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+# A LEARNING-BASED ITERATIVE METHOD FOR SOLVING VEHICLE ROUTING PROBLEMS
+
+Hao Lu ∗ Princeton University, Princeton, NJ 08540 {haolu}@princeton.edu
+
+Xingwen Zhang ∗ & Shuang Yang Ant Financial Services Group, San Mateo, CA 94402 {xingwen.zhang,shuang.yang}@antfin.com
+
+# ABSTRACT
+
+This paper is concerned with solving combinatorial optimization problems, in particular, the capacitated vehicle routing problems (CVRP). Classical Operations Research (OR) algorithms such as LKH3 (Helsgaun, 2017) are inefficient and difficult to scale to larger-size problems. Machine learning based approaches have recently shown to be promising, partly because of their efficiency (once trained, they can perform solving within minutes or even seconds). However, there is still a considerable gap between the quality of a machine learned solution and what OR methods can offer (e.g., on CVRP-100, the best result of learned solutions is between 16.10-16.80, significantly worse than LKH3’s 15.65). In this paper, we present “Learn to Improve” (L2I), the first learning based approach for CVRP that is efficient in solving speed and at the same time outperforms OR methods. Starting with a random initial solution, L2I learns to iteratively refine the solution with an improvement operator, selected by a reinforcement learning based controller. The improvement operator is selected from a pool of powerful operators that are customized for routing problems. By combining the strengths of the two worlds, our approach achieves the new state-of-the-art results on CVRP, e.g., an average cost of 15.57 on CVRP-100.
+
+# 1 INTRODUCTION
+
+In this paper, we focus on an important class of combinatorial optimization, vehicle routing problems (VRP), which have a wide range of applications in logistics. Capacitated vehicle routing problem (CVRP) is a basic variant of VRP, aiming to find a set of routes with minimal cost to fulfill the demands of a set of customers without violating vehicle capacity constraints. The CVRP is NP-hard (Dantzig & Ramser, 1959), and both exact and heuristic methods have been developed to solve it (Fukasawa et al., 2006; Golden et al., 2008; Kumar & Panneerselvam, 2012; Toth & Vigo, 2014).
+
+In recent years, especially after the seminal work of Pointer Networks (Vinyals et al., 2015), researchers start to develop new deep learning and reinforcement learning (RL) frameworks to solve combinatorial optimization problems (Bello et al., 2016; Mao et al., 2016; Khalil et al., 2017; Bengio et al., 2018; Kool et al., 2019; Chen & Tian, 2019). For the CVRP itself, a number of RLbased methods have been proposed in the literature (Nazari et al., 2018; Kool et al., 2019; Chen & Tian, 2019). The learning based methods are trained on a huge number of problem instances, and have been shown to be extremely fast in producing solutions of reasonably good quality. However, when tested with the same benchmark instances, these learning-based methods cannot outperform the state-of-the-art method LKH3 (Helsgaun, 2017), which is a penalty-function-based extension of classical Lin-Kernighan heuristic (Lin & Kernighan, 1973; Helsgaun, 2000). For example, on CVRP with 100 customers, LKH3 is able to produce an average cost of 15.65. This line of research motivated us to study a framework that combines the strength of Operations Research (OR) heuristics with learning capabilities of machine learning (RL in particular). Machine learning can learn to solve a class of problem instances fast, when test instances are generated from the same distribution as training instances. Classical approaches like search algorithms are effective but may need heavy computation, which is time-consuming. Our research interest is in fusing the strengths of these two worlds. Another related line of research is hyper-heuristics, which is “a search method or learning mechanism for selecting or generating heuristics to solve computational search problems” (Burke et al., 2013). Instead of developing a high-level methodology without the need of knowing the details of low-level heuristics, we are primarily interested in a closely integrated system that best utilizes the strength of OR operators and learning capability.
+
+Our Contributions. Instead of directly constructing a solution from the problem instance (Graves et al., 2014; Sutskever et al., 2014; Vinyals et al., 2015), we propose a framework that iteratively searches among solutions, until a certain termination condition is satisfied. Our main contributions are as follows:
+
+• We present a learning-based algorithm for solving CVRP, achieving new state-of-the-art results. The recent line of works using RL to solve CVRP shows the potential of machine learning algorithms. They can solve CVRP faster, but cannot beat classical OR solvers like LKH3 in term of solution quality. Our algorithm is the first machine learning framework that outperforms LKH3 on CVRP.   
+We propose a novel hierarchical framework. Instead of putting all operators in one action pool, we separate heuristic operators into two classes, namely improvement operators and perturbation operators. At each state, we choose the class first and then choose operators within the class. Learning from the current solution is made easier by focusing RL on the improvement operators only.   
+• We propose an ensemble method, which trains several RL policies at the same time, but with different state input features. The ensemble method is shown to produce superior results than individual policies with an equivalent amount of computation.
+
+Related Work. In recent years, there have been many studies using deep learning and RL to solve combinatorial optimization problems (Smith, 1999; Mao et al., 2016; Lodi & Zarpellon, 2017; Velickovi ˇ c et al., 2017; Lombardi & Milano, 2018; Bengio et al., 2018). Routing problems, es- ´ pecially traveling salesman problems (TSP) and VRP, have been explored by a sequence of works (Vinyals et al., 2015; Bello et al., 2016; Khalil et al., 2017; Li et al., 2018; Deudon et al., 2018; Kaempfer & Wolf, 2018; Nazari et al., 2018; Kool et al., 2019; Chen & Tian, 2019). Most of these works, with the exception of Chen & Tian (2019), follow an end-to-end approach, which is directly constructing a solution from scratch. Vinyals et al. (2015) first introduce the Pointer Network, inspired by sequence-to-sequence models, to solve TSP. They use an attention model to learn the order of different nodes in a supervised fashion. Later Bello et al. (2016) develop an RL algorithm to train the Pointer Network. Their framework learns the optimal policy from problem instances and needs no supervised solutions. Nazari et al. (2018) improve the Pointer Network with a new design, making the model invariant with respect to the input sequence, and extend it to solve VRP. Kool et al. (2019) propose a model based on attention layers, and an RL algorithm to train this model with a simple but effective baseline. Chen & Tian (2019) propose a NeuRewriter model for VRP. They define a rewriting rule set and train two policy networks, a region-picking policy and a rule-picking policy, to obtain the next state. Given an initial solution, their goal is to find a sequence of steps towards the solution with minimal cost.
+
+# 1.1 NOTATIONS
+
+In CVRP, there is a depot and a set of $N$ customers. Each customer $i$ , $i \in \{ 1 , \ldots , N \}$ , has a demand $d _ { i }$ to be satisfied. A vehicle, which always starts at and ends at the depot, can serve a set of customers as long as the total customer demand does not exceed the capacity of the vehicle. The traveling cost $c _ { i , j }$ is the cost of a vehicle going from node $i$ to $j$ , with $i , j \in \{ 0 , 1 , \ldots , N \}$ (where the depot is denoted by node 0 for convenience). The objective is to find a routing plan with minimal cost that serves all customers without violating vehicle capacity constraints. Figure 1 gives an illustration of CVRP, while a mathematical formulation is given in the Appendix.
+
+Route. A route is a sequence of nodes visited in order, with the depot (node 0) as the starting and ending node. For example, [0, 1, 3, 4, 0] is a traveling plan that starts at the depot, visits node 1, 3 and 4 sequentially, and returns to the depot.
+
+Solution. A solution is a set of routes such that each customer is visited exactly once, and the total demand along each route is less than the vehicle capacity.
+
+Operator. An operator is a mapping from one solution to another. In this paper, instead of directly constructing a solution from scratch, we improve or perturb the solution iteratively using operators.
+
+![](images/f63b654f7eb5408c622a747a930a1b2dfc862a82d0f0901a48d1d9b3a103db60.jpg)  
+Figure 1: An illustration of CVRP. Here we provide a problem instance. The red one is a sample route, and three routes consist of a solution for this problem instance. After applying an operator, current solution changes to a new solution with dashed lines replaced by blue lines.
+
+# 2 LEARN TO IMPROVE
+
+In this section we formally introduce our “Learn to Improve” (L2I) framework, including main components of the system as well as the design intuitions behind individual components. Figure 2 illustrates the overall framework. As mentioned previously, the framework is iterative in nature, that is, we always start with a feasible solution, continuously improving the solution or perturbing it. Along the way, all constraints remain satisfied. By always maintaining the feasibility of the solution,
+
+we are exploring the space of feasible solutions and any of them found by our search process could potentially be a good solution. After $T$ (a parameter set in advance) steps the algorithm stops, and we choose the one with the minimum traveling cost as our final solution. Our framework has a few distinct components (e.g., how to improve the solution, when and how to perturb it), which could be rule-based, learned by machine learning or mixed, and thus allows us to experiment with different ways of integrating machine learning and OR, some of which could lead to superior methods either in terms of better solution quality or in terms of computational efficiency. It is worthwhile to point out that, as contrast to the work of Chen & Tian (2019), the solution space we explore is much larger since we are employing a rich set of improvement and perturbation operators.
+
+![](images/6e285bf0ea8b32c6e74ebe548072e52ab2b5e2b3cbfc206a872784e86fd9b8a4.jpg)  
+Figure 2: Our hierarchy framework. Given a problem instance, our algorithm first generates a feasible solution. Then it iteratively updates the solution with an improvement operator selected by an RL-based controller or with a perturbation operator chosen by a rule-based controller. After a certain number of steps, we choose the best one among all visited solutions.
+
+For this research work, we implemented and experimented with a number of design choices, and ended up with a method that is both computationally efficient and able to produce state-of-the-art results. The details of the main components of our method are as follows. Given a history of most recent solutions, our method uses a threshold-based rule to decide whether we should continue to improve the current solution, or should perturb it and restart with the perturbed solution. If it decides that the current solution could still be improved, it will use an RL-based controller to choose one of the improvement operators, and try to improve the solution with the chosen operator. We have a rich set of improvement operators (the list of improvement operators and their details are given in the Appendix), where intra-route ones attempt to reduce the cost of current solution by moving customers to different positions in individual routes, while inter-route ones attempt to reduce the cost by moving customers among different routes. Given that the improvement operators are of distinct characteristics, it is not straightforward to know in advance which operators are most effective for the problem under investigation. It is also difficult to know a pre-defined ordering of the operators that is best for the problem. Thus, an RL-based controller is a good choice to learn the set of improvement operators that is more effective for the problem (or more specifically, the problem and the data distribution from which the training and test instances are sampled). As will be shown in Section 3, our RL model is able to differentiate more useful improvement operators from less useful ones for CVRP, as well as generate an implicit ordering on how the operators will be applied.
+
+On the other hand, upon reaching a local minimum the perturbation controller randomly chooses a perturbation operator that destroys (completely or partially) and reconstructs a number of routes to generate a new starting solution (Table 6 in the Appendix gives the list of perturbation operators and their details). Specifically, if no cost reduction has been made for $L$ improvement steps, we perturb the solution and restart the improvement iteration (where for ease of explanation a maximum consecutive sequence of improvement operators applied before perturbation is called an improvement iteration). As a perturbation changes the solution quite dramatically (by producing a substantially different solution that is usually worse than the current one), we found that it is useful to start a new improvement iteration with a reasonably good starting point (e.g. by filtering out the restarting solutions that are significantly worse than current solution or currently best solution). It is clear that we purposely separate improvement operators from perturbation ones, and an alternative design would be to mix them all together and have a single controller deciding which operator to apply next. However, the improvement operators are of different nature from the perturbation ones, and their impacts are different since the perturbation operators have long-lasting effect by affecting an entire improvement iteration. Our experience also suggests that learning is made easier by focusing RL on the improvement operators only. Lastly, it is worthwhile to point out that although the rulebased perturbation controller is shown to be effective, we do not rule out the possibility that it can also be learning-based.
+
+The framework described above provides a way of combining the strength of OR operators, which are powerful since they are custom-made for routing problems, with learning capabilities of RL, which is flexible and can be adapted to a given problem and its associated data distribution. Having described our overall framework, we are now ready to present the details of the improvement controller and operators.
+
+# 2.1 IMPROVEMENT CONTROLLER AND OPERATORS
+
+The improvement controller starts with an initial solution, which is either constructed randomly (for the first improvement iteration)1 or produced by a perturbation operator (for subsequent iterations), and then tries to improve it, i.e., reducing the total traveling distance without violating any constraints, by selectively applying an improvement operator in Table 5. For the RL model, the set of improvement operators constitute our action space. These operators change the solution locally and most are computationally light. With the current state as input, a neural network produces a vector of action probabilities, and the weights of the network are trained with policy gradient. Figure 6 illustrates the components of our RL model, and their details are given as follows.
+
+# 2.1.1 STATES
+
+Each state includes features from the problem instance, the solution and the running history. Stationary features, such as the location and demand of each customer, are considered as problem-specific since they are invariant across solutions. Solution-specific features are based on the current traveling plan. For example, given a solution, for each customer we compute its neighboring nodes that are visited before and afterwards, as well as the relevant distances. Following Odling (2018), the ¨ running history includes the actions that are recently taken as well as their effects. For example, $a _ { t - h }$ , $1 \leq h \leq H$ , is the action taken $h$ steps before current step $t$ , and its effect $e _ { t - h }$ is $+ 1$ if the action led to a reduction of total distance, $^ { - 1 }$ otherwise. A complete description of state features is given in Table 4, where $i ^ { - }$ and $i ^ { + }$ denote the node visited before and after node $; 1 \leq i \leq N$ , in the solution, respectively.
+
+# 2.1.2 ACTIONS
+
+The actions can be classified into two classes, intra-route operators and inter-route operators. An intra-route operator attempts to reduce traveling distance of an individual route, while inter-route operators aim at reducing total traveling distance by moving customers between more than one route. The details of the operators are given in Table 5. It is worthwhile to point out that the same operator with different parameters are considered as different actions. For example, Symmetric-exchange(2) with $m = 1 , 2 , 3$ are considered as three distinct actions. For a given problem (or even solution), the same operator with different parameters may perform differently and thus it is appropriate to treat them as separate actions and let the RL model learn how to best use them.
+
+# 2.1.3 POLICY NETWORK
+
+We use the well-known REINFORCE algorithm (Williams, 1992) to update the gradient
+
+$$
+\nabla _ { \theta } J ( \theta | s ) = \mathbb { E } _ { \pi \sim p _ { \theta } ( . | s ) } \big [ \big ( L ( \pi | s ) - b ( s ) \big ) \nabla _ { \theta } \log p _ { \theta } ( \pi | s ) \big ]
+$$
+
+of policy with a baseline function $b ( s )$ . Given a state as described in Section 2.1.1, a policy network outputs a list of action probabilities, one for each action described. As illustrated in Figure 6, problem- and solution-specific input features are transformed into an embedding of length $D$ (we use $D = 6 4$ ), which is fed into an attention network (Vaswani et al., 2017) (we use an attention layer with 8 heads and 64 output units). The output of the attention network is concatenated with a sequence of recent actions and their effects (when $H > 0$ ). Lastly, the concatenated values are fed into a network of two fully connected layers, where the first layer uses 64 units and a Relu activation function and the second layer uses Softmax, producing $| A |$ action probabilities where $A$ is the set of actions.
+
+# 2.1.4 REWARDS
+
+We have experimented with a number of reward designs, two of which are producing satisfactory results as well as distinct patterns of operator sequences. The first reward function (denoted by RF1) focuses on the intermediate impact of the improvement operators. Specifically, the reward is $+ 1$ if the operator improves the current solution, -1 otherwise. The second reward function (denoted by RF2) is advantage-based. The total distance achieved for the problem instance during the first improvement iteration is taken as a baseline. For each subsequent iteration, all operators applied during this iteration received a reward equal to the difference between the distance achieved during the iteration and the baseline. We observed that an operator is often able to achieve a large distance reduction for a freshly perturbed solution, while it becomes harder and harder to do so in later improvement steps. In particular, the likelihood of distance reduction as well as the magnitude of such reduction, both decrease as the iteration proceeds. Therefore, it seemed unfair to give early improvement operators a larger reward. The observation suggested that operators used in the same improvement iteration should be rewarded equally and there would be no discounting (or equivalently, the discount factor $\gamma$ is 1).
+
+To conclude the methodology section, we restart the improvement iteration until reaching a maximum number $T$ of rollout (either improvement or perturbation) steps. Following a common practice of encouraging exploration, we use $\epsilon$ -greedy (Sutton & Barto, 2018) such that with a probability of 0.05 the RL controller will choose a random improvement action. Lastly, we also experimented with ensembling by training 6 different policies with $H = 1 , 2 , \ldots , 6$ (while keeping other components of the policy network identical). Ensembling facilitates learning of a diverse set of policies, as well as reducing wall-clock running time.
+
+# 3 EXPERIMENTS AND ANALYSES
+
+In this section, we present our experiment results. First we introduce a detailed setup of CVRP and hyper-parameters used. Then we compare our performance with prior neural network approaches, i.e., Nazari et al. (2018), Kool et al. (2019), Chen & Tian (2019), and a classic state-of-the-art heuristic algorithm. At last we provide detailed analysis of our framework2.
+
+Setup and hyper-parameters. We follow the same settings as previous works (Nazari et al., 2018; Kool et al., 2019; Chen & Tian, 2019) for CVRP. We consider three sub-problems with number of customers $N = 2 0 , 5 0 , 1 0 0$ , respectively. The location $( x _ { i } , y _ { i } )$ of each customer, as well as of the depot, is uniformly sampled from unit square (specifically, $x _ { i }$ and $y _ { i }$ are uniformly distributed in the interval $[ 0 , 1 ]$ , respectively), and the traveling cost between two locations $c _ { i , j }$ is simply the corresponding Euclidean distance. The demand $d _ { i }$ of each customer is uniformly sampled from the discrete set $\{ \bar { 1 } , 2 , \dots , 9 \}$ . The capacity of a vehicle is 20, 30, 40 for $N = 2 0 , 5 0 , 1 0 0$ , respectively. After $L \ = \ 6$ consecutive step of no improvement, we perturb the solution. To train the policy network, we use ADAM with a learning rate of 0.001. Unless otherwise stated, for a problem instance and a given policy we randomly initiate a feasible solution, and then iteratively update the solution $T = 4 0 0 0 0$ times following the policy. In the following section we will discuss the performance of a policy with different rollout steps (recall that a rollout step is either an improvement step or a perturbation one). We choose the best one among all 40000 visited solutions as the final solution for a given problem instance. In the ensemble model we use different number of historical actions and effects to train a set of diverse policies (recall the policy network in Figure 6), and for any problem instance we choose the best solution produced by these policies. Unless otherwise stated, all reported metrics, such as the final traveling cost and the running time, are always computed as the average over 2000 random samples. Lastly, our method was implemented in Python, and the experiments were run computer nodes, each with a single Nvidia Tesla T4 GPU.
+
+# 3.1 PERFORMANCE COMPARISON
+
+In Table 1 we compare the performance of our algorithm with prior neural network approaches mentioned above, Google OR-tools (Google, 2019), and classic state-of-the-art heuristic algorithm LKH3 (Helsgaun, 2017). Our ensemble method chose the solution with minimum traveling cost among those produced by Policy $i$ (with $1 \leq i = h \leq 6$ ). As shown in Table 1, our algorithm outperforms the prior approaches in terms of average traveling cost. In particular, the average distance achieved by our algorithm is significantly shorter than prior neural network approaches. Thus, our algorithm is producing state-of-the-art results for CVRP, and is the first learning-based framework that outperforms the well-known classic heuristic algorithm.
+
+Table 1: Comparison of our experiment results with those reported in the literature   
+
+<table><tr><td></td><td>N=20 Obj.</td><td>N=50 Obj.</td><td>N = 100 Obj.</td></tr><tr><td>Google OR Tools</td><td>6.43</td><td>11.31</td><td>17.16</td></tr><tr><td>Nazari et al. (2018)</td><td>6.40</td><td>11.15</td><td>16.96</td></tr><tr><td>AM greedy (Kool et al., 2019)</td><td>6.40</td><td>10.98</td><td>16.80</td></tr><tr><td>AM sampling (Kool et al., 2019)</td><td>6.25</td><td>10.62</td><td>16.23</td></tr><tr><td>Chen&amp; Tian (2019)</td><td>6.16</td><td>10.51</td><td>16.10</td></tr><tr><td>LKH (Helsgaun, 2017)</td><td>6.14</td><td>10.38</td><td>15.65</td></tr><tr><td>L2I</td><td>6.12</td><td>10.35</td><td>15.57</td></tr></table>
+
+# 3.2 ANALYSIS OF THE ENSEMBLE METHOD
+
+Recall that we train 6 different policies with $H \in \{ 1 , 2 , \ldots , 6 \}$ . To illustrate the motivation of the ensemble method, we randomly pick 10 problem instances and show the traveling cost under different policies for these problem instances in Table 2. We can see that Policy 1 did the best for the first problem instance, but it performed the worst for the second one. Furthermore, there is no clear winning policy, which is the reason we propose an ensemble method.
+
+Table 2: Motivating examples: traveling cost under different policies for random problem instances   
+
+<table><tr><td rowspan="2"></td><td colspan="10">Problem instances</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td>Policy 1</td><td>18.65</td><td>16.67</td><td>15.41</td><td>15.11</td><td>16.69</td><td>14.97</td><td>16.88</td><td>16.37</td><td>13.72</td><td>15.56</td></tr><tr><td>Policy 2</td><td>18.86</td><td>16.61</td><td>15.39</td><td>15.10</td><td>16.55</td><td>14.75</td><td>16.73</td><td>16.39</td><td>13.77</td><td>15.56</td></tr><tr><td>Policy 3</td><td>18.81</td><td>16.63</td><td>15.43</td><td>15.06</td><td>16.82</td><td>14.72</td><td>16.70</td><td>16.37</td><td>13.73</td><td>15.79</td></tr><tr><td>Policy 4</td><td>18.90</td><td>16.60</td><td>15.37</td><td>15.04</td><td>16.66</td><td>14.93</td><td>16.83</td><td>16.32</td><td>13.70</td><td>15.62</td></tr><tr><td>Policy 5</td><td>18.73</td><td>16.56</td><td>15.43</td><td>15.12</td><td>16.66</td><td>15.09</td><td>16.80</td><td>16.53</td><td>13.72</td><td>15.56</td></tr><tr><td>Policy 6</td><td>18.92</td><td>16.60</td><td>15.43</td><td>15.09</td><td>16.73</td><td>14.65</td><td>16.59</td><td>16.55</td><td>13.73</td><td>15.56</td></tr></table>
+
+![](images/624ae57c3fc32697f30820a4fa695016252a2639b07ba367b61d1639788307d1.jpg)  
+Figure 3: Average traveling cost of different policies with different rollout steps
+
+In Figure 3(a), (b) and (c), we plot the average traveling cost over 2000 problem instances for CVRP with $N = 2 0$ , 50, 100, respectively. It is worthwhile to point out that the same hyper-parameters are used for different $N$ values. The top blue line is for a random policy, the bottom red line is for our ensemble method, and lines in between are for Policy 1, 2, . . . , 6. The plots show that, regardless of the number of rollout steps, our trained RL policies consistently outperform the random policy. Figure 4(c) shows that the gap between the random policy and the ensemble method first increases, peaks at about 5000 rollout steps, and then gets smaller and smaller. It is encouraging that we get the maximum gap fast, which will be helpful when computational time is limited. Lastly, in Figure 4(d) we show three policies, the random policy, the ensemble method, and the best RL policy (Policy 3), for CVRP-100. To match the performance of the random policy with 40000 rollout steps, the ensemble method would take 2500 rollout steps, while it takes 5000 rollout steps to match Policy 3 with 40000 rollout steps. The outperformance of the ensemble method becomes more obvious when $N = 2 0 , 5 0$ . These analysis shows that naively increasing rollout steps of a policy provides less marginal gains than ensembling a diverse set of policies.
+
+Table 3: Improvement operators mostly used   
+
+<table><tr><td rowspan=1 colspan=1>Class</td><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Details</td></tr><tr><td rowspan=2 colspan=1>Intra-route</td><td rowspan=1 colspan=1>2-0pt</td><td rowspan=1 colspan=1>Remove two edges andreconnect their endpoints</td></tr><tr><td rowspan=1 colspan=1>Relocate(1)</td><td rowspan=1 colspan=1>Move a customer in the route toa new location</td></tr><tr><td rowspan=3 colspan=1>Inter-route</td><td rowspan=1 colspan=1>Cross(2)</td><td rowspan=1 colspan=1>Exchange the tails of two routes</td></tr><tr><td rowspan=1 colspan=1>Symmetric-exchange(2)</td><td rowspan=1 colspan=1>Exchange segments of length m（m = 1) between two routes</td></tr><tr><td rowspan=1 colspan=1>Relocate(2)</td><td rowspan=1 colspan=1>Move a segment of length m(m = 1) from a route to another</td></tr></table>
+
+# 3.3 ANALYSIS OF OPERATOR USAGES
+
+As we mentioned before, the RL model is able to differentiate more useful improvement operators from less useful ones for CVRP. In our experiments, we count the usage of different operators for different policies as training epochs grow. When the myopic reward function RF1 in Section 2.1.4 is used, our experimental results show that the policy converges to use a fixed subset of improvement operators (for detailed operators, see Table 3). This subset of operators are also preferred by all policies when we use RF2. However, the pattern of operator usages varies among the policies. For example, Figure 4(a) and (b) illustrate different patterns of operator usages for Policy 1 and Policy 2, respectively.
+
+![](images/ce3d9514a6179351fd7a273c5743c85971fba32406e8a6dbf149bfd524a33d35.jpg)  
+Figure 4: Pattern of operator usages as training epoch grows
+
+# 3.4 ANALYSIS OF PERTURBATION MAGNITUDE
+
+From our experimental study we also observed that the solution quality is sensitive to the magnitude of perturbation. To illustrate the impact of perturbation, in Figure 5 we plot the performance of two run configurations, one with Random-permute applied to all routes (called Random-permute-all), while the other with Random-permute applied to two routes only (called Random-permute-2). As seen from Figure 5, Random-permute-2 significantly outperformed Random-permute-all. Intuitively, when the magnitude of perturbation is too large, the resulting solution generally becomes much worse and it will take our algorithm a large number of improvement steps to remedy the deterioration.
+
+![](images/b959c7ff23f3e84dd13b04beacf1d87591b63b639cdcbb60b9dae11d17059099.jpg)  
+Figure 5: Impact of perturbation magnitude
+
+# 4 CONCLUSION
+
+In this paper we propose “Learn to Improve” for solving VRP, which starts with an initial solution and iteratively updates the solution with an improvement operator selected by an RL-based controller or with a perturbation operator chosen by a rule-based controller. We also propose an ensemble method that trains several RL policies and chooses the best solution produced by the policies. Our method achieved new state-of-the-art results for CVRP instances.
+
+Our work provides a way of combining the strength of OR with learning capabilities of RL. For future work, we would like to apply the solution framework to solve other variants of the VRP, such as vehicle routing problems with time windows (VRPTW), as well as other combinatorial problems, such as maximum independent set problems and graph coloring problems. Furthermore, it is interesting to investigate whether allowing temporary constraint violations in our framework will help improve solution quality or not.
+
+# 5 ACKNOWLEDGEMENT
+
+We would like to thank our colleagues at Ant Financial, Wei Yan and Junping Zhao, for allocating GPU resources needed for the computational experiments.
+
+# REFERENCES
+
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+
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+
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+
+# A AN INTEGER PROGRAMMING FORMULATION OF THE CVRP
+
+Formally, there is a depot and a set of $N$ customers in the CVRP. Each customer $i$ , $i \in \{ 1 , \ldots , N \}$ , has a demand $d _ { i }$ to be satisfied. A vehicle, which always starts at and ends at the depot, can serve a set of customers as long as the total customer demand does not exceed the capacity of the vehicle $C$ . The traveling cost $c _ { i , j }$ is the cost of a vehicle going from node $i$ to $j$ , with $i , j \in V = \{ 0 , 1 , . . . , N \}$ (where the depot is denoted by node 0 for convenience). The objective is to find a routing plan with minimal cost that serves all customers without violating vehicle capacity constraints. An integer programming formulation of the CVRP (Toth & Vigo, 2002) is given below.
+
+$$
+\begin{array} { r l } & { \displaystyle \operatorname* { m i n } _ { \tau \to \tau } \sum _ { \ell \in \mathcal { N } _ { i } } \sum _ { \ell \in \mathcal { N } _ { i } } \alpha _ { i , j } } \\ & { \ : \ : \ : \sum _ { \ell \in \mathcal { N } _ { i } } \alpha _ { i , j } \le C _ { i , j } \le V \setminus \{ 0 \} } \\ & { \ : \ : \ : \delta \cdot \sum _ { \ell \in \mathcal { N } _ { i } } \gamma _ { \ell } = 1 , \ : \forall j \in V \setminus \{ 0 \} } \\ & { \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \sum _ { \ell \in \mathcal { N } _ { i } } \gamma _ { \ell , j } = 1 , \ : \forall i \in V \setminus \{ 0 \} } \\ & { \ : \ : \ : \ : \sum _ { \ell \in \mathcal { N } _ { i } \setminus \{ 0 \} } \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \widetilde { \mathcal { M } } _ { i } = K , } \\ & { \ : \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \ : \ : \mathcal { M } _ { i } \leq C \cdot \ : \ : \forall \ : \ : \ : \ : \mathcal { M } _ { i } , \ : \ : \ : \ : \ : \ : \ : \mathcal { M } _ { i } \neq \mathcal { I } \setminus \{ \left\{ \left[ \mathcal { R } \right] , \ : \ : \ : \ : \mu _ { i } \right\} } = \mathcal { A } \le \mathcal { L }  \\ & { \ : \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \ : } \\ & { \ : \ : \ : \ : \ : \ : \ : \ : } \end{array}
+$$
+
+where $K$ is the number of vehicles available (w.l.o.g., it is assumed that $K = N$ for the CVRP we consider). Constraints (1) and (2) specify that each customer is visited exactly once, while constraints (3) and (4) specify the in and out degree of the depot, respectively. Constraints (5) and (6) impose the vehicle capacity requirements.
+
+Table 4: State features   
+
+<table><tr><td rowspan=1 colspan=1>Type</td><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Details</td></tr><tr><td rowspan=8 colspan=1>Problem-andsolution-specific</td><td rowspan=1 colspan=1>Ci</td><td rowspan=1 colspan=1>Demand of customer i</td></tr><tr><td rowspan=1 colspan=1>Ci</td><td rowspan=1 colspan=1>Free capacity of the route containing customer i</td></tr><tr><td rowspan=1 colspan=1>(xi，yi)</td><td rowspan=1 colspan=1>Location of customer i</td></tr><tr><td rowspan=1 colspan=1>(xi-,yi-)</td><td rowspan=1 colspan=1>Location of node visited before i</td></tr><tr><td rowspan=1 colspan=1>(xi+,yi+)</td><td rowspan=1 colspan=1>Location of node visited after i</td></tr><tr><td rowspan=1 colspan=1>di-i</td><td rowspan=1 colspan=1>Distance from i- to i</td></tr><tr><td rowspan=1 colspan=1>dii+</td><td rowspan=1 colspan=1>Distance from i to i+</td></tr><tr><td rowspan=1 colspan=1>di-i+</td><td rowspan=1 colspan=1>Distance from i- to i+</td></tr><tr><td rowspan=2 colspan=1>History-related</td><td rowspan=1 colspan=1>at-h</td><td rowspan=1 colspan=1>Action taken h steps before</td></tr><tr><td rowspan=1 colspan=1>et-h</td><td rowspan=1 colspan=1>Effect of at-h</td></tr></table>
+
+# B DETAILS OF STATES AND OPERATORS
+
+We list the details of our state features in Table 4, and of operators in Table 5 and 6.
