 +5 +6 +6 +6 +6 +6 +6 +7 +6 +6 +6 +6 +5 +5 +6 +6 +6 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +4 +4 +4 +4 +5 +5 +5 +5 +6 +5 +5 +5 +7 +6 +6 +6 +5 +5 +5 +5 +5 +5 +6 +6 +6 +6 +6 +6 +7 +6 +6 +6 +6 +5 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +4 +4 +4 +4 +5 +5 +5 +5 +5 +5 +5 +7 +7 +6 +6 +6 +6 +6 +6 +5 +5 +5 +6 +6 +5 +5 +5 +6 +6 +6 +6 +6 +5 +6 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +4 +4 +4 +4 +5 +5 +5 +5 +5 +5 +7 +7 +6 +6 +6 +6 +6 +6 +6 +5 +5 +4 +5 +5 +4 +5 +4 +5 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +5 +5 +4 +4 +4 +5 +4 +5 +5 +5 +5 +4 +4 +4 +4 +4 +5 +5 +5 +5 +5 +7 +7 +6 +6 +5 +6 +6 +6 +6 +5 +5 +5 +5 +5 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +5 +6 +6 +5 +5 +4 +4 +4 +4 +5 +5 +5 +5 +4 +5 +4 +4 +4 +4 +5 +5 +5 +5 +7 +7 +7 +7 +6 +5 +6 +6 +5 +5 +6 +5 +5 +5 +6 +6 +6 +6 +6 +6 +7 +6 +6 +6 +6 +6 +6 +6 +6 +5 +5 +5 +4 +4 +4 +5 +5 +5 +5 +5 +5 +5 +4 +4 +4 +4 +5 +5 +5 +7 +7 +7 +6 +6 +6 +4 +5 +5 +5 +6 +6 +5 +5 +5 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +5 +4 +5 +5 +5 +5 +5 +5 +5 +5 +4 +4 +4 +4 +4 +4 +4 +5 +5 +7 +7 +7 +7 +6 +6 +6 +3 +3 +4 +5 +5 +6 +6 +5 +4 +5 +6 +6 +6 +6 +5 +6 +5 +5 +5 +6 +6 +6 +6 +6 +5 +4 +5 +5 +5 +5 +5 +5 +5 +5 +4 +4 +3 +4 +4 +4 +4 +5 +7 +7 +7 +7 +7 +6 +6 +6 +3 +2 +2 +3 +4 +5 +5 +4 +3 +4 +5 +5 +5 +5 +4 +4 +4 +4 +5 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +4 +4 +4 +4 +4 +4 +7 +7 +7 +7 +7 +7 +6 +6 +6 +5 +5 +4 +5 +5 +5 +5 +4 +5 +6 +6 +6 +5 +5 +5 +5 +4 +5 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +5 +5 +4 +5 +5 +5 +5 +5 +5 +4 +4 +4 +5 +6 +6 +7 +7 +7 +7 +6 +6 +6 +5 +6 +6 +6 +6 +6 +6 +5 +5 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +4 +5 +4 +4 +4 +4 +5 +6 +6 +7 +6 +7 +6 +6 +6 +5 +6 +6 +6 +6 +6 +6 +5 +5 +5 +5 +6 +6 +6 +6 +6 +6 +6 +6 +6 +5 +4 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +3 +4 +4 +5 +4 +3 +4 +4 +5 +5 +5 +5 +5 +5 +5 +6 +5 +5 +6 +6 +5 +6 +5 +5 +4 +5 +4 +5 +5 +5 +4 +5 +5 +6 +5 +5 +5 +4 +4 +4 +4 +5 +5 +5 +4 +4 +5 +4 +4 +3 +3 +3 +5 +4 +4 +4 +4 +4 +2 +2 +2 +3 +1 +1 +2 +2 +1 +5 +5 +4 +3 +2 +2 +3 +4 +3 +3 +3 +3 +4 +3 +4 +4 +4 +3 +4 +3 +3 +3 +4 +4 +4 +5 +4 +4 +4 +4 +3 +3 +4 +4 +4 +4 +3 +5 +5 +5 +4 +4 +3 +3 +2 +4 +5 +4 +3 +4 +5 +6 +5 +4 +4 +5 +5 +4 +5 +4 +4 +4 +4 +5 +4 +3 +3 +3 +3 +3 +3 +5 +4 +5 +4 +4 +4 +4 +4 +3 +4 +4 +4 +5 +4 +3 +4 +5 +5 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +5 +6 +6 +6 +6 +6 +5 +5 +5 +6 +6 +6 +6 +6 +6 +5 +4 +4 +3 +4 +5 +5 +5 +6 +5 +4 +4 +4 +4 +4 +5 +4 +5 +5 +4 +4 +5 +6 +6 +6 +6 +7 +6 +7 +7 +7 +7 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +5 +5 +6 +6 +5 +6 +6 +6 +5 +5 +4 +5 +5 +6 +6 +6 +5 +4 +4 +4 +4 +5 +5 +5 +5 +5 +5 +5 +5 +5 +6 +6 +6 +6 +6 +6 +7 +7 +7 +7 +6 +6 +7 +6 +6 +6 +6 +6 +6 +6 +5 +5 +6 +5 +5 +5 +6 +6 +6 +6 +5 +6 +6 +5 +5 +5 +4 +4 +4 +4 +4 +4 +5 +5 +5 +5 +5 +5 +5 +5 +6 +6 +6 +6 +6 +6 +7 +7 +7 +7 +6 +6 +6 +5 +6 +6 +5 +5 +5 +5 +4 +5 +4 +4 +4 +5 +6 +6 +6 +6 +6 +6 +6 +5 +6 +5 +5 +5 +4 +3 +4 +4 +5 +5 +4 +5 +5 +5 +5 +5 +6 +6 +6 +5 +6 +6 +7 +7 +7 +7 +6 +6 +6 +5 +6 +5 +4 +5 +4 +3 +3 +4 +4 +4 +4 +5 +5 +6 +6 +6 +6 +6 +5 +6 +5 +5 +5 +5 +4 +3 +4 +5 +4 +4 +4 +5 +5 +4 +5 +5 +6 +5 +4 +4 +5 +6 +7 +7 +7 +7 +6 +6 +6 +5 +6 +5 +6 +6 +5 +5 +5 +5 +4 +3 +4 +4 +5 +5 +6 +6 +5 +5 +5 +5 +5 +5 +5 +4 +4 +4 +4 +4 +4 +4 +5 +5 +5 +5 +5 +5 +4 +5 +5 +4 +6 +7 +7 +7 +7 +7 +7 +7 +7 +5 +6 +6 +6 +6 +6 +5 +5 +5 +5 +4 +4 +3 +4 +4 +6 +5 +5 +5 +5 +4 +5 +4 +2 +4 +4 +5 +4 +4 +4 +5 +4 +5 +5 +4 +5 +4 +5 +5 +5 +5 +7 +7 +7 +7 +7 +7 +7 +7 +6 +5 +6 +6 +6 +6 +5 +5 +5 +5 +5 +6 +4 +4 +4 +4 +5 +5 +5 +5 +5 +4 +4 +3 +4 +3 +4 +4 +4 +5 +5 +5 +5 +5 +5 +5 +4 +4 +6 +5 +5 +6 +7 +7 +7 +7 +7 +7 +6 +6 +6 +5 +6 +5 +5 +4 +4 +5 +4 +5 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +4 +5 +4 +4 +4 +4 +4 +5 +5 +5 +5 +5 +5 +5 +5 +5 +5 +4 +5 +6 +6 +6 +7 +7 +7 +7 +7 +7 +6 +6 +6 +4 +5 +5 +4 +4 +4 +4 +5 +5 +6 +6 +6 +6 +6 +6 +6 +5 +4 +4 +4 +4 +5 +5 +5 +4 +4 +4 +5 +5 +5 +5 +5 +5 +5 +5 +4 +5 +6 +6 +6 +5 +6 +7 +7 +7 +7 +7 +6 +5 +5 +5 +6 +5 +5 +5 +5 +4 +5 +5 +5 +6 +5 +5 +5 +4 +3 +4 +4 +4 +4 +5 +5 +5 +5 +5 +4 +5 +5 +5 +5 +5 +5 +5 +4 +4 +5 +5 +6 +6 +6 +6 +7 +7 +7 +7 +7 +7 +6 +6 +6 +5 +6 +6 +6 +6 +5 +5 +4 +4 +5 +6 +4 +3 +4 +3 +3 +5 +5 +5 +5 +5 +6 +5 +5 +5 +5 +5 +5 +5 +5 +5 +4 +4 +4 +5 +5 +5 +6 +6 +6 +6 +7 +7 +7 +7 +7 +7 +6 +6 +6 +5 +6 +6 +6 +6 +6 +4 +4 +4 +5 +5 +4 +6 +5 +6 +6 +6 +6 +5 +5 +5 +5 +5 +5 +5 +5 +4 +5 +5 +5 +4 +4 +4 +5 +5 +5 +5 +6 +6 +6 +6 +7 +7 +7 +7 +7 +7 +6 +6 +6 +5 +6 +6 +6 +6 +6 +5 +5 +5 +5 +6 +6 +6 +6 +6 +6 +6 +5 +5 +6 +5 +5 +4 +5 +4 +4 +5 +4 +4 +4 +4 +4 +5 +5 +5 +5 +5 +Figure 7. Visualization of discrete probability grid. We use K-means clustering to quantize the probability space to convert continuous +synchronization probability S(i, j) to discrete probability bin ˆS(i, j). Then we build a autoregressive Transformer [100] model on the +probability grid. +distribution. Then we decompose pθ(x1, x2, · · · , xN): +pθ(x1, x2, · · · , xN) = +N−1 +� +i=0 +2τ+1 +� +q=1 +pθ(xi+1,q|x1, · · · , xi; xi+1,1, · · · , xi+1,q−1). +(8) +We utilize similar loss function in PixelCNN [91,99] to su- +pervise the autoregressive model. +E. Implementation Details +Audio-visual synchronization model. +Following prior +work [6, 18, 60], we utilize curriculum learning to train the +audio-visual synchronization model. Specifically, we use +the two-stage training procedure. During the first phase, +the negatives are from different videos, while for the sec- +ond stage, the negatives are sampled within the same videos +randomly (the starting time steps are sampled randomly). +Anomaly detection model. +We process each the input +vector xi with an affine transformation, before passing it +into the autoregressive model. This projects the input (e.g., +a time delay distribution Si ∈ R31) to R256. +Also, we +add an affine transformation for the to project the embed- +ding back into the feature set space, e.g., R256 → R31 for +the time delay distribution. We use a learnable positional +encoding for both the synchronization and autoregressive +models. +Hyperparameters. +For the synchronization model, we +use Adam [56] with a learning rate of 1 × 10−4, with a +batch size of 16 for the first phase and 40 for the second. +Original +Block-wise +Compression +Contrast +Gaussian Noise +JPEG +Saturation +Gaussian Blur +Figure 8. Visualization of corrupted images. Examples of the +corruptions taken into account at intensity level 5. The set of cor- +ruptions is introduced in Jiang et al. [50], and it consists of color +saturation, local block-wise distortion, color contrast, Gaussian +blur, white Gaussian noise, JPEG compression, and video com- +pression rate change. +During the second stage, we sample four 5-frame short clips +per video. For the autoregressive model (anomaly detection +model), we use the the Adam optimizer [56] with 1 × 10−3 +learning rate, weight decay of 1 × 10−6 and warm-up and +cosine learning rate decay [69] strategies. We use batch size +of 16 and the dropout [95] rate of 0.1. We train the synchro- +nization model on 8 NVIDIA A40 GPUs, and use a single +GeForce RTX 2080 Ti GPU for the autoregressive model. +F. Visualization of perturbed images +We visualize some images used for unseen perturbations +robustness test in Fig. 8. +15 + diff --git a/N9AzT4oBgHgl3EQfzP7F/content/tmp_files/load_file.txt b/N9AzT4oBgHgl3EQfzP7F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bab3dad896d0c5b838f27565b8de99ac0d06124b --- /dev/null +++ b/N9AzT4oBgHgl3EQfzP7F/content/tmp_files/load_file.txt @@ -0,0 +1,2636 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf,len=2635 +page_content='Self-Supervised Video Forensics by Audio-Visual Anomaly Detection Chao Feng Ziyang Chen Andrew Owens University of Michigan Abstract Manipulated videos often contain subtle inconsistencies between their visual and audio signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We propose a video forensics method, based on anomaly detection, that can identify these inconsistencies, and that can be trained solely using real, unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We train an autoregres- sive model to generate sequences of audio-visual features, using feature sets that capture the temporal synchronization between video frames and sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' At test time, we then flag videos that the model assigns low probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Despite be- ing trained entirely on real videos, our model obtains strong performance on the task of detecting manipulated speech videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Project site: https://cfeng16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='io/ audio-visual-forensics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Introduction Supervised learning underlies today’s most successful methods for image and video forensics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' However, the diffi- culty of collecting large, labeled datasets that fully capture all of the possible manipulations that one might encounter in the wild places significant limitations on this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' A longstanding goal of the forensics community has been to design methods that, instead, learn to detect manipula- tions using cues discovered by analyzing large amounts of real data through self-supervision [28,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We propose a method that identifies manipulated video through anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our model learns how audio and visual data temporally co-occur by training on large amounts of real, unlabeled video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' At test time, we can then flag videos that our model assigns low probability, such as those whose video and audio streams are inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' One might expect that this problem could be posed as simply detecting out-of-sync examples, such as by finding cases in which a speaker’s mouth does not open precisely at the onset of a spoken word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Unfortunately, videos in the wild are often “naturally” misaligned due to errors in en- coding or recording, such as by having a single, consistent shift by a few frames [2,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Instead, we pose the problem as detecting anomalies in what we call synchronization features: audio-visual features Input video Time (frame) Input audio Ziyang’s version (remove this text) Time delay Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Audio-visual anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We identify fake videos by finding anomalies in their audio-visual features, using generative models trained entirely on real videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In one variation of our model (shown here), we use the time delay between the two modalities as our feature set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=', temporal misalignment between each video frame and the audio stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We learn the distribution of these sequences, then flag sequences with low probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' that are designed to convey the temporal alignment between vision and sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We evaluate several feature sets, each ex- tracted from a model that has been trained to temporally align audio and visual streams of a video [18, 24, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In Figure 1, we show one such feature set: the amount of time that each video frame appears to be temporally offset from its corresponding sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' To detect anomalies, we fit an au- toregressive generative model [85,100] to sequences of syn- chronization features extracted from real videos, and iden- tify low probability examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' A key advantage of formulation is that it does not require any manipulated examples for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' It also does not re- quire the speakers in the test set to already be present in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' This is contrast to previous audio-visual foren- sics approaches, which either require finetuning on datasets of manipulated video [41], or which are based on verifying that the speaker’s voice matches previously observed exam- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='01767v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='CV] 4 Jan 2023 +15 15ples [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We evaluate our model on videos that have manipulated a person’s speech and face, using datasets of lip-synced and audio-driven face reenactment videos, some of which are also manipulated by faceswap techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our model obtains strong performance on the FakeAVCeleb [54] and KoDF [62] datasets, despite the fact that it is trained en- tirely on real examples obtained from other video datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our model generalizes to other spoken languages without retraining and obtains robustness to a variety of postpro- cessing operations, such as compression and blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We show through our experiments that: Video forensics can be posed as an audio-visual anomaly detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Synchronization features convey information about video manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our model can successfully detect fake videos, while training solely on real videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our model generalizes to many types of image postpro- cessing operations and to speech videos from spoken lan- guages not observed during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Related Work Audio-visual forensics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In early work, Malik and Farid [72] detected audio manipulations by finding incon- sistencies in reverberation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Recent work has focused on detecting manipulated speech videos using audio-visual in- consistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Several approaches have directly trained audio-visual networks through supervised learning, using labels indicating whether a video is manipulated [21, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' A variety of methods have recently used audio-visual self- supervision for pretraining supervised models, which are finetuned with “real or fake” labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' [106] used local and global contrastive learning methods to learn video features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Haliassos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' [41] jointly solved a negative- less contrastive learning problem [39] and a forensics task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Zhou and Lim [114] used audio-visual synchronization sig- nal implicitly, and proposed a dataset for audio-visual deep- fake detection1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Other work [42] pretrains using lip-reading data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In contrast to these methods, our approach is trained entirely using real data and does not require any labels or examples of fake videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Other work has used speaker ver- ification [26] and phoneme-viseme mismatches [7] to de- tect fake videos and it also detects face swap manipula- tions, which preserve the synchronization between modal- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In contrast, our approach detects misaligned images and sounds and does not require that examples from the speaker be present in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Audio-visual representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' A variety of methods have been proposed to learn audio-visual repre- sentation from videos via self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Researchers 1Their dataset is not publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' have leveraged the natural semantic correspondence in the videos between frames and audio tracks [10, 76, 106] to learn multi-modal features and applied them to downstream tasks such as sound localization [9,46,75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Other work stud- ies temporal synchronization between audio and visual sig- nals to learn audio-visual features [24, 60, 79], which can be used for active speaker detection [8,59,96], source sep- aration [37, 71, 112], lip reading [5, 70, 73] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our method uses the off-the-shelf audio-visual synchronization model to perform anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Visual face forensics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' A major focus of the forensics field has been on the problem of detecting manipulated videos of human faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In recent years, a variety of visual face manipulation datasets are proposed, such as FaceForen- sics++ [88], VideoForensicsHQ [35] and FFIW10K [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Meanwhile, many methods are proposed to detect syn- thetic contents to fight against their potential threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Some work [12,40,64] has proposed to use hand-crafted features to capture inconsistent visual or JPEG artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Other work has proposed to use deep learning to inspect specific arti- facts, such as blending [63], frequency domain [32,36,84], or texture [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' A variety of methods have studied the gen- eralization between detection classifiers [16,101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' A variety of methods have learned a distribution, then flagged unusual examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' These ex- amples are often considered anomalies [67,92,105,115] or outliers [82, 90], and are used as part of open-set recog- nition [58, 107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We formulate video forensics as the task of detecting anomalies, using a feature set that con- veys information that would be hard for a forger to cre- ate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' There have been a variety of methods proposed for learning this distribution, such as GAN discriminators [38, 58, 67, 82, 90, 92, 105, 115], flow-based models [107], and autoregressive models [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Similarly, our model is based on an autoregressive generative model [11,85], since they have achieved strong performance at modeling com- plex distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Other work addresses goals similar to anomaly detection by creating methods that model uncer- tainty [65] or that perform outlier exposure [31, 45, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Some work [13, 27, 29, 48, 51, 55, 80] has used special- purpose anomaly detection methods for image/video foren- sics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Other work [34,47,102,110] uses supervised learning to find anomalous patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In contrast, our method builds the likelihood function entirely on real videos and views low-probability examples as fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Method We formulate the problem of detecting manipulated videos as an anomaly detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We model the dis- tribution of audio-visual examples, then flag examples that have low probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' If we were to fit a model on the raw data, then this would be a very challenging learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2 Audio-visual Synchronization Model A V Autoregressive Prediction Time Likelihood Target features ··· ℒ Predicted features ··· 1 +1 ··· 2 Discrete time delays Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Distribution over delays V A Feature activations (a) Synchronization feature extraction (b) Anomaly detection Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Audio-visual anomaly detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' (a) We extract a feature set from an audio-visual synchronization network: the number of frames of delay between video frames and sound, the distribution over delays at each frame, and feature activations from the audio and visual subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' (b) We train an autoregressive Transformer model to assign probabilities to synchronization features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' At test time, we flag low probability examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Instead, we learn the distribution over a feature set that con- veys subtle properties that are unlikely to be accurately cap- tured in manipulated video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Estimating audio-visual synchronization We obtain our feautres from a network that performs audio-visual synchronization [18, 24, 25, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use the model of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We learn a function φ(Vi, Aj) that indicates how likely video clip Vi temporally co-occurs with audio clip Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We estimate the synchronization score S(i, j) of all audio-visual pairs in a temporal window: S(i, j) = exp (φ(Vi, Aj)) �i+τ k=i−τ exp (φ(Vi, Ak)) , (1) where τ is maximum time difference between two streams, and φ(Vi, Aj) = h (gv(Vi), ga(Aj)) is calculated using late fusion by a visual encoder gv, audio encoder ga and the fusion module h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We also interpret S(i, j) as synchroniza- tion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We maximize the synchronization of true audio-visual pairs (Vi, Ai) using the InfoNCE loss [78]: Lsync = − 1 T T � i=1 log S(i, i), (2) for a video of length T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We provide details about the archi- tecture and training procedure in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' After training, we can use the learned model to obtain a feature set for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' For example, we can use the rows of S, which provide a probability distribution over possible alignments between video clips and audio clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Audio-visual anomaly detection We use our learned model to obtain a feature set for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We learn the distribution of these fea- tures on a training set of real videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Then at test time, videos with low probability will be flagged as potential fakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We now explore two key design decisions that go into such a system: what feature set to use, and how the distribution is learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Given features for each frame, we learn a distribution pθ(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' , xN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We generally use autoregressive mod- els to learn this distribution, given their success in model- ing complex distributions [14,104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' These models take the form: pθ(x1, x2, · · · , xN) = N−1 � i=0 pθ(xi+1|x1, · · · , xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' (3) We train a model ˆxi+1 = fθ(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' , xi) that estimates the features of the next frame, given all of the features from the previous frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Maximizing the log probability can be posed as minimizing a per-frame loss, L: L = N � i=1 L(ˆxi, xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' (4) We now describe different formulations of the loss func- tion L, the feature representation xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In each case, we im- plement fθ as a Transformer [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Discrete time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We first consider a simple model that uses discrete time delay estimates as our feature rep- resentation, following the success of autoregressive mod- 3 els for fitting discrete data [33, 87, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Taking inspira- tion from work on time delay estimation [19, 57], for ev- ery video frame, we estimate how far ahead (or behind) it appears to be from the audio signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' For each frame, we set xi to be the time delay with the highest probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=', xi = arg maxj(S(i, j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We then set L to be cross en- tropy loss between the ground truth and predicted time de- lay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' This amounts to solving a categorization problem with 2τ + 1 possible labels for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Distributions over delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' While discrete time delays are straightforward to represent in the model, they discard im- portant information, such as when there is ambiguity in the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We therefore propose a model that directly predicts the entries of the time delay distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We set the features xi to be the rows of S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' the probability of each possible delay, and use cross entropy loss: L(ˆxi, xi) = − 2τ+1 � j xi,j log(ˆxi,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' (5) We constrain the predictions made by our model fθ to sum to 1 by applying a softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Audio-visual network activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The feature activa- tions within the audio-visual synchronization network con- vey information about the time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We, therefore, ask whether these activations can be directly used as features for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We concatenate the representations of the visual and audio subnetworks, gv and ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' To pro- vide a straightforward comparison with the time delay dis- tribution model, we reduce the dimensionality of the fea- tures by projecting them onto the top 2τ + 1 principal components, following other work in autoregressive mod- els of features [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use squared distance as our loss: L (ˆxi, xi) = ∥xi − ˆxi∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Results We evaluate the different variations of our model on a variety of video forensics tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Implementation details Synchronization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Following Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' [18], we use ResNet-18 2D+3D [43, 44] as the visual encoder, us- ing 5 frames frames (25 fps) as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The audio en- coder uses VGG-M [17] and extracts features from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2s audio clips (16kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We fuse audio and visual data us- ing a Transformer that has 3 standard Transformer encoder blocks [100], 4 attention heads, and 512 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We train using the cropped faces provided by each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Please see Appendix E for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Anomaly detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use a decoder-only au- toregressive Transformer [33, 66, 85] to learn the distri- bution over synchronization features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use 2 decoder blocks [100], each with 16 attention heads with 256 chan- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' For models that use time delay, we set the maximum delay to be τ = 15 frames, resulting in the distribution Si ∈ R31 for each video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use sequences of length N = 50 from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0s video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' All videos are resampled to 25 fps and 16kHz mono audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We represent audio segments as the mel spectrogram of size 21×80 via Short-Term Fourier Transform (STFT) with 80 mel filter banks, the hop length of 160, and the window size of 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Please see more details in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Dataset We train our model on real, unlabeled speech video, and evaluate it on forensics datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We train our models on Lip Reading Sentences 2 (LRS2, 97k videos) [3] and Lip Reading Sen- tences 3 (LRS3, 120k videos) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The videos in each con- tain tightly cropped face tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We divide each dataset into 3 splits and train the audio-visual synchronization model and the autoregressive model on different splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Evaluation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We evaluate on two video foren- sics datasets, spanning several different types of manip- ulations that change the speech and face of a human speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' FakeAVCeleb [54], which is derived from Vox- Celeb2 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' This dataset contains 500 real videos and 19,500 fake videos manipulated by Faceswap [61], FS- GAN [77], and Wav2Lip [83], and fake sounds that are generated by SV2TTS [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The examples in the dataset contain different combinations of these manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use the dataset’s provided face crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We sample 2400 videos (400 real videos and 2000 fake videos) as train/val splits and 600 videos (100 real videos and 500 fake videos) as test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We note that our method does not use any videos from train/val splits, since it is trained from an- other dataset (LRS2 or LRS3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Second, we evaluate on KoDF [62], a large-scale Korean-language deepfake detec- tion dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' It contains 62,166 real videos and 175,776 fake videos, where fake videos are generated by 6 synthesized methods: FaceSwap [1], FSGAN [77], DeepFaceLab [81], FOMM [93], ATFHP [103] and Wav2Lip [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We extract faces by using face detection [108] and alignment [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Evaluation methods Following common practice [16,41,42,54,62,63,84,88, 101, 111], we evaluate using average precision (AP) and AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' These evaluation metrics are widely used for cross- dataset generalization and unsupervised models since they avoid the need to threshold the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We compare our approach to both supervised and self-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Unless otherwise stated, we use time delay distributions as our feature set (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4 Pretrained dataset Category Method Modality RVFA FVRA-WL FVFA-FS FVFA-GAN FVFA-WL AVG-FV AP AUC AP AUC AP AUC AP AUC AP AUC AP AUC Supervised Xception [88] V ImageNet [30] – – 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 LipForensics [42] V LRW [23] – – 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 AD DFD [114] AV Kinetics [53] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 FTCN [111] V – – – 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 RealForensics [41] V LRW [23] – – 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 Unsupervised AVBYOL [39,41] AV LRW [23] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 VQ-GAN [33] V FFHQ [52] – – 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 Ours AV LRS2 [3] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 Ours AV LRS3 [4] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Manipulation detection on FakeAVCeleb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We report AP scores (%) and AUC scores (%), following the evaluation protocol of Haliassos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' [42], in which supervised methods are evaluated on unseen manipulation types (unsupervised methods are not trained with labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We report results with combinations of real/fake video/audio, using different generation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We report the average performance over four fake video (FV) categories in AVG-FV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We retrained all models, since there is no standard dataset split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' For supervised methods, we train several state-of-the-art detectors on the two datasets: 1) Xception [88]: a popular baseline for forensics detec- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2) LipForensics [42]: a detector is built on high- level semantic embeddings of mouth and targets irregu- larities in mouth movements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3) AD DFD [114]: a mul- timodal detector with audio and video branches, utilizes audio-visual synchronization signal implicitly for detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4) FTCN [111]: a video forensics detector leverages tem- poral incoherence to boost generalization capability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 5) Re- alForensics [41]: it first pretrains the network by audio- visual BYOL [39] framework and then finetunes the pre- trained model and forensics datasets by multi-task learning to obtain robust and general face forgery detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Since there is no standard split for our forensics tasks, we retrain each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Self-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Since we are not aware of any existing methods that consider self-supervised speech video forensics, we adapt two existing methods to the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' First, we consider an audio-visual contrastive learning model, which we call AVBYOL, that learns to determine whether the visual and audio streams of a video do (or do not) match, an approach that has been used as a part of other audio- visual forensics models [26, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We adapt the model of Haliassos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' [41], which uses BYOL [39] to learn a joint audio-visual embedding for pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Instead of pre- training, we directly use the model’s audio-visual similarity score to flag fake examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Second, we use an off-the- shelf generative model, VQGAN [33], for anomaly detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' VQGAN converts an image into a sequence of discrete codes, then uses an autoregressive Transformer to learn the distribution of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use the code’s log likelihood, av- eraged over each video frame, for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Evaluation In real-world scenarios, the deployed detectors are ex- pected to recognize fake videos manipulated by unseen techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Thus, following the standard procedure used in [41,42,111,114], we conduct the experiment to evaluate the cross-manipulation generalization ability of our model on the FakeAVCeleb dataset [54] which videos are manip- ulated in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Since our approach and other self- supervised baselines learn from real, unmanipulated videos and perform zero-shot fake video detection, all the fake videos during evaluation are considered as manipulated by unseen methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We split FakeAVCeleb dataset [54] into five cate- gories based on the manipulation methods and manipu- lated modalities: 1) RVFA: real video with fake audio by SV2TTS [49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2) FVRA-WL: real audio with fake video by Wav2Lip [83];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3) FVFA-WL: fake video by Wav2Lip [83], and fake audio by SV2TTS [49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4) FVFA- FS: fake video by Faceswap [61] and Wav2Lip [83], and fake audio by SV2TTS [49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 5) FVFA-GAN: fake video by FaceswapGAN [77] and Wav2Lip [83], and fake audio by SV2TTS [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' For the supervised methods, we hold out the evaluated category and train the models on the four remain- ing categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Note that some approaches are only able to detect the manipulation on a certain modality, we do not report their performance on the categories with the manip- ulation only on the other modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We show our results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our method substantially outperforms both self-supervised methods AVBYOL [39, 41] and VQGAN [33] on each category by a large mar- gin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' More importantly, our method works on par with or outperforms some supervised methods on certain cat- egories, especially FVFA-GAN, even though our method does not use any labeled supervision or fake examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 5 Method Modality KoDF [62] AP AUC Supervised (transfer) Xception [88] V 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 LipForensics [42] V 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 AD DFD [114] AV 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 FTCN [111] V 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 RealForensics [41] V 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 Unsupervised AVBYOL [39,41] AV 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 VQ-GAN [33] V 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 Ours AV 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Generalization to Korean speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' AP scores (%) and AUC scores (%) are reported on KoDF dataset [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Supervised methods are trained on FakeAVCeleb dataset [54] and transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Ours is trained on LRS2 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Best results are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Moreover, our method has quite consistent performances and it can achieve AP over 90% on the most of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' While Xception [88], LipForensics [42], AD DFD [114] and FTCN [111] work well on 75% of the settings, there are settings where performance collapses to near-chance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=', AD DFD [114] on FVFA-GAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Interestingly, the two self- supervised baselines struggle to detect fake videos, perhaps because both models do not necessarily capture the sub- tle information that would be needed to detect manipula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In addition, VQGAN [33] compresses the visual sig- nal using a codebook, which might drop the artifact clues and harm the detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Moreover, our model trained on LRS2 [3] works on par with the one trained on LRS3 [4], indicating that our method’s performance is not tied to a single training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Cross-dataset generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We also evaluate the gen- eralization capability of our model by evaluating it on the KoDF dataset [62], following [41,42,111,114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We focus on the audio-driven synthesis examples in the dataset, where videos are manipulated by ATFHP [103] or Wav2Lip [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We train the supervised models on FakeAVCeleb [54] to evaluate their generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Many of these training videos share the same technique used in KoDF for synthe- sis [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' As the results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2, our approach ob- tains a comparable performance to many supervised meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Although our system is trained on the English speech datasets, it still generalizes to KoDF [62] dataset of Korean speech, perhaps because it learns low-level lip motion cues that are broadly useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We provide more results in Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We visualize the ground truth and predicted time delay distributions generated by our au- toregressive continuous time delay model (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use the four main categories from FakeAVCeleb dataset [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' For each one, we display a heat map indicat- Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Time delay predictions for real vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' fake examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We visualize the time delay distributions from the synchronization model and predicted results generated by the autoregressive model for four random samples from different categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Synchroniza- tion probabilities are in a range from 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We show the predictions of the autoregressive model when feeding it ground truth observations of the previous timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We show cumulative prediction error (indicating the probability of being fake) for each sample over time steps in the last row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' ing the predicted time delay, using a model that obtains the ground truth delays of the previous frames as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We also plot the cumulative prediction loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4) over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3, we can see that our autoregressive model accurately predicts the ground truth for real video, which results in a lower score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' For fake videos, we can find clear differences between ground truth and predicted time delay distribution, leading to higher prediction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Robustness to unseen perturbations When the fake video is redistributed, it may undergo many types of postprocessing that result in corruption, mak- ing detection more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Thus, it is important for forensics models to be robust to the types of postprocess- ing operations they may encounter in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Follow- ing [41, 42, 111], we use the set of visual perturbations proposed in [50]: 1) Color saturation change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2) Block- wise distortion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3) Color contrast change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4) Gaussian blur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 6 50 60 70 80 90 100 AUC (%) Saturation JPEG Block-wise Gaussian Noise Gaussian Blur 0 2 4 Intensity 50 60 70 80 90 100 AUC (%) Compression 0 2 4 Intensity Contrast 0 2 4 Intensity Average 0 2 4 Intensity Adversarial Chance FTCN Xcepetion AD DFD RealForensics Ours Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Robustness to unseen perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' AUC scores (%) of different detectors as a function of perturbation intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' There are 6 intensity levels in total from [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' “Average” represents the average over 7 perturbations under each intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' “Adversarial” means we pick the worst performance across 7 perturbations under each intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 5) Gaussian noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 6) JPEG compression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 7) Video com- pression rate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We set the intensity levels from 0 to 5 for each perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We compare our model with four supervised meth- ods XceptionNet [88] FTCN [114], AD DFD [114] and RealForensics [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our proposed self-supervised model is overall more robust to unseen visual perturbations on average compared with these supervised methods, with the exception of RealForen- sics [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' This is also true when we consider “worst case” performance, by taking the minimum performance over all types of augmentation of a given intensity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Interest- ingly, we obtain this performance even though our model is trained in a very different way from other works, suggesting that the feature set continues to convey useful information to the anomaly detection model, even in the presence of sig- nificant corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Feature set analysis We evaluate the effectiveness of different feature sets used by our anomaly detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2, we start with discrete time delays as our feature representation and optimize the model with cross entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Then, we use continuous time delay distributions for representations instead, where we optimize models with different objective functions: 1) Soft CE: we use the time delay distribution as target (akin to a “soft” label) and use cross entropy loss (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2) CE: we map each distri- bution into one-hot encoding as target by using arg max and employ cross entropy loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3) BCE: we use the dis- tribution as the target while treating each synchronization score S(i, j) (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1) within the same time step indepen- Model Feature set L AVG-ALL AVG-FV AP AUC AP AUC Bayes 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 Ours discrete delay CE 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 distribution CE 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 distribution BCE 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 distribution Soft CE 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 activation-AV MSE 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='-AV+dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' MSE 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 activation-V MSE – – 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 discrete prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' – 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Feature set analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' AP (%) and AUC (%) are reported on FakeAVCeleb [54] when using different feature sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Best re- sults are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' AVG-ALL means the average over all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' AVG-FV represents the average over four fake video categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' dently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use the sigmoid function and binary cross en- tropy loss to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We also use our network’s fea- ture activations as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2: 1) audio-visual feature ac- tivations (activation-AV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2) visual-only feature activations (activation-V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Besides, we consider using a combination of different feature sets where we concatenate continuous time delay distributions and audio-visual feature activa- tions (Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='-AV+dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=') as a new feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Similar to audio- visual feature activations as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2, we use squared dis- tance as the loss for the concatenation of these two types of feature sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We also compare with a simple model based on Naive Bayes and discrete time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' This model assumes that each frame’s time delay is independent, and obtains a prob- ability for the entire sequence by multiplying the probability 7 Synchronization dataset Auto-regression dataset AVG AP AUC LRS2 [3] LRS2 [3] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 LRS3 [4] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 LRS2 [3]+LRS3 [4] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='9 LRS3 [4] LRS2 [3] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 LRS3 [4] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 LRS2 [3]+LRS3 [4] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Dataset ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' AP scores (%) and AUC scores (%) are reported on FakeAVCeleb [54] dataset by using different datasets to train synchronization model or atuoregressive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Best re- sults are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' of each frame’s time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' This amounts to simply detect- ing large misalignments, since in practice the Naive Bayes model will assign probability solely based on the magnitude of each delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Finally, we consider a version of the model that autore- gressively predicts the entire distribution of time delays, in- spired by autoregresive models, such as PixelCNN [97] that generate images in a raster scan order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We autoregressively predict each element of the 2D matrix ˆS(i, j), where ˆS(i, j) is created by vector quantizing the entries of the synchro- nization probability S(i, j) using k-means (see Appendix D for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We evaluate each variant on FakeAVCeleb [54] and report results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' These results suggest that all formulations achieve performance significantly better than chance, indicating that these feature sets are useful for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' As in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 3, the time delay distribu- tion model outperforms the discrete time delay model, sug- gesting that there is important information conveyed in the probability of unlikely delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The autoregressive model that uses distribution as input and soft labels (soft CE) per- forms best, since it forces the output prediction to match the distribution from the synchronization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Interestingly, the model that uses audio-visual feature activations obtains performance close to that of the soft CE model, indicating that the networks’ audio-visual features convey useful infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Finally, the multimodal activation-AV model sig- nificantly outperforms the visual-only activation-V model, suggesting that having access to both modalities is useful for our anomaly detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Ablation study Different training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We ask how the choice of dataset affects the quality of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' To test this, we train our synchronization and autoregressive models on dif- ferent datasets to analyze the generalization abilities of each component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=', training the synchronization model on LRS2/LRS3 and training the autoregressive model on 10 20 30 40 50 60 Sequence length 74 77 80 83 86 89 92 Average AP AUC 11 21 31 41 51 Distribution length 80 82 84 86 88 90 92 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Hyperparameter ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We evaluate with dif- ferent input sequence lengths for our autoregressive model on FakeAVCeleb (left), and study the effect of the time delay dis- tribution’s maximum temporal offset (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' LRS3/LRS2 or LRS2+LRS3 with the same hyperparame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4, there is no significant performance change when we train these two components on different combinations of datasets, including when they are trained on the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' This suggests that the distribution of time delay predictions may be stable between these speech video datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Influence of sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' To explore the influence of input sequence length for autoregressive model, we sam- ple the same amount of training videos for sequence length N of 10, 20, 30, 40, 50, and 60, and keep other hyperpa- rameters the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We test these models on FakeAVCeleb dataset [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 5 shows that as the sequence length in- creases, the performance increases with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Effect of time delay distribution maximum offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We also study how the length of time delay distribution would affect the performance of the autoregressive model with dis- tribution over delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We experiment with maximum offset τ ∈ {5, 10, 15, 20, 25} resulting in the delay distribution length of {11, 21, 31, 41, 51}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We test these models on the FakeAVCeleb dataset [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 5 shows that as distribu- tion length increases, the performance first increases, after which point results plateau or slightly decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' This may be due to the fact that when considering larger ranges of offsets, the distribution spreads over a large number of un- likely possibilities, making important information less ap- parent after normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Conclusion We have proposed a method for detecting video manipu- lation by self-supervised anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' To do this, we create novel feature sets that convey audio-visual synchro- nization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We then show that fake videos can be detected by flagging examples with unlikely sequences of these fea- tures, according to a learned distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our model ob- tains strong performance on the FakeAVCeleb and KoDF datasets, despite the fact that it was trained only on real 8 video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' It also obtains robustness to visual postprocessing operations and to videos containing other spoken languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We see our work as opening in two directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The first is in posing forensics as an anomaly detection problem with a self-supervised feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' While we have proposed one such model, based on autoregressive sequence models, the field of anomaly detection offers many possible future ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The second direction is in developing new feature sets that are well-suited to forensics problems, beyond the synchronization features used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We will release code, models, and dataset splits upon acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Limitations and Broader Impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our work provides methods that can potentially be applied to detecting mali- cious video manipulations and disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' While we have shown that our model is capable of detecting several types of fake video, there may be other techniques that our model fails to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In particular, due to the design of our use of synchronization-based features, our model is not well suited to detecting manipulations that leave the syn- chronization between motion and sound relatively consis- tent, such as those that change a speaker’s appearance with- out significantly changing the motion of their mouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We thank David Fouhey, Richard Higgins, Sarah Jabbour, Yuexi Du, Mandela Patrick, Deva Ramanan, Haochen Wang, and Aayush Bansal for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' This work was supported in part by DARPA Semafor and Cisco Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The views, opinions and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Government.