+
+Table 5: Improvement operators   
+
+<table><tr><td rowspan=1 colspan=1>Class</td><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Details</td></tr><tr><td rowspan=3 colspan=1>Intra-route</td><td rowspan=1 colspan=1>2-0pt</td><td rowspan=1 colspan=1>Remove two edges andreconnect their endpoints</td></tr><tr><td rowspan=1 colspan=1>Symmetric-exchange(1)</td><td rowspan=1 colspan=1>Exchange two customers in the route</td></tr><tr><td rowspan=1 colspan=1>Relocate(1)</td><td rowspan=1 colspan=1>Move a customer in the route toa new location</td></tr><tr><td rowspan=6 colspan=1>Inter-route</td><td rowspan=1 colspan=1>Cross(2)</td><td rowspan=1 colspan=1>Exchange the tails of two routes</td></tr><tr><td rowspan=1 colspan=1>Reverse-cross(2)</td><td rowspan=1 colspan=1>Reverse one of two routes and thenexchange their tails</td></tr><tr><td rowspan=1 colspan=1>Symmetric-exchange(2)</td><td rowspan=1 colspan=1>Exchange segments of length m(m =1,2,3) between two routes</td></tr><tr><td rowspan=1 colspan=1>Asymmetric-exchange(2)</td><td rowspan=1 colspan=1>Exchange segments of length m and n(m=1,2,3,n=1,2,3,m/ n)between two routes</td></tr><tr><td rowspan=1 colspan=1>Relocate(2)</td><td rowspan=1 colspan=1>Movea segment oflength m(m =1,2,3) from a route to another</td></tr><tr><td rowspan=1 colspan=1>Cyclic-exchange(3)</td><td rowspan=1 colspan=1>Exchange cyclically one customerbetween three routes</td></tr></table>
+
+Table 6: Perturbation operators   
+
+<table><tr><td rowspan=1 colspan=1>Class</td><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Details</td></tr><tr><td rowspan=3 colspan=1>Inter-route perturbation</td><td rowspan=1 colspan=1>Random-permute</td><td rowspan=1 colspan=1>Randomly destroy m routes andre-construct routes by visiting affectedcustomers in a random order</td></tr><tr><td rowspan=1 colspan=1>Random-exchange(2)</td><td rowspan=1 colspan=1>Randomly exchange m pairs of nearbycustomers between two routes</td></tr><tr><td rowspan=1 colspan=1>Cyclic-exchange</td><td rowspan=1 colspan=1>Exchange cyclically customersbetween multiple routes</td></tr></table>
+
+# C POLICY NETWORK
+
+Figure 6 shows the structure of our policy network.
+
+# D RESULTS ON TSP
+
+To quickly produce results for the TSP, we slightly modified our code for the CVRP by always using the first point in a TSP input as the depot in the CVRP formulation and enforcing that there is exactly one route in a solution. The capacity of any point is naturally assumed to be zero. Lastly, we implemented a simple perturbation operator by randomly permuting 20 points for TSP-50 and TSP-100 (10 points for TSP20). Figure 7 shows the trends of individual polices, as well as the ensemble one, for TSP-100. Again we observe that the ensemble method out-performed individual policies, while producing a total distance slightly above the optimal value of 7.76 as reported in the literature. The results for TSP-20 and TSP-50 are similar and thus omitted.
+
+# E SCALABILITY ANALYSIS
+
+We also tested our method on larger CVRP instances, CVRP-200 and CVRP- $1 0 0 0 ^ { 3 }$ , using the same experimental settings as CVRP-100 (e.g. the capacity of a vehicle is fixed at 50). The results are
+
+![](images/783b019a672d88743284734b64dfad47db4f6d8b3db3b15d09e89e7f16446ad1.jpg)  
+Figure 6: Policy network. The dash-line box is the state embedding part of policy network, which contains problem- and solution-specific input features, an attention network, and a sequence of historical actions and effects. The concatenated values are fed into a network of two fully connected layers, producing a vector of action probabilities.
+
+![](images/81b0bcc5f4fef4dedcd2eb460defc19884151001d2cbb85a7763085b32dc779e.jpg)  
+Figure 7: Results for TSP-100
+
+given in Figure 8, which shows that our ensemble method scales well as the number of customers increases. In particular, the running time of our method increases less dramatically than LKH3.
+
+![](images/03699e7afe1a49a3e3cab690e41519f06213daed8067e598acc25ce915494521.jpg)  
+Figure 8: Traveling cost and average computation time as the number of customers increases
+
+# F SENSITIVITY ANALYSIS
+
+![](images/ddd526c3add62c990e1eeba084d6cf3fc80402e68a557ffb9a214e29bf86f950.jpg)  
+Figure 9: CVRP-100 results under different data distributions by Policy 3
+
+![](images/964326a33b9b570479b09dfc1088bc5ff3594bb5df94cdf739f3834fe467e1cb.jpg)  
+Figure 10: Generalization results using Policy 3
+
+Following the same protocol of data generation as in Uchoa et al. (2017) (the authors of the paper were the creators and owners of the CVRPLib website4), we tested our method for four additional scenarios, namely, central depot positioning, eccentric depot positioning, clustered customer positioning, and random-clustered customer positioning. It is worthwhile to point out that the data distribution used in our initial paper corresponds to random depot and random customer positioning. Figure 9 (a) shows the impact of depot positioning (while using random customer positioning), and (b) shows the effect of customer positioning (while using random depot positioning). To make results from different data distributions comparable, we normalized the distance by the minimal distance achieved for each data distribution (denoted by “Ratio” as in the Figure), respectively. The plots show that our method works across different data distributions, and the decreasing trends of distance look similar.
+
+Furthermore, we also used the model trained for CVRP-100 to solve CVRP-50 and CVRP-20. Similarly, we also tested a trained CVRP-50 model for CVRP-20. Figure 10 shows that current implementation of our method can be applied to problems of different sizes.

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+    {
+        "type": "text",
+        "text": "A LEARNING-BASED ITERATIVE METHOD FOR SOLVING VEHICLE ROUTING PROBLEMS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Hao Lu ∗ Princeton University, Princeton, NJ 08540 {haolu}@princeton.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Xingwen Zhang ∗ & Shuang Yang Ant Financial Services Group, San Mateo, CA 94402 {xingwen.zhang,shuang.yang}@antfin.com ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            311,
+            544,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper is concerned with solving combinatorial optimization problems, in particular, the capacitated vehicle routing problems (CVRP). Classical Operations Research (OR) algorithms such as LKH3 (Helsgaun, 2017) are inefficient and difficult to scale to larger-size problems. Machine learning based approaches have recently shown to be promising, partly because of their efficiency (once trained, they can perform solving within minutes or even seconds). However, there is still a considerable gap between the quality of a machine learned solution and what OR methods can offer (e.g., on CVRP-100, the best result of learned solutions is between 16.10-16.80, significantly worse than LKH3’s 15.65). In this paper, we present “Learn to Improve” (L2I), the first learning based approach for CVRP that is efficient in solving speed and at the same time outperforms OR methods. Starting with a random initial solution, L2I learns to iteratively refine the solution with an improvement operator, selected by a reinforcement learning based controller. The improvement operator is selected from a pool of powerful operators that are customized for routing problems. By combining the strengths of the two worlds, our approach achieves the new state-of-the-art results on CVRP, e.g., an average cost of 15.57 on CVRP-100. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            601,
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+            617
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we focus on an important class of combinatorial optimization, vehicle routing problems (VRP), which have a wide range of applications in logistics. Capacitated vehicle routing problem (CVRP) is a basic variant of VRP, aiming to find a set of routes with minimal cost to fulfill the demands of a set of customers without violating vehicle capacity constraints. The CVRP is NP-hard (Dantzig & Ramser, 1959), and both exact and heuristic methods have been developed to solve it (Fukasawa et al., 2006; Golden et al., 2008; Kumar & Panneerselvam, 2012; Toth & Vigo, 2014). ",
+        "bbox": [
+            174,
+            632,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In recent years, especially after the seminal work of Pointer Networks (Vinyals et al., 2015), researchers start to develop new deep learning and reinforcement learning (RL) frameworks to solve combinatorial optimization problems (Bello et al., 2016; Mao et al., 2016; Khalil et al., 2017; Bengio et al., 2018; Kool et al., 2019; Chen & Tian, 2019). For the CVRP itself, a number of RLbased methods have been proposed in the literature (Nazari et al., 2018; Kool et al., 2019; Chen & Tian, 2019). The learning based methods are trained on a huge number of problem instances, and have been shown to be extremely fast in producing solutions of reasonably good quality. However, when tested with the same benchmark instances, these learning-based methods cannot outperform the state-of-the-art method LKH3 (Helsgaun, 2017), which is a penalty-function-based extension of classical Lin-Kernighan heuristic (Lin & Kernighan, 1973; Helsgaun, 2000). For example, on CVRP with 100 customers, LKH3 is able to produce an average cost of 15.65. This line of research motivated us to study a framework that combines the strength of Operations Research (OR) heuristics with learning capabilities of machine learning (RL in particular). Machine learning can learn to solve a class of problem instances fast, when test instances are generated from the same distribution as training instances. Classical approaches like search algorithms are effective but may need heavy computation, which is time-consuming. Our research interest is in fusing the strengths of these two worlds. Another related line of research is hyper-heuristics, which is “a search method or learning mechanism for selecting or generating heuristics to solve computational search problems” (Burke et al., 2013). Instead of developing a high-level methodology without the need of knowing the details of low-level heuristics, we are primarily interested in a closely integrated system that best utilizes the strength of OR operators and learning capability. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our Contributions. Instead of directly constructing a solution from the problem instance (Graves et al., 2014; Sutskever et al., 2014; Vinyals et al., 2015), we propose a framework that iteratively searches among solutions, until a certain termination condition is satisfied. Our main contributions are as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• We present a learning-based algorithm for solving CVRP, achieving new state-of-the-art results. The recent line of works using RL to solve CVRP shows the potential of machine learning algorithms. They can solve CVRP faster, but cannot beat classical OR solvers like LKH3 in term of solution quality. Our algorithm is the first machine learning framework that outperforms LKH3 on CVRP.   \nWe propose a novel hierarchical framework. Instead of putting all operators in one action pool, we separate heuristic operators into two classes, namely improvement operators and perturbation operators. At each state, we choose the class first and then choose operators within the class. Learning from the current solution is made easier by focusing RL on the improvement operators only.   \n• We propose an ensemble method, which trains several RL policies at the same time, but with different state input features. The ensemble method is shown to produce superior results than individual policies with an equivalent amount of computation. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Related Work. In recent years, there have been many studies using deep learning and RL to solve combinatorial optimization problems (Smith, 1999; Mao et al., 2016; Lodi & Zarpellon, 2017; Velickovi ˇ c et al., 2017; Lombardi & Milano, 2018; Bengio et al., 2018). Routing problems, es- ´ pecially traveling salesman problems (TSP) and VRP, have been explored by a sequence of works (Vinyals et al., 2015; Bello et al., 2016; Khalil et al., 2017; Li et al., 2018; Deudon et al., 2018; Kaempfer & Wolf, 2018; Nazari et al., 2018; Kool et al., 2019; Chen & Tian, 2019). Most of these works, with the exception of Chen & Tian (2019), follow an end-to-end approach, which is directly constructing a solution from scratch. Vinyals et al. (2015) first introduce the Pointer Network, inspired by sequence-to-sequence models, to solve TSP. They use an attention model to learn the order of different nodes in a supervised fashion. Later Bello et al. (2016) develop an RL algorithm to train the Pointer Network. Their framework learns the optimal policy from problem instances and needs no supervised solutions. Nazari et al. (2018) improve the Pointer Network with a new design, making the model invariant with respect to the input sequence, and extend it to solve VRP. Kool et al. (2019) propose a model based on attention layers, and an RL algorithm to train this model with a simple but effective baseline. Chen & Tian (2019) propose a NeuRewriter model for VRP. They define a rewriting rule set and train two policy networks, a region-picking policy and a rule-picking policy, to obtain the next state. Given an initial solution, their goal is to find a sequence of steps towards the solution with minimal cost. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.1 NOTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            795,
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+            809
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In CVRP, there is a depot and a set of $N$ customers. Each customer $i$ , $i \\in \\{ 1 , \\ldots , N \\}$ , has a demand $d _ { i }$ to be satisfied. A vehicle, which always starts at and ends at the depot, can serve a set of customers as long as the total customer demand does not exceed the capacity of the vehicle. The traveling cost $c _ { i , j }$ is the cost of a vehicle going from node $i$ to $j$ , with $i , j \\in \\{ 0 , 1 , \\ldots , N \\}$ (where the depot is denoted by node 0 for convenience). The objective is to find a routing plan with minimal cost that serves all customers without violating vehicle capacity constraints. Figure 1 gives an illustration of CVRP, while a mathematical formulation is given in the Appendix. ",
+        "bbox": [
+            174,
+            823,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Route. A route is a sequence of nodes visited in order, with the depot (node 0) as the starting and ending node. For example, [0, 1, 3, 4, 0] is a traveling plan that starts at the depot, visits node 1, 3 and 4 sequentially, and returns to the depot. ",
+        "bbox": [
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+            104,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Solution. A solution is a set of routes such that each customer is visited exactly once, and the total demand along each route is less than the vehicle capacity. ",
+        "bbox": [
+            173,
+            208,
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+            279
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Operator. An operator is a mapping from one solution to another. In this paper, instead of directly constructing a solution from scratch, we improve or perturb the solution iteratively using operators. ",
+        "bbox": [
+            173,
+            285,
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/f63b654f7eb5408c622a747a930a1b2dfc862a82d0f0901a48d1d9b3a103db60.jpg",
+        "image_caption": [
+            "Figure 1: An illustration of CVRP. Here we provide a problem instance. The red one is a sample route, and three routes consist of a solution for this problem instance. After applying an operator, current solution changes to a new solution with dashed lines replaced by blue lines. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "2 LEARN TO IMPROVE ",
+        "text_level": 1,
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+        "text": "In this section we formally introduce our “Learn to Improve” (L2I) framework, including main components of the system as well as the design intuitions behind individual components. Figure 2 illustrates the overall framework. As mentioned previously, the framework is iterative in nature, that is, we always start with a feasible solution, continuously improving the solution or perturbing it. Along the way, all constraints remain satisfied. By always maintaining the feasibility of the solution, ",
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+        "type": "text",
+        "text": "we are exploring the space of feasible solutions and any of them found by our search process could potentially be a good solution. After $T$ (a parameter set in advance) steps the algorithm stops, and we choose the one with the minimum traveling cost as our final solution. Our framework has a few distinct components (e.g., how to improve the solution, when and how to perturb it), which could be rule-based, learned by machine learning or mixed, and thus allows us to experiment with different ways of integrating machine learning and OR, some of which could lead to superior methods either in terms of better solution quality or in terms of computational efficiency. It is worthwhile to point out that, as contrast to the work of Chen & Tian (2019), the solution space we explore is much larger since we are employing a rich set of improvement and perturbation operators. ",
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+            "Figure 2: Our hierarchy framework. Given a problem instance, our algorithm first generates a feasible solution. Then it iteratively updates the solution with an improvement operator selected by an RL-based controller or with a perturbation operator chosen by a rule-based controller. After a certain number of steps, we choose the best one among all visited solutions. "
+        ],
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+        "type": "text",
+        "text": "For this research work, we implemented and experimented with a number of design choices, and ended up with a method that is both computationally efficient and able to produce state-of-the-art results. The details of the main components of our method are as follows. Given a history of most recent solutions, our method uses a threshold-based rule to decide whether we should continue to improve the current solution, or should perturb it and restart with the perturbed solution. If it decides that the current solution could still be improved, it will use an RL-based controller to choose one of the improvement operators, and try to improve the solution with the chosen operator. We have a rich set of improvement operators (the list of improvement operators and their details are given in the Appendix), where intra-route ones attempt to reduce the cost of current solution by moving customers to different positions in individual routes, while inter-route ones attempt to reduce the cost by moving customers among different routes. Given that the improvement operators are of distinct characteristics, it is not straightforward to know in advance which operators are most effective for the problem under investigation. It is also difficult to know a pre-defined ordering of the operators that is best for the problem. Thus, an RL-based controller is a good choice to learn the set of improvement operators that is more effective for the problem (or more specifically, the problem and the data distribution from which the training and test instances are sampled). As will be shown in Section 3, our RL model is able to differentiate more useful improvement operators from less useful ones for CVRP, as well as generate an implicit ordering on how the operators will be applied. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "On the other hand, upon reaching a local minimum the perturbation controller randomly chooses a perturbation operator that destroys (completely or partially) and reconstructs a number of routes to generate a new starting solution (Table 6 in the Appendix gives the list of perturbation operators and their details). Specifically, if no cost reduction has been made for $L$ improvement steps, we perturb the solution and restart the improvement iteration (where for ease of explanation a maximum consecutive sequence of improvement operators applied before perturbation is called an improvement iteration). As a perturbation changes the solution quite dramatically (by producing a substantially different solution that is usually worse than the current one), we found that it is useful to start a new improvement iteration with a reasonably good starting point (e.g. by filtering out the restarting solutions that are significantly worse than current solution or currently best solution). It is clear that we purposely separate improvement operators from perturbation ones, and an alternative design would be to mix them all together and have a single controller deciding which operator to apply next. However, the improvement operators are of different nature from the perturbation ones, and their impacts are different since the perturbation operators have long-lasting effect by affecting an entire improvement iteration. Our experience also suggests that learning is made easier by focusing RL on the improvement operators only. Lastly, it is worthwhile to point out that although the rulebased perturbation controller is shown to be effective, we do not rule out the possibility that it can also be learning-based. ",
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+    {
+        "type": "text",
+        "text": "The framework described above provides a way of combining the strength of OR operators, which are powerful since they are custom-made for routing problems, with learning capabilities of RL, which is flexible and can be adapted to a given problem and its associated data distribution. Having described our overall framework, we are now ready to present the details of the improvement controller and operators. ",
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+    {
+        "type": "text",
+        "text": "2.1 IMPROVEMENT CONTROLLER AND OPERATORS",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The improvement controller starts with an initial solution, which is either constructed randomly (for the first improvement iteration)1 or produced by a perturbation operator (for subsequent iterations), and then tries to improve it, i.e., reducing the total traveling distance without violating any constraints, by selectively applying an improvement operator in Table 5. For the RL model, the set of improvement operators constitute our action space. These operators change the solution locally and most are computationally light. With the current state as input, a neural network produces a vector of action probabilities, and the weights of the network are trained with policy gradient. Figure 6 illustrates the components of our RL model, and their details are given as follows. ",
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+    {
+        "type": "text",
+        "text": "2.1.1 STATES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Each state includes features from the problem instance, the solution and the running history. Stationary features, such as the location and demand of each customer, are considered as problem-specific since they are invariant across solutions. Solution-specific features are based on the current traveling plan. For example, given a solution, for each customer we compute its neighboring nodes that are visited before and afterwards, as well as the relevant distances. Following Odling (2018), the ¨ running history includes the actions that are recently taken as well as their effects. For example, $a _ { t - h }$ , $1 \\leq h \\leq H$ , is the action taken $h$ steps before current step $t$ , and its effect $e _ { t - h }$ is $+ 1$ if the action led to a reduction of total distance, $^ { - 1 }$ otherwise. A complete description of state features is given in Table 4, where $i ^ { - }$ and $i ^ { + }$ denote the node visited before and after node $; 1 \\leq i \\leq N$ , in the solution, respectively. ",
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+    {
+        "type": "text",
+        "text": "2.1.2 ACTIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The actions can be classified into two classes, intra-route operators and inter-route operators. An intra-route operator attempts to reduce traveling distance of an individual route, while inter-route operators aim at reducing total traveling distance by moving customers between more than one route. The details of the operators are given in Table 5. It is worthwhile to point out that the same operator with different parameters are considered as different actions. For example, Symmetric-exchange(2) with $m = 1 , 2 , 3$ are considered as three distinct actions. For a given problem (or even solution), the same operator with different parameters may perform differently and thus it is appropriate to treat them as separate actions and let the RL model learn how to best use them. ",
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+    {
+        "type": "text",
+        "text": "2.1.3 POLICY NETWORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We use the well-known REINFORCE algorithm (Williams, 1992) to update the gradient ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/31523f468fffca19d15184b24c1e6a722e998feac0ed883391490510c46a461b.jpg",
+        "text": "$$\n\\nabla _ { \\theta } J ( \\theta | s ) = \\mathbb { E } _ { \\pi \\sim p _ { \\theta } ( . | s ) } \\big [ \\big ( L ( \\pi | s ) - b ( s ) \\big ) \\nabla _ { \\theta } \\log p _ { \\theta } ( \\pi | s ) \\big ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "of policy with a baseline function $b ( s )$ . Given a state as described in Section 2.1.1, a policy network outputs a list of action probabilities, one for each action described. As illustrated in Figure 6, problem- and solution-specific input features are transformed into an embedding of length $D$ (we use $D = 6 4$ ), which is fed into an attention network (Vaswani et al., 2017) (we use an attention layer with 8 heads and 64 output units). The output of the attention network is concatenated with a sequence of recent actions and their effects (when $H > 0$ ). Lastly, the concatenated values are fed into a network of two fully connected layers, where the first layer uses 64 units and a Relu activation function and the second layer uses Softmax, producing $| A |$ action probabilities where $A$ is the set of actions. ",
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+    {
+        "type": "text",
+        "text": "2.1.4 REWARDS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We have experimented with a number of reward designs, two of which are producing satisfactory results as well as distinct patterns of operator sequences. The first reward function (denoted by RF1) focuses on the intermediate impact of the improvement operators. Specifically, the reward is $+ 1$ if the operator improves the current solution, -1 otherwise. The second reward function (denoted by RF2) is advantage-based. The total distance achieved for the problem instance during the first improvement iteration is taken as a baseline. For each subsequent iteration, all operators applied during this iteration received a reward equal to the difference between the distance achieved during the iteration and the baseline. We observed that an operator is often able to achieve a large distance reduction for a freshly perturbed solution, while it becomes harder and harder to do so in later improvement steps. In particular, the likelihood of distance reduction as well as the magnitude of such reduction, both decrease as the iteration proceeds. Therefore, it seemed unfair to give early improvement operators a larger reward. The observation suggested that operators used in the same improvement iteration should be rewarded equally and there would be no discounting (or equivalently, the discount factor $\\gamma$ is 1). ",
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+    {
+        "type": "text",
+        "text": "To conclude the methodology section, we restart the improvement iteration until reaching a maximum number $T$ of rollout (either improvement or perturbation) steps. Following a common practice of encouraging exploration, we use $\\epsilon$ -greedy (Sutton & Barto, 2018) such that with a probability of 0.05 the RL controller will choose a random improvement action. Lastly, we also experimented with ensembling by training 6 different policies with $H = 1 , 2 , \\ldots , 6$ (while keeping other components of the policy network identical). Ensembling facilitates learning of a diverse set of policies, as well as reducing wall-clock running time. ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS AND ANALYSES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we present our experiment results. First we introduce a detailed setup of CVRP and hyper-parameters used. Then we compare our performance with prior neural network approaches, i.e., Nazari et al. (2018), Kool et al. (2019), Chen & Tian (2019), and a classic state-of-the-art heuristic algorithm. At last we provide detailed analysis of our framework2. ",
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+    {
+        "type": "text",
+        "text": "Setup and hyper-parameters. We follow the same settings as previous works (Nazari et al., 2018; Kool et al., 2019; Chen & Tian, 2019) for CVRP. We consider three sub-problems with number of customers $N = 2 0 , 5 0 , 1 0 0$ , respectively. The location $( x _ { i } , y _ { i } )$ of each customer, as well as of the depot, is uniformly sampled from unit square (specifically, $x _ { i }$ and $y _ { i }$ are uniformly distributed in the interval $[ 0 , 1 ]$ , respectively), and the traveling cost between two locations $c _ { i , j }$ is simply the corresponding Euclidean distance. The demand $d _ { i }$ of each customer is uniformly sampled from the discrete set $\\{ \\bar { 1 } , 2 , \\dots , 9 \\}$ . The capacity of a vehicle is 20, 30, 40 for $N = 2 0 , 5 0 , 1 0 0$ , respectively. After $L \\ = \\ 6$ consecutive step of no improvement, we perturb the solution. To train the policy network, we use ADAM with a learning rate of 0.001. Unless otherwise stated, for a problem instance and a given policy we randomly initiate a feasible solution, and then iteratively update the solution $T = 4 0 0 0 0$ times following the policy. In the following section we will discuss the performance of a policy with different rollout steps (recall that a rollout step is either an improvement step or a perturbation one). We choose the best one among all 40000 visited solutions as the final solution for a given problem instance. In the ensemble model we use different number of historical actions and effects to train a set of diverse policies (recall the policy network in Figure 6), and for any problem instance we choose the best solution produced by these policies. Unless otherwise stated, all reported metrics, such as the final traveling cost and the running time, are always computed as the average over 2000 random samples. Lastly, our method was implemented in Python, and the experiments were run computer nodes, each with a single Nvidia Tesla T4 GPU. ",
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+    {
+        "type": "text",
+        "text": "3.1 PERFORMANCE COMPARISON ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In Table 1 we compare the performance of our algorithm with prior neural network approaches mentioned above, Google OR-tools (Google, 2019), and classic state-of-the-art heuristic algorithm LKH3 (Helsgaun, 2017). Our ensemble method chose the solution with minimum traveling cost among those produced by Policy $i$ (with $1 \\leq i = h \\leq 6$ ). As shown in Table 1, our algorithm outperforms the prior approaches in terms of average traveling cost. In particular, the average distance achieved by our algorithm is significantly shorter than prior neural network approaches. Thus, our algorithm is producing state-of-the-art results for CVRP, and is the first learning-based framework that outperforms the well-known classic heuristic algorithm. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 1: Comparison of our experiment results with those reported in the literature "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>N=20 Obj.</td><td>N=50 Obj.</td><td>N = 100 Obj.</td></tr><tr><td>Google OR Tools</td><td>6.43</td><td>11.31</td><td>17.16</td></tr><tr><td>Nazari et al. (2018)</td><td>6.40</td><td>11.15</td><td>16.96</td></tr><tr><td>AM greedy (Kool et al., 2019)</td><td>6.40</td><td>10.98</td><td>16.80</td></tr><tr><td>AM sampling (Kool et al., 2019)</td><td>6.25</td><td>10.62</td><td>16.23</td></tr><tr><td>Chen&amp; Tian (2019)</td><td>6.16</td><td>10.51</td><td>16.10</td></tr><tr><td>LKH (Helsgaun, 2017)</td><td>6.14</td><td>10.38</td><td>15.65</td></tr><tr><td>L2I</td><td>6.12</td><td>10.35</td><td>15.57</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.2 ANALYSIS OF THE ENSEMBLE METHOD ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Recall that we train 6 different policies with $H \\in \\{ 1 , 2 , \\ldots , 6 \\}$ . To illustrate the motivation of the ensemble method, we randomly pick 10 problem instances and show the traveling cost under different policies for these problem instances in Table 2. We can see that Policy 1 did the best for the first problem instance, but it performed the worst for the second one. Furthermore, there is no clear winning policy, which is the reason we propose an ensemble method. ",
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+        "text": "",
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+    {
+        "type": "table",
+        "img_path": "images/62ec0f973d219a6466f71b3d860b87ae210ef77986c8dcd70d46314dab60a4d8.jpg",
+        "table_caption": [
+            "Table 2: Motivating examples: traveling cost under different policies for random problem instances "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"10\">Problem instances</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td>Policy 1</td><td>18.65</td><td>16.67</td><td>15.41</td><td>15.11</td><td>16.69</td><td>14.97</td><td>16.88</td><td>16.37</td><td>13.72</td><td>15.56</td></tr><tr><td>Policy 2</td><td>18.86</td><td>16.61</td><td>15.39</td><td>15.10</td><td>16.55</td><td>14.75</td><td>16.73</td><td>16.39</td><td>13.77</td><td>15.56</td></tr><tr><td>Policy 3</td><td>18.81</td><td>16.63</td><td>15.43</td><td>15.06</td><td>16.82</td><td>14.72</td><td>16.70</td><td>16.37</td><td>13.73</td><td>15.79</td></tr><tr><td>Policy 4</td><td>18.90</td><td>16.60</td><td>15.37</td><td>15.04</td><td>16.66</td><td>14.93</td><td>16.83</td><td>16.32</td><td>13.70</td><td>15.62</td></tr><tr><td>Policy 5</td><td>18.73</td><td>16.56</td><td>15.43</td><td>15.12</td><td>16.66</td><td>15.09</td><td>16.80</td><td>16.53</td><td>13.72</td><td>15.56</td></tr><tr><td>Policy 6</td><td>18.92</td><td>16.60</td><td>15.43</td><td>15.09</td><td>16.73</td><td>14.65</td><td>16.59</td><td>16.55</td><td>13.73</td><td>15.56</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/624ae57c3fc32697f30820a4fa695016252a2639b07ba367b61d1639788307d1.jpg",
+        "image_caption": [
+            "Figure 3: Average traveling cost of different policies with different rollout steps "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Figure 3(a), (b) and (c), we plot the average traveling cost over 2000 problem instances for CVRP with $N = 2 0$ , 50, 100, respectively. It is worthwhile to point out that the same hyper-parameters are used for different $N$ values. The top blue line is for a random policy, the bottom red line is for our ensemble method, and lines in between are for Policy 1, 2, . . . , 6. The plots show that, regardless of the number of rollout steps, our trained RL policies consistently outperform the random policy. Figure 4(c) shows that the gap between the random policy and the ensemble method first increases, peaks at about 5000 rollout steps, and then gets smaller and smaller. It is encouraging that we get the maximum gap fast, which will be helpful when computational time is limited. Lastly, in Figure 4(d) we show three policies, the random policy, the ensemble method, and the best RL policy (Policy 3), for CVRP-100. To match the performance of the random policy with 40000 rollout steps, the ensemble method would take 2500 rollout steps, while it takes 5000 rollout steps to match Policy 3 with 40000 rollout steps. The outperformance of the ensemble method becomes more obvious when $N = 2 0 , 5 0$ . These analysis shows that naively increasing rollout steps of a policy provides less marginal gains than ensembling a diverse set of policies. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6946c218f62521a394c44a857f0d0be9245ff7cf8f3da45000b9762e7abbc440.jpg",
+        "table_caption": [
+            "Table 3: Improvement operators mostly used "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Class</td><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Details</td></tr><tr><td rowspan=2 colspan=1>Intra-route</td><td rowspan=1 colspan=1>2-0pt</td><td rowspan=1 colspan=1>Remove two edges andreconnect their endpoints</td></tr><tr><td rowspan=1 colspan=1>Relocate(1)</td><td rowspan=1 colspan=1>Move a customer in the route toa new location</td></tr><tr><td rowspan=3 colspan=1>Inter-route</td><td rowspan=1 colspan=1>Cross(2)</td><td rowspan=1 colspan=1>Exchange the tails of two routes</td></tr><tr><td rowspan=1 colspan=1>Symmetric-exchange(2)</td><td rowspan=1 colspan=1>Exchange segments of length m（m = 1) between two routes</td></tr><tr><td rowspan=1 colspan=1>Relocate(2)</td><td rowspan=1 colspan=1>Move a segment of length m(m = 1) from a route to another</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.3 ANALYSIS OF OPERATOR USAGES",
+        "text_level": 1,
+        "bbox": [
+            176,
+            321,
+            444,
+            335
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As we mentioned before, the RL model is able to differentiate more useful improvement operators from less useful ones for CVRP. In our experiments, we count the usage of different operators for different policies as training epochs grow. When the myopic reward function RF1 in Section 2.1.4 is used, our experimental results show that the policy converges to use a fixed subset of improvement operators (for detailed operators, see Table 3). This subset of operators are also preferred by all policies when we use RF2. However, the pattern of operator usages varies among the policies. For example, Figure 4(a) and (b) illustrate different patterns of operator usages for Policy 1 and Policy 2, respectively. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/ce3d9514a6179351fd7a273c5743c85971fba32406e8a6dbf149bfd524a33d35.jpg",
+        "image_caption": [
+            "Figure 4: Pattern of operator usages as training epoch grows "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            478,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.4 ANALYSIS OF PERTURBATION MAGNITUDE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            703,
+            508,
+            715
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "From our experimental study we also observed that the solution quality is sensitive to the magnitude of perturbation. To illustrate the impact of perturbation, in Figure 5 we plot the performance of two run configurations, one with Random-permute applied to all routes (called Random-permute-all), while the other with Random-permute applied to two routes only (called Random-permute-2). As seen from Figure 5, Random-permute-2 significantly outperformed Random-permute-all. Intuitively, when the magnitude of perturbation is too large, the resulting solution generally becomes much worse and it will take our algorithm a large number of improvement steps to remedy the deterioration. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/b959c7ff23f3e84dd13b04beacf1d87591b63b639cdcbb60b9dae11d17059099.jpg",
+        "image_caption": [
+            "Figure 5: Impact of perturbation magnitude "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            318,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper we propose “Learn to Improve” for solving VRP, which starts with an initial solution and iteratively updates the solution with an improvement operator selected by an RL-based controller or with a perturbation operator chosen by a rule-based controller. We also propose an ensemble method that trains several RL policies and chooses the best solution produced by the policies. Our method achieved new state-of-the-art results for CVRP instances. ",
+        "bbox": [
+            174,
+            133,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our work provides a way of combining the strength of OR with learning capabilities of RL. For future work, we would like to apply the solution framework to solve other variants of the VRP, such as vehicle routing problems with time windows (VRPTW), as well as other combinatorial problems, such as maximum independent set problems and graph coloring problems. Furthermore, it is interesting to investigate whether allowing temporary constraint violations in our framework will help improve solution quality or not. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 ACKNOWLEDGEMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            315,
+            387,
+            332
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank our colleagues at Ant Financial, Wei Yan and Junping Zhao, for allocating GPU resources needed for the computational experiments. ",
+        "bbox": [
+            173,
+            348,
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+        ],
+        "page_idx": 8
+    },
+    {
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+    {
+        "type": "text",
+        "text": "A AN INTEGER PROGRAMMING FORMULATION OF THE CVRP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            191,
+            702,
+            207
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Formally, there is a depot and a set of $N$ customers in the CVRP. Each customer $i$ , $i \\in \\{ 1 , \\ldots , N \\}$ , has a demand $d _ { i }$ to be satisfied. A vehicle, which always starts at and ends at the depot, can serve a set of customers as long as the total customer demand does not exceed the capacity of the vehicle $C$ . The traveling cost $c _ { i , j }$ is the cost of a vehicle going from node $i$ to $j$ , with $i , j \\in V = \\{ 0 , 1 , . . . , N \\}$ (where the depot is denoted by node 0 for convenience). The objective is to find a routing plan with minimal cost that serves all customers without violating vehicle capacity constraints. An integer programming formulation of the CVRP (Toth & Vigo, 2002) is given below. ",
+        "bbox": [
+            173,
+            220,
+            825,
+            319
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ffb127c2bedd0cf8ff014640269ed4dfe33b14376ba84d9e19b68b27bfb5b4a2.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\operatorname* { m i n } _ { \\tau \\to \\tau } \\sum _ { \\ell \\in \\mathcal { N } _ { i } } \\sum _ { \\ell \\in \\mathcal { N } _ { i } } \\alpha _ { i , j } } \\\\ & { \\ : \\ : \\ : \\sum _ { \\ell \\in \\mathcal { N } _ { i } } \\alpha _ { i , j } \\le C _ { i , j } \\le V \\setminus \\{ 0 \\} } \\\\ & { \\ : \\ : \\ : \\delta \\cdot \\sum _ { \\ell \\in \\mathcal { N } _ { i } } \\gamma _ { \\ell } = 1 , \\ : \\forall j \\in V \\setminus \\{ 0 \\} } \\\\ & { \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\sum _ { \\ell \\in \\mathcal { N } _ { i } } \\gamma _ { \\ell , j } = 1 , \\ : \\forall i \\in V \\setminus \\{ 0 \\} } \\\\ & { \\ : \\ : \\ : \\ : \\sum _ { \\ell \\in \\mathcal { N } _ { i } \\setminus \\{ 0 \\} } \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\widetilde { \\mathcal { M } } _ { i } = K , } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\mathcal { M } _ { i } \\leq C \\cdot \\ : \\ : \\forall \\ : \\ : \\ : \\ : \\mathcal { M } _ { i } , \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\mathcal { M } _ { i } \\neq \\mathcal { I } \\setminus \\{ \\left\\{ \\left[ \\mathcal { R } \\right] , \\ : \\ : \\ : \\ : \\mu _ { i } \\right\\} } = \\mathcal { A } \\le \\mathcal { L }  \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ & { \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            321,
+            745,
+            553
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $K$ is the number of vehicles available (w.l.o.g., it is assumed that $K = N$ for the CVRP we consider). Constraints (1) and (2) specify that each customer is visited exactly once, while constraints (3) and (4) specify the in and out degree of the depot, respectively. Constraints (5) and (6) impose the vehicle capacity requirements. ",
+        "bbox": [
+            176,
+            554,
+            823,
+            611
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/ae1113fb124271a667bbe89fe44bc749e2a5a4bdc7d0715e42bffc7fca8f68c2.jpg",
+        "table_caption": [
+            "Table 4: State features "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Type</td><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Details</td></tr><tr><td rowspan=8 colspan=1>Problem-andsolution-specific</td><td rowspan=1 colspan=1>Ci</td><td rowspan=1 colspan=1>Demand of customer i</td></tr><tr><td rowspan=1 colspan=1>Ci</td><td rowspan=1 colspan=1>Free capacity of the route containing customer i</td></tr><tr><td rowspan=1 colspan=1>(xi，yi)</td><td rowspan=1 colspan=1>Location of customer i</td></tr><tr><td rowspan=1 colspan=1>(xi-,yi-)</td><td rowspan=1 colspan=1>Location of node visited before i</td></tr><tr><td rowspan=1 colspan=1>(xi+,yi+)</td><td rowspan=1 colspan=1>Location of node visited after i</td></tr><tr><td rowspan=1 colspan=1>di-i</td><td rowspan=1 colspan=1>Distance from i- to i</td></tr><tr><td rowspan=1 colspan=1>dii+</td><td rowspan=1 colspan=1>Distance from i to i+</td></tr><tr><td rowspan=1 colspan=1>di-i+</td><td rowspan=1 colspan=1>Distance from i- to i+</td></tr><tr><td rowspan=2 colspan=1>History-related</td><td rowspan=1 colspan=1>at-h</td><td rowspan=1 colspan=1>Action taken h steps before</td></tr><tr><td rowspan=1 colspan=1>et-h</td><td rowspan=1 colspan=1>Effect of at-h</td></tr></table>",
+        "bbox": [
+            220,
+            655,
+            772,
+            840
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B DETAILS OF STATES AND OPERATORS",
+        "text_level": 1,
+        "bbox": [
+            174,
+            878,
+            526,
+            895
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We list the details of our state features in Table 4, and of operators in Table 5 and 6. ",
+        "bbox": [
+            174,
+            909,
+            722,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/2e53331f0f10e420a7ffbf700071a8984b82b5832c2e4dad7e25708f1eff9587.jpg",
+        "table_caption": [
+            "Table 5: Improvement operators "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Class</td><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Details</td></tr><tr><td rowspan=3 colspan=1>Intra-route</td><td rowspan=1 colspan=1>2-0pt</td><td rowspan=1 colspan=1>Remove two edges andreconnect their endpoints</td></tr><tr><td rowspan=1 colspan=1>Symmetric-exchange(1)</td><td rowspan=1 colspan=1>Exchange two customers in the route</td></tr><tr><td rowspan=1 colspan=1>Relocate(1)</td><td rowspan=1 colspan=1>Move a customer in the route toa new location</td></tr><tr><td rowspan=6 colspan=1>Inter-route</td><td rowspan=1 colspan=1>Cross(2)</td><td rowspan=1 colspan=1>Exchange the tails of two routes</td></tr><tr><td rowspan=1 colspan=1>Reverse-cross(2)</td><td rowspan=1 colspan=1>Reverse one of two routes and thenexchange their tails</td></tr><tr><td rowspan=1 colspan=1>Symmetric-exchange(2)</td><td rowspan=1 colspan=1>Exchange segments of length m(m =1,2,3) between two routes</td></tr><tr><td rowspan=1 colspan=1>Asymmetric-exchange(2)</td><td rowspan=1 colspan=1>Exchange segments of length m and n(m=1,2,3,n=1,2,3,m/ n)between two routes</td></tr><tr><td rowspan=1 colspan=1>Relocate(2)</td><td rowspan=1 colspan=1>Movea segment oflength m(m =1,2,3) from a route to another</td></tr><tr><td rowspan=1 colspan=1>Cyclic-exchange(3)</td><td rowspan=1 colspan=1>Exchange cyclically one customerbetween three routes</td></tr></table>",
+        "bbox": [
+            220,
+            137,
+            772,
+            400
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/7c62abb095f32f82e80d381838d98e152d28bf12d22bb2737124553d3f5d92f4.jpg",
+        "table_caption": [
+            "Table 6: Perturbation operators "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Class</td><td rowspan=1 colspan=1>Name</td><td rowspan=1 colspan=1>Details</td></tr><tr><td rowspan=3 colspan=1>Inter-route perturbation</td><td rowspan=1 colspan=1>Random-permute</td><td rowspan=1 colspan=1>Randomly destroy m routes andre-construct routes by visiting affectedcustomers in a random order</td></tr><tr><td rowspan=1 colspan=1>Random-exchange(2)</td><td rowspan=1 colspan=1>Randomly exchange m pairs of nearbycustomers between two routes</td></tr><tr><td rowspan=1 colspan=1>Cyclic-exchange</td><td rowspan=1 colspan=1>Exchange cyclically customersbetween multiple routes</td></tr></table>",
+        "bbox": [
+            192,
+            455,
+            802,
+            573
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C POLICY NETWORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            599,
+            366,
+            617
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 shows the structure of our policy network. ",
+        "bbox": [
+            173,
+            632,
+            509,
+            647
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D RESULTS ON TSP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            665,
+            356,
+            683
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To quickly produce results for the TSP, we slightly modified our code for the CVRP by always using the first point in a TSP input as the depot in the CVRP formulation and enforcing that there is exactly one route in a solution. The capacity of any point is naturally assumed to be zero. Lastly, we implemented a simple perturbation operator by randomly permuting 20 points for TSP-50 and TSP-100 (10 points for TSP20). Figure 7 shows the trends of individual polices, as well as the ensemble one, for TSP-100. Again we observe that the ensemble method out-performed individual policies, while producing a total distance slightly above the optimal value of 7.76 as reported in the literature. The results for TSP-20 and TSP-50 are similar and thus omitted. ",
+        "bbox": [
+            173,
+            696,
+            825,
+            808
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "E SCALABILITY ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            829,
+            411,
+            844
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We also tested our method on larger CVRP instances, CVRP-200 and CVRP- $1 0 0 0 ^ { 3 }$ , using the same experimental settings as CVRP-100 (e.g. the capacity of a vehicle is fixed at 50). The results are ",
+        "bbox": [
+            176,
+            859,
+            821,
+            888
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/783b019a672d88743284734b64dfad47db4f6d8b3db3b15d09e89e7f16446ad1.jpg",
+        "image_caption": [
+            "Figure 6: Policy network. The dash-line box is the state embedding part of policy network, which contains problem- and solution-specific input features, an attention network, and a sequence of historical actions and effects. The concatenated values are fed into a network of two fully connected layers, producing a vector of action probabilities. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            338,
+            101,
+            660,
+            570
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/81b0bcc5f4fef4dedcd2eb460defc19884151001d2cbb85a7763085b32dc779e.jpg",
+        "image_caption": [
+            "Figure 7: Results for TSP-100 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            334,
+            672,
+            661,
+            824
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "given in Figure 8, which shows that our ensemble method scales well as the number of customers increases. In particular, the running time of our method increases less dramatically than LKH3. ",
+        "bbox": [
+            173,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/03699e7afe1a49a3e3cab690e41519f06213daed8067e598acc25ce915494521.jpg",
+        "image_caption": [
+            "Figure 8: Traveling cost and average computation time as the number of customers increases "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            99,
+            812,
+            251
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "F SENSITIVITY ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            303,
+            403,
+            318
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/ddd526c3add62c990e1eeba084d6cf3fc80402e68a557ffb9a214e29bf86f950.jpg",
+        "image_caption": [
+            "Figure 9: CVRP-100 results under different data distributions by Policy 3 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            338,
+            808,
+            502
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/964326a33b9b570479b09dfc1088bc5ff3594bb5df94cdf739f3834fe467e1cb.jpg",
+        "image_caption": [
+            "Figure 10: Generalization results using Policy 3 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            549,
+            818,
+            705
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Following the same protocol of data generation as in Uchoa et al. (2017) (the authors of the paper were the creators and owners of the CVRPLib website4), we tested our method for four additional scenarios, namely, central depot positioning, eccentric depot positioning, clustered customer positioning, and random-clustered customer positioning. It is worthwhile to point out that the data distribution used in our initial paper corresponds to random depot and random customer positioning. Figure 9 (a) shows the impact of depot positioning (while using random customer positioning), and (b) shows the effect of customer positioning (while using random depot positioning). To make results from different data distributions comparable, we normalized the distance by the minimal distance achieved for each data distribution (denoted by “Ratio” as in the Figure), respectively. The plots show that our method works across different data distributions, and the decreasing trends of distance look similar. ",
+        "bbox": [
+            173,
+            748,
+            825,
+            901
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we also used the model trained for CVRP-100 to solve CVRP-50 and CVRP-20. Similarly, we also tested a trained CVRP-50 model for CVRP-20. Figure 10 shows that current implementation of our method can be applied to problems of different sizes. ",
+        "bbox": [
+            174,
+            103,
+            821,
+            146
+        ],
+        "page_idx": 14
+    }
+]

parse/train/Hk6WhagRW/Hk6WhagRW.md

@@ -0,0 +1,297 @@
+# EMERGENT COMMUNICATION THROUGH NEGOTIATION
+
+Kris Cao∗   
+Department of Computer Science and Technology,   
+University of Cambridge, UK   
+Angeliki Lazaridou, Marc Lanctot, Joel Z Leibo, Karl Tuyls, Stephen Clark   
+DeepMind,   
+London, UK
+
+# ABSTRACT
+
+Multi-agent reinforcement learning offers a way to study how communication could emerge in communities of agents needing to solve specific problems. In this paper, we study the emergence of communication in the negotiation environment, a semi-cooperative model of agent interaction. We introduce two communication protocols – one grounded in the semantics of the game, and one which is a priori ungrounded and is a form of cheap talk. We show that self-interested agents can use the pre-grounded communication channel to negotiate fairly, but are unable to effectively use the ungrounded channel. However, prosocial agents do learn to use cheap talk to find an optimal negotiating strategy, suggesting that cooperation is necessary for language to emerge. We also study communication behaviour in a setting where one agent interacts with agents in a community with different levels of prosociality and show how agent identifiability can aid negotiation.
+
+# 1 INTRODUCTION
+
+How can communication emerge? A necessary prerequisite is a task that requires coordination between multiple agents to solve, and some communication protocol for the agents to exchange messages through (see a review by Wagner et al. (2003) on earlier work on emergent communication as well as recent deep reinforcement learning methods by Foerster et al. (2016) and Sukhbaatar et al. (2016)). Given these basic requirements, an interesting question to ask is what task structures aid the emergence of communication and how different communication protocols affect task success.
+
+In the context of linguistic communication, previous work on this subject has mainly studied the emergence of communication in co-operative games like referential games, variants of the Lewis signaling game (Lewis, 1969), where messages are used to disambiguate between different possible referents (Goldman et al., 2007; Lazaridou et al., 2016; Evtimova et al., 2017). Human language, though, is not merely a referential tool. Amongst other things, we communicate private information and thoughts, discuss plans, ask questions and tell jokes. Moreover, many human interactions are not fully cooperative, yet we can still successfully use language to communicate in these situations.
+
+In this paper, we study communication in the negotiation game (see Figure 1), an established model of non-cooperative games in classical game theory (Nash, 1950b; Neumann & Morgenstern, 1944; Nash, 1950a; 1951; Schelling, 1960; Binmore et al., 1986; Peters, 2008). In this game, agents are asked to establish a mutually acceptable division of a common pool of items while having their own hidden utilities for each of them. Effective communication is crucial in this game, as the agents need to exchange strategic information about their desires, infer their opponent’s desires from communication, and balance between the two.
+
+Work in classical game theory on negotiation typically uses simple forms of offer / counter-offer bargaining games that do not explicitly address the question of emergent communication (Rubinstein, 1982). Recent work on deep multi-agent reinforcement learning (MARL) has shown great success in teaching agents complex behaviour without a complex environment simulator or demonstration data (Panait & Luke, 2005; Busoniu et al., 2008; Tuyls & Weiss, 2012; Silver et al., 2017; Leibo et al., 2017). By repeatedly interacting with other agents learning at the same time, agents can gradually bootstrap complex behaviour, including motor skills (Bansal et al., 2017) and linguistic communication (Lazaridou et al., 2016; Havrylov & Titov, 2017).
+
+![](images/77e7b9b7e1fb927342b9e14cc564b4ad8fff40a371b3635dd58b8b08ed9dd949.jpg)  
+Figure 1: High-level overview of the negotiation environment that we implement. Agent A consistently refers to the agent who goes first.
+
+We apply techniques from the MARL literature and train agents to negotiate using task success as the only supervision signal.1 We show that, when communicating via a task-specific communication channel with inherent semantics, selfish agents can learn to negotiate fairly and divide up the item pool to the agents’ mutual satisfaction. However, when communicating via cheap talk (Crawford & Sobel, 1982; Farrell & Rabin, 1996), a task-independent communication channel consisting of sequences of arbitrary symbols similar to language, selfish agents fail to exhibit negotiating behaviour at all. On the other hand, we show that cheap talk can facilitate effective negotiation in prosocial agents, which take the other agent’s reward into consideration, providing experimental evidence that cooperation is necessary for language emergence (Nowak & Krakauer, 1999).
+
+The above results are obtained from paired agents interacting exclusively with each other. In more realistic multi-agent scenarios, agents may interact with many other agents within a society. In these cases, cheap talk can have a significant effect on the evolutionary dynamics of the population (Robson, 1990) as well as the equilibria, stability, and basins of attractions (Skyrms, 2002). Furthermore, it is well-known that, unless trained in a diverse environment, agents overfit to the their specific opponent or teammate (Lanctot et al., 2017). Inspired by these considerations, we perform experiments where agents interact with many agents having different prosociality levels, and find that being able to identify and model other agents’ beliefs aids the negotiation success. This is consistent with experiments using models based on Theory of Mind: boundedly rational agents can collectively benefit by making inferences about the sophistication levels and beliefs of their opponents, and there is evidence that this occurs in human behavior (Yoshida et al., 2008).
+
+# 2 GAME SETTING
+
+# 2.1 NEGOTIATION ENVIRONMENT
+
+The negotiation task is based on the set-up of Lewis et al. (2017), itself based on DeVault et al. (2015). Agents are presented with three types of items: peppers, cherries and strawberries. At each round (i) an item pool is sampled uniformly, instantiating a quantity (between 0 and 5) for each of the types and represented as a vector $i \in \{ 0 . . . 5 \} ^ { 3 }$ and (ii) each agent $j$ receives a utility function sampled uniformly, which specifies how rewarding one unit of each item is (with item rewards between 0 and 10, and with the constraint that there is at least one item with non-zero utility), represented as a vector $u _ { j } \in \{ 0 . . . 1 0 \} ^ { 3 }$ . These utilities are hidden, i.e., each agent only has access to its own utilities. Following these assignments, the agents start “negotiating” for $N$ timesteps by exchanging messages $m _ { t } ^ { j }$ and proposals $p _ { t } ^ { j } \in \{ 0 . . . 5 \} ^ { 3 }$ for each timestep $1 \leq t \leq N$ . Agents alternate between timesteps, meaning that agent A always acts on the odd turns while agent B on the even turns.
+
+Either agent can terminate the negotiation at any timestep $t$ with a special action, which signifies agreement with the most recent proposal made by the other agent. For example, if agent B terminates the negotiation at $t = 4$ , agent A gets rewarded by $R _ { A } = u _ { A } \cdot ( p _ { t - 1 } ^ { A } )$ while agent $\mathbf { B }$ receives reward $R _ { B } = u _ { B } \cdot ( i - p _ { t - 1 } ^ { A } )$ , where $\cdot$ denotes the dot product of vectors. Note that, if an agent makes an invalid proposal, such as asking for more items than exist in the item pool, and this is accepted, then both agents receive $R _ { A } = R _ { B } = 0$ .
+
+We impose an upper limit, $N$ , on the number of negotiation turns allowed. If the agents get to the limit without agreement, then both agents receive no reward. We experimented with having a fixed upper limit of 10 turns. However, this led to a “first-mover” advantage: if the agents know in advance how many turns the negotiation lasts, there is a degenerate policy in which agent A can wait until the last timestep $N - 1$ before giving a lopsided offer, which agent B is compelled to accept, a situation similar to the ultimatum game (Guth et al., 1982). To eliminate this effect, we sample N ¨ between 4 and 10 at each round, according to a truncated Poisson distribution with mean 7.
+
+# 2.2 COMMUNICATION CHANNELS
+
+To achieve negotiation, the agents need to communicate. One obvious communication protocol is to directly transmit the proposed division of items $p _ { j }$ . We refer to this as the proposal channel. This communication channel is task-specific and pertains to the action space of the agents used to derive their pay-offs. Consequently, the information bandwidth is restricted and grounded in the action space of the game. Motivated by work on emergent communication, we also give our agents the ability of transmitting strings of arbitrary symbols with no a priori grounding. We call this the linguistic channel, a specific instantiation of the more general concept of cheap talk. This channel differs from the proposal channel in two key properties:
+
+• Non-bindingness: Messages sent via this channel do not commit the sender to any course of action, unlike directly transmitting a proposal which binds the sender to the proposal. • Unverifiability: There is no inherent link between the linguistic utterance and the private proposal made, meaning that the agents could potentially lie.
+
+We are interested in whether and under what circumstances the agents can make use of this channel to facilitate negotiation and establish a common ground for the symbols.
+
+In our experiments, we consider 4 separate communication configurations: only the proposal channel, only the linguistic channel, both channels open, and no communication at all.2 If a channel is closed, we replace any messages in that channel with a fixed dummy symbol to ensure no information is transmitted. If the proposal channel is closed, we still use the decoded proposal to calculate the item division, but this information is not revealed to the opponent agent.
+
+# 2.3 AGENT SOCIALITY AND REWARD SCHEMES
+
+While purely self-interested agents may learn to divide up items so that only each individual is satisfied, this may not lead to an optimal joint allocation of items, where each item goes to the agent with the highest utility for that item. We introduce a “prosocial” reward $R$ that is shared across A and B, which is the sum of the agents’ individual (selfish) rewards, i.e., $R = R _ { A } + R _ { B }$ . Prosocial agents are incentivised to communicate, as finding the optimal joint allocation requires communicating the hidden values between the agents. These different reward schemes can be seen as specific instantiations of a more general reward formula of the form $R = \alpha R _ { A } + \beta R _ { B }$ (Peysakhovich & Lerer, 2017). For the selfish agent, $\alpha = 1$ and $\beta = 0$ , while for the prosocial agent $\alpha \overset { \cdot } { = } \beta = 1$ .3
+
+# 2.4 AGENT ARCHITECTURE AND LEARNING
+
+At each timestep $t$ , the proposer receives three inputs:
+
+• The item context $c ^ { j } = [ i ; u _ { j } ]$ , a concatenation of the item pool and the proposer’s utilities.   
+• The utterance $m _ { t - 1 }$ produced by the other agent in the previous timestep $t - 1$ . If the linguistic channel is closed, this is simply a dummy message.   
+• The proposal $p _ { t - 1 }$ made by the other agent in the previous timestep $t - 1$ . If the proposal channel is closed, this is again a dummy proposal.
+
+First, the discrete inputs are turned into dense vectors through an embedding table. We use two embedding tables, one for the item context and the previous proposal, and a separate one for the previous utterance. This is essential as the item context and the previous proposal have predefined numeric semantics, distinct from the linguistic semantics of the utterances’ symbols.
+
+Then, each of the input sequences is encoded using an LSTM (Hochreiter & Schmidhuber, 1997), one for each input. This results in 3 fixed-size vectors: $h _ { t } ^ { c }$ , $h _ { t } ^ { m }$ and $h _ { t } ^ { p }$ . These vectors are concatenated and fed through a feedforward layer, followed by a ReLU non-linearity (Nair & Hinton, 2010), to give $h _ { t }$ , the hidden state of the agent at timestep $t$ . This hidden state is then used to initialise the policies for the actions the agent can take:
+
+• $\pi _ { t e r m }$ is the policy for the termination action. If this action is taken by an agent, both agents receive reward according to the last proposal made by the other agent. This is a binary decision, and we parametrise $\pi _ { t e r m }$ as a single feedforward layer, with the hidden state as input, followed by a sigmoid function, to represent the probability of termination. • $\pi _ { u t t }$ is the policy for the linguistic utterances. This is parametrised by an LSTM, which takes the hidden state of the agent as the initial hidden state. For the first timestep, a dummy symbol is fed in as input; subsequently, the model prediction from the previous timestep is fed in as input at the next timestep, in order to predict the next symbol. $\pi _ { p r o p }$ is the policy for the proposals the agent generates. This is parametrised by 3 separate feedforward neural networks, one for each item type, which each take as input $h _ { t }$ and output a distribution over $\{ 0 . . . 5 \}$ indicating the proposal for that item.
+
+The overall policy for the agent, $\pi ^ { * }$ , is a combination of the separate policies. The action that the agent takes at turn $t$ can be summarised by a triple $( e _ { t } , m _ { t } , p _ { t } )$ , where $e _ { t }$ is a binary variable indicating whether the agent took the termination action, $m _ { t }$ is a sequence of symbols produced from $\pi _ { u t t }$ , and $p _ { t }$ is a proposal produced by $\pi _ { p r o p }$ . A negotiation can then be thought of as a sequence of triples $\tau = \langle ( e _ { t } , u _ { t } , p _ { t } ) | 1 \leq t \leq N \rangle$ , with the agents alternating turns.
+
+During training, each agent $i$ tries to independently find the policy $\pi _ { i } ^ { * }$ that maximises the following objective function:
+
+$$
+\pi _ { i } ^ { * } = \underset { \pi _ { i } } { \arg \operatorname* { m a x } } \ \underset { \tau \sim ( \pi _ { A } , \pi _ { B } ) } { \mathbb { E } } [ R _ { i } ( \tau ) ] + \lambda H ( \pi _ { i } )
+$$
+
+where $R _ { i } ( \tau )$ is the reward agent $i$ receives from the trajectory $\tau$ and $H ( \pi _ { i } )$ is an entropy regularisation term to encourage the agent to explore during training. Note that each agent has its own objective function, and the objectives of both agents are coupled by the trajectory sampled from both agents’ policies. At test time, instead of sampling from the policy $\pi$ , we take the action with the highest probability. Parameters are updated using the REINFORCE (Williams, 1992) update rule with an exponentially smoothed mean baseline. Full hyperparameter details are in the appendix.
+
+![](images/3cb1863a75996c1449c459dda4732af0ba7982b86aebbe33fea181b97ad0bac6.jpg)  
+Figure 2: a) Training curves for self-interested agents learning to negotiate under the various communication channels. The results show the mean across 20 different random seeds, as well as bootstrapped confidence intervals via shading (only visible for the linguistic communication case) b) The optimality of the proposed item division as negotiation proceeds for two selfish agents communicating via the proposal channel, shown with error bars for interquartile range.
+
+# 3 EXPERIMENT 1: CAN SELF-INTERESTED AGENTS LEARN TO NEGOTIATE?
+
+# 3.1 EXPERIMENT DESCRIPTION
+
+In our first experiment, we test whether purely self-interested agents can learn to negotiate and divide up items fairly, and investigate the effect of the various communication channels on negotiation success. We train self-interested agents to negotiate for 500k episodes. Each episode corresponds to a batch of 128 games, each with item pools and hidden utilities generated as described in Section 2.1. We also hold out 5 batches worth of games as a test set, and test the agents every 50 episodes, in order to measure the robustness of the negotiating behaviour to unseen environments.
+
+# 3.2 RESULTS
+
+Self-interested agents can learn to negotiate fairly Figure 2a shows that self-interested agents learn to divide up the items fairly when they exchange proposals directly. We see that the proportion of total utility each agent receives is roughly equal, and above $50 \%$ , suggesting that the agents have learnt to keep items with higher utility, while giving away items with low utility.4 In addition, the agents seem to show evidence of compromise: there is a drop of 0.24 between the value of the proposal that agent 1 makes at the beginning of negotiation, and the final reward it receives; for agent 2, the corresponding drop is 0.18. Furthermore, Figure 2b shows the mean optimality of the proposal made at each turn of the negotiation averaged over 1280 games, calculated as the sum of the reward of both agents divided by the maximum possible reward. This value increases with more rounds of negotiation, suggesting that the agents are adjusting their proposals based on information from the other agent.
+
+Table 1 presents an illustrative example of two self-interested agents negotiating using the proposal channel. While the initial proposals made by both agents are over-optimistic, over the course of the negotiation, they compromise to less items. This allows them to achieve higher joint reward, as seen in Figure 2b, a clear example of “negotiation”.
+
+Self-interested agents do not appear to ground cheap talk When using the linguistic channel, agents do not negotiate optimally. Instead, the agents randomly alternate taking all the items, which is borne out by the oscillations in Figure 2a. Indeed, examination of the messages exchanged by self-interested agents show they predominantly consist of a single symbol – see Figures 5a and 5b in the appendix for more details. This result suggests that the self-interested agents we implement find it difficult to cheap talk to exchange meaningful information about how to divide up the items, which is in line with the theoretical analysis of cheap talk in Crawford & Sobel (1982), who find that, once agent interests diverge by a finite amount, no communication is to be expected. Our findings also corroborate the hypothesis found in the language evolution literature that cooperation is a prerequisite for the emergence of language (Nowak & Krakauer, 1999).
+
+Item pool: [5, 5, 1] A utilities: [8, 7, 1] B utilities: [8, 3, 2]   
+Table 1: Actual transcript of two self-interested agents negotiating using the proposal channel. In this setting, no linguistic message is decoded at all.   
+
+<table><tr><td>Turn</td><td>Agent</td><td>Proposal</td></tr><tr><td>1</td><td>A</td><td>[3,4,4]</td></tr><tr><td>2</td><td>B</td><td>[4,2,0]</td></tr><tr><td>3</td><td>A</td><td>[3, 4,0]</td></tr><tr><td>4</td><td>B</td><td>[4,1,0]</td></tr></table>
+
+Table 2: Joint reward success and average number of turns taken for paired agents negotiating with random game termination, varying the agent reward scheme and communication channel. The results are averaged across 20 seeds, with 128 games per seed. We also report the standard deviation as the $\pm$ number and the interquartile range.   
+
+<table><tr><td>Agent sociality</td><td></td><td>Proposal</td><td>Linguistic</td><td>Both</td><td>None</td></tr><tr><td>Self-interested</td><td>Fraction of joint reward 25th&amp;75th percentiles</td><td>0.87 ± 0.11 [0.81,0.95]</td><td>0.75± 0.22 [0.61, 0.94]</td><td>0.87±0.12 [0.81, 0.95]</td><td>0.77 ± 0.22 [0.62, 0.97]</td></tr><tr><td>Prosocial</td><td>Turns taken Fraction of joint reward 25th &amp; 75th percentiles Turns taken</td><td>3.55 ± 1.12 0.93 ± 0.10 [0.89,1.0] 3.10 ± 0.99</td><td>5.36 ± 1.20 0.99 ± 0.02 [1.0, 1.0] 3.71 ± 0.58</td><td>3.43 ± 1.10 0.92 ± 0.11 [0.88,1.0] 2.98 ± 0.97</td><td>3.00 ± 0.13 0.95 ± 0.11 [0.93,1.0] 2.27 ± 0.69</td></tr></table>
+
+We conjecture that this finding is partially due to the fact that the agents operate in a non-iterated environment (i.e., they have no explicit memory of previous interactions). In the classic prisoner’s dilemma game, rational agents playing a single-shot version of the game learn to always defect, whereas rational agents playing iterated prisoner’s dilemma can learn more mutually beneficial cooperative strategies under the right conditions (Sandholm & Crites, 1996; Wunder et al., 2010). More sophisticated players will often have an incentive to take a short-term loss in reward to teach or punish another player for long-term gain in the repeated setting (Fudenberg & Levine, 1998). We suspect that, by iterating the game, there is an incentive for the cheap talk to be more verifiable, as there is a more credible threat of punishment if the accepted proposal and the linguistic utterance do not correspond.