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 5778–5788, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2 [114] Yipin Zhou and Ser-Nam Lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Joint audio-visual deep- fake detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Inter- national Conference on Computer Vision, pages 14800– 14809, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2, 5, 6, 7, 14 [115] Bo Zong, Qi Song, Martin Renqiang Min, Wei Cheng, Cristian Lumezanu, Daeki Cho, and Haifeng Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Deep autoencoding gaussian mixture model for unsupervised anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' In International conference on learn- ing representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 2 13 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Video Results We provide some qualitative video results of some ran- dom samples from the FakeAVCeleb dataset [54] in our webpage with audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We show “ground truth” outputs from the synchronization model and autoregressive predictions over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We also show a score indicating the probabil- ity that the example is fake (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' (4), main paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use the time delay distribution as our feature set, and display the predictions as a heat map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Each step xt of the predicted out- puts were obtained by providing the ground truth features from times x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=', xt−1 into the autoregressive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Cross-dataset Generalization We also experiment with the another state-of-the-art audio-driven video editing method VideoReTalking [20] which takes audio-visual synchronization into considera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We test on a small dataset consisting of 37 real videos and 37 fake videos, which are based on the HDTF dataset [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We show results on Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Our method leverages AV feature activations as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 (number of principal components is set to 512) and obtains a compa- rable performance to many supervised methods although it is only trained on real videos, indicating that our method possesses certain generalization capability to different ma- nipulation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Method Modality VideoReTalking [20] AP AUC Supervised (transfer) Xception [88] V 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='4 LipForensics [42] V 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 AD DFD [114] AV 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1 FTCN [111] V 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='6 RealForensics [41] V 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Unsupervised AVBYOL [39,41] AV 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 VQ-GAN [33] V 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='0 Ours (activation-AV) AV 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='8 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Generalization to VideoReTalking method [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' AP scores (%) and AUC scores (%) are reported on HDTF dataset [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Fake videos are manipulated by VideoReTalk- ing [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Supervised methods are trained on FakeAVCeleb dataset [54] and transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Ours is trained on LRS2 [3] and uses the feature set of AV activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Best results are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Ablation Study Feature activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We study the influence of the num- ber of principal components D for the variation of the anomaly detection model that projects the audio-visual fea- ture activations into a lower dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We set D to 11, 31, 32, 64, 128, 256, 512 and keep other hyperparame- ters the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 6 shows that as the D increases, the accu- racy of the anomaly detection model decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The reason 11 31 32 64 128 256 512 Number of principal components 80 81 82 83 84 85 86 87 88 Average AP AUC Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Number of PCA projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We evaluate with differ- ent number of principal components for the model that obtains a feature set by projecting the audio-visual feature activations to a lower dimensional space using PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We report average results for the metrics in the FakeAVCeleb dataset [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' might be that the higher dimensional prediction problem is also more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Feature Set Variations We describe more details about building the autoregres- sive model on some of our feature sets in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Binary cross entropy (BCE) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We assume that the 2τ + 1 possible time delays of each time step are indepen- dent, and use and BCE loss for probability S(i, j) of each time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Thus, we can decompose pθ(x1, x2, · · · , xN) as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' (6): pθ(x1, x2, · · · , xN) = N−1 � i=0 2τ+1 � q=1 pθ (xi+1,q|x1, · · · , xi) , (6) We then maximize pθ (xi+1,q|x1, · · · , xi) using BCE loss: L(ˆxi, xi) = − 2τ+1 � j=1 xi,j log(ˆxi,j)+(1−xi,j) log(1− ˆxi,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' (7) We constrain the prediction ˆxi,j to the range of [0, 1] via a sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Discrete 2D probability model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The discrete prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' model generates the entire 2D frame/time delay matrix au- toregressively in “raster scan” order, similar to models such as PixelCNN [91, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We unroll the time delay distribu- tion sequence into 2D grid like a grayscale image as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use K-means (K = 8) clustering to quan- tize eacn entry and assume each probability ˆS(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' j) of time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='delay not only depends on the previous time delay distri- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='butions but also the previous probabilities within the same ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='14 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Visualization of discrete probability grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use K-means clustering to quantize the probability space to convert continuous synchronization probability S(i, j) to discrete probability bin ˆS(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Then we build a autoregressive Transformer [100] model on the probability grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Then we decompose pθ(x1, x2, · · · , xN): pθ(x1, x2, · · · , xN) = N−1 � i=0 2τ+1 � q=1 pθ(xi+1,q|x1, · · · , xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' xi+1,1, · · · , xi+1,q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' (8) We utilize similar loss function in PixelCNN [91,99] to su- pervise the autoregressive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Implementation Details Audio-visual synchronization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Following prior work [6, 18, 60], we utilize curriculum learning to train the audio-visual synchronization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Specifically, we use the two-stage training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' During the first phase, the negatives are from different videos, while for the sec- ond stage, the negatives are sampled within the same videos randomly (the starting time steps are sampled randomly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Anomaly detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We process each the input vector xi with an affine transformation, before passing it into the autoregressive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' This projects the input (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=', a time delay distribution Si ∈ R31) to R256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Also, we add an affine transformation for the to project the embed- ding back into the feature set space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=', R256 → R31 for the time delay distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use a learnable positional encoding for both the synchronization and autoregressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' For the synchronization model, we use Adam [56] with a learning rate of 1 × 10−4, with a batch size of 16 for the first phase and 40 for the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Original Block-wise Compression Contrast Gaussian Noise JPEG Saturation Gaussian Blur Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Visualization of corrupted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Examples of the corruptions taken into account at intensity level 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' The set of cor- ruptions is introduced in Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' [50], and it consists of color saturation, local block-wise distortion, color contrast, Gaussian blur, white Gaussian noise, JPEG compression, and video com- pression rate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' During the second stage, we sample four 5-frame short clips per video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' For the autoregressive model (anomaly detection model), we use the the Adam optimizer [56] with 1 × 10−3 learning rate, weight decay of 1 × 10−6 and warm-up and cosine learning rate decay [69] strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We use batch size of 16 and the dropout [95] rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' We train the synchro- nization model on 8 NVIDIA A40 GPUs, and use a single GeForce RTX 2080 Ti GPU for the autoregressive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' Visualization of perturbed images We visualize some images used for unseen perturbations robustness test in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf'} diff --git a/NdAyT4oBgHgl3EQfs_n5/vector_store/index.pkl b/NdAyT4oBgHgl3EQfs_n5/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..0941e3ba80fd4c5f9f1608094fa55f1472de9aa5 --- /dev/null +++ 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+Accenture Labs, +Sophia Antipolis, France +Inria, Universit´e Cˆote d’Azur, +Sophia Antipolis, France +Accenture Labs +Sophia Antipolis, France +Accenture Labs +Sophia Antipolis, France +Abstract +Federated learning (FL) is an effective solution to train ma- +chine learning models on the increasing amount of data +generated by IoT devices and smartphones while keep- +ing such data localized. Most previous work on federated +learning assumes that clients operate on static datasets col- +lected before training starts. This approach may be inef- +ficient because 1) it ignores new samples clients collect +during training, and 2) it may require a potentially long +preparatory phase for clients to collect enough data. More- +over, learning on static datasets may be simply impossible +in scenarios with small aggregate storage across devices. +It is, therefore, necessary to design federated algorithms +able to learn from data streams. In this work, we formu- +late and study the problem of federated learning for data +streams. We propose a general FL algorithm to learn from +data streams through an opportune weighted empirical risk +minimization. Our theoretical analysis provides insights to +configure such an algorithm, and we evaluate its perfor- +mance on a wide range of machine learning tasks. +1 +Introduction +Federated learning (McMahan et al., 2017) usually involves +the minimization of an objective function, which is only +available through unbiased estimates of its gradients (Bot- +tou et al., 2018). The objective function is either the ex- +pected risk, when clients can sample new data points at ev- +ery iteration, or the empirical risk, when they rely on a fixed +dataset. +Most previous works on federated learning, e.g., (McMa- +han et al., 2017; Koneˇcny et al., 2016), focus on the second +case, i.e., the minimization of the empirical risk. They as- +sume that every client collects and stores all the samples +before training starts. Learning on static datasets can be +sub-optimal (or even impossible) in many cases, because +(1) new samples collected during training are ignored, and +(2) clients may have limited memory capacities, and cannot +store a large number of data samples. For example, nodes +in a sensor network may continuously collect new measure- +ments, but may be able to store only a few of them in the +local memory (De Francisci Morales et al., 2016). +Our contributions. In this work, we formulate and study +the problem of learning from separate data streams. We +propose and theoretically analyze a general federated al- +gorithm targeting this goal. +Our analysis shows a bias- +optimization trade-off: by controlling the relative impor- +tance of older samples in comparison to newer ones, one +can speed training up at the cost of a larger bias of the +learned model, or reduce the bias at the cost of a longer +training time. The analysis also provides insights to opti- +mally configure our federated algorithm. We demonstrate +the relevance of our theoretical results through simulations +spanning a wide range of machine learning tasks. In par- +ticular, experiments show that “reasonable” ways to extend +FedAvg McMahan et al. (2017) to data streams may lead +to poor learned models, while our configuration rule con- +sistently leads to almost-optimal performance. +Paper outline. The rest of the paper is organized as fol- +lows. Section 2 provides a review of related work. Sec- +tion 3 formulates the problem of federated learning for data +streams. +Section 4 describes our FL algorithm for data +streams and states its convergence results. Section 5 studies +a scenario of practical interest and exploits the theoretical +results in Section 4 to provide configuration rules for our +algorithm. Finally, we provide experimental results in Sec- +tion 6 before concluding in Section 7. +2 +Related Work +Since its introduction in the seminal works (Koneˇcny et al., +2016; McMahan et al., 2017), federated learning has re- +ceived increasing attention as a promising large-scale dis- +tributed learning framework and has been applied to a wide +range of tasks, including language modeling (Yang et al., +2018), automatic speech recognition (Gao et al., 2022), +medical imaging (Courtiol et al., 2019; Silva et al., 2019), +and recommender systems (Yang et al., 2020). +Our fo- +cus on data streams is a key difference with respect to +most of the FL literature, which assumes clients have static +datasets. In particular, this assumption is shared by the the- +oretical work studying FL algorithms’ convergence on non- +arXiv:2301.01542v1 [cs.LG] 4 Jan 2023 + +Federated Learning for Data Streams +iid data and under partial clients’ participation (Li et al., +2019), PAC learning bounds (Mohri et al., 2019), privacy +guarantees (Wei et al., 2020), or resilience to Byzantine +faults (Blanchard et al., 2017). +Learning from a data stream enjoys an extensive litera- +ture with applications, for example, to the financial sec- +tor (Zhu and Shasha, 2002), network monitoring (Babu and +Widom, 2001), and sensor networks (De Francisci Morales +et al., 2016). In this field, we can roughly distinguish three +main lines of research corresponding to different assump- +tions about the data process. The first focuses on the case +where samples in the data stream are drawn independently +from some fixed unknown distribution; this setting can be +analyzed through stochastic approximation (Moulines and +Bach, 2011). The second line allows the data distribution +to change over time and falls then in the context of con- +tinual learning, where a model is trained on a sequence of +tasks and each task can correspond to a different data dis- +tribution (Thrun, 1994; Kumar and Daum´e III, 2012; Ru- +volo and Eaton, 2013; Kirkpatrick et al., 2017; Schwarz +et al., 2018). +Finally, the third line drops any assump- +tion about the data stream, which may be thought to be +generated by an adversary. +This setting can be studied +in the framework of online learning with regret guaran- +tees (Zinkevich, 2003). Our paper considers that data at +each client is drawn from the same distribution. Learning +from multiple data streams with different samples’ gener- +ation rates and clients’ memory sizes sets our work apart +from the papers mentioned above. +There is almost no work formalizing the problem of feder- +ated learning for data streams and providing a theoretical +analysis. To the best of our knowledge, the only exceptions +are (Chen et al., 2020), (Yoon et al., 2021), and (Odeyomi +and Zaruba, 2021). Chen et al. (2020) propose ASO-Fed, +an asynchronous FL algorithm to minimize the empiri- +cal loss computed over the aggregation of clients’ data +streams. Their analysis requires that all clients have the +same optimal model and that updates at any time t are +consistent with new samples arriving in the future (more +details in Appendix A). On the contrary, the theoretical +analysis in our paper holds under statistical heterogene- +ity across clients’ local data distributions and accounts for +the bias due to the need to work with samples currently +stored by clients. Moreover, we provide statistical learning +guarantees for our algorithm. Yoon et al. (2021) propose +FedWeIT, which extends regularization-based algorithms +for continual learning to the FL setting. The main goal of +FedWeIT is to minimize interference between incompati- +ble tasks while allowing positive knowledge transfer across +clients during learning, but no generalization guarantee is +provided. Odeyomi and Zaruba (2021) consider the prob- +lem of online federated learning under constraints on the +amount of resources consumed over the whole time hori- +zon and proposes an online mirror descent-based algorithm +with regret guarantees. Differently from our contribution, +both (Odeyomi and Zaruba, 2021) and (Yoon et al., 2021) +assume each client can only use the most recent data. Our +experiments show that reusing as little as 5% of the col- +lected samples may be highly beneficial. +Federated learning from temporally shifting distributions +(Zhu et al., 2022; Eichner et al., 2019; Ding et al., 2020; +Guo et al., 2021) is a related, yet different, problem to +learning from a data stream. These papers assume the shift +is due to changes in the set of available clients (e.g., be- +cause of diurnal patterns), but clients’ local datasets do not +change. The only exception is (Guo et al., 2021), which +can capture a setting where clients keep collecting data dur- +ing training without storage constraints. Theoretical results +assume that new data is drawn from a client-independent +distribution (see Appendix A). Instead, our analysis takes +into account both memory constraints and statistical het- +erogeneity across clients’ local data distributions. +Finally, we mention a number of papers studying different +variants of “online federated learning” problems, mostly +focusing on dynamic resource allocation. Many of them +are discussed in the recent survey (Dai and Meng, 2022). +Among these papers, Damaskinos et al. (2020) propose +Fleet, a middleware between the edge device operating +system and the machine learning application, which can be +used to learn on data streams. The middleware is designed +with the device’s energy minimization as the main concern. +Jin et al. (2020) propose an online algorithm to dynam- +ically select the participating clients and their number of +local gradient iterations at each communication round to +minimize the cumulative resource usage over time under +a constraint on the quality of the final model. Zhou et al. +(2020) study a similar problem. They include the possi- +bility of discarding new data points or distributing them to +clients with more resources and propose a resource allo- +cation algorithm based on Lyapunov optimization (Neely, +2010). Both Jin et al. (2020) and Zhou et al. (2020) ignore +the possibility of reusing samples across multiple commu- +nication rounds. +3 +Problem Formulation +In this work, we use [M] ≜ {1, . . . , M} to denote the set +of positive integers up to M. We consider M > 0 clients; +each of them corresponds to a potentially different learn- +ing task. +We associate to each client m ∈ [M]: 1) a +probability distribution Pm over a domain Z = X × Y, +2) a counting process N (t) +m , t ≥ 0, and 3) a dynamic +memory/cache M(t) +m , t > 0 of capacity Cm > 0. +At +time step t > 0, client m ∈ [M] receives a batch +B(t) +m += +� +z(t,i) +m += +� +x(t,i) +m , y(t,i) +m +� +, i ∈ [b(t) +m ] +� +containing +b(t) +m ≜ N (t) +m −N (t−1) +m +samples drawn i.i.d. from Pm. Client +m ∈ [M] can cache a sub-part of the samples in its local + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +memory, without exceeding the capacity Cm. Without loss +of generality we suppose that 1 ≤ b(t) +m ≤ Cm. We consider +a finite time horizon T > 0, and we let Nm ≜ N (T ) +m +and +Sm ≜ �T +t=1 B(t) +m denote the number and the set of samples +gathered by client m up to the time horizon T. We write +Sm = +� +z(i) +m , i ∈ [Nm] +� +, where we arbitrarily ordered the +elements of Sm. We define I(t) +m ⊂ [Nm] to be the set of the +indices of samples present at memory M(t) +m , i.e., j ∈ I(t) +m +if and only if z(j) +m +∈ M(t) +m . Finally, S ≜ �M +m=1 Sm de- +notes the training dataset (aggregated across clients and +across time) with size N ≜ �M +m=1 Nm. The relative size +of client-m’s dataset is nm ≜ Nm/N. +Let HΘ = +� +hθ : X �→ Y, θ ∈ Θ ⊂ Rd� +be a set of para- +metric hypotheses/models mapping X to Y, and ℓ : Θ × +Z +�→ R+ be a loss function. +We define LP (θ) ≜ +Ez∼P [ℓ(θ; z)] to be the true (expected) risk of hypothesis +hθ ∈ HΘ under a generic probability distribution P over +Z and we define LS (θ) = +1 +|S| +� +(x,y)∈S ℓ(θ; z) to be the +empirical risk of model (hypothesis) hθ ∈ HΘ on a generic +dataset S of samples from Z. +In federated learning, clients, usually, collaborate to solve +minimize +θ∈Θ +LP(α) (θ) = +M +� +m=1 +αmLPm (θ) , +(1) +where P(α) ≜ �M +m=1 αm · Pm and α ≜ (αm)1≤m≤M +with αm ≥ 0 and ∥α∥1 = 1. Common choices for α +are αm = nm and αm = +1 +M . The first one corresponds +to minimizing the empirical loss over the aggregate train- +ing dataset S = �M +m=1 Sm, which gives the same impor- +tance to each sample. The second choice instead targets +per-client fairness, by giving the same importance to each +client. +In standard federated learning, local datasets {Sm}m∈[M] +are available since the beginning of the training and the fol- +lowing empirical risk minimization problem is considered +as a proxy for Problem 1: +minimize +θ∈Θ +M +� +m=1 +αm · LSm (θ) . +(2) +Our goal is to design a potentially randomized algorithm A +solving, in a federated fashion, Problem 1 using clients’ +data streams and taking into account clients’ memory con- +straints. +4 +Federated Learning Meta-Algorithm for +Data Streams +When learning from a data stream, every client only has ac- +cess to samples currently present in its local memory. Due +Algorithm 1: Meta Algorithm for Federated Learning +from Data Streams +Input : Nbr of local epochs E; mini-batch size K; +local learning rate η > 0; sample weights +λ = +� +λ(t,j) +m +; m ∈ [M], t ∈ [T], j ∈ I(t) +m +� +Output: ¯θ(T ) = �T +t=1 q(t)θ(t) +1 for t = 1, . . . , T do +2 +Server selects a subset S(t) ⊆ [M] of clients; +3 +for m ∈ S(t) (in parallel) do +4 +θ(t,1) +m +← θ(t); +5 +Sample B(t) +m = {z(t,1) +m +, . . . z(t,b(t) +m ) +m +} ∼ Pb(t) +m +m ; +6 +M(t) +m ← Update +� +M(t−1) +m +, B(t) +m +� +; +7 +for e = 1, . . . , E do +8 +Sample min +� +K, |I(t) +m | +� +indices ξ(t,e) +m +uniformly from I(t) +m ; +9 +g(t,e) +m +← +|I(t) +m | +|ξ(t,e) +m +| +� +j∈ξ(t,e) +m +λ(t,j) +m +� +j′∈I(t) +m +λ(t,j′) +m +· +∇ℓ(θ(t,e) +m +; z(t,j) +m +) ; +10 +θ(t,e+1) +m +← θ(t,e) +m +− η · g(t,e) +m +; +11 +end +12 +end +13 +∆(t) ← �M +m=1 p(t) +m · +� +θ(t,E+1) +m +− θ(t)� +; +14 +θ(t+1) ← ΠΘ +� +θ(t) + ∆(t)� +; +15 end +to the limited storage capacity at each client and to the vari- +ability in the number of new samples arriving across time, +samples may spend different amounts of time in mem- +ory and then be used a different number of times dur- +ing training. In order to potentially compensate for such +heterogeneity, we allow samples to be weighted differ- +ently over time and across clients. In particular, we de- +note by λ(t,j) +m +≥ 0 the weight assigned at time t to sam- +ple j stored in client m’s memory (then j ∈ I(t) +m ), and +by λ ≜ +� +λ(t,j) +m +; m ∈ [M], t ∈ [T], j ∈ I(t) +m +� +the set of all +weights. We define the weighted local objective associated +to client-m’s local memory at time step t ∈ [T] as +L(λ) +M(t) +m (θ) ≜ +� +j∈I(t) +m λ(t,j) +m +ℓ +� +θ, z(j) +m +� +� +j∈I(t) +m λ(t,j) +m +, +(3) +and similarly the global weighted empirical risk as +L(λ) +S (θ) ≜ +�M +m=1 +�T +t=1 +� +j∈I(t) +m λ(t,j) +m +· ℓ +� +θ; z(j) +m +� +�M +m=1 +�T +t=1 +� +j∈I(t) +m λ(t,j) +m +. +(4) + +Federated Learning for Data Streams +We additionally define client-m’s aggregation weight as +p(t) +m ≜ +� +j∈I(t) +m λ(t,j) +m +�M +m′=1 +� +j∈I(t) +m′ λ(t,j) +m′ +, +(5) +and +q(t) ≜ +�M +m=1 +� +j∈I(t) +m λ(t,j) +m +�T +s=1 +�M +m′=1 +� +j∈I(s) +m′ λ(s,j) +m′ +. +(6) +In this work we consider a meta-algorithm similar to vanilla +FedAvg (McMahan et al., 2017) to minimize the weighted +empirical risk (4). +Algorithm 1 operates in an iterative +fashion: at time step t ∈ [T] (also called communication +round), the central server broadcasts the global model θ(t) +to a subset of clients (line 4). Then every selected client, +say it m, receives a new batch of data (line 5) that is used +to update the client’s local memory M(t) +m (line 6). The +selected clients perform E local stochastic gradient steps +(line 10), where the stochastic gradient g(t,e) +m +is an unbiased +estimator of ∇L(λ) +M(t) +m +� +θ(t,e) +m +� +computed using at most K +samples (line 9). After E local steps, clients send back their +models to the central server for aggregation (line 13, 14). +The update at time step t can also written as follows +θ(t+1) = Π +Θ +� +θ(t) − η · +M +� +m=1 +p(t) +m +E +� +e=1 +g(t,e) +m +� +, +(7) +where ΠΘ(·) denotes the projection over the set Θ. +Note that the output of Algorithm 1 depends on the actual +sample arrival sequences at clients, on the memory update +rule, and on the weights λ. In particular, the memory up- +date rule determines which samples can be considered at +a given time step and then which weights can be different +from zero. Nevertheless, for the sake of simplicity, we de- +note the output simply as A(λ)(S). +In this paper, we restrict our analysis to the case where +both the memory update rule and the weight selection +rule are deterministic and do not depend on the features +or the labels of the samples in the memory. +More for- +mally, given a particular instance of the counting process +N (t) +m , the weights {λ(t,i) +m }t∈[T ] of sample z(i) +m ∈ Sm re- +main unchanged if z(i) +m += +� +x(i) +m , y(i) +m +� +is replaced by +z(i) +m = +� +˜x(i) +m , ˜y(i) +m +� +with ˜x(i) +m ̸= x(i) +m or ˜y(i) +m ̸= y(i) +m . +For a given sample arrival sequence and memory update +rule, the quality of the algorithm is evaluated through the +true error +ϵtrue ≜ EA(λ),S +� +LP(α) +� +A(λ) (S) +�� +− min +θ∈Θ LP(α) (θ) , +(8) +where the expectation is taken over the potential random- +ness of algorithm A(λ), i.e., clients’ (line 2) and batches’ +(line 8) sampling processes, and the samples collected. +4.1 +General Analysis +The true error ϵtrue of our meta-algorithm in (8) can be +bounded as follows (see proof in Appendix B.1) +ϵtrue ≤ +E +S,A(λ) +� +L(λ) +S +� +A(λ)� +S(T )�� +− min +θ∈Θ L(λ) +S (θ) +� +� +�� +� +≜ϵopt ++ 2 E +S +� +sup +θ∈Θ +���LP(α)(θ) − L(λ) +S (θ) +��� +� +� +�� +� +≜ϵgen +. +(9) +The generalization error ϵgen is the expected value of the +representativeness of the dataset S, which is the maxi- +mal distance between the true risk LP(α) and the empirical +risk L(λ) +S . Intuitively, the smaller the generalization error, +the better we can approach the minimum of LP(α) by min- +imizing L(λ) +S . +The optimization error ϵopt measures how well Algo- +rithm 1 approaches the minimizer of the weighted empir- +ical risk L(λ) +S . +In the rest of this section, we first provide bounds for for +the generalization error ϵgen (Theorem 4.1) and for the op- +timization error ϵopt (Theorem 4.3) and and then combine +them to bound the overall error ϵtrue (Theorem 4.4). Our +results rely on the following assumptions: +Assumption 1. (Bounded loss) The loss function is +bounded, i.e., ∀θ ∈ Θ, z ∈ Z, ℓ(θ; z) ∈ [0, B]. +Assumption 2. (Bounded domain) We suppose that Θ is +convex, closed and bounded with diameter D. +Assumption 3. (Convexity) For all z ∈ Z, the function +θ �→ ℓ(θ; z) is convex on Rd. +Assumption 4. (Smoothness) For all z ∈ Z, the function +θ �→ ℓ(θ; z) is L-smooth on Rd. +Assumption 1 is a standard assumption in statistical learn- +ing theory (e.g., (Mohri et al., 2018) and (Shalev-Shwartz +and Ben-David, 2014)). Assumptions 2–4 are common as- +sumptions in the analysis of (stochastic) gradient methods +(see for example (Bubeck et al., 2015) and (Bottou et al., +2018)) and online convex optimization (Hazan, 2019). +Remark 1. Assumptions 1 and 4 imply that (it follows from +Lemma B.2 in Appendix B.2) +σ2 +0 ≜ max +m +E +z∼Pm +� +sup +θ∈Θ +∥∇ℓ(θ; z) − ∇LPm (θ)∥2 +� +(10) +≤ +� +2 · +√ +2LB +�2 +, +(11) +and (it follows from Lemma B.3 in Appendix B.2) +ζ ≜ max +m,m′ sup +θ∈Θ +��∇LPm′ (θ) − ∇LPm (θ) +�� +(12) +≤ 2 · +√ +2LB. +(13) + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +These properties are similar to the stochastic gradients’ +bounded variance, and the clients’ bounded dissimilarity +assumptions usually employed in the analysis of federated +learning algorithms (Wang et al., 2021a). +4.2 +Bounding the Generalization Error +Theorem 4.1 (proof in Appendix B.3) quantifies the gener- +alization error and in particular how the weighted empiri- +cal risk L(λ) +S +differs from the target expected risk LP(α) for +the minimizer of the first one, i.e., it bounds |LP(α)(θ′) − +L(λ) +S (θ′)| for θ′ ∈ arg minθ∈Θ L(λ) +S (θ). The bound differs +from classic statistical learning results (as those in (Shalev- +Shwartz and Ben-David, 2014)) because L(λ) +S +is a weighted +empirical risk and its expected value does not necessar- +ily coincide with LP(α). +We recall that the label dis- +crepancy associated to a hypothesis class H quantifies the +distance between two distributions P and P′ as follows +discH (P, P′) ≜ maxh∈H |LP (h) − LP′ (h)| (Mansour +et al., 2020). +Theorem 4.1. Suppose that Assumption 1 holds, when us- +ing Algorithm 1 with weights λ, it follows that +ϵgen ≤ discH +� +P(α), P(p)� ++ ˜O +� +� +� +VCdim (H) +Neff +� +� , +(14) +where Neff = +��M +m=1 +�Nm +i=1 p2 +m,i +�−1 +, +pm,i = +�T +t=1 +� +j∈I(t) +m 1 {j = i} · λ(t,j) +m +�M +m′=1 +�T +t=1 +� +j∈I(t) +m′ λ(t,j) +m′ +, +i ∈ Nm, +(15) +and p = +��Nm +i=1 pm,i +� +1≤m≤M. +The coefficient pm,i represents the relative importance +given, during the whole training period, to sample i with +respect to all the samples collected by all clients and +pm = �Nm +i=1 pm,i represents the relative importance given +to client m during training. Note that pm = �T +t=1 q(t)p(t) +m +and the p(t) +m coincides with the relative importance pm, +when p(t) +m is constant over time. +In general, there is an inconsistency between the impor- +tance we should give to clients (quantified by α in (1)) +and the one we actually give them during training (quan- +tified by p). The first term on the RHS of (14) captures +the mismatch between the target distribution P(α) and the +“effective distribution” P(p) = �M +m=1 pmPm through the +discrepancy. +The second term in the RHS of (14) is similar in shape +to the usual bounds observed in statistical learning the- +ory, e.g., (Shalev-Shwartz and Ben-David, 2014), which +are proportional to the square root of the ratio of the VC +dimension of the hypotheses class and the total number of +samples N. In our case, Neff plays the role of the effective +number of samples and Lemma 4.2 (proof in Appendix B.4) +shows that, as expected, Neff is at most N, and reaches this +value when each sample is given the same importance. +Lemma 4.2. It holds Neff ≤ N and the bound is attained +when each sample has the same relative importance, i.e., +pm,i = pm,j, for each i, j ∈ [Nm]. +The generalization error ϵgen decreases the closer α and p +are and the larger Neff is. When αm = nm (remember +that nm = Nm/N), the choice pm,i = 1/N minimizes the +bound, as it leads both to p = n = α and to Neff = N. +In our streaming learning setting, pm,i += +1/N can +be obtained by different combinations of memory up- +date rules and sample weight selection rules. For exam- +ple, this is the case when clients’ memories only con- +tain the samples received during the current round (i.e., +Update(M(t−1) +m +, B(t) +m ) = B(t) +m in line 6 of Alg. 1) and +all samples currently in the memory get weight 1 (i.e., +λ(t,j) +m += 1 for each j ∈ I(t) +m ). +But it is also the case +when the memory update rule lets samples stay in mem- +ory for multiple consecutive rounds (e.g., τ (j) +m rounds for +sample j at client m) and samples receive a weight in- +versely proportional to the number of consecutive rounds +(i.e., λ(t,j) +m += 1/τ (j) +m ). In what follows, we refer to any +combination of memory update rules and weight selection +rules leading to pm,i = 1/N as a Uniform strategy. +While a Uniform strategy minimizes the bound for the +generalization error ϵgen when α = n, it is in general sub- +optimal in terms of the optimization error ϵopt, as we are +going to show in the next section. +4.3 +Bounding the Optimization Error +We provide our bound on ϵopt under full clients participa- +tion (S(t) = [M]) with full batch (K ≥ |I(t) +m |). Under +mini-batch gradients an additional vanishing error term ap- +pears. The proof is provided in Appendix B.5. +Theorem 4.3. Suppose that Assumptions 1–4 hold, the +sequence +� +q(t)� +t is non increasing, and verifies q(1) = +O (1/T), and η ∝ 1/ +√ +T · min{1, 1/¯σ (λ)}. Under full +clients participation (S(t) = [M]) with full batch (K ≥ +|I(t) +m |), we have +ϵopt ≤ O +� +¯σ (λ) +� ++ O +� ¯σ (λ) +√ +T +� ++ O +� 1 +√ +T +� +, +(16) + +Federated Learning for Data Streams +where, +¯σ2 (λ) ≜ +T +� +t=1 +q(t)× +E +S +� +sup +θ∈Θ +�����∇L(λ) +S (θ) − +M +� +m=1 +p(t) +m ∇L(λ) +M(t) +m (θ) +����� +2 � +. (17) +Moreover, there exist a data arrival process and a loss func- +tion ℓ, such that, under FIFO memory update rule,1 for any +choice of weights λ, ϵopt = Ω (¯σ (λ)). +The coefficient ¯σ2 (λ) quantifies the variability of the gra- +dient considered in the update at round t w.r.t. the gradi- +ent of the global objective L(λ) +S +and, as shown by Theo- +rem 4.3, it prevents the optimization error to vanish when +T diverges. Lemma B.4 provides a general upper bound +for ¯σ2 (λ) in terms of stochastic gradients’ variance and +clients’ dissimilarity. +The optimization error ϵopt is smaller the closer ¯σ2(λ) +is to zero. +In our streaming learning setting, ¯σ2(λ) = +0 may be obtained if the memory is never updated +(Update(M(t−1) +m +, B(t) +m ) = M(t−1) +m +, ∀t ≥ 1) and the ag- +gregation weights are constant over time (p(t) +m = pm, ∀t ∈ +[T]). +It is indeed easy to check that under these con- +ditions L(λ) +S (θ) = �M +m=1 p(t) +m L(λ) +M(t) +m (θ) (and they equal +�M +m=1 pmL(λ) +M(0) +m (θ)). Any set of time-independent sample +weights leads to constant aggregation weights, but, among +them, the choice λ(t,j) +m += 1 reduces the generalization +bound ϵgen. We refer to these memory update and weight +selection rules as the Historical strategy. +The Historical strategy minimizes the optimization +bound by ignoring all the samples collected during train- +ing. It is in sharp contrast with the Uniform strategy, +which assigns the same relative importance to all collected +samples. +4.4 +Main Result +The tension between the two error components ϵgen and ϵopt +is evident from our discussion above. One can minimize +ϵgen by considering at each time only the most recent sam- +ples, and, at the opposite, ϵopt by ignoring those samples. +By combining Theorems 4.1 and 4.3, Theorem 4.4 formally +quantifies this trade-off and provides a bound on ϵtrue. +Theorem 4.4. Under the same assumptions as in Theo- +1The FIFO (First-In-First-Out) update rule evicts the oldest +samples in the memory to store the most recent ones. +10 +1 +100 +101 +c2/c1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p* +hist +Nhist/N = 95.0% +Nhist/N = 70.0% +Nhist/N = 40.0% +Nhist/N = 20.0% +Nhist/N = 5.0% +Figure 1: Effect of c2/c1 on the historical clients rela- +tive importance p∗ +hist for different values of Nhist/N, when +M = 50 and Mhist = 25. The dashed vertical line cor- +responds to our estimation of c2/c1 on CIFAR-10 experi- +ments (ˆc2/ˆc1 = 0.15). +rem 4.1 and Theorem 4.3, +ϵtrue ≤O +� 1 +√ +T +� ++ O +� +¯σ (λ) +� ++ 2discH +� +P(α), P(p)� ++ ˜O +� +� +� +VCdim (H) +Neff +� +� . +(18) +5 +Case Study +In fog computing environments, IoT devices, edge servers, +and cloud servers can jointly participate to train an ML +model (Bonomi et al., 2012). IoT devices keep generating +new data, but may not be able to store them permanently +due to sever memory constraints. Instead, edge servers may +contribute with larger static datasets (Hosseinalipour et al., +2020; Wang et al., 2021b). Motivated by this scenario, we +consider two groups of clients: Mhist clients with “histor- +ical” datasets, which do not change during training, and +M − Mhist clients, who collect “fresh” samples with con- +stant rates {bm > 0, m ∈ �Mhist + 1, M�} and only store +the most recent bm samples due to memory constraints (i.e., +Cm = bm).2 We refer to these two categories as historical +clients and fresh clients, respectively. Fresh clients can also +capture the setting where clients are available during a sin- +gle communication round—see details in Appendix C.1. +At each client all samples are used the same number of +times (T and 1 at historical and fresh clients, respec- +tively). Then, one can prove that each client, say it m, +should assign the same weight to any sample currently +available at its local memory, i.e., λ(t,j) +m += λ(t) +m . +For +simplicity, we consider stationary weights, i.e., λ(t) +m += +λm, and we want then to determine per-client sample +weights (λm)m∈[M] leading to the best guarantees in terms +of ϵtrue.3 Equivalently, we want to determine the clients’ +2Note that we are implicitly selecting FIFO as memory update +rule. +3Restricting the weights to be stationary, i.e., λ(t) +m += λm, +might be suboptimal. + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +0.2 +0.4 +0.6 +0.8 +Nhist/N +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +( +hist +*)/c2 +log(c2/c1) = +1.5 +log(c2/c1) = +1.3 +log(c2/c1) = +1.0 +log(c2/c1) = +0.5 +log(c2/c1) = 2.0 +0.2 +0.4 +0.6 +0.8 +Nhist/N +0 +1 +2 +3 +4 +( +unif +*)/c2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Nhist/N +4 +3 +2 +1 +0 +1 +2 +3 +( +hist +unif)/c2 +Figure 2: The differences ψhist − ψ∗ (left), ψuniform − ψ∗ +(center), and ψhist −ψuniform (right) as a function of Nhist/N +for different values of c2/c1, on CIFAR-10 dataset (N = +5 × 105) when M = 50 and Mhist = 25. +relative importance values p = (pm)m∈[M], where pm = +λmNm/ +��M +m′=1 λm′Nm′ +� +. Note that in this setting ag- +gregation weights and relative importance values coincide +(i.e., p(t) +m = pm). Corollary 5.1′ (Appendix C) bounds ϵtrue +as a function of p in this scenario. For the sake of simplic- +ity, we provide here the bound for the case αm = nm, m ∈ +[M] (which we assume to hold in the rest of this section): +Corollary 5.1. Consider the scenario with Mhist historical +clients, and M − Mhist fresh clients. Suppose that the same +assumptions of Theorem 4.4 hold, that α = n, and that +Algorithm 1 is used with clients’ aggregation weights p = +(pm)m∈[M] ∈ ∆M−1, then +ϵtrue ≤ ψ(p; c) ≜ +c0 + c1 · +� +� +� +� +M +� +m=Mhist+1 +p2m + c2 · +� +� +� +� +M +� +m=1 +p2m +nm +, +(19) +where c = (c0, c1, c2) are non-negative constants not de- +pending on p, given as: +c0 = (C1 + C3) + C2 +T − 2 · max +m,m′ disc (Pm, Pm′) +(20) +c1 = σ0 +� +M − M0 · +� +D + +2 +√ +T +� +(21) +c2 = 4 · +� +1 + log +� +N +VCdim (H) +� +· +� +VCdim (H) +N ++ 2 · max +m,m′ disc (Pm, Pm′) +(22) +and C1, C2, and C3 are the constants defined in the proof +of Theorem 4.3, and σ0 is defined in Remark 1. +The second term in (19) captures the gradient variability +(second term in (18)), while the third term in (19) cap- +tures both contributions to the generalization error, i.e., the +distribution discrepancy and the effective number of sam- +ples (third and fourth terms in (19)). In particular, it holds +�M +m=1 +p2 +m +nm ∝ 1/Neff. +The minimization of ψ over the unitary simplex is a con- +vex optimization problem (proof in Appendix C.4), which +can then be solved efficiently with, for example, projected +gradient descent. We use ψ∗, p∗, and p∗ +hist to denote the +minimum of ψ, its minimizer, and the aggregate relative +importance given to historical clients (p∗ +hist ≜ �Mhist +m=1 p∗ +m), +respectively. +The solution p∗ depends on the value of n—in particu- +lar on the fraction of historical samples Nhist/N (where +Nhist ≜ �Mhist +m=1 Nm)—and on the ratio c2/c1. The ratio +c2/c1 only depends on the intrinsic properties of the learn- +ing problem (VCdim (H), D, B, and σ0), and the total +number of samples N (see Appendix C.3). +Figure 1 illustrates how the optimal clients’ importance val- +ues change as a function of the ratio c2/c1 and the fraction +of historical samples Nhist/N (other results are in Figure 4). +Beside the specific numerical values, one can distinguish +two corner cases. When c2/c1 ≫ 1, the optimal solution +corresponds to minimize �M +m=1 p2 +m/nm, i.e., to maximize +the effective number of samples. The optimal strategy is +then the Uniform one and the aggregate relative impor- +tance for historical clients is p∗ +hist = Nhist/N. On the con- +trary, when c2/c1 ≪ 1, the optimal solution corresponds to +minimize � +m>Mhist p2 +m, i.e., the gradient variability. The +Historical strategy is then optimal and corresponds to +p∗ +m = Nm/Nhist = +N +Nhist nm for m ∈ [Mhist] and p∗ +hist = 1. +For general values of c2/c1, the optimal strategy to +assign clients’ importance values—or equivalently sam- +ple weights—differs from both the Uniform and the +Historical ones. +We propose then the following +heuristic, which we evaluate in the next section. At the +beginning of training, clients cooperatively estimate c2/c1 +using a fraction of their historical samples, as ˆc2/ˆc1 ≈ +B+√ +d/N +GD√M−Mhist (see details in Appendix C.6). Then, clients’ +importance values are selected minimizing the bound in +(19), i.e., ˆp∗ = arg min ψ (·, ˆc). +Beside +providing +configuration +rules +for +our +meta- +algorithm, our analysis allows us also to evaluate how the +performances of different strategies like Uniform and +Historical depend on the different parameters as in +Figure 2. Our experimental results in the next section con- +firm these theoretical predictions. +6 +Experimental Results +Datasets and models. +We considered different ma- +chine +learning +tasks +on +five +federated +benchmark +datasets: image classification (CIFAR-10 and CIFAR-100 +(Krizhevsky, 2009)), handwritten character recognition +(FEMNIST (Caldas et al., 2018)), language modeling +(Shakespeare (Caldas et al., 2018; McMahan et al., +2017)), and logistic regression on a synthetic dataset +described in Appendix D.1. Table 1 summarizes datasets, +models, and the total number of clients. +Details on +the datasets, +models, +and hyperparameters selection + +Federated Learning for Data Streams +Table 1: Datasets and models. +DATASET +CLIENTS +TOTAL SAMPLES +MODEL +SYNTHETIC +11 +200 +LINEAR MODEL +CIFAR-10 / 100 +50 +50, 000 +2 CNN + 2 FC +FEMNIST +3, 597 +817, 851 +2 FC +SHAKESPEARE +916 +3, 436, 096 +STACKED-LSTM +are provided in Appendix D. The code is available at +https://github.com/omarfoq/streaming-fl. +Arrival process. +For the synthetic dataset and CIFAR- +10/100 we adopted common strategies to split the datasets +across clients and divided clients into two groups as in Sec- +tion 5 with Mhist = 10 and Mhist = 25, respectively. For +FEMNIST and Shakespeare datasets, we adopted their nat- +ural partitions and set Mhist such that Mhist/M = 5%, 20%, +and 50%, but allowed fresh clients to participate to train- +ing for a few rounds. Experimental results for these two +datasets suggest that our analysis is robust to departures +from the setting considered in Section 5. Details are in Ap- +pendix D.3. +Baselines. +We compared our strategy to select clients’ +importance values, (see Sec. 5), with three baselines: the +Uniform and Historical strategies described above +as well as the Fresh strategy which only considers fresh +clients. We observe that under our samples’ arrival pro- +cess and α = n, there could be two natural ways to extend +the classic FedAvg’s aggregation rule (McMahan et al., +2017): set each client’s aggregation weight proportional to +(1) the number of samples collected by the client over the +whole time-horizon, or (2) the number of samples currently +in the client’s memory. The first aggregation rule coincides +with the Uniform strategy, the second one leads in all set- +tings we considered to very small aggregation weights for +fresh clients so that it is practically indistinguishable from +the Historical strategy. Interestingly, both these rules +are in general suboptimal, motivating the practical interest +of our study and of the strategy we propose. +Main Results. +Table 2 reports the test accuracy when +Nhist/N = 20% for the different strategies together with +the optimal test accuracy obtained selecting the value of +phist = �Mhist +m=1 pm in the grid {0, 0.2, 0.5, 0.8, 1.0}. Our +observations are confirmed for other values of Nhist/N +(see Table 4 and Table 5 in Appendix E). A first remark +is that working only with new data (as Fresh does) is +never optimal, not even when historical data account for +just 5% of the total dataset (Table 4). Second, neither of +the two “reasonable” ways to extend FedAvg consistently +achieves good accuracy: Historical performs poorly +over Synthetic and Uniform over FEMNIST and Shake- +speare. On the contrary, our method always performs at +least as well as the best baseline and it often achieves a +test accuracy similar to the (estimated) optimal one. +In +particular, it correctly sets weights as Uniform over Syn- +thetic and as Historical over FEMNIST and Shake- +speare. +We observe that our analysis also helps to ex- +plain the counter-intuitive conclusion that, on FEMNIST +and Shakespeare, it is beneficial to ignore new collected +samples (even for Nhist/N = 5%, see Table 4). Our strat- +egy correctly sets ˆp∗ +hist = 1, because it estimates that, for +these two datasets, the ratio of the number of parameters to +the aggregate training dataset size (d/N) is much smaller +than the gradients’ norm (G)—numerical values are pro- +vided in Appendix D.4. This information suggests that we +can use a small subset of the original dataset to identify a +good model in the selected hypotheses class, and in particu- +lar we can rely only on historical data avoiding the potential +noise introduced by new samples. +Figure 3 shows the effect of p on CIFAR-10 test accuracy +for different values of the ratio Nhist/N—similar figures for +other datasets are provided in Appendix E. It confirms that +performances in terms of final test accuracy match the pre- +dictions of our model on the bound ψ illustrated in Figure 2. +First, Figure 3 shows that the performance gap between +Historical and the optimal assignment p∗ decreases +when Nhist/N increases (as predicted in Figure 2 (left)): the +gap is 15.5±0.30, 7.9±1.17, and 5.3±2.8 pp when Nhist/N +is 5%, 20%, and 50%, respectively. +Second, Figure 3 +confirms that the performance gap between Uniform and +the optimal assignment first increases and then decreases, +when Nhist/N increases (as in Figure 2 (center)): the gap is +3.0 ± 0.57, 6.2 ± 0.55, and 4.3 ± 0.35 pp when Nhist/N is +5%, 20%, and 50%, respectively. Finally, Figure 3 shows +that the relative ranking of Uniform and Historical +changes, with Uniform being a better option for smaller +values of Nhist/N and Historical becoming slightly +better for larger values. Again, this behavior is predicted +by our analysis. Indeed, in this experiment, our estimation +for the ratio c2/c1 is ˆc2/ˆc1 ≈ 0.15 ∈ [10−1.3, 10−0.5] cor- +responding to a setting for which ψhist − ψunif changes sign +in Figure 2 (right). +7 +Conclusion +In this paper, we formalized the problem of federated learn- +ing for data streams and highlighted a new source of hetero- +geneity resulting from local datasets’ variability over time. +We proposed a general federated algorithm to learn in this +setting and studied its theoretical guarantees. Our analy- +sis reveals a new bias-optimization trade-off controlled by +the relative importance of older samples in comparison to +newer ones and leads to practical guidelines to configure +such importance in our algorithm. Experiments show that +our configuration rule outperforms natural ways to extend +the usual FedAvg aggregation rule in the presence of data +streams. Moreover, experimental results confirm other the- +oretical conclusions, despite the theoretical assumptions +and the mismatch in the corresponding performance met- + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +Table 2: Average test accuracy across clients for different datasets in the settings when Nhist/N = 20%. +DATASET +ˆc2/ˆc1 +ˆp∗ +HIST +TEST ACCURACY +FRESH +HISTORICAL +UNIFORM +OURS +OPTIMAL +SYNTHETIC +0.092 +0.20 +84.7 ± 1.44 +77.3 ± 3.15 +85.5 ± 1.60 +85.5 ± 1.60 +85.5 ± 1.60 +CIFAR-10 +0.150 +0.45 +59.6 ± 0.94 +59.8 ± 2.16 +61.5 ± 0.63 +66.9 ± 0.81 +67.7 ± 0.91 +CIFAR-100 +0.284 +0.32 +22.4 ± 0.57 +22.6 ± 0.50 +25.3 ± 0.43 +28.5 ± 0.57 +31.5 ± 0.25 +FEMNIST +0.001 +1.00 +53.3 ± 1.85 +66.1 ± 0.20 +55.4 ± 0.80 +66.1 ± 0.20 +66.1 ± 0.80 +SHAKESPEARE +0.064 +1.00 +38.4 ± 0.43 +49.0 ± 0.26 +39.3 ± 0.38 +49.0 ± 0.26 +49.0 ± 0.26 +0 +200 +400 +600 +800 +Time step +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Test accuracy +phist=0.00 +phist=0.05 +p* +hist=0.12 +phist=0.20 +phist=0.50 +phist=0.80 +phist=1.00 +0 +200 +400 +600 +800 +Time step +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Test accuracy +phist=0.00 +phist=0.20 +p* +hist=0.45 +phist=0.50 +phist=0.80 +phist=1.00 +0 +200 +400 +600 +800 +Time step +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Test accuracy +phist=0.00 +phist=0.20 +phist=0.50 +phist=0.80 +p* +hist=0.95 +phist=1.00 +Figure 3: Evolution of the test accuracy when using different values of phist for CIFAR-10 (left) dataset, when Nhist/N = +5% (left), 20% (center), and 50% (right). The setting phist = Nhist/N corresponds to Uniform strategy. +rics (e.g., test accuracy versus a loss bound). +To the best of our knowledge, this work is the first to frame +the problem of federated learning for data streams. It high- +lights new challenges and—we believe—lays the founda- +tions for further research. For example, part of our results +are restricted to the important, but still quite specific, sce- +nario where some clients have static datasets and others +process new samples at each step. In this setting, samples +are used a different number of times across clients but ex- +actly the same number of times at a given client, simpli- +fying the analysis. But what happens if heterogeneity in +samples’ availability also appears at the level of a single +client? How do different memory update rules affect such +heterogeneity, and how can we design such policies to min- +imize the total error of the final model? Finally, how do our +results change if local data distributions change over time? +Acknowledgments +This research was supported in part by the Groupe La +Poste, sponsor of the Inria Foundation, in the framework +of the FedMalin Inria Challenge, and in part by the French +government, through the 3IA Cˆote d’Azur Investments +in the Future project managed by the National Research +Agency (ANR) with the reference number ANR-19-P3IA- +0002. The authors are grateful to the OPAL infrastructure +from Universit´e Cˆote d’Azur for providing computational +resources and technical support. +References +Brendan McMahan, Eider Moore, Daniel Ramage, Seth +Hampson, and Blaise Aguera y Arcas. Communication- +efficient learning of deep networks from decentralized +data. +In Artificial Intelligence and Statistics, pages +1273–1282. PMLR, 2017. +L´eon Bottou, Frank E Curtis, and Jorge Nocedal. Opti- +mization methods for large-scale machine learning. 