+
+4 EXPERIMENT 2 - CAN PROSOCIAL AGENTS LEARN TO COORDINATE?
+
+# 4.1 EXPERIMENT DESCRIPTION
+
+In the previous experiment, we showed that self-interested agents can divide up items fairly such that each agent gets equal reward. However, this does not correspond to an optimal joint allocation where items go to the agent with the higher utility for that item (equivalently, maximising the sum of both agents’ rewards). The prosocial reward scheme aligns exactly with this optimal outcome, as learning to negotiate equates to finding the optimal joint item allocation. In this setting, there is an in-built bias towards communication, as solving the game optimally requires pooling hidden utilities. Table 2 reports the turns taken as well as the joint reward optimality, i.e., the sum of both agents’ rewards, divided by the maximum possible reward given the item context, where the latter corresponds to each agent getting all the items for which they have a higher utility.
+
+<table><tr><td colspan="2">Item pool: [4, 5,1]</td><td>A utilities: [1, 7, 9]</td><td>B utilities: [7, 0, 10]</td></tr><tr><td>Turn</td><td>Agent</td><td>Linguistic utterance</td><td>Proposal (hidden)</td></tr><tr><td>1</td><td>A</td><td>[3,3, 3, 3, 3, 3]</td><td>[0,5,1]</td></tr><tr><td>2</td><td>B</td><td>[6, 6, 6, 4, 7, 4]</td><td>[4,0, 1]</td></tr><tr><td>3</td><td>A</td><td>[3, 3, 3, 3, 3, 3]</td><td>[0,5,0]</td></tr></table>
+
+Table 3: Actual transcript of two prosocial agents negotiating using the linguistic channel. At each turn, the agents decode a proposal but this is hidden and not revealed to the other agent.
+
+# 4.2 RESULTS
+
+Cheap talk helps agents coordinate For prosocial agents with aligned interests, theoretical results suggest that communication using cheap talk is a Nash equilibrium. This is indeed what we observe, as the linguistic channel results in much better task success than any other communication scheme. The information bandwidth of the proposal channel is limited by the type and number of items, whereas the linguistic channel’s bandwidth is unconstrained and can (in theory) be arbitrarily large, helping the effective exchange of task-specific information. In addition, the variance in joint optimality is much lower for the linguistic channel, suggesting it is more robust. We suspect that difficulties in optimisation cause the results for both channels open to be similar to those for only the proposal channel open, as the proposal channel is pre-grounded, which poses a strong local maxima. With better optimisation, we believe that the performance of both channels open will more closely match those of just the linguistic channel open.
+
+Table 3 illustrates an example of negotiation between the two prosocial agents. Interestingly, we observe that agent A over the course of the negotiation game adjusts its proposal (which is however hidden and not revealed) based on the utterances of agent B.
+
+Prosocial agents can still use the proposal channel to co-ordinate With random termination, agents communicating with the proposal channel do even worse than the no-communication baseline. However, when given the full 10 turns to exchange information (see Figure 7 in the appendix), prosocial agents manage to outperform the no-communication baseline. This suggests that the agents are slowly learning to repurpose the proposal channel to transmit information, i.e., they communicate directly on the action space. Reusing task-specific actions to transmit information has precedent, such as in bridge bidding (Simon, 1949), and it is interesting to observe that agents learn to develop a codebook without the need for prior agreement.
+
+# 5 ANALYSIS OF LINGUISTIC COMMUNICATION
+
+# 5.1 SYMBOL USAGE
+
+The symbol unigram and bigram distributions of messages exchanged by prosocial agents (see Figures 3a and 3b respectively) show that agent A, the agent who initiates the negotiation, is not transmitting any information using the linguistic channel. On the other hand, agent B uses a diversity of symbols resulting in a long-tailed bigram symbol distribution, reminiscent of the Zipfian distribution found in natural languages. This suggests that, even though the task is symmetric, the agents differentiate into speaker and listener roles, similar to other co-ordination games of communication, such as the reference game. Thus, they have adopted a simple strategy for solving the game: an agent shares its utilities (the speaker), while the other (the listener) compares the shared utilities with their own and generates the optimal proposal.
+
+In contrast, the selfish agents do not show evidence of grounded symbol usage. The unigram and bigram statistics show that most messages consist solely of strings of a single fixed symbol, regardless of the item context, and hence no information is being exchanged.
+
+![](images/c84936bfe3d71051d14d48a9d7cfbd42b061d7e0e5c92d3f48d62b69e50844d7.jpg)  
+Figure 3: a) Unigram statistics of symbol usage broken down by turn and by position within the utterance for prosocial agents communicating via the linguistic channel. b) Bigram counts for prosocial agents communicating via the linguistic channel, sorted by frequency.
+
+<table><tr><td>Agent types</td><td>% agent A value correct</td><td>% agent B value correct</td><td>% proposal correct</td></tr><tr><td>Selfish-Selfish</td><td>21</td><td>25</td><td>94 (94)</td></tr><tr><td>Prosocial-Prosocial</td><td>26</td><td>81</td><td>80 (57)</td></tr><tr><td>Random baseline</td><td>20</td><td>20</td><td>17</td></tr></table>
+
+Table 4: Accuracy of predicting individual elements of the agents’ hidden utilities, as well as the final proposal that was accepted. Numbers in brackets indicate accuracy predicting the proposal from just the item pool. Random baseline numbers are obtained by predicting from a message transcript and item pool of all 0’s.
+
+# 5.2 CONTENT OF THE MESSAGES
+
+To interpret the information that is transmitted through the linguistic channel, we use the message transcript and the item pool to create probe classifiers that predict the hidden utilities of each agent and the (hidden) accepted proposal. We use an LSTM to encode the sequence of symbols belonging to the message transcript into a vector, and another LSTM to encode the item pool into a vector. The resulting two vectors are then concatenated and used to predict the hidden utility functions and the accepted proposal using 9 linear classifiers, one for each item (3 each for agent A and agent B’s utilities and 3 for the proposal). We conduct 10-fold cross-validation and report averaged per-item accuracy (see Table 4).
+
+We also include two baselines. The first, reported as the “random baseline” in Table 4, attempts to predict the accepted proposal and agent utilities from a message transcript and item pool of all 0’s (indicating chance performance at the task). The other predicts the accepted proposal from the item pool and a message transcript of all $\mathrm { 0 ^ { \circ } s }$ . This shows how much additional information about the proposal is contained in the message transcript.
+
+We can see from Table 4 that not only is there variety in the symbols used, but the messages also have semantic content: they contain information about the hidden utilities of agent B, and about the proposal that was made. This shows that our agents have learned to give meaning to the symbols, and can use them to transmit information. However, the purely self-interested agents do not seem to transmit meaningful information using the linguistic channel, and thus do not appear to ground the symbols.5
+
+# 6 EXPERIMENT 3: A SOCIETY OF AGENTS
+
+In more realistic scenarios, we learn and practice negotiation in environments with diverse agent populations and levels of prosociality. In this setting, maximising one’s reward requires identifying which agents are most useful in achieving one’s aims. In practice, this involves identifying which agents are the most prosocial, as they are both the most exploitable for self-interested agents, and the most cooperative for other prosocial agents.
+
+# 6.1 EXPERIMENTAL PROTOCOL
+
+We trained a fixed agent against a community of 10 agents, with varying proportions of selfish and prosocial agents. For each training episode, we randomly sample one agent from the community, play a batch of negotiation games, and then update both agents. We do this for both fixed agent A (the agent who goes first) and for fixed agent B (the agent who goes second). We experiment with either 1 or 5 prosocial agents, which communication channels are open, and also whether the agents are identifiable.
+
+At test time, we care primarily about whether self-interested agents can exploit prosocial agents and whether prosocial agents can cooperate with other prosocial agents. Hence, we only test the fixed agent against the prosocial agents in the community. We generate 10 batches of 128 games each batch, and play each game with every pair of fixed agent and community prosocial agent. We average the results across all pairs and all games, and show the mean and standard deviations of the rewards obtained in this experiment in Table 5.
+
+If opposing agents are identifiable (i.e. each agent has a name-tag), the fixed agent receives this information as a one-hot vector, which is then used to look up an embedding table with trainable parameters with one embedding per opposing agent. This opponent embedding is then concatenated with the encoded inputs, before being passed through the feedforward layer to get the agent’s hidden state. Intuitively, the induced agent embedding, that takes the form of machine theory of mind (Rabinowitz et al., 2018), should capture any information about the opponent’s behaviour (e.g., negotiating strategy) that, when taken into account, can help an agent maximize their reward.
+
+# 6.2 ANALYSIS
+
+Identifying agents Our results show that providing agent ID information does help the fixed agent achieve its aims. This effect is particularly pronounced for a selfish fixed agent; here, providing ID information uniformly improves performance. For cooperative agents, the results mixed; for a fixed agent A communicating via the linguistic channel, having agent ID information harms performance. Indeed, the only prosocial agents who beat the no communication baseline (which achieved an average joint optimal reward of 0.95) was when agent A was fixed and ID information was not provided.
+
+We also computed the 2D PCA projections of the learnt opponent embeddings for a fixed agent A under a variety of channels and prosocial levels (see Figure 4). Even in cases where the agent ID does not aid negotiation, we find that the embeddings cluster according to the reward scheme of the other agent. This shows that the agents can distinguish different agents by their reward schemes from purely observing negotiating behaviour.
+
+Community linguistic phenomena In one of our experimental settings, a community of prosocial agents developed a language and were able to use this to achieve better negotiation success. This is the outcome in the starred cell in Table 5. For a visualisation of the bigram usage statistics in
+
+![](images/380ae6cff0a30b2f8548ea3da3931ae0f678430a5ad69e7e9776ee49de2221c3.jpg)  
+Figure 4: PCA plot of whitened opponent ID embeddings that was learnt by a fixed agent 1 for a variety of reward schemes and communication channels.
+
+<table><tr><td colspan="3"></td><td colspan="3">Fixed agent A</td><td colspan="3">Fixed agent B</td></tr><tr><td>Fixed # agent type</td><td>cial agents</td><td>IDs proso- given</td><td>Proposal</td><td>Linguistic</td><td>Both</td><td>Proposal</td><td>Linguistic</td><td>Both</td></tr><tr><td rowspan="2">Self</td><td>1</td><td>False True</td><td>0.81 ± 0.17 0.96 ± 0.06</td><td>0.97 ± 0.05 0.97 ± 0.10</td><td>0.87 ± 0.11 0.97 ± 0.04</td><td>0.76 ± 0.18 0.95 ± 0.08</td><td>1.0 ± 0.0 1.0 ± 0.0</td><td>0.76 ± 0.16 1.0 ± 0.0</td></tr><tr><td>5</td><td>False True</td><td>0.82 ±0.17 1.0 ± 0.01</td><td>1.0 ± 0.0 1.0 ± 0.0</td><td>0.98 ± 0.07 1.0 ± 0.01</td><td>0.88 ± 0.14 1.0 ± 0.0</td><td>1.0 ± 0.0 1.0 ± 0.0</td><td>0.88 ± 0.14 1.0 ± 0.02</td></tr><tr><td rowspan="2">Pros.</td><td>1</td><td>False True</td><td>0.91 ± 0.11 0.95 ± 0.10</td><td>0.95 ± 0.09 0.93 ± 0.11</td><td>0.91 ± 0.12 0.95 ± 0.09</td><td>0.93 ± 0.11 0.92 ± 0.01</td><td>0.95 ± 0.10 0.96 ± 0.08</td><td>0.92 ± 0.11 0.92 ± 0.10</td></tr><tr><td>5</td><td>False True</td><td>0.90 ± 0.13 0.94 ± 0.09</td><td>0.98*± 0.06 0.97 ± 0.06</td><td>0.93 ± 0.10 0.94 ± 0.01</td><td>0.92 ± 0.12 0.93 ± 0.10</td><td>0.95 ± 0.08 0.95 ± 0.08</td><td>0.90 ± 0.16 0.92 ± 0.10</td></tr></table>
+
+Table 5: Fixed agent return vs opposing prosocial agents, for different numbers of opposing prosocial agents and whether IDs are revealed across the communication channels. For the self-interested fixed agent, the numbers report personal reward, and for the prosocial fixed agent, the numbers report joint reward. The starred result indicates when a community has made use of the linguistic channel to transmit information.
+
+each interaction pair in this community, see Figure 6. Interestingly, the only situation when agents developed a language was when ID information was not provided. We suspect this result is due to a poor local optimum; since the prosocial agents can learn to use IDs to distinguish prosocial agents from the selfish agents they can ignore the linguistic utterance.
+
+Moreover, we find that when prosocial agents make use of the linguistic channel, the communication protocol differs within the community. We calculate the Spearman correlation $\rho$ between bigram ranks of different pairs of agents, all of them with the same fixed agent A, and show the results in Table 6. Even though all pairs of agents share the same fixed agent A, different pairs of agents learn to use bigrams in a different way, resulting in relatively low correlations between -0.22 and 0.27.
+
+# 7 DISCUSSION
+
+We showed that by communicating through a verifiable and binding communication channel, selfinterested agents can learn to negotiate fairly by reinforcement learning, using only task success as the reward signal. Moreover, cheap talk facilitated negotiation in prosocial but not in self-interested agents, corroborating theoretical results from the game theory literature (Crawford & Sobel, 1982). An interesting future direction of research would be to investigate whether cheap talk can be made to emerge out of self-interested agents interacting. Recent encouraging results by Crandall et al. (2018) show that communication can help agents cooperate. However, their signalling mechanism is heavily engineered: the speech acts are predefined and the consequences of the speech acts on the observed behaviour are deterministically hard-coded. It would be interesting to see whether a learning algorithm, such as Foerster et al. (2017), can discover the same result.
+
+A related paper from Lewis et al. (2017) takes a top-down approach to learning to negotiate by leveraging dialogue data. We demonstrated a bottom up alternative towards learning communicative behaviours directly from interaction with peers. This opens up the exciting possibility of learning domain-specific reasoning capabilities from interaction, while having a general-purpose language layer at the top producing natural language.
+
+# ACKNOWLEDGEMENTS
+
+We would like to thank Mike Johanson for his insightful comments on an earlier version of this paper, as well as Karl Moritz Hermann and the rest of the DeepMind language team for many fruitful discussions over the course of the project.
+
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+
+![](images/418c9f4828ae6a0beb60ecb00c68ebd99a6c3bbcd379aa63b3834726f66f8f57.jpg)  
+Figure 5: a) Unigram statistics of symbol usage broken down by turn and by position within the utterance for selfish agents communicating via the linguistic channel. b) Bigram counts for selfish agents communicating via the linguistic channel, sorted by frequency.
+
+![](images/f4ed06bb39797d0baa769d4d6258761decdb4762cc69c667efb71130fc79a81d.jpg)  
+Figure 6: Bigram usage in all interaction pairs between a fixed prosocial agent A and a community of 5 prosocial agent Bs. This is the only case where the average joint optimality reward of the negotiating agents is higher than the no communication baseline.
+
+Table 6: Correlation between bigram usage between the different responders in the community.   
+
+<table><tr><td></td><td>Agent 2 ID 1</td><td>Agent 2 ID 2</td><td>Agent 2 ID 3</td><td>Agent 2 ID 4</td></tr><tr><td>Agent 2 ID 0</td><td>-0.118</td><td>0.132</td><td>0.101</td><td>0.057</td></tr><tr><td>Agent 2 ID 1</td><td></td><td>-0.019</td><td>-0.228</td><td>0.269</td></tr><tr><td>Agent 2 ID 2</td><td></td><td></td><td>0.178</td><td>-0.165</td></tr><tr><td>Agent 2 ID 3</td><td></td><td></td><td></td><td>-0.067</td></tr></table>
+
+<table><tr><td colspan="2">Agent sociality</td><td>Proposal</td><td>Linguistic</td><td>Both</td><td>None</td></tr><tr><td rowspan="2">Self-interested</td><td rowspan="2">Fraction of joint reward 25h&amp;75h percentiles</td><td>0.76± 0.27</td><td>0.74 ± 0.23</td><td>0.76±0.28</td><td>0.23 ± 0.06</td></tr><tr><td>[0.66,0.96]</td><td>[0.59, 0.96]</td><td>[0.68, 0.96]</td><td>[0.19,0.29]</td></tr><tr><td rowspan="2">Prosocial</td><td>Turns taken Fraction of joint reward</td><td>9.99 ± 0.35</td><td>4.29 ± 2.13</td><td>10.0 ± 0.25</td><td>7.90 ± 3.21</td></tr><tr><td>25th &amp;75th percentiles</td><td>0.96 ± 0.07</td><td>0.95 ± 0.10</td><td>0.96 ± 0.07</td><td>0.85 ± 0.31</td></tr><tr><td rowspan="2"></td><td>Turns taken</td><td>[0.95,1.0]</td><td>[0.91,1.0]</td><td>[0.94, 1.0]</td><td>[0.90, 1.0]</td></tr><tr><td></td><td>9.06 ± 2.50</td><td>4.56± 2.83</td><td>9.06 ± 2.41</td><td>3.30 ± 2.50</td></tr></table>
+
+Table 7: Joint reward success and average number of turns taken for paired agents negotiating when allowed the full 10 turns, varying the agent reward scheme and communication channel. The results are averaged across 20 seeds, with 128 games per seed. We also report the standard deviation as the $\pm$ number and the quartiles.
+
+# B HYPERPARAMETER DETAILS
+
+Embedding sizes, and all neural network hidden states, had dimension 100. We used the ADAM optimizer (Kingma & Ba, 2014), with default parameter settings, to optimize the parameters of each agent. Each agent had a separate optimizer. We used a separate value of $\lambda$ , the entropy regularisation weight hyperparameter, for each policy. For $\pi _ { t e r m }$ and $\pi _ { p r o p }$ , $\lambda = 0 . 0 5$ ; for $\pi _ { u t t }$ , $\lambda = 0 . 0 0 1$ . The symbol vocabulary size was 11, and the agents were allowed to generate utterances of up to length 6. The smoothing constant for the exponential moving average baseline was 0.7 (i.e. if the old baseline value was $b _ { o l d }$ , and the current reward is $R$ , then the new estimate of the baseline is $b _ { n e w } = 0 . 7 b _ { o l d } + 0 . 3 R )$ .

parse/train/Hk6WhagRW/Hk6WhagRW_content_list.json

@@ -0,0 +1,1674 @@
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+    {
+        "type": "text",
+        "text": "EMERGENT COMMUNICATION THROUGH NEGOTIATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kris Cao∗   \nDepartment of Computer Science and Technology,   \nUniversity of Cambridge, UK   \nAngeliki Lazaridou, Marc Lanctot, Joel Z Leibo, Karl Tuyls, Stephen Clark   \nDeepMind,   \nLondon, UK ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
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+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            304,
+            544,
+            319
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Multi-agent reinforcement learning offers a way to study how communication could emerge in communities of agents needing to solve specific problems. In this paper, we study the emergence of communication in the negotiation environment, a semi-cooperative model of agent interaction. We introduce two communication protocols – one grounded in the semantics of the game, and one which is a priori ungrounded and is a form of cheap talk. We show that self-interested agents can use the pre-grounded communication channel to negotiate fairly, but are unable to effectively use the ungrounded channel. However, prosocial agents do learn to use cheap talk to find an optimal negotiating strategy, suggesting that cooperation is necessary for language to emerge. We also study communication behaviour in a setting where one agent interacts with agents in a community with different levels of prosociality and show how agent identifiability can aid negotiation. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            529,
+            336,
+            545
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "How can communication emerge? A necessary prerequisite is a task that requires coordination between multiple agents to solve, and some communication protocol for the agents to exchange messages through (see a review by Wagner et al. (2003) on earlier work on emergent communication as well as recent deep reinforcement learning methods by Foerster et al. (2016) and Sukhbaatar et al. (2016)). Given these basic requirements, an interesting question to ask is what task structures aid the emergence of communication and how different communication protocols affect task success. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the context of linguistic communication, previous work on this subject has mainly studied the emergence of communication in co-operative games like referential games, variants of the Lewis signaling game (Lewis, 1969), where messages are used to disambiguate between different possible referents (Goldman et al., 2007; Lazaridou et al., 2016; Evtimova et al., 2017). Human language, though, is not merely a referential tool. Amongst other things, we communicate private information and thoughts, discuss plans, ask questions and tell jokes. Moreover, many human interactions are not fully cooperative, yet we can still successfully use language to communicate in these situations. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we study communication in the negotiation game (see Figure 1), an established model of non-cooperative games in classical game theory (Nash, 1950b; Neumann & Morgenstern, 1944; Nash, 1950a; 1951; Schelling, 1960; Binmore et al., 1986; Peters, 2008). In this game, agents are asked to establish a mutually acceptable division of a common pool of items while having their own hidden utilities for each of them. Effective communication is crucial in this game, as the agents need to exchange strategic information about their desires, infer their opponent’s desires from communication, and balance between the two. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Work in classical game theory on negotiation typically uses simple forms of offer / counter-offer bargaining games that do not explicitly address the question of emergent communication (Rubinstein, 1982). Recent work on deep multi-agent reinforcement learning (MARL) has shown great success in teaching agents complex behaviour without a complex environment simulator or demonstration data (Panait & Luke, 2005; Busoniu et al., 2008; Tuyls & Weiss, 2012; Silver et al., 2017; Leibo et al., 2017). By repeatedly interacting with other agents learning at the same time, agents can gradually bootstrap complex behaviour, including motor skills (Bansal et al., 2017) and linguistic communication (Lazaridou et al., 2016; Havrylov & Titov, 2017). ",
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+    {
+        "type": "image",
+        "img_path": "images/77e7b9b7e1fb927342b9e14cc564b4ad8fff40a371b3635dd58b8b08ed9dd949.jpg",
+        "image_caption": [
+            "Figure 1: High-level overview of the negotiation environment that we implement. Agent A consistently refers to the agent who goes first. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We apply techniques from the MARL literature and train agents to negotiate using task success as the only supervision signal.1 We show that, when communicating via a task-specific communication channel with inherent semantics, selfish agents can learn to negotiate fairly and divide up the item pool to the agents’ mutual satisfaction. However, when communicating via cheap talk (Crawford & Sobel, 1982; Farrell & Rabin, 1996), a task-independent communication channel consisting of sequences of arbitrary symbols similar to language, selfish agents fail to exhibit negotiating behaviour at all. On the other hand, we show that cheap talk can facilitate effective negotiation in prosocial agents, which take the other agent’s reward into consideration, providing experimental evidence that cooperation is necessary for language emergence (Nowak & Krakauer, 1999). ",
+        "bbox": [
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+            457,
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "The above results are obtained from paired agents interacting exclusively with each other. In more realistic multi-agent scenarios, agents may interact with many other agents within a society. In these cases, cheap talk can have a significant effect on the evolutionary dynamics of the population (Robson, 1990) as well as the equilibria, stability, and basins of attractions (Skyrms, 2002). Furthermore, it is well-known that, unless trained in a diverse environment, agents overfit to the their specific opponent or teammate (Lanctot et al., 2017). Inspired by these considerations, we perform experiments where agents interact with many agents having different prosociality levels, and find that being able to identify and model other agents’ beliefs aids the negotiation success. This is consistent with experiments using models based on Theory of Mind: boundedly rational agents can collectively benefit by making inferences about the sophistication levels and beliefs of their opponents, and there is evidence that this occurs in human behavior (Yoshida et al., 2008). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 GAME SETTING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            762,
+            334,
+            777
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 NEGOTIATION ENVIRONMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            794,
+            419,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The negotiation task is based on the set-up of Lewis et al. (2017), itself based on DeVault et al. (2015). Agents are presented with three types of items: peppers, cherries and strawberries. At each round (i) an item pool is sampled uniformly, instantiating a quantity (between 0 and 5) for each of the types and represented as a vector $i \\in \\{ 0 . . . 5 \\} ^ { 3 }$ and (ii) each agent $j$ receives a utility function sampled uniformly, which specifies how rewarding one unit of each item is (with item rewards between 0 and 10, and with the constraint that there is at least one item with non-zero utility), represented as a vector $u _ { j } \\in \\{ 0 . . . 1 0 \\} ^ { 3 }$ . These utilities are hidden, i.e., each agent only has access to its own utilities. Following these assignments, the agents start “negotiating” for $N$ timesteps by exchanging messages $m _ { t } ^ { j }$ and proposals $p _ { t } ^ { j } \\in \\{ 0 . . . 5 \\} ^ { 3 }$ for each timestep $1 \\leq t \\leq N$ . Agents alternate between timesteps, meaning that agent A always acts on the odd turns while agent B on the even turns. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Either agent can terminate the negotiation at any timestep $t$ with a special action, which signifies agreement with the most recent proposal made by the other agent. For example, if agent B terminates the negotiation at $t = 4$ , agent A gets rewarded by $R _ { A } = u _ { A } \\cdot ( p _ { t - 1 } ^ { A } )$ while agent $\\mathbf { B }$ receives reward $R _ { B } = u _ { B } \\cdot ( i - p _ { t - 1 } ^ { A } )$ , where $\\cdot$ denotes the dot product of vectors. Note that, if an agent makes an invalid proposal, such as asking for more items than exist in the item pool, and this is accepted, then both agents receive $R _ { A } = R _ { B } = 0$ . ",
+        "bbox": [
+            174,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We impose an upper limit, $N$ , on the number of negotiation turns allowed. If the agents get to the limit without agreement, then both agents receive no reward. We experimented with having a fixed upper limit of 10 turns. However, this led to a “first-mover” advantage: if the agents know in advance how many turns the negotiation lasts, there is a degenerate policy in which agent A can wait until the last timestep $N - 1$ before giving a lopsided offer, which agent B is compelled to accept, a situation similar to the ultimatum game (Guth et al., 1982). To eliminate this effect, we sample N ¨ between 4 and 10 at each round, according to a truncated Poisson distribution with mean 7. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 COMMUNICATION CHANNELS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To achieve negotiation, the agents need to communicate. One obvious communication protocol is to directly transmit the proposed division of items $p _ { j }$ . We refer to this as the proposal channel. This communication channel is task-specific and pertains to the action space of the agents used to derive their pay-offs. Consequently, the information bandwidth is restricted and grounded in the action space of the game. Motivated by work on emergent communication, we also give our agents the ability of transmitting strings of arbitrary symbols with no a priori grounding. We call this the linguistic channel, a specific instantiation of the more general concept of cheap talk. This channel differs from the proposal channel in two key properties: ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "• Non-bindingness: Messages sent via this channel do not commit the sender to any course of action, unlike directly transmitting a proposal which binds the sender to the proposal. • Unverifiability: There is no inherent link between the linguistic utterance and the private proposal made, meaning that the agents could potentially lie. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We are interested in whether and under what circumstances the agents can make use of this channel to facilitate negotiation and establish a common ground for the symbols. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we consider 4 separate communication configurations: only the proposal channel, only the linguistic channel, both channels open, and no communication at all.2 If a channel is closed, we replace any messages in that channel with a fixed dummy symbol to ensure no information is transmitted. If the proposal channel is closed, we still use the decoded proposal to calculate the item division, but this information is not revealed to the opponent agent. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 AGENT SOCIALITY AND REWARD SCHEMES ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            779,
+            511,
+            791
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While purely self-interested agents may learn to divide up items so that only each individual is satisfied, this may not lead to an optimal joint allocation of items, where each item goes to the agent with the highest utility for that item. We introduce a “prosocial” reward $R$ that is shared across A and B, which is the sum of the agents’ individual (selfish) rewards, i.e., $R = R _ { A } + R _ { B }$ . Prosocial agents are incentivised to communicate, as finding the optimal joint allocation requires communicating the hidden values between the agents. These different reward schemes can be seen as specific instantiations of a more general reward formula of the form $R = \\alpha R _ { A } + \\beta R _ { B }$ (Peysakhovich & Lerer, 2017). For the selfish agent, $\\alpha = 1$ and $\\beta = 0$ , while for the prosocial agent $\\alpha \\overset { \\cdot } { = } \\beta = 1$ .3 ",
+        "bbox": [
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+            804,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 AGENT ARCHITECTURE AND LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            165,
+            490,
+            179
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "At each timestep $t$ , the proposer receives three inputs: ",
+        "bbox": [
+            174,
+            190,
+            526,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• The item context $c ^ { j } = [ i ; u _ { j } ]$ , a concatenation of the item pool and the proposer’s utilities.   \n• The utterance $m _ { t - 1 }$ produced by the other agent in the previous timestep $t - 1$ . If the linguistic channel is closed, this is simply a dummy message.   \n• The proposal $p _ { t - 1 }$ made by the other agent in the previous timestep $t - 1$ . If the proposal channel is closed, this is again a dummy proposal. ",
+        "bbox": [
+            215,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "First, the discrete inputs are turned into dense vectors through an embedding table. We use two embedding tables, one for the item context and the previous proposal, and a separate one for the previous utterance. This is essential as the item context and the previous proposal have predefined numeric semantics, distinct from the linguistic semantics of the utterances’ symbols. ",
+        "bbox": [
+            174,
+            315,
+            825,
+            371
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Then, each of the input sequences is encoded using an LSTM (Hochreiter & Schmidhuber, 1997), one for each input. This results in 3 fixed-size vectors: $h _ { t } ^ { c }$ , $h _ { t } ^ { m }$ and $h _ { t } ^ { p }$ . These vectors are concatenated and fed through a feedforward layer, followed by a ReLU non-linearity (Nair & Hinton, 2010), to give $h _ { t }$ , the hidden state of the agent at timestep $t$ . This hidden state is then used to initialise the policies for the actions the agent can take: ",
+        "bbox": [
+            176,
+            377,
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+            448
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• $\\pi _ { t e r m }$ is the policy for the termination action. If this action is taken by an agent, both agents receive reward according to the last proposal made by the other agent. This is a binary decision, and we parametrise $\\pi _ { t e r m }$ as a single feedforward layer, with the hidden state as input, followed by a sigmoid function, to represent the probability of termination. • $\\pi _ { u t t }$ is the policy for the linguistic utterances. This is parametrised by an LSTM, which takes the hidden state of the agent as the initial hidden state. For the first timestep, a dummy symbol is fed in as input; subsequently, the model prediction from the previous timestep is fed in as input at the next timestep, in order to predict the next symbol. $\\pi _ { p r o p }$ is the policy for the proposals the agent generates. This is parametrised by 3 separate feedforward neural networks, one for each item type, which each take as input $h _ { t }$ and output a distribution over $\\{ 0 . . . 5 \\}$ indicating the proposal for that item. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The overall policy for the agent, $\\pi ^ { * }$ , is a combination of the separate policies. The action that the agent takes at turn $t$ can be summarised by a triple $( e _ { t } , m _ { t } , p _ { t } )$ , where $e _ { t }$ is a binary variable indicating whether the agent took the termination action, $m _ { t }$ is a sequence of symbols produced from $\\pi _ { u t t }$ , and $p _ { t }$ is a proposal produced by $\\pi _ { p r o p }$ . A negotiation can then be thought of as a sequence of triples $\\tau = \\langle ( e _ { t } , u _ { t } , p _ { t } ) | 1 \\leq t \\leq N \\rangle$ , with the agents alternating turns. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "During training, each agent $i$ tries to independently find the policy $\\pi _ { i } ^ { * }$ that maximises the following objective function: ",
+        "bbox": [
+            174,
+            717,
+            821,
+            744
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2b9604f93e3d4c0bdf5ee558a0af4ec5f1f307450c1495ffe627f6d52fe5e38d.jpg",
+        "text": "$$\n\\pi _ { i } ^ { * } = \\underset { \\pi _ { i } } { \\arg \\operatorname* { m a x } } \\ \\underset { \\tau \\sim ( \\pi _ { A } , \\pi _ { B } ) } { \\mathbb { E } } [ R _ { i } ( \\tau ) ] + \\lambda H ( \\pi _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            770
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $R _ { i } ( \\tau )$ is the reward agent $i$ receives from the trajectory $\\tau$ and $H ( \\pi _ { i } )$ is an entropy regularisation term to encourage the agent to explore during training. Note that each agent has its own objective function, and the objectives of both agents are coupled by the trajectory sampled from both agents’ policies. At test time, instead of sampling from the policy $\\pi$ , we take the action with the highest probability. Parameters are updated using the REINFORCE (Williams, 1992) update rule with an exponentially smoothed mean baseline. Full hyperparameter details are in the appendix. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/3cb1863a75996c1449c459dda4732af0ba7982b86aebbe33fea181b97ad0bac6.jpg",
+        "image_caption": [
+            "Figure 2: a) Training curves for self-interested agents learning to negotiate under the various communication channels. The results show the mean across 20 different random seeds, as well as bootstrapped confidence intervals via shading (only visible for the linguistic communication case) b) The optimality of the proposed item division as negotiation proceeds for two selfish agents communicating via the proposal channel, shown with error bars for interquartile range. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENT 1: CAN SELF-INTERESTED AGENTS LEARN TO NEGOTIATE? ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            401,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 EXPERIMENT DESCRIPTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            433,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In our first experiment, we test whether purely self-interested agents can learn to negotiate and divide up items fairly, and investigate the effect of the various communication channels on negotiation success. We train self-interested agents to negotiate for 500k episodes. Each episode corresponds to a batch of 128 games, each with item pools and hidden utilities generated as described in Section 2.1. We also hold out 5 batches worth of games as a test set, and test the agents every 50 episodes, in order to measure the robustness of the negotiating behaviour to unseen environments. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Self-interested agents can learn to negotiate fairly Figure 2a shows that self-interested agents learn to divide up the items fairly when they exchange proposals directly. We see that the proportion of total utility each agent receives is roughly equal, and above $50 \\%$ , suggesting that the agents have learnt to keep items with higher utility, while giving away items with low utility.4 In addition, the agents seem to show evidence of compromise: there is a drop of 0.24 between the value of the proposal that agent 1 makes at the beginning of negotiation, and the final reward it receives; for agent 2, the corresponding drop is 0.18. Furthermore, Figure 2b shows the mean optimality of the proposal made at each turn of the negotiation averaged over 1280 games, calculated as the sum of the reward of both agents divided by the maximum possible reward. This value increases with more rounds of negotiation, suggesting that the agents are adjusting their proposals based on information from the other agent. ",
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+    {
+        "type": "text",
+        "text": "Table 1 presents an illustrative example of two self-interested agents negotiating using the proposal channel. While the initial proposals made by both agents are over-optimistic, over the course of the negotiation, they compromise to less items. This allows them to achieve higher joint reward, as seen in Figure 2b, a clear example of “negotiation”. ",
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+    {
+        "type": "text",
+        "text": "Self-interested agents do not appear to ground cheap talk When using the linguistic channel, agents do not negotiate optimally. Instead, the agents randomly alternate taking all the items, which is borne out by the oscillations in Figure 2a. Indeed, examination of the messages exchanged by self-interested agents show they predominantly consist of a single symbol – see Figures 5a and 5b in the appendix for more details. This result suggests that the self-interested agents we implement find it difficult to cheap talk to exchange meaningful information about how to divide up the items, which is in line with the theoretical analysis of cheap talk in Crawford & Sobel (1982), who find that, once agent interests diverge by a finite amount, no communication is to be expected. Our findings also corroborate the hypothesis found in the language evolution literature that cooperation is a prerequisite for the emergence of language (Nowak & Krakauer, 1999). ",
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+    {
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+        "img_path": "images/e6a6a35d312e01dcd29a5ba243ba870c88b81002bbc033bdba8920f6a46b4f22.jpg",
+        "table_caption": [
+            "Item pool: [5, 5, 1] A utilities: [8, 7, 1] B utilities: [8, 3, 2] ",
+            "Table 1: Actual transcript of two self-interested agents negotiating using the proposal channel. In this setting, no linguistic message is decoded at all. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Turn</td><td>Agent</td><td>Proposal</td></tr><tr><td>1</td><td>A</td><td>[3,4,4]</td></tr><tr><td>2</td><td>B</td><td>[4,2,0]</td></tr><tr><td>3</td><td>A</td><td>[3, 4,0]</td></tr><tr><td>4</td><td>B</td><td>[4,1,0]</td></tr></table>",
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+        "table_caption": [
+            "Table 2: Joint reward success and average number of turns taken for paired agents negotiating with random game termination, varying the agent reward scheme and communication channel. The results are averaged across 20 seeds, with 128 games per seed. We also report the standard deviation as the $\\pm$ number and the interquartile range. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Agent sociality</td><td></td><td>Proposal</td><td>Linguistic</td><td>Both</td><td>None</td></tr><tr><td>Self-interested</td><td>Fraction of joint reward 25th&amp;75th percentiles</td><td>0.87 ± 0.11 [0.81,0.95]</td><td>0.75± 0.22 [0.61, 0.94]</td><td>0.87±0.12 [0.81, 0.95]</td><td>0.77 ± 0.22 [0.62, 0.97]</td></tr><tr><td>Prosocial</td><td>Turns taken Fraction of joint reward 25th &amp; 75th percentiles Turns taken</td><td>3.55 ± 1.12 0.93 ± 0.10 [0.89,1.0] 3.10 ± 0.99</td><td>5.36 ± 1.20 0.99 ± 0.02 [1.0, 1.0] 3.71 ± 0.58</td><td>3.43 ± 1.10 0.92 ± 0.11 [0.88,1.0] 2.98 ± 0.97</td><td>3.00 ± 0.13 0.95 ± 0.11 [0.93,1.0] 2.27 ± 0.69</td></tr></table>",
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+        "type": "text",
+        "text": "We conjecture that this finding is partially due to the fact that the agents operate in a non-iterated environment (i.e., they have no explicit memory of previous interactions). In the classic prisoner’s dilemma game, rational agents playing a single-shot version of the game learn to always defect, whereas rational agents playing iterated prisoner’s dilemma can learn more mutually beneficial cooperative strategies under the right conditions (Sandholm & Crites, 1996; Wunder et al., 2010). More sophisticated players will often have an incentive to take a short-term loss in reward to teach or punish another player for long-term gain in the repeated setting (Fudenberg & Levine, 1998). We suspect that, by iterating the game, there is an incentive for the cheap talk to be more verifiable, as there is a more credible threat of punishment if the accepted proposal and the linguistic utterance do not correspond. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENT 2 - CAN PROSOCIAL AGENTS LEARN TO COORDINATE? ",
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENT DESCRIPTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In the previous experiment, we showed that self-interested agents can divide up items fairly such that each agent gets equal reward. However, this does not correspond to an optimal joint allocation where items go to the agent with the higher utility for that item (equivalently, maximising the sum of both agents’ rewards). The prosocial reward scheme aligns exactly with this optimal outcome, as learning to negotiate equates to finding the optimal joint item allocation. In this setting, there is an in-built bias towards communication, as solving the game optimally requires pooling hidden utilities. Table 2 reports the turns taken as well as the joint reward optimality, i.e., the sum of both agents’ rewards, divided by the maximum possible reward given the item context, where the latter corresponds to each agent getting all the items for which they have a higher utility. ",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Item pool: [4, 5,1]</td><td>A utilities: [1, 7, 9]</td><td>B utilities: [7, 0, 10]</td></tr><tr><td>Turn</td><td>Agent</td><td>Linguistic utterance</td><td>Proposal (hidden)</td></tr><tr><td>1</td><td>A</td><td>[3,3, 3, 3, 3, 3]</td><td>[0,5,1]</td></tr><tr><td>2</td><td>B</td><td>[6, 6, 6, 4, 7, 4]</td><td>[4,0, 1]</td></tr><tr><td>3</td><td>A</td><td>[3, 3, 3, 3, 3, 3]</td><td>[0,5,0]</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 3: Actual transcript of two prosocial agents negotiating using the linguistic channel. At each turn, the agents decode a proposal but this is hidden and not revealed to the other agent. ",
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+    {
+        "type": "text",
+        "text": "4.2 RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Cheap talk helps agents coordinate For prosocial agents with aligned interests, theoretical results suggest that communication using cheap talk is a Nash equilibrium. This is indeed what we observe, as the linguistic channel results in much better task success than any other communication scheme. The information bandwidth of the proposal channel is limited by the type and number of items, whereas the linguistic channel’s bandwidth is unconstrained and can (in theory) be arbitrarily large, helping the effective exchange of task-specific information. In addition, the variance in joint optimality is much lower for the linguistic channel, suggesting it is more robust. We suspect that difficulties in optimisation cause the results for both channels open to be similar to those for only the proposal channel open, as the proposal channel is pre-grounded, which poses a strong local maxima. With better optimisation, we believe that the performance of both channels open will more closely match those of just the linguistic channel open. ",
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+    {
+        "type": "text",
+        "text": "Table 3 illustrates an example of negotiation between the two prosocial agents. Interestingly, we observe that agent A over the course of the negotiation game adjusts its proposal (which is however hidden and not revealed) based on the utterances of agent B. ",
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+    {
+        "type": "text",
+        "text": "Prosocial agents can still use the proposal channel to co-ordinate With random termination, agents communicating with the proposal channel do even worse than the no-communication baseline. However, when given the full 10 turns to exchange information (see Figure 7 in the appendix), prosocial agents manage to outperform the no-communication baseline. This suggests that the agents are slowly learning to repurpose the proposal channel to transmit information, i.e., they communicate directly on the action space. Reusing task-specific actions to transmit information has precedent, such as in bridge bidding (Simon, 1949), and it is interesting to observe that agents learn to develop a codebook without the need for prior agreement. ",
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+    {
+        "type": "text",
+        "text": "5 ANALYSIS OF LINGUISTIC COMMUNICATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 SYMBOL USAGE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The symbol unigram and bigram distributions of messages exchanged by prosocial agents (see Figures 3a and 3b respectively) show that agent A, the agent who initiates the negotiation, is not transmitting any information using the linguistic channel. On the other hand, agent B uses a diversity of symbols resulting in a long-tailed bigram symbol distribution, reminiscent of the Zipfian distribution found in natural languages. This suggests that, even though the task is symmetric, the agents differentiate into speaker and listener roles, similar to other co-ordination games of communication, such as the reference game. Thus, they have adopted a simple strategy for solving the game: an agent shares its utilities (the speaker), while the other (the listener) compares the shared utilities with their own and generates the optimal proposal. ",
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+    {
+        "type": "text",
+        "text": "In contrast, the selfish agents do not show evidence of grounded symbol usage. The unigram and bigram statistics show that most messages consist solely of strings of a single fixed symbol, regardless of the item context, and hence no information is being exchanged. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c84936bfe3d71051d14d48a9d7cfbd42b061d7e0e5c92d3f48d62b69e50844d7.jpg",
+        "image_caption": [
+            "Figure 3: a) Unigram statistics of symbol usage broken down by turn and by position within the utterance for prosocial agents communicating via the linguistic channel. b) Bigram counts for prosocial agents communicating via the linguistic channel, sorted by frequency. "
+        ],
+        "image_footnote": [],
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+        "table_caption": [],
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+        "table_body": "<table><tr><td>Agent types</td><td>% agent A value correct</td><td>% agent B value correct</td><td>% proposal correct</td></tr><tr><td>Selfish-Selfish</td><td>21</td><td>25</td><td>94 (94)</td></tr><tr><td>Prosocial-Prosocial</td><td>26</td><td>81</td><td>80 (57)</td></tr><tr><td>Random baseline</td><td>20</td><td>20</td><td>17</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 4: Accuracy of predicting individual elements of the agents’ hidden utilities, as well as the final proposal that was accepted. Numbers in brackets indicate accuracy predicting the proposal from just the item pool. Random baseline numbers are obtained by predicting from a message transcript and item pool of all 0’s. ",
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+    {
+        "type": "text",
+        "text": "5.2 CONTENT OF THE MESSAGES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To interpret the information that is transmitted through the linguistic channel, we use the message transcript and the item pool to create probe classifiers that predict the hidden utilities of each agent and the (hidden) accepted proposal. We use an LSTM to encode the sequence of symbols belonging to the message transcript into a vector, and another LSTM to encode the item pool into a vector. The resulting two vectors are then concatenated and used to predict the hidden utility functions and the accepted proposal using 9 linear classifiers, one for each item (3 each for agent A and agent B’s utilities and 3 for the proposal). We conduct 10-fold cross-validation and report averaged per-item accuracy (see Table 4). ",
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+    {
+        "type": "text",
+        "text": "We also include two baselines. The first, reported as the “random baseline” in Table 4, attempts to predict the accepted proposal and agent utilities from a message transcript and item pool of all 0’s (indicating chance performance at the task). The other predicts the accepted proposal from the item pool and a message transcript of all $\\mathrm { 0 ^ { \\circ } s }$ . This shows how much additional information about the proposal is contained in the message transcript. ",
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+    {
+        "type": "text",
+        "text": "We can see from Table 4 that not only is there variety in the symbols used, but the messages also have semantic content: they contain information about the hidden utilities of agent B, and about the proposal that was made. This shows that our agents have learned to give meaning to the symbols, and can use them to transmit information. However, the purely self-interested agents do not seem to transmit meaningful information using the linguistic channel, and thus do not appear to ground the symbols.5 ",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENT 3: A SOCIETY OF AGENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In more realistic scenarios, we learn and practice negotiation in environments with diverse agent populations and levels of prosociality. In this setting, maximising one’s reward requires identifying which agents are most useful in achieving one’s aims. In practice, this involves identifying which agents are the most prosocial, as they are both the most exploitable for self-interested agents, and the most cooperative for other prosocial agents. ",
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+    {
+        "type": "text",
+        "text": "6.1 EXPERIMENTAL PROTOCOL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We trained a fixed agent against a community of 10 agents, with varying proportions of selfish and prosocial agents. For each training episode, we randomly sample one agent from the community, play a batch of negotiation games, and then update both agents. We do this for both fixed agent A (the agent who goes first) and for fixed agent B (the agent who goes second). We experiment with either 1 or 5 prosocial agents, which communication channels are open, and also whether the agents are identifiable. ",
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+    {
+        "type": "text",
+        "text": "At test time, we care primarily about whether self-interested agents can exploit prosocial agents and whether prosocial agents can cooperate with other prosocial agents. Hence, we only test the fixed agent against the prosocial agents in the community. We generate 10 batches of 128 games each batch, and play each game with every pair of fixed agent and community prosocial agent. We average the results across all pairs and all games, and show the mean and standard deviations of the rewards obtained in this experiment in Table 5. ",
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+    {
+        "type": "text",
+        "text": "If opposing agents are identifiable (i.e. each agent has a name-tag), the fixed agent receives this information as a one-hot vector, which is then used to look up an embedding table with trainable parameters with one embedding per opposing agent. This opponent embedding is then concatenated with the encoded inputs, before being passed through the feedforward layer to get the agent’s hidden state. Intuitively, the induced agent embedding, that takes the form of machine theory of mind (Rabinowitz et al., 2018), should capture any information about the opponent’s behaviour (e.g., negotiating strategy) that, when taken into account, can help an agent maximize their reward. ",
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+    {
+        "type": "text",
+        "text": "6.2 ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Identifying agents Our results show that providing agent ID information does help the fixed agent achieve its aims. This effect is particularly pronounced for a selfish fixed agent; here, providing ID information uniformly improves performance. For cooperative agents, the results mixed; for a fixed agent A communicating via the linguistic channel, having agent ID information harms performance. Indeed, the only prosocial agents who beat the no communication baseline (which achieved an average joint optimal reward of 0.95) was when agent A was fixed and ID information was not provided. ",
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+    {
+        "type": "text",
+        "text": "We also computed the 2D PCA projections of the learnt opponent embeddings for a fixed agent A under a variety of channels and prosocial levels (see Figure 4). Even in cases where the agent ID does not aid negotiation, we find that the embeddings cluster according to the reward scheme of the other agent. This shows that the agents can distinguish different agents by their reward schemes from purely observing negotiating behaviour. ",
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+    {
+        "type": "text",
+        "text": "Community linguistic phenomena In one of our experimental settings, a community of prosocial agents developed a language and were able to use this to achieve better negotiation success. This is the outcome in the starred cell in Table 5. For a visualisation of the bigram usage statistics in ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/380ae6cff0a30b2f8548ea3da3931ae0f678430a5ad69e7e9776ee49de2221c3.jpg",
+        "image_caption": [
+            "Figure 4: PCA plot of whitened opponent ID embeddings that was learnt by a fixed agent 1 for a variety of reward schemes and communication channels. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/77268d9e18740f184c793513d3e7d793f4f5bcb24592f9f94a4bbfcb0e998e49.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\"></td><td colspan=\"3\">Fixed agent A</td><td colspan=\"3\">Fixed agent B</td></tr><tr><td>Fixed # agent type</td><td>cial agents</td><td>IDs proso- given</td><td>Proposal</td><td>Linguistic</td><td>Both</td><td>Proposal</td><td>Linguistic</td><td>Both</td></tr><tr><td rowspan=\"2\">Self</td><td>1</td><td>False True</td><td>0.81 ± 0.17 0.96 ± 0.06</td><td>0.97 ± 0.05 0.97 ± 0.10</td><td>0.87 ± 0.11 0.97 ± 0.04</td><td>0.76 ± 0.18 0.95 ± 0.08</td><td>1.0 ± 0.0 1.0 ± 0.0</td><td>0.76 ± 0.16 1.0 ± 0.0</td></tr><tr><td>5</td><td>False True</td><td>0.82 ±0.17 1.0 ± 0.01</td><td>1.0 ± 0.0 1.0 ± 0.0</td><td>0.98 ± 0.07 1.0 ± 0.01</td><td>0.88 ± 0.14 1.0 ± 0.0</td><td>1.0 ± 0.0 1.0 ± 0.0</td><td>0.88 ± 0.14 1.0 ± 0.02</td></tr><tr><td rowspan=\"2\">Pros.</td><td>1</td><td>False True</td><td>0.91 ± 0.11 0.95 ± 0.10</td><td>0.95 ± 0.09 0.93 ± 0.11</td><td>0.91 ± 0.12 0.95 ± 0.09</td><td>0.93 ± 0.11 0.92 ± 0.01</td><td>0.95 ± 0.10 0.96 ± 0.08</td><td>0.92 ± 0.11 0.92 ± 0.10</td></tr><tr><td>5</td><td>False True</td><td>0.90 ± 0.13 0.94 ± 0.09</td><td>0.98*± 0.06 0.97 ± 0.06</td><td>0.93 ± 0.10 0.94 ± 0.01</td><td>0.92 ± 0.12 0.93 ± 0.10</td><td>0.95 ± 0.08 0.95 ± 0.08</td><td>0.90 ± 0.16 0.92 ± 0.10</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 5: Fixed agent return vs opposing prosocial agents, for different numbers of opposing prosocial agents and whether IDs are revealed across the communication channels. For the self-interested fixed agent, the numbers report personal reward, and for the prosocial fixed agent, the numbers report joint reward. The starred result indicates when a community has made use of the linguistic channel to transmit information. ",
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+    {
+        "type": "text",
+        "text": "each interaction pair in this community, see Figure 6. Interestingly, the only situation when agents developed a language was when ID information was not provided. We suspect this result is due to a poor local optimum; since the prosocial agents can learn to use IDs to distinguish prosocial agents from the selfish agents they can ignore the linguistic utterance. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Moreover, we find that when prosocial agents make use of the linguistic channel, the communication protocol differs within the community. We calculate the Spearman correlation $\\rho$ between bigram ranks of different pairs of agents, all of them with the same fixed agent A, and show the results in Table 6. Even though all pairs of agents share the same fixed agent A, different pairs of agents learn to use bigrams in a different way, resulting in relatively low correlations between -0.22 and 0.27. ",
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+    },
+    {
+        "type": "text",
+        "text": "7 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We showed that by communicating through a verifiable and binding communication channel, selfinterested agents can learn to negotiate fairly by reinforcement learning, using only task success as the reward signal. Moreover, cheap talk facilitated negotiation in prosocial but not in self-interested agents, corroborating theoretical results from the game theory literature (Crawford & Sobel, 1982). An interesting future direction of research would be to investigate whether cheap talk can be made to emerge out of self-interested agents interacting. Recent encouraging results by Crandall et al. (2018) show that communication can help agents cooperate. However, their signalling mechanism is heavily engineered: the speech acts are predefined and the consequences of the speech acts on the observed behaviour are deterministically hard-coded. It would be interesting to see whether a learning algorithm, such as Foerster et al. (2017), can discover the same result. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A related paper from Lewis et al. (2017) takes a top-down approach to learning to negotiate by leveraging dialogue data. We demonstrated a bottom up alternative towards learning communicative behaviours directly from interaction with peers. This opens up the exciting possibility of learning domain-specific reasoning capabilities from interaction, while having a general-purpose language layer at the top producing natural language. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Mike Johanson for his insightful comments on an earlier version of this paper, as well as Karl Moritz Hermann and the rest of the DeepMind language team for many fruitful discussions over the course of the project. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/418c9f4828ae6a0beb60ecb00c68ebd99a6c3bbcd379aa63b3834726f66f8f57.jpg",
+        "image_caption": [
+            "Figure 5: a) Unigram statistics of symbol usage broken down by turn and by position within the utterance for selfish agents communicating via the linguistic channel. b) Bigram counts for selfish agents communicating via the linguistic channel, sorted by frequency. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/f4ed06bb39797d0baa769d4d6258761decdb4762cc69c667efb71130fc79a81d.jpg",
+        "image_caption": [
+            "Figure 6: Bigram usage in all interaction pairs between a fixed prosocial agent A and a community of 5 prosocial agent Bs. This is the only case where the average joint optimality reward of the negotiating agents is higher than the no communication baseline. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c3b0c17eb9f1418deeec335936d63c39ae2fa88e8a6ded0b7b517a52e5e1e27a.jpg",
+        "table_caption": [
+            "Table 6: Correlation between bigram usage between the different responders in the community. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Agent 2 ID 1</td><td>Agent 2 ID 2</td><td>Agent 2 ID 3</td><td>Agent 2 ID 4</td></tr><tr><td>Agent 2 ID 0</td><td>-0.118</td><td>0.132</td><td>0.101</td><td>0.057</td></tr><tr><td>Agent 2 ID 1</td><td></td><td>-0.019</td><td>-0.228</td><td>0.269</td></tr><tr><td>Agent 2 ID 2</td><td></td><td></td><td>0.178</td><td>-0.165</td></tr><tr><td>Agent 2 ID 3</td><td></td><td></td><td></td><td>-0.067</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Agent sociality</td><td>Proposal</td><td>Linguistic</td><td>Both</td><td>None</td></tr><tr><td rowspan=\"2\">Self-interested</td><td rowspan=\"2\">Fraction of joint reward 25h&amp;75h percentiles</td><td>0.76± 0.27</td><td>0.74 ± 0.23</td><td>0.76±0.28</td><td>0.23 ± 0.06</td></tr><tr><td>[0.66,0.96]</td><td>[0.59, 0.96]</td><td>[0.68, 0.96]</td><td>[0.19,0.29]</td></tr><tr><td rowspan=\"2\">Prosocial</td><td>Turns taken Fraction of joint reward</td><td>9.99 ± 0.35</td><td>4.29 ± 2.13</td><td>10.0 ± 0.25</td><td>7.90 ± 3.21</td></tr><tr><td>25th &amp;75th percentiles</td><td>0.96 ± 0.07</td><td>0.95 ± 0.10</td><td>0.96 ± 0.07</td><td>0.85 ± 0.31</td></tr><tr><td rowspan=\"2\"></td><td>Turns taken</td><td>[0.95,1.0]</td><td>[0.91,1.0]</td><td>[0.94, 1.0]</td><td>[0.90, 1.0]</td></tr><tr><td></td><td>9.06 ± 2.50</td><td>4.56± 2.83</td><td>9.06 ± 2.41</td><td>3.30 ± 2.50</td></tr></table>",
+        "bbox": [
+            178,
+            219,
+            821,
+            333
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Joint reward success and average number of turns taken for paired agents negotiating when allowed the full 10 turns, varying the agent reward scheme and communication channel. The results are averaged across 20 seeds, with 128 games per seed. We also report the standard deviation as the $\\pm$ number and the quartiles. ",
+        "bbox": [
+            173,
+            343,
+            825,
+            400
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B HYPERPARAMETER DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            424,
+            444,
+            440
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Embedding sizes, and all neural network hidden states, had dimension 100. We used the ADAM optimizer (Kingma & Ba, 2014), with default parameter settings, to optimize the parameters of each agent. Each agent had a separate optimizer. We used a separate value of $\\lambda$ , the entropy regularisation weight hyperparameter, for each policy. For $\\pi _ { t e r m }$ and $\\pi _ { p r o p }$ , $\\lambda = 0 . 0 5$ ; for $\\pi _ { u t t }$ , $\\lambda = 0 . 0 0 1$ . The symbol vocabulary size was 11, and the agents were allowed to generate utterances of up to length 6. The smoothing constant for the exponential moving average baseline was 0.7 (i.e. if the old baseline value was $b _ { o l d }$ , and the current reward is $R$ , then the new estimate of the baseline is $b _ { n e w } = 0 . 7 b _ { o l d } + 0 . 3 R )$ . ",
+        "bbox": [
+            173,
+            455,
+            825,
+            568
+        ],
+        "page_idx": 14
+    }
+]

parse/train/SJgf6Z-0W/SJgf6Z-0W.md

@@ -0,0 +1,233 @@
+# PREDICTING MULTIPLE ACTIONS FOR STOCHASTIC CONTINUOUS CONTROL
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We introduce a new approach to estimate continuous actions using actor-critic algorithms for reinforcement learning problems. Policy gradient methods usually predict one continuous action estimate or parameters of a presumed distribution (most commonly Gaussian) for any given state which might not be optimal as it may not capture the complete description of the target distribution. Our approach instead predicts $M$ actions with the policy network (actor) and then uniformly sample one action during training as well as testing at each state. This allows the agent to learn a simple stochastic policy that has an easy to compute expected return. In all experiments, this facilitates better exploration of the state space during training and converges to a better policy.
+
+# 1 INTRODUCTION
+
+Reinforcement learning is a traditional branch of machine learning which focuses on learning complex tasks by assigning rewards to agents that interact with their environment. It has recently gained momentum thanks to the combination of novel algorithms for continuous control with deep learning models, sometimes even matching human performance in tasks such as playing video games and manipulating objects Mnih et al. (2015); Silver et al. (2016). Recent methods for continuous control problems like Deep Deterministic Policy Gradient (DDPG) Lillicrap et al. (2016), Asynchronous Advantage Actor Critic (A3C) Mnih et al. (2016) use actor-critic architectures, where an action function is learned by mapping states to actions. DDPG works well on many tasks, but it does not model the uncertainty in actions as it produces a point estimate of the action distribution over states. The actor is forced to deterministically choose an action for every state. A3C and other stochastic policy gradient algorithms output distribution parameters (e.g. Gaussian distributions) instead of point estimate, which can be sampled for action values.
+
+As a simple example where this is sub-optimal, consider the inverted pendulum task, where a pendulum is attached to a cart and the agent needs to control the one dimensional movement of the cart to balance the pendulum upside down. A deterministic agent chooses a single action for every state. This breaks the inherent symmetry of the task. When the cart in not moving and the pendulum is hanging down, two actions are equally promising: either moving left or right. The distribution parameter estimation (e.g. A3C) might work better in this case as there are only two good options, but in cases when there are more than two good actions to select, this will not be optimal. In our approach we allow the agent to suggest multiple actions, which enables it to resolve cases like this easily.
+
+Further, we observe that a deterministic behavior of DDPG can lead to sub-optimal convergence during training. The main limitation is that, especially in the beginning of the learning procedure, the actor favors actions that lead to a good immediate reward but might end up being far from the globally optimal choice.
+
+This work is based on the intuition that if the actor is allowed to suggest, at each time step, multiple actions rather than a single one, this can render the resulting policy non-deterministic, leading to a better exploration of the entire solution space as well as a final solution of potentially higher quality. This can also eliminate the external exploration mechanisms required during training e.g. OrnsteinUhlenbeck process noise Uhlenbeck & Ornstein (1930), parameter noise Plappert et al. (2017) or differential entropy of normal distribution.
+
+Here, we introduce an algorithm, which we refer to as Multiple Action Policy Gradients (MAPG), that models a stochastic policy with several point estimates and allows to predict a pre-defined number $M$ of actions at each time step, extending any policy gradient algorithm with little overhead. We will demonstrate the working of this algorithm by adapting DDPG Lillicrap et al. (2016) to use MAPG.
+
+Another benefit of the proposed method is that the variance of the predicted actions can give additional insights into the decision process during runtime. A low variance usually implies that the model only sees one way to act in a certain situation. A wider or even multi-modal distribution suggests that there exist several possibilities given the current state.
+
+We evaluate the proposed method on six continuous control problems of the OpenAI Gym Brockman et al. (2016) as well as a deep driving scenario using the TORCS car simulator Wymann et al. (2014). For a fair evaluation we directly compare DDPG to our MAPG without changing hyper-parameters or modifying the training scheme. In all experiments, we show an improved performance using MAPG over DDPG. To verify if MAPG helps in better exploration during training, we also analyze MAPG under no external exploration policy.
+
+# 2 RELATED WORK
+
+There is currently a wide adoption of deep neural networks for reinforcement learning. Deep Q Networks (DQN) Mnih et al. (2015) directly learn the action-value function with a deep neural network. Although this method can handle very high dimensional inputs, such as images, it can only deal well with discrete and low dimensional action spaces. Guided Policy Search Levine & Koltun (2013) can exploit high and low dimensional state descriptions by concatenating the low dimensional state to a fully connected layer inside the network.
+
+Recent methods for continuous control problems come in two flavours, vanilla policy gradient methods which directly optimize the policy and actor-critic methods which also approximate state-value function in addition to policy optimization. Trust Region Policy Optimization (TRPO) Schulman et al. (2015) and Proximal Policy Optimization Algorithms Schulman et al. (2017) can be used as vanilla policy gradient as well as actor-critic methods. Whereas, Deep Deterministic Policy Gradient (DDPG) Lillicrap et al. (2016) and Asynchronous Advantage Actor Critic (A3C) Mnih et al. (2016) use actor-critic architectures, where state-action function is learned to calculate policy gradients.
+
+Stochastic Value Gradients (SVG) Heess et al. (2015), Generalized Advantage Estimation (GAE) Schulman et al. (2015), A3C, TRPO all use stochastic policy gradients and predict action probability distribution parameters. The action values are then sampled from the predicted distribution. A parametrized normal distribution is most commonly used as action distribution. This means that this formulation models a kind of action noise instead of the true action distribution. For example a distribution with two modes cannot be modeled with a Gaussian.
+
+DDPG Lillicrap et al. (2016) which extends DPG Silver et al. (2014) uses deterministic policy gradients and achieves stability when using neural networks to learn the actor-critic functions. The limitation of DDPG is that it always gives a points which may not be desired in stochastic action problems.
+
+Lazaric et al. (2007) estimate stochastic action values using a sequential Monte Carlo method (SMC). SMC has actor and critic models where the actor is represented by Monte Carlo sampling weights instead of a general function approximator like a neural network. SMC learning works well in small state space problems, but cannot be extended directly to high dimensional non-linear action space problems.