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Although some convergence results are stated in the paper, their interest and +applicability are questionable, as the analysis requires that all clients have the same optimal model and that updates at +any time t are consistent with new samples arriving in the future. Indeed, the paper mentions that clients can receive new +samples during training (see Fig. 2), but also requires that, at any time t and for any client k, the expected value of the +update ∇ζk(w) has a non-null component in the direction of the gradient of the global empirical loss F, which depends +on samples arriving after time t (see Assumption 1). Moreover, the bounded gradient dissimilarity assumption implies +that the minimizer of F (F is assumed to be strongly-convex) is also a stationary point of each local objective function fk +(consider β = 0 and λ = 0). On the contrary, the theoretical analysis in our paper holds under statistical heterogeneity +across clients’ local data distributions and accounts for the bias due to working with samples currently stored at clients. +Moreover, we provide statistical learning guarantees for our algorithm. +The model considered in (Guo et al., 2021) can capture a setting where clients keep collecting data during training without +storage constraints. Indeed, clients track the dynamic objective in (Guo et al., 2021, Eq. (2)) which depends on data samples +received until the current time. Theoretical results assume that new data is drawn from a client-independent distribution. +This is shown by (Guo et al., 2021, Eq. (5)), which requires that local gradients computed on new data samples are unbiased +estimators of the gradient of the global objective function. Instead, our analysis takes into account both memory constraints +and statistical heterogeneity across clients’ local data distributions. + +Federated Learning for Data Streams +B +Proofs +We remind that all our results rely on the following assumptions: +Assumption 1. (Bounded loss) The loss function is bounded, i.e., ∀θ ∈ Θ, z ∈ Z, ℓ(θ; z) ∈ [0, B] +Assumption 2. (Bounded domain) We suppose that Θ is convex, closed and bounded; we use D to denote its diameter, +i.e., ∀θ, θ′ ∈ Θ, ∥θ − θ′∥ ≤ D. +Assumption 3. (Convexity) For all z ∈ Z, the function θ �→ ℓ(θ; z) is convex on Rd. +Assumption 4. (Smoothness) For all z ∈ Z, the function θ �→ ℓ(θ; z) is L-smooth on Rd. +In what follows, we use ∆D−1 to denote the unitary simplex of dimension D − 1, i.e., ∆D−1 = +� +f ∈ RD ++, �D +i=1 fi = 1 +� +B.1 +Proof of (9) +ϵtrue = +E +S,A(λ) +� +LP(α) +� +A(λ) (S) +� +− L(λ) +S +� +A(λ) (S) +�� ++ +E +S,A(λ) +� +L(λ) +S +� +A(λ) (S) +� +− min +θ∈Θ L(λ) +S +(θ) +� ++ E +S +� +min +θ∈Θ L(λ) +S +(θ) +� +− min +θ∈Θ LP(α) (θ) +(23) +≤ 2 E +S +� +sup +θ∈Θ +���LP(α) (θ) − L(λ) +S +(θ) +��� +� +� +�� +� +≜ϵgen ++ +E +S,A(λ) +� +L(λ) +S +� +A(λ) (S) +� +− min +θ∈Θ L(λ) +S +(θ) +� +� +�� +� +≜ϵopt +, +(24) +where we exploited the fact that minx∈X f(x) − minx∈X g(x) ≤ supx∈X |f(x) − g(x)|. +B.2 +Properties +Lemma B.1. Let f be an L-smooth function taking values in [0, B], then ∥∇f∥ ≤ +√ +2LB. +Proof. Let θ ∈ Θ, then using the definition of the L-smoothness of f with θ′ = θ − 1 +L∇f (θ), we have +f(θ′) = f(θ − 1 +L∇f (θ)) ≤ f (θ) − 1 +L⟨∇f (θ) , ∇f (θ)⟩ + L +2 +���� +1 +L∇f (θ) +���� +2 +(25) += f (θ) − 1 +2L ∥∇f (θ)∥2 . +(26) +If follows that, +∥∇f (θ)∥2 ≤ 2L (f (θ) − f (θ′)) ≤ 2LB. +(27) +Lemma B.2. Suppose that Assumptions 1, and 4 hold. For all +sup +θ∈Θ +∥∇ℓ(θ; z) − ∇LPm (θ)∥2 ≤ +� +2 +√ +2LB +�2 +(28) +. +Proof. Let z ∈ Z, and m ∈ [M]. Both ℓ (·, z), and LPmare L-smooth and bounded within [0, B]. +For θ ∈ Θ, we have +∥∇ℓ(θ; z) − ∇LPm (θ)∥2 ≤ 2 ∥∇ℓ(θ; z)∥2 + 2 ∥∇LPm (θ)∥2 +(29) +≤ 2 · 2LB + 2 · 2LB +(30) += 8LB = +� +2 +√ +2LB +�2 +, +(31) +where we used Lemma B.1 to obtain the last inequality. + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +Lemma B.3. Suppose that Assumptions 1, and 4 hold. For all z ∈ Z, we have +max +m,m′ sup +θ∈Θ +��∇LPm′ (θ) − ∇LPm (θ) +�� ≤ 2 +√ +2LB. +(32) +. +Proof. The proof follows using the triangular inequality and Lemma B.1. +B.3 +Proof of Theorem 4.1 +Theorem 4.1. Suppose that Assumption 1 holds, when using Algorithm 1 with weights λ, it follows that +ϵgen ≤ discH +� +P(α), P(p)� ++ ˜O +� +� +� +VCdim (H) +Neff +� +� , +where Neff = +��M +m=1 +�Nm +i=1 p2 +m,i +�−1 +, +pm,i = +�T +t=1 +� +j∈I(t) +m 1 {j = i} · λ(t,j) +m +�M +m′=1 +�T +t=1 +� +j∈I(t) +m′ λ(t,j) +m′ +, +i ∈ Nm, +and p = +��Nm +i=1 pm,i +� +1≤m≤M. +Proof. For client, m ∈ [M], we remind that pm ≜ �Nm +i=1 pm,i is the relative importance of client m in comparison to the +other clients. We define +LS,p = +M +� +m=1 +Nm +� +i=1 +pm,i · ℓ(·; z(i) +m ). +(33) +Note that LS,p = L(λ) +S , and ES [LS,p (θ)] = � +m pmLPm (θ) = LP (p) (θ) for any θ ∈ Θ, where P(p) = � +m pmPm. We +have +ϵgen = E +S +� +sup +h∈H +|LP(α) (h) − LS,p (h)| +� +(34) += E +S +� +sup +h∈H +|LP(α) (h) − LP(p) (h) + LP(p) (h) − LS,p (h)| +� +(35) +≤ E +S +� +sup +h∈H +|LP(α) (h) − LP(p) (h)| +� ++ E +S +� +sup +h∈H +|LP(p) (h) − LS,p (h)| +� +(36) +≤ discH +� +P(α), P(p)� ++ E +S +� +sup +h∈H +|LP(p) (h) − LS,p (h)| +� +. +(37) +We bound now the second term in the right-hand side of Eq. (37). Note that, for h ∈ H, we can write LP(p) (h) = +ES′ [LS′,p (h)], where S′ = �M +m=1 S′m and S′m ∼ PNm +m +is a dataset of Nm samples drawn i.i.d. from Pm such that +Sm = +� +z(i) +m , i ∈ [Nm] +� +and S′m = +� +z′(i) +m , i ∈ [Nm] +� +. Using triangular inequality, it follows that +E +S +� +sup +h∈H +|LP(p) (h) − LS,p (h)| +� +≤ E +S,S′ +� +sup +h∈H +|LS′,p (h) − LS,p (h)| +� +(38) += E +S,S′ +� +sup +h∈H +����� +M +� +m=1 +Nm +� +i=1 +pm,i +� +ℓ(h; z(i) +m ) − ℓ(h; z′(i) +m ) +������ +� +(39) += E +S,S′ E +σ +� +sup +h∈H +����� +M +� +m=1 +Nm +� +i=1 +σ(i) +m · pm,i +� +ℓ(h; z(i) +m ) − ℓ(h; z′(i) +m ) +������ +� +, +(40) + +Federated Learning for Data Streams +where σ(i) +m , m ∈ [M], i ∈ [Nm] is a random variable drawn from uniform distribution over {±1}. Fix S and S′ and let C +be the instances appearing in S and S′, and HC be the restriction of H to C, as defined in (Shalev-Shwartz and Ben-David, +2014, Defintion 6.2). It follows that +E +S +� +sup +h∈H +|LP(p) (h) − LS,p (h)| +� +≤ +E +S′,S′ E +σ +� +sup +h∈HC +����� +M +� +m=1 +Nm +� +i=1 +σ(i) +m · pm,i +� +ℓ(h; z(i) +m ) − ℓ(h; z′(i) +m ) +������ +� +. +(41) +Fix some h ∈ HC and denote γ(i) +m += σ(i) +m · pm,i +� +ℓ(h; z(i) +m ) − ℓ(h; z′(i) +m ) +� +for m ∈ [M] and i ∈ [Nm]. We have +that E +� +γ(i) +m +� += 0 and from Assumption 1, we have that γ(i) +m +∈ [−pm,i · B, pm,i · B]. Since the random variables +� +γ(i) +m , m ∈ [M], i ∈ [Nm] +� +are independent, using Hoeffding inequality it follows that, for all ρ ≥ 0, we have +P +������ +M +� +m=1 +Nm +� +i=1 +σ(i) +m · pm,i +� +ℓ(h; z(i) +m ) − ℓ(h; z′(i) +m ) +������ ≥ ρ +� +≤ 2 exp +� +−2B2Neffρ2� +, +(42) +where Neff = +��M +m=1 +�Nm +i=1 (pm,i)2�−1 +. Applying the union bound over h ∈ HC and using (Shalev-Shwartz and Ben- +David, 2014, Lemma A.4), it follows that +E +� +sup +h∈HC +����� +M +� +m=1 +Nm +� +i=1 +σ(i) +m · pm,i +� +ℓ(h; z(i) +m ) − ℓ(h; z′(i) +m ) +������ +� +≤ 4 + +� +log (|HC|) +√2NeffB +. +(43) +It follows that, +E +� +sup +h∈HC +����� +M +� +m=1 +Nm +� +i=1 +σ(i) +m · pm,i +� +ℓ(h; z(i) +m ) − ℓ(h; z′(i) +m ) +������ +� +≤ +4 + +� +log +� +τH +� +N (T )�� +√2NeffB +, +(44) +where τH is the growth function of H as defined in (Shalev-Shwartz and Ben-David, 2014, Definition 6.9). Using Sauer’s +Lemma (Shalev-Shwartz and Ben-David, 2014, Lemma 6.10) and following the same steps as in the proof of (Marfoq +et al., 2022, Lemma A.1) we have +E +S +� +sup +h∈H +|LP(p) (h) − LS,p (h)| +� +≤ 2 +� +VCdim (H) +Neff +· +� +1 + log +� +N +VCdim (H) +� +, +(45) +where δ1 and δ2 are non-negative constants. Thus, +E +S +� +sup +h∈H +|LP(p) (h) − LS,p (h)| +� +≤ ˜O +� +� +� +VCdim (H) +Neff +� +� , +(46) +thus, +ϵgen ≤ ˜O +� +� +� +VCdim (H) +Neff +� +� + discH +� +P(α), P(p)� +. +(47) +B.4 +Proof of Lemma 4.2 +Lemma 4.2. With the same notation as in Theorem 4.1, Neff ≤ N and this bound is attained when p is uniform. +Proof. We remind that +Neff = +� M +� +m=1 +Nm +� +i=1 +(pm,i)2 +�−1 +. +(48) + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +Let u ∈ ∆N be the vector obtained by concatenating all the values pm,i for m ∈ [M] and i ∈ [Nm]. It follows that +Neff = +� N +� +n=1 +u2 +n +�−1 += ∥u∥−2 +2 +. +(49) +Let u∗ ≜ 1/N, it is clear that u∗ ∈ ∆N, and ∥u∗∥2 +2 = 1/N. Let u ∈ ∆N, using Cauchy-Shwartz inequality, we have +1 = +N +� +n=1 +un = +N +� +n=1 +(un × 1) ≤ +� +� +� +� +N +� +n=1 +u2n · +� +� +� +� +N +� +n=1 +1 = ∥u∥2 · +√ +N. +(50) +Thus, ∥u∥−2 +2 +≤ N, which concludes the proof. +B.5 +Proof of Theorem 4.3 +Theorem 4.3. Suppose that Assumptions 1–4 hold, the sequence +� +q(t)� +t is non increasing, and verifies q(1) = O (1/T), +and η ∝ 1/ +√ +T · min{1, 1/¯σ (λ)}. Under full clients participation (S(t) = [M]) with full batch (K ≥ |I(t) +m |), we have +ϵopt ≤ O +� +¯σ (λ) +� ++ O +� ¯σ (λ) +√ +T +� ++ O +� 1 +√ +T +� +, +where, +¯σ2 (λ) ≜ +T +� +t=1 +q(t) × E +S +� +sup +θ∈Θ +�����∇L(λ) +S (θ) − +M +� +m=1 +p(t) +m ∇L(λ) +M(t) +m (θ) +����� +2 � +. +Moreover, there exist a data arrival process and a loss function ℓ, such that, under FIFO memory update rule, for any +choice of weights λ, ϵopt = Ω (¯σ (λ)). +Proof. We remind that +p(t) +m = +� +j∈I(t) +m λ(t,j) +m +�M +m′=1 +� +j∈I(t) +m′ λ(t,j) +m′ +, +(51) +and +q(t) = +�M +m=1 +� +j∈I(t) +m λ(t,j) +m +�T +s=1 +�M +m=1 +� +j∈I(s) +m′ λ(s,j) +m′ +. +(52) +For ease of notation we introduce the following functions defined on Θ; +f (t) +m ≜ L(λ) +M(t) +m , +(53) +F (t) ≜ +M +� +m=1 +p(t) +m · L(λ) +M(t) +m = +M +� +m=1 +p(t) +m · f (t) +m , +(54) +F ≜ L(λ) +S += +T +� +t=1 +q(t) · F (t). +(55) +Note that this notation hides the dependence of the functions f (t) +m , F (t) and F on the samples S and the parameters λ. In +this proof we simply use E to refer to the expectation of the samples S, e.g., E [∇F(θ)] = ES +� +∇L(λ) +S +(θ) +� +. + +Federated Learning for Data Streams +We remind that +∆(t) = +M +� +m=1 +p(t) +m · +� +θ(t,E+1) +m +− θ(t)� += −η · +E +� +e=1 +M +� +m=1 +p(t) +m · ∇f (t) +m +� +θ(t,e) +m +� +. +(56) +We define ˜η ≜ ηE > 0 and ˜∇(t) ≜ − ∆(t) +˜η +∈ Rd. The coefficient ˜η and the vector ˜∇(t) can be seen as the efficient learning +rate and the pseudo-gradient used at global iteration t ∈ [T], respectively (Wang et al., 2021a, Section 2). With this set of +notation, the update rule of Algorithm 1 can be summarized as +˜∇(t) = 1 +E +E +� +e=1 +M +� +m=1 +p(t) +m · ∇f (t) +m +� +θ(t,e) +m +� +(57) +θ(t+1) = Π +Θ +� +θ(t) − ˜η · ˜∇(t)� +(58) +Under Assumptions 3–4, the functions f (t) +m , F (t), and F are bounded, convex and L-smooth as convex combinations of +bounded, convex and L-smooth functions. +Let θ∗ be a minimizer of F over Θ, and F ∗ ≜ F (θ∗) (note that θ∗ and F ∗ depend on S). By convexity of F, we have +− +� +∇F(θ), θ − θ∗� +≤ − (F(θ) − F ∗) . +(59) +Lemma B.1 and Jensen inequality imply that +max +����∇f (t,e) +m +(θ) +��� , +���∇F (t) (θ) +��� , ∥∇F (θ)∥ , +��� ˜∇(t)��� +� +≤ G, +(60) +where G ≜ +√ +2LB. +For convenience, we quantify the variance between the current and global functions’ gradients with +σt = sup +θ∈Θ +���∇F(θ) − ∇F (t) (θ) +��� . +(61) +We define σ2 (λ) ≜ �T +t=1 q(t)σ2 +t . Therefore, ¯σ2 (λ) = E +� +σ2 (λ) +� +. +The idea of the proof it to bound the distance between the pseudo-gradient ˜∇(t) and the correct gradient, ∇F +� +θ(t)� +, that +should have been used at iteration t > 0. One can write +E +����θ(t+1)−θ∗��� +2 +� += E +����Π +Θ +� +θ(t) − ˜η ˜∇ +� +− θ∗��� +2� +(62) +≤ E +����θ(t) − ˜η ˜∇ − θ∗��� +2� +(63) += E +����θ(t) − ˜η∇F +� +θ(t)� +− θ∗ + ˜η +� +∇F +� +θt� +− ˜∇(t)���� +2� +(64) += E +� ���θ(t) − ˜η∇F +� +θ(t)� +− θ∗��� +2 +� +�� +� +≜T1 +� ++ ˜η2 E +� ���∇F +� +θ(t)� +− ˜∇(t)��� +2 +� +�� +� +≜T2 +� ++ 2˜η E +� � +∇F +� +θ(t)� +− ˜∇(t), θ(t) − ˜η∇F +� +θ(t)� +− θ∗� +� +�� +� +≜T3 +� +. +(65) +Bound T1. +We have, +T1 = +���θ(t) − ˜η∇F +� +θ(t)� +− θ∗��� +2 +(66) += +���θ(t) − θ∗��� +2 ++ ˜η2 ���∇F +� +θ(t)���� +2 +− 2˜η · +� +∇F +� +θ(t)� +, θ(t) − θ∗� +(67) +≤ +���θ(t) − θ∗��� +2 ++ ˜η2G2 − 2˜η +� +F +� +θ(t)� +− F ∗� +, +(68) +where we used (59) and (60) to obtain the last inequality. + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +Bound T2. +Let α > 0, we have, +T2 = +���∇F +� +θt� +− ˜∇(t)��� +2 +(69) += +�����∇F +� +θ(t)� +− +M +� +m=1 +p(t) +m ∇f (t) +m +� +θ(t)� ++ +M +� +m=1 +p(t) +m ∇f (t) +m +� +θ(t)� +− ˜∇(t) +����� +2 +(70) +≤ (1 + α) +���∇F +� +θ(t)� +− ∇F (t) � +θ(t)���� +2 ++ (1 + α−1) +����� +M +� +m=1 +p(t) +m ∇f (t) +m +� +θ(t)� +− ˜∇(t) +����� +2 +, +(71) +where we used the fact that for any two vectors a, b ∈ Rd and a coefficient α > 0, it holds that ∥a + b∥2 ≤ (1+α) ∥a∥2 + +(1 + α−1) ∥b∥2, with the particular choice a = ∇F +� +θ(t)� +− ∇F (t) � +θ(t)� +, and b = �M +m=1 p(t) +m ∇f (t) +m +� +θ(t)� +− ˜∇(t). +We remind that, +˜∇ = −∆(t) +ηE = +E +� +e=1 +M +� +m=1 +p(t) +m +E g(t,e) +m += +E +� +e=1 +M +� +m=1 +p(t) +m +E ∇f (t) +m +� +θ(t,e) +m +� +. +(72) +Thus, +��� +M +� +m=1 +p(t) +m ∇f (t) +m +� +θ(t)� +− ˜∇(t)��� +2 += +����� +E +� +e=1 +M +� +m=1 +p(t) +m +E +� +∇f (t) +m +� +θ(t)� +− ∇f (t) +m +� +θ(t,e)������� +2 +(73) +≤ +E +� +e=1 +M +� +m=1 +p(t) +m +E +���∇f (t) +m +� +θ(t)� +− ∇f (t) +m +� +θ(t,e) +m +���� +2 +(74) += +E +� +e=1 +M +� +m=1 +p(t) +m +E +���∇f (t) +m +� +θ(t,1) +m +� +− ∇f (t) +m +� +θ(t,e) +m +���� +2 +(75) +≤ L2 +E +� +e=1 +M +� +m=1 +p(t) +m +E +���θ(t,1) +m +− θ(t,e) +m +��� +2 +(76) += L2 +E +� +e=1 +M +� +m=1 +p(t) +m +E +����� +e−1 +� +e′=1 +θ(t,e′) +m +− θ(t,e′+1) +m +����� +2 +(77) += ˜η2L2 +E3 +M +� +m=1 +p(t) +m +E +� +e=1 +����� +e−1 +� +e′=1 +∇f (t) +m +� +θ(t,e′) +m +������ +2 +(78) +≤ ˜η2L2 +E3 +M +� +m=1 +p(t) +m +E +� +e=1 +(e − 1) +e−1 +� +e′=1 +���∇f (t) +m +� +θ(t,e′) +m +���� +2 +(79) +≤ ˜η2L2G2 +E3 +E +� +e=1 +(e − 1)2 +(80) +≤ 2˜η2L2G2(1 − E−1), +(81) +where we used Jensen inequality to obtain (74) and (79), the L-smoothness of f (t) +m to obtain (76), and (60) to obtain (80). +Replacing (81) in (71) and using σt defined in (61), we have +T2 ≤ (1 + α) σ2 +t + 2 +� +1 + α−1� +˜η2L2G2(1 − E−1). +(82) +With the particular choice α = ˜ηLG +σt · +� +2 (1 − E−1), it follows that +T2 ≤ +� +σt + ˜ηLG +� +2 (1 − E−1) +�2 +≤ 2σ2 +t + 4˜η2L2G2 � +1 − E−1� +(83) +Our bound ((83)) shows that, as expected, the term T2, measuring the deviation between the true gradient ∇F +� +θ(t)� +and +the pseudo-gradient ˜∇(t), is equal to zero when E = 1 and σt = 0. This scenario corresponds exactly to the centralized +version of gradient descent. + +Federated Learning for Data Streams +Bound T3. +We have +T3 = +� +∇F +� +θ(t)� +− ˜∇(t), θ(t) − ˜η∇F +� +θ(t)� +− θ∗� +(84) += +� +∇F +� +θ(t)� +− ∇F (t) � +θ(t)� +, θ(t) − θ∗� ++ +� +∇F (t) � +θ(t)� +− ˜∇(t), θ(t) − θ∗� +− ˜η +� +∇F +� +θ(t)� +− ˜∇(t), ∇F +� +θ(t)� � +. +(85) +We remind that Θ is bounded and that D is its diameter. Using Cauchy-Schwarz inequality, we have +� +∇F (t) � +θ(t)� +− ˜∇(t), θ(t) − θ∗� +≤ +���∇F (t) � +θ(t)� +− ˜∇(t)��� · +���θ(t) − θ∗��� +(86) += +����� +M +� +m=1 +p(t) +m ∇f (t) +m +� +θ(t)� +− ˜∇(t) +����� · +���θ(t) − θ∗��� +(87) +≤ ˜ηLDG +� +2 (1 − E−1), +(88) +where we used (81) to obtain the last inequality. Using Cauchy-Shwartz inequality again and the fact that gradients are +bounded ((60)), we have +−˜η +� +∇F +� +θ(t)� +− ˜∇(t), ∇F +� +θ(t)� � +≤ ˜η +���∇F +� +θ(t)� +− ˜∇(t)��� · +���∇F +� +θ(t)���� ≤ 2˜η · G2. +(89) +Finally using Cauchy-Shwartz inequality and the boundedness of Θ, we have +� +∇F +� +θ(t)� +− ∇F (t) � +θ(t)� +, θ(t) − θ∗� +≤ σ(t) · D. +(90) +Replacing (88), (89), and (90) in (85), we have +T3 ≤ σ(t) · D + ˜ηG +� +2G + LD +� +2 (1 − E−1) +� +(91) +Bound ϵopt. +Replacing (68), (83), and (91) in (65), we have +E +����θ(t+1)−θ∗��� +2 +� += E +����θ(t) − θ∗��� +2 +� +− 2˜η · E +� +F +� +θ(t)� +− F ∗ +� ++ 2˜η · ¯σ(t)D ++ ˜η2 · +� +2¯σ2 +t + G +� +5G + 2LD +� +2 (1 − E−1) +�� ++ 4˜η4 · L2G2 � +1 − E−1� +, +(92) +where ¯σ2 +t = E +� +σ2 +t +� += E +� +supθ∈Θ +��∇F(θ) − ∇F (t) (θ) +��2� +. +The sequence +� +q(t)� +t is non increasing, i.e., for t ∈ [T] q(t+1) ≤ q(t). It follows from (92) that, for t > 0, we have +q(t+1) E +����θ(t+1)−θ∗��� +2 +� +≤ q(t) E +����θ(t+1) − θ∗��� +2 +� +(93) +≤ q(t) E +����θ(t) − θ∗��� +2 +� +− 2˜ηq(t) E +� +F +� +θ(t)� +− F ∗ +� ++ 2˜η · q(t)¯σ(t)D ++ 2˜η2 · q(t)¯σ2 +t + 2˜η2q(t) · C1 + 2˜η4q(t) · C2, +(94) +where C1 = G +� +5 +2G + LD +� +2 (1 − E−1) +� +, and C2 = 2L2G2 � +1 − E−1� +. Rearranging the terms and summing over +t ∈ {1, . . . , T}, we have +T +� +t=1 +q(t) E +� +F +� +θ(t)� +− F ∗ +� +≤ +� T +� +t=1 +q(t)¯σt +� +· D + Tq(1) · D2 +2˜ηT + ˜η · +� T +� +t=1 +q(t)¯σ2 +t +� ++ ˜η · +� +C1 + ˜η2C2 +� +(95) + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +We remind that ¯σ2 (λ) = �T +t=1 q(t)¯σ2 +t . Using the concavity of the function √·, it follows that ¯σ (λ) ≥ �T +t=1 q(t)¯σt. It +follows that +E +� +F +� +¯θ(t)� +− F ∗ +� +≤ ¯σ (λ) · D + Tq(1) · D2 +2˜ηT + ˜η · ¯σ2 (λ) + ˜ηC1 + ˜η3C2. +(96) +The final results is obtained by using O +� +Tq(1)� += 1. We have +E +� +F +� +¯θ(t)� +− F ∗ +� +≤ ¯σ (λ) · D + ¯σ (λ) +√ +T ++ C1 + C3 +√ +T ++ C2 +√ +T 3 , +(97) +where C3 is a constant proportional to D2. +Lower Bound. +In the rest of this proof, we use θ to denote the model parameters, and θ1, and θ2 its components. +We artificially construct a simple problem and a particular arrival process, such that the output of Algorithm 1, with M = 1, +C1 = 1, FIFO update rule, and η = Ω +� +1/ +√ +T +� +, verifies limT →∞ F +�¯θ(T )� +− F ∗ ≥ c · ¯σ2 (λ), where c > 0 is a constant. +We consider a setting with Θ = [−1, 1]2, Z = {1, 2}, and a loss function defined for θ ∈ Θ with +ℓ(θ; 1) ≜ (θ1 + 1)2 + 1 +2(θ1 + θ2 + 1)2, +(98) +and +ℓ (θ; 2) ≜ 1 +2 (θ1 − 1)2 + 1 +2(θ1 + θ2 − 1)2. +(99) +We observe that the minimizer of ℓ(·; 1) (resp. ℓ(·; 2)) is θ∗ +1 = (−1, 0) (resp. θ∗ +2 = (1, 0)). +For time horizon T, we consider the arrival process, where one sample, say z1, is drawn uniformly at random from Z +at time step t1 = 1, and a second sample, z2, is drawn uniformly at random from Z a time step t2 = T/2. We define +q ≜ �T/2 +t=1 q(t). Since +� +q(t)� +t≥1 is non increasing, then q ≥ 1/2. We remark that, in this setting, the trajectory of +Algorithm 1 is only determined by the values of z1 and z2, i.e., the values taken by the sequence +� +θ(t)� +t≥1 are only +determined by the values of z1 and z2. +We have +ϵopt = E +S +� +L(λ) +S +� +¯θ(T )� +− min +θ∈Θ L(λ) +S (θ) +� +(100) += 1 +2 E +S +� +L(λ) +S +� +¯θ(T )� +− min +θ∈Θ L(λ) +S (θ) +��S = {1, 2} +� ++ 1 +4 E +S +� +L(λ) +S +� +¯θ(T )� +− min +θ∈Θ L(λ) +S (θ) +��S = {1} +� +(101) ++ 1 +4 E +S +� +L(λ) +S +� +¯θ(T )� +− min +θ∈Θ L(λ) +S (θ) +��S = {2} +� +(102) +≥ 1 +2 E +S +� +L(λ) +S +� +¯θ(T )� +− min +θ∈Θ L(λ) +S (θ) +��S = {1, 2} +� +, +(103) +and +¯σ2(λ) = q (1 − q) E +S +� +max +θ∈Θ ∥∇ℓ(θ; z1) − ∇ℓ(θ; z2)∥2 +� +(104) +≤ q(1 − q) +2 +· max +θ∈Θ ∥∇ℓ(θ; 1) − ∇ℓ(θ; 2)∥2 +(105) +≤ 20 · q (1 − q) . +(106) +We consider the case when z1 = 1, and z2 = 2. Thus +L(λ) +S (θ) = q · ℓ(θ; 1) + (1 − q) · ℓ(θ; 2). +(107) +Let θ∗ be a minimizer of L(λ) +S , then +θ∗ +1 = 1 − 3q +1 + q +and +θ∗ +2 = 1 − 2q − 1 − 3q +1 + q . +(108) + +Federated Learning for Data Streams +Moreover, one can prove that +min +θ∈[−1,1] L(λ) +S +((θ, 0)) − min +θ∈Θ L(λ) +S +(θ) ≥ 6 · q(1 − q) +(109) +For ϵ > 0, it exists E ≥ 1, and T0 ≥ 1, such that for any T ≥ T0, we have +���¯θ(T ) +2 +��� ≤ ϵ. Therefore, +L(λ) +S +� +¯θ(T )� +− min +θ∈Θ L(λ) +S +(θ) ∼ϵ→0 L(λ) +S +� +(θ(T ) +1 +, 0) +� +− min +θ∈Θ L(λ) +S +(θ) +(110) +≥ +min +θ∈[−1,1] L(λ) +S +((θ, 0)) − min +θ∈Θ L(λ) +S +(θ) +(111) +≥ 6 · q(1 − q) +(112) += 3 +10 ¯σ2 (λ) +(113) +The same holds when z1 = 2, and z2 = 1. It follows that +ϵopt ≥ 3 +20 ¯σ2 (λ) . +(114) +B.6 +Bound ¯σ2(λ) +We remind, from Remark 1, that +σ2 +0 ≜ max +m +E +z∼Pm +� +sup +θ∈Θ +∥∇ℓ(θ; z) − ∇LPm (θ)∥2 +� +, +(115) +and +ζ ≜ max +m,m′ sup +θ∈Θ +��∇LPm′ (θ) − ∇LPm (θ) +�� . +(116) +Lemma B.4. For any memory update rule and any choice of memory parameters λ we have +¯σ2 (λ) = O +� +σ2 +0 + ζ2 · +T +� +t=1 +q(t) +M +� +m=1 +� +pm − p(t) +m +�2 +� +. +(117) +Proof. We remind that +¯σ2 (λ) = +T +� +t=1 +q(t) E +S +� +�sup +θ∈Θ +�����∇L(λ) +S +(θ) − +M +� +m=1 +p(t) +m ∇L(λ) +M(t) +m (θ) +����� +2� +� , +(118) +and, for m ∈ [M], we define +L(λ) +Sm (·) ≜ +�T +t=1 +� +j∈I(t) +m λ(t,j) +m +ℓ +� +·, z(j) +m +� +�T +s=1 +� +i∈I(s) +m λ(s,i) +m +, +(119) +and we remind (see Theorem 4.1) that +pm = +�T +t=1 +� +j∈I(t) +m λ(t,j) +m +�M +m′=1 +�T +s=1 +� +i∈I(s) +m λ(s,i) +m +. +(120) +L(λ) +Sm and pm represent client m’s weighted empirical risk of client m and its relative importance, respectively. We remark +that +L(λ) +S += +M +� +m=1 +pmL(λ) +Sm, +(121) + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +and +pm = +T +� +t=1 +q(t)p(t) +m . +(122) +For t ∈ [T] and θ ∈ Θ, we have +���∇L(λ) +S +(θ) − +M +� +m=1 +p(t) +m ∇L(λ) +M(t) +m (θ) +��� +2 += +���∇L(λ) +S +(θ) − +M +� +m=1 +p(t) +m ∇L(λ) +Sm (θ) + +M +� +m=1 +p(t) +m ∇L(λ) +Sm (θ) − +M +� +m=1 +p(t) +m ∇L(λ) +M(t) +m (θ) +��� +2 +(123) +≤ 2 +���∇L(λ) +S +(θ) − +M +� +m=1 +p(t) +m ∇L(λ) +Sm (θ) +��� +2 ++ 2 +��� +M +� +m=1 +p(t) +m ∇L(λ) +Sm (θ) − +M +� +m=1 +p(t) +m ∇L(λ) +M(t) +m (θ) +��� +2 +(124) += 2 +����� +M +� +m=1 +p(t) +m +� +∇L(λ) +Sm (θ) − ∇L(λ) +M(t) +m (θ) +� ����� +2 +� +�� +� +≜T1 ++2 +��� +M +� +m=1 +� +pm − p(t) +m +� +· ∇L(λ) +Sm (θ) +��� +2 +� +�� +� +≜T2 +. +(125) +Bound T1. +We have +T1 = +����� +M +� +m=1 +p(t) +m +� +∇L(λ) +Sm (θ) − ∇L(λ) +M(t) +m (θ) +� ����� +2 +(126) +≤ +M +� +m=1 +p(t) +m +���∇L(λ) +Sm (θ) − ∇L(λ) +M(t) +m (θ) +��� +2 +(127) += +M +� +m=1 +p(t) +m +���∇L(λ) +Sm (θ) − ∇LPm (θ) + ∇LPm (θ) − ∇L(λ) +M(t) +m (θ) +��� +2 +(128) +≤ 2 +M +� +m=1 +p(t) +m +���∇L(λ) +Sm (θ) − ∇LPm (θ) +��� +2 ++ 2 +M +� +m=1 +p(t) +m +���∇LPm (θ) − ∇L(λ) +M(t) +m (θ) +��� +2 +. +(129) +Bound T2. +For m′ ∈ [m], we have +T2 = +��� +M +� +m=1 +� +pm − p(t) +m +� +· ∇L(λ) +Sm (θ) +��� +2 +(130) += +��� +M +� +m=1 +� +pm − p(t) +m +� +· +� +∇L(λ) +Sm (θ) − ∇L(λ) +Sm′ (θ) +� ��� +2 +(131) +≤ +M +� +m=1 +� +pm − p(t) +m +�2 +· +M +� +m=1 +���∇L(λ) +Sm (θ) − ∇L(λ) +Sm′ (θ) +��� +2 +(132) += +M +� +m=1 +� +pm − p(t) +m +�2 +· +M +� +m=1 +���∇L(λ) +Sm (θ) − ∇LPm (θ) + ∇LPm (θ) − ∇LPm′ (θ) + ∇LPm′ (θ) − ∇L(λ) +Sm′ (θ) +��� +2 +(133) +≤ 3 +M +� +m=1 +� +pm − p(t) +m +�2 +· +� +M +� +m=1 +���∇L(λ) +Sm (θ) − ∇LPm (θ) +��� +2 ++ +���∇L(λ) +Sm′ (θ) − ∇LPm′ (θ) +��� +2 +� ++ 3 +M +� +m=1 +� +pm − p(t) +m +�2 +· +M +� +m=1 +��∇LPm (θ) − ∇LPm′ (θ) +��2 . +(134) +≤ 3 +M +� +m=1 +� +pm − p(t) +m +�2 +· +� +M +� +m=1 +���∇L(λ) +Sm (θ) − ∇LPm (θ) +��� +2 ++ +���∇L(λ) +Sm′ (θ) − ∇LPm′ (θ) +��� +2 +� + +Federated Learning for Data Streams ++ 3Mζ2 +M +� +m=1 +� +pm − p(t) +m +�2 +. +(135) +We observe that +∇L(λ) +Sm (θ) = +Nm +� +i=1 +˜pm,i∇ℓ(θ; z(i) +m ), +(136) +where, for i ∈ Nm, +˜pm,i = +�T +t=1 +� +j∈Im 1 {j = i} · λ(t,j) +m +�T +t=1 +� +j∈I(t) +m λ(t,j) +m +. +(137) +Thus, +E +S +����∇L(λ) +Sm (θ) − ∇LPm (θ) +��� +2� += E +Sm +����∇L(λ) +Sm (θ) − ∇LPm (θ) +��� +2� +(138) += E +Sm +� +� +����� +Nm +� +i=1 +˜pm,i∇ℓ(θ; z(i) +m ) − ∇LPm (θ) +����� +2� +� +(139) += E +Sm +������ +Nm +� +i=1 +˜pm,i +� +∇ℓ(θ; z(i) +m ) − ∇LPm (θ) +��� +2 +�� +(140) +≤ +Nm +� +i=1 +˜pm,i E +Sm +����∇ℓ(θ; z(i) +m ) − ∇LPm (θ) +��� +2� +(141) += +Nm +� +i=1 +˜pm,i E +z(i) +m +����∇ℓ(θ; z(i) +m ) − ∇LPm (θ) +��� +2� +(142) +≤ +Nm +� +i=1 +˜pm,iσ2 +0 +(143) += σ2 +0. +(144) +In the same way we prove that +E +S +���∇LPm (θ) − ∇L(λ) +M(t) +m (θ) +��� +2 +≤ σ2 +0. +(145) +We conclude by combining (125), (129), (135), (144), and (145). +B.7 +Proof of Theorem 4.4 +Theorem 4.4. Under the same assumptions as in Theorem 4.1 and Theorem 4.3, +ϵtrue ≤O +� 1 +√ +T +� ++ O +� +¯σ (λ) +� ++ 2discH +� +P(α), P(p)� ++ ˜O +� +� +� +VCdim (H) +Neff +� +� . +Proof. This result is an immediate implication of Theorem 4.1 and Theorem 4.3 using (9). + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +C +Case Study +C.1 +Intermittent Client Availability +In Section 5, we considered the scenario with two groups of clients: +Mhist clients with “historical” datasets, +which do not change during training, and M − Mhist clients, who collect “fresh” samples with constant rates +{bm > 0, m ∈ �Mhist + 1, M�} and only store the most recent bm samples due to memory constraints (i.e., Cm = bm). +Fresh clients can also capture the setting where clients are available during a single communication round: we would +then have Mhist “permanent” clients, which are are always available and do not change during training, and M − Mhist +“intermittent” clients, each of them available during one or a few consecutive communication rounds. +In the settings of Section 5, every client assigns the same weight to all the samples present in its memory independently +from the time; let λm be the weight assigned by client m ∈ [M] to the samples currently present in ts memory, i.e., +λ(t,j) +m += λm for every t ∈ [T] and j ∈ I(t) +m . +We remind that the total number of samples collected by client m ∈ [M] is Nm. For a fresh client, say it m > Mhist, +Nm = bmT. +C.2 +General Case +Corollary 5.1′. Consider the scenario with Mhist historical clients, and M − Mhist fresh clients. Suppose that the same +assumption of Theorem 4.4 hold, and that Algorithm 1 is used with with clients’ aggregation weights p = (pm)m∈[M] ∈ +∆M−1, then +ϵtrue ≤ (C1 + C3) +√ +T ++ C2 +√ +T 3 + +� +D + +2 +√ +T +� +σ0 +� +M − Mhist +� +� +� +� +M +� +m=Mhist+1 +p2m + 2 · max +m,m′ disc (Pm, Pm′) · ∥α − p∥1 ++ 4 · +� +1 + log +� +N +VCdim (H) +� +· +� +VCdim (H) +N +· +� +� +� +� +M +� +m=1 +p2m +nm +, +(146) +where C1, C2 and C3 are constants defined in the proof of Theorem 4.3, and σ0 is defined in Remark 1. +Proof. We remind that +pm,i = +�T +t=1 +� +j∈I(t) +m 1 {j = i} · λ(t,j) +m +�M +m′=1 +�T +t=1 +� +j∈I(t) +m′ λ(t,j) +m′ +, +i ∈ N (T ) +m , +(147) +and +p(t) +m = +� +j∈I(t) +m λ(t,j) +m +�M +m′=1 +� +j∈I(t) +m′ λ(t,j) +m′ +, +t ∈ [T]. +(148) +Replacing λ(t,j) +m += λm, we have +pm,i = +λm · �T +t=1 +� +j∈I(t) +m 1 {j = i} +�M +m′=1 λm′ �T +t=1 +���I(t) +m′ +��� +, +(149) +and, +p(t) +m = +λm +���I(t) +m +��� +�M +m′=1 λm′ +���I(t) +m′ +��� +. +(150) +In the settings of Corollary 5.1′, we have +I(t) +m = +� +{1, . . . , Nm} +, +m ∈ {1, . . . , Mhist} +{(t − 1) · bm + 1, . . . , t · bm − 1} +, +m ∈ {Mhist + 1, . . . , M} . +(151) + +Federated Learning for Data Streams +Thus, +p(t) +m = Nmλm · 1 {m ∈ �1, Mhist�} + bmλm · 1 {m ∈ �Mhist + 1, M�} +�Mhist +m′=1 Nm′λm′ + �M +m′=Mhist+1 bm′λm′ +, +(152) +and +pm,i = λmT · 1 {m ∈ �1, Mhist�} + λm · 1 {m ∈ �Mhist + 1, M�} +�M +m′=1 Nm′λm′ +. +(153) +Therefore, pm,i = pm +Nm , for every sample i ∈ [Nm]. +Bound discH +� +P(α), P(p)� +Let m′ ∈ [M], we have +discH +� +P(α), P(p)� += sup +θ∈Θ +����� +M +� +m=1 +(αm − pm) · LPm (θ) +����� +(154) += sup +θ∈Θ +����� +M +� +m=1 +(αm − pm) · +� +LPm (θ) − LPm′ (θ) +� +����� , +(155) +where the last equality follows from the fact that �M +m=1 αm = �M +m=1 pm = 1. For all m ∈ [M], we have +(αm − pm) · +� +LPm (θ) − LPm′ (θ) +� +≤ |αm − pm| · +��LPm (θ) − LPm′ (θ) +�� +(156) +≤ |αm − pm| · sup +θ∈Θ +��LPm (θ) − LPm′ (θ) +�� +(157) += |αm − pm| · discH (Pm, Pm′) +(158) +≤ |αm − pm| max +m,m′ discH (Pm, Pm′) . +(159) +Combining (155), and (159), we have +discH +� +P(α), P(p)� +≤ +M +� +m=1 +|αm − pm| · max +m,m′ discH (Pm, Pm′) +(160) += ∥α − p∥1 · max +m,m′ discH (Pm, Pm′) . +(161) +Compute N −1 +eff +We have N −1 +eff = �M +m=1 +�Nm +i=1 +� +pm +Nm +�2 += �M +m=1 +p2 +m +Nm = 1 +N +�M +m=1 +p2 +m +nm . +Bound ¯σ (λ) +We have +¯σ2 (λ) = +T +� +t=1 +q(t) E +S +� +�sup +θ∈Θ +�����∇L(λ) +S +(θ) − +M +� +m=1 +p(t) +m ∇L(λ) +M(t) +m (θ) +����� +2� +� . +(162) +In the settings of Corollary 5.1′, q(t) = 1/T, and p(t) +m = pm, thus +¯σ2 (λ) = 1 +T +T +� +t=1 +E +S +� +�sup +θ∈Θ +�����∇L(λ) +S +(θ) − +M +� +m=1 +pm∇LM(t) +m (θ) +����� +2� +� , +(163) +where LM(t) +m = � +j∈I(t) +m ℓ +� +·, z(j) +m +� +/ +���I(t) +m +���. Moreover, it is easy to check that, in this setting, +L(λ) +S += 1 +T +T +� +t=1 +M +� +m=1 +pm · LM(t) +m . +(164) +Moreover, M(t) +m = M(1) +m for m ∈ [Mhist], thus for θ ∈ Θ, +∇L(λ) +S +(θ) − +M +� +m=1 +pm∇LM(t) +m (θ) = +M +� +m=Mhist+1 +pm · 1 +T +T +� +s=1 +� +∇LM(s) +m (θ) − ∇LM(t) +m (θ) +� +. +(165) + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +It follows that, +���∇L(λ) +S +(θ) − +M +� +m=1 +pm∇LM(t) +m (θ) +��� +2 += +����� +M +� +m=Mhist+1 +pm · 1 +T +T +� +s=1 +� +∇LM(s) +m (θ) − ∇LM(t) +m (θ) +������ +2 +(166) +≤ (M − Mhist) +M +� +m=Mhist+1 +p2 +m +����� +1 +T +T +� +s=1 +� +∇LM(s) +m (θ) − ∇LM(t) +m (θ) +������ +2 +(167) +≤ (M − Mhist) +M +� +m=Mhist+1 +p2 +m +T +T +� +t=1 +���∇LM(s) +m (θ) − ∇LM(t) +m (θ) +��� +2 +. +(168) +For the fresh clients, i.e., for m > M0, we have LM(t) +m (θ) = �bm +i=1 ℓ(θ, z(t,i) +m +)/bm, thus +E +S +���∇LM(s) +m (θ) − ∇LM(t) +m (θ) +��� +2 +≤ E +S +����� +1 +bm +bm +� +i=1 +∇ℓ +� +θ; z(t,i) +m +� +− ∇ℓ +� +θ; z(s,i) +m +������ +2 +(169) +≤ 1 +bm +bm +� +i=1 +E +S +���∇ℓ +� +θ; z(t,i) +m +� +− ∇ℓ +� +θ; z(s,i) +m +���� +2 +(170) +≤ σ2 +0. +(171) +Thus, +E +S +���∇L(λ) +S +(θ) − +M +� +m=1 +pm∇LM(t) +m (θ) +��� +2 +≤ σ2 +0 (M − Mhist) · +M +� +m=1 +p2 +m +(172) +Conclusion +We conclude the proof by precising that: ˜c0 = (C1 + C3)/ +√ +T + C2/ +√ +T 3, where C1, C2, and C3 are the +constant introduced in the proof of Theorem 4.3. +The third term of (146) originates from the variability of the gradients across time as captured by ¯σ2 (λ) in (18). In +particular, it only depends on the weights of the fresh clients (as there is no gradient variability for the historical clients). +The fourth term in (146) corresponds to the discrepancy between the target distribution, P(α), and the effective distribution +P(p) in (18). As expected, it vanishes when all clients have the same distribution, and, for a given heterogeneity of +the local distributions, it is smaller the closer the target relative importance of clients and the effective one are (i.e., the +closer α and p are). Finally, the fifth term in (146), corresponds to the term ˜O +�� +VCdim (H) /Neff +� +in (18), as Neff = +N/ +��M +m=1 p2 +m/nm +� +in this setting. +C.3 +Proof of Corollary 5.1 +Corollary 5.1. Consider the scenario with Mhist historical clients, and M − Mhist fresh clients. Suppose that the same +assumptions of Theorem 4.4 hold, that α = n, and that Algorithm 1 is used with clients’ aggregation weights p = +(pm)m∈[M] ∈ ∆M−1, then +ϵtrue ≤ ψ(p; c) ≜ +c0 + c1 · +� +� +� +� +M +� +m=Mhist+1 +p2m + c2 · +� +� +� +� +M +� +m=1 +p2m +nm +, +where c = (c0, c1, c2) are non-negative constants not depending on p, given as: +c0 = (C1 + C3) + C2 +T +c1 = σ0 +� +M − Mhist · +� +D + +2 +√ +T +� + +Federated Learning for Data Streams +c2 = 4 · +� +1 + log +� +N +VCdim (H) +� +· +� +VCdim (H) +N ++ 2 · max +m,m′ disc (Pm, Pm′) +and C1, C2, and C3 are the constants defined in the proof of Theorem 4.3, and σ0 is defined in Remark 1. +Proof. We remind that Corollary 5.1′ implies that +ϵtrue ≤ (C1 + C3) +√ +T ++ C2 +√ +T 3 + +� +D + +2 +√ +T +� +σ0 +� +M − Mhist +� +� +� +� +M +� +m=Mhist+1 +p2m + 2 · max +m,m′ disc (Pm, Pm′) · ∥n − p∥1 ++ 4 · +� +1 + log +� +N +VCdim (H) +� +· +� +VCdim (H) +N +· +� +� +� +� +M +� +m=1 +p2m +nm +. +(173) +The result follows using the fact that ∥p − n∥1 ≤ +��M +m=1 p2m/nm − 1, which we prove below. +∥p − n∥1 = +M +� +m=1 +|pm − nm| +(174) += +M +� +m=1 +|pm − nm| +√nm +· √nm +(175) +≤ +� +� +� +� +M +� +m=1 +(pm − nm)2 +nm +· +M +� +m=1 +nm +(176) += +� +� +� +� +M +� +m=1 +(pm − nm)2 +nm +(177) += +� +� +� +� +M +� +m=1 +p2m +nm +− 2 +M +� +m=1 +pmnm +nm ++ +M +� +m=1 +n2m +nm +(178) += +� +� +� +� +M +� +m=1 +p2m +nm +− 1, +(179) +where we used Cauchy-Schwarz inequality to bound �M +m=1 +|pm−nm| +√nm +· √nm. +C.4 +Proof of the Convexity of ψ +We remind that for p ∈ ∆M−1, and c ∈ R3 ++, we have +ψ(p; c) = c0 +√ +T ++ c1 · +� +� +� +� +M +� +m=Mhist+1 +p2m + c2 · +� +� +� +� +M +� +m=1 +p2m +nm +. +(180) +In order to prove the convexity of p �→ +��M +m=1 +p2m +nm , and p �→ +��M +m=Mhist p2m, it is sufficient to prove that the function +ϕβ : p �→ +��M +m=1 βmp2m is convex for any vector β ∈ RM ++ . Let β ∈ RM ++ , p, ˜p ∈ ∆M, and γ ∈ [0, 1], we have +ϕ2 +β +� +γ · p + (1 − γ) · ˜p +� += +M +� +m=1 +βm · +� +γ · pm + (1 − γ) · ˜pm +�2 +(181) += γ2 · +M +� +m=1 +βmp2 +m + (1 − γ)2 · +M +� +m=1 +βm˜p2 +m + 2γ(1 − γ) · +M +� +m=1 +βmpm˜pm +(182) + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +10 +1 +100 +101 +c2/c1 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +M +m = 1(p* +m)2/nm +Nhist/N = 95.0% +Nhist/N = 70.0% +Nhist/N = 40.0% +Nhist/N = 20.0% +Nhist/N = 5.0% +10 +1 +100 +101 +c2/c1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +M +m = Mhist + 1(p* +m)2 +×10 +2 +10 +1 +100 +101 +c2/c1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p* +hist +Figure 4: From left to the right: effect of c2/c1 on the effective number of samples, the normalized gradient noise, and +the historical clients relative importance p∗ +hist for CIFAR-10 dataset (N = 5 × 105) and different values of Nhist/N, when +M = 50, and Mhist = 25. The dashed vertical line corresponds to our estimation of c2/c1 on CIFAR-10 experiments +(ˆc2/ˆc1 = 0.15). +≤ γ2 · +M +� +m=1 +βmp2 +m + (1 − γ)2 · +M +� +m=1 +βm˜p2 +m + 2γ(1 − γ) · +� +� +� +� +M +� +m=1 +βmp2m · +� +� +� +� +M +� +m=1 +βm˜p2m +(183) += +� +�γ · +� +� +� +� +M +� +m=1 +βmp2m + (1 − γ) · +� +� +� +� +M +� +m=1 +βm˜p2m +� +� +2 +(184) += (γ · ϕβ(p) + (1 − γ) · ϕβ(˜p))2 , +(185) +where we use Cauchy-Shwartz inequality to bound �M +m=1 βmpm˜pm, as follows +M +� +m=1 +βmpm˜pm = +M +� +m=1 +� +pm +� +βm +� +· +� +˜pm +� +βm˜pm +� +≤ +� +� +� +� +M +� +m=1 +βmp2m · +� +� +� +� +M +� +m=1 +βm˜p2m. +(186) +Since ϕβ is a non-negative function, we have +ϕβ +� +γ · p + (1 − γ) · p +� +≤ γ · ϕβ(p) + (1 − γ) · ϕβ(˜p), +(187) +proving that ϕβ is convex. +C.5 +Numerical Study of Bound Minimization +Figure 4 illustrates how the solution and important system quantities change as a function of the ratio c2/c1, and fraction +of historical samples Nhist/N, in the particular setting when M = 50 and Mhist = 25. Beside the specific numerical values, +one can distinguish two corner cases. When c2/c1 ≫ 1, the optimal solution corresponds to minimize �M +m=1 p2 +m/nm, +i.e., to maximize the effective number of samples, and then � +m (p∗ +m)2 /nm. The optimal aggregation vector p∗ is then the +Uniform one: each sample is assigned the same importance during the whole training and each client a relative importance +proportional to its number of samples (p∗ +m = nm). In particular, the aggregate relative importance for historical clients +is p∗ +hist = Nhist/N. On the contrary, when c2/c1 ≪ 1, the optimal solution corresponds to minimize � +m>Mhist pm, i.e., +the gradient variability. The Historical strategy is then optimal: fresh clients are ignored and historical clients receive +a relative importance proportional to the size of their local dataset (i.e., p∗ +m = Nm/Nhist = +N +Nhist nm for m ∈ [Mhist] and +p∗ +hist = 1). Figure 4 confirms these qualitative considerations, but also shows that the transition between these two regimes +depends on Nhist/N, with the transition occurring at smaller values of c2/c1 for smaller values of the Nhist/N. + +Federated Learning for Data Streams +C.6 +Details on the Estimation of the c2/c1 +Using the expression of c1 and c2 from Corollary 5.1, we have +c2 +c1 += 2 · +maxm,m′ disc (Pm, Pm′) + 2 · +� +1 + log +� +N +VCdim(H) +� +· +� +VCdim(H) +N +σ0 +√M − Mhist · +� +D + +2 +√ +T +� +. +(188) +We use the approximations +� +1 + log +� +N +VCdim (H) +� +≈ 1, +(189) +D + +2 +√ +T +≈ D, +(190) +4VCdim (H) ≈ d, +(191) +where d is the number of parameters of the model θ ∈ Θ ⊂ Rd (see Section 3). We remind the definition of σ0 from +Remark 1: +σ0 = +� +max +m +E +z∼Pm +� +sup +θ∈Θ +∥∇ℓ(θ; z) − ∇LPm (θ)∥2 +� +≤ 2 +√ +2 · LB = 2G, +(192) +where G was defined in (60). We use the approximation σ0 ≈ 2G. Finally, we remark that maxm,m′ disc (Pm, Pm′) ≤ B, +therefore, we approximate c2/c1 as +ˆc2 +ˆc1 +≈ +B + +� +d/N +GD√M − Mhist +. +(193) +In our experiments, clients cooperatively estimate ˆc2/ˆc1 using a fraction of their historical samples, with the particularity +that D is estimated as ˆD = maxM +m=1 +���ˆθ∗ +m − θ(1)���, where ˆθ∗ +m is the model obtained after few iterations of stochastic +gradient descent using a fraction of the historical data of client m ∈ [M]. + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +Table 3: Average test accuracy across clients for different datasets in the settings when Nhist/N = 50%. +DATASET +D +G +B +d +SYNTHETIC +1.9 +0.4 +0.7 +21 +CIFAR-10 +1.0 +5.5 +2.3 +3, 353, 034 +CIFAR-100 +1.0 +4.7 +4.6 +3, 537, 444 +FEMNIST +5.9 +12.9 +3.5 +867, 390 +SHAKESPEARE +2.6 +1.4 +6.1 +226, 180 +D +Details on Experimental Setup +D.1 +Datasets and Models +In this section, we provide detailed description of the datasets and models used in our experiments. We considered +five federated benchmark datasets with different machine learning tasks: image classification (CIFAR10 and CIFAR100 +(Krizhevsky, 2009)), handwritten character recognition (FEMNIST (Caldas et al., 2018)), and language modeling (Shake- +speare (Caldas et al., 2018; McMahan et al., 2017)), as well as a synthetic dataset described in Appendix D.1.1. For +Shakespeare and FEMNIST datasets there is a natural way to partition data through clients (by character and by writer, +respectively). We relied on common approaches in the literature to sample heterogeneous local datasets from CIFAR-10 +and CIFAR-100. Below, we give a detailed description of the datasets and the models / tasks considered for each of them. +D.1.1 +Synthetic Dataset +Our synthetic dataset has been generated as follows: +1. Sample θ0 ∈ Rd ∼ N(0, Id), from the multivariate normal distribution of dimension d, with zero mean and unitary +variance +2. Sample θm ∈ Rd ∼ N(θ0, ε2Id), m ∈ [M] from from the multivariate normal distribution of dimension d, centered +around θ0 and variance equal to ε2 +3. For m ∈ [M] and i ∈ [Nm], sample x(i) +m ∼ U +� +[−1, 1]d� +from a uniform distribution over [−1, 1]d +4. For m ∈ [M] and i ∈ [Nm], sample y(i) +m ∼ B +� +sigmoid +� +⟨x(i) +t , θm⟩ +�� +, where B is the standard Bernoulli distribution +D.1.2 +CIFAR-10 / CIFAR-100 +We created federated versions of CIFAR-10 by distributing samples with the same label across the clients according to a +symmetric Dirichlet distribution with parameter 0.4, as in (Wang et al., 2020). For CIFAR100, we exploited the availability +of “coarse” and “fine” labels, using a two-stage Pachinko allocation method (Li and McCallum, 2006) to distribute the +samples across the clients, as in (Reddi et al., 2021). We train a shallow convolutional neural network for CIFAR-10/100 +datasets. +D.1.3 +FEMNIST +FEMNIST (Federated Extended MNIST) is a 62-class image classification dataset built by partitioning the data of Extended +MNIST based on the writer of the digits/characters. We train two-layer fully connected neural network for FEMNIST +dataset +D.1.4 +Shakespeare +Shakespeare is a language modeling dataset built from the collective works of William Shakespeare. In this dataset, each +client corresponds to a speaking role with at least two lines. The task is next character prediction. We use an RNN that first +takes a series of characters as input and embeds each of them into a learned 8-dimensional space. The embedded characters +are then passed through 2 RNN layers, each with 256 nodes, followed by a densely connected softmax output layer. We +split the lines of each speaking role into into sequences of 80 characters, padding if necessary. + +Federated Learning for Data Streams +D.2 +Training Details. +In all experiments, the learning rate was tuned via grid search on the grid {10−3.5, 10−3, 10−2.5, 10−2, 10−1.5, 10−1} using +the validation set. Once the learning rate had been selected, we retrained the models on the concatenation of the training +and validation sets. Each experiment was repeated for three different seeds for the random number generator; we report +the mean value and the 95% confidence bound. +D.3 +Arrival Process +For CIFAR-10/100 datasets, we consider an arrival process with Mhist = 25 clients with “historical” datasets, which +do not change during training, and M − Mhist += +25 clients, who collect “fresh” samples with constant rates +{bm > 0, m ∈ �Mhist + 1, M�} and only store the most recent bm samples due to memory constraints (i.e., Cm = bm). +For a given value of Nhist/N, we split the train part of the original CIFAR-10/100 into two groups, historical and fresh, +with Nhist and N −Nhist samples, respectively. We then distribute the samples from the historical (resp. fresh) group across +Mhist historical (resp. M − Mhist fresh) clients. A symmetric Dirichlet distribution is employed in the case of CIFAR-10, +and a Pachinko allocation method is employed in the case of CIFAR-100. +Shakespeare and FEMNIST datasets have a natural partition across clients—by character and by writer, respectively. In our +experiments, we split the natural clients of FEMNIST and Shakespeare into two groups, historical and fresh, with Mhist and +M − Mhist clients, respectively. The historical clients participate to every communication round, while each fresh client is +only available in a single communication round in the case of FEMNIST and for at most two consecutive communication +rounds for Shakespeare dataset. +D.4 +Numerical Values for ˆc2/ˆc1 +Table 3 provide the values of D, G, B, and d and used for the estimation of th ratio ˆc2/ˆc1. + +Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal +Table 4: Average test accuracy across clients for different datasets in the settings when Nhist/N = 5%. +DATASET +ˆc2/ˆc1 +p∗ +HIST +TEST ACCURACY +FRESH +HISTORICAL +UNIFORM +OURS +OPTIMAL +SYNTHETIC +0.094 +0.06 +82.4 ± 1.89 +68.1 ± 2.39 +82.7 ± 1.94 +82.7 ± 1.90 +82.9 ± 2.17 +CIFAR-10 +0.150 +0.12 +59.5 ± 0.77 +48.2 ± 0.21 +60.7 ± 0.58 +61.0 ± 0.42 +63.7 ± 0.57 +CIFAR-100 +0.284 +0.08 +23.5 ± 0.65 +13.5 ± 0.41 +24.4 ± 0.54 +25.2 ± 0.66 +27.8 ± 0.39 +FEMNIST +0.001 +1.00 +55.2 ± 1.79 +65.7 ± 0.09 +58.4 ± 1.80 +65.7 ± 0.09 +65.7 ± 0.09 +SHAKESPEARE +0.064 +1.00 +40.2 ± 0.34 +49.0 ± 0.06 +41.0 ± 1.33 +49.0 ± 0.06 +49.0 ± 0.06 +Table 5: Average test accuracy across clients for different datasets in the settings when Nhist/N = 50%. +DATASET +ˆc2/ˆc1 +pHIST +TEST ACCURACY +FRESH +HISTORICAL +UNIFORM +OURS +OPTIMAL +SYNTHETIC +0.085 +0.50 +84.2 ± 1.27 +84.8 ± 1.58 +86.5 ± 1.20 +86.5 ± 1.20 +86.5 ± 1.20 +CIFAR-10 +0.150 +0.95 +52.1 ± 2.98 +64.1 ± 5.60 +65.1 ± 0.66 +68.7 ± 0.37 +69.4 ± 0.25 +CIFAR-100 +0.284 +0.69 +17.5 ± 0.57 +29.4 ± 1.40 +29.7 ± 0.55 +34.4 ± 0.31 +34.4 ± 0.31 +FEMNIST +0.001 +1.00 +48.3 ± 2.98 +66.2 ± 0.23 +57.8 ± 1.93 +66.2 ± 0.23 +66.2 ± 0.23 +SHAKESPEARE +0.095 +1.00 +30.9 ± 0.51 +44.1 ± 0.27 +41.1 ± 0.56 +44.1 ± 0.27 +44.1 ± 0.27 +E +Additional Experimental Results + +Federated Learning for Data Streams +0 +25 +50 +75 +100 +125 +150 +Time step +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +Test accuracy +phist=0.00 +p* +hist=0.05 +phist=0.20 +phist=0.50 +phist=0.80 +phist=1.00 +0 +25 +50 +75 +100 +125 +150 +Time step +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +Test accuracy +phist=0.00 +p* +hist=0.20 +phist=0.50 +phist=0.80 +phist=1.00 +0 +25 +50 +75 +100 +125 +150 +Time step +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +Test accuracy +phist=0.00 +phist=0.20 +p* +hist=0.50 +phist=0.80 +phist=1.00 +Figure 5: Evolution of the test accuracy when using different values of phist for the synthetic dataset, when Nhist/N = 5% +(left), 20% (center), and 50% (right). +0 +200 +400 +600 +800 +Time step +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Test accuracy +phist=0.00 +phist=0.05 +p* +hist=0.08 +phist=0.20 +phist=0.50 +phist=0.80 +phist=1.00 +0 +200 +400 +600 +800 +Time step +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Test accuracy +phist=0.00 +phist=0.20 +p* +hist=0.32 +phist=0.50 +phist=0.80 +phist=1.00 +0 +200 +400 +600 +800 +Time step +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Test accuracy +phist=0.00 +phist=0.20 +phist=0.50 +p* +hist=0.69 +phist=0.80 +phist=1.00 +Figure 6: Evolution of the test accuracy when using different values of phist for CIFAR-100 dataset, when Nhist/N = 5% +(left), 20% (center), and 50% (right). +0 +200 +400 +600 +800 +Time step +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Test accuracy +phist=0.00 +phist=0.20 +phist=0.50 +phist=0.80 +p* +hist=1.00 +0 +200 +400 +600 +800 +Time step +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Test accuracy +phist=0.00 +phist=0.20 +phist=0.50 +phist=0.80 +p* +hist=1.00 +0 +200 +400 +600 +800 +Time step +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Test accuracy +phist=0.00 +phist=0.20 +phist=0.50 +phist=0.80 +p* +hist=1.00 +Figure 7: Evolution of the test accuracy when using different values of phist for FEMNIST dataset, when Mhist/M = 5% +(left), 20% (center), and 50% (right). +0 +200 +400 +600 +800 +1000 +1200 +1400 +Time step +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Test accuracy +phist=0.00 +phist=0.20 +phist=0.50 +phist=0.80 +p* +hist=1.00 +0 +200 +400 +600 +800 +1000 +1200 +Time step +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Test accuracy +phist=0.00 +phist=0.20 +phist=0.50 +phist=0.80 +p* +hist=1.00 +0 +100 +200 +300 +400 +500 +Time step +0.0 +0.1 +0.2 +0.3 +0.4 +Test accuracy +phist=0.00 +phist=0.20 +phist=0.50 +phist=0.80 +p* +hist=1.00 +Figure 8: Evolution of the test accuracy when using different values of phist for Shakespeare dataset, when Mhist/M = 5% +(left), 20% (center), and 50% (right). + diff --git a/ONAzT4oBgHgl3EQfk_3j/content/tmp_files/load_file.txt b/ONAzT4oBgHgl3EQfk_3j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..60bc54638a92ecace6e81a090bf6a76ca5d53796 --- /dev/null +++ b/ONAzT4oBgHgl3EQfk_3j/content/tmp_files/load_file.txt @@ -0,0 +1,1836 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf,len=1835 +page_content='Federated Learning for Data Streams Othmane Marfoq Giovanni Neglia Laetitia Kameni Richard Vidal Inria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Universit´e Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Accenture Labs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Sophia Antipolis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' France Inria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Universit´e Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Sophia Antipolis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' France Accenture Labs Sophia Antipolis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' France Accenture Labs Sophia Antipolis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' France Abstract Federated learning (FL) is an effective solution to train ma- chine learning models on the increasing amount of data generated by IoT devices and smartphones while keep- ing such data localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Most previous work on federated learning assumes that clients operate on static datasets col- lected before training starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' This approach may be inef- ficient because 1) it ignores new samples clients collect during training, and 2) it may require a potentially long preparatory phase for clients to collect enough data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' More- over, learning on static datasets may be simply impossible in scenarios with small aggregate storage across devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It is, therefore, necessary to design federated algorithms able to learn from data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In this work, we formu- late and study the problem of federated learning for data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We propose a general FL algorithm to learn from data streams through an opportune weighted empirical risk minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our theoretical analysis provides insights to configure such an algorithm, and we evaluate its perfor- mance on a wide range of machine learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 1 Introduction Federated learning (McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2017) usually involves the minimization of an objective function, which is only available through unbiased estimates of its gradients (Bot- tou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The objective function is either the ex- pected risk, when clients can sample new data points at ev- ery iteration, or the empirical risk, when they rely on a fixed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Most previous works on federated learning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', (McMa- han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Koneˇcny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2016), focus on the second case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', the minimization of the empirical risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' They as- sume that every client collects and stores all the samples before training starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Learning on static datasets can be sub-optimal (or even impossible) in many cases, because (1) new samples collected during training are ignored, and (2) clients may have limited memory capacities, and cannot store a large number of data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For example, nodes in a sensor network may continuously collect new measure- ments, but may be able to store only a few of them in the local memory (De Francisci Morales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In this work, we formulate and study the problem of learning from separate data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We propose and theoretically analyze a general federated al- gorithm targeting this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our analysis shows a bias- optimization trade-off: by controlling the relative impor- tance of older samples in comparison to newer ones, one can speed training up at the cost of a larger bias of the learned model, or reduce the bias at the cost of a longer training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The analysis also provides insights to opti- mally configure our federated algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We demonstrate the relevance of our theoretical results through simulations spanning a wide range of machine learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In par- ticular, experiments show that “reasonable” ways to extend FedAvg McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2017) to data streams may lead to poor learned models, while our configuration rule con- sistently leads to almost-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Paper outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The rest of the paper is organized as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Section 2 provides a review of related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Sec- tion 3 formulates the problem of federated learning for data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Section 4 describes our FL algorithm for data streams and states its convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Section 5 studies a scenario of practical interest and exploits the theoretical results in Section 4 to provide configuration rules for our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Finally, we provide experimental results in Sec- tion 6 before concluding in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 2 Related Work Since its introduction in the seminal works (Koneˇcny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2017), federated learning has re- ceived increasing attention as a promising large-scale dis- tributed learning framework and has been applied to a wide range of tasks, including language modeling (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2018), automatic speech recognition (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2022), medical imaging (Courtiol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2019), and recommender systems (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our fo- cus on data streams is a key difference with respect to most of the FL literature, which assumes clients have static datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In particular, this assumption is shared by the the- oretical work studying FL algorithms’ convergence on non- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='01542v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='LG] 4 Jan 2023 Federated Learning for Data Streams iid data and under partial clients’ participation (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2019), PAC learning bounds (Mohri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2019), privacy guarantees (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2020), or resilience to Byzantine faults (Blanchard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Learning from a data stream enjoys an extensive litera- ture with applications, for example, to the financial sec- tor (Zhu and Shasha, 2002), network monitoring (Babu and Widom, 2001), and sensor networks (De Francisci Morales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In this field, we can roughly distinguish three main lines of research corresponding to different assump- tions about the data process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The first focuses on the case where samples in the data stream are drawn independently from some fixed unknown distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' this setting can be analyzed through stochastic approximation (Moulines and Bach, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The second line allows the data distribution to change over time and falls then in the context of con- tinual learning, where a model is trained on a sequence of tasks and each task can correspond to a different data dis- tribution (Thrun, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Kumar and Daum´e III, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Ru- volo and Eaton, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Finally, the third line drops any assump- tion about the data stream, which may be thought to be generated by an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' This setting can be studied in the framework of online learning with regret guaran- tees (Zinkevich, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our paper considers that data at each client is drawn from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Learning from multiple data streams with different samples’ gener- ation rates and clients’ memory sizes sets our work apart from the papers mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' There is almost no work formalizing the problem of feder- ated learning for data streams and providing a theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' To the best of our knowledge, the only exceptions are (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2020), (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021), and (Odeyomi and Zaruba, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2020) propose ASO-Fed, an asynchronous FL algorithm to minimize the empiri- cal loss computed over the aggregation of clients’ data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Their analysis requires that all clients have the same optimal model and that updates at any time t are consistent with new samples arriving in the future (more details in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' On the contrary, the theoretical analysis in our paper holds under statistical heterogene- ity across clients’ local data distributions and accounts for the bias due to the need to work with samples currently stored by clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Moreover, we provide statistical learning guarantees for our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2021) propose FedWeIT, which extends regularization-based algorithms for continual learning to the FL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The main goal of FedWeIT is to minimize interference between incompati- ble tasks while allowing positive knowledge transfer across clients during learning, but no generalization guarantee is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Odeyomi and Zaruba (2021) consider the prob- lem of online federated learning under constraints on the amount of resources consumed over the whole time hori- zon and proposes an online mirror descent-based algorithm with regret guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Differently from our contribution, both (Odeyomi and Zaruba, 2021) and (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021) assume each client can only use the most recent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our experiments show that reusing as little as 5% of the col- lected samples may be highly beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Federated learning from temporally shifting distributions (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Eichner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021) is a related, yet different, problem to learning from a data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' These papers assume the shift is due to changes in the set of available clients (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', be- cause of diurnal patterns), but clients’ local datasets do not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The only exception is (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021), which can capture a setting where clients keep collecting data dur- ing training without storage constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Theoretical results assume that new data is drawn from a client-independent distribution (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Instead, our analysis takes into account both memory constraints and statistical het- erogeneity across clients’ local data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Finally, we mention a number of papers studying different variants of “online federated learning” problems, mostly focusing on dynamic resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Many of them are discussed in the recent survey (Dai and Meng, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Among these papers, Damaskinos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2020) propose Fleet, a middleware between the edge device operating system and the machine learning application, which can be used to learn on data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The middleware is designed with the device’s energy minimization as the main concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2020) propose an online algorithm to dynam- ically select the participating clients and their number of local gradient iterations at each communication round to minimize the cumulative resource usage over time under a constraint on the quality of the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2020) study a similar problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' They include the possi- bility of discarding new data points or distributing them to clients with more resources and propose a resource allo- cation algorithm based on Lyapunov optimization (Neely, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Both Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2020) and Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2020) ignore the possibility of reusing samples across multiple commu- nication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 3 Problem Formulation In this work, we use [M] ≜ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' , M} to denote the set of positive integers up to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We consider M > 0 clients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' each of them corresponds to a potentially different learn- ing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We associate to each client m ∈ [M]: 1) a probability distribution Pm over a domain Z = X × Y, 2) a counting process N (t) m , t ≥ 0, and 3) a dynamic memory/cache M(t) m , t > 0 of capacity Cm > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' At time step t > 0, client m ∈ [M] receives a batch B(t) m = � z(t,i) m = � x(t,i) m , y(t,i) m � , i ∈ [b(t) m ] � containing b(t) m ≜ N (t) m −N (t−1) m samples drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' from Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Client m ∈ [M] can cache a sub-part of the samples in its local Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal memory, without exceeding the capacity Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Without loss of generality we suppose that 1 ≤ b(t) m ≤ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We consider a finite time horizon T > 0, and we let Nm ≜ N (T ) m and Sm ≜ �T t=1 B(t) m denote the number and the set of samples gathered by client m up to the time horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We write Sm = � z(i) m , i ∈ [Nm] � , where we arbitrarily ordered the elements of Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We define I(t) m ⊂ [Nm] to be the set of the indices of samples present at memory M(t) m , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', j ∈ I(t) m if and only if z(j) m ∈ M(t) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Finally, S ≜ �M m=1 Sm de- notes the training dataset (aggregated across clients and across time) with size N ≜ �M m=1 Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The relative size of client-m’s dataset is nm ≜ Nm/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Let HΘ = � hθ : X �→ Y, θ ∈ Θ ⊂ Rd� be a set of para- metric hypotheses/models mapping X to Y, and ℓ : Θ × Z �→ R+ be a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We define LP (θ) ≜ Ez∼P [ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z)] to be the true (expected) risk of hypothesis hθ ∈ HΘ under a generic probability distribution P over Z and we define LS (θ) = 1 |S| � (x,y)∈S ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) to be the empirical risk of model (hypothesis) hθ ∈ HΘ on a generic dataset S of samples from Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In federated learning, clients, usually, collaborate to solve minimize θ∈Θ LP(α) (θ) = M � m=1 αmLPm (θ) , (1) where P(α) ≜ �M m=1 αm · Pm and α ≜ (αm)1≤m≤M with αm ≥ 0 and ∥α∥1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Common choices for α are αm = nm and αm = 1 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The first one corresponds to minimizing the empirical loss over the aggregate train- ing dataset S = �M m=1 Sm, which gives the same impor- tance to each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The second choice instead targets per-client fairness, by giving the same importance to each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In standard federated learning, local datasets {Sm}m∈[M] are available since the beginning of the training and the fol- lowing empirical risk minimization problem is considered as a proxy for Problem 1: minimize θ∈Θ M � m=1 αm · LSm (θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2) Our goal is to design a potentially randomized algorithm A solving, in a federated fashion, Problem 1 using clients’ data streams and taking into account clients’ memory con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 4 Federated Learning Meta-Algorithm for Data Streams When learning from a data stream, every client only has ac- cess to samples currently present in its local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Due Algorithm 1: Meta Algorithm for Federated Learning from Data Streams Input : Nbr of local epochs E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' mini-batch size K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' local learning rate η > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' sample weights λ = � λ(t,j) m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' m ∈ [M], t ∈ [T], j ∈ I(t) m � Output: ¯θ(T ) = �T t=1 q(t)θ(t) 1 for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' , T do 2 Server selects a subset S(t) ⊆ [M] of clients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 3 for m ∈ S(t) (in parallel) do 4 θ(t,1) m ← θ(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 5 Sample B(t) m = {z(t,1) m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(t,b(t) m ) m } ∼ Pb(t) m m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 6 M(t) m ← Update � M(t−1) m , B(t) m � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 7 for e = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' , E do 8 Sample min � K, |I(t) m | � indices ξ(t,e) m uniformly from I(t) m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 9 g(t,e) m ← |I(t) m | |ξ(t,e) m | � j∈ξ(t,e) m λ(t,j) m � j′∈I(t) m λ(t,j′) m ∇ℓ(θ(t,e) m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(t,j) m ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 10 θ(t,e+1) m ← θ(t,e) m − η · g(t,e) m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 11 end 12 end 13 ∆(t) ← �M m=1 p(t) m · � θ(t,E+1) m − θ(t)� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 14 θ(t+1) ← ΠΘ � θ(t) + ∆(t)� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 15 end to the limited storage capacity at each client and to the vari- ability in the number of new samples arriving across time, samples may spend different amounts of time in mem- ory and then be used a different number of times dur- ing training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In order to potentially compensate for such heterogeneity, we allow samples to be weighted differ- ently over time and across clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In particular, we de- note by λ(t,j) m ≥ 0 the weight assigned at time t to sam- ple j stored in client m’s memory (then j ∈ I(t) m ), and by λ ≜ � λ(t,j) m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' m ∈ [M], t ∈ [T], j ∈ I(t) m � the set of all weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We define the weighted local objective associated to client-m’s local memory at time step t ∈ [T] as L(λ) M(t) m (θ) ≜ � j∈I(t) m λ(t,j) m ℓ � θ, z(j) m � � j∈I(t) m λ(t,j) m , (3) and similarly the global weighted empirical risk as L(λ) S (θ) ≜ �M m=1 �T t=1 � j∈I(t) m λ(t,j) m ℓ � θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(j) m � �M m=1 �T t=1 � j∈I(t) m λ(t,j) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (4) Federated Learning for Data Streams We additionally define client-m’s aggregation weight as p(t) m ≜ � j∈I(t) m λ(t,j) m �M m′=1 � j∈I(t) m′ λ(t,j) m′ , (5) and q(t) ≜ �M m=1 � j∈I(t) m λ(t,j) m �T s=1 �M m′=1 � j∈I(s) m′ λ(s,j) m′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (6) In this work we consider a meta-algorithm similar to vanilla FedAvg (McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2017) to minimize the weighted empirical risk (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Algorithm 1 operates in an iterative fashion: at time step t ∈ [T] (also called communication round), the central server broadcasts the global model θ(t) to a subset of clients (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Then every selected client, say it m, receives a new batch of data (line 5) that is used to update the client’s local memory M(t) m (line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The selected clients perform E local stochastic gradient steps (line 10), where the stochastic gradient g(t,e) m is an unbiased estimator of ∇L(λ) M(t) m � θ(t,e) m � computed using at most K samples (line 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' After E local steps, clients send back their models to the central server for aggregation (line 13, 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The update at time step t can also written as follows θ(t+1) = Π Θ � θ(t) − η · M � m=1 p(t) m E � e=1 g(t,e) m � , (7) where ΠΘ(·) denotes the projection over the set Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Note that the output of Algorithm 1 depends on the actual sample arrival sequences at clients, on the memory update rule, and on the weights λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In particular, the memory up- date rule determines which samples can be considered at a given time step and then which weights can be different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Nevertheless, for the sake of simplicity, we de- note the output simply as A(λ)(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In this paper, we restrict our analysis to the case where both the memory update rule and the weight selection rule are deterministic and do not depend on the features or the labels of the samples in the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' More for- mally, given a particular instance of the counting process N (t) m , the weights {λ(t,i) m }t∈[T ] of sample z(i) m ∈ Sm re- main unchanged if z(i) m = � x(i) m , y(i) m � is replaced by z(i) m = � ˜x(i) m , ˜y(i) m � with ˜x(i) m ̸= x(i) m or ˜y(i) m ̸= y(i) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For a given sample arrival sequence and memory update rule, the quality of the algorithm is evaluated through the true error ϵtrue ≜ EA(λ),S � LP(α) � A(λ) (S) �� − min θ∈Θ LP(α) (θ) , (8) where the expectation is taken over the potential random- ness of algorithm A(λ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', clients’ (line 2) and batches’ (line 8) sampling processes, and the samples collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 General Analysis The true error ϵtrue of our meta-algorithm in (8) can be bounded as follows (see proof in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1) ϵtrue ≤ E S,A(λ) � L(λ) S � A(λ)� S(T )�� − min θ∈Θ L(λ) S (θ) � � �� � ≜ϵopt + 2 E S � sup θ∈Θ ���LP(α)(θ) − L(λ) S (θ) ��� � � �� � ≜ϵgen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (9) The generalization error ϵgen is the expected value of the representativeness of the dataset S, which is the maxi- mal distance between the true risk LP(α) and the empirical risk L(λ) S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Intuitively, the smaller the generalization error, the better we can approach the minimum of LP(α) by min- imizing L(λ) S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The optimization error ϵopt measures how well Algo- rithm 1 approaches the minimizer of the weighted empir- ical risk L(λ) S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In the rest of this section, we first provide bounds for for the generalization error ϵgen (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1) and for the op- timization error ϵopt (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3) and and then combine them to bound the overall error ϵtrue (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our results rely on the following assumptions: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (Bounded loss) The loss function is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', ∀θ ∈ Θ, z ∈ Z, ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) ∈ [0, B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (Bounded domain) We suppose that Θ is convex, closed and bounded with diameter D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (Convexity) For all z ∈ Z, the function θ �→ ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) is convex on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (Smoothness) For all z ∈ Z, the function θ �→ ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) is L-smooth on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Assumption 1 is a standard assumption in statistical learn- ing theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', (Mohri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2018) and (Shalev-Shwartz and Ben-David, 2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Assumptions 2–4 are common as- sumptions in the analysis of (stochastic) gradient methods (see for example (Bubeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2015) and (Bottou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2018)) and online convex optimization (Hazan, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Assumptions 1 and 4 imply that (it follows from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2) σ2 0 ≜ max m E z∼Pm � sup θ∈Θ ∥∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) − ∇LPm (θ)∥2 � (10) ≤ � 2 · √ 2LB �2 , (11) and (it follows from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2) ζ ≜ max m,m′ sup θ∈Θ ��∇LPm′ (θ) − ∇LPm (θ) �� (12) ≤ 2 · √ 2LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (13) Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal These properties are similar to the stochastic gradients’ bounded variance, and the clients’ bounded dissimilarity assumptions usually employed in the analysis of federated learning algorithms (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 Bounding the Generalization Error Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 (proof in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3) quantifies the gener- alization error and in particular how the weighted empiri- cal risk L(λ) S differs from the target expected risk LP(α) for the minimizer of the first one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', it bounds |LP(α)(θ′) − L(λ) S (θ′)| for θ′ ∈ arg minθ∈Θ L(λ) S (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The bound differs from classic statistical learning results (as those in (Shalev- Shwartz and Ben-David, 2014)) because L(λ) S is a weighted empirical risk and its expected value does not necessar- ily coincide with LP(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We recall that the label dis- crepancy associated to a hypothesis class H quantifies the distance between two distributions P and P′ as follows discH (P, P′) ≜ maxh∈H |LP (h) − LP′ (h)| (Mansour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Suppose that Assumption 1 holds, when us- ing Algorithm 1 with weights λ, it follows that ϵgen ≤ discH � P(α), P(p)� + ˜O � � � VCdim (H) Neff � � , (14) where Neff = ��M m=1 �Nm i=1 p2 m,i �−1 , pm,i = �T t=1 � j∈I(t) m 1 {j = i} · λ(t,j) m �M m′=1 �T t=1 � j∈I(t) m′ λ(t,j) m′ , i ∈ Nm, (15) and p = ��Nm i=1 pm,i � 1≤m≤M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The coefficient pm,i represents the relative importance given, during the whole training period, to sample i with respect to all the samples collected by all clients and pm = �Nm i=1 pm,i represents the relative importance given to client m during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Note that pm = �T t=1 q(t)p(t) m and the p(t) m coincides with the relative importance pm, when p(t) m is constant over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In general, there is an inconsistency between the impor- tance we should give to clients (quantified by α in (1)) and the one we actually give them during training (quan- tified by p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The first term on the RHS of (14) captures the mismatch between the target distribution P(α) and the “effective distribution” P(p) = �M m=1 pmPm through the discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The second term in the RHS of (14) is similar in shape to the usual bounds observed in statistical learning the- ory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', (Shalev-Shwartz and Ben-David, 2014), which are proportional to the square root of the ratio of the VC dimension of the hypotheses class and the total number of samples N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In our case, Neff plays the role of the effective number of samples and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 (proof in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4) shows that, as expected, Neff is at most N, and reaches this value when each sample is given the same importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It holds Neff ≤ N and the bound is attained when each sample has the same relative importance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', pm,i = pm,j, for each i, j ∈ [Nm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The generalization error ϵgen decreases the closer α and p are and the larger Neff is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' When αm = nm (remember that nm = Nm/N), the choice pm,i = 1/N minimizes the bound, as it leads both to p = n = α and to Neff = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In our streaming learning setting, pm,i = 1/N can be obtained by different combinations of memory up- date rules and sample weight selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For exam- ple, this is the case when clients’ memories only con- tain the samples received during the current round (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', Update(M(t−1) m , B(t) m ) = B(t) m in line 6 of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 1) and all samples currently in the memory get weight 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', λ(t,j) m = 1 for each j ∈ I(t) m ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' But it is also the case when the memory update rule lets samples stay in mem- ory for multiple consecutive rounds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', τ (j) m rounds for sample j at client m) and samples receive a weight in- versely proportional to the number of consecutive rounds (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', λ(t,j) m = 1/τ (j) m ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In what follows, we refer to any combination of memory update rules and weight selection rules leading to pm,i = 1/N as a Uniform strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' While a Uniform strategy minimizes the bound for the generalization error ϵgen when α = n, it is in general sub- optimal in terms of the optimization error ϵopt, as we are going to show in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 Bounding the Optimization Error We provide our bound on ϵopt under full clients participa- tion (S(t) = [M]) with full batch (K ≥ |I(t) m |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Under mini-batch gradients an additional vanishing error term ap- pears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The proof is provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Suppose that Assumptions 1–4 hold, the sequence � q(t)� t is non increasing, and verifies q(1) = O (1/T), and η ∝ 1/ √ T · min{1, 1/¯σ (λ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Under full clients participation (S(t) = [M]) with full batch (K ≥ |I(t) m |), we have ϵopt ≤ O � ¯σ (λ) � + O � ¯σ (λ) √ T � + O � 1 √ T � , (16) Federated Learning for Data Streams where, ¯σ2 (λ) ≜ T � t=1 q(t)× E S � sup θ∈Θ �����∇L(λ) S (θ) − M � m=1 p(t) m ∇L(λ) M(t) m (θ) ����� 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (17) Moreover, there exist a data arrival process and a loss func- tion ℓ, such that, under FIFO memory update rule,1 for any choice of weights λ, ϵopt = Ω (¯σ (λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The coefficient ¯σ2 (λ) quantifies the variability of the gra- dient considered in the update at round t w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' the gradi- ent of the global objective L(λ) S and, as shown by Theo- rem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3, it prevents the optimization error to vanish when T diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 provides a general upper bound for ¯σ2 (λ) in terms of stochastic gradients’ variance and clients’ dissimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The optimization error ϵopt is smaller the closer ¯σ2(λ) is to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In our streaming learning setting, ¯σ2(λ) = 0 may be obtained if the memory is never updated (Update(M(t−1) m , B(t) m ) = M(t−1) m , ∀t ≥ 1) and the ag- gregation weights are constant over time (p(t) m = pm, ∀t ∈ [T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It is indeed easy to check that under these con- ditions L(λ) S (θ) = �M m=1 p(t) m L(λ) M(t) m (θ) (and they equal �M m=1 pmL(λ) M(0) m (θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Any set of time-independent sample weights leads to constant aggregation weights, but, among them, the choice λ(t,j) m = 1 reduces the generalization bound ϵgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We refer to these memory update and weight selection rules as the Historical strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The Historical strategy minimizes the optimization bound by ignoring all the samples collected during train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It is in sharp contrast with the Uniform strategy, which assigns the same relative importance to all collected samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 Main Result The tension between the two error components ϵgen and ϵopt is evident from our discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' One can minimize ϵgen by considering at each time only the most recent sam- ples, and, at the opposite, ϵopt by ignoring those samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' By combining Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 formally quantifies this trade-off and provides a bound on ϵtrue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Under the same assumptions as in Theo- 1The FIFO (First-In-First-Out) update rule evicts the oldest samples in the memory to store the most recent ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 10 1 100 101 c2/c1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 p* hist Nhist/N = 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0% Nhist/N = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0% Nhist/N = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0% Nhist/N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0% Nhist/N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0% Figure 1: Effect of c2/c1 on the historical clients rela- tive importance p∗ hist for different values of Nhist/N, when M = 50 and Mhist = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The dashed vertical line cor- responds to our estimation of c2/c1 on CIFAR-10 experi- ments (ˆc2/ˆc1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' rem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3, ϵtrue ≤O � 1 √ T � + O � ¯σ (λ) � + 2discH � P(α), P(p)� + ˜O � � � VCdim (H) Neff � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (18) 5 Case Study In fog computing environments, IoT devices, edge servers, and cloud servers can jointly participate to train an ML model (Bonomi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' IoT devices keep generating new data, but may not be able to store them permanently due to sever memory constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Instead, edge servers may contribute with larger static datasets (Hosseinalipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Motivated by this scenario, we consider two groups of clients: Mhist clients with “histor- ical” datasets, which do not change during training, and M − Mhist clients, who collect “fresh” samples with con- stant rates {bm > 0, m ∈ �Mhist + 1, M�} and only store the most recent bm samples due to memory constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', Cm = bm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 We refer to these two categories as historical clients and fresh clients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Fresh clients can also capture the setting where clients are available during a sin- gle communication round—see details in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' At each client all samples are used the same number of times (T and 1 at historical and fresh clients, respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Then, one can prove that each client, say it m, should assign the same weight to any sample currently available at its local memory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', λ(t,j) m = λ(t) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For simplicity, we consider stationary weights, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', λ(t) m = λm, and we want then to determine per-client sample weights (λm)m∈[M] leading to the best guarantees in terms of ϵtrue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 Equivalently, we want to determine the clients’ 2Note that we are implicitly selecting FIFO as memory update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 3Restricting the weights to be stationary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', λ(t) m = λm, might be suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8 Nhist/N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ( hist )/c2 log(c2/c1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 log(c2/c1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 log(c2/c1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 log(c2/c1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 log(c2/c1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8 Nhist/N 0 1 2 3 4 ( unif )/c2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 Nhist/N 4 3 2 1 0 1 2 3 ( hist unif)/c2 Figure 2: The differences ψhist − ψ∗ (left), ψuniform − ψ∗ (center), and ψhist −ψuniform (right) as a function of Nhist/N for different values of c2/c1, on CIFAR-10 dataset (N = 5 × 105) when M = 50 and Mhist = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' relative importance values p = (pm)m∈[M], where pm = λmNm/ ��M m′=1 λm′Nm′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Note that in this setting ag- gregation weights and relative importance values coincide (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', p(t) m = pm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1′ (Appendix C) bounds ϵtrue as a function of p in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For the sake of simplic- ity, we provide here the bound for the case αm = nm, m ∈ [M] (which we assume to hold in the rest of this section): Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Consider the scenario with Mhist historical clients, and M − Mhist fresh clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Suppose that the same assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 hold, that α = n, and that Algorithm 1 is used with clients’ aggregation weights p = (pm)m∈[M] ∈ ∆M−1, then ϵtrue ≤ ψ(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' c) ≜ c0 + c1 · � � � � M � m=Mhist+1 p2m + c2 · � � � � M � m=1 p2m nm , (19) where c = (c0, c1, c2) are non-negative constants not de- pending on p, given as: c0 = (C1 + C3) + C2 T − 2 · max m,m′ disc (Pm, Pm′) (20) c1 = σ0 � M − M0 · � D + 2 √ T � (21) c2 = 4 · � 1 + log � N VCdim (H) � � VCdim (H) N + 2 · max m,m′ disc (Pm, Pm′) (22) and C1, C2, and C3 are the constants defined in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3, and σ0 is defined in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The second term in (19) captures the gradient variability (second term in (18)), while the third term in (19) cap- tures both contributions to the generalization error, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', the distribution discrepancy and the effective number of sam- ples (third and fourth terms in (19)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In particular, it holds �M m=1 p2 m nm ∝ 1/Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The minimization of ψ over the unitary simplex is a con- vex optimization problem (proof in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4), which can then be solved efficiently with, for example, projected gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We use ψ∗, p∗, and p∗ hist to denote the minimum of ψ, its minimizer, and the aggregate relative importance given to historical clients (p∗ hist ≜ �Mhist m=1 p∗ m), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The solution p∗ depends on the value of n—in particu- lar on the fraction of historical samples Nhist/N (where Nhist ≜ �Mhist m=1 Nm)—and on the ratio c2/c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The ratio c2/c1 only depends on the intrinsic properties of the learn- ing problem (VCdim (H), D, B, and σ0), and the total number of samples N (see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Figure 1 illustrates how the optimal clients’ importance val- ues change as a function of the ratio c2/c1 and the fraction of historical samples Nhist/N (other results are in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Beside the specific numerical values, one can distinguish two corner cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' When c2/c1 ≫ 1, the optimal solution corresponds to minimize �M m=1 p2 m/nm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', to maximize the effective number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The optimal strategy is then the Uniform one and the aggregate relative impor- tance for historical clients is p∗ hist = Nhist/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' On the con- trary, when c2/c1 ≪ 1, the optimal solution corresponds to minimize � m>Mhist p2 m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', the gradient variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The Historical strategy is then optimal and corresponds to p∗ m = Nm/Nhist = N Nhist nm for m ∈ [Mhist] and p∗ hist = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For general values of c2/c1, the optimal strategy to assign clients’ importance values—or equivalently sam- ple weights—differs from both the Uniform and the Historical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We propose then the following heuristic, which we evaluate in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' At the beginning of training, clients cooperatively estimate c2/c1 using a fraction of their historical samples, as ˆc2/ˆc1 ≈ B+√ d/N GD√M−Mhist (see details in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Then, clients’ importance values are selected minimizing the bound in (19), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', ˆp∗ = arg min ψ (·, ˆc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Beside providing configuration rules for our meta- algorithm, our analysis allows us also to evaluate how the performances of different strategies like Uniform and Historical depend on the different parameters as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our experimental results in the next section con- firm these theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 6 Experimental Results Datasets and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We considered different ma- chine learning tasks on five federated benchmark datasets: image classification (CIFAR-10 and CIFAR-100 (Krizhevsky, 2009)), handwritten character recognition (FEMNIST (Caldas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2018)), language modeling (Shakespeare (Caldas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2017)), and logistic regression on a synthetic dataset described in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Table 1 summarizes datasets, models, and the total number of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Details on the datasets, models, and hyperparameters selection Federated Learning for Data Streams Table 1: Datasets and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' DATASET CLIENTS TOTAL SAMPLES MODEL SYNTHETIC 11 200 LINEAR MODEL CIFAR-10 / 100 50 50, 000 2 CNN + 2 FC FEMNIST 3, 597 817, 851 2 FC SHAKESPEARE 916 3, 436, 096 STACKED-LSTM are provided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='com/omarfoq/streaming-fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Arrival process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For the synthetic dataset and CIFAR- 10/100 we adopted common strategies to split the datasets across clients and divided clients into two groups as in Sec- tion 5 with Mhist = 10 and Mhist = 25, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For FEMNIST and Shakespeare datasets, we adopted their nat- ural partitions and set Mhist such that Mhist/M = 5%, 20%, and 50%, but allowed fresh clients to participate to train- ing for a few rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Experimental results for these two datasets suggest that our analysis is robust to departures from the setting considered in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Details are in Ap- pendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We compared our strategy to select clients’ importance values, (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 5), with three baselines: the Uniform and Historical strategies described above as well as the Fresh strategy which only considers fresh clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We observe that under our samples’ arrival pro- cess and α = n, there could be two natural ways to extend the classic FedAvg’s aggregation rule (McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2017): set each client’s aggregation weight proportional to (1) the number of samples collected by the client over the whole time-horizon, or (2) the number of samples currently in the client’s memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The first aggregation rule coincides with the Uniform strategy, the second one leads in all set- tings we considered to very small aggregation weights for fresh clients so that it is practically indistinguishable from the Historical strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Interestingly, both these rules are in general suboptimal, motivating the practical interest of our study and of the strategy we propose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Main Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Table 2 reports the test accuracy when Nhist/N = 20% for the different strategies together with the optimal test accuracy obtained selecting the value of phist = �Mhist m=1 pm in the grid {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our observations are confirmed for other values of Nhist/N (see Table 4 and Table 5 in Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' A first remark is that working only with new data (as Fresh does) is never optimal, not even when historical data account for just 5% of the total dataset (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Second, neither of the two “reasonable” ways to extend FedAvg consistently achieves good accuracy: Historical performs poorly over Synthetic and Uniform over FEMNIST and Shake- speare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' On the contrary, our method always performs at least as well as the best baseline and it often achieves a test accuracy similar to the (estimated) optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In particular, it correctly sets weights as Uniform over Syn- thetic and as Historical over FEMNIST and Shake- speare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We observe that our analysis also helps to ex- plain the counter-intuitive conclusion that, on FEMNIST and Shakespeare, it is beneficial to ignore new collected samples (even for Nhist/N = 5%, see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our strat- egy correctly sets ˆp∗ hist = 1, because it estimates that, for these two datasets, the ratio of the number of parameters to the aggregate training dataset size (d/N) is much smaller than the gradients’ norm (G)—numerical values are pro- vided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' This information suggests that we can use a small subset of the original dataset to identify a good model in the selected hypotheses class, and in particu- lar we can rely only on historical data avoiding the potential noise introduced by new samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Figure 3 shows the effect of p on CIFAR-10 test accuracy for different values of the ratio Nhist/N—similar figures for other datasets are provided in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It confirms that performances in terms of final test accuracy match the pre- dictions of our model on the bound ψ illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' First, Figure 3 shows that the performance gap between Historical and the optimal assignment p∗ decreases when Nhist/N increases (as predicted in Figure 2 (left)): the gap is 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='30, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='17, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8 pp when Nhist/N is 5%, 20%, and 50%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Second, Figure 3 confirms that the performance gap between Uniform and the optimal assignment first increases and then decreases, when Nhist/N increases (as in Figure 2 (center)): the gap is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='57, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='55, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='35 pp when Nhist/N is 5%, 20%, and 50%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Finally, Figure 3 shows that the relative ranking of Uniform and Historical changes, with Uniform being a better option for smaller values of Nhist/N and Historical becoming slightly better for larger values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Again, this behavior is predicted by our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Indeed, in this experiment, our estimation for the ratio c2/c1 is ˆc2/ˆc1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='15 ∈ [10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3, 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5] cor- responding to a setting for which ψhist − ψunif changes sign in Figure 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 7 Conclusion In this paper, we formalized the problem of federated learn- ing for data streams and highlighted a new source of hetero- geneity resulting from local datasets’ variability over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We proposed a general federated algorithm to learn in this setting and studied its theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Our analy- sis reveals a new bias-optimization trade-off controlled by the relative importance of older samples in comparison to newer ones and leads to practical guidelines to configure such importance in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Experiments show that our configuration rule outperforms natural ways to extend the usual FedAvg aggregation rule in the presence of data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Moreover, experimental results confirm other the- oretical conclusions, despite the theoretical assumptions and the mismatch in the corresponding performance met- Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal Table 2: Average test accuracy across clients for different datasets in the settings when Nhist/N = 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' DATASET ˆc2/ˆc1 ˆp∗ HIST TEST ACCURACY FRESH HISTORICAL UNIFORM OURS OPTIMAL SYNTHETIC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='20 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='44 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='15 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='60 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='60 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='60 CIFAR-10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='45 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='94 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='16 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='63 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='81 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='91 CIFAR-100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='32 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='57 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='50 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='43 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='57 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='25 FEMNIST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='85 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='20 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='80 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='20 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='80 SHAKESPEARE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='064 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='43 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='26 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='38 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='26 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='26 0 200 400 600 800 Time step 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 Test accuracy phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='05 p* hist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='12 phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='20 phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='50 phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='80 phist=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 0 200 400 600 800 Time step 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 Test accuracy phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='20 p* hist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='45 phist=0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 Test accuracy phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='20 phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='50 phist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='80 p* hist=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='95 phist=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 Figure 3: Evolution of the test accuracy when using different values of phist for CIFAR-10 (left) dataset, when Nhist/N = 5% (left), 20% (center), and 50% (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The setting phist = Nhist/N corresponds to Uniform strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' rics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', test accuracy versus a loss bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' To the best of our knowledge, this work is the first to frame the problem of federated learning for data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It high- lights new challenges and—we believe—lays the founda- tions for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For example, part of our results are restricted to the important, but still quite specific, sce- nario where some clients have static datasets and others process new samples at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In this setting, samples are used a different number of times across clients but ex- actly the same number of times at a given client, simpli- fying the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' But what happens if heterogeneity in samples’ availability also appears at the level of a single client?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' How do different memory update rules affect such heterogeneity, and how can we design such policies to min- imize the total error of the final model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Finally, how do our results change if local data distributions change over time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Acknowledgments This research was supported in part by the Groupe La Poste, sponsor of the Inria Foundation, in the framework of the FedMalin Inria Challenge, and in part by the French government, through the 3IA Cˆote d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA- 0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The authors are grateful to the OPAL infrastructure from Universit´e Cˆote d’Azur for providing computational resources and technical support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' References Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Fed- erated learning with matched averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In In- ternational Conference on Learning Representations, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' URL https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' id=BkluqlSFDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Wei Li and Andrew McCallum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Pachinko allocation: Dag- structured mixture models of topic correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In Proceedings of the 23rd International Conference on Machine Learning, ICML ’06, page 577–584, New York, NY, USA, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Association for Computing Ma- chinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' ISBN 1595933832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1145/1143844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 1143917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1145/ 1143844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1143917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Sashank J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Reddi, Zachary Charles, Manzil Zaheer, Zachary Garrett, Keith Rush, Jakub Koneˇcn´y, Sanjiv Kumar, and Hugh Brendan McMahan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Adaptive fed- erated optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In International Conference on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' URL https:// openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='id=LkFG3lB13U5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal A Related Work In this section we provide more details about some related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2020) propose ASO-Fed, an asynchronous FL algorithm to minimize the empirical loss computed over the aggregation of clients’ data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Although some convergence results are stated in the paper, their interest and applicability are questionable, as the analysis requires that all clients have the same optimal model and that updates at any time t are consistent with new samples arriving in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Indeed, the paper mentions that clients can receive new samples during training (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 2), but also requires that, at any time t and for any client k, the expected value of the update ∇ζk(w) has a non-null component in the direction of the gradient of the global empirical loss F, which depends on samples arriving after time t (see Assumption 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Moreover, the bounded gradient dissimilarity assumption implies that the minimizer of F (F is assumed to be strongly-convex) is also a stationary point of each local objective function fk (consider β = 0 and λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' On the contrary, the theoretical analysis in our paper holds under statistical heterogeneity across clients’ local data distributions and accounts for the bias due to working with samples currently stored at clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Moreover, we provide statistical learning guarantees for our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The model considered in (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021) can capture a setting where clients keep collecting data during training without storage constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Indeed, clients track the dynamic objective in (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (2)) which depends on data samples received until the current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Theoretical results assume that new data is drawn from a client-independent distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' This is shown by (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (5)), which requires that local gradients computed on new data samples are unbiased estimators of the gradient of the global objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Instead, our analysis takes into account both memory constraints and statistical heterogeneity across clients’ local data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Federated Learning for Data Streams B Proofs We remind that all our results rely on the following assumptions: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (Bounded loss) The loss function is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', ∀θ ∈ Θ, z ∈ Z, ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) ∈ [0, B] Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (Bounded domain) We suppose that Θ is convex, closed and bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' we use D to denote its diameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', ∀θ, θ′ ∈ Θ, ∥θ − θ′∥ ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (Convexity) For all z ∈ Z, the function θ �→ ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) is convex on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (Smoothness) For all z ∈ Z, the function θ �→ ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) is L-smooth on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In what follows, we use ∆D−1 to denote the unitary simplex of dimension D − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', ∆D−1 = � f ∈ RD +, �D i=1 fi = 1 � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 Proof of (9) ϵtrue = E S,A(λ) � LP(α) � A(λ) (S) � − L(λ) S � A(λ) (S) �� + E S,A(λ) � L(λ) S � A(λ) (S) � − min θ∈Θ L(λ) S (θ) � + E S � min θ∈Θ L(λ) S (θ) � − min θ∈Θ LP(α) (θ) (23) ≤ 2 E S � sup θ∈Θ ���LP(α) (θ) − L(λ) S (θ) ��� � � �� � ≜ϵgen + E S,A(λ) � L(λ) S � A(λ) (S) � − min θ∈Θ L(λ) S (θ) � � �� � ≜ϵopt , (24) where we exploited the fact that minx∈X f(x) − minx∈X g(x) ≤ supx∈X |f(x) − g(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 Properties Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Let f be an L-smooth function taking values in [0, B], then ∥∇f∥ ≤ √ 2LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Let θ ∈ Θ, then using the definition of the L-smoothness of f with θ′ = θ − 1 L∇f (θ), we have f(θ′) = f(θ − 1 L∇f (θ)) ≤ f (θ) − 1 L⟨∇f (θ) , ∇f (θ)⟩ + L 2 ���� 1 L∇f (θ) ���� 2 (25) = f (θ) − 1 2L ∥∇f (θ)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (26) If follows that, ∥∇f (θ)∥2 ≤ 2L (f (θ) − f (θ′)) ≤ 2LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (27) Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Suppose that Assumptions 1, and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For all sup θ∈Θ ∥∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) − ∇LPm (θ)∥2 ≤ � 2 √ 2LB �2 (28) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Let z ∈ Z, and m ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Both ℓ (·, z), and LPmare L-smooth and bounded within [0, B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For θ ∈ Θ, we have ∥∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) − ∇LPm (θ)∥2 ≤ 2 ∥∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z)∥2 + 2 ∥∇LPm (θ)∥2 (29) ≤ 2 · 2LB + 2 · 2LB (30) = 8LB = � 2 √ 2LB �2 , (31) where we used Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 to obtain the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Suppose that Assumptions 1, and 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For all z ∈ Z, we have max m,m′ sup θ∈Θ ��∇LPm′ (θ) − ∇LPm (θ) �� ≤ 2 √ 2LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (32) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The proof follows using the triangular inequality and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Suppose that Assumption 1 holds, when using Algorithm 1 with weights λ, it follows that ϵgen ≤ discH � P(α), P(p)� + ˜O � � � VCdim (H) Neff � � , where Neff = ��M m=1 �Nm i=1 p2 m,i �−1 , pm,i = �T t=1 � j∈I(t) m 1 {j = i} · λ(t,j) m �M m′=1 �T t=1 � j∈I(t) m′ λ(t,j) m′ , i ∈ Nm, and p = ��Nm i=1 pm,i � 1≤m≤M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For client, m ∈ [M], we remind that pm ≜ �Nm i=1 pm,i is the relative importance of client m in comparison to the other clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We define LS,p = M � m=1 Nm � i=1 pm,i · ℓ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (33) Note that LS,p = L(λ) S , and ES [LS,p (θ)] = � m pmLPm (θ) = LP (p) (θ) for any θ ∈ Θ, where P(p) = � m pmPm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We have ϵgen = E S � sup h∈H |LP(α) (h) − LS,p (h)| � (34) = E S � sup h∈H |LP(α) (h) − LP(p) (h) + LP(p) (h) − LS,p (h)| � (35) ≤ E S � sup h∈H |LP(α) (h) − LP(p) (h)| � + E S � sup h∈H |LP(p) (h) − LS,p (h)| � (36) ≤ discH � P(α), P(p)� + E S � sup h∈H |LP(p) (h) − LS,p (h)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (37) We bound now the second term in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Note that, for h ∈ H, we can write LP(p) (h) = ES′ [LS′,p (h)], where S′ = �M m=1 S′m and S′m ∼ PNm m is a dataset of Nm samples drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' from Pm such that Sm = � z(i) m , i ∈ [Nm] � and S′m = � z′(i) m , i ∈ [Nm] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Using triangular inequality, it follows that E S � sup h∈H |LP(p) (h) − LS,p (h)| � ≤ E S,S′ � sup h∈H |LS′,p (h) − LS,p (h)| � (38) = E S,S′ � sup h∈H ����� M � m=1 Nm � i=1 pm,i � ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z′(i) m ) ������ � (39) = E S,S′ E σ � sup h∈H ����� M � m=1 Nm � i=1 σ(i) m · pm,i � ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z′(i) m ) ������ � , (40) Federated Learning for Data Streams where σ(i) m , m ∈ [M], i ∈ [Nm] is a random variable drawn from uniform distribution over {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Fix S and S′ and let C be the instances appearing in S and S′, and HC be the restriction of H to C, as defined in (Shalev-Shwartz and Ben-David, 2014, Defintion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It follows that E S � sup h∈H |LP(p) (h) − LS,p (h)| � ≤ E S′,S′ E σ � sup h∈HC ����� M � m=1 Nm � i=1 σ(i) m · pm,i � ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z′(i) m ) ������ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (41) Fix some h ∈ HC and denote γ(i) m = σ(i) m · pm,i � ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z′(i) m ) � for m ∈ [M] and i ∈ [Nm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We have that E � γ(i) m � = 0 and from Assumption 1, we have that γ(i) m ∈ [−pm,i · B, pm,i · B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Since the random variables � γ(i) m , m ∈ [M], i ∈ [Nm] � are independent, using Hoeffding inequality it follows that, for all ρ ≥ 0, we have P ������ M � m=1 Nm � i=1 σ(i) m · pm,i � ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z′(i) m ) ������ ≥ ρ � ≤ 2 exp � −2B2Neffρ2� , (42) where Neff = ��M m=1 �Nm i=1 (pm,i)2�−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Applying the union bound over h ∈ HC and using (Shalev-Shwartz and Ben- David, 2014, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4), it follows that E � sup h∈HC ����� M � m=1 Nm � i=1 σ(i) m · pm,i � ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z′(i) m ) ������ � ≤ 4 + � log (|HC|) √2NeffB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (43) It follows that, E � sup h∈HC ����� M � m=1 Nm � i=1 σ(i) m · pm,i � ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ℓ(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z′(i) m ) ������ � ≤ 4 + � log � τH � N (T )�� √2NeffB , (44) where τH is the growth function of H as defined in (Shalev-Shwartz and Ben-David, 2014, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Using Sauer’s Lemma (Shalev-Shwartz and Ben-David, 2014, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='10) and following the same steps as in the proof of (Marfoq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2022, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1) we have E S � sup h∈H |LP(p) (h) − LS,p (h)| � ≤ 2 � VCdim (H) Neff � 1 + log � N VCdim (H) � , (45) where δ1 and δ2 are non-negative constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Thus, E S � sup h∈H |LP(p) (h) − LS,p (h)| � ≤ ˜O � � � VCdim (H) Neff � � , (46) thus, ϵgen ≤ ˜O � � � VCdim (H) Neff � � + discH � P(α), P(p)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (47) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' With the same notation as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1, Neff ≤ N and this bound is attained when p is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We remind that Neff = � M � m=1 Nm � i=1 (pm,i)2 �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (48) Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal Let u ∈ ∆N be the vector obtained by concatenating all the values pm,i for m ∈ [M] and i ∈ [Nm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It follows that Neff = � N � n=1 u2 n �−1 = ∥u∥−2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (49) Let u∗ ≜ 1/N, it is clear that u∗ ∈ ∆N, and ∥u∗∥2 2 = 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Let u ∈ ∆N, using Cauchy-Shwartz inequality, we have 1 = N � n=1 un = N � n=1 (un × 1) ≤ � � � � N � n=1 u2n · � � � � N � n=1 1 = ∥u∥2 · √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (50) Thus, ∥u∥−2 2 ≤ N, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Suppose that Assumptions 1–4 hold, the sequence � q(t)� t is non increasing, and verifies q(1) = O (1/T), and η ∝ 1/ √ T · min{1, 1/¯σ (λ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Under full clients participation (S(t) = [M]) with full batch (K ≥ |I(t) m |), we have ϵopt ≤ O � ¯σ (λ) � + O � ¯σ (λ) √ T � + O � 1 √ T � , where, ¯σ2 (λ) ≜ T � t=1 q(t) × E S � sup θ∈Θ �����∇L(λ) S (θ) − M � m=1 p(t) m ∇L(λ) M(t) m (θ) ����� 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Moreover, there exist a data arrival process and a loss function ℓ, such that, under FIFO memory update rule, for any choice of weights λ, ϵopt = Ω (¯σ (λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We remind that p(t) m = � j∈I(t) m λ(t,j) m �M m′=1 � j∈I(t) m′ λ(t,j) m′ , (51) and q(t) = �M m=1 � j∈I(t) m λ(t,j) m �T s=1 �M m=1 � j∈I(s) m′ λ(s,j) m′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (52) For ease of notation we introduce the following functions defined on Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' f (t) m ≜ L(λ) M(t) m , (53) F (t) ≜ M � m=1 p(t) m · L(λ) M(t) m = M � m=1 p(t) m · f (t) m , (54) F ≜ L(λ) S = T � t=1 q(t) · F (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (55) Note that this notation hides the dependence of the functions f (t) m , F (t) and F on the samples S and the parameters λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In this proof we simply use E to refer to the expectation of the samples S, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', E [∇F(θ)] = ES � ∇L(λ) S (θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Federated Learning for Data Streams We remind that ∆(t) = M � m=1 p(t) m · � θ(t,E+1) m − θ(t)� = −η · E � e=1 M � m=1 p(t) m · ∇f (t) m � θ(t,e) m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (56) We define ˜η ≜ ηE > 0 and ˜∇(t) ≜ − ∆(t) ˜η ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The coefficient ˜η and the vector ˜∇(t) can be seen as the efficient learning rate and the pseudo-gradient used at global iteration t ∈ [T], respectively (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021a, Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' With this set of notation, the update rule of Algorithm 1 can be summarized as ˜∇(t) = 1 E E � e=1 M � m=1 p(t) m · ∇f (t) m � θ(t,e) m � (57) θ(t+1) = Π Θ � θ(t) − ˜η · ˜∇(t)� (58) Under Assumptions 3–4, the functions f (t) m , F (t), and F are bounded, convex and L-smooth as convex combinations of bounded, convex and L-smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Let θ∗ be a minimizer of F over Θ, and F ∗ ≜ F (θ∗) (note that θ∗ and F ∗ depend on S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' By convexity of F, we have − � ∇F(θ), θ − θ∗� ≤ − (F(θ) − F ∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (59) Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 and Jensen inequality imply that max ����∇f (t,e) m (θ) ��� , ���∇F (t) (θ) ��� , ∥∇F (θ)∥ , ��� ˜∇(t)��� � ≤ G, (60) where G ≜ √ 2LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For convenience, we quantify the variance between the current and global functions’ gradients with σt = sup θ∈Θ ���∇F(θ) − ∇F (t) (θ) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (61) We define σ2 (λ) ≜ �T t=1 q(t)σ2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Therefore, ¯σ2 (λ) = E � σ2 (λ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The idea of the proof it to bound the distance between the pseudo-gradient ˜∇(t) and the correct gradient, ∇F � θ(t)� , that should have been used at iteration t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' One can write E ����θ(t+1)−θ∗��� 2 � = E ����Π Θ � θ(t) − ˜η ˜∇ � − θ∗��� 2� (62) ≤ E ����θ(t) − ˜η ˜∇ − θ∗��� 2� (63) = E ����θ(t) − ˜η∇F � θ(t)� − θ∗ + ˜η � ∇F � θt� − ˜∇(t)���� 2� (64) = E � ���θ(t) − ˜η∇F � θ(t)� − θ∗��� 2 � �� � ≜T1 � + ˜η2 E � ���∇F � θ(t)� − ˜∇(t)��� 2 � �� � ≜T2 � + 2˜η E � � ∇F � θ(t)� − ˜∇(t), θ(t) − ˜η∇F � θ(t)� − θ∗� � �� � ≜T3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (65) Bound T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We have, T1 = ���θ(t) − ˜η∇F � θ(t)� − θ∗��� 2 (66) = ���θ(t) − θ∗��� 2 + ˜η2 ���∇F � θ(t)���� 2 − 2˜η · � ∇F � θ(t)� , θ(t) − θ∗� (67) ≤ ���θ(t) − θ∗��� 2 + ˜η2G2 − 2˜η � F � θ(t)� − F ∗� , (68) where we used (59) and (60) to obtain the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal Bound T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Let α > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' T2 = ���∇F � θt� − ˜∇(t)��� 2 (69) = �����∇F � θ(t)� − M � m=1 p(t) m ∇f (t) m � θ(t)� + M � m=1 p(t) m ∇f (t) m � θ(t)� − ˜∇(t) ����� 2 (70) ≤ (1 + α) ���∇F � θ(t)� − ∇F (t) � θ(t)���� 2 + (1 + α−1) ����� M � m=1 p(t) m ∇f (t) m � θ(t)� − ˜∇(t) ����� 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (71) where we used the fact that for any two vectors a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' b ∈ Rd and a coefficient α > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' it holds that ∥a + b∥2 ≤ (1+α) ∥a∥2 + (1 + α−1) ∥b∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' with the particular choice a = ∇F � θ(t)� − ∇F (t) � θ(t)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' and b = �M m=1 p(t) m ∇f (t) m � θ(t)� − ˜∇(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We remind that, ˜∇ = −∆(t) ηE = E � e=1 M � m=1 p(t) m E g(t,e) m = E � e=1 M � m=1 p(t) m E ∇f (t) m � θ(t,e) m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (72) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' ��� M � m=1 p(t) m ∇f (t) m � θ(t)� − ˜∇(t)��� 2 = ����� E � e=1 M � m=1 p(t) m E � ∇f (t) m � θ(t)� − ∇f (t) m � θ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e)������� 2 (73) ≤ E � e=1 M � m=1 p(t) m E ���∇f (t) m � θ(t)� − ∇f (t) m � θ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e) m ���� 2 (74) = E � e=1 M � m=1 p(t) m E ���∇f (t) m � θ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1) m � − ∇f (t) m � θ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e) m ���� 2 (75) ≤ L2 E � e=1 M � m=1 p(t) m E ���θ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1) m − θ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e) m ��� 2 (76) = L2 E � e=1 M � m=1 p(t) m E ����� e−1 � e′=1 θ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e′) m − θ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e′+1) m ����� 2 (77) = ˜η2L2 E3 M � m=1 p(t) m E � e=1 ����� e−1 � e′=1 ∇f (t) m � θ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e′) m ������ 2 (78) ≤ ˜η2L2 E3 M � m=1 p(t) m E � e=1 (e − 1) e−1 � e′=1 ���∇f (t) m � θ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e′) m ���� 2 (79) ≤ ˜η2L2G2 E3 E � e=1 (e − 1)2 (80) ≤ 2˜η2L2G2(1 − E−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (81) where we used Jensen inequality to obtain (74) and (79),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' the L-smoothness of f (t) m to obtain (76),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' and (60) to obtain (80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Replacing (81) in (71) and using σt defined in (61), we have T2 ≤ (1 + α) σ2 t + 2 � 1 + α−1� ˜η2L2G2(1 − E−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (82) With the particular choice α = ˜ηLG σt · � 2 (1 − E−1), it follows that T2 ≤ � σt + ˜ηLG � 2 (1 − E−1) �2 ≤ 2σ2 t + 4˜η2L2G2 � 1 − E−1� (83) Our bound ((83)) shows that, as expected, the term T2, measuring the deviation between the true gradient ∇F � θ(t)� and the pseudo-gradient ˜∇(t), is equal to zero when E = 1 and σt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' This scenario corresponds exactly to the centralized version of gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Federated Learning for Data Streams Bound T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We have T3 = � ∇F � θ(t)� − ˜∇(t), θ(t) − ˜η∇F � θ(t)� − θ∗� (84) = � ∇F � θ(t)� − ∇F (t) � θ(t)� , θ(t) − θ∗� + � ∇F (t) � θ(t)� − ˜∇(t), θ(t) − θ∗� − ˜η � ∇F � θ(t)� − ˜∇(t), ∇F � θ(t)� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (85) We remind that Θ is bounded and that D is its diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Using Cauchy-Schwarz inequality, we have � ∇F (t) � θ(t)� − ˜∇(t), θ(t) − θ∗� ≤ ���∇F (t) � θ(t)� − ˜∇(t)��� · ���θ(t) − θ∗��� (86) = ����� M � m=1 p(t) m ∇f (t) m � θ(t)� − ˜∇(t) ����� · ���θ(t) − θ∗��� (87) ≤ ˜ηLDG � 2 (1 − E−1), (88) where we used (81) to obtain the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Using Cauchy-Shwartz inequality again and the fact that gradients are bounded ((60)), we have −˜η � ∇F � θ(t)� − ˜∇(t), ∇F � θ(t)� � ≤ ˜η ���∇F � θ(t)� − ˜∇(t)��� · ���∇F � θ(t)���� ≤ 2˜η · G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (89) Finally using Cauchy-Shwartz inequality and the boundedness of Θ, we have � ∇F � θ(t)� − ∇F (t) � θ(t)� , θ(t) − θ∗� ≤ σ(t) · D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (90) Replacing (88), (89), and (90) in (85), we have T3 ≤ σ(t) · D + ˜ηG � 2G + LD � 2 (1 − E−1) � (91) Bound ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Replacing (68), (83), and (91) in (65), we have E ����θ(t+1)−θ∗��� 2 � = E ����θ(t) − θ∗��� 2 � − 2˜η · E � F � θ(t)� − F ∗ � + 2˜η · ¯σ(t)D + ˜η2 · � 2¯σ2 t + G � 5G + 2LD � 2 (1 − E−1) �� + 4˜η4 · L2G2 � 1 − E−1� , (92) where ¯σ2 t = E � σ2 t � = E � supθ∈Θ ��∇F(θ) − ∇F (t) (θ) ��2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The sequence � q(t)� t is non increasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', for t ∈ [T] q(t+1) ≤ q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It follows from (92) that, for t > 0, we have q(t+1) E ����θ(t+1)−θ∗��� 2 � ≤ q(t) E ����θ(t+1) − θ∗��� 2 � (93) ≤ q(t) E ����θ(t) − θ∗��� 2 � − 2˜ηq(t) E � F � θ(t)� − F ∗ � + 2˜η · q(t)¯σ(t)D + 2˜η2 · q(t)¯σ2 t + 2˜η2q(t) · C1 + 2˜η4q(t) · C2, (94) where C1 = G � 5 2G + LD � 2 (1 − E−1) � , and C2 = 2L2G2 � 1 − E−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Rearranging the terms and summing over t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' , T}, we have T � t=1 q(t) E � F � θ(t)� − F ∗ � ≤ � T � t=1 q(t)¯σt � D + Tq(1) · D2 2˜ηT + ˜η · � T � t=1 q(t)¯σ2 t � + ˜η · � C1 + ˜η2C2 � (95) Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal We remind that ¯σ2 (λ) = �T t=1 q(t)¯σ2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Using the concavity of the function √·, it follows that ¯σ (λ) ≥ �T t=1 q(t)¯σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It follows that E � F � ¯θ(t)� − F ∗ � ≤ ¯σ (λ) · D + Tq(1) · D2 2˜ηT + ˜η · ¯σ2 (λ) + ˜ηC1 + ˜η3C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (96) The final results is obtained by using O � Tq(1)� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We have E � F � ¯θ(t)� − F ∗ � ≤ ¯σ (λ) · D + ¯σ (λ) √ T + C1 + C3 √ T + C2 √ T 3 , (97) where C3 is a constant proportional to D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Lower Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In the rest of this proof, we use θ to denote the model parameters, and θ1, and θ2 its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We artificially construct a simple problem and a particular arrival process, such that the output of Algorithm 1, with M = 1, C1 = 1, FIFO update rule, and η = Ω � 1/ √ T � , verifies limT →∞ F �¯θ(T )� − F ∗ ≥ c · ¯σ2 (λ), where c > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We consider a setting with Θ = [−1, 1]2, Z = {1, 2}, and a loss function defined for θ ∈ Θ with ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 1) ≜ (θ1 + 1)2 + 1 2(θ1 + θ2 + 1)2, (98) and ℓ (θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 2) ≜ 1 2 (θ1 − 1)2 + 1 2(θ1 + θ2 − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (99) We observe that the minimizer of ℓ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' ℓ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 2)) is θ∗ 1 = (−1, 0) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' θ∗ 2 = (1, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For time horizon T, we consider the arrival process, where one sample, say z1, is drawn uniformly at random from Z at time step t1 = 1, and a second sample, z2, is drawn uniformly at random from Z a time step t2 = T/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We define q ≜ �T/2 t=1 q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Since � q(t)� t≥1 is non increasing, then q ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We remark that, in this setting, the trajectory of Algorithm 1 is only determined by the values of z1 and z2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', the values taken by the sequence � θ(t)� t≥1 are only determined by the values of z1 and z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We have ϵopt = E S � L(λ) S � ¯θ(T )� − min θ∈Θ L(λ) S (θ) � (100) = 1 2 E S � L(λ) S � ¯θ(T )� − min θ∈Θ L(λ) S (θ) ��S = {1, 2} � + 1 4 E S � L(λ) S � ¯θ(T )� − min θ∈Θ L(λ) S (θ) ��S = {1} � (101) + 1 4 E S � L(λ) S � ¯θ(T )� − min θ∈Θ L(λ) S (θ) ��S = {2} � (102) ≥ 1 2 E S � L(λ) S � ¯θ(T )� − min θ∈Θ L(λ) S (θ) ��S = {1, 2} � , (103) and ¯σ2(λ) = q (1 − q) E S � max θ∈Θ ∥∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z1) − ∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z2)∥2 � (104) ≤ q(1 − q) 2 max θ∈Θ ∥∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 1) − ∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 2)∥2 (105) ≤ 20 · q (1 − q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (106) We consider the case when z1 = 1, and z2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Thus L(λ) S (θ) = q · ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 1) + (1 − q) · ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (107) Let θ∗ be a minimizer of L(λ) S , then θ∗ 1 = 1 − 3q 1 + q and θ∗ 2 = 1 − 2q − 1 − 3q 1 + q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (108) Federated Learning for Data Streams Moreover, one can prove that min θ∈[−1,1] L(λ) S ((θ, 0)) − min θ∈Θ L(λ) S (θ) ≥ 6 · q(1 − q) (109) For ϵ > 0, it exists E ≥ 1, and T0 ≥ 1, such that for any T ≥ T0, we have ���¯θ(T ) 2 ��� ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Therefore, L(λ) S � ¯θ(T )� − min θ∈Θ L(λ) S (θ) ∼ϵ→0 L(λ) S � (θ(T ) 1 , 0) � − min θ∈Θ L(λ) S (θ) (110) ≥ min θ∈[−1,1] L(λ) S ((θ, 0)) − min θ∈Θ L(λ) S (θ) (111) ≥ 6 · q(1 − q) (112) = 3 10 ¯σ2 (λ) (113) The same holds when z1 = 2, and z2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' It follows that ϵopt ≥ 3 20 ¯σ2 (λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (114) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 Bound ¯σ2(λ) We remind, from Remark 1, that σ2 0 ≜ max m E z∼Pm � sup θ∈Θ ∥∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) − ∇LPm (θ)∥2 � , (115) and ζ ≜ max m,m′ sup θ∈Θ ��∇LPm′ (θ) − ∇LPm (θ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (116) Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For any memory update rule and any choice of memory parameters λ we have ¯σ2 (λ) = O � σ2 0 + ζ2 · T � t=1 q(t) M � m=1 � pm − p(t) m �2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (117) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We remind that ¯σ2 (λ) = T � t=1 q(t) E S � �sup θ∈Θ �����∇L(λ) S (θ) − M � m=1 p(t) m ∇L(λ) M(t) m (θ) ����� 2� � , (118) and, for m ∈ [M], we define L(λ) Sm (·) ≜ �T t=1 � j∈I(t) m λ(t,j) m ℓ � , z(j) m � �T s=1 � i∈I(s) m λ(s,i) m , (119) and we remind (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1) that pm = �T t=1 � j∈I(t) m λ(t,j) m �M m′=1 �T s=1 � i∈I(s) m λ(s,i) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (120) L(λ) Sm and pm represent client m’s weighted empirical risk of client m and its relative importance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We remark that L(λ) S = M � m=1 pmL(λ) Sm, (121) Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal and pm = T � t=1 q(t)p(t) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (122) For t ∈ [T] and θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='���∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='(θ) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='p(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='= ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm (θ) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='p(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ∇L(λ) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='(θ) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='p(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm (θ) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='(124) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='= 2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm (θ) − ∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='≜T1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='pm − p(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='≜T2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (125) Bound T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We have T1 = ����� M � m=1 p(t) m � ∇L(λ) Sm (θ) − ∇L(λ) M(t) m (θ) � ����� 2 (126) ≤ M � m=1 p(t) m ���∇L(λ) Sm (θ) − ∇L(λ) M(t) m (θ) ��� 2 (127) = M � m=1 p(t) m ���∇L(λ) Sm (θ) − ∇LPm (θ) + ∇LPm (θ) − ∇L(λ) M(t) m (θ) ��� 2 (128) ≤ 2 M � m=1 p(t) m ���∇L(λ) Sm (θ) − ∇LPm (θ) ��� 2 + 2 M � m=1 p(t) m ���∇LPm (θ) − ∇L(λ) M(t) m (θ) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (129) Bound T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For m′ ∈ [m],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='T2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='pm − p(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='(130) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='pm − p(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm (θ) − ∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm′ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='(131) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='pm − p(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='�2 M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='���∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm (θ) − ∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm′ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='(132) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='pm − p(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='�2 M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='���∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm (θ) − ∇LPm (θ) + ∇LPm (θ) − ∇LPm′ (θ) + ∇LPm′ (θ) − ∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm′ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='(133) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='≤ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='pm − p(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='�2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='���∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm (θ) − ∇LPm (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='���∇L(λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='Sm′ (θ) − ∇LPm′ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='+ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='pm − p(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='�2 M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='m=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��∇LPm (θ) − ∇LPm′ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (134) ≤ 3 M � m=1 � pm − p(t) m �2 � M � m=1 ���∇L(λ) Sm (θ) − ∇LPm (θ) ��� 2 + ���∇L(λ) Sm′ (θ) − ∇LPm′ (θ) ��� 2 � Federated Learning for Data Streams + 3Mζ2 M � m=1 � pm − p(t) m �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (135) We observe that ∇L(λ) Sm (θ) = Nm � i=1 ˜pm,i∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ), (136) where, for i ∈ Nm, ˜pm,i = �T t=1 � j∈Im 1 {j = i} · λ(t,j) m �T t=1 � j∈I(t) m λ(t,j) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (137) Thus, E S ����∇L(λ) Sm (θ) − ∇LPm (θ) ��� 2� = E Sm ����∇L(λ) Sm (θ) − ∇LPm (θ) ��� 2� (138) = E Sm � � ����� Nm � i=1 ˜pm,i∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ∇LPm (θ) ����� 2� � (139) = E Sm ������ Nm � i=1 ˜pm,i � ∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ∇LPm (θ) ��� 2 �� (140) ≤ Nm � i=1 ˜pm,i E Sm ����∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ∇LPm (θ) ��� 2� (141) = Nm � i=1 ˜pm,i E z(i) m ����∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(i) m ) − ∇LPm (θ) ��� 2� (142) ≤ Nm � i=1 ˜pm,iσ2 0 (143) = σ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (144) In the same way we prove that E S ���∇LPm (θ) − ∇L(λ) M(t) m (θ) ��� 2 ≤ σ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (145) We conclude by combining (125), (129), (135), (144), and (145).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Under the same assumptions as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3, ϵtrue ≤O � 1 √ T � + O � ¯σ (λ) � + 2discH � P(α), P(p)� + ˜O � � � VCdim (H) Neff � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' This result is an immediate implication of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 using (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal C Case Study C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 Intermittent Client Availability In Section 5, we considered the scenario with two groups of clients: Mhist clients with “historical” datasets, which do not change during training, and M − Mhist clients, who collect “fresh” samples with constant rates {bm > 0, m ∈ �Mhist + 1, M�} and only store the most recent bm samples due to memory constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', Cm = bm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Fresh clients can also capture the setting where clients are available during a single communication round: we would then have Mhist “permanent” clients, which are are always available and do not change during training, and M − Mhist “intermittent” clients, each of them available during one or a few consecutive communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In the settings of Section 5, every client assigns the same weight to all the samples present in its memory independently from the time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' let λm be the weight assigned by client m ∈ [M] to the samples currently present in ts memory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', λ(t,j) m = λm for every t ∈ [T] and j ∈ I(t) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We remind that the total number of samples collected by client m ∈ [M] is Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For a fresh client, say it m > Mhist, Nm = bmT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 General Case Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Consider the scenario with Mhist historical clients, and M − Mhist fresh clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Suppose that the same assumption of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 hold, and that Algorithm 1 is used with with clients’ aggregation weights p = (pm)m∈[M] ∈ ∆M−1, then ϵtrue ≤ (C1 + C3) √ T + C2 √ T 3 + � D + 2 √ T � σ0 � M − Mhist � � � � M � m=Mhist+1 p2m + 2 · max m,m′ disc (Pm, Pm′) · ∥α − p∥1 + 4 · � 1 + log � N VCdim (H) � � VCdim (H) N � � � � M � m=1 p2m nm , (146) where C1, C2 and C3 are constants defined in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3, and σ0 is defined in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We remind that pm,i = �T t=1 � j∈I(t) m 1 {j = i} · λ(t,j) m �M m′=1 �T t=1 � j∈I(t) m′ λ(t,j) m′ , i ∈ N (T ) m , (147) and p(t) m = � j∈I(t) m λ(t,j) m �M m′=1 � j∈I(t) m′ λ(t,j) m′ , t ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (148) Replacing λ(t,j) m = λm, we have pm,i = λm · �T t=1 � j∈I(t) m 1 {j = i} �M m′=1 λm′ �T t=1 ���I(t) m′ ��� , (149) and, p(t) m = λm ���I(t) m ��� �M m′=1 λm′ ���I(t) m′ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (150) In the settings of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1′, we have I(t) m = � {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' , Nm} , m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' , Mhist} {(t − 1) · bm + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' , t · bm − 1} , m ∈ {Mhist + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' , M} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (151) Federated Learning for Data Streams Thus, p(t) m = Nmλm · 1 {m ∈ �1, Mhist�} + bmλm · 1 {m ∈ �Mhist + 1, M�} �Mhist m′=1 Nm′λm′ + �M m′=Mhist+1 bm′λm′ , (152) and pm,i = λmT · 1 {m ∈ �1, Mhist�} + λm · 1 {m ∈ �Mhist + 1, M�} �M m′=1 Nm′λm′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (153) Therefore, pm,i = pm Nm , for every sample i ∈ [Nm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Bound discH � P(α), P(p)� Let m′ ∈ [M], we have discH � P(α), P(p)� = sup θ∈Θ ����� M � m=1 (αm − pm) · LPm (θ) ����� (154) = sup θ∈Θ ����� M � m=1 (αm − pm) · � LPm (θ) − LPm′ (θ) � ����� , (155) where the last equality follows from the fact that �M m=1 αm = �M m=1 pm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For all m ∈ [M], we have (αm − pm) · � LPm (θ) − LPm′ (θ) � ≤ |αm − pm| · ��LPm (θ) − LPm′ (θ) �� (156) ≤ |αm − pm| · sup θ∈Θ ��LPm (θ) − LPm′ (θ) �� (157) = |αm − pm| · discH (Pm, Pm′) (158) ≤ |αm − pm| max m,m′ discH (Pm, Pm′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (159) Combining (155), and (159), we have discH � P(α), P(p)� ≤ M � m=1 |αm − pm| · max m,m′ discH (Pm, Pm′) (160) = ∥α − p∥1 · max m,m′ discH (Pm, Pm′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (161) Compute N −1 eff We have N −1 eff = �M m=1 �Nm i=1 � pm Nm �2 = �M m=1 p2 m Nm = 1 N �M m=1 p2 m nm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Bound ¯σ (λ) We have ¯σ2 (λ) = T � t=1 q(t) E S � �sup θ∈Θ �����∇L(λ) S (θ) − M � m=1 p(t) m ∇L(λ) M(t) m (θ) ����� 2� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (162) In the settings of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1′, q(t) = 1/T, and p(t) m = pm, thus ¯σ2 (λ) = 1 T T � t=1 E S � �sup θ∈Θ �����∇L(λ) S (θ) − M � m=1 pm∇LM(t) m (θ) ����� 2� � , (163) where LM(t) m = � j∈I(t) m ℓ � , z(j) m � / ���I(t) m ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Moreover, it is easy to check that, in this setting, L(λ) S = 1 T T � t=1 M � m=1 pm · LM(t) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (164) Moreover, M(t) m = M(1) m for m ∈ [Mhist], thus for θ ∈ Θ, ∇L(λ) S (θ) − M � m=1 pm∇LM(t) m (θ) = M � m=Mhist+1 pm · 1 T T � s=1 � ∇LM(s) m (θ) − ∇LM(t) m (θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (165) Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal It follows that, ���∇L(λ) S (θ) − M � m=1 pm∇LM(t) m (θ) ��� 2 = ����� M � m=Mhist+1 pm · 1 T T � s=1 � ∇LM(s) m (θ) − ∇LM(t) m (θ) ������ 2 (166) ≤ (M − Mhist) M � m=Mhist+1 p2 m ����� 1 T T � s=1 � ∇LM(s) m (θ) − ∇LM(t) m (θ) ������ 2 (167) ≤ (M − Mhist) M � m=Mhist+1 p2 m T T � t=1 ���∇LM(s) m (θ) − ∇LM(t) m (θ) ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (168) For the fresh clients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', for m > M0, we have LM(t) m (θ) = �bm i=1 ℓ(θ, z(t,i) m )/bm, thus E S ���∇LM(s) m (θ) − ∇LM(t) m (θ) ��� 2 ≤ E S ����� 1 bm bm � i=1 ∇ℓ � θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(t,i) m � − ∇ℓ � θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(s,i) m ������ 2 (169) ≤ 1 bm bm � i=1 E S ���∇ℓ � θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(t,i) m � − ∇ℓ � θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z(s,i) m ���� 2 (170) ≤ σ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (171) Thus, E S ���∇L(λ) S (θ) − M � m=1 pm∇LM(t) m (θ) ��� 2 ≤ σ2 0 (M − Mhist) · M � m=1 p2 m (172) Conclusion We conclude the proof by precising that: ˜c0 = (C1 + C3)/ √ T + C2/ √ T 3, where C1, C2, and C3 are the constant introduced in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The third term of (146) originates from the variability of the gradients across time as captured by ¯σ2 (λ) in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In particular, it only depends on the weights of the fresh clients (as there is no gradient variability for the historical clients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The fourth term in (146) corresponds to the discrepancy between the target distribution, P(α), and the effective distribution P(p) in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' As expected, it vanishes when all clients have the same distribution, and, for a given heterogeneity of the local distributions, it is smaller the closer the target relative importance of clients and the effective one are (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', the closer α and p are).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Finally, the fifth term in (146), corresponds to the term ˜O �� VCdim (H) /Neff � in (18), as Neff = N/ ��M m=1 p2 m/nm � in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Consider the scenario with Mhist historical clients, and M − Mhist fresh clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Suppose that the same assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 hold, that α = n, and that Algorithm 1 is used with clients’ aggregation weights p = (pm)m∈[M] ∈ ∆M−1, then ϵtrue ≤ ψ(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' c) ≜ c0 + c1 · � � � � M � m=Mhist+1 p2m + c2 · � � � � M � m=1 p2m nm , where c = (c0, c1, c2) are non-negative constants not depending on p, given as: c0 = (C1 + C3) + C2 T c1 = σ0 � M − Mhist · � D + 2 √ T � Federated Learning for Data Streams c2 = 4 · � 1 + log � N VCdim (H) � � VCdim (H) N + 2 · max m,m′ disc (Pm, Pm′) and C1, C2, and C3 are the constants defined in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3, and σ0 is defined in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We remind that Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1′ implies that ϵtrue ≤ (C1 + C3) √ T + C2 √ T 3 + � D + 2 √ T � σ0 � M − Mhist � � � � M � m=Mhist+1 p2m + 2 · max m,m′ disc (Pm, Pm′) · ∥n − p∥1 + 4 · � 1 + log � N VCdim (H) � � VCdim (H) N � � � � M � m=1 p2m nm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (173) The result follows using the fact that ∥p − n∥1 ≤ ��M m=1 p2m/nm − 1, which we prove below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' ∥p − n∥1 = M � m=1 |pm − nm| (174) = M � m=1 |pm − nm| √nm √nm (175) ≤ � � � � M � m=1 (pm − nm)2 nm M � m=1 nm (176) = � � � � M � m=1 (pm − nm)2 nm (177) = � � � � M � m=1 p2m nm − 2 M � m=1 pmnm nm + M � m=1 n2m nm (178) = � � � � M � m=1 p2m nm − 1, (179) where we used Cauchy-Schwarz inequality to bound �M m=1 |pm−nm| √nm √nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 Proof of the Convexity of ψ We remind that for p ∈ ∆M−1, and c ∈ R3 +, we have ψ(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' c) = c0 √ T + c1 · � � � � M � m=Mhist+1 p2m + c2 · � � � � M � m=1 p2m nm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (180) In order to prove the convexity of p �→ ��M m=1 p2m nm , and p �→ ��M m=Mhist p2m, it is sufficient to prove that the function ϕβ : p �→ ��M m=1 βmp2m is convex for any vector β ∈ RM + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Let β ∈ RM + , p, ˜p ∈ ∆M, and γ ∈ [0, 1], we have ϕ2 β � γ · p + (1 − γ) · ˜p � = M � m=1 βm · � γ · pm + (1 − γ) · ˜pm �2 (181) = γ2 · M � m=1 βmp2 m + (1 − γ)2 · M � m=1 βm˜p2 m + 2γ(1 − γ) · M � m=1 βmpm˜pm (182) Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal 10 1 100 101 c2/c1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 M m = 1(p* m)2/nm Nhist/N = 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0% Nhist/N = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0% Nhist/N = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0% Nhist/N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0% Nhist/N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0% 10 1 100 101 c2/c1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 M m = Mhist + 1(p* m)2 ×10 2 10 1 100 101 c2/c1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 p* hist Figure 4: From left to the right: effect of c2/c1 on the effective number of samples, the normalized gradient noise, and the historical clients relative importance p∗ hist for CIFAR-10 dataset (N = 5 × 105) and different values of Nhist/N, when M = 50, and Mhist = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The dashed vertical line corresponds to our estimation of c2/c1 on CIFAR-10 experiments (ˆc2/ˆc1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' ≤ γ2 · M � m=1 βmp2 m + (1 − γ)2 · M � m=1 βm˜p2 m + 2γ(1 − γ) · � � � � M � m=1 βmp2m · � � � � M � m=1 βm˜p2m (183) = � �γ · � � � � M � m=1 βmp2m + (1 − γ) · � � � � M � m=1 βm˜p2m � � 2 (184) = (γ · ϕβ(p) + (1 − γ) · ϕβ(˜p))2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (185) where we use Cauchy-Shwartz inequality to bound �M m=1 βmpm˜pm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' as follows M � m=1 βmpm˜pm = M � m=1 � pm � βm � � ˜pm � βm˜pm � ≤ � � � � M � m=1 βmp2m · � � � � M � m=1 βm˜p2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (186) Since ϕβ is a non-negative function, we have ϕβ � γ · p + (1 − γ) · p � ≤ γ · ϕβ(p) + (1 − γ) · ϕβ(˜p), (187) proving that ϕβ is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 Numerical Study of Bound Minimization Figure 4 illustrates how the solution and important system quantities change as a function of the ratio c2/c1, and fraction of historical samples Nhist/N, in the particular setting when M = 50 and Mhist = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Beside the specific numerical values, one can distinguish two corner cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' When c2/c1 ≫ 1, the optimal solution corresponds to minimize �M m=1 p2 m/nm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', to maximize the effective number of samples, and then � m (p∗ m)2 /nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The optimal aggregation vector p∗ is then the Uniform one: each sample is assigned the same importance during the whole training and each client a relative importance proportional to its number of samples (p∗ m = nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In particular, the aggregate relative importance for historical clients is p∗ hist = Nhist/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' On the contrary, when c2/c1 ≪ 1, the optimal solution corresponds to minimize � m>Mhist pm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', the gradient variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The Historical strategy is then optimal: fresh clients are ignored and historical clients receive a relative importance proportional to the size of their local dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', p∗ m = Nm/Nhist = N Nhist nm for m ∈ [Mhist] and p∗ hist = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Figure 4 confirms these qualitative considerations, but also shows that the transition between these two regimes depends on Nhist/N, with the transition occurring at smaller values of c2/c1 for smaller values of the Nhist/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Federated Learning for Data Streams C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 Details on the Estimation of the c2/c1 Using the expression of c1 and c2 from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1, we have c2 c1 = 2 · maxm,m′ disc (Pm, Pm′) + 2 · � 1 + log � N VCdim(H) � � VCdim(H) N σ0 √M − Mhist · � D + 2 √ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (188) We use the approximations � 1 + log � N VCdim (H) � ≈ 1, (189) D + 2 √ T ≈ D, (190) 4VCdim (H) ≈ d, (191) where d is the number of parameters of the model θ ∈ Θ ⊂ Rd (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We remind the definition of σ0 from Remark 1: σ0 = � max m E z∼Pm � sup θ∈Θ ∥∇ℓ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' z) − ∇LPm (θ)∥2 � ≤ 2 √ 2 · LB = 2G, (192) where G was defined in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We use the approximation σ0 ≈ 2G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Finally, we remark that maxm,m′ disc (Pm, Pm′) ≤ B, therefore, we approximate c2/c1 as ˆc2 ˆc1 ≈ B + � d/N GD√M − Mhist .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' (193) In our experiments, clients cooperatively estimate ˆc2/ˆc1 using a fraction of their historical samples, with the particularity that D is estimated as ˆD = maxM m=1 ���ˆθ∗ m − θ(1)���, where ˆθ∗ m is the model obtained after few iterations of stochastic gradient descent using a fraction of the historical data of client m ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal Table 3: Average test accuracy across clients for different datasets in the settings when Nhist/N = 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' DATASET D G B d SYNTHETIC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 21 CIFAR-10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 3, 353, 034 CIFAR-100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 3, 537, 444 FEMNIST 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 867, 390 SHAKESPEARE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 226, 180 D Details on Experimental Setup D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 Datasets and Models In this section, we provide detailed description of the datasets and models used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We considered five federated benchmark datasets with different machine learning tasks: image classification (CIFAR10 and CIFAR100 (Krizhevsky, 2009)), handwritten character recognition (FEMNIST (Caldas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2018)), and language modeling (Shake- speare (Caldas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' McMahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2017)), as well as a synthetic dataset described in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For Shakespeare and FEMNIST datasets there is a natural way to partition data through clients (by character and by writer, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We relied on common approaches in the literature to sample heterogeneous local datasets from CIFAR-10 and CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Below, we give a detailed description of the datasets and the models / tasks considered for each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 Synthetic Dataset Our synthetic dataset has been generated as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Sample θ0 ∈ Rd ∼ N(0, Id), from the multivariate normal distribution of dimension d, with zero mean and unitary variance 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Sample θm ∈ Rd ∼ N(θ0, ε2Id), m ∈ [M] from from the multivariate normal distribution of dimension d, centered around θ0 and variance equal to ε2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For m ∈ [M] and i ∈ [Nm], sample x(i) m ∼ U � [−1, 1]d� from a uniform distribution over [−1, 1]d 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For m ∈ [M] and i ∈ [Nm], sample y(i) m ∼ B � sigmoid � ⟨x(i) t , θm⟩ �� , where B is the standard Bernoulli distribution D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 CIFAR-10 / CIFAR-100 We created federated versions of CIFAR-10 by distributing samples with the same label across the clients according to a symmetric Dirichlet distribution with parameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4, as in (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For CIFAR100, we exploited the availability of “coarse” and “fine” labels, using a two-stage Pachinko allocation method (Li and McCallum, 2006) to distribute the samples across the clients, as in (Reddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We train a shallow convolutional neural network for CIFAR-10/100 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 FEMNIST FEMNIST (Federated Extended MNIST) is a 62-class image classification dataset built by partitioning the data of Extended MNIST based on the writer of the digits/characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We train two-layer fully connected neural network for FEMNIST dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 Shakespeare Shakespeare is a language modeling dataset built from the collective works of William Shakespeare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In this dataset, each client corresponds to a speaking role with at least two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The task is next character prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We use an RNN that first takes a series of characters as input and embeds each of them into a learned 8-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The embedded characters are then passed through 2 RNN layers, each with 256 nodes, followed by a densely connected softmax output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We split the lines of each speaking role into into sequences of 80 characters, padding if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Federated Learning for Data Streams D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 Training Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In all experiments, the learning rate was tuned via grid search on the grid {10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5, 10−3, 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5, 10−2, 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5, 10−1} using the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Once the learning rate had been selected, we retrained the models on the concatenation of the training and validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Each experiment was repeated for three different seeds for the random number generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' we report the mean value and the 95% confidence bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 Arrival Process For CIFAR-10/100 datasets, we consider an arrival process with Mhist = 25 clients with “historical” datasets, which do not change during training, and M − Mhist = 25 clients, who collect “fresh” samples with constant rates {bm > 0, m ∈ �Mhist + 1, M�} and only store the most recent bm samples due to memory constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=', Cm = bm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' For a given value of Nhist/N, we split the train part of the original CIFAR-10/100 into two groups, historical and fresh, with Nhist and N −Nhist samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' We then distribute the samples from the historical (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' fresh) group across Mhist historical (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' M − Mhist fresh) clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' A symmetric Dirichlet distribution is employed in the case of CIFAR-10, and a Pachinko allocation method is employed in the case of CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Shakespeare and FEMNIST datasets have a natural partition across clients—by character and by writer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' In our experiments, we split the natural clients of FEMNIST and Shakespeare into two groups, historical and fresh, with Mhist and M − Mhist clients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' The historical clients participate to every communication round, while each fresh client is only available in a single communication round in the case of FEMNIST and for at most two consecutive communication rounds for Shakespeare dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 Numerical Values for ˆc2/ˆc1 Table 3 provide the values of D, G, B, and d and used for the estimation of th ratio ˆc2/ˆc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' Othmane Marfoq, Giovanni Neglia, Laetitia Kameni, Richard Vidal Table 4: Average test accuracy across clients for different datasets in the settings when Nhist/N = 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' DATASET ˆc2/ˆc1 p∗ HIST TEST ACCURACY FRESH HISTORICAL UNIFORM OURS OPTIMAL SYNTHETIC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='06 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='89 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='39 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='94 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='90 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='17 CIFAR-10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='12 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='77 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='21 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='58 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='42 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='57 CIFAR-100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='08 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='65 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='41 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='54 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='66 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='39 FEMNIST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='79 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='09 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='80 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='09 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='09 SHAKESPEARE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='064 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='34 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='06 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='33 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='06 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='06 Table 5: Average test accuracy across clients for different datasets in the settings when Nhist/N = 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content=' DATASET ˆc2/ˆc1 pHIST TEST ACCURACY FRESH HISTORICAL UNIFORM OURS OPTIMAL SYNTHETIC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='50 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='27 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='58 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='20 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='20 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='20 CIFAR-10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='95 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='98 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='60 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='66 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='37 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='25 CIFAR-100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='69 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='57 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='40 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='55 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='31 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='31 FEMNIST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='98 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='23 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='93 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='23 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='23 SHAKESPEARE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='095 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='00 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='51 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='27 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='56 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf'} +page_content='1 ± 0.' metadata={'source': 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Ly-alpha_tomography +©ESO 2023 +January 5, 2023 +Extended Lyman-α emission towards the SPT2349-56 protocluster +at z = 4.3 +Yordanka Apostolovski1, Manuel Aravena2, Timo Anguita3,4, Matthieu Bethermin5, James Burgoyne6, Scott +Chapman7, Carlos De Breuck8, Anthony Gonzalez9, Max Gronke10, Lucia Guaita3, Yashar Hezaveh11,12, Ryley Hill7, +Sreevani Jarugula13, Evelyn Johnston2, Matt Malkan14, Desika Narayanan9, Cassie Reuter13, Manuel Solimano2, Justin +Spilker15, Nikolaus Sulzenauer16, Joaquin Vieira13, David Vizgan13, and Axel Weiß16 +1 Instituto de Física y Astronomía, Universidad de Valparaíso, Avda. Gran Bretaña 1111, Valparaíso, Chile +e-mail: yordanka.apostolovski@gmail.com +2 Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejército 441, Santiago, Chile +3 Instituto de Astrofisica, Facultad de Ciencias Exactas, Universidad Andres Bello, Fernandez Concha 700, Santiago, Chile +4 Millennium Institute of Astrophysics, Monseñor Nuncio Sotero Sanz 100, Oficina 104, Santiago, Chile +5 Aix Marseille Univ, CNRS, LAM, Laboratoire d’Astrophysique de Marseille, Marseille, France +6 Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada +7 Department of Physics and Atmospheric Science, Dalhousie University, Halifax, NS, B3H 4R2, Canada +8 European Southern Observatory, Karl Schwarzschild Straße 2, 85748 Garching bei München, Germany +9 Department of Astronomy, University of Florida, 211 Bryant Space Sciences Center, Gainesville, FL, 32611, USA +10 Max Planck Institut fur Astrophysik, Karl-Schwarzschild-Straße 1, D-85748 Garching bei München, Germany +11 Département de Physique, Université de Montréal, Montreal, Quebec, H3T 1J4, Canada +12 Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY, 10010, USA +13 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green St., Urbana, IL, 61801, USA +14 Department of Physics and Astronomy, University of California, Los Angeles, CA, 90095-1547, USA +15 Department of Physics and Astronomy and George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astron- +omy, Texas A&M University, 4242 TAMU, College Station, TX 77843-4242, USA +16 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121, Bonn, Germany +January 5, 2023 +ABSTRACT +Context. Deep spectroscopic surveys with the Atacama Large Millimeter/submillimeter Array (ALMA) have revealed that some of +the brightest infrared sources in the sky correspond to concentrations of dusty star-forming galaxies (DSFG) at high redshift. Among +these, the SPT2349-56 protocluster system at z = 4.304 is amongst the most extreme examples due to its high source density and +integrated star formation rate. +Aims. We conducted a deep Lyman-α line emission survey around SPT2349-56 using the Multi-Unit Spectroscopic Explorer (MUSE) +at Very Large Telescope (VLT) in order to characterize this uniquely dense environment. +Methods. Taking advantage of the deep three-dimensional nature of this survey, we performed a sensitive search for Lyman-α emitters +(LAEs) toward the core and northern extension of the protocluster, which correspond to the brightest infrared regions in this field. +Using a smoothed narrowband image extracted from the MUSE datacube around the protocluster redshift, we searched for possible +extended structures. +Results. We identify only three LAEs at z = 4.3 in this field, in concordance with expectations for blank-fields, and an extended +Lyman-α structure spatially associated with core of the protocluster. All the previously-identified DSFGs in this field are undetected +in Lyman-α emission, consistent with the conspicuous dust obscuration in these systems. We find an extended Lyman-α structure, +about 60 × 60 kpc2 in size, and located 56 kpc west of the protocluster core. Three DSFGs coincide spatially with the location of +this structure. We conclude that either the three co-spatial DSFGs or the protocluster core itself are feeding ionizing photons to the +Lyman-α structure. +Key words. galaxies – formation: galaxies – intergalactic medium +1. Introduction +Studies of massive galaxies at the peak of their star-formation +activity and their relation to the densest protocluster systems are +key to understanding the hierarchical formation of the most mas- +sive galaxy structures in the early Universe. Current studies seek +to understand the role of active galactic nuclei (AGN) feedback +(Pike et al. 2014; Smolˇci´c et al. 2017), or the relation between +downsizing and star formation (Magliocchetti et al. 2013; Miller +et al. 2015) during the active growth phases of such forming +structures. +Cosmological simulations show that cold dark matter (CDM) +haloes merge and form a web-like network traced by young +galaxies and reionized gas. A protocluster will form at the high- +est overdensity regions within this filamentary structure at early +cosmic times (z ∼ 4 − 6; Baugh et al. 1998; De Lucia & Blaizot +2007), eventually