+
+Similar to our idea of predicting multiple instead of one output, but originating from the domain of supervised learning, is Multiple Hypothesis Prediction Rupprecht et al. (2017), which in turn is closely related to Multiple Choice Learning Lee et al. (2016) and Lee et al. (2017). In this line of work, the model is trained to predict multiple possible answers for the given task. Specific care has to be taken since often in supervised datasets not all possible outcomes are labeled, this leading to loss functions that contain an arg min-like term and, as such, are hard to differentiate.
+
+# 3 THE MULTIPLE ACTION POLICY GRADIENT ALGORITHM
+
+In this section we will describe in detail how multiple action policy gradients can be derived and compare it to DDPG. We will then analyze the differences to understand the performance gain.
+
+# 3.1 BACKGROUND
+
+We investigate a typical reinforcement learning setup Sutton & Barto (1998) where an agent interacts with an environment $E$ . At discrete time steps $t$ , the agent observes the full state $s _ { t } \in \mathcal { S } \subset \mathbb { R } ^ { c }$ , and after taking action $a _ { t } \in \mathcal { A } \subset \mathbb { R } ^ { d }$ , it receives the reward $r _ { t } \in \mathbb { R }$ . We are interested in learning a policy $\pi : S  { \mathcal { P } } ( A )$ , that produces a probability distribution over actions for each state. Similarly to other algorithms, we model the environment as a Markov Decision Process (MDP) with a probabilistic transition between states $p ( s _ { t + 1 } | s _ { t } , a _ { t } )$ and the rewards $r ( s _ { t } , a _ { t } )$ .
+
+We associate a state with its current and (discounted with $\gamma \in [ 0 , 1 ]$ ) future rewards by using
+
+$$
+R _ { t } = \sum _ { i = 1 } ^ { T } \gamma ^ { i - t } r ( s _ { i } , a _ { i } ) .
+$$
+
+Since $\pi$ and $E$ are stochastic, it is more meaningful to investigate the expected reward instead. Thus, the agent tries to find a policy that maximizes the expected discounted reward from the starting state distribution $p ( s _ { 1 } )$ .
+
+$$
+J = \mathbb { E } _ { r _ { i } , s _ { i } \sim E , a _ { i } \sim \pi } ( R _ { 1 } )
+$$
+
+Here, it is useful to investigate the recursive Bellman equation that associates a value to a state-action pair:
+
+$$
+Q ^ { \pi } \big ( s _ { t } , a _ { t } \big ) = \mathbb { E } _ { r _ { t } , s _ { t + 1 } \sim E } \big [ r \big ( s _ { t } , a _ { t } \big ) + \gamma \mathbb { E } _ { a _ { t + 1 } \sim \pi } \big [ Q ^ { \pi } \big ( s _ { t + 1 } , a _ { t + 1 } \big ) \big ] \big ]
+$$
+
+Methods such as (D)DPG use a deterministic policy where each state is deterministically mapped to an action using a function $\mu : { \mathcal { S } }  A$ which simplifies Equation 3 to
+
+$$
+Q ^ { \mu } ( s _ { t } , a _ { t } ) = \mathbb { E } _ { r _ { t } , s _ { t + 1 } \sim E } [ r ( s _ { t } , a _ { t } ) + \gamma Q ^ { \mu } ( s _ { t + 1 } , \mu ( s _ { t + 1 } ) ) ] .
+$$
+
+In Q-learning Watkins & Dayan (1992), $\mu$ selects the highest value action for the current state:
+
+$$
+\mu ( s _ { t } ) = \operatorname * { a r g m a x } _ { a _ { t } } ( Q ( s _ { t } , a _ { t } ) )
+$$
+
+The $Q$ value of an action is approximated by a critic network which estimates $Q ^ { \mu } ( s _ { t } , a _ { t } )$ for the action chosen by the actor network.
+
+# 3.2 ALGORITHM
+
+The key idea behind predicting multiple actions is that it is possible to learn a stochastic policy as long as the inner expectation remains tractable. Multiple action prediction achieves this by predicting a fixed number $M$ of actions $\rho : \mathcal { S }  \mathcal { A } ^ { M }$ and uniformly sampling from them. The expected value is then the mean over all $M$ state-action pairs. The state-action value can then be defined as
+
+$$
+Q ^ { \rho } ( s _ { t } , a _ { t } ) = \mathbb { E } _ { r _ { t } , s _ { t + 1 } \sim E } \left[ r ( s _ { t } , a _ { t } ) + \gamma \frac { 1 } { M } \sum _ { m = 1 } ^ { M } Q ^ { \rho } ( s _ { t + 1 } , \rho _ { m } ( s _ { t + 1 } ) ) \right] .
+$$
+
+This is beneficial since we not only enable the agent to employ a stochastic policy when necessary, but we also approximate the action distribution of the policy with multiple samples instead of one.
+
+There exists an intuitive proof that the outer expectation in Equation 6 will be maximal if and only if the inner $Q ^ { \rho }$ are all equal. The idea is based on the following argument: let us assume $\rho$ as an optimal policy maximizing Equation 2. Further, one of the $M$ actions $\rho _ { j } \big ( s _ { t + 1 } \big ) ,$ ) for a state $s _ { t + 1 }$ has a lower expected return than another action $k$ .
+
+$$
+Q ^ { \rho } ( s _ { t + 1 } , \rho _ { j } ( s _ { t + 1 } ) ) < Q ^ { \rho } ( s _ { t + 1 } , \rho _ { k } ( s _ { t + 1 } ) )
+$$
+
+Then there exists a policy $\rho ^ { * }$ that would score higher than $\rho$ that is exactly the same as rho exept that it predicts action $k$ instead of $j$ : $\rho _ { j } ^ { * } ( s _ { t + 1 } ) : = \bar { \rho _ { k } } ( s _ { t + 1 } )$ . However, this contradicts the assumption
+
+# Algorithm 1 MAPG algorithm
+
+<table><tr><td>Modify actor network μ(s|0μ) to output M actions, At = {p1(st),..., pM(st)}. Randomly initialize actor μ(s|θμ) and critic Q(s|0q) network weights. Initialize target actor μ&#x27; and critic Q&#x27; networks,0’μ ← 0μ and 0Q ←0Q. for episode = 1 to N do Initialize random process  for exploration. Receive initial observation/state S1.</td></tr><tr><td>fort=1to Tdo Predict M action proposals At = {pi(st),...,ρm(st)}= μ&#x27;(st|0μ).</td></tr><tr><td>Uniformly sample an action j from At: α² = ρj(st) + Nt.</td></tr><tr><td>Execute action α and observe reward rt and state St+1·</td></tr><tr><td>Store transition (St,α²,rt, St+1) to replay buffer R.</td></tr><tr><td>Sample a random batch of size B from R. Set yi=ri+Q&#x27;(Si+1,μ&#x27;(si+1l0μ)|0Q).</td></tr><tr><td>Update critic by minimizing the loss,</td></tr><tr><td>L=B∑i(yi-Q(si,a|Q)）²</td></tr><tr><td>Update all actor weights connected to αt .</td></tr><tr><td>VθuJ~B∑iVQ(s,a|0Q)ls=sia=a²Vθ-θ{1.M)+j μ(s|0)|si Update the target networks:</td></tr><tr><td>0μ←T0μ+(1-T）θμ</td></tr><tr><td>←T0Q+(1-T）Q end for end for</td></tr></table>
+
+that we had learned an optimal policy beforehand. Thus in an optimal policy all $M$ action proposals will have the same expected return. More informal, this can also be seen as a derivation from the training procedure. If we always select a random action from the $M$ proposals, they should all be equally good since the actor cannot decide which action should be executed.
+
+This result has several interesting implications. From the proof, it directly follows that it is possible - and sometimes necessary - that all proposed actions are identical. This is the case in situations where there is just one single right action to take. When the action proposals do not collapse into one, there are two possibilities: either it does not matter what action is currently performed, or all proposed actions lead to a desired outcome.
+
+Naturally, the set of stochastic policies includes all deterministic policies, since a deterministic policy is a stochastic policy with a single action having probability density equal to one. This means that in theory we expect the multiple action version of a deterministic algorithm to perform better or equally well, since it could always learn a deterministic policy by predicting $M$ identical actions for every state.
+
+Algorithm 1 outlines the MAPG technique. The main change is that the actor is modified to produce $M$ instead of one output. For every timestep one action $j$ is then selected. When updating the actor network, a gradient is only applied to the action (head) that was selected during sampling. Over time each head will be selected equally often, thus every head will be updated and learned during training.
+
+# 4 EXPERIMENTS
+
+In this section we will investigate and analyze the performance of MAPG in different aspects. First, we compare scores between DDPG, A3C and MAPG on six different tasks. Second, we analyze the influence of the number of actions on the performance by training agents with different $M$ on five tasks. Further, to understand the benefit of multiple action prediction, we observe the variance over actions of a trained agent: the goal is to analyze for which states the predicted actions greatly differ from each other and for which ones they collapse into a single choice instead. Finally, we compare the performance of DDPG and MAPG without any external noise for exploration during training.
+
+Table 1: Tasks used for evaluation   
+Table 2: Average score $\pm 3 \sigma$ over 100 episodes for Mujoco tasks with different $M$ . For better readability we denote the highest mean score for each task in bold. Corresponding boxplots can be found in Figure 1a and 1b and the appendix.   
+
+<table><tr><td>TASK</td><td>ACTION DIMENSION</td><td>STATE DIMENSION</td><td>DESCRIPTION</td></tr><tr><td>PENDULUM</td><td>1</td><td>3</td><td>PENDULUM ON A CART.</td></tr><tr><td>HOPPER</td><td>3</td><td>11</td><td>ONELEGGED ROBOT.</td></tr><tr><td>WALKER2D</td><td>6</td><td>17</td><td>TWO DIMENSIONAL BIPEDAL ROBOT.</td></tr><tr><td>HUMANOID</td><td>17</td><td>376</td><td>THREE DIMENSIONAL BIPEDAL ROBOT.</td></tr><tr><td>HALFCHEETAH</td><td>6</td><td>17</td><td>TWO LEG ROBOT.</td></tr><tr><td>SWIMMER</td><td>2</td><td>6</td><td>THREE JOINT SWIMMING ROBOT.</td></tr><tr><td>TORCS</td><td>3</td><td>29</td><td>CONTROL CAR IN3D SIMULATION.</td></tr></table>
+
+<table><tr><td>ENVIRONMENT</td><td>DDPG</td><td>A3C</td><td>M=10</td><td>M= 20</td><td>M=50</td></tr><tr><td>HOPPER-V1</td><td>603 ±76</td><td>532 ±105</td><td>824±94</td><td>923±90</td><td>732 ±34</td></tr><tr><td>WALKER2D-V1</td><td>960 ±72</td><td>764 ±112</td><td>1297 ± 70</td><td>1319 ± 50</td><td>1589 ± 45</td></tr><tr><td>HUMANOID-V1</td><td>1091 ± 65</td><td>281±40</td><td>1248 ±115</td><td>1112 ±75</td><td>1212 ±110</td></tr><tr><td>HALFCHEETAH-V1</td><td>4687 ±455</td><td>3803 ±125</td><td>6659 ± 570</td><td>4116± 85</td><td>4333 ± 70</td></tr><tr><td>SWIMMER-V1</td><td>38±7</td><td>33±10</td><td>51±6</td><td>41±4</td><td>40±2</td></tr></table>
+
+# 4.1 SETUP
+
+In all our experiments, we use five continuous control tasks from the Mujoco Simulator Todorov et al. (2012) and a driving task for The Open Racing Car Simulator (TORCS). A detailed description about the tasks is given in Table 1. We use the OpenAI Gym Brockman et al. (2016) and OpenAI baselines Hesse et al. (2017) for evaluating our experiments.
+
+The base actor and critic networks are fixed in all experiments. Each network has two fully connected hidden layers with 64 units each. Each fully-connected layer is followed by a ReLU nonlinearity. The actor network takes the current observed state $s _ { t }$ as input and produces $M$ actions $a _ { t } ^ { ( m ) } \in [ - 1 , 1 ] ^ { d }$ by applying tanh. From $M$ actions $a _ { t } ^ { ( m ) }$ , a single action $a _ { t }$ is randomly chosen with equal probability. The critic uses the current state $s _ { t }$ and action $a _ { t }$ as input and outputs a scalar value ( $Q$ -value). In the critic network, the action value is concatenated with the output of the first layer followed by one hidden layer and an output layer with one unit.
+
+The critic network is trained by minimizing the mean square loss between the calculated discounted reward and the computed $Q$ value. The actor network is trained by computing the policy gradient from the $Q$ -value of the chosen action. The network weights of the last layer are only updated for the selected action. Ornstein-Uhlenbeck process noise is added to the action values from the actor for exploration. The training is done for a total of two million steps in all tasks.
+
+For A3C training, we use same actor-critic networks as for earlier experiment. The output of actor network is a mean vector $( \mu _ { a } )$ (one for each action value) and a scalar standard deviation $( \sigma ^ { 2 }$ , shared for all actions). The actions values are sampled from the normal distribution $( \mathcal { N } ( \mu _ { a } , \sigma ^ { 2 } ) )$ . We used differential entropy of normal distribution to encourage exploration with weight $1 0 ^ { - } 4$ . In our experiments, A3C performed poorly than DDPG in all tasks and was not able to learn a good policy for Humanoid task.
+
+# 4.2 MUJOCO EXPERIMENTS
+
+For more meaningful quantitative results, we report the average reward over 100 episodes with different values of $M$ for various tasks in 2. For all environments except HUMANOID we already score higher with $M = 5$ . The lower performance in the HUMANOID task might be explained by the drastically higher dimensionality of the world state in this task which makes it more difficult to observe.
+
+The scores of policy based reinforcement learning algorithms can vary a lot depending on network hyper-parameters, reward function and codebase/framework as outlined in Henderson et al. (2017). To minimize the variation in score due to these factors, we fixed all parameters of different algorithms and only studied changes on score by varying $M$ . Our metric for performance in each task is average reward over 100 episodes by an agent trained for 2 million steps. This evaluation hinders actors with high $M$ since in every training step only a single out of the $M$ actions will be updated per state. Thus, in general actors with higher number of action proposals, will need a longer time to learn a meaningful distribution of action.
+
+We show a plot for the scores in the HOPPER and WALKER2D environments in Figure 1a and 1b, where we can see that the overall score increases with $M$ .
+
+![](images/ffe57f760e0c5ecdb7ea599b0962f79cd49c64ec3736c8d18bb717038446a420.jpg)  
+Figure 1: Variation in score of Hopper and Walker2d with different values of $M$
+
+In Figure 2, we studied the variance in action values for $M = 1 0$ during training together with the achieved reward. The standard deviation of actions generated by MAPG decreases with time. As the network converges to a good policy (increase in expected reward) the variation in action values is reduced. However there are some spikes in standard deviation even when network is converged to a better policy. It shows that there are situations in which the policy sees multiple good actions (with high $Q$ -value) which can exploited using MAPG.
+
+# 4.3 VARIANCE ANALYSIS
+
+We use the simple Pendulum environment to analyze the variance during one episode. The task is the typical inverted pendulum task, where a cart has to be moved such that it balances a pendulum in an inverted position. Figure 3 plots standard deviation and the angle of the pendulum. Some interesting relationships can be observed. The variance exhibits two strong spikes that coincide with an angle of 0 degrees. This indicates that the agent has learned that there are two ways it can swing up the pole: either by swinging it clockwise or counter clockwise. A deterministic agent would need to pick one over the other instead of deciding randomly. Further, once the target inverted pose (at 180 degrees) is reached the variance does not go down to 0. This means that for the agent a slight jitter seems to be the best way to keep the pendulum from gaining momentum in one or the other direction.
+
+With this analysis we could show that a MAPG agent can learn meaningful policies. The variance over predicted actions can give additional insight into the learned policy and results in a more diverse agent that can for example swing up the pole in two different directions instead of picking one.
+
+# 4.4 EFFECT ON EXPLORATION
+
+Here, we study the effect of MAPG on exploration during training. We compare the performance of DDPG and MAPG during training with and without any external noise on Pendulum and HalfCheetah environments. Figure 4 shows the average reward during training with DDPG and MAPG $M = 1 0$ . The policy trained using MAPG converges to better average reward than DDPG in both cases. Moreover, the performance of MAPG without any external exploration is comparable to DDPG with added exploration noise. This means MAPG can explore the state space enough to find a good policy.
+
+![](images/a9d050997bbcb518de69f3dba1f71d448790a5383d4d0ada0e41aa1b314ad36a.jpg)  
+Figure 2: Standard deviation and reward with $M = 1 0$ for the Pendulum task during training.
+
+![](images/485c98f8b8e6c185987979292333c3f05ce8976314d05d00ad968a6e39eb0c28.jpg)  
+Figure 3: Standard deviation and angle during one episode of the Pendulum environment. An angle of $\pm 1 8 0$ is the target inverted pose. 0 is hanging downwards.
+
+In the Half Cheetah environment we can see that using exploration creates a much bigger performance difference between DDPG and MAPG than without. The difference sets in after about 500 epochs. This is an indication that in the beginning of training the actions predicted by MAPG are similar to the one from DDPG. The noise later helps to pull the $M$ actions apart such that they find individual loss minima, leading to a more diverse policy with better reward.
+
+# 4.5 TORCS
+
+TORCS (The Open Racing Car Simulator) is an open source 3D car racing simulator. It provides an interface for agents to drive the cars. During training, the reward was set proportional to component of car velocity along direction of road $v * \cos ( \alpha )$ , where $\alpha$ is the angle between the velocity vector and the center line of the track. This reward encourages forward motion. The car’s sensor data (velocity, distance from road edges etc.) is used as input state and steer, brake, accelerate as actions at each time step.
+
+![](images/0ae2acc456a9194837f9a54b5a8e40a09a2f8f1e6309b5daa2bd63cfef746875.jpg)  
+Figure 4: Performance curves for two environments with and without external exploration noise on DDPG and MAPG: original DDPG with OU process noise (green), DDPG without any exploration noise (blue), MAPG $\mathbf { \Gamma } ( \mathbf { M } { = } 1 0$ ) with OU process noise (red) and MAPG $( \mathbf { M } { = } 1 0 _ { , }$ ) without OU process noise (orange).
+
+In our experiments, MAPG with $M = 1 0$ was able to complete multiple laps of the track, whereas the DDPG based agent could not complete even one lap of track. The average distance traveled over 100 episodes by DDPG is 807 and 5882 (both in meters) for MAPG agent.
+
+Similar to our other experiments we find that MAPG agents explore more possibilities due to their stochastic nature and can then learn more stable and better policies.
+
+# 5 CONCLUSION
+
+In this paper, we have proposed MAPG, a technique that leverages multiple action prediction to learn better policies in continuous control problems. The proposed method enables a better exploration of the state space and shows improved performance over DDPG. As indicated by exploration experiments, it can also be a used as a standalone exploration technique, although more work needs to be done in this direction. Last but not least, we conclude with interesting insights gained from the action variance. There are several interesting directions which we would like to investigate in the future. The number of actions $M$ is a hyper-parameter in our model that needs to be selected and seems to be task specific. In general, the idea of predicting multiple action proposals can be extended to other on- or off-policy algorithms, such as NAF Gu et al. (2016) or TRPO. Evaluating MA-NAF and MA-TRPO will enable studying the generality of the proposed approach.
+
+# ACKNOWLEDGMENTS
+
+Will be added after anonymous review.
+
+# REFERENCES
+
+Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym, 2016.
+
+Shixiang Gu, Timothy P. Lillicrap, Ilya Sutskever, and Sergey Levine. Continuous deep q-learning with model-based acceleration. In Proc. International Conference on Machine Learning (ICML), 2016.
+
+Nicolas Heess, Gregory Wayne, David Silver, Timothy P. Lillicrap, Tom Erez, and Yuval Tassa. Learning continuous control policies by stochastic value gradients. In Advances in Neural Information Processing Systems, pp. 2944–2952, 2015.
+
+Peter Henderson, Riashat Islam, Philip Bachman, Joelle Pineau, Doina Precup, and David Meger. Deep reinforcement learning that matters. arXiv preprint arXiv:1709.06560, 2017.
+
+Christopher Hesse, Matthias Plappert, Alec Radford, John Schulman, Szymon Sidor, and Yuhuai Wu. Openai baselines. https://github.com/openai/baselines, 2017.
+
+Alessandro Lazaric, Marcello Restelli, and Andrea Bonarini. Reinforcement learning in continuous action spaces through sequential monte carlo methods. In Advances in Neural Information Processing Systems, 2007.
+
+Kimin Lee, Changho Hwang, KyoungSoo Park, and Jinwoo Shin. Confident multiple choice learning. arXiv preprint arXiv:1706.03475, 2017.
+
+Stefan Lee, Senthil Purushwalkam Shiva Prakash, Michael Cogswell, Viresh Ranjan, David Crandall, and Dhruv Batra. Stochastic multiple choice learning for training diverse deep ensembles. In Advances in Neural Information Processing Systems, pp. 2119–2127, 2016.
+
+Sergey Levine and Vladlen Koltun. Guided policy search. In Proceedings of the 30th International Conference on Machine Learning (ICML-13), pp. 1–9, 2013.
+
+Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. In Proc. International Conference on Learning Representations (ICLR), 2016.
+
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness, Marc G. Bellemare, Alex Graves, Martin Riedmiller, Andreas K. Fidjeland, Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. Nature, 518:529–533, 2015.
+
+Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In Maria Florina Balcan and Kilian Q. Weinberger (eds.), Proceedings of The $3 3 r d$ International Conference on Machine Learning, volume 48 of Proceedings of Machine Learning Research, pp. 1928–1937. PMLR, 20–22 Jun 2016.
+
+Matthias Plappert, Rein Houthooft, Prafulla Dhariwal, Szymon Sidor, Richard Y. Chen, Xi Chen, Tamim Asfour, Pieter Abbeel, and Marcin Andrychowicz. Parameter space noise for exploration. CoRR, abs/1706.01905, 2017.
+
+Christian Rupprecht, Iro Laina, Robert DiPietro, Maximilian Baust, Federico Tombari, Nassir Navab, and Gregory D. Hager. Learning in an uncertain world: Representing ambiguity through multiple hypotheses. International Conference on Computer Vision (ICCV 2017), Venice, Italy, October 2017, 2017.
+
+John Schulman, Sergey Levine, Philipp Moritz, Michael Jordan, and Pieter Abbeel. Trust region policy optimization. In Proceedings of the 32Nd International Conference on International Conference on Machine Learning - Volume 37, ICML’15, pp. 1889–1897. JMLR.org, 2015.
+
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv, abs/1707.06347, 2017.
+
+David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In Proceedings of the 31st International Conference on Machine Learning (ICML-14), pp. 387–395, 2014.
+
+David Silver, Aja Huang, Chris J. Maddison, Arthur Guez, Laurent Sifre, George van den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, Sander Dieleman, Dominik Grewe, John Nham, Nal Kalchbrenner, Ilya Sutskever, Timothy Lillicrap, Madeleine Leach, Koray Kavukcuoglu, Thore Graepel, and Demis Hassabis. Mastering the game of Go with deep neural networks and tree search. Nature, 529(7587):484–489, jan 2016.
+
+Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction, volume 1. MIT press Cambridge, 1998.
+
+Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International Conference on, pp. 5026– 5033. IEEE, 2012.
+
+George E Uhlenbeck and Leonard S Ornstein. On the theory of the brownian motion. In Physical review, volume 36.5, pp. 823, 1930.
+
+Christopher JCH Watkins and Peter Dayan. Q-learning. Machine learning, 8(3-4):279–292, 1992.
+
+Bernhard Wymann, Eric Espie, Christophe Guionneau, Christos Dimitrakakis, R ´ emi Coulom, and ´ Andrew Sumner. TORCS, The Open Racing Car Simulator. http://www.torcs.org, 2014.
+
+# APPENDIX
+
+In the following we display the box plots similar to Figure 1a and 1b for the remaining tasks.
+
+![](images/2e8b8a6804b5105cd7b02ee0884259f7fa00f5f30052aefa2717e6865f3763eb.jpg)  
+Figure 5: Variation in score of (from top left) Swimmer, Humanoid and HalfCheetah with different values of $M$ .

parse/train/SJgf6Z-0W/SJgf6Z-0W_content_list.json

@@ -0,0 +1,1232 @@
+[
+    {
+        "type": "text",
+        "text": "PREDICTING MULTIPLE ACTIONS FOR STOCHASTIC CONTINUOUS CONTROL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce a new approach to estimate continuous actions using actor-critic algorithms for reinforcement learning problems. Policy gradient methods usually predict one continuous action estimate or parameters of a presumed distribution (most commonly Gaussian) for any given state which might not be optimal as it may not capture the complete description of the target distribution. Our approach instead predicts $M$ actions with the policy network (actor) and then uniformly sample one action during training as well as testing at each state. This allows the agent to learn a simple stochastic policy that has an easy to compute expected return. In all experiments, this facilitates better exploration of the state space during training and converges to a better policy. ",
+        "bbox": [
+            233,
+            268,
+            764,
+            407
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            438,
+            336,
+            454
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning is a traditional branch of machine learning which focuses on learning complex tasks by assigning rewards to agents that interact with their environment. It has recently gained momentum thanks to the combination of novel algorithms for continuous control with deep learning models, sometimes even matching human performance in tasks such as playing video games and manipulating objects Mnih et al. (2015); Silver et al. (2016). Recent methods for continuous control problems like Deep Deterministic Policy Gradient (DDPG) Lillicrap et al. (2016), Asynchronous Advantage Actor Critic (A3C) Mnih et al. (2016) use actor-critic architectures, where an action function is learned by mapping states to actions. DDPG works well on many tasks, but it does not model the uncertainty in actions as it produces a point estimate of the action distribution over states. The actor is forced to deterministically choose an action for every state. A3C and other stochastic policy gradient algorithms output distribution parameters (e.g. Gaussian distributions) instead of point estimate, which can be sampled for action values. ",
+        "bbox": [
+            174,
+            470,
+            825,
+            637
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As a simple example where this is sub-optimal, consider the inverted pendulum task, where a pendulum is attached to a cart and the agent needs to control the one dimensional movement of the cart to balance the pendulum upside down. A deterministic agent chooses a single action for every state. This breaks the inherent symmetry of the task. When the cart in not moving and the pendulum is hanging down, two actions are equally promising: either moving left or right. The distribution parameter estimation (e.g. A3C) might work better in this case as there are only two good options, but in cases when there are more than two good actions to select, this will not be optimal. In our approach we allow the agent to suggest multiple actions, which enables it to resolve cases like this easily. ",
+        "bbox": [
+            174,
+            645,
+            825,
+            770
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Further, we observe that a deterministic behavior of DDPG can lead to sub-optimal convergence during training. The main limitation is that, especially in the beginning of the learning procedure, the actor favors actions that lead to a good immediate reward but might end up being far from the globally optimal choice. ",
+        "bbox": [
+            174,
+            776,
+            823,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This work is based on the intuition that if the actor is allowed to suggest, at each time step, multiple actions rather than a single one, this can render the resulting policy non-deterministic, leading to a better exploration of the entire solution space as well as a final solution of potentially higher quality. This can also eliminate the external exploration mechanisms required during training e.g. OrnsteinUhlenbeck process noise Uhlenbeck & Ornstein (1930), parameter noise Plappert et al. (2017) or differential entropy of normal distribution. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Here, we introduce an algorithm, which we refer to as Multiple Action Policy Gradients (MAPG), that models a stochastic policy with several point estimates and allows to predict a pre-defined number $M$ of actions at each time step, extending any policy gradient algorithm with little overhead. We will demonstrate the working of this algorithm by adapting DDPG Lillicrap et al. (2016) to use MAPG. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            172
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another benefit of the proposed method is that the variance of the predicted actions can give additional insights into the decision process during runtime. A low variance usually implies that the model only sees one way to act in a certain situation. A wider or even multi-modal distribution suggests that there exist several possibilities given the current state. ",
+        "bbox": [
+            174,
+            180,
+            823,
+            236
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the proposed method on six continuous control problems of the OpenAI Gym Brockman et al. (2016) as well as a deep driving scenario using the TORCS car simulator Wymann et al. (2014). For a fair evaluation we directly compare DDPG to our MAPG without changing hyper-parameters or modifying the training scheme. In all experiments, we show an improved performance using MAPG over DDPG. To verify if MAPG helps in better exploration during training, we also analyze MAPG under no external exploration policy. ",
+        "bbox": [
+            174,
+            243,
+            825,
+            327
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            369,
+            343,
+            386
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "There is currently a wide adoption of deep neural networks for reinforcement learning. Deep Q Networks (DQN) Mnih et al. (2015) directly learn the action-value function with a deep neural network. Although this method can handle very high dimensional inputs, such as images, it can only deal well with discrete and low dimensional action spaces. Guided Policy Search Levine & Koltun (2013) can exploit high and low dimensional state descriptions by concatenating the low dimensional state to a fully connected layer inside the network. ",
+        "bbox": [
+            174,
+            415,
+            825,
+            498
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recent methods for continuous control problems come in two flavours, vanilla policy gradient methods which directly optimize the policy and actor-critic methods which also approximate state-value function in addition to policy optimization. Trust Region Policy Optimization (TRPO) Schulman et al. (2015) and Proximal Policy Optimization Algorithms Schulman et al. (2017) can be used as vanilla policy gradient as well as actor-critic methods. Whereas, Deep Deterministic Policy Gradient (DDPG) Lillicrap et al. (2016) and Asynchronous Advantage Actor Critic (A3C) Mnih et al. (2016) use actor-critic architectures, where state-action function is learned to calculate policy gradients. ",
+        "bbox": [
+            174,
+            506,
+            825,
+            603
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Stochastic Value Gradients (SVG) Heess et al. (2015), Generalized Advantage Estimation (GAE) Schulman et al. (2015), A3C, TRPO all use stochastic policy gradients and predict action probability distribution parameters. The action values are then sampled from the predicted distribution. A parametrized normal distribution is most commonly used as action distribution. This means that this formulation models a kind of action noise instead of the true action distribution. For example a distribution with two modes cannot be modeled with a Gaussian. ",
+        "bbox": [
+            174,
+            609,
+            823,
+            693
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "DDPG Lillicrap et al. (2016) which extends DPG Silver et al. (2014) uses deterministic policy gradients and achieves stability when using neural networks to learn the actor-critic functions. The limitation of DDPG is that it always gives a points which may not be desired in stochastic action problems. ",
+        "bbox": [
+            174,
+            700,
+            823,
+            756
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Lazaric et al. (2007) estimate stochastic action values using a sequential Monte Carlo method (SMC). SMC has actor and critic models where the actor is represented by Monte Carlo sampling weights instead of a general function approximator like a neural network. SMC learning works well in small state space problems, but cannot be extended directly to high dimensional non-linear action space problems. ",
+        "bbox": [
+            174,
+            763,
+            823,
+            833
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Similar to our idea of predicting multiple instead of one output, but originating from the domain of supervised learning, is Multiple Hypothesis Prediction Rupprecht et al. (2017), which in turn is closely related to Multiple Choice Learning Lee et al. (2016) and Lee et al. (2017). In this line of work, the model is trained to predict multiple possible answers for the given task. Specific care has to be taken since often in supervised datasets not all possible outcomes are labeled, this leading to loss functions that contain an arg min-like term and, as such, are hard to differentiate. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 THE MULTIPLE ACTION POLICY GRADIENT ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            101,
+            679,
+            118
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section we will describe in detail how multiple action policy gradients can be derived and compare it to DDPG. We will then analyze the differences to understand the performance gain. ",
+        "bbox": [
+            173,
+            132,
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+            162
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            178,
+            316,
+            193
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We investigate a typical reinforcement learning setup Sutton & Barto (1998) where an agent interacts with an environment $E$ . At discrete time steps $t$ , the agent observes the full state $s _ { t } \\in \\mathcal { S } \\subset \\mathbb { R } ^ { c }$ , and after taking action $a _ { t } \\in \\mathcal { A } \\subset \\mathbb { R } ^ { d }$ , it receives the reward $r _ { t } \\in \\mathbb { R }$ . We are interested in learning a policy $\\pi : S  { \\mathcal { P } } ( A )$ , that produces a probability distribution over actions for each state. Similarly to other algorithms, we model the environment as a Markov Decision Process (MDP) with a probabilistic transition between states $p ( s _ { t + 1 } | s _ { t } , a _ { t } )$ and the rewards $r ( s _ { t } , a _ { t } )$ . ",
+        "bbox": [
+            173,
+            204,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We associate a state with its current and (discounted with $\\gamma \\in [ 0 , 1 ]$ ) future rewards by using ",
+        "bbox": [
+            169,
+            294,
+            779,
+            310
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cd028423ca08942013563765d4023a34b0d09ea4f2aed94cfe8d5c5fd87e002.jpg",
+        "text": "$$\nR _ { t } = \\sum _ { i = 1 } ^ { T } \\gamma ^ { i - t } r ( s _ { i } , a _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            315,
+            578,
+            359
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Since $\\pi$ and $E$ are stochastic, it is more meaningful to investigate the expected reward instead. Thus, the agent tries to find a policy that maximizes the expected discounted reward from the starting state distribution $p ( s _ { 1 } )$ . ",
+        "bbox": [
+            173,
+            363,
+            825,
+            406
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/91c2eafb54d106487bbc08f30d5dea6f4c2c1ba4946af120e56ad3130f02078e.jpg",
+        "text": "$$\nJ = \\mathbb { E } _ { r _ { i } , s _ { i } \\sim E , a _ { i } \\sim \\pi } ( R _ { 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            418,
+            405,
+            580,
+            421
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here, it is useful to investigate the recursive Bellman equation that associates a value to a state-action pair: ",
+        "bbox": [
+            171,
+            431,
+            823,
+            458
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e0420204d6cb00f21dc31142d39c20e7e3692fc215e7be49c96d5b29c409df5.jpg",
+        "text": "$$\nQ ^ { \\pi } \\big ( s _ { t } , a _ { t } \\big ) = \\mathbb { E } _ { r _ { t } , s _ { t + 1 } \\sim E } \\big [ r \\big ( s _ { t } , a _ { t } \\big ) + \\gamma \\mathbb { E } _ { a _ { t + 1 } \\sim \\pi } \\big [ Q ^ { \\pi } \\big ( s _ { t + 1 } , a _ { t + 1 } \\big ) \\big ] \\big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            458,
+            714,
+            476
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Methods such as (D)DPG use a deterministic policy where each state is deterministically mapped to an action using a function $\\mu : { \\mathcal { S } }  A$ which simplifies Equation 3 to ",
+        "bbox": [
+            174,
+            477,
+            823,
+            506
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e0b9338234880e4f454a62e36a4751d970f25f4f041330f3949af813a674a693.jpg",
+        "text": "$$\nQ ^ { \\mu } ( s _ { t } , a _ { t } ) = \\mathbb { E } _ { r _ { t } , s _ { t + 1 } \\sim E } [ r ( s _ { t } , a _ { t } ) + \\gamma Q ^ { \\mu } ( s _ { t + 1 } , \\mu ( s _ { t + 1 } ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            511,
+            694,
+            530
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In Q-learning Watkins & Dayan (1992), $\\mu$ selects the highest value action for the current state: ",
+        "bbox": [
+            171,
+            541,
+            794,
+            556
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/25ccc3fbbe5690701257458bd0f7269aacae9b822e05abc336b964b83dc62731.jpg",
+        "text": "$$\n\\mu ( s _ { t } ) = \\operatorname * { a r g m a x } _ { a _ { t } } ( Q ( s _ { t } , a _ { t } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            561,
+            594,
+            588
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The $Q$ value of an action is approximated by a critic network which estimates $Q ^ { \\mu } ( s _ { t } , a _ { t } )$ for the action chosen by the actor network. ",
+        "bbox": [
+            174,
+            594,
+            825,
+            622
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            640,
+            303,
+            654
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The key idea behind predicting multiple actions is that it is possible to learn a stochastic policy as long as the inner expectation remains tractable. Multiple action prediction achieves this by predicting a fixed number $M$ of actions $\\rho : \\mathcal { S }  \\mathcal { A } ^ { M }$ and uniformly sampling from them. The expected value is then the mean over all $M$ state-action pairs. The state-action value can then be defined as ",
+        "bbox": [
+            173,
+            665,
+            825,
+            722
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/18aa48def13c1aa381a6d2941d24f96568db8651351eb3dc57c97f8afb5e2a0f.jpg",
+        "text": "$$\nQ ^ { \\rho } ( s _ { t } , a _ { t } ) = \\mathbb { E } _ { r _ { t } , s _ { t + 1 } \\sim E } \\left[ r ( s _ { t } , a _ { t } ) + \\gamma \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } Q ^ { \\rho } ( s _ { t + 1 } , \\rho _ { m } ( s _ { t + 1 } ) ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            727,
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+    {
+        "type": "text",
+        "text": "This is beneficial since we not only enable the agent to employ a stochastic policy when necessary, but we also approximate the action distribution of the policy with multiple samples instead of one. ",
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+    {
+        "type": "text",
+        "text": "There exists an intuitive proof that the outer expectation in Equation 6 will be maximal if and only if the inner $Q ^ { \\rho }$ are all equal. The idea is based on the following argument: let us assume $\\rho$ as an optimal policy maximizing Equation 2. Further, one of the $M$ actions $\\rho _ { j } \\big ( s _ { t + 1 } \\big ) ,$ ) for a state $s _ { t + 1 }$ has a lower expected return than another action $k$ . ",
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+    {
+        "type": "equation",
+        "img_path": "images/e2ae844e55a5dad7af50940c76f8ca0130ab2dc265bbc4ecd3790667e50553f1.jpg",
+        "text": "$$\nQ ^ { \\rho } ( s _ { t + 1 } , \\rho _ { j } ( s _ { t + 1 } ) ) < Q ^ { \\rho } ( s _ { t + 1 } , \\rho _ { k } ( s _ { t + 1 } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Then there exists a policy $\\rho ^ { * }$ that would score higher than $\\rho$ that is exactly the same as rho exept that it predicts action $k$ instead of $j$ : $\\rho _ { j } ^ { * } ( s _ { t + 1 } ) : = \\bar { \\rho _ { k } } ( s _ { t + 1 } )$ . However, this contradicts the assumption ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 MAPG algorithm ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/de52201c521f20ceaf7be056562f9b4ec1672a4f7305029c008a7421c19f5cd4.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Modify actor network μ(s|0μ) to output M actions, At = {p1(st),..., pM(st)}. Randomly initialize actor μ(s|θμ) and critic Q(s|0q) network weights. Initialize target actor μ&#x27; and critic Q&#x27; networks,0’μ ← 0μ and 0Q ←0Q. for episode = 1 to N do Initialize random process  for exploration. Receive initial observation/state S1.</td></tr><tr><td>fort=1to Tdo Predict M action proposals At = {pi(st),...,ρm(st)}= μ&#x27;(st|0μ).</td></tr><tr><td>Uniformly sample an action j from At: α² = ρj(st) + Nt.</td></tr><tr><td>Execute action α and observe reward rt and state St+1·</td></tr><tr><td>Store transition (St,α²,rt, St+1) to replay buffer R.</td></tr><tr><td>Sample a random batch of size B from R. Set yi=ri+Q&#x27;(Si+1,μ&#x27;(si+1l0μ)|0Q).</td></tr><tr><td>Update critic by minimizing the loss,</td></tr><tr><td>L=B∑i(yi-Q(si,a|Q)）²</td></tr><tr><td>Update all actor weights connected to αt .</td></tr><tr><td>VθuJ~B∑iVQ(s,a|0Q)ls=sia=a²Vθ-θ{1.M)+j μ(s|0)|si Update the target networks:</td></tr><tr><td>0μ←T0μ+(1-T）θμ</td></tr><tr><td>←T0Q+(1-T）Q end for end for</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "that we had learned an optimal policy beforehand. Thus in an optimal policy all $M$ action proposals will have the same expected return. More informal, this can also be seen as a derivation from the training procedure. If we always select a random action from the $M$ proposals, they should all be equally good since the actor cannot decide which action should be executed. ",
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+    {
+        "type": "text",
+        "text": "This result has several interesting implications. From the proof, it directly follows that it is possible - and sometimes necessary - that all proposed actions are identical. This is the case in situations where there is just one single right action to take. When the action proposals do not collapse into one, there are two possibilities: either it does not matter what action is currently performed, or all proposed actions lead to a desired outcome. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Naturally, the set of stochastic policies includes all deterministic policies, since a deterministic policy is a stochastic policy with a single action having probability density equal to one. This means that in theory we expect the multiple action version of a deterministic algorithm to perform better or equally well, since it could always learn a deterministic policy by predicting $M$ identical actions for every state. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Algorithm 1 outlines the MAPG technique. The main change is that the actor is modified to produce $M$ instead of one output. For every timestep one action $j$ is then selected. When updating the actor network, a gradient is only applied to the action (head) that was selected during sampling. Over time each head will be selected equally often, thus every head will be updated and learned during training. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section we will investigate and analyze the performance of MAPG in different aspects. First, we compare scores between DDPG, A3C and MAPG on six different tasks. Second, we analyze the influence of the number of actions on the performance by training agents with different $M$ on five tasks. Further, to understand the benefit of multiple action prediction, we observe the variance over actions of a trained agent: the goal is to analyze for which states the predicted actions greatly differ from each other and for which ones they collapse into a single choice instead. Finally, we compare the performance of DDPG and MAPG without any external noise for exploration during training. ",
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+    {
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+        "img_path": "images/65c5bb58f4ebb3c41b6cc2d4acfc4c26c3779ea64fb7250fd6c053573e7a2e61.jpg",
+        "table_caption": [
+            "Table 1: Tasks used for evaluation ",
+            "Table 2: Average score $\\pm 3 \\sigma$ over 100 episodes for Mujoco tasks with different $M$ . For better readability we denote the highest mean score for each task in bold. Corresponding boxplots can be found in Figure 1a and 1b and the appendix. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>TASK</td><td>ACTION DIMENSION</td><td>STATE DIMENSION</td><td>DESCRIPTION</td></tr><tr><td>PENDULUM</td><td>1</td><td>3</td><td>PENDULUM ON A CART.</td></tr><tr><td>HOPPER</td><td>3</td><td>11</td><td>ONELEGGED ROBOT.</td></tr><tr><td>WALKER2D</td><td>6</td><td>17</td><td>TWO DIMENSIONAL BIPEDAL ROBOT.</td></tr><tr><td>HUMANOID</td><td>17</td><td>376</td><td>THREE DIMENSIONAL BIPEDAL ROBOT.</td></tr><tr><td>HALFCHEETAH</td><td>6</td><td>17</td><td>TWO LEG ROBOT.</td></tr><tr><td>SWIMMER</td><td>2</td><td>6</td><td>THREE JOINT SWIMMING ROBOT.</td></tr><tr><td>TORCS</td><td>3</td><td>29</td><td>CONTROL CAR IN3D SIMULATION.</td></tr></table>",
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/a21bd86b045b0dcc5b5dd011b727f4ebdfacfe3f3982d1a5a614721eb1eaddd1.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>ENVIRONMENT</td><td>DDPG</td><td>A3C</td><td>M=10</td><td>M= 20</td><td>M=50</td></tr><tr><td>HOPPER-V1</td><td>603 ±76</td><td>532 ±105</td><td>824±94</td><td>923±90</td><td>732 ±34</td></tr><tr><td>WALKER2D-V1</td><td>960 ±72</td><td>764 ±112</td><td>1297 ± 70</td><td>1319 ± 50</td><td>1589 ± 45</td></tr><tr><td>HUMANOID-V1</td><td>1091 ± 65</td><td>281±40</td><td>1248 ±115</td><td>1112 ±75</td><td>1212 ±110</td></tr><tr><td>HALFCHEETAH-V1</td><td>4687 ±455</td><td>3803 ±125</td><td>6659 ± 570</td><td>4116± 85</td><td>4333 ± 70</td></tr><tr><td>SWIMMER-V1</td><td>38±7</td><td>33±10</td><td>51±6</td><td>41±4</td><td>40±2</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In all our experiments, we use five continuous control tasks from the Mujoco Simulator Todorov et al. (2012) and a driving task for The Open Racing Car Simulator (TORCS). A detailed description about the tasks is given in Table 1. We use the OpenAI Gym Brockman et al. (2016) and OpenAI baselines Hesse et al. (2017) for evaluating our experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The base actor and critic networks are fixed in all experiments. Each network has two fully connected hidden layers with 64 units each. Each fully-connected layer is followed by a ReLU nonlinearity. The actor network takes the current observed state $s _ { t }$ as input and produces $M$ actions $a _ { t } ^ { ( m ) } \\in [ - 1 , 1 ] ^ { d }$ by applying tanh. From $M$ actions $a _ { t } ^ { ( m ) }$ , a single action $a _ { t }$ is randomly chosen with equal probability. The critic uses the current state $s _ { t }$ and action $a _ { t }$ as input and outputs a scalar value ( $Q$ -value). In the critic network, the action value is concatenated with the output of the first layer followed by one hidden layer and an output layer with one unit. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The critic network is trained by minimizing the mean square loss between the calculated discounted reward and the computed $Q$ value. The actor network is trained by computing the policy gradient from the $Q$ -value of the chosen action. The network weights of the last layer are only updated for the selected action. Ornstein-Uhlenbeck process noise is added to the action values from the actor for exploration. The training is done for a total of two million steps in all tasks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For A3C training, we use same actor-critic networks as for earlier experiment. The output of actor network is a mean vector $( \\mu _ { a } )$ (one for each action value) and a scalar standard deviation $( \\sigma ^ { 2 }$ , shared for all actions). The actions values are sampled from the normal distribution $( \\mathcal { N } ( \\mu _ { a } , \\sigma ^ { 2 } ) )$ . We used differential entropy of normal distribution to encourage exploration with weight $1 0 ^ { - } 4$ . In our experiments, A3C performed poorly than DDPG in all tasks and was not able to learn a good policy for Humanoid task. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2 MUJOCO EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For more meaningful quantitative results, we report the average reward over 100 episodes with different values of $M$ for various tasks in 2. For all environments except HUMANOID we already score higher with $M = 5$ . The lower performance in the HUMANOID task might be explained by the drastically higher dimensionality of the world state in this task which makes it more difficult to observe. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The scores of policy based reinforcement learning algorithms can vary a lot depending on network hyper-parameters, reward function and codebase/framework as outlined in Henderson et al. (2017). To minimize the variation in score due to these factors, we fixed all parameters of different algorithms and only studied changes on score by varying $M$ . Our metric for performance in each task is average reward over 100 episodes by an agent trained for 2 million steps. This evaluation hinders actors with high $M$ since in every training step only a single out of the $M$ actions will be updated per state. Thus, in general actors with higher number of action proposals, will need a longer time to learn a meaningful distribution of action. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We show a plot for the scores in the HOPPER and WALKER2D environments in Figure 1a and 1b, where we can see that the overall score increases with $M$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/ffe57f760e0c5ecdb7ea599b0962f79cd49c64ec3736c8d18bb717038446a420.jpg",
+        "image_caption": [
+            "Figure 1: Variation in score of Hopper and Walker2d with different values of $M$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Figure 2, we studied the variance in action values for $M = 1 0$ during training together with the achieved reward. The standard deviation of actions generated by MAPG decreases with time. As the network converges to a good policy (increase in expected reward) the variation in action values is reduced. However there are some spikes in standard deviation even when network is converged to a better policy. It shows that there are situations in which the policy sees multiple good actions (with high $Q$ -value) which can exploited using MAPG. ",
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+    {
+        "type": "text",
+        "text": "4.3 VARIANCE ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We use the simple Pendulum environment to analyze the variance during one episode. The task is the typical inverted pendulum task, where a cart has to be moved such that it balances a pendulum in an inverted position. Figure 3 plots standard deviation and the angle of the pendulum. Some interesting relationships can be observed. The variance exhibits two strong spikes that coincide with an angle of 0 degrees. This indicates that the agent has learned that there are two ways it can swing up the pole: either by swinging it clockwise or counter clockwise. A deterministic agent would need to pick one over the other instead of deciding randomly. Further, once the target inverted pose (at 180 degrees) is reached the variance does not go down to 0. This means that for the agent a slight jitter seems to be the best way to keep the pendulum from gaining momentum in one or the other direction. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "With this analysis we could show that a MAPG agent can learn meaningful policies. The variance over predicted actions can give additional insight into the learned policy and results in a more diverse agent that can for example swing up the pole in two different directions instead of picking one. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.4 EFFECT ON EXPLORATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, we study the effect of MAPG on exploration during training. We compare the performance of DDPG and MAPG during training with and without any external noise on Pendulum and HalfCheetah environments. Figure 4 shows the average reward during training with DDPG and MAPG $M = 1 0$ . The policy trained using MAPG converges to better average reward than DDPG in both cases. Moreover, the performance of MAPG without any external exploration is comparable to DDPG with added exploration noise. This means MAPG can explore the state space enough to find a good policy. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "image",
+        "img_path": "images/a9d050997bbcb518de69f3dba1f71d448790a5383d4d0ada0e41aa1b314ad36a.jpg",
+        "image_caption": [
+            "Figure 2: Standard deviation and reward with $M = 1 0$ for the Pendulum task during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/485c98f8b8e6c185987979292333c3f05ce8976314d05d00ad968a6e39eb0c28.jpg",
+        "image_caption": [
+            "Figure 3: Standard deviation and angle during one episode of the Pendulum environment. An angle of $\\pm 1 8 0$ is the target inverted pose. 0 is hanging downwards. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "In the Half Cheetah environment we can see that using exploration creates a much bigger performance difference between DDPG and MAPG than without. The difference sets in after about 500 epochs. This is an indication that in the beginning of training the actions predicted by MAPG are similar to the one from DDPG. The noise later helps to pull the $M$ actions apart such that they find individual loss minima, leading to a more diverse policy with better reward. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "4.5 TORCS ",
+        "text_level": 1,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "TORCS (The Open Racing Car Simulator) is an open source 3D car racing simulator. It provides an interface for agents to drive the cars. During training, the reward was set proportional to component of car velocity along direction of road $v * \\cos ( \\alpha )$ , where $\\alpha$ is the angle between the velocity vector and the center line of the track. This reward encourages forward motion. The car’s sensor data (velocity, distance from road edges etc.) is used as input state and steer, brake, accelerate as actions at each time step. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/0ae2acc456a9194837f9a54b5a8e40a09a2f8f1e6309b5daa2bd63cfef746875.jpg",
+        "image_caption": [
+            "Figure 4: Performance curves for two environments with and without external exploration noise on DDPG and MAPG: original DDPG with OU process noise (green), DDPG without any exploration noise (blue), MAPG $\\mathbf { \\Gamma } ( \\mathbf { M } { = } 1 0$ ) with OU process noise (red) and MAPG $( \\mathbf { M } { = } 1 0 _ { , }$ ) without OU process noise (orange). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, MAPG with $M = 1 0$ was able to complete multiple laps of the track, whereas the DDPG based agent could not complete even one lap of track. The average distance traveled over 100 episodes by DDPG is 807 and 5882 (both in meters) for MAPG agent. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Similar to our other experiments we find that MAPG agents explore more possibilities due to their stochastic nature and can then learn more stable and better policies. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have proposed MAPG, a technique that leverages multiple action prediction to learn better policies in continuous control problems. The proposed method enables a better exploration of the state space and shows improved performance over DDPG. As indicated by exploration experiments, it can also be a used as a standalone exploration technique, although more work needs to be done in this direction. Last but not least, we conclude with interesting insights gained from the action variance. There are several interesting directions which we would like to investigate in the future. The number of actions $M$ is a hyper-parameter in our model that needs to be selected and seems to be task specific. In general, the idea of predicting multiple action proposals can be extended to other on- or off-policy algorithms, such as NAF Gu et al. (2016) or TRPO. Evaluating MA-NAF and MA-TRPO will enable studying the generality of the proposed approach. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Will be added after anonymous review. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            736,
+            285,
+            751
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym, 2016. ",
+        "bbox": [
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+            758,
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+            786
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            264,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In the following we display the box plots similar to Figure 1a and 1b for the remaining tasks. ",
+        "bbox": [
+            171,
+            133,
+            781,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/2e8b8a6804b5105cd7b02ee0884259f7fa00f5f30052aefa2717e6865f3763eb.jpg",
+        "image_caption": [
+            "Figure 5: Variation in score of (from top left) Swimmer, Humanoid and HalfCheetah with different values of $M$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            172,
+            784,
+            453
+        ],
+        "page_idx": 10
+    }
+]

parse/train/SknC0bW0-/SknC0bW0-.md

@@ -0,0 +1,249 @@
+# CONTINUOUS-FIDELITY BAYESIAN OPTIMIZATION WITH KNOWLEDGE GRADIENT
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+While Bayesian optimization (BO) has achieved great success in optimizing expensive-to-evaluate black-box functions, especially tuning hyperparameters of neural networks, methods such as random search (Li et al., 2016) and multifidelity BO (e.g. Klein et al. (2017)) that exploit cheap approximations, e.g. training on a smaller training data or with fewer iterations, can outperform standard BO approaches that use only full-fidelity observations. In this paper, we propose a novel Bayesian optimization algorithm, the continuous-fidelity knowledge gradient (cfKG) method, that can be used when fidelity is controlled by one or more continuous settings such as training data size and the number of training iterations. cfKG characterizes the value of the information gained by sampling a point at a given fidelity, choosing to sample at the point and fidelity with the largest value per unit cost. Furthermore, cfKG can be generalized, following Wu et al. (2017), to settings where derivatives are available in the optimization process, e.g. large-scale kernel learning, and where more than one point can be evaluated simultaneously. Numerical experiments show that cfKG outperforms state-of-art algorithms when optimizing synthetic functions, tuning convolutional neural networks (CNNs) on CIFAR-10 and SVHN, and in large-scale kernel learning.
+
+# 1 INTRODUCTION
+
+In hyperparameter tuning of machine learning models, we seek to find a set of hyperparameters $x$ in some set A to minimize the validation error $f ( x )$ , i.e., to solve
+
+$$
+\operatorname* { m i n } _ { x \in \mathbb { A } } f ( x )
+$$
+
+Evaluating $f ( x )$ can take substantial time and computational power (Bergstra & Bengio, 2012), and may not provide gradient evaluations. Thus, machine learning practitioners have turned to Bayesian optimization for solving (1.1) (Snoek et al., 2012) because it tends to find good solutions with few function evaluations (Jones et al., 1998).
+
+As the computational expense of training and testing a modern deep neural network for a single set of hyperparameters has grown as long as days or weeks, it has become natural to seek ways to solve (1.1) more quickly by supplanting some evaluations of $f ( x )$ with computationally inexpensive lowfidelity approximations. Indeed, when training a neural network or most other machine learning models, we can approximate $f ( x )$ by training on less than the full training data, or using fewer training iterations. Both of these controls on fidelity can be set to achieve either better accuracy or lower computational cost across a range of values reasonably modeled as continuous.
+
+In this paper, we consider optimization with evaluations of multiple fidelities and costs where the fidelity is controlled by one or more continuous parameters. We model these evaluations by a realvalued function $g ( x , s )$ where $f ( x ) : = g ( x , 1 _ { m } )$ and $s \in [ 0 , 1 ] ^ { m }$ denotes the $m$ fidelity-control parameters. $g ( x , s )$ can be evaluated, optionally with noise, at a cost that depends on $x$ and $s$ . In the context of hyperparameter tuning, we may take $m = 2$ and let $g ( x , s _ { 1 } , s _ { 2 } )$ denote the loss on the validation set when training using hyperparameters $x$ with a fraction $s _ { 1 }$ of the training data and a fraction $s _ { 2 }$ of some maximum allowed number of training iterations. We may also set $m = 1$ and let $s$ index either training data or training iterations. We assume A is a compact connected uncountable set into which it is easy to project, such as a hyperrectangle.
+
+This problem setting also appears outside of hyperparameter tuning, in any application where the objective is expensive to evaluate and we may observe cheap low-fidelity approximations parameterized by a continuous vector. For example, when optimizing a system evaluated via a Monte Carlo simulator, we can evaluate a system configuration approximately by running with fewer replications. Also, when optimizing an engineering system modeled by a partial differential equation (PDE), we can evaluate a system configuration approximately by solving the PDE using a coarse grid.
+
+Given this problem setting, we use the knowledge gradient approach (Frazier et al., 2009) to design an algorithm to adaptively select the hyperparameter configuration and fidelity to evaluate, to best support solving (1.1). By generalizing a computational technique based on the envelope theorem first developed in Wu et al. (2017), our algorithm supports parallel function evaluations, and also can take advantage of derivative observations when they are available. This algorithm chooses the point or set of points to evaluate next that maximizes the ratio of the value of information from evaluation against its cost.
+
+Unlike most existing work on discrete- and continuous-fidelity Bayesian optimization, our approach considers the impact of our measurement on the future posterior distribution over the full feasible domain, while existing expected-improvement-based approaches consider its impact at only the point evaluated. One exception is the entropy-search-based method [10], which also considers the impact over the full posterior. Our approach differs from entropy search in that it chooses points to sample to directly minimize expected simple regret, while entropy search seeks to minimize the entropy of the location or value of the global optimizer, indirectly reducing simple regret.
+
+We summarize our contributions as follows.
+
+# Contributions of this paper:
+
+• We develop a novel Bayesian Optimization algorithm, the continuous-fidelity knowledge gradient (cfKG) method, which chooses the point and fidelity to sample next that maximizes the ratio of the value of information to its cost; • After first developing this algorithm in the sequential derivative-free setting, and inspired by Wu et al. (2017), we generalize to settings where function evaluations may be performed in parallel, and where we can access possibly noisy and biased gradient information. • We show that our algorithm outperforms a number of start-of-art benchmark algorithms when optimizing common synthetic functions, tuning convolutional neural networks on CIFAR-10 and SVHN, and in a large-scale kernel learning example.
+
+The rest of the paper is organized as follows. Sect. 2 reviews related work. Sect. 3 presents the cfKG method. Sect. 4 tests cfKG on benchmarks including synthetic functions and hyperparameter tuning for deep learning and kernel learning. Sect. 5 concludes.
+
+# 2 RELATED WORK
+
+Algorithms exploiting inexpensive low-fidelity approximations for hyperparameter tuning have been proposed both within and outside the field of Bayesian optimization.
+
+Outside of Bayesian optimization, Li et al. (2016) develops an early-stopping method called Hyperband that can outperform traditional Bayesian optimization when tuning hyperparameters within a random search framework by adaptively allocating a single predefined resource that can be taken to be the number of training iterations or the amount of training data. In contrast to Hyperband we allow more than one fidelity control, support derivative observations, and use one-step optimality in terms of value vs. cost to select the point or points to sample. We compare against Hyperband in our numerical experiments, and show improved performance training a convolutional neural network.
+
+Using a modified expected improvement (EI) acquisition function, Huang et al. (2006) and Lam et al. (2015) develop methods for multi-fidelity settings with a finite discrete set of low-fidelity approximations. We call this the “discrete-fidelity setting” to contrast it with the continuous-fidelity setting we consider here. We do not compare against discrete-fidelity methods in our numerical experiments as both Kandasamy et al. (2017) and Klein et al. (2017) show that continuous-fidelity methods can find good solutions in continuous-fidelity problems as much as an order of magnitude faster than discrete-fidelity ones.
+
+In the specific context of hyperparameter tuning, Domhan et al. (2015) proposes a simple early stopping criterion combined with SMAC (an EI acquisition function with a random forest based statistical model) in which evaluation of a configuration is stopped if its performance is predicted to be worse than the current best configuration.
+
+Kandasamy et al. (2016) generalizes the upper confidence bound (UCB) criteria to the discretefidelity setting, and further generalizes to continuous fidelities in Kandasamy et al. (2017). We compare against Kandasamy et al. (2017) in our numerical experiments, finding that cfKG provides improved performance when the budget is sufficiently large. This may be because the UCB criteria was originally designed for minimizing the cumulative regret and not simple regret, with the simple regret bounds resulting from theoretical analysis of cumulative regret being loose enough to be consistent with the empirical performance gap we observe in our experiments. It may also be because both Kandasamy et al. (2016) and Kandasamy et al. (2017) use a two-stage process in which the point to evaluate is selected without considering the fidelity, and the fidelity only selecting afterward. In contrast, cfKG selects the point and fidelity jointly, and considers the impact of fidelity choice on the best point to sample.
+
+Entropy search has also been extended to the discrete-fidelity setting by Swersky et al. (2013). Considering training iterations as a one-dimensional continuous fidelity control, Swersky et al. (2014) generalizes Swersky et al. (2013) to this specific continuous-fidelity setting. Klein et al. (2017) adapts entropy search to settings where the size of the training data is a one-dimensional continuous fidelity control. McLeod et al. (2017) extends predictive entropy search to settings with one fidelity control and provides a faster way to compute the acquisition function.
+
+Poloczek et al. (2017) develops a knowledge gradient method for the discrete-fidelity setting, but does not consider the continuous-fidelity setting. Our computational techniques are quite different from the ones developed there, as necessitated by our consideration of continuous fidelities. As an advantageous byproduct, our computational techniques also avoid the need to discrete A.
+
+This paper proposes the first knowledge gradient method for continuous-fidelity settings, and show how to generalize it to the batch and derivative-enabled settings by generalizing the computational technique developed in Wu et al. (2017). To the best of our knowledge, this is the first multi-fidelity batch Bayesian optimization algorithm. The relationship of this paper to the existing multi-fidelity BO literature is summarized in Table. 1.
+
+Table 1: Summary of the multi-fidelity BO literature.   
+
+<table><tr><td rowspan=1 colspan=1>Literature summary</td><td rowspan=1 colspan=1>EI/SMAC</td><td rowspan=1 colspan=1>Entropy Search</td><td rowspan=1 colspan=1>UCB</td><td rowspan=1 colspan=1>KnowledgeGradient</td></tr><tr><td rowspan=1 colspan=1>Discrete-fidelity</td><td rowspan=1 colspan=1>Huang et al. (2006);Lam et al. (2015)</td><td rowspan=1 colspan=1>Swersky et al. (2013)</td><td rowspan=1 colspan=1>Kandasamyet al. (2016)</td><td rowspan=1 colspan=1>Poloczeket al. (2017)</td></tr><tr><td rowspan=1 colspan=1>Continuous-fidelity</td><td rowspan=1 colspan=1>Domhan et al. (2015)</td><td rowspan=1 colspan=1>Swersky et al. (2014);Klein et al.(2017)；McLeod et al. (2017)</td><td rowspan=1 colspan=1>Kandasamyet al. (2017)</td><td rowspan=1 colspan=1>This paper</td></tr></table>
+
+Our method is also related to knowledge gradient methods for single-fidelity BO. In particular, the way in which we extend cfKG from the sequential (one evaluation at a time) derivative-free setting to allow batches of points and gradient observations follows work extending single-fidelity knowledge-gradient methods to the batch and derivative settings in Wu & Frazier (2016) and $\mathrm { W u }$ et al. (2017). We also generalize the envelope-theorem based computational technique developed for the single-fidelity setting in Wu et al. (2017) to continuous fidelities. We compare with single-fidelity knowledge gradient methods in our numerical experiments, and show that cfKG is able to levarage inexpensive low-fidelity observations to provide improved performance in both derivative-free and derivative-enabled settings.
+
+# 3 CONTINUOUS-FIDELITY KNOWLEDGE GRADIENT
+
+In this section, we propose the continuous-fidelity knowledge gradient (cfKG), a novel Bayesian optimization algorithm that exploits inexpensive low-fidelity approximations. This algorithm, like most Bayesian optimization algorithms, consists of a statistical model and an acquisition function. cfKG proceeds iteratively by fitting the statistical model (described below in Section 3.1) to all previously sampled (point, fidelty) pairs, and then finding and sampling the (point, fidelity) pair that maximizes the acquisition function. Once the budget is exhausted, at some iteration $N$ , cfKG returns as its final solution the point with the lowest estimated $g ( x , 1 _ { m } )$ .