becoming a massive virialized cluster by z < 1 +(e.g.; Overzier 2016). These cosmological simulations indicate +Article number, page 1 of 14 +arXiv:2301.01328v1 [astro-ph.GA] 3 Jan 2023 + +A&A proofs: manuscript no. Ly-alpha_tomography +Table 1. Results of the blind line, narrow band and prior selected sample search in the MUSE data cubes. +ID +RA +DEC +∆v‡ +FWHM +S Lyα +EW +. . . +(J2000) +(J2000) +(km s−1) +(km s−1) +(10−20 erg cm−2 s−1) +(km s−1) +LAB +23:49:43.39 +−56:38:23.49 +200 ± 19 +760 ± 40 +3660 ± 1030 +1530 +LAE1 +23:49:41.28 +−56:37:58.20 +−99 ± 9 +330 ± 20 +510 ± 150 +540 +LAE3 +23:49:42.18 +−56:38:10.48 +70 ± 9 +330 ± 50 +2220 ± 280 +1230 +LAE8 +23:49:40.03 +−56:37:34.11 +1689 ± 9 +330 ± 30 +2300 ± 430 +1300 +Tentative candidates† +NL3 +23:49:44.73 +−56:38:39.99 +−1818 ± 39 +270 ± 90 +210 ± 150 +1705 +LAE2 +23:49:44.26 +−56:38:40.90 +739 ± 19 +330 ± 50 +190 ± 170 +570 +LAE4 +23:49:39.90 +−56:38:12.22 +360 ± 19 +380 ± 30 +240 ± 170 +830 +LAE5 +23:49:43.47 +−56:37:02.37 +0 ± 19 +440 ± 50 +370 ± 240 +670 +LAE6 +23:49:40.96 +−56:37:09.18 +669 ± 19 +310 ± 50 +330 ± 200 +560 +LAE7 +23:49:45.41 +−56:37:28.70 +709 ± 9 +320 ± 30 +320 ± 220 +1170 +Notes: † List of Lyman-α line candidates, which showed (snr)det > 5 as computed in the LSDCat detection cube. Despite the high +snr obtained in the “maximal” LSDCat extraction, these detections are considered tentative based on their low significance +measured in the original cube through homogeneous 1′′ radii aperture measurements. ‡ Velocity offset relative to the protocluster’s +mean [Cii]-derived redshift, z = 4.304. +that galaxies within galaxy protoclusters experience a luminous +starburst-phase (Miley & De Breuck 2008). +To identify and study these starbursting protocluster sys- +tems, several observational methods have been used. One of +them corresponds to sub/millimeter wavelength observations, +which allow one to pinpoint the obscured star-formation activ- +ity in young protocluster members (e.g., Chapman et al. 2009; +Daddi et al. 2009; Aravena et al. 2010; Capak et al. 2011; Casey +et al. 2015; Miller et al. 2018; Oteo et al. 2018). Similarly, low- +frequency radio observations are typically used to search for +radio-loud quasars sitting in the centers of dense protocluster +fields (Galametz et al. 2013; Rigby et al. 2014). In these radio- +selected protoclusters, Lyman-α emitters (LAEs), star-forming +galaxies selected through their significant UV rest-frame Lyman- +α emission line (λrest = 1215.67Å), show overdensity factors +3 − 5 times larger than the field at the same redshift (Venemans +et al. 2005, 2007). Given this ubiquity of LAE overdensities in +radio-selected fields, deep searches for these sources have been +performed to confirm the redshifts of protocluster galaxy candi- +dates using 8m-class optical/IR telescopes (e.g., Pentericci et al. +1997; Kurk et al. 2000; Venemans et al. 2002, 2004, 2005, 2007; +Croft et al. 2005). +The velocity dispersions of radio-selected galaxy protoclus- +ters are typically found in the ∼ 300−1000 km s−1 range centered +at the mean velocity of the radio galaxies. Although these sys- +tems are not yet virialized, such large velocity dispersions sug- +gest that these systems have large halo masses, possibly evolving +into the most massive cluster systems in the local Universe. +Narrowband image surveys in protocluster fields have iden- +tified a population of LAEs with luminosities larger than 1043.4 +erg s−1 and large spatial extensions (40 − 150 kpc). These struc- +tures are often referred to as Lyman-α blobs (LABs; Steidel et al. +2000; Matsuda et al. 2004). The origin of the emission of these +structures can be explained by different scenarios such as the +presence of AGN or massive star-forming galaxies. The produc- +tion of Lyman-α photons in these objects could be associated +with different processes such as recombination radiation, contin- +uum pumping, or collisional excitation (see; Cantalupo 2017). +The large-scale millimeter survey covering 2500 squares de- +grees of the sky conducted with the South Pole Telescope (SPT; +Carlstrom et al. 2011) discovered a population of millimeter- +bright sources (S 1.4mm > 20 mJy; Vieira et al. 2010, 2013; +Everett et al. 2020; Reuter et al. 2020). Follow-up observa- +tions with the Atacama Large Millimeter/submillimeter Array +(ALMA) showed that the majority of these sources are gravi- +tationally lensed submillimeter galaxies (> 90%; SMGs; also +known as dusty star-forming galaxies, or DSFGs) with magni- +fications µ870µm ∼ 5 − 20 (median µ870µm = 6.3 Spilker et al. +2016). The remaining sources show no evidence of gravita- +tional lensing, being either intrinsically bright or composed of +fainter multiple-component SMGs or groups of SMGs (Heza- +veh et al. 2013; Spilker et al. 2016). The high number of SMGs +spread over a small area of the sky (< 1′) found in these fields +strongly suggests the existence of (sub)millimeter bright proto- +cluster fields (Wang et al. 2021). +Among the sample of SPT protoclusters, the SPT2349-56 +system stands out due to its exceptionally high surface density +of SMGs. SPT2349-56 is located at z = 4.304 and has a surface +density of more than ten times the average blank-field value and +a volume density 1000 times the average (Miller et al. 2018; Hill +et al. 2020). This system could represent the core of a massive +galaxy cluster and is one of the most massive structures known to +date in the early Universe. The SPT2349-56 system has two main +infrared (IR) bright structures as seen in the APEX/LABOCA +870 µm maps, following the north-south direction (Figure 1). +The southern component comprises by itself a flux density of +S870 ≈ 77 mJy, whereas the northern component contributes with +S870 ≈ 33 mJy. For reference, a typical unlensed SMG has a flux +density of around 5–10 mJy at 870 µm. Higher-resolution deep +ALMA spectroscopy yielded a total of 24 [Cii] and 16 CO(4-3) +line emitters in the southern and northern extensions of the clus- +ter (e.g. Miller et al. 2018; Hill et al. 2020). Several components +of this system have SFR estimates of ∼ 1000 M⊙ yr−1, while +the full protocluster system is estimated to have a SFR of about +6.6 × 104 M⊙ yr−1 (Hill et al. 2020). Similarly, the dynamical +mass of the core region is estimated to be ∼ 9 × 1012 M⊙, while +the total halo mass of the whole structure is ∼ 2.5 × 1013 M⊙ +(Hill et al. 2020). +The physical properties of these sources indicate that this +protocluster already harbors massive galaxies that are rapidly +forming stars from an abundant gas supply. The large number +of SMGs in this system pushes and challenges theoretical mod- +els seeking to explain the origin and evolution of protoclusters +(Chiang et al. 2013). +Due to the proximity of the SMG members in the core of +the protocluster (the diameter is about 130 kpc), it is likely that +Article number, page 2 of 14 + +Yordanka Apostolovski et al.: Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.3 +23h49m48s +44s +40s +36s +-56°37'00" +30" +38'00" +30" +39'00" +RA [J2000] +DEC [J2000] +Fig. 1. Deep IRAC mosaic obtained toward the SPT2349-56 system. +Red contours show the ALMA [Cii] coverage. Blue squares show the +observed MUSE footprint, where we used two pointings to cover the +full IR bright region previously detected with LABOCA. +its component galaxies will merge to form a massive elliptical +galaxy at the core of a lower-redshift Coma-like galaxy cluster +(Miller et al. 2018; Hill et al. 2020). +A recent search for Lyman Break Galaxies (LBGs) in the +extended SPT2349-56 environment found 4 LBGs in the south- +ern part of the protocluster (Rotermund et al. 2021), indicating +that most of the SMGs are inconspicuous at optical wavelengths, +with only one of the 4 LBGs coinciding with a previously re- +ported SMG. +Motivated by the significant overdensities found in radio- +selected protocluster fields, we conducted an independent cen- +sus of star-forming galaxies in the SPT2349-56 field through +a sensitive search for Lyman-α emission using deep optical +spectroscopy obtained with the Multi-Object Spectroscopy Unit +(MUSE) at the Very Large Telescope (VLT). In Section 2 we de- +scribe the observations and reduction of the MUSE data towards +SPT2349-56, and summarize previous observations. In Section +3 we present the detection of Lyman-α emission through both a +blind search and narrowband imaging. In Section 4 we analyze +the nature of the extended Lyman-α emission along with its con- +nection with LAEs and the structure of the protocluster. Section +5 summarizes and presents the conclusions of this work. +Hereafter, we adopt a flat ΛCDM cosmology with h = 0.677, +Ωm = 0.307 and ΩΛ = 0.693 (Planck Collaboration et al. 2016). +2. Observations +In this section, we describe details of the Lyman-α line observa- +tions in the SPT2349-56 protocluster at z = 4.3. +2.1. MUSE observations +Observations with MUSE at the VLT UT4 were performed in +two separate pointings targeting the north and south extensions +of the SPT2349-56 protocluster system (Figure 1). MUSE covers +the wavelength range 480-930 nm. Each pointing covers roughly +1 square arcmin (60′′ × 60′′). These observations were carried +out in the wide field mode (WFM) in service-mode observing +as a part of projects 0100.A-0437(A) and 0100.A-0437(B) (PI: +M. Aravena) during dark-time. Each pointing was observed for 5 +hours (a total of 10 hours) between November 2017 and Septem- +ber 2018. Each pointing consisted of a set of exposures of 680 +seconds each, with individual exposures rotated by 90 degrees +with respect to each other. +The average seeing of these observations was 0.97 and 0.98 +arcsec for the southern and northern pointings, respectively, af- +ter correction of air-mass. Weather conditions were classified by +ESO as clear (CL; 55%), with high wind (CL-WI; 11%) and pho- +tometric conditions (PH; 33%) for all observing blocks (OBs). +We reduced the data using the MUSE pipeline v2.6 (Weil- +bacher et al. 2014) for bias subtraction, flat-fielding, and wave- +length and flux calibration, resulting in a single data cube per +each of the 5 OBs per field. We combined the five OB data +cubes per field using the MUSE Python Data Analysis Frame- +work (MPDAF; Bacon et al. 2016). The data cubes were merged +using a sigma-clipped mean with σclip = 5. Since the field is +relatively sparse (especially in Lyman-α at z = 4.304), we used +Zurich Atmosphere Purge (ZAP; Soto et al. 2016) to perform +a sky subtraction through principal component analysis (PCA). +For this process, we used a mask in order to avoid spaxels that +contained obvious continuum sources. +2.2. Previous ALMA observations +In this study, we used as reference the images, cubes and location +of protocluster members previously identified through ALMA +Cycle 5 and 6 observations. These observations and the corre- +sponding data reduction and source identification are described +in detail by Miller et al. (2018) and Hill et al. (2020) and we refer +the reader to those papers for full details. +In brief, observations of the redshifted [Cii]158µm fine struc- +ture line towards the SPT2349-56 system were obtained us- +ing ALMA in Band 7. These were centered at a frequency of +νobs = 358.4 GHz, yielding an average synthesized beam size of +0.43′′ × 0.34′′ and 3σ sensitivities of ≈ 0.3 mJy beam−1. These +observations, which cover the full IR-bright region, led to the +identification of 24 [Cii] emitters in the field. The MUSE ob- +servations described above fully cover the region observed by +ALMA in Band 7 at with uniform sensitivity (Fig. 1). Based on +the identified [Cii] sources, the mean redshift of the system was +determined to be at z = 4.304 (Miller et al. 2018; Hill et al. +2020). +2.3. HST imaging +We used HST/Wide Field Camera 3 (WFC3)-IR images under +program 15701 (PI: S. Chapman). The target was assigned 2 or- +bits for the F110W filter and three orbits for the F160W filter +in the infrared channels. Dithering was implemented for maxi- +mum resolution. The data was reduced using the standard HST +pipeline. The pixel size in the WFC3 images is 0.075′′. +3. Results +We used the MUSE observations obtained toward SPT2349-56 +to perform a systematic search of Lyman-α emission with three +methods: a blind automatic search in the cube, creating a nar- +row band image around the known protocluster redshift and a +search for Lyman-α emission in (dusty) sources that had previ- +Article number, page 3 of 14 + +A&A proofs: manuscript no. Ly-alpha_tomography +70 kpc +Fig. 2. Left: Lyman-α emission toward the SPT2349-56 protocluster system at z = 4.3. The MUSE covered area is shown, with the HST F160W +image in the background in grey-scale, and red contours representing a rendered Lyman-α image. The later is obtained as the average of each +individual line map of the detected LAEs and the LAB, in steps of 2, 5 and 7 σ, where σ is the rms noise level in the average image. The blue +circles highlight the location of the ALMA [Cii] and CO(4-3) line detections in the field (Miller et al. 2018; Hill et al. 2020). Green squares show +the location of the LBGs in the field (Rotermund et al. 2021). Right: The map of ALMA [Cii] line emission toward the identified Lyman-α blob +(LAB) is shown in the background, with blue circles representing the location of the previously identified [Cii] line emitters C10, C14, C17 (see +Table 2; Hill et al. 2020). Red contours show the Lyman-α emission of the LAB at 2, 4, 6 and 8σ. +ously been identified in this field. Below, we describe each of +these searches. +3.1. Blind Search +We performed a blind search for Lyman-α emission in the +MUSE data cubes, using the Line Source Detection and Cata- +loguing Tool (LSDCat; Herenz & Wisotzki 2017). For this, we +focused on a 4000 km s−1 band centered on the red-shifted (z = +4.304) Lyman-α wavelength (λred = 6444.2 Å). The LSDCat rou- +tine detects emission lines through an spatial and spectral filter- +ing (3D matched-filtering) approach and sorts them into discrete +objects. This method is used to maximise the signal-to-noise ra- +tio (snr) of the entire cube, and thus creating a snr detection cube. +To determine an appropriate threshold for detection in the snr +cube, we conducted an unbiased line search also in the negative +version of our original cube (multiplied by −1). Assuming that +the noise in this reduced velocity range of the cube is symmetric +around 0 and roughly follows a Gaussian distribution, the detec- +tions obtained in the negative cube will set the maximum level +at which we expect line features produced by noise. From this, +we find that the most significant feature in the negative cube is +found at (snr)det ∼ 5, thus yielding our detection threshold. +This process yields a significant number of positive fea- +tures located at the edges of the independent channel images, +and with linewidths of one or two channels only, which we re- +move from our catalogue as they are unphysically narrow. To +filter the Lyman-α line candidates from spurious positive fea- +tures, we constrain the full-width at half maximum (FWHM) of +the detected lines to the range of widths found for the [Cii] and +CO(4-3) lines for sources in the field (Hill et al. 2020), which +correspond to 50 − 600 km s−1. After this selection process, we +identified eight LAE candidates, four in the northern and four in +the southern pointing (Figure 2). +We extracted the spectra of each of the LAE candidates using +apertures with radii of 1 arcsec. Due to the more extended spatial +nature compared to the other LAEs, we extracted the spectra of +sources LAE3 and LAE8 using apertures of 2 arcsec radius (Fig- +Article number, page 4 of 14 + +Yordanka Apostolovski et al.: Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.3 +LAB +LAE8 +LAE3 +LAE1 +LAE7 +LAE6 +LAE5 +LAE4 +LAE2 +Fig. 3. Continuum subtracted spectra of the Lyman-α emission line +identified significantly within the MUSE footprint around SPT2359-56. +For comparison purposes, the vertical axis has been normalized. The +measured fluxes are given in Table 1. Red dotted line shows the cen- +tral velocity of the protocluster (expected redshifted Lyman-α line at +λ = 6444.2Å ). The nomenclature of the LAEs does not follow the snr +of the emission lines. The blue tag name denotes secure detections while +the red tag names denote tentative detections. +ure 3). Based on the significance of each of the line candidates +measured in these apertures (see Fig. 3 and Table 1), we split the +sample in secured LAEs and tentative candidate sources. +Only three sources are securely detected in this fashion +(LAE1, LAE3 and LAE8), and the remaining five sources are +thus considered tentative detections. All the spectra were manu- +ally inspected and searched for other lines that would point to a +lower redshift possibility. However, all line detections were con- +sistent with Lyman-α at z ∼ 4.3. +LAE1 is associated with detections in all available broad- +band images, including g, r, i through KS band (see Appendix +B in Hill et al. 2022). However, if the galaxy is at z ∼ 4.3, +we would expect it to be faint in the g band due to the Lyman +break. Inspection of the HST F110W image (see Fig. B.1) sug- +gest that the excess g-band emission comes from a foreground +object along the line of sight. Indeed, due to the mismatch be- +tween MUSE Lyman-α position and the optical broadband po- +sition, Hill et al. (2022) lists its photometry as upper limits (see +their Table 1). The significance of the detected line and the lack +of other line features in the MUSE spectrum strongly favour the +z ∼ 4.3 spectroscopic confirmation. As a precedent, note that +ALMA source C1 (source ‘A’) appears to be well detected in +g-band, since there is a foreground z = 2.5 galaxy as shown in +Rotermund et al. (2021). +LAE3 and LAE8 are both undetected in the g-band and have +faint detections in the deep r and HST F110W images (see Fig. +B.1 and Appendix B in Hill et al. 2022). They are significantly +detected in Lyman-α without other line identifications, and have +considerable EWs compared to the other sources identified in +this field. +Based on the redshift implied by the identified Lyman-α +lines, we computed a median velocity offset for all the LAEs in +the field with respect to the protocluster redshift z = 4.304 (ob- +tained from previous [Cii] and CO identifications; Miller et al. +2018; Hill et al. 2020). The northern and southern LAEs are +found to have velocity offsets of ∆v = 930 and ∆v = 430 km +s−1 respectively, indicating that the Lyman-α emissions are sys- +tematically redshifted from the center of the protocluster. +3.2. Narrowband Image +To independently search for line emission in the field, we pro- +duced a continuum-subtracted narrowband image using a spec- +trally and spatially smoothed version of the MUSE datacube +with LSDCat. We selected as a central wavelength for the im- +age of the Lyman-α line redshifted to z = 4.304 and a width of +2000 km s−1 (i.e. 6401.3 − 6487.2Å). As such, this procedure +was specifically designed to search for extended emission. As +a result, we found an extended Lyman-α structure towards the +east of the protocluster core, which we associate with a so-called +“Lyman-α blob” (LAB, see Figure 2). The Lyman-α emission +of the LAB subtends a roughly circular region with an area of +10′′ × 10.4′′ in the sky (≈ 70 ×70), and is located about 56 kpc +east of the center of SPT2349-56. With a radius of ≈ 5′′ (34.4 +kpc), this yields an area, πr2 = 3720 kpc2 or ∼ 60 × 60 kpc2. +To obtain a spectrum of this extended emission, we draw a +polygon around this source, containing all pixels detected above +2σ in the narrowband image (see Figures 2 and 3). Based on the +Lyman-α profile, we find that the extended feature shows a line +with a FWHM of about 760 km s−1 and a velocity offset with +respect to the [Cii]/CO protocluster redshift of ∆v = 365 km s−1. +After integrating along the full width of the line emission we +obtain a flux of S Lyα = 3663 × 10−20 erg cm−2 s−1. +Article number, page 5 of 14 + +A&A proofs: manuscript no. Ly-alpha_tomography +0 +100 +200 +300 +400 +500 +600 +Distance (kpc) +3000 +2000 +1000 +0 +1000 +2000 +3000 +3v (km s +1) +North +South +LAB +ALMA +Fig. 4. Velocity offset scaled by +√ +3 as an estimate for 3-dimensional velocity, centered at the center of the protocluster (Hill et al. 2020) versus +projected distance from the 850 µm-weighted centre of the protocluster. Orange circles show detections of [Cii] and CO(4-3) with ALMA (Miller +et al. 2018; Hill et al. 2020), green square and triangles show Lyman-α emitters detected and candidates respectively in the extended emission +showed by LABOCA observations (Figure 1), blue squares and triangles show the Lyman-α emitters detected and candidates respectively in the +southern pointing and the red star shows the Lyman-α blob inside the 90 kpc defined as the effective radius. Black lines show the escape velocity +from the protocluster. The measured velocity offset uncertainties are negligible, and thus errorbars are smaller than the size of the symbols. +A comparison of the position of the LAB with the location of +the previously-identified SMGs in this field (Miller et al. 2018; +Hill et al. 2020) shows that three sources overlap spatially with +the western part of the blob. These sources, called C10, C14 and +C17 using the nomenclature of Hill et al. (2020), were identified +based on their bright [Cii] line emission, with fluxes of 2.96, +1.70 and 0.93 Jy km s−1, respectively. These galaxies are not the +most luminous in the sample of [Cii] emitters in the SPT2349- +56 system, and are located at about 65 kpc from the center of the +protocluster. +3.3. Previously Known Protocluster Members +In addition to the independent searches described above, we +searched the MUSE datacubes for Lyman-α emission at the posi- +tions of the previously-identified SMGs in the SPT2349-56 sys- +tem at z = 4.3. All of these sources have confirmed systemic red- +shifts based on the identification of the [Cii] and CO(4-3) lines +with ALMA (Hill et al. 2020). +For each of these sources, we extracted a spectrum using +apertures with radii of 2′′ centered at the location of either the +[Cii] and CO(4-3) detections (Figure A.1). Inside the range of +6000 km s−1 centered at z = 4.304 we do not find significant +Lyman-α emission in any of the previously-confirmed SMGs in +the field. However, we do find a tentative detection of Lyman-α +emission from one of the ALMA continuum sources in this field +for which no redshift confirmation was possible using the [Cii] or +CO(4-3) lines, source NL3. This source shows possible Lyman- +α emission at a velocity of -1600 km s−1 from the cluster core +redshift (z = 4.304). This velocity is covered by the ALMA CO +observations but not by the [Cii] ones. Thus, while the ALMA +[Cii] observations missed the line, it is possible that either the +source is too faint in CO(4-3) emission or the tentative Lyman-α +feature is not real. We thus tag this as a tentative candidate here. +We note that stacking of the MUSE spectra to yield a con- +strain on the average Lyman-α emission in these undetected +SMGs is difficult. Several studies have demonstrated that the +Lyman-α emission line does not always trace the galaxies’ sys- +temic redshifts due to IGM scattering and absorption (e.g. Shap- +ley et al. 2003; Song et al. 2014; Hashimoto et al. 2015). There- +fore, aligning the MUSE spectra at the [Cii] or CO-derived sys- +temic redshifts or correcting them to the rest-frame will yield a +diluted Lyman-α stack signal. While corrections for the Lyman- +α-derived to the systemic redshifts (or velocities) as a function of +equivalent width have been calibrated, these are statistical in na- +ture and will not yield the precise redshift/velocities necessary +for stacking. We do use such corrections in the check to see if +the identified LAEs are gravitationally bound to the protocluster +core (see next section). +4. Analysis and discussion +4.1. Lyman-α emitters +As we showed in the results section, we have found 8 blindly +selected LAEs and one LAB. Three of the blind identifications +are considered secure and 5 are tentative based on their low sig- +nificance (see Table 1). As shown in Fig. 2, the LAB and four of +Article number, page 6 of 14 + +Yordanka Apostolovski et al.: Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.3 +1000 +500 +0 +500 +1000 +1500 +Velocity (km s +1) +0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +S[CII] (mJy) +C10 +C14 +C17 +Fig. 5. Observed [Cii] line profiles obtained with ALMA for sources +C10, C14 and C17 (see, Hill et al. 2020), which overlap spatially with +the Lyman-α Blob obtained with MUSE (see Fig.2). The scaled Lyman- +α profile is shown in orange, for reference. The velocity scale refers to +z = 4.304 for all emission lines. +the identified LAEs are located in the southern structure, while +the rest are located in the north. The locations of the LAEs and +candidates range from 95 to 580 kpc from the core of the proto- +cluster and the implied velocities for Lyman-α appear redshifted +with respect to its systemic redshift. +The significant mass of the central core of the protocluster +is ∼ 9 × 1012M⊙, making it possible that most of the galaxies +identified in the core neighbourhood are gravitationally bound as +already shown by Hill et al. (2020). To test whether the identi- +fied Lyman-α emitters are also bound, which may yield insights +into the future evolution of the LAEs within the SPT2349-56 +system, we compare the offset velocities of each of the proto- +cluster members as a function of distance from the protocluster +core (Fig. 4). For the ALMA-identified protocluster members +we show their [Cii]/CO-based velocities, while for the LAEs, we +show their Lyman-α-based velocities. For the later, we obtain +a statistical correction to the systemic redshift of each galaxy +using the relation described in Verhamme et al. (2018). Galax- +ies that have velocities lower than the the escape velocity enve- +lope at a given radius from the protocluster core are expected +to be bound to the structure. The southern LAEs appear to be +consistently associated to the protocluster core, including the +LAB. Only one of the identified LAEs in the southern struc- +ture (a tentative candidate) resides outside the escape velocity +vesc = √2GM/R envelope. Conversely, most of the northern +LAEs, including the secure LAE8 identification, appear to be +unbound and redshifted with respect to the protocluster core yet +following the trend of the other [Cii] identified sources. The fact +that the northern LAEs follow the velocity offset of the ALMA +[Cii] sources in the north, bolsters the interpretation of the north- +ern structure as an unbound/infalling sub-halo (e.g., Miller et al. +2018; Hill et al. 2020). In addition, this supports the idea that the +southern structure, which includes the protocluster core is likely +already reaching a virialized form. +We compare the number of Lyman-α emitters found in the +SPT2349-56 field at z = 4.3 with the field counts using the ultra- +deep MUSE observations in the Hubble Ultra Deep Field (In- +ami et al. 2017, HUDF mosaic area of 3 × 3 arcmin2). Down +to a Lyman-α luminosity of log(LLyα) = 41.0, corresponding +to the depth reached by our SPT2349-56 observations, Drake +et al. (2017) finds 144 LAEs in the redshift range z = 4.0 − 5.0. +The MUSE HUDF observations are > 2× deeper than our point- +ings, and thus are complete to this depth. Considering the red- +shift range used to search for LAEs in the SPT2349-56 (z = +4.25 − 4.36) and the MUSE covered area (2 arcmin2), we would +thus expect to have found 3−4 LAEs. This indicates that the de- +tection of 3 secure sources in the SPT2349-56 field are consistent +with blank-field counts, and strengthens the case that most of the +emission output and mass in this system is associated to heavily +dust obscured sources. In this scenario, the existence of a LAB +in such complex obscured environment appears as a rare case, +where the Lyman-α emission is able to escape in a preferential, +less obscured direction. +2 +4 +6 +8 +10 +12 +14 +Position (arcsec) +2000 +1000 +0 +1000 +2000 +Velocity (km s +1) +C10 +C14 +C17 +Fig. 6. Position-velocity diagram towards the Lyman-α Blob, extracted +from MUSE cube at 0 degrees of inclination towards the east. Blue, +red and green circles show the observed [Cii] emissions (C10, C14 and +C17) from Hill et al. (2020). This shows a spatial connection between +the LAB and two DSFGs members of the PC. +4.2. Insights on the nature of the extended Lyman-α +emission +As mentioned previously, the extended Lyman-α emission found +in the MUSE-based narrowband image broadly coincides with +the position of three SMGs that were identified as part of the pro- +tocluster structure. This suggests a possible physical relationship +between them as indicated by previous studies of LABs at high +redshift (e.g., Chapman et al. 2001; Umehata et al. 2015; Geach +et al. 2016; Oteo et al. 2018). +Cen & Zheng (2013) constructed a model for the origin of +extended Lyman-α emission in the context of the cold dark mat- +ter framework, in which the LAB are produced due to starburst +activity. The model incorporates AGN feedback, although it is +expected that it should have a subdominant contribution (e.g., +Webb et al. 2009). For extended Lyman-α emission in a proto- +cluster, each galaxy member contributes to the whole Lyman- +α emission yielding a variety of sizes and geometries typically +found within a contiguous structure. The relative contribution of +these DSFGs depends on the dust attenuation of Lyman-α pho- +tons and the propagation and diffusion process through the cir- +cumgalactic medium (CGM) and intergalactic medium (IGM) of +each member. +In Figure 5 we compare the Lyman-α spectra of the LAB +with the [Cii] emission line spectra of the three DSFGs spa- +tially coincident to it: C10, C14 and C17. The [Cii] line emis- +sion is expected to trace the kinematics of each host galaxy, and +thus probe the galaxies’ systemic velocities and geometries (ro- +tation, merger, etc). Due to obscuration and absorption by the +intergalactic medium, the Lyman-α spectrum is expected to be +Article number, page 7 of 14 + +A&A proofs: manuscript no. Ly-alpha_tomography +redshifted with respect to the galaxies’ systemic velocities, and +thus with respect to the [Cii] lines. In this case, the Lyman-α +spectrum of the LAB is found ∼ 300 km s−1 redward of the pro- +tocluster core velocity (v = 0 km/s) and ∼ 100 km s−1 of the +C14 and C17 SMGs. A much larger velocity difference is seen +between the LAB and the systemic velocity of the C10 galaxy. +The inconsistency in velocities for C10 suggest this source might +be unrelated to the Lyman-α emission. We explore this issue in +more detail in the next sections. +4.3. The protocluster core as the origin for the LAB? +The LAB is located ∼ 56 kpc to the east of the of the proto- +cluster core, thus being within the protocluster effective radius +defined by Hill et al. (2020). Along with the velocity connection +between the LAB and the SMGs, this spatial coincidence suggest +a physical link between the protocluster core and the LAB. It is +thus possible that the powering source of the extended Lyman- +α emission is star formation or AGN activity in the starbursting +SMGs at the protocluster center, where the Lyman-α photons are +produced in a photon-ionized medium. +In this scenario, it is possible that most of the Lyman-α pho- +tons along our line of sight are not absorbed and/or scattered +but are instead able to escape toward the eastern part of the pro- +tocluster core. Indeed, Vernet et al. (2017) observed similar re- +gions with offsets of ∼ 100 kpc in the haloes of high redshift +AGN-host galaxies, invoking similar arguments. +Following Furlanetto et al. (2005), the Lyman-α emission +can be used to yield an estimate of the underlying SFR from +the powering source. For star formation episodes following a +Salpeter initial mass function (Salpeter 1959) and that two thirds +of the ionizing photons are absorbed in the dense ISM, we have: +LLyα = 1042( SFR/[M⊙ yr−1]) erg s−1 +(1) +Taking the value of LLyα = 1.32 ± 0.37 × 1042 erg s−1, we +obtain a SFR for the extended emission of 1.32 ± 0.37 M⊙ yr−1, +which is orders of magnitude lower than the SFR estimates for +any of the SMGs in the field. This is consistent with the idea that +most (99%) of the UV radiation is obscured by dust within the +SMGs. +Recent radio imaging of the SPT2349-56 field using the +Australia Telescope Compact Array (ATCA) and the Australian +Square Kilometer Array Pathfinder (ASKAP) found strong radio +emission from the protocluster core complex (Chapman et al. in +preparation). The steep radio spectrum found clearly indicates +that at least one of the three central sources (B, C and G in the +nomenclature used by Miller et al., or C3, C6 and C13 following +Hill et al.) host a radio AGN. This finding supports the idea that +enhanced Lyman-α emission at the LAB location is produced by +AGN activity at the protocluster core (e.g. Vito et al. 2020). +Figure 6 shows a position-velocity (PV) diagram of the +Lyman-α emission of the LAB, extracted along the x-axis (0 de- +grees of inclination) towards the west of the MUSE datacube, +with a slit width of 14.2 arcsec. Here we note a widespread emis- +sion along the central velocity with an extension of 5 arcsec. At +the edges, for the more distant structure, the emission goes to- +wards bluer velocities. On the other hand, in the edge closer to +the cluster, we have a structure that shifts to positive velocities. +Another important issue is the behavior of the luminosity in the +PV diagram. We can divide the structure into two different blobs: +with the western being brighter than the eastern. This result is in +agreement with the spectral line of the LAB, where we observe +6380 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength (Å) +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +Flux Density (10 +16 erg cm +2 s +1) +Model 0 +Model 1 +Model 2 +Model 3 +Fig. 7. Lyman-α spectrum of the LAB compared to the best-fit mod- +els that assume different systemic redshifts for the emitting source. The +models are described in the text and their best-fit parameters are listed +in Table 2. The best description of the observed Lyman-α spectrum is +given by models 1-3, suggesting that the emission is produced by pho- +toionization from either DSFGs C14, C17 or the protocluster core. +that the reddest emission is strongest and wider than the bluest +emission (Figure 5). +4.4. Modeling the Lyman-α spectrum of the extended +emission +To further explore the origin of the LAB, we test the idea that +either the protocluster core or the galaxies spatially coincident +with the LAB are the source of the Lyman-α emission. +For this, we use the Lyman-α line profile of the LAB and the +Lyman-α Monte Carlo Radiate Transfer code tlac (Gronke & +Dijkstra 2014; Gronke et al. 2015). We utilize a expanding shell +model, which has been widely used in several studies to success- +fully reproduce the Lyman-α profiles of galaxies in different red- +shifts and environments. The model assumes an homogeneous, +spherical shell that expands radially outwards, with uniformly +mixed neutral gas (HI) and dust (Verhamme et al. 2006), and +the emitting source located at the center of the shell. The shell +model is defined by a set of seven parameters including the ex- +panding velocity (vexp), the HI column density (NHI), the dust op- +tical depth (τd), the effective temperature of the gas (T), the sys- +temic redshift of the emitter (zsys), the intrinsic equivalent width +of the Lyman-α line (EW(Lyman-α)) and the intrinsic FWHM of +the Lyman-α line (FWHM(Lyman-α)). For more details on these +parameters, we refer the reader to Gronke et al. (2015). Given a +constraint for the systemic redshift of the source and the input +Lyman-α spectrum, the code yields the most likely set of param- +eters that reproduce the observed spectrum under the assumed +geometry. +Since we are interested in learning which of the underlying +starburst galaxies might be producing the Lyman-α emission, we +constrain zsys using the [Cii]-based redshifts of each of the pos- +sible sources of the Lyman-α line: C10 (model 0), C14 (model +1) and C17 galaxies (model 2), and the protocluster core (model +3). Instead of simply fixing zsys, we allow for a range in red- +shift given by the 3σ uncertainty around measured [Cii] redshift +in each case. Since these ranges overlap, some of the solutions +found are similar between each other. The results of this proce- +dure are shown in Figure 7 and listed in Table 2. +We find that the model 0 does not converge into a proper fit +to the data, mostly due to the significant difference between the +[Cii] redshift and that of the Lyman-α line. This forces the model +to a high expansion velocity (∼ 500 km s−1) and low dust optical +Article number, page 8 of 14 + +Yordanka Apostolovski et al.: Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.3 +Table 2. Results from the radiative transfer modeling of the LAB line profile +Model +Source† +zsys +vexp +log(NHI) +τd +log(T) +EW(Lyman-α) +σ(Lyman-α)‡ +(km s−1) +(cm−2) +(K) +(Å) +(km s−1) +0 +C10 +4.2895 ± 0.0019 +480+9 +−14 +19.85+0.17 +−0.11 +0.01+0.02 +−0.01 +4.2+0.4 +−0.6 +5.1+1.3 +−1.2 +798+15 +−31 +1 +C14 +4.3057 ± 0.0020 +377+28 +−46 +16.19+0.43 +−0.22 +3.7+0.9 +−1.6 +3.1+0.4 +−0.2 +6.6+0.9 +−0.8 +328+33 +−22 +2 +C17 +4.3049 ± 0.0020 +375+17 +−31 +16.15+0.43 +−0.19 +2.8+1.1 +−1.3 +3.15+0.35 +−0.23 +6.6+1.0 +−0.8 +330+30 +−16 +3 +Core +4.3040 ± 0.0020 +310+36 +−22 +16.65+0.39 +−0.50 +1.7+1.8 +−1.2 +3.10.3 +−0.2 +6.6+1.1 +−0.9 +346+42 +−35 +Notes: † Source assumed to be producing the Lyman-α emission. Its [Cii] redshift is assumed to be the systemic redshift of the +system for each model. ‡ σ = FWHM/2.35. +depth. This solution is less preferred, since the assumed emitting +source (the C10 galaxy) is a gas-rich, dusty galaxy contrary to +the result of a low dust optical depth. +The best fits are produced when using higher systemic red- +shifts (models 1−3), which are more consistent with the redshift +of the Lyman-α line. This is the case for sources C14 and C17, +and the protocluster core. In these cases, the solutions are simi- +lar, yielding high outflow velocities (∼ 300−400 km s−1) and yet +very low HI column densities (log(N(HI))∼ 16). In these cases, +the opacities appear to be moderate (τ > 1.5−3.0), yet more con- +sistent with the dusty nature of the purported emission sources. +Based on these results alone it is hard to disentangle the origin of +the LAB. However, if the protocluster core starbursting galaxies +are producing the Lyman-α emission it would require a complex +patchy geometry where some of the UV radiation escapes and +illuminates the HI gas in the LAB direction. While this is a plau- +sible scenario, supported by the moderate optical depth of this +solution (model 3), such solution is less likely than the scenario +where the UV radiation is produced in-situ by either the C14, the +C17 an/or both galaxies. +5. Summary and conclusions +We presented a census of Lyman-α emission toward the IR- +bright protocluster SPT2349-56 at z = 4.3 obtained using MUSE +observations. Through a blind search of Lyman-α emission to- +wards the protocluster core and northern extension, we found +three LAEs at distances > 90 kpc from the protocluster core. +The LAEs are bound to the 9 × 1012M⊙ protocluster core and +all of them are redshifted relative to SPT2349-56. Only one of +the ALMA SMGs previously identified in this field is tentatively +detected in Lyman-α. +Using a continuum-subtracted narrowband image we detect +extended Lyman-α emission, which we refer to as a LAB, with +a size of about 70 kpc across, located at ∼ 56 kpc to the east of +the protocluster core. The bulk of the LAB emission is also red- +shifted with respect to the core of the protocluster, in agreement +with a red-skewed asymmetric profile. +Two of the spatially overlapping DSFGs C14 and C17, are +found to also coincide spectrally, when comparing their [Cii] +emission lines with that of the Lyman-α emission from the LAB +(Miller et al. 2018; Hill et al. 2020). This observation could be +explained by the high star-formation activity seen in the DSFG +protocluster members. Based on their locations and redshifts, the +main suspects to be producing the ionizing photons and thus the +Lyman-α emission are the C14 and C17 DSFGs, or the proto- +cluster core. In the later case, the geometry of the dust distribu- +tion should allow the Lyman-α photons to get scattered from the +core such that the photons find a region to escape to the east. +Such scenarios are supported by radiative transfer modeling of +the Lyman-α line profile of the LAB. +We do not find an overdensity of LAEs, or a source density +comparable to what we might have expected from the number of +[CII] and submillimeter continuum sources found in this field. +We interpret this as a structure that is still heavily dust obscured +and dominated by submm-detected galaxies. +Acknowledgements. This paper makes use of the following ALMA data: +ADS/JAO.ALMA#2017.1.00273.S; +and +ADS/JAO.ALMA#2018.1.00058.S. +ALMA is a partnership of ESO (representing its member states), NSF (USA) +and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), +and KASI (Republic of Korea), in cooperation with the Republic of Chile. The +Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. Y.A. +acknowledges partial support from Comité Mixto ESO - Gobierno de Chile. +MA acknowledges support from FONDECYT grant 1211951, CONICYT ++ PCI + INSTITUTO MAX PLANCK DE ASTRONOMIA MPG190030 +and CONICYT+PCI+REDES 190194. This work was partially funded by +the ANID BASAL project FB210003. T.A. acknowledges support from the +Millennium Science Initiative ICN12_009. D.N. acknowledges support from the +US NSF under grant 1715206 and Space Telescope Science Institute under grant +AR-15043.0001. J.D.V. and S.J. acknowledge support from the US NSF under +grants AST-1715213 and AST-1716127. S.J. acknowledge support from the US +NSF NRAO under grants SOSPA5-001 and SOSPA7-006, and SOSPA4-007, +respectively. J.D.V. acknowledges support from an A. P. Sloan Foundation +Fellowship. 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Forshay, 451 +Article number, page 10 of 14 + +Yordanka Apostolovski et al.: Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.3 +Appendix A: Lyman-α spectra toward the SPT2349-56 DSFGs +The following figures show the observed MUSE spectra toward all the DSFGs in the SPT2349-56 system, centered at the expected +location of the Lyman-α line emission. +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +30 +20 +10 +0 +10 +20 +30 +40 +(10 +20)erg s +1cm +2Å) +C1 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +30 +20 +10 +0 +10 +20 +30 +(10 +20)erg s +1cm +2Å) +C2 +6380 +6400 +6420 +6440 +6460 +6480 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +20 +0 +20 +40 +(10 +20)erg s +1cm +2Å) +C3 +6380 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +20 +0 +20 +40 +(10 +20)erg s +1cm +2Å) +C4 +6380 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +30 +20 +10 +0 +10 +20 +30 +(10 +20)erg s +1cm +2Å) +C5 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +30 +20 +10 +0 +10 +20 +30 +40 +(10 +20)erg s +1cm +2Å) +C6 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +30 +20 +10 +0 +10 +20 +(10 +20)erg s +1cm +2Å) +C7 +6400 +6420 +6440 +6460 +6480 +6500 +6520 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +30 +20 +10 +0 +10 +20 +30 +40 +(10 +20)erg s +1cm +2Å) +C8 +6380 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +30 +20 +10 +0 +10 +20 +30 +(10 +20)erg s +1cm +2Å) +C9 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +20 +0 +20 +40 +(10 +20)erg s +1cm +2Å) +C10 +6380 +6400 +6420 +6440 +6460 +6480 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +30 +20 +10 +0 +10 +20 +30 +40 +(10 +20)erg s +1cm +2Å) +C11 +6380 +6400 +6420 +6440 +6460 +6480 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +20 +0 +20 +40 +(10 +20)erg s +1cm +2Å) +C12 +6380 +6400 +6420 +6440 +6460 +6480 +Observed Wavelength(Å) +Article number, page 11 of 14 + +A&A proofs: manuscript no. Ly-alpha_tomography +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +30 +20 +10 +0 +10 +20 +30 +40 +50 +(10 +20)erg s +1cm +2Å) +C13 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +20 +0 +20 +40 +(10 +20)erg s +1cm +2Å) +C14 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +20 +0 +20 +40 +(10 +20)erg s +1cm +2Å) +C15 +6380 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +20 +0 +20 +40 +60 +(10 +20)erg s +1cm +2Å) +C16 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +30 +20 +10 +0 +10 +20 +30 +40 +(10 +20)erg s +1cm +2Å) +C17 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +30 +20 +10 +0 +10 +20 +30 +40 +50 +(10 +20)erg s +1cm +2Å) +C18 +6380 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +30 +20 +10 +0 +10 +20 +30 +40 +(10 +20)erg s +1cm +2Å) +C19 +6380 +6400 +6420 +6440 +6460 +6480 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +30 +20 +10 +0 +10 +20 +30 +(10 +20)erg s +1cm +2Å) +C20 +6380 +6400 +6420 +6440 +6460 +6480 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +30 +20 +10 +0 +10 +20 +30 +40 +(10 +20)erg s +1cm +2Å) +C21 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +20 +0 +20 +40 +60 +(10 +20)erg s +1cm +2Å) +C22 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +40 +20 +0 +20 +40 +60 +(10 +20)erg s +1cm +2Å) +C23 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +3000 +2000 +1000 +0 +1000 +2000 +3000 +Velocity (km s +1) +60 +40 +20 +0 +20 +40 +(10 +20)erg s +1cm +2Å) +NL3 +6380 +6400 +6420 +6440 +6460 +6480 +6500 +Observed Wavelength(Å) +Fig. A.1. MUSE spectra of all the DSFGs previously detected toward the SPT2349-56 system at z = 4.304. The spectra are centered at the expected +wavelength for Lyman-α line emission. The red vertical line highlights the location of the Lyman-α emission line expected from the previous [Cii] +or CO-based redshift measurement (Miller et al. 2018; Hill et al. 2020). None of the DSFGs are formally detected in Lyman-α emission, and only +mild evidence for such line is seen in some of these spectra. +Article number, page 12 of 14 + +Yordanka Apostolovski et al.: Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.3 +Appendix B: Detected and confirmed Lyman-α emitters maps at different wavelength +23h49m41.6s41.4s +41.2s +41.0s +-56°37'56" +58" +38'00" +RA [J2000] +DEC [J2000] +HST +LAE1 +IRAC +MUSE +23h49m44.6s44.4s +44.2s +44.0s +-56°38'38" +40" +42" +44" +RA [J2000] +DEC [J2000] +HST +LAE2 +IRAC +MUSE +23h49m42.6s42.4s +42.2s +42.0s +41.8s +-56°38'08" +10" +12" +RA [J2000] +DEC [J2000] +HST +LAE3 +IRAC +MUSE +23h49m40.2s40.0s +39.8s +39.6s +-56°38'10" +12" +14" +RA [J2000] +DEC [J2000] +HST +LAE4 +IRAC +MUSE +Article number, page 13 of 14 + +A&A proofs: manuscript no. Ly-alpha_tomography +23h49m43.8s43.6s +43.4s +43.2s +-56°37'00" +02" +04" +RA [J2000] +DEC [J2000] +HST +LAE5 +IRAC +MUSE +23h49m41.2s41.0s +40.8s +40.6s +-56°37'06" +08" +10" +12" +RA [J2000] +DEC [J2000] +HST +LAE6 +IRAC +MUSE +23h49m45.8s45.6s +45.4s +45.2s +45.0s +-56°37'26" +28" +30" +32" +RA [J2000] +DEC [J2000] +HST +LAE7 +IRAC +MUSE +23h49m40.4s40.2s +40.0s +39.8s +-56°37'32" +34" +36" +RA [J2000] +DEC [J2000] +HST +LAE8 +IRAC +MUSE +23h49m45.0s44.8s +44.6s +44.4s +-56°38'38" +40" +42" +RA [J2000] +DEC [J2000] +HST +NL3 +IRAC +MUSE +Fig. B.1. Maps centered of the detected and y tentative Lyman-α emitters. Left: HST F160W. Center: Ultra-deep IRAC mosaic. Right: Moment 0 +of MUSE. +Article number, page 14 of 14 + diff --git a/U9AzT4oBgHgl3EQfX_y1/content/tmp_files/load_file.txt b/U9AzT4oBgHgl3EQfX_y1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..55f6e10849a2d78049c6033ac341a52c1b1aa857 --- /dev/null +++ b/U9AzT4oBgHgl3EQfX_y1/content/tmp_files/load_file.txt @@ -0,0 +1,1824 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf,len=1823 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Ly-alpha_tomography ©ESO 2023 January 5, 2023 Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 Yordanka Apostolovski1, Manuel Aravena2, Timo Anguita3,4, Matthieu Bethermin5, James Burgoyne6, Scott Chapman7, Carlos De Breuck8, Anthony Gonzalez9, Max Gronke10, Lucia Guaita3, Yashar Hezaveh11,12, Ryley Hill7, Sreevani Jarugula13, Evelyn Johnston2, Matt Malkan14, Desika Narayanan9, Cassie Reuter13, Manuel Solimano2, Justin Spilker15, Nikolaus Sulzenauer16, Joaquin Vieira13, David Vizgan13, and Axel Weiß16 1 Instituto de Física y Astronomía, Universidad de Valparaíso, Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Gran Bretaña 1111, Valparaíso, Chile e-mail: yordanka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='apostolovski@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='com 2 Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias, Universidad Diego 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' USA 13 Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 1002 West Green St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Urbana, IL, 61801, USA 14 Department of Physics and Astronomy, University of California, Los Angeles, CA, 90095-1547, USA 15 Department of Physics and Astronomy and George P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' and Cynthia Woods Mitchell Institute for Fundamental Physics and Astron- omy, Texas A&M University, 4242 TAMU, College Station, TX 77843-4242, USA 16 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121, Bonn, Germany January 5, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Deep spectroscopic surveys with the Atacama Large Millimeter/submillimeter Array (ALMA) have revealed that some of the brightest infrared sources in the sky correspond to concentrations of dusty star-forming galaxies (DSFG) at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Among these, the SPT2349-56 protocluster system at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304 is amongst the most extreme examples due to its high source density and integrated star formation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We conducted a deep Lyman-α line emission survey around SPT2349-56 using the Multi-Unit Spectroscopic Explorer (MUSE) at Very Large Telescope (VLT) in order to characterize this uniquely dense environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Taking advantage of the deep three-dimensional nature of this survey, we performed a sensitive search for Lyman-α emitters (LAEs) toward the core and northern extension of the protocluster, which correspond to the brightest infrared regions in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Using a smoothed narrowband image extracted from the MUSE datacube around the protocluster redshift, we searched for possible extended structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We identify only three LAEs at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 in this field, in concordance with expectations for blank-fields, and an extended Lyman-α structure spatially associated with core of the protocluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' All the previously-identified DSFGs in this field are undetected in Lyman-α emission, consistent with the conspicuous dust obscuration in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We find an extended Lyman-α structure, about 60 × 60 kpc2 in size, and located 56 kpc west of the protocluster core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Three DSFGs coincide spatially with the location of this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We conclude that either the three co-spatial DSFGs or the protocluster core itself are feeding ionizing photons to the Lyman-α structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' galaxies – formation: galaxies – intergalactic medium 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Introduction Studies of massive galaxies at the peak of their star-formation activity and their relation to the densest protocluster systems are key to understanding the hierarchical formation of the most mas- sive galaxy structures in the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Current studies seek to understand the role of active galactic nuclei (AGN) feedback (Pike et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Smolˇci´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2017), or the relation between downsizing and star formation (Magliocchetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2015) during the active growth phases of such forming structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Cosmological simulations show that cold dark matter (CDM) haloes merge and form a web-like network traced by young galaxies and reionized gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' A protocluster will form at the high- est overdensity regions within this filamentary structure at early cosmic times (z ∼ 4 − 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Baugh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' De Lucia & Blaizot 2007), eventually becoming a massive virialized cluster by z < 1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Overzier 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' These cosmological simulations indicate Article number, page 1 of 14 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='01328v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='GA] 3 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Ly-alpha_tomography Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Results of the blind line, narrow band and prior selected sample search in the MUSE data cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' ID RA DEC ∆v‡ FWHM S Lyα EW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (J2000) (J2000) (km s−1) (km s−1) (10−20 erg cm−2 s−1) (km s−1) LAB 23:49:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='39 −56:38:23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='49 200 ± 19 760 ± 40 3660 ± 1030 1530 LAE1 23:49:41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='28 −56:37:58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='20 −99 ± 9 330 ± 20 510 ± 150 540 LAE3 23:49:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='18 −56:38:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='48 70 ± 9 330 ± 50 2220 ± 280 1230 LAE8 23:49:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='03 −56:37:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='11 1689 ± 9 330 ± 30 2300 ± 430 1300 Tentative candidates† NL3 23:49:44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='73 −56:38:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='99 −1818 ± 39 270 ± 90 210 ± 150 1705 LAE2 23:49:44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='26 −56:38:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='90 739 ± 19 330 ± 50 190 ± 170 570 LAE4 23:49:39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='90 −56:38:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='22 360 ± 19 380 ± 30 240 ± 170 830 LAE5 23:49:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='47 −56:37:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='37 0 ± 19 440 ± 50 370 ± 240 670 LAE6 23:49:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='96 −56:37:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='18 669 ± 19 310 ± 50 330 ± 200 560 LAE7 23:49:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='41 −56:37:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='70 709 ± 9 320 ± 30 320 ± 220 1170 Notes: † List of Lyman-α line candidates, which showed (snr)det > 5 as computed in the LSDCat detection cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Despite the high snr obtained in the “maximal” LSDCat extraction, these detections are considered tentative based on their low significance measured in the original cube through homogeneous 1′′ radii aperture measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' ‡ Velocity offset relative to the protocluster’s mean [Cii]-derived redshift, z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' that galaxies within galaxy protoclusters experience a luminous starburst-phase (Miley & De Breuck 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' To identify and study these starbursting protocluster sys- tems, several observational methods have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' One of them corresponds to sub/millimeter wavelength observations, which allow one to pinpoint the obscured star-formation activ- ity in young protocluster members (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Aravena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Capak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Oteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Similarly, low- frequency radio observations are typically used to search for radio-loud quasars sitting in the centers of dense protocluster fields (Galametz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In these radio- selected protoclusters, Lyman-α emitters (LAEs), star-forming galaxies selected through their significant UV rest-frame Lyman- α emission line (λrest = 1215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='67Å), show overdensity factors 3 − 5 times larger than the field at the same redshift (Venemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2005, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Given this ubiquity of LAE overdensities in radio-selected fields, deep searches for these sources have been performed to confirm the redshifts of protocluster galaxy candi- dates using 8m-class optical/IR telescopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Pentericci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Kurk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Venemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2002, 2004, 2005, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Croft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The velocity dispersions of radio-selected galaxy protoclus- ters are typically found in the ∼ 300−1000 km s−1 range centered at the mean velocity of the radio galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Although these sys- tems are not yet virialized, such large velocity dispersions sug- gest that these systems have large halo masses, possibly evolving into the most massive cluster systems in the local Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Narrowband image surveys in protocluster fields have iden- tified a population of LAEs with luminosities larger than 1043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4 erg s−1 and large spatial extensions (40 − 150 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' These struc- tures are often referred to as Lyman-α blobs (LABs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Matsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The origin of the emission of these structures can be explained by different scenarios such as the presence of AGN or massive star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The produc- tion of Lyman-α photons in these objects could be associated with different processes such as recombination radiation, contin- uum pumping, or collisional excitation (see;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Cantalupo 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The large-scale millimeter survey covering 2500 squares de- grees of the sky conducted with the South Pole Telescope (SPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Carlstrom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2011) discovered a population of millimeter- bright sources (S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4mm > 20 mJy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Vieira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2010, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Everett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Reuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Follow-up observa- tions with the Atacama Large Millimeter/submillimeter Array (ALMA) showed that the majority of these sources are gravi- tationally lensed submillimeter galaxies (> 90%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' SMGs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' also known as dusty star-forming galaxies, or DSFGs) with magni- fications µ870µm ∼ 5 − 20 (median µ870µm = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 Spilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The remaining sources show no evidence of gravita- tional lensing, being either intrinsically bright or composed of fainter multiple-component SMGs or groups of SMGs (Heza- veh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Spilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The high number of SMGs spread over a small area of the sky (< 1′) found in these fields strongly suggests the existence of (sub)millimeter bright proto- cluster fields (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Among the sample of SPT protoclusters, the SPT2349-56 system stands out due to its exceptionally high surface density of SMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' SPT2349-56 is located at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304 and has a surface density of more than ten times the average blank-field value and a volume density 1000 times the average (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This system could represent the core of a massive galaxy cluster and is one of the most massive structures known to date in the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The SPT2349-56 system has two main infrared (IR) bright structures as seen in the APEX/LABOCA 870 µm maps, following the north-south direction (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The southern component comprises by itself a flux density of S870 ≈ 77 mJy, whereas the northern component contributes with S870 ≈ 33 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For reference, a typical unlensed SMG has a flux density of around 5–10 mJy at 870 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Higher-resolution deep ALMA spectroscopy yielded a total of 24 [Cii] and 16 CO(4-3) line emitters in the southern and northern extensions of the clus- ter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Several components of this system have SFR estimates of ∼ 1000 M⊙ yr−1, while the full protocluster system is estimated to have a SFR of about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6 × 104 M⊙ yr−1 (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Similarly, the dynamical mass of the core region is estimated to be ∼ 9 × 1012 M⊙, while the total halo mass of the whole structure is ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='5 × 1013 M⊙ (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The physical properties of these sources indicate that this protocluster already harbors massive galaxies that are rapidly forming stars from an abundant gas supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The large number of SMGs in this system pushes and challenges theoretical mod- els seeking to explain the origin and evolution of protoclusters (Chiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Due to the proximity of the SMG members in the core of the protocluster (the diameter is about 130 kpc), it is likely that Article number, page 2 of 14 Yordanka Apostolovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' : Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 23h49m48s 44s 40s 36s 56°37\'00" 30" 38\'00" 30" 39\'00" RA [J2000] DEC [J2000] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Deep IRAC mosaic obtained toward the SPT2349-56 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Red contours show the ALMA [Cii] coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Blue squares show the observed MUSE footprint, where we used two pointings to cover the full IR bright region previously detected with LABOCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' its component galaxies will merge to form a massive elliptical galaxy at the core of a lower-redshift Coma-like galaxy cluster (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' A recent search for Lyman Break Galaxies (LBGs) in the extended SPT2349-56 environment found 4 LBGs in the south- ern part of the protocluster (Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2021), indicating that most of the SMGs are inconspicuous at optical wavelengths, with only one of the 4 LBGs coinciding with a previously re- ported SMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Motivated by the significant overdensities found in radio- selected protocluster fields, we conducted an independent cen- sus of star-forming galaxies in the SPT2349-56 field through a sensitive search for Lyman-α emission using deep optical spectroscopy obtained with the Multi-Object Spectroscopy Unit (MUSE) at the Very Large Telescope (VLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In Section 2 we de- scribe the observations and reduction of the MUSE data towards SPT2349-56, and summarize previous observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In Section 3 we present the detection of Lyman-α emission through both a blind search and narrowband imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In Section 4 we analyze the nature of the extended Lyman-α emission along with its con- nection with LAEs and the structure of the protocluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Section 5 summarizes and presents the conclusions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hereafter, we adopt a flat ΛCDM cosmology with h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='677, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='307 and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='693 (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Observations In this section, we describe details of the Lyman-α line observa- tions in the SPT2349-56 protocluster at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' MUSE observations Observations with MUSE at the VLT UT4 were performed in two separate pointings targeting the north and south extensions of the SPT2349-56 protocluster system (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' MUSE covers the wavelength range 480-930 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Each pointing covers roughly 1 square arcmin (60′′ × 60′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' These observations were carried out in the wide field mode (WFM) in service-mode observing as a part of projects 0100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='A-0437(A) and 0100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='A-0437(B) (PI: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Aravena) during dark-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Each pointing was observed for 5 hours (a total of 10 hours) between November 2017 and Septem- ber 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Each pointing consisted of a set of exposures of 680 seconds each, with individual exposures rotated by 90 degrees with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The average seeing of these observations was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='97 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='98 arcsec for the southern and northern pointings, respectively, af- ter correction of air-mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Weather conditions were classified by ESO as clear (CL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 55%), with high wind (CL-WI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 11%) and pho- tometric conditions (PH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 33%) for all observing blocks (OBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We reduced the data using the MUSE pipeline v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6 (Weil- bacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2014) for bias subtraction, flat-fielding, and wave- length and flux calibration, resulting in a single data cube per each of the 5 OBs per field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We combined the five OB data cubes per field using the MUSE Python Data Analysis Frame- work (MPDAF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The data cubes were merged using a sigma-clipped mean with σclip = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Since the field is relatively sparse (especially in Lyman-α at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304), we used Zurich Atmosphere Purge (ZAP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Soto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2016) to perform a sky subtraction through principal component analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For this process, we used a mask in order to avoid spaxels that contained obvious continuum sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Previous ALMA observations In this study, we used as reference the images, cubes and location of protocluster members previously identified through ALMA Cycle 5 and 6 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' These observations and the corre- sponding data reduction and source identification are described in detail by Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2018) and Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2020) and we refer the reader to those papers for full details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In brief, observations of the redshifted [Cii]158µm fine struc- ture line towards the SPT2349-56 system were obtained us- ing ALMA in Band 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' These were centered at a frequency of νobs = 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4 GHz, yielding an average synthesized beam size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='43′′ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='34′′ and 3σ sensitivities of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' These observations, which cover the full IR-bright region, led to the identification of 24 [Cii] emitters in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The MUSE ob- servations described above fully cover the region observed by ALMA in Band 7 at with uniform sensitivity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Based on the identified [Cii] sources, the mean redshift of the system was determined to be at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304 (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' HST imaging We used HST/Wide Field Camera 3 (WFC3)-IR images under program 15701 (PI: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Chapman).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The target was assigned 2 or- bits for the F110W filter and three orbits for the F160W filter in the infrared channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Dithering was implemented for maxi- mum resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The data was reduced using the standard HST pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The pixel size in the WFC3 images is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='075′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Results We used the MUSE observations obtained toward SPT2349-56 to perform a systematic search of Lyman-α emission with three methods: a blind automatic search in the cube, creating a nar- row band image around the known protocluster redshift and a search for Lyman-α emission in (dusty) sources that had previ- Article number, page 3 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Ly-alpha_tomography 70 kpc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Left: Lyman-α emission toward the SPT2349-56 protocluster system at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The MUSE covered area is shown, with the HST F160W image in the background in grey-scale, and red contours representing a rendered Lyman-α image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The later is obtained as the average of each individual line map of the detected LAEs and the LAB, in steps of 2, 5 and 7 σ, where σ is the rms noise level in the average image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The blue circles highlight the location of the ALMA [Cii] and CO(4-3) line detections in the field (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Green squares show the location of the LBGs in the field (Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Right: The map of ALMA [Cii] line emission toward the identified Lyman-α blob (LAB) is shown in the background, with blue circles representing the location of the previously identified [Cii] line emitters C10, C14, C17 (see Table 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Red contours show the Lyman-α emission of the LAB at 2, 4, 6 and 8σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' ously been identified in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Below, we describe each of these searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Blind Search We performed a blind search for Lyman-α emission in the MUSE data cubes, using the Line Source Detection and Cata- loguing Tool (LSDCat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Herenz & Wisotzki 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For this, we focused on a 4000 km s−1 band centered on the red-shifted (z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304) Lyman-α wavelength (λred = 6444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The LSDCat rou- tine detects emission lines through an spatial and spectral filter- ing (3D matched-filtering) approach and sorts them into discrete objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This method is used to maximise the signal-to-noise ra- tio (snr) of the entire cube, and thus creating a snr detection cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' To determine an appropriate threshold for detection in the snr cube, we conducted an unbiased line search also in the negative version of our original cube (multiplied by −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Assuming that the noise in this reduced velocity range of the cube is symmetric around 0 and roughly follows a Gaussian distribution, the detec- tions obtained in the negative cube will set the maximum level at which we expect line features produced by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' From this, we find that the most significant feature in the negative cube is found at (snr)det ∼ 5, thus yielding our detection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This process yields a significant number of positive fea- tures located at the edges of the independent channel images, and with linewidths of one or two channels only, which we re- move from our catalogue as they are unphysically narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' To filter the Lyman-α line candidates from spurious positive fea- tures, we constrain the full-width at half maximum (FWHM) of the detected lines to the range of widths found for the [Cii] and CO(4-3) lines for sources in the field (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020), which correspond to 50 − 600 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' After this selection process, we identified eight LAE candidates, four in the northern and four in the southern pointing (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We extracted the spectra of each of the LAE candidates using apertures with radii of 1 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Due to the more extended spatial nature compared to the other LAEs, we extracted the spectra of sources LAE3 and LAE8 using apertures of 2 arcsec radius (Fig- Article number, page 4 of 14 Yordanka Apostolovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' : Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 LAB LAE8 LAE3 LAE1 LAE7 LAE6 LAE5 LAE4 LAE2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Continuum subtracted spectra of the Lyman-α emission line identified significantly within the MUSE footprint around SPT2359-56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For comparison purposes, the vertical axis has been normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The measured fluxes are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Red dotted line shows the cen- tral velocity of the protocluster (expected redshifted Lyman-α line at λ = 6444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2Å ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The nomenclature of the LAEs does not follow the snr of the emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The blue tag name denotes secure detections while the red tag names denote tentative detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' ure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Based on the significance of each of the line candidates measured in these apertures (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 3 and Table 1), we split the sample in secured LAEs and tentative candidate sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Only three sources are securely detected in this fashion (LAE1, LAE3 and LAE8), and the remaining five sources are thus considered tentative detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' All the spectra were manu- ally inspected and searched for other lines that would point to a lower redshift possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' However, all line detections were con- sistent with Lyman-α at z ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' LAE1 is associated with detections in all available broad- band images, including g, r, i through KS band (see Appendix B in Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' However, if the galaxy is at z ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3, we would expect it to be faint in the g band due to the Lyman break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Inspection of the HST F110W image (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1) sug- gest that the excess g-band emission comes from a foreground object along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Indeed, due to the mismatch be- tween MUSE Lyman-α position and the optical broadband po- sition, Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2022) lists its photometry as upper limits (see their Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The significance of the detected line and the lack of other line features in the MUSE spectrum strongly favour the z ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 spectroscopic confirmation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' As a precedent, note that ALMA source C1 (source ‘A’) appears to be well detected in g-band, since there is a foreground z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='5 galaxy as shown in Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' LAE3 and LAE8 are both undetected in the g-band and have faint detections in the deep r and HST F110W images (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1 and Appendix B in Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' They are significantly detected in Lyman-α without other line identifications, and have considerable EWs compared to the other sources identified in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Based on the redshift implied by the identified Lyman-α lines, we computed a median velocity offset for all the LAEs in the field with respect to the protocluster redshift z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304 (ob- tained from previous [Cii] and CO identifications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The northern and southern LAEs are found to have velocity offsets of ∆v = 930 and ∆v = 430 km s−1 respectively, indicating that the Lyman-α emissions are sys- tematically redshifted from the center of the protocluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Narrowband Image To independently search for line emission in the field, we pro- duced a continuum-subtracted narrowband image using a spec- trally and spatially smoothed version of the MUSE datacube with LSDCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We selected as a central wavelength for the im- age of the Lyman-α line redshifted to z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304 and a width of 2000 km s−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 6401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 − 6487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' As such, this procedure was specifically designed to search for extended emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' As a result, we found an extended Lyman-α structure towards the east of the protocluster core, which we associate with a so-called “Lyman-α blob” (LAB, see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The Lyman-α emission of the LAB subtends a roughly circular region with an area of 10′′ × 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4′′ in the sky (≈ 70 ×70), and is located about 56 kpc east of the center of SPT2349-56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' With a radius of ≈ 5′′ (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4 kpc), this yields an area, πr2 = 3720 kpc2 or ∼ 60 × 60 kpc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' To obtain a spectrum of this extended emission, we draw a polygon around this source, containing all pixels detected above 2σ in the narrowband image (see Figures 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Based on the Lyman-α profile, we find that the extended feature shows a line with a FWHM of about 760 km s−1 and a velocity offset with respect to the [Cii]/CO protocluster redshift of ∆v = 365 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' After integrating along the full width of the line emission we obtain a flux of S Lyα = 3663 × 10−20 erg cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Article number, page 5 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Ly-alpha_tomography 0 100 200 300 400 500 600 Distance (kpc) 3000 2000 1000 0 1000 2000 3000 3v (km s 1) North South LAB ALMA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Velocity offset scaled by √ 3 as an estimate for 3-dimensional velocity, centered at the center of the protocluster (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020) versus projected distance from the 850 µm-weighted centre of the protocluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Orange circles show detections of [Cii] and CO(4-3) with ALMA (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020), green square and triangles show Lyman-α emitters detected and candidates respectively in the extended emission showed by LABOCA observations (Figure 1), blue squares and triangles show the Lyman-α emitters detected and candidates respectively in the southern pointing and the red star shows the Lyman-α blob inside the 90 kpc defined as the effective radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Black lines show the escape velocity from the protocluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The measured velocity offset uncertainties are negligible, and thus errorbars are smaller than the size of the symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' A comparison of the position of the LAB with the location of the previously-identified SMGs in this field (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020) shows that three sources overlap spatially with the western part of the blob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' These sources, called C10, C14 and C17 using the nomenclature of Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2020), were identified based on their bright [Cii] line emission, with fluxes of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='96, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='70 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='93 Jy km s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' These galaxies are not the most luminous in the sample of [Cii] emitters in the SPT2349- 56 system, and are located at about 65 kpc from the center of the protocluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Previously Known Protocluster Members In addition to the independent searches described above, we searched the MUSE datacubes for Lyman-α emission at the posi- tions of the previously-identified SMGs in the SPT2349-56 sys- tem at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' All of these sources have confirmed systemic red- shifts based on the identification of the [Cii] and CO(4-3) lines with ALMA (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For each of these sources, we extracted a spectrum using apertures with radii of 2′′ centered at the location of either the [Cii] and CO(4-3) detections (Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Inside the range of 6000 km s−1 centered at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304 we do not find significant Lyman-α emission in any of the previously-confirmed SMGs in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' However, we do find a tentative detection of Lyman-α emission from one of the ALMA continuum sources in this field for which no redshift confirmation was possible using the [Cii] or CO(4-3) lines, source NL3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This source shows possible Lyman- α emission at a velocity of -1600 km s−1 from the cluster core redshift (z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This velocity is covered by the ALMA CO observations but not by the [Cii] ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Thus, while the ALMA [Cii] observations missed the line, it is possible that either the source is too faint in CO(4-3) emission or the tentative Lyman-α feature is not real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We thus tag this as a tentative candidate here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We note that stacking of the MUSE spectra to yield a con- strain on the average Lyman-α emission in these undetected SMGs is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Several studies have demonstrated that the Lyman-α emission line does not always trace the galaxies’ sys- temic redshifts due to IGM scattering and absorption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Shap- ley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hashimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' There- fore, aligning the MUSE spectra at the [Cii] or CO-derived sys- temic redshifts or correcting them to the rest-frame will yield a diluted Lyman-α stack signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' While corrections for the Lyman- α-derived to the systemic redshifts (or velocities) as a function of equivalent width have been calibrated, these are statistical in na- ture and will not yield the precise redshift/velocities necessary for stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We do use such corrections in the check to see if the identified LAEs are gravitationally bound to the protocluster core (see next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Analysis and discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Lyman-α emitters As we showed in the results section, we have found 8 blindly selected LAEs and one LAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Three of the blind identifications are considered secure and 5 are tentative based on their low sig- nificance (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2, the LAB and four of Article number, page 6 of 14 Yordanka Apostolovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' : Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 1000 500 0 500 1000 1500 Velocity (km s 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='20 S[CII] (mJy) C10 C14 C17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Observed [Cii] line profiles obtained with ALMA for sources C10, C14 and C17 (see, Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020), which overlap spatially with the Lyman-α Blob obtained with MUSE (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The scaled Lyman- α profile is shown in orange, for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The velocity scale refers to z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304 for all emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' the identified LAEs are located in the southern structure, while the rest are located in the north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The locations of the LAEs and candidates range from 95 to 580 kpc from the core of the proto- cluster and the implied velocities for Lyman-α appear redshifted with respect to its systemic redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The significant mass of the central core of the protocluster is ∼ 9 × 1012M⊙, making it possible that most of the galaxies identified in the core neighbourhood are gravitationally bound as already shown by Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' To test whether the identi- fied Lyman-α emitters are also bound, which may yield insights into the future evolution of the LAEs within the SPT2349-56 system, we compare the offset velocities of each of the proto- cluster members as a function of distance from the protocluster core (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For the ALMA-identified protocluster members we show their [Cii]/CO-based velocities, while for the LAEs, we show their Lyman-α-based velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For the later, we obtain a statistical correction to the systemic redshift of each galaxy using the relation described in Verhamme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Galax- ies that have velocities lower than the the escape velocity enve- lope at a given radius from the protocluster core are expected to be bound to the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The southern LAEs appear to be consistently associated to the protocluster core, including the LAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Only one of the identified LAEs in the southern struc- ture (a tentative candidate) resides outside the escape velocity vesc = √2GM/R envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Conversely, most of the northern LAEs, including the secure LAE8 identification, appear to be unbound and redshifted with respect to the protocluster core yet following the trend of the other [Cii] identified sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The fact that the northern LAEs follow the velocity offset of the ALMA [Cii] sources in the north, bolsters the interpretation of the north- ern structure as an unbound/infalling sub-halo (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In addition, this supports the idea that the southern structure, which includes the protocluster core is likely already reaching a virialized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We compare the number of Lyman-α emitters found in the SPT2349-56 field at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 with the field counts using the ultra- deep MUSE observations in the Hubble Ultra Deep Field (In- ami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2017, HUDF mosaic area of 3 × 3 arcmin2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Down to a Lyman-α luminosity of log(LLyα) = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0, corresponding to the depth reached by our SPT2349-56 observations, Drake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2017) finds 144 LAEs in the redshift range z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The MUSE HUDF observations are > 2× deeper than our point- ings, and thus are complete to this depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Considering the red- shift range used to search for LAEs in the SPT2349-56 (z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='25 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='36) and the MUSE covered area (2 arcmin2), we would thus expect to have found 3−4 LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This indicates that the de- tection of 3 secure sources in the SPT2349-56 field are consistent with blank-field counts, and strengthens the case that most of the emission output and mass in this system is associated to heavily dust obscured sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In this scenario, the existence of a LAB in such complex obscured environment appears as a rare case, where the Lyman-α emission is able to escape in a preferential, less obscured direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2 4 6 8 10 12 14 Position (arcsec) 2000 1000 0 1000 2000 Velocity (km s 1) C10 C14 C17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Position-velocity diagram towards the Lyman-α Blob, extracted from MUSE cube at 0 degrees of inclination towards the east.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Blue, red and green circles show the observed [Cii] emissions (C10, C14 and C17) from Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This shows a spatial connection between the LAB and two DSFGs members of the PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Insights on the nature of the extended Lyman-α emission As mentioned previously, the extended Lyman-α emission found in the MUSE-based narrowband image broadly coincides with the position of three SMGs that were identified as part of the pro- tocluster structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This suggests a possible physical relationship between them as indicated by previous studies of LABs at high redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Umehata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Oteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Cen & Zheng (2013) constructed a model for the origin of extended Lyman-α emission in the context of the cold dark mat- ter framework, in which the LAB are produced due to starburst activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The model incorporates AGN feedback, although it is expected that it should have a subdominant contribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Webb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For extended Lyman-α emission in a proto- cluster, each galaxy member contributes to the whole Lyman- α emission yielding a variety of sizes and geometries typically found within a contiguous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The relative contribution of these DSFGs depends on the dust attenuation of Lyman-α pho- tons and the propagation and diffusion process through the cir- cumgalactic medium (CGM) and intergalactic medium (IGM) of each member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In Figure 5 we compare the Lyman-α spectra of the LAB with the [Cii] emission line spectra of the three DSFGs spa- tially coincident to it: C10, C14 and C17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The [Cii] line emis- sion is expected to trace the kinematics of each host galaxy, and thus probe the galaxies’ systemic velocities and geometries (ro- tation, merger, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Due to obscuration and absorption by the intergalactic medium, the Lyman-α spectrum is expected to be Article number, page 7 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Ly-alpha_tomography redshifted with respect to the galaxies’ systemic velocities, and thus with respect to the [Cii] lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In this case, the Lyman-α spectrum of the LAB is found ∼ 300 km s−1 redward of the pro- tocluster core velocity (v = 0 km/s) and ∼ 100 km s−1 of the C14 and C17 SMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' A much larger velocity difference is seen between the LAB and the systemic velocity of the C10 galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The inconsistency in velocities for C10 suggest this source might be unrelated to the Lyman-α emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We explore this issue in more detail in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The protocluster core as the origin for the LAB?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The LAB is located ∼ 56 kpc to the east of the of the proto- cluster core, thus being within the protocluster effective radius defined by Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Along with the velocity connection between the LAB and the SMGs, this spatial coincidence suggest a physical link between the protocluster core and the LAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' It is thus possible that the powering source of the extended Lyman- α emission is star formation or AGN activity in the starbursting SMGs at the protocluster center, where the Lyman-α photons are produced in a photon-ionized medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In this scenario, it is possible that most of the Lyman-α pho- tons along our line of sight are not absorbed and/or scattered but are instead able to escape toward the eastern part of the pro- tocluster core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Indeed, Vernet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2017) observed similar re- gions with offsets of ∼ 100 kpc in the haloes of high redshift AGN-host galaxies, invoking similar arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Following Furlanetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2005), the Lyman-α emission can be used to yield an estimate of the underlying SFR from the powering source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For star formation episodes following a Salpeter initial mass function (Salpeter 1959) and that two thirds of the ionizing photons are absorbed in the dense ISM, we have: LLyα = 1042( SFR/[M⊙ yr−1]) erg s−1 (1) Taking the value of LLyα = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='37 × 1042 erg s−1, we obtain a SFR for the extended emission of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='37 M⊙ yr−1, which is orders of magnitude lower than the SFR estimates for any of the SMGs in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This is consistent with the idea that most (99%) of the UV radiation is obscured by dust within the SMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Recent radio imaging of the SPT2349-56 field using the Australia Telescope Compact Array (ATCA) and the Australian Square Kilometer Array Pathfinder (ASKAP) found strong radio emission from the protocluster core complex (Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The steep radio spectrum found clearly indicates that at least one of the three central sources (B, C and G in the nomenclature used by Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', or C3, C6 and C13 following Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=') host a radio AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This finding supports the idea that enhanced Lyman-α emission at the LAB location is produced by AGN activity at the protocluster core (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Vito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Figure 6 shows a position-velocity (PV) diagram of the Lyman-α emission of the LAB, extracted along the x-axis (0 de- grees of inclination) towards the west of the MUSE datacube, with a slit width of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Here we note a widespread emis- sion along the central velocity with an extension of 5 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' At the edges, for the more distant structure, the emission goes to- wards bluer velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' On the other hand, in the edge closer to the cluster, we have a structure that shifts to positive velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Another important issue is the behavior of the luminosity in the PV diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We can divide the structure into two different blobs: with the western being brighter than the eastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This result is in agreement with the spectral line of the LAB, where we observe 6380 6400 6420 6440 6460 6480 6500 Observed Wavelength (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='030 Flux Density (10 16 erg cm 2 s 1) Model 0 Model 1 Model 2 Model 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Lyman-α spectrum of the LAB compared to the best-fit mod- els that assume different systemic redshifts for the emitting source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The models are described in the text and their best-fit parameters are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The best description of the observed Lyman-α spectrum is given by models 1-3, suggesting that the emission is produced by pho- toionization from either DSFGs C14, C17 or the protocluster core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' that the reddest emission is strongest and wider than the bluest emission (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Modeling the Lyman-α spectrum of the extended emission To further explore the origin of the LAB, we test the idea that either the protocluster core or the galaxies spatially coincident with the LAB are the source of the Lyman-α emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For this, we use the Lyman-α line profile of the LAB and the Lyman-α Monte Carlo Radiate Transfer code tlac (Gronke & Dijkstra 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Gronke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We utilize a expanding shell model, which has been widely used in several studies to success- fully reproduce the Lyman-α profiles of galaxies in different red- shifts and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The model assumes an homogeneous, spherical shell that expands radially outwards, with uniformly mixed neutral gas (HI) and dust (Verhamme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2006), and the emitting source located at the center of the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The shell model is defined by a set of seven parameters including the ex- panding velocity (vexp), the HI column density (NHI), the dust op- tical depth (τd), the effective temperature of the gas (T), the sys- temic redshift of the emitter (zsys), the intrinsic equivalent width of the Lyman-α line (EW(Lyman-α)) and the intrinsic FWHM of the Lyman-α line (FWHM(Lyman-α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' For more details on these parameters, we refer the reader to Gronke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Given a constraint for the systemic redshift of the source and the input Lyman-α spectrum, the code yields the most likely set of param- eters that reproduce the observed spectrum under the assumed geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Since we are interested in learning which of the underlying starburst galaxies might be producing the Lyman-α emission, we constrain zsys using the [Cii]-based redshifts of each of the pos- sible sources of the Lyman-α line: C10 (model 0), C14 (model 1) and C17 galaxies (model 2), and the protocluster core (model 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Instead of simply fixing zsys, we allow for a range in red- shift given by the 3σ uncertainty around measured [Cii] redshift in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Since these ranges overlap, some of the solutions found are similar between each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The results of this proce- dure are shown in Figure 7 and listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We find that the model 0 does not converge into a proper fit to the data, mostly due to the significant difference between the [Cii] redshift and that of the Lyman-α line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This forces the model to a high expansion velocity (∼ 500 km s−1) and low dust optical Article number, page 8 of 14 Yordanka Apostolovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' : Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Results from the radiative transfer modeling of the LAB line profile Model Source† zsys vexp log(NHI) τd log(T) EW(Lyman-α) σ(Lyman-α)‡ (km s−1) (cm−2) (K) (Å) (km s−1) 0 C10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2895 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0019 480+9 −14 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2 798+15 −31 1 C14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3057 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0020 377+28 −46 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='19+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8 328+33 −22 2 C17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3049 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0020 375+17 −31 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8 330+30 −16 3 Core 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3040 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0020 310+36 −22 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='9 346+42 −35 Notes: † Source assumed to be producing the Lyman-α emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Its [Cii] redshift is assumed to be the systemic redshift of the system for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' ‡ σ = FWHM/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This solution is less preferred, since the assumed emitting source (the C10 galaxy) is a gas-rich, dusty galaxy contrary to the result of a low dust optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The best fits are produced when using higher systemic red- shifts (models 1−3), which are more consistent with the redshift of the Lyman-α line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This is the case for sources C14 and C17, and the protocluster core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In these cases, the solutions are simi- lar, yielding high outflow velocities (∼ 300−400 km s−1) and yet very low HI column densities (log(N(HI))∼ 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In these cases, the opacities appear to be moderate (τ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='5−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0), yet more con- sistent with the dusty nature of the purported emission sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Based on these results alone it is hard to disentangle the origin of the LAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' However, if the protocluster core starbursting galaxies are producing the Lyman-α emission it would require a complex patchy geometry where some of the UV radiation escapes and illuminates the HI gas in the LAB direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' While this is a plau- sible scenario, supported by the moderate optical depth of this solution (model 3), such solution is less likely than the scenario where the UV radiation is produced in-situ by either the C14, the C17 an/or both galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Summary and conclusions We presented a census of Lyman-α emission toward the IR- bright protocluster SPT2349-56 at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 obtained using MUSE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Through a blind search of Lyman-α emission to- wards the protocluster core and northern extension, we found three LAEs at distances > 90 kpc from the protocluster core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The LAEs are bound to the 9 × 1012M⊙ protocluster core and all of them are redshifted relative to SPT2349-56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Only one of the ALMA SMGs previously identified in this field is tentatively detected in Lyman-α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Using a continuum-subtracted narrowband image we detect extended Lyman-α emission, which we refer to as a LAB, with a size of about 70 kpc across, located at ∼ 56 kpc to the east of the protocluster core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The bulk of the LAB emission is also red- shifted with respect to the core of the protocluster, in agreement with a red-skewed asymmetric profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Two of the spatially overlapping DSFGs C14 and C17, are found to also coincide spectrally, when comparing their [Cii] emission lines with that of the Lyman-α emission from the LAB (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This observation could be explained by the high star-formation activity seen in the DSFG protocluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Based on their locations and redshifts, the main suspects to be producing the ionizing photons and thus the Lyman-α emission are the C14 and C17 DSFGs, or the proto- cluster core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' In the later case, the geometry of the dust distribu- tion should allow the Lyman-α photons to get scattered from the core such that the photons find a region to escape to the east.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Such scenarios are supported by radiative transfer modeling of the Lyman-α line profile of the LAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We do not find an overdensity of LAEs, or a source density comparable to what we might have expected from the number of [CII] and submillimeter continuum sources found in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' We interpret this as a structure that is still heavily dust obscured and dominated by submm-detected galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This paper makes use of the following ALMA data: ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='ALMA#2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='00273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' and ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='ALMA#2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='00058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' acknowledges partial support from Comité Mixto ESO - Gobierno de Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' MA acknowledges support from FONDECYT grant 1211951, CONICYT + PCI + INSTITUTO MAX PLANCK DE ASTRONOMIA MPG190030 and CONICYT+PCI+REDES 190194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' This work was partially funded by the ANID BASAL project FB210003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' acknowledges support from the Millennium Science Initiative ICN12_009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' acknowledges support from the US NSF under grant 1715206 and Space Telescope Science Institute under grant AR-15043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' acknowledge support from the US NSF under grants AST-1715213 and AST-1716127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' acknowledge support from the US NSF NRAO under grants SOSPA5-001 and SOSPA7-006, and SOSPA4-007, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' J.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2014, MNRAS, 437, 1882 Rotermund, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Chapman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2014, ApJ, 791, 3 Soto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Lilly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Bacon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Richard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', & Conseil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2016, MNRAS, 458, 3210 Spilker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Marrone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Aravena, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2016, ApJ, 826, 112 Steidel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Adelberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Shapley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2000, ApJ, 532, 170 Umehata, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Tamura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Kohno, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2015, ApJ, 815, L8 Venemans, B.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Röttgering, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Miley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2005, A&A, 431, 793 Venemans, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Röttgering, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Miley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} 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E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2010, ApJ, 719, 763 Vieira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Marrone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2021, MNRAS, 508, 3754 Webb, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Yamada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2009, ApJ, 692, 1561 Weilbacher, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Streicher, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', Urrutia, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2014, Astronomical Society of the Pacific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 485, The MUSE Data Reduction Pipeline: Status after Preliminary Acceptance Europe, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Manset & P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Forshay, 451 Article number, page 10 of 14 Yordanka Apostolovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' : Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 Appendix A: Lyman-α spectra toward the SPT2349-56 DSFGs The following figures show the observed MUSE spectra toward all the DSFGs in the SPT2349-56 system, centered at the expected location of the Lyman-α line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6420 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6440 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6460 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6480 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='Observed Wavelength(Å) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' MUSE spectra of all the DSFGs previously detected toward the SPT2349-56 system at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The spectra are centered at the expected wavelength for Lyman-α line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' The red vertical line highlights the location of the Lyman-α emission line expected from the previous [Cii] or CO-based redshift measurement (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' None of the DSFGs are formally detected in Lyman-α emission, and only mild evidence for such line is seen in some of these spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Article number, page 12 of 14 Yordanka Apostolovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' : Extended Lyman-α emission towards the SPT2349-56 protocluster at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='3 Appendix B: Detected and confirmed Lyman-α emitters maps at different wavelength 23h49m41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6s41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4s 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2s 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0s 56°37\'56" 58" 38\'00" RA [J2000] DEC [J2000] HST LAE1 IRAC MUSE 23h49m44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6s44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4s 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2s 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0s 56°38\'38" 40" 42" 44" RA [J2000] DEC [J2000] HST LAE2 IRAC MUSE 23h49m42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6s42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4s 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2s 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0s 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8s 56°38\'08" 10" 12" RA [J2000] DEC [J2000] HST LAE3 IRAC MUSE 23h49m40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2s40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6s 56°38\'10" 12" 14" RA [J2000] DEC [J2000] HST LAE4 IRAC MUSE Article number, page 13 of 14 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' Ly-alpha_tomography 23h49m43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8s43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6s 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4s 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2s 56°37\'00" 02" 04" RA [J2000] DEC [J2000] HST LAE5 IRAC MUSE 23h49m41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2s41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0s 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8s 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6s 56°37\'06" 08" 10" 12" RA [J2000] DEC [J2000] HST LAE6 IRAC MUSE 23h49m45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8s45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6s 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4s 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2s 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0s 56°37\'26" 28" 30" 32" RA [J2000] DEC [J2000] HST LAE7 IRAC MUSE 23h49m40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4s40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='2s 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8s 56°37\'32" 34" 36" RA [J2000] DEC [J2000] HST LAE8 IRAC MUSE 23h49m45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='0s44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='8s 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='6s 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content='4s 56°38\'38" 40" 42" RA [J2000] DEC [J2000] HST NL3 IRAC MUSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQfX_y1/content/2301.01328v1.pdf'} +page_content=' B.' metadata={'source': 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The result of going long +in a daily leveraged ETF for more than one day is less clear. Here, bounds +are given for the log-returns of a leveraged ETF when going long for more +than just one day. The bounds are quadratic in the daily log-returns of +the underlying benchmark index, and they are used to find sufficient con- +ditions for outperformance and underperformance of a leveraged ETF +in relation to its underlying benchmark index. Results show that if the +underlying benchmark index drops 10+% over the course of 63 consecu- +tive trading days, and the standard deviation of the benchmark index’s +daily log-returns is no more than .015, then going long in a -3x lever- +aged ETF during that period gives a log-return of at least 1.5 times +the log-return of a short position in the underlying benchmark index. +Results also show promise for a 2x daily leveraged S&P 500 ETF. If the +average annual log-return of the S&P 500 index continues to be at least +.0658, as it has been in the past, and the standard deviation of daily +S&P 500 log-returns is under .0125, then a 2x daily leveraged S&P 500 +ETF will perform at least as well as the S&P 500 index in the long-run. +Keywords: Leveraged ETFs; Leveraged exchange traded funds; Inverse +leveraged ETFs, Returns estimation +1 Introduction +A daily leveraged exchange traded fund amplifies the daily return of its under- +lying benchmark index between adjusted closing prices (adjusted closing prices +account for stock splits and dividends). For example, consider an underly- +ing benchmark index returning 1% between two consecutive adjusted closing +∗Department of Mathematics and Statistics, University of Nevada, Reno +Email address: haydenb@nevada.unr.edu +ORCID: 0000-0002-2975-2711 +1 +arXiv:2301.03186v1 [q-fin.MF] 9 Jan 2023 + +2 +Long-Term Returns Estimation of Leveraged Indexes and ETFs +prices. Then a daily leveraged ETF with leverage multiple 2x or -3x would +return roughly 2% or -3%, respectively, depending on the expense ratio, fund +management and whether it is trading at a premium or discount on the close. +The results presented here aim at adressing the viability of going long in a daily +leveraged ETF for more than one day. Their advantage over existing results +in the literature is that they bound the leng-term log-return of a daily lever- +aged ETF, and they depend only on the daily leveraged ETF’s expense ratio +and the bounds, mean and standard deviation of the underlying benchmark +index’s daily log-returns. +It is now commonplace for investors to buy-and-hold an ETF tracking a +large market index like the S&P 500. Over a long period of time, like 40 years, +investors are confident that such an ETF will provide a positive log-return that +is favorable over most other available ETFs. But can the same be said about +a daily leveraged S&P 500 ETF? One of the main goals here is to determine +when a daily leveraged ETF will outperform its underlying benchmark index +in the long-run. +When a stock or ETF looks overvalued, investors may take on a short +position, hoping the price will drop in the near future. Alternatively, investors +could buy-and-hold an inverse daily leveraged ETF during that period. Here, +conditions are given indicating when the latter option offers a superior return. +There are toy examples where a portfolio that invests L% in a particular +stock or ETF and (1 − L)% in cash outperforms a portfolio that goes all in +the stock or ETF. In general, this outperformance occurs when volatility is +high enough. The results presented here show when this outperformance is +impossible, with the goal being to validate the long-term portfolio that invests +100% in a S&P 500 ETF over a portfolio that reduces exposure to the S&P +500. +1.1 Literature review +Several empirical studies have measured the returns of daily leveraged ETFs +over longer time spans than just one day. The general consensus is that lever- +aged ETFs track the leveraged multiple of their benchmark indexes’ returns +well in the short term but deviate in the long term. Since the deviations can be +markedly negative, there is considerable risk associated with a long position +in a leveraged ETF. +Over time spans of 1, 3, 5 and 10 years, a 2x daily leveraged S&P 500 +ETF offers a moderate increase to expected return at the cost of a significant +increase in standard deviation Trainor Jr and Baryla Jr (2008). Over time +spans no longer than one month, 2x and -2x daily leveraged ETFs generally +provide 2x and -2x, respectively, the return of the underlying benchmark index +Lu et al (2009). For time spans longer than one month, serious deviations start +to happen. Those deviations are attributed, in part, to the quadratic variation +of the underlying benchmark index. On the other hand, Bansal and Marshall +(2015) show that, for investment horizons of 1 calendar year from 1964 to +2013, the average difference between a leveraged S&P 500 ETF’s return and + +Long-Term Returns Estimation of Leveraged Indexes and ETFs +3 +the underlying benchmark index’s return multiplied by the leverage amount is +greater than 0. So there is clearly potential for a daily leveraged S&P 500 ETF +to provide significant amplification of return over a calendar year. The results +presented here complement these empirical findings by estimating returns of +a daily leveraged ETF for investment horizons having any number of days. +Most theoretical results address a long-term position in a continuously +leveraged ETF. Provided the underlying benchmark index follows a geometric +Brownian motion, leveraged ETFs appear to cause value destruction in the +long-run Cheng and Madhavan (2009). At a minimum, leveraged ETFs do not +achieve their leverage multiple in the long-run Jarrow (2010). The risk of a +leveraged ETF is measured in Leung and Santoli (2012), and admissible lever- +age multiples are given accordingly. Using continuous leverage, Giese (2010) +shows that dynamic adjustment of the leverage multiple based on market con- +ditions leads to outperformance of the underlying benchmark index in the +long-run. In reality, daily leveraged ETFs do not implement continuous lever- +age, so the forementioned results cannot be applied without assuming some +level of error. The results presented here do not use continuous leverage to +avoid this error. +Other theoretical results address a long-term position in an ETF that is +leveraged discretely in time. An approximation to the long-term return of a +daily leveraged ETF is given by Avellaneda and Zhang (2010) for investment +horizons of less than one year. It is based on the leverage multiple and the +mean and variance of the underlying index’s daily returns. Empirically, this +approximation has been shown to be very accurate for quarterly horizons. +However, it is not an upper or lower bound. The approximations presented here +are advantageous because they are upper and lower bounds, which facilitates +the provision of sufficient conditions for outperformance and underperformance +of a daily leveraged ETF relative to its underlying benchmark index. +During the financial crisis from 2008 to 2009, daily leveraged ETFs did not +generally meet their target multiple of daily returns, even on a daily basis Shum +and Kang (2013). Similar finding are in Tang and Xu (2013). These errors can +be attributed to management and trading premiums/discounts, and the effect +is a reduction in the magnification of daily returns. For example, 2x and -2x +daily leveraged S&P 500 ETFs were more like 1.9x and -1.9x daily leveraged +ETFs during the financial crisis. These errors are not considered here because +their randomness is difficult to incorporate into theoretical results. However, +the results can account for such errors, to some extent, with an increased +expense ratio. +Based on simulation of 3x and -3x daily leveraged S&P 500 ETFs, it +appears that a combination of volatility and market condition (sideways, +upward-trending or downward-trending) of the S&P 500 index determines +long-term performance of the leveraged ETF Charupat et al (2022). The results +presented here provide a theoretical foundation for these simulation-based +findings. + +4 +Long-Term Returns Estimation of Leveraged Indexes and ETFs +Daily leveraged ETFs are certainly popular, but it appears that their +present use by institutions is leading to poor performance relative to portfolios +that avoid daily leveraged ETFs DeVault et al (2021). In other words, recent +attempts by institutions to time the market with their leverage ETF hold- +ings are backfiring. The results presented here are aimed at providing further +guidance on when a leveraged ETF is worth having in a portfolio. +1.2 Main results +Lower and upper bounds are given for the log-return of a daily leveraged index +over n consecutive trading days. The bounds are expressed quadratically in +terms of the daily log-returns of the underlying benchmark index. In particular, +the bounds are of the form n(am2 +bm1 +c), where m2 is the average squared +daily log-return of the benchmark index, m1 is the average daily log-return of +the benchmark index, and a, b and c are constants. The results cover a range +of leverage multiples, including the popular -3x, -2x, 2x and 3x. +1.3 Applications +Sufficient conditions are given for the log-return of a daily leveraged index or +ETF to be some multiple, L0, of the log-return of its underlying benchmark +index, over n consecutive trading days. Here, ETFs are distinguished from +indexes because they have expense ratios. To simplify notation, let RL +n,r denote +the log-return of a daily leveraged ETF after n consecutive trading days, with +leverage multiple L and expense ratio r. Now the goal of applications can be +expressed more concisely: to provide sufficient conditions for RL +n,r to be at least +or at most L0R1 +n,0. +First, thresholds are given for m1/m2, indicating when RL +n,r is at least or +at most L0R1 +n,0. Here, m1 and m2 are as in Section 1.2. Special attention is +given to the thresholds for 1 < L because 2x and 3x leverage multiples are so +popular. +Let s denote the standard deviation of the underlying benchmark index’s +daily log-returns. For L > 1 and L0 < L, an upper bound is given on s, +indicating when RL +n,r ≥ L0R1 +n,0. Taking L = 2, 3 and L0 = 0, 1 is especially +important for practical reasons, because the upper bound on s indicates when +a 2x or 3x daily leveraged ETF will have a non-negative log-return or perform +at least as well as its underlying benchmark index. If the average annual log- +return of the S&P 500 index continues to be at least .0658, as it has been in +the past, daily percentage changes between adjusted closing prices are at least +-20%, and the standard deviation of daily log-returns is under .0125, then a +2x daily leveraged ETF will perform at least as well as the S&P 500 index in +the long-run. +For L < L0 < 0, an upper bound is given on s, indicating when RL +n,r ≥ +L0R1 +n,0. The focus here is on L0 = −1 because then the latter inequality +indicates when a daily inverse leveraged ETF performs at least as well as a +short position in its underlying benchmark index. For example, results show + +Long-Term Returns Estimation of Leveraged Indexes and ETFs +5 +that if the benchmark index drops 10+% over the course of 63 consecutive +trading days, daily percentage changes between adjusted closing prices are at +most 15%, and s ≤ .015, then going long in a -3x leveraged ETF during that +period gives a log-return of at least 1.5 times the log-return of a short position +in the benchmark index. Furthermore, if the 10+% drop happens faster, then +the 1.5 multiple of log-returns can be achieved with even larger s. +For 0 < L < 1, an upper bound is given on s, indicating when RL +n,0 ≤ R1 +n,0. +Note that a log-return of RL +n,0 can be achieved via daily rebalancing with +L% in the benchmark index and (1 − L)% in cash. Interestingly, this theory +easily extends from daily leverage to longer periods like weekly leverage, where +rebalancing occurs weekly, and quarterly leverage, where rebalancing occurs +quarterly. If the S&P 500 continues to have an average annual log-return of at +least .0658, then the standard deviation of its daily, weekly, monthly, quarterly, +semi-annual or annual log-returns would have to exceed .02, .04, .08, .15, .2 or +.35, respectively, for a portfolio rebalancing daily, weekly, monthly, quarterly, +semi-annually or annually, respectively, with .64 < L < 1, to outperform the +benchmark L = 1 in the long-run. It seems unlikely for this level of volatility to +persist in the long-run, so maintaining a L : (1−L) portfolio in the benchmark +and cash with 64 < L < 1 is not advised under any standard rebalancing +schedule. +1.4 Organization +Section 2 lays out the notation and framework for the returns of daily leveraged +indexes and ETFs. Section 3 provides main results, and Section 4 applies those +results. Data used in applications is described in Section 4.1. Section 5 provides +closing remarks, including a discussion of related future research ideas. Last, +A provides proofs of the theorems stated in Section 3. +2 Preliminaries +Let Ci denote the adjusted closing price of trading day i for a particular +stock market index I. Then {Ci}n +i=0 is a sequence of adjusted closing prices +for n + 1 consecutive trading days. Note that adjusted closing prices account +for dividends and stock splits, but not inflation. For example, suppose a stock +has a closing price of $100 on day 1, a closing price of $98 on day 2, and a +dividend distribution of $1 on day 2. Then the adjusted closing prices will be +$100 on day 1 and $99 on day 2. Suppose there is a 2-for-1 stock split on day +3 and the closing price on day 3 is $49. Then the adjusted closing price of day +3 will be $99 = 2 · $49 + $1. +Let Xi = Ci/Ci−1 − 1 for i = 1, ..., n. Then {100 · Xi}n +i=1 is the sequence +of n percentage changes between adjusted closing prices. Observe that +n +� +i=1 +(1 + Xi) = Cn +C0 +. + +6 +Long-Term Returns Estimation of Leveraged Indexes and ETFs +Denote the daily leveraged version of I as LxI, where L indicates the amount +of leverage. For example, 3xI indicates the index tracking I with 3x daily +leverage. The closing prices of LxI are given by +CL +i := C0 · +i� +k=1 +(1 + LXk), +i = 0, ..., n. +So the log-returns realized by going long in LxI from the close of trading day +0 to the close of trading day n are given by +log CL +n +C0 += +n +� +i=1 +log(1 + LXi). +Note that here, log refers to the natural logarithm. Let Yi = log(1 + Xi) for +i = 1, ..., n. Then Yi is the log-return for day i, and +log Cn +C0 += +n +� +i=1 +Yi, +log CL +n +C0 += +n +� +i=1 +log(1 + L(exp Yi − 1)). +To shorten notation, let +m1 = 1 +n +n +� +i=1 +Yi, +m2 = 1 +n +n +� +i=1 +Y 2 +i , +s = +��n +i=1(Yi − m1)2 +n +. +Denote the ETF version of LxI as LxIr, where r is the annual expense +ratio, compounded on a daily basis. Assuming 252 trading days in a year, the +log-return of LxIr after n days is given by +RL +n,r := log CL +n +C0 +− n log +� +1 + +r +252 +� +. +3 Main Results +Theorems 1, 2, 3 and 4 provide lower and upper bounds for the log-return of +LxI. The bounds are expressed quadratically in terms of the Yi. Theorem 1 +covers L > 1, Theorems 2 and 3 cover 0 < L < 1, and Theorem 4 covers L < 0. +Theorem 1 Fix L > 1 and log(1 − L−1) < y0 < y1. Then, provided y0 ≤ Yi ≤ y1 +for i = 1, ..., n, log-returns of LxI from the close of trading day 0 to the close of +trading day n are bounded as follows: +sup +y0 L−1 for each i. This, in turn, makes each daily +leveraged return 1 + LXi well-defined (i.e. non-negative). +Remark 2 In Theorem 1, having y = 0 and y0 < 0 < y1 simplifies the expressions +for ak, bk, and ck considerably. In particular, +ak = 1 +yk +� +log(1 + L(exp yk − 1)) +yk +− L +� +, +bk = L, +ck = 0. +Theorem 2 Fix 0 < L < 1 and y0 < y1 < log(L−1−1). Then, provided y0 ≤ Yi ≤ y1 +for i = 1, ..., n, log-returns of LxI from the close of trading day 0 to the close of +trading day n are bounded as follows: +sup +y0 L +L log(1 − L−1). +Thus, the log-return of LxI can never exceed L times the log-return of I. Note that +this upper bound follows the fact that log(1+Lx) ≤ L log(1+x) where the logarithms +are well-defined. +For L ∈ [0, 1], the upper bound just described is a lower bound, i.e. +L log Cn +C0 += Lnm1 ≤ log CL +n +C0 +. +4 Applications +Applications first use Theorem 1 and Remark 2 to provide thresholds for +m1/m2 indicating when LxI has a log-return that is at least or at most L0 +times the log-return of I. The focus is on 1 < L and L0 < L, but similar +thresholds exist for arbitrary L and L0 based on Theorems 2, 3 and 4. Note +that m1/m2 is akin to a Sharpe ratio, since m1 and m2 denote means of the +Yi and Y 2 +i , respectively. +Next, main results are used to to provide sufficient conditions for the log- +return of LxIr to be at least L0 times the log-return of I. For practical reasons, +the focus is on two cases: (1 < L, L0 < L) and (L < L0 < 0). The former case +is aimed at indicating when a leveraged ETF like 2xIr or 3xIr outperforms +I. The latter case is aimed at indicating when an inverse leveraged ETF like +-2xIr or -3xIr outperforms a short position in I. +Last, Theorems 2 and 3 are used to provide sufficient conditions for the log- +return of LxI to be at most the log-return of I. Here, the focus is on 0 < L < 1 +because then the log-return of LxI can be achieved by maintaining a portfolio +with L% in I and (1 − L)% in cash. Interestingly, this theory easily extends +from daily leverage to longer periods like weekly leverage, where rebalancing +occurs weekly, and quarterly leverage, where rebalancing occurs quarterly. +4.1 Data +Applications use the average annual real log-return of the S&P Compos- +ite Index from 1871 to 2020, which is .0658. Here, S&P Composite Index +refers to three indexes: Cowles and Associates from 1871 to 1926, Stan- +dard & Poor 90 from 1926 to 1957 and Standard & Poor 500 from 1957 +to 2020. The Cowles and Associates and S&P 90 indexes are backward +extensions of the S&P 500 index used to extrapolate a longer term aver- +age annual real log-return of the S&P 500 index. The data was taken +from http://www.econ.yale.edu/∼shiller/data.html and is collected for easy +access at https://github.com/HaydenBrown/Investing. For an overview of + +Long-Term Returns Estimation of Leveraged Indexes and ETFs +9 +Table 1 Data variable descriptions +Notation +Description +P +average monthly close of the S&P composite index +D +dividend per share of the S&P composite index +J +January consumer price index +the S&P 500, see https://www.spglobal.com/spdji/en/indices/equity/sp-500/. +Relevant variables from the data are described below. Inflation and dividend +adjusted (i.e. real) annual returns are computed using the consumer price +index, the S&P Composite Index price and the S&P Composite Index divi- +dend. Use the subscript k to denote the kth year of J, P and D. Then the real +return for year k is given by ((Pk+1 +Dk)/Pk)·(Jk/Jk+1). Note that the aver- +age annual total log-return (adjustment for dividends but not inflation) of the +S&P Composite Index from 1871 to 1926 is greater than .0658, since inflation +has been far more common than deflation in the past. +4.2 Thresholds for m1 +m2 +Let 1 < L, L0 < L and log(1−L−1) < y0 < y1. Observe that L0 log(Cn/C0) = +L0nm1. It follows from Theorem 1 and Remark 2 that L0 log(Cn/C0) ≤ +log(CL +n /C0), provided L0m1 ≤ a0m2 + Lm1 and y0 ≤ Yi ≤ y1 for i = 1, ..., n. +If at least one Yi is non-zero, then m2 is positive and +L0m1 ≤ a0m2 + Lm1 ⇐⇒ +−a0 +L − L0 +≤ m1 +m2 +. +(1) +Of special interest are the cases L0 = 0 and L0 = 1. When L0 = 0, satisfaction +of (1) indicates LxI has a non-negative log-return. When L0 = 1, satisfaction of +(1) indicates LxI has a log-return that is at least the log-return of I. Moreover, it +is not hard to see that when L0 = 1, satisfaction of (1) indicates the log-return +of LxIr is at least the log-return of 1xIr. Figure 1 illustrates the threshold +−a0(L − L0)−1 for various y0, L and L0. The horizontal axis is in terms of +100(exp(y0)−1) for the sake of interpretation. Observe that Yi ≥ y0 if and only +if 100Xi ≥ 100(exp(y0)−1). So if Yi ≥ y0 for i = 1, ..., n, then 100(exp(y0)−1) +indicates the lower bound on the daily percentage changes between adjusted +closing prices. +Using similar logic, L0 log(Cn/C0) ≥ log(CL +n /C0), provided y0 ≤ Yi ≤ y1 +for i = 1, ..., n and +−a1 +L − L0 +≥ m1 +m2 +. +Using Theorems 2, 3 and 4, similar thresholds for m1/m2 can be made for +arbitrary L and L0, indicating when L0 log(Cn/C0) is at least or at most +log(CL +n /C0). + +10 +Long-Term Returns Estimation of Leveraged Indexes and ETFs +Fig. 1 Illustrates +−a0 +L−L0 for various y0, L and L0. The dashed line represents L = 2, and +the solid line represents L = 3. +4.3 Outperformance of Daily Leveraged Indexes and +ETFs +In general, Theorems 1, 2, 3 and 4 can be used to provide sufficient conditions +for the log-return of LxIr to be at least or at most some multiple, denoted L0, +of the log-returns of I, i.e. +L0 log(Cn/C0) ≤ RL +n,r +or +L0 log(Cn/C0) ≥ RL +n,r. +Rather than detail said sufficient conditions for the many cases that arise when +considering general L and L0, it is more worthwhile to focus on a few cases of +practical interest. +The goal here is to provide sufficient conditions for the log-return of LxIr +to be at least L0 times the log-return of I, i.e. +L0 log(Cn/C0) ≤ RL +n,r. +(2) +Furthermore, only two cases of L and L0 are considered for practical interest: +1. 1 < L and L0 < L, +2. L < L0 < 0. +4.3.1 Case (i) +Fix log(1 − L−1) < y0 < y1, and suppose y0 ≤ Yi ≤ y1 for i = 1, ..., n. Recall +that L0 log(Cn/C0) = L0nm1. By Theorem 1, (2) follows, provided +L0m1 ≤ sup +y0 y0 +with a fine mesh. Interestingly, the y that produces the approximate supre- +mum is close to 0 in each case. Recall that the lower bound of Theorem 1 is +constructed to be especially close to the actual leveraged log-return when the +Yi are close to y. Having y ≈ 0 makes the lower bound especially accurate +when the Yi are close to 0. If a quadratic lower bound is selected from Theorem +1, but the y is selected based on intuition rather than taking the supremum, +then it makes sense to choose y ≈ 0, because daily log-returns should have a +mean close to 0 in the long-run. +Figure 4 shows that, for y0 = log(1−.2), L0 ≥ 1.6 and r = .0095 , L = 3 has +a larger upper bound for s. If one is trying to achieve at least 1.6 times the log- +return of I with a leveraged ETF, it appears that 3xIr can do so while allowing +a higher standard deviation of the daily log-returns of I, when compared to +2xIr. Note, the word “appears” is used because the main results only provide +sufficient conditions for outperformance. For L0 < 1.6, the situation can be +reversed, depending on the value of 252m1. For example, if one is trying to +achieve 1.4 times the log-return of I, and 252m1 > .04, then it appears that +2xIr can do so while allowing a higher standard deviation of the daily log- +returns of I. Furthermore, if one is trying to achieve at least the log-return of +I, then it appears that 2xIr is far more forgiving than 3xIr, in terms of the +allowed standard deviation of the daily log-returns of I. +Based on Section 4.1, the average annual real log-return of the S&P Com- +posite Index from 1871 to 2020 is .0658 (recall that real log-return indicates +adjustment for dividends and inflation). Assuming this trend continues, infla- +tion occurs in the long-run, daily percentage changes between adjusted closing +prices are at least -20%, and the standard deviation of daily log-returns is +under .0125, Figure 2 indicates that a 2x daily leveraged S&P 500 ETF with +a .95% expense ratio will perform at least as good as the benchmark S&P +500 index in the long-run. Moreover, the situation only improves when there + +12 +Long-Term Returns Estimation of Leveraged Indexes and ETFs +Fig. 2 Illustrates (4) for various y0, L, L0 and r. The dashed line represents y0 = log(1−.2) +(minimum daily percentage change is -20%), and the solid line represents y0 = log(1 − .1) +(minimum daily percentage change is -10%). Black indicates r = 0 and red indicates r = +.0095 (expense ratio is .95%). +is persistent inflation, in which case the upper bound on the standard devia- +tion of daily log-returns increases. On the other hand, if one takes the position +that the benchmark S&P 500 index is unbeatable in the long-run, then it is +necessary for the standard deviation of daily log-returns to be at least .0125, +provided an average annual real log-return of 0.0658, persistent inflation, and +daily percentage changes between adjusted closing prices of at least -20%. This +means that if the benchmark S&P 500 index is unbeatable in the long-run, +average annual log-returns continue to be at least 0.0658, and daily percent- +age changes between adjusted closing prices are not too extreme, then daily +log-returns must be quite volatile. + +Lo = 0, L = 2 +Lo = 0, L = 3 +0.025 +0.015 +0.020 +0.010 +900'0 +0.05 +0.10 +0.15 +0.20 +0.05 +0.10 +0.15 +0.20 +252m1 +252m +Lo = 1, L = 2 +Lo = 1, L = 3 +0.020 +0.020 +0.015 +0.010 +0.010 +900'0 +0.05 +0.10 +0.15 +0.20 +0.05 +0.10 +0.15 +0.20 +252m1 +252mLong-Term Returns Estimation of Leveraged Indexes and ETFs +13 +Fig. 3 Illustrates (4) for various y0, L, L0 and r. The dashed line represents y0 = log(1−.2) +(minimum daily percentage change is -20%), and the solid line represents y0 = log(1 − .1) +(minimum daily percentage change is -10%). Black indicates r = 0 and red indicates r = +.0095 (expense ratio is .95%). +4.3.2 Case (ii) +Fix y0 < y1 < log(1 − L−1), and suppose y0 ≤ Yi ≤ y1 for i = 1, ..., n. Recall +that L0 log(Cn/C0) = L0nm1. By Theorem 4, (2) follows, provided +L0m1 ≤ sup +y 0) reveals that (8) is equivalent to +s ≤ inf +y 0) reveals that (10) is equivalent to +s ≤ inf +y0 y0. The values +for a0, b0 and c0 given in the statement of the theorem are found by solving +the system f(y0) = p(y0), f(y) = p(y) and df +dx(y) = dp +dx(y). From here, the goal +is to show p(x) ≤ f(x) for all x ∈ [y0, ∞). Then the result will follow because +n(am2 + bm1 + c) = +n +� +i=1 +p(Yi), +log CL +n +C0 += +n +� +i=1 +f(Yi). +By Rolle’s Theorem, there is a x∗ ∈ (y0, y) such that df +dx(x∗) − dp +dx(x∗) = 0. +Next observe that +d3 +dx3 [f(x) − p(x)] = (1 − L)L(1 − L(exp x + 1)) exp x +(1 + L(exp x − 1))3 +. +It follows that +d +dx[f(x) − p(x)] is strictly convex, since L > 1 and x ≥ y0 > +log(1 − L−1). By strict convexity, the only zeros of +d +dx[f(x) − p(x)] are x∗ and +y. Moreover, x∗ is a local max, and y is a local min. Combining this with the +fact that y0 and y are zeros of f − p reveals that f − p ≥ 0. +For the upper bound, redefine p, f : (log(1−L−1), y1] → R such that p(x) = +a1x2 + b1x + c1 and f(x) = log(1 + L(exp x − 1)). Let y ∈ (log(1 − L−1), y1). +The values for a1, b1 and c1 given in the statement of the theorem are found +by solving the system f(y1) = p(y1), f(y) = p(y) and +df +dx(y) = dp +dx(y). From + +22 +Long-Term Returns Estimation of Leveraged Indexes and ETFs +here, the goal is to show p(x) ≥ f(x) for all x ∈ (log(1 − L−1), y1]. The rest of +the proof follows similar logic as was used to verify the lower bound. +□ +Proof of Theorems 2 and 3 +The proof is very similar to that of Theorem 1. Just observe that now the +function f(x) = log(1+L(exp x−1)) is well-defined on R, and it has df +dx strictly +convex on (−∞, log(L−1 − 1)) and strictly concave on (log(L−1 − 1), ∞). +□ +Proof of Theorem 4 +The proof is very similar to that of Theorem 1. Just observe that now the +function f(x) = log(1 + L(exp x − 1)) is well-defined on (−∞, log(1 − L−1)), +and it has df +dx strictly concave. +□ +References +Avellaneda M, Zhang S (2010) Path-dependence of leveraged etf returns. SIAM +Journal on Financial Mathematics 1(1):586–603 +Bansal VK, Marshall JF (2015) A tracking error approach to leveraged etfs: +Are they really that bad? Global Finance Journal 26:47–63 +Charupat N, Ma Z, Miu P (2022) Understanding leveraged etfs’ compounding +effect. Managerial Finance (ahead-of-print) +Cheng M, Madhavan A (2009) The dynamics of leveraged and inverse +exchange-traded funds. Journal of investment management 16(4):43 +DeVault L, Turtle HJ, Wang K (2021) Blessing or curse? institutional +investment in leveraged etfs. Journal of Banking & Finance 129:106,169 +Giese G (2010) On the risk-return profile of leveraged and inverse etfs. Journal +of Asset Management 11(4):219–228 +Jarrow RA (2010) Understanding the risk of leveraged etfs. Finance Research +Letters 7(3):135–139 +Leung T, Santoli M (2012) Leveraged etfs: Admissible leverage and risk +horizon. Journal of Investment Strategies 2(1):39–61 +Lu L, Wang J, Zhang G (2009) Long term performance of leveraged etfs. +Available at SSRN 1344133 + +Long-Term Returns Estimation of Leveraged Indexes and ETFs +23 +Shum PM, Kang J (2013) Leveraged and inverse etf performance during the +financial crisis. Managerial Finance +Tang H, Xu XE (2013) Solving the return deviation conundrum of lever- +aged exchange-traded funds. Journal of Financial and Quantitative Analysis +48(1):309–342 +Trainor Jr WJ, Baryla Jr EA (2008) Leveraged etfs: A risky double that doesn’t +multiply by two. Journal of Financial Planning 21(5) + diff --git a/ZtE1T4oBgHgl3EQfcgT-/content/tmp_files/load_file.txt b/ZtE1T4oBgHgl3EQfcgT-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df5c13cf4c969f6c1b89e2402b040a36074740be --- /dev/null +++ b/ZtE1T4oBgHgl3EQfcgT-/content/tmp_files/load_file.txt @@ -0,0 +1,787 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf,len=786 +page_content='Long-Term Returns Estimation of Leveraged Indexes and ETFs Hayden Brown∗ Abstract Daily leveraged exchange traded funds amplify gains and losses of their underlying benchmark indexes on a daily basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The result of going long in a daily leveraged ETF for more than one day is less clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Here, bounds are given for the log-returns of a leveraged ETF when going long for more than just one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The bounds are quadratic in the daily log-returns of the underlying benchmark index, and they are used to find sufficient con- ditions for outperformance and underperformance of a leveraged ETF in relation to its underlying benchmark index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Results show that if the underlying benchmark index drops 10+% over the course of 63 consecu- tive trading days, and the standard deviation of the benchmark index’s daily log-returns is no more than .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='015, then going long in a -3x lever- aged ETF during that period gives a log-return of at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='5 times the log-return of a short position in the underlying benchmark index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Results also show promise for a 2x daily leveraged S&P 500 ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' If the average annual log-return of the S&P 500 index continues to be at least .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='0658, as it has been in the past, and the standard deviation of daily S&P 500 log-returns is under .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='0125, then a 2x daily leveraged S&P 500 ETF will perform at least as well as the S&P 500 index in the long-run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Keywords: Leveraged ETFs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Leveraged exchange traded funds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Inverse leveraged ETFs, Returns estimation 1 Introduction A daily leveraged exchange traded fund amplifies the daily return of its under- lying benchmark index between adjusted closing prices (adjusted closing prices account for stock splits and dividends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' For example, consider an underly- ing benchmark index returning 1% between two consecutive adjusted closing ∗Department of Mathematics and Statistics, University of Nevada, Reno Email address: haydenb@nevada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='unr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='edu ORCID: 0000-0002-2975-2711 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='03186v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='MF] 9 Jan 2023 2 Long-Term Returns Estimation of Leveraged Indexes and ETFs prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Then a daily leveraged ETF with leverage multiple 2x or -3x would return roughly 2% or -3%, respectively, depending on the expense ratio, fund management and whether it is trading at a premium or discount on the close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The results presented here aim at adressing the viability of going long in a daily leveraged ETF for more than one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Their advantage over existing results in the literature is that they bound the leng-term log-return of a daily lever- aged ETF, and they depend only on the daily leveraged ETF’s expense ratio and the bounds, mean and standard deviation of the underlying benchmark index’s daily log-returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' It is now commonplace for investors to buy-and-hold an ETF tracking a large market index like the S&P 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Over a long period of time, like 40 years, investors are confident that such an ETF will provide a positive log-return that is favorable over most other available ETFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' But can the same be said about a daily leveraged S&P 500 ETF?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' One of the main goals here is to determine when a daily leveraged ETF will outperform its underlying benchmark index in the long-run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' When a stock or ETF looks overvalued, investors may take on a short position, hoping the price will drop in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Alternatively, investors could buy-and-hold an inverse daily leveraged ETF during that period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Here, conditions are given indicating when the latter option offers a superior return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' There are toy examples where a portfolio that invests L% in a particular stock or ETF and (1 − L)% in cash outperforms a portfolio that goes all in the stock or ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' In general, this outperformance occurs when volatility is high enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The results presented here show when this outperformance is impossible, with the goal being to validate the long-term portfolio that invests 100% in a S&P 500 ETF over a portfolio that reduces exposure to the S&P 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='1 Literature review Several empirical studies have measured the returns of daily leveraged ETFs over longer time spans than just one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The general consensus is that lever- aged ETFs track the leveraged multiple of their benchmark indexes’ returns well in the short term but deviate in the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Since the deviations can be markedly negative, there is considerable risk associated with a long position in a leveraged ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Over time spans of 1, 3, 5 and 10 years, a 2x daily leveraged S&P 500 ETF offers a moderate increase to expected return at the cost of a significant increase in standard deviation Trainor Jr and Baryla Jr (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Over time spans no longer than one month, 2x and -2x daily leveraged ETFs generally provide 2x and -2x, respectively, the return of the underlying benchmark index Lu et al (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' For time spans longer than one month, serious deviations start to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Those deviations are attributed, in part, to the quadratic variation of the underlying benchmark index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' On the other hand, Bansal and Marshall (2015) show that, for investment horizons of 1 calendar year from 1964 to 2013, the average difference between a leveraged S&P 500 ETF’s return and Long-Term Returns Estimation of Leveraged Indexes and ETFs 3 the underlying benchmark index’s return multiplied by the leverage amount is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' So there is clearly potential for a daily leveraged S&P 500 ETF to provide significant amplification of return over a calendar year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The results presented here complement these empirical findings by estimating returns of a daily leveraged ETF for investment horizons having any number of days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Most theoretical results address a long-term position in a continuously leveraged ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Provided the underlying benchmark index follows a geometric Brownian motion, leveraged ETFs appear to cause value destruction in the long-run Cheng and Madhavan (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' At a minimum, leveraged ETFs do not achieve their leverage multiple in the long-run Jarrow (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The risk of a leveraged ETF is measured in Leung and Santoli (2012), and admissible lever- age multiples are given accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Using continuous leverage, Giese (2010) shows that dynamic adjustment of the leverage multiple based on market con- ditions leads to outperformance of the underlying benchmark index in the long-run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' In reality, daily leveraged ETFs do not implement continuous lever- age, so the forementioned results cannot be applied without assuming some level of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The results presented here do not use continuous leverage to avoid this error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Other theoretical results address a long-term position in an ETF that is leveraged discretely in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' An approximation to the long-term return of a daily leveraged ETF is given by Avellaneda and Zhang (2010) for investment horizons of less than one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' It is based on the leverage multiple and the mean and variance of the underlying index’s daily returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Empirically, this approximation has been shown to be very accurate for quarterly horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' However, it is not an upper or lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The approximations presented here are advantageous because they are upper and lower bounds, which facilitates the provision of sufficient conditions for outperformance and underperformance of a daily leveraged ETF relative to its underlying benchmark index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' During the financial crisis from 2008 to 2009, daily leveraged ETFs did not generally meet their target multiple of daily returns, even on a daily basis Shum and Kang (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Similar finding are in Tang and Xu (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' These errors can be attributed to management and trading premiums/discounts, and the effect is a reduction in the magnification of daily returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' For example, 2x and -2x daily leveraged S&P 500 ETFs were more like 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='9x and -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='9x daily leveraged ETFs during the financial crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' These errors are not considered here because their randomness is difficult to incorporate into theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' However, the results can account for such errors, to some extent, with an increased expense ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Based on simulation of 3x and -3x daily leveraged S&P 500 ETFs, it appears that a combination of volatility and market condition (sideways, upward-trending or downward-trending) of the S&P 500 index determines long-term performance of the leveraged ETF Charupat et al (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The results presented here provide a theoretical foundation for these simulation-based findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' 4 Long-Term Returns Estimation of Leveraged Indexes and ETFs Daily leveraged ETFs are certainly popular, but it appears that their present use by institutions is leading to poor performance relative to portfolios that avoid daily leveraged ETFs DeVault et al (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' In other words, recent attempts by institutions to time the market with their leverage ETF hold- ings are backfiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The results presented here are aimed at providing further guidance on when a leveraged ETF is worth having in a portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='2 Main results Lower and upper bounds are given for the log-return of a daily leveraged index over n consecutive trading days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The bounds are expressed quadratically in terms of the daily log-returns of the underlying benchmark index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' In particular, the bounds are of the form n(am2 +bm1 +c), where m2 is the average squared daily log-return of the benchmark index, m1 is the average daily log-return of the benchmark index, and a, b and c are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The results cover a range of leverage multiples, including the popular -3x, -2x, 2x and 3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='3 Applications Sufficient conditions are given for the log-return of a daily leveraged index or ETF to be some multiple, L0, of the log-return of its underlying benchmark index, over n consecutive trading days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Here, ETFs are distinguished from indexes because they have expense ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' To simplify notation, let RL n,r denote the log-return of a daily leveraged ETF after n consecutive trading days, with leverage multiple L and expense ratio r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Now the goal of applications can be expressed more concisely: to provide sufficient conditions for RL n,r to be at least or at most L0R1 n,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' First, thresholds are given for m1/m2, indicating when RL n,r is at least or at most L0R1 n,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Here, m1 and m2 are as in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Special attention is given to the thresholds for 1 < L because 2x and 3x leverage multiples are so popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Let s denote the standard deviation of the underlying benchmark index’s daily log-returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' For L > 1 and L0 < L, an upper bound is given on s, indicating when RL n,r ≥ L0R1 n,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Taking L = 2, 3 and L0 = 0, 1 is especially important for practical reasons, because the upper bound on s indicates when a 2x or 3x daily leveraged ETF will have a non-negative log-return or perform at least as well as its underlying benchmark index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' If the average annual log- return of the S&P 500 index continues to be at least .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='0658, as it has been in the past, daily percentage changes between adjusted closing prices are at least 20%, and the standard deviation of daily log-returns is under .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='0125, then a 2x daily leveraged ETF will perform at least as well as the S&P 500 index in the long-run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' For L < L0 < 0, an upper bound is given on s, indicating when RL n,r ≥ L0R1 n,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The focus here is on L0 = −1 because then the latter inequality indicates when a daily inverse leveraged ETF performs at least as well as a short position in its underlying benchmark index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' For example, results show Long-Term Returns Estimation of Leveraged Indexes and ETFs 5 that if the benchmark index drops 10+% over the course of 63 consecutive trading days, daily percentage changes between adjusted closing prices are at most 15%, and s ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='015, then going long in a -3x leveraged ETF during that period gives a log-return of at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='5 times the log-return of a short position in the benchmark index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Furthermore, if the 10+% drop happens faster, then the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='5 multiple of log-returns can be achieved with even larger s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' For 0 < L < 1, an upper bound is given on s, indicating when RL n,0 ≤ R1 n,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Note that a log-return of RL n,0 can be achieved via daily rebalancing with L% in the benchmark index and (1 − L)% in cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Interestingly, this theory easily extends from daily leverage to longer periods like weekly leverage, where rebalancing occurs weekly, and quarterly leverage, where rebalancing occurs quarterly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' If the S&P 500 continues to have an average annual log-return of at least .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='0658, then the standard deviation of its daily, weekly, monthly, quarterly, semi-annual or annual log-returns would have to exceed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='02, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='04, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='08, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='15, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='2 or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='35, respectively, for a portfolio rebalancing daily, weekly, monthly, quarterly, semi-annually or annually, respectively, with .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='64 < L < 1, to outperform the benchmark L = 1 in the long-run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' It seems unlikely for this level of volatility to persist in the long-run, so maintaining a L : (1−L) portfolio in the benchmark and cash with 64 < L < 1 is not advised under any standard rebalancing schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='4 Organization Section 2 lays out the notation and framework for the returns of daily leveraged indexes and ETFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Section 3 provides main results, and Section 4 applies those results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Data used in applications is described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Section 5 provides closing remarks, including a discussion of related future research ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Last, A provides proofs of the theorems stated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' 2 Preliminaries Let Ci denote the adjusted closing price of trading day i for a particular stock market index I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Then {Ci}n i=0 is a sequence of adjusted closing prices for n + 1 consecutive trading days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Note that adjusted closing prices account for dividends and stock splits, but not inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' For example, suppose a stock has a closing price of $100 on day 1, a closing price of $98 on day 2, and a dividend distribution of $1 on day 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Then the adjusted closing prices will be $100 on day 1 and $99 on day 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Suppose there is a 2-for-1 stock split on day 3 and the closing price on day 3 is $49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Then the adjusted closing price of day 3 will be $99 = 2 · $49 + $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Let Xi = Ci/Ci−1 − 1 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Then {100 · Xi}n i=1 is the sequence of n percentage changes between adjusted closing prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Observe that n � i=1 (1 + Xi) = Cn C0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' 6 Long-Term Returns Estimation of Leveraged Indexes and ETFs Denote the daily leveraged version of I as LxI, where L indicates the amount of leverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' For example, 3xI indicates the index tracking I with 3x daily leverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The closing prices of LxI are given by CL i := C0 · i� k=1 (1 + LXk), i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' So the log-returns realized by going long in LxI from the close of trading day 0 to the close of trading day n are given by log CL n C0 = n � i=1 log(1 + LXi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Note that here, log refers to the natural logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Let Yi = log(1 + Xi) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Then Yi is the log-return for day i, and log Cn C0 = n � i=1 Yi, log CL n C0 = n � i=1 log(1 + L(exp Yi − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' To shorten notation, let m1 = 1 n n � i=1 Yi, m2 = 1 n n � i=1 Y 2 i , s = ��n i=1(Yi − m1)2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Denote the ETF version of LxI as LxIr, where r is the annual expense ratio, compounded on a daily basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Assuming 252 trading days in a year, the log-return of LxIr after n days is given by RL n,r := log CL n C0 − n log � 1 + r 252 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' 3 Main Results Theorems 1, 2, 3 and 4 provide lower and upper bounds for the log-return of LxI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' The bounds are expressed quadratically in terms of the Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Theorem 1 covers L > 1, Theorems 2 and 3 cover 0 < L < 1, and Theorem 4 covers L < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Theorem 1 Fix L > 1 and log(1 − L−1) < y0 < y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=' Then, provided y0 ≤ Yi ≤ y1 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfcgT-/content/2301.03186v1.pdf'} +page_content=', n, log-returns of LxI from the close of trading day 0 to the close of trading day n are bounded as follows: sup y0