+
+To describe cfKG in detail, Sect. 3.1 first describes Gaussian process regression for modeling both $g ( x , s )$ and its cost of evaluation. This approach is standard, with the novelty arising in cfKG’s acquisition function and how we optimize it. Then, Sect. 3.2 presents the cfKG acquisition function, which values sampling a (point, fidelity) pair according to the ratio of the value of the information gained from sampling that point at that fidelity, to the cost of doing so. Sect. 3.3 generalizes an envelope-theorem based computational technique developed in $\mathrm { W u }$ et al. (2017) to efficiently maximize this acquisition function. We discuss extensions to the derivative-enabled setting in Sect. 3.4.
+
+# 3.1 GAUSSIAN PROCESSES
+
+We put a Gaussian process (GP) prior (Rasmussen & Williams, 2006) on the function $g$ or its logarithm. We describe this procedure placing the prior on $g$ directly, and then discuss below when we recommend instead placing it on $( { \bar { x } } , s ) \mapsto \log { \bar { g } } ( x , s )$ . The GP prior is defined by its mean function $\mu ^ { ( 0 ) } : \mathbb { A } \times [ 0 , 1 ] ^ { m } \mapsto \mathbb { R }$ and kernel function $K ^ { ( 0 ) } : \{ \mathbb { A } \times [ 0 , 1 ] ^ { m } \} \times \{ \mathbb { A } \times [ 0 , 1 ] ^ { m } \} \mapsto \mathbb { R }$ . These mean and kernel functions have hyperparameters, whose inference we discuss below.
+
+We assume that evaluations of $g ( x , s )$ are subject to additive independent normally distributed noise with common variance $\sigma ^ { 2 }$ . We treat the parameter $\sigma ^ { 2 }$ as a hyperparameter of our model, and also discuss its inference below. Our assumption of normally distributed noise with constant variance is common in the BO literature (Klein et al., 2017).
+
+The posterior distribution of $g$ after $n$ function evaluations at points $\begin{array} { r l r } { z ^ { ( 1 : n ) } } & { { } : = } & { } \end{array}$ $\{ ( x ^ { ( 1 ) } , s ^ { ( 1 ) } ) , ( x ^ { ( 2 ) } , s ^ { ( 2 ) } ) , \cdot \cdot \cdot , ( x ^ { ( n ) } , s ^ { ( n ) } ) \}$ with observed values $y ^ { ( 1 : n ) } : = \{ y ^ { ( 1 ) } , y ^ { ( 2 ) } , \cdot \cdot \cdot , y ^ { ( n ) } \}$ remains a Gaussian process (Rasmussen & Williams, 2006), and $g \mid z ^ { ( 1 : n ) } , y ^ { ( 1 : n ) } \sim \mathbf { G P } ( \mu ^ { n } , K ^ { ( n ) } )$ with $\mu ^ { n }$ and $K ^ { ( n ) }$ evaluated at a point $z = ( x , s )$ (or pair of points $z$ , $\tilde { z } = ( \tilde { x } , \tilde { s } ) )$ given as follows
+
+$$
+\begin{array} { r l r } & { } & { \mu ^ { ( n ) } ( z ) = \mu ^ { ( 0 ) } \left( z \right) + K ^ { ( 0 ) } \left( z , z ^ { ( 1 : n ) } \right) \left( K ^ { ( 0 ) } ( z ^ { ( 1 : n ) } , z ^ { ( 1 : n ) } ) + \sigma ^ { 2 } I \right) ^ { - 1 } \left( y ^ { ( 1 : n ) } - \mu ( z ^ { ( 1 : n ) } ) \right) , } \\ & { } & { K ^ { ( n ) } \left( z , \tilde { z } \right) = K ^ { ( 0 ) } ( z , \tilde { z } ) - K ^ { ( 0 ) } \left( z , z ^ { ( 1 : n ) } \right) \left( K ^ { ( 0 ) } ( z ^ { ( 1 : n ) } , z ^ { ( 1 : n ) } ) + \sigma ^ { 2 } I \right) ^ { - 1 } K ^ { ( 0 ) } \left( z ^ { ( 1 : n ) } , \tilde { z } \right) . } \end{array}
+$$
+
+This statistical approach contains several hyperparameters: the variance $\sigma ^ { 2 }$ , and any parameters in the mean and kernel functions. We treat these hyperparameters in a Bayesian way as proposed in Snoek et al. (2012).
+
+When $g$ is the validation error in a hyperparameter optimization problem, we recommend putting a GP prior on $\log g ( x , s )$ , rather than on $g ( x , s )$ directly, because (1) $g ( x , s )$ is nonnegative and will be allowed to be negative after log scaling, better matching the range of values assumed by the GP, and (2) because $g ( x , s )$ can climb steeply over several orders of magnitude as we move away from the optimal $x$ , making $\log g ( x , s )$ easier to model.
+
+We analogously train a separate GP on the logarithm of the cost of evaluating $g ( x , s )$
+
+# 3.2 THE CFKG ACQUISITION FUNCTION
+
+cfKG samples the point and fidelity that jointly maximize an acquisition function, which we define in this section by adopting the knowledge gradient concept (Frazier et al., 2009) in the continuousfidelity setting to value the information gained through one additional sample.
+
+If we were to stop sampling after $n$ samples, we would select as our solution to (1.1) a point $x$ with minimum estimated validation error $\mu ^ { ( n ) } ( x , 1 _ { m } )$ , and this point would have a conditional expected validation error of $\begin{array} { r } { \operatorname* { m i n } _ { x \in \mathbb { A } } \mu ^ { ( n ) } ( x , 1 _ { m } ) } \end{array}$ under the posterior. If instead we took an additional sample at $x ^ { ( n + 1 ) }$ with the fidelity $s ^ { ( n + 1 ) }$ , then the minimum expected validation error under the resulting posterior would become minx∈A µ(n+1)(x, 1m). This quantity depends on x(n+1) and s(n+1) through the dependence of $\boldsymbol { \mu } ^ { ( n + 1 ) } ( x , \boldsymbol { 1 } _ { m } )$ on the point and fidelity sampled, and is random under the posterior at iteration $n$ because $\boldsymbol { \mu } ^ { ( n + 1 ) } ( x , \boldsymbol { 1 } _ { m } )$ depends on the observation $y ^ { ( n + 1 ) }$ . We discuss this dependence explicitly in Sect. 3.3.
+
+The value of the information gained by sampling at $x ^ { ( n + 1 ) }$ with the fidelity $s ^ { ( n + 1 ) }$ conditioned on any particular outcome $y ^ { ( n + 1 ) }$ is thus the difference of these two expected validation errors $\begin{array} { r } { \operatorname* { m i n } _ { x \in \mathbb { A } } \mu ^ { ( n ) } ( x , 1 _ { m } ) - \operatorname* { m i n } _ { x \in \mathbb { A } } \mu ^ { ( n + 1 ) } ( x , 1 _ { m } ) } \end{array}$ . We then take the expectation of this difference, over the random outcome $y ^ { ( n + 1 ) }$ , to obtain the (unconditional) value of the information gained, and take the ratio of this value with the cost of obtaining it to obtain the cfKG acquistion function,
+
+$$
+\operatorname { c f K G } ( x , s ) = { \frac { \operatorname* { m i n } _ { x ^ { \prime } \in \mathbb { A } } \mu ^ { ( n ) } ( x ^ { \prime } , 1 _ { m } ) - \mathbb { E } _ { n } \left[ \operatorname* { m i n } _ { x ^ { \prime } \in \mathbb { A } } \mu ^ { ( n + 1 ) } ( x ^ { \prime } , 1 _ { m } ) \mid x ^ { ( n + 1 ) } = x , s ^ { ( n + 1 ) } = s \right] } { \cot ^ { ( n ) } ( x , s ) } } ,
+$$
+
+where $\mathrm { c o s t } ^ { ( n ) } ( x , s )$ is the estimated cost of evaluating at $x$ with the fidelity $s$ based on the observations available at iteration $n$ , according to the GP described in Sect. 3.1, and $\mathbb { E } _ { n }$ indicates the expectation taken with respect to the posterior given $\boldsymbol { x } ^ { ( 1 : n ) } , \boldsymbol { s } ^ { ( 1 : n ) } , \boldsymbol { y } ^ { ( 1 : n ) }$ .
+
+The cfKG algorithm chooses to sample at the point and fidelity that jointly maximize the cfKG acquistion function
+
+$$
+\operatorname* { m a x } _ { ( x , s ) \in \mathbb { A } \times [ 0 , 1 ] ^ { m } } { \mathrm { c f K G } } ( x , s ) .
+$$
+
+Although this acquisition function considers the expected value of an improvement due to sampling, it differs from expected improvement approaches such as Lam et al. (2015) because the point at which an improvement occurs, $\operatorname { a r g m a x } _ { x \in \mathbb { A } } \mu ^ { ( n + 1 ) } ( x , 1 _ { m } )$ may differ from the point sampled. Moreover, this acquisition function allows joint valuation of both the point $x$ and the fidelity $s$ , while approaches such as Lam et al. (2015) require valuing a point $x$ assuming it will be evaluated at full fidelity and then choose the fidelity in a second stage.
+
+cfKG generalizes naturally to batch settings where we can evaluate multiple (point, fidelity) pairs at once. We value joint evaluation of $q \geq 1$ points $x _ { 1 : q }$ at fidelities $s _ { 1 : q }$ , where $z _ { 1 : q } =$ $\mathsf { \bar { ( } } ( x _ { 1 } , s _ { 1 } ) , \ldots , ( x _ { q } , s _ { q } ) )$ , by
+
+$$
+\mathsf { q \mathrm { - } \mathrm { c f K G } } ( z _ { 1 : q } ) = \frac { \operatorname* { m i n } _ { x ^ { \prime } \in \mathbb { A } } \mu ^ { ( n ) } ( x ^ { \prime } , 1 _ { m } ) - \mathbb { E } _ { n } \left[ \operatorname* { m i n } _ { x ^ { \prime } \in \mathbb { A } } \mu ^ { ( n + q ) } ( x ^ { \prime } , 1 _ { m } ) \mid z ^ { ( n + 1 : n + q ) } = z _ { 1 : q } \right] } { \operatorname* { m a x } _ { 1 \leq i \leq q } \mathrm { c o s t } ^ { ( n ) } ( z ^ { ( n + i ) } ) } ,
+$$
+
+We then modify (3.3) by sampling at the batch of points and fidelities that maximize
+
+$$
+\operatorname* { m a x } _ { z _ { 1 : q } \subset \mathbb { A } \times [ 0 , 1 ] ^ { m } } { \bf q } \mathrm { - c f K G } ( z _ { 1 : q } )
+$$
+
+Although we have defined the cfKG algorithm’s sampling decision theoretically, (3.3) or (3.5) are challenging optimization problems and naive brute-force approaches are unlikely to produce highquality results with a reasonable amount of computation. Thus, in the next section, we discuss efficient computational methods for solving (3.3) and (3.5).
+
+# 3.3 ENVELOPE-THEOREM-BASED COMPUTATIONAL METHOD
+
+In this section, we describe computational methods for solving (3.3) and (3.5). We describe our method in the context of (3.5), and observe that (3.3) is a special case.
+
+We generalize a recently proposed envelope-theorem based computational method developed for single-fidelity optimization in Wu et al. (2017), which is used to provide unbiased estimators of both $\mathbf { q }$ -cfKG and its gradient. We then use stochastic gradient ascent to optimize the q-cfKG acquistion function, optionally with multiple starts.
+
+# 3.3.1 ESTIMATING Q-CFKG
+
+To support computation, we express $\mu ^ { ( n + q ) } ( x , 1 _ { m } )$ that results from a chosen batch of points and fidelities $z ^ { ( n + 1 : n + q ) } = z _ { 1 : q }$ as
+
+$$
+\begin{array} { l l l } { \iota ^ { ( n + q ) } ( x , 1 _ { m } ) } & { = } & { \mu ^ { ( n ) } ( x , 1 _ { m } ) + } \\ & & { K ^ { ( n ) } ( ( x , 1 _ { m } ) , z _ { 1 : q } ) \left( K ^ { ( n ) } ( z _ { 1 : q } , z _ { 1 : q } ) + \sigma ^ { 2 } I \right) ^ { - 1 } \left( y ^ { ( n + 1 : n + q ) } - \mu ^ { ( n ) } ( z _ { 1 : q } ) \right) . } \end{array}
+$$
+
+Because $y ^ { ( n + 1 : n + q ) } \ - \ \mu ^ { ( n ) } ( z _ { 1 : q } )$ is normally distributed with zero mean and covariance matrix $( K ^ { ( n ) } ( z _ { 1 : q } , z _ { 1 : q } ) + \sigma ^ { 2 } I )$ with respect to the posterior after $n$ observations, we can rewrite $\boldsymbol { \mu } ^ { ( n + q ) } \left( \boldsymbol { x } , \boldsymbol { 1 } _ { m } \right)$ as
+
+$$
+\begin{array} { r l r } { \mu ^ { ( n + q ) } \left( x , 1 _ { m } \right) } & { = } & { \mu ^ { ( n ) } \left( x , 1 _ { m } \right) + \tilde { \sigma } _ { n } \left( x , z _ { 1 : q } \right) W _ { q } , } \end{array}
+$$
+
+where $W _ { q }$ is a standard $q$ -dimensional normal random vector, and
+
+$$
+\begin{array} { r c l } { \tilde { \sigma } _ { n } ( x , z _ { ( 1 : q ) } ) } & { = } & { K ^ { ( n ) } \left( ( x , 1 _ { m } ) , z _ { 1 : q } \right) \left( D ^ { ( n ) } \left( z _ { 1 : q } \right) ^ { T } \right) ^ { - 1 } , } \end{array}
+$$
+
+where $D ^ { ( n ) } \left( z _ { 1 : q } \right)$ is the Cholesky factor of the covariance matrix $K ^ { ( n ) } \left( z _ { 1 : q } , z _ { 1 : q } \right) + \sigma ^ { 2 } I .$
+
+Thus, to provide an unbiased Monte Carlo estimator of the expectation within (3.4), we may sample $W _ { q }$ , and then calculate $\begin{array} { r } { \operatorname* { m i n } _ { x ^ { \prime } \in \mathbb { A } } \mu ^ { ( n + q ) } ( x ^ { \prime } , 1 _ { m } ) = \operatorname* { m i n } _ { x ^ { \prime } \in \mathbb { A } } \mu ^ { ( n ) } \left( x , 1 _ { m } \right) + \tilde { \sigma } _ { n } \left( x , z _ { 1 : q } \right) W _ { q } } \end{array}$ . To do this optimization, we use a second-order continuous optimization method, where the gradient and Hessian of $\boldsymbol { \mu } ^ { ( n + q ) } \left( \boldsymbol { x } , \boldsymbol { 1 } _ { m } \right)$ with respect to $x$ in (3.6) can be computed by calculating the gradient and Hessian of $\boldsymbol { \mu } ^ { ( n ) } \left( \boldsymbol { x } , \boldsymbol { 1 } _ { m } \right)$ and $\tilde { \sigma } _ { n } \left( x , z _ { 1 : q } \right)$ . We can then compute the $\mathbf { q }$ -cfKG acquistion function to arbitrary accuracy by averaging many such independent Monte Carlo estimates.
+
+# 3.3.2 ESTIMATING THE GRADIENT OF Q-CFKG
+
+To solve (3.5), we generalize a recently proposed computational method based on the envelope theorem from $\mathrm { W u }$ et al. (2017) to provide an unbiased estimator of the gradient of the $\mathsf { q }$ -cfKG acquisition function, and then use stochastic gradient ascent.
+
+Exploiting (3.6), the q-cfKG acquisition function can be expressed as
+
+$$
+\mathsf { q } \mathsf { - } \mathsf { c f K G } ( z _ { 1 : q } ) = \frac { \operatorname* { m i n } _ { x \in \mathbb { A } } \mu ^ { ( n ) } \left( x , 1 _ { m } \right) - \mathbb { E } _ { n } \left[ \operatorname* { m i n } _ { x \in \mathbb { A } } \left( \mu ^ { ( n ) } \left( x , 1 _ { m } \right) + \tilde { \sigma } _ { n } \left( x , z _ { 1 : q } \right) W _ { q } \right) \right] } { \operatorname* { m a x } _ { 1 \leq i \leq q } \mathsf { c o s t } ^ { ( n ) } \left( x _ { i } , s _ { i } \right) } ,
+$$
+
+where $W _ { q }$ is a standard $q$ -dimensional normal random vector, $\begin{array} { r l r } { \tilde { \sigma } _ { n } ( x , z _ { ( 1 : q ) } ) } & { { } = } & { } \end{array}$ $K ^ { ( n ) } ( ( x , 1 _ { m } ) , z _ { 1 : q } ) ( D ^ { ( n ) } ( z _ { 1 : q } ) ^ { T } ) ^ { - 1 }$ , and $D ^ { ( n ) } \left( z _ { 1 : q } \right)$ is the Cholesky factor of the covariance matrix $K ^ { ( n ) } \left( z _ { 1 : q } , z _ { 1 : q } \right) ~ + ~ \sigma ^ { 2 } I .$ $\nabla \mathbf { q } \mathbf { - } \mathbf { c f K G } ( z _ { 1 : q } )$ can be computed from $\begin{array} { r l } {  { - \nabla \mathbb { E } _ { n } [ \operatorname* { m i n } _ { x \in \mathbb { A } } ( \mu ^ { ( n ) } ( x , 1 _ { m } ) + \tilde { \sigma } _ { n } ( x , z _ { 1 : q } ) W _ { q } ) ] } \quad } & { { } } \end{array}$ and $\begin{array} { r } { \nabla \operatorname* { m a x } _ { 1 \leq i \leq q } \operatorname { c o s t } ^ { ( n ) } \big ( z ^ { ( n + i ) } \big ) } \end{array}$ , where differentiability of $\mathrm { c o s t } ^ { ( n ) } ( \cdot )$ implies $\operatorname* { m a x } _ { 1 \leq i \leq q } \cos t ^ { ( n ) } ( x _ { i } , s _ { i } )$ is differentiable almost everywhere. To compute the first term, under sufficient regularity conditions (L’Ecuyer, 1990) that we conjecture hold in most applications to hyperparameter tuning, one can interchange the gradient and expectation operators,
+
+$$
+\nabla \mathbb { E } _ { n } \left[ \operatorname* { m i n } _ { x \in \mathbb { A } } \left( \mu ^ { ( n ) } \left( x , 1 _ { m } \right) + \widetilde { \sigma } _ { n } \left( x , z _ { 1 : q } \right) W _ { q } \right) \right] = \mathbb { E } _ { n } \left[ \nabla \operatorname* { m i n } _ { x \in \mathbb { A } } \left( \mu ^ { ( n ) } ( x , 1 _ { m } ) + \widetilde { \sigma } _ { n } \left( x , z _ { 1 : q } \right) W _ { q } \right) \right] .
+$$
+
+This technique is called infinitesimal perturbation analysis (IPA) (L’Ecuyer, 1990).
+
+Since multiplication, matrix inversion (when the inverse exists), and Cholesky factorization (Smith, 1995) preserve continuous differentiability, $( x , z _ { 1 : q } ) \mapsto \left( \mu ^ { ( n ) } ( x , 1 _ { m } ) + \tilde { \sigma } _ { n } ( x , z _ { 1 : q } ) W _ { q } \right)$ is continuously differentiable under mild regularity conditions. When this function is continuously differentiable and A is compact, the envelope theorem (Milgrom $\&$ Segal, 2002, Corollary 4) implies
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { n } \left[ \nabla \underset { x \in \mathbb { A } } { \mathrm { m i n } } \left( \mu ^ { ( n ) } ( x , 1 _ { m } ) + \tilde { \sigma } _ { n } \left( x , z _ { 1 : q } \right) W _ { q } \right) \right] } \\ & { = \mathbb { E } _ { n } \left[ \nabla \left( \mu ^ { ( n ) } \left( x ^ { * } ( W _ { q } ) , 1 _ { m } \right) + \tilde { \sigma } _ { n } \left( x ^ { * } ( W _ { q } ) , z _ { 1 : q } \right) \cdot W _ { q } \right) \right] , } \\ & { = \mathbb { E } _ { n } \left[ \nabla \tilde { \sigma } _ { n } \left( x ^ { * } ( W _ { q } ) , z _ { 1 : q } \right) \cdot W _ { q } \right] , } \end{array}
+$$
+
+where $\boldsymbol { x } ^ { * } ( W _ { q } ) \in \arg \operatorname* { m i n } _ { \boldsymbol { x } \in \mathbb { A } } \left( \mu ^ { ( n ) } ( \boldsymbol { x } , \boldsymbol { 1 } _ { m } ) + \tilde { \sigma } _ { n } ( \boldsymbol { x } , \boldsymbol { z } _ { 1 : q } ) W _ { q } \right)$ . We can use this unbiased gradient estimator within stochastic gradient ascent (Harold et al., 2003) to solve the optimization problem (3.5).
+
+# 3.4 EXTENSIONS TO DERIVATIVE-ENABLED SETTINGS
+
+Following $\mathbf { W } \mathbf { u }$ et al. (2017), which developed single-fidelity BO methods for use when gradient information is available, we develop a version of the cfKG algorithm, called derivative-enabled cfKG, that can be used when gradients are available in the continuous-fidelity setting. This can be used to reduce the number of function evaluations required to find a high-quality solution to (1.1).
+
+First, observe that a $G P ( \mu ^ { n } , K ^ { n } )$ prior on $g ( x , s )$ implies a multi-output $G P ( \tilde { \mu } ^ { n } , \tilde { K } ^ { n } )$ prior on $( g , \nabla _ { x } g )$ . Then, observe that the same reasoning we used to develop the cfKG acquistion function in (3.2) can be used when when we observe gradients to motivate the acquisition function,
+
+$$
+\mathtt { c f - d - K G } ( x , s ) = \frac { \operatorname* { m i n } _ { x \in \mathbb { A } } \tilde { \mu } _ { 1 } ^ { ( n ) } ( x , 1 _ { m } ) - \mathbb { E } _ { n } \left[ \operatorname* { m i n } _ { x \in \mathbb { A } } \tilde { \mu } _ { 1 } ^ { ( n + 1 ) } ( x , 1 _ { m } ) \mid x ^ { ( n + 1 ) } = x , s ^ { ( n + 1 ) } ) = s \right] } { \cos \mathfrak { c } ^ { ( n ) } ( x , s ) } ,
+$$
+
+where $\cos \mathbf { t } ^ { ( n ) } ( x , s )$ is now our estimate after $n$ samples of the cost of evaluating both $g$ and its gradient with respect to $x$ , and $\tilde { \mu } _ { 1 } ^ { ( n ) } ( x , s )$ is the posterior mean on $g ( x , s )$ in the multi-output GP. The conditional expectation is taken with respect to both the observed function value and gradient. A batch version of the derivative-enabled cfKG acquistion function can be defined analogously. We use techniques similar to those described in Sect. 3.3 to optimize these acquisition functions.
+
+# 4 NUMERICAL EXPERIMENTS
+
+In this section, we compare sequential, batch, and derivative-enabled cfKG with several benchmark algorithms on four synthetic functions, tuning convolutional neural networks on CIFAR-10 and SVHN, and on large-scale kernel learning. Our benchmarks include the traditional Bayesian optimization algorithms KG (Wu & Frazier, 2016) and EI (Jones et al., 1998), and the multi-fidelity Bayesian optimization with continuous approximation algorithm (BOCA) (Kandasamy et al., 2017). We also compare with Hyperband (Li et al., 2016) in the CIFRA-10 experiment, and with derivativeenabled KG (Wu et al., 2017) in the kernel-learning experiment. We use squared-exponential kernels with constant mean functions and integrate out the GP hyperparameters by sampling $M = 1 0$ sets of hyperparameters using the emcee package (Foreman-Mackey et al., 2013).
+
+# 4.1 OPTIMIZING SYNTHETIC FUNCTIONS
+
+Inspired by numerical experiments in Kandasamy et al. (2017), we modify four standard synthetic test functions, 2-d Branin, 3-d Rosenbrock, 3-d Hartmann, and 6-d Hartmann, by adding 1 fidelity control, as described in detail in Appendix. A. We use cost $\textstyle \operatorname { c o s t } ( x , s ) : = \prod _ { i = 1 } ^ { \bar { m } } ( 0 . 0 1 + s _ { i } )$ to mimic the linear cost of training a neural network with more data and longer iterations, and $\begin{array} { r } { \mathrm { c o s t } ( z _ { 1 : q } ) : = \mathrm { m a x } _ { 1 \leq j \leq q } \mathrm { c o s t } ( z _ { j } ) } \end{array}$ to mimic wall-clock time for training $q$ neural networks in parallel. Fig. 1 summarizes the results. cfKG achieves lower simple regret than competitors at the same cost.
+
+# 4.2 TUNING CONVOLUTIONAL NEURAL NETWORKS ON CIFAR-10 AND SVHN
+
+We use sequential and batch cfKG to tune convolution neural networks (CNNs) on CIFAR-10 and SVHN. Our CNN consists of 3 convolutional blocks and a softmax classification layer. Each convolutional block consists of two convolutional layers with the same number of filters followed by a max-pooling layer. There is no dropout or batch-normalization layer. We split the CIFAR-10 dataset into 40000 training samples, 10000 validation samples and 10000 test samples. We split the SVHN training dataset into 67235 training samples and 6000 validation samples, and use the standard 26032 test samples. We apply standard data augmentation: horizontal and vertical shifts, and horizontal flips. We optimize 5 hyperparameters to minimize the classification error on the validation set: the learning rate, batch size, and number of filters in each convolutional block. cfKG and $\mathbf { q }$ -cfKG use two fidelity controls: the size of the training set and the number of training iterations. Hyperband uses the size of the training set as its resource (it can use only one resource or fidelity), using a bracket size of $s _ { \operatorname* { m a x } } = 4$ as in Li et al. (2016) and the maximum resource allowed by a single configuration set to 40000. We set the maximum number of training epochs for all algorithms to 50 for CIFAR-10 and 40 for SVHN. Fig. 2 shows the performance of cfKG relative to several benchmarks. cfKG successfully exploits the cheap approximations and find a good solution much faster than KG and Hyperband. When we train using optimized hyperparameters on the full training dataset for 200 epochs, test data classification error is $\sim 1 2 \%$ for CIFAR-10 and $\sim 5 \%$ for SVHN.
+
+![](images/08281fbda24d24ce93e41224a3219369ec0988b387e4f0eb1f4f7b0c2407a9e0.jpg)  
+Figure 1: Optimizing 4 synthetic functions: 2-d Branin, 3-d Rosenbrock, 3-d Hartmann, and 6-d Hartmann with 20 independent runs. Fidelity is 1-d.
+
+# 4.3 LARGE-SCALE KERNEL LEARNING
+
+We use derivative-enabled cfKG (cf-d-KG) in a large-scale kernel learning example obtained by modifying the 1-d demo example for KISS-GP (Wilson & Nickisch, 2015) on the GPML website (Rasmussen & Nickisch, 2016). In this example, we optimize 3 hyperparameters (the alpha, the length scale and the variance of the noise) of a GP with an RBF kernel on 1 million training points to maximize the log marginal likelihood. We evaluate both the log marginal likelihood and its gradient using the KISS-GP framework. We use two fidelity controls: the number of training points and the number of inducing points in the KISS-GP framework. We set the maximum number of inducing points to $m = 1 0 0 0$ . We compare cf-d-KG to the derivative-enabled knowledge gradient (d-KG) (Wu et al., 2017) both with batch size $q = 8$ . Fig. 2 shows that cf-d-KG successfully utilizes the inexpensive function and gradient evaluations to find a good solution more quickly.
+
+![](images/580470d835d17f9de2683c6601f0aeb8d69bb5c35eaa24f6276dd39aa59cf605.jpg)  
+Figure 2: From left to right: Tuning convolutional neural networks on CIFAR-10 and SVHN with 6 independent runs, and kernel learning under the KISS-GP framework with 10 independent runs.
+
+# 5 CONCLUSION
+
+We propose a novel continuous-fidelity BO algorithm, cfKG, which generalizes naturally to batch and derivative settings. This algorithm can find good solutions to global optimization problems with less cost than state-of-art algorithms in applications including deep learning and kernel learning.
+
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+
+# A ADDITIONAL EXPERIMENTAL DETAILS
+
+Here we define in detail the synthetic test functions on which we perform numerical experiments described in Sect. 4.1. The test functions are:
+
+$$
+\begin{array} { r c l } { \mathrm { a u g m e n t e d - B r a m i n } ( x , s ) } & { = } & { \displaystyle \left( x _ { 2 } - \left( \frac { 5 . 1 } { 4 \pi ^ { 2 } } - 0 . 0 0 1 * ( 1 - s _ { 1 } ) \right) x _ { 1 } ^ { 2 } + \frac { 5 } { \pi } x _ { 1 } - 6 \right) ^ { 2 } } \\ & & { \displaystyle + 1 0 * \left( 1 - \frac { 1 } { 8 \pi } \right) \cos ( x _ { 1 } ) + 1 0 } \\ { \mathrm { a g m e n t e d - H a r t m a m } ( x , s ) } & { = } & { \displaystyle ( \alpha _ { 1 } - 0 . 0 1 * ( 1 - s _ { 1 } ) ) \exp \left( - \sum _ { j = 1 } ^ { d } A _ { i j } ( x _ { j } - P _ { \mathrm { i j } } ) ^ { 2 } \right) } \\ & & { \displaystyle + \sum _ { i = 2 } ^ { d } \alpha _ { i } \exp \left( - \sum _ { j = 1 } ^ { d } A _ { i j } ( x _ { j } - P _ { \mathrm { i j } } ) ^ { 2 } \right) } \\ { \mathrm { b a r e n t e d - R o s e n b r o c k } ( x , s ) } & { = } & { \displaystyle \sum _ { i = 1 } ^ { 2 } \left( 1 0 0 * ( x _ { i + 1 } - x _ { i } ^ { 2 } + 0 . 0 0 1 * ( 1 - s _ { 1 } ) ) ^ { 2 } + ( x _ { i } - 1 ) ^ { 2 } \right) . } \end{array}
+$$

parse/train/SknC0bW0-/SknC0bW0-_content_list.json

@@ -0,0 +1,1162 @@
+[
+    {
+        "type": "text",
+        "text": "CONTINUOUS-FIDELITY BAYESIAN OPTIMIZATION WITH KNOWLEDGE GRADIENT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            250
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While Bayesian optimization (BO) has achieved great success in optimizing expensive-to-evaluate black-box functions, especially tuning hyperparameters of neural networks, methods such as random search (Li et al., 2016) and multifidelity BO (e.g. Klein et al. (2017)) that exploit cheap approximations, e.g. training on a smaller training data or with fewer iterations, can outperform standard BO approaches that use only full-fidelity observations. In this paper, we propose a novel Bayesian optimization algorithm, the continuous-fidelity knowledge gradient (cfKG) method, that can be used when fidelity is controlled by one or more continuous settings such as training data size and the number of training iterations. cfKG characterizes the value of the information gained by sampling a point at a given fidelity, choosing to sample at the point and fidelity with the largest value per unit cost. Furthermore, cfKG can be generalized, following Wu et al. (2017), to settings where derivatives are available in the optimization process, e.g. large-scale kernel learning, and where more than one point can be evaluated simultaneously. Numerical experiments show that cfKG outperforms state-of-art algorithms when optimizing synthetic functions, tuning convolutional neural networks (CNNs) on CIFAR-10 and SVHN, and in large-scale kernel learning. ",
+        "bbox": [
+            233,
+            266,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            531,
+            334,
+            547
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In hyperparameter tuning of machine learning models, we seek to find a set of hyperparameters $x$ in some set A to minimize the validation error $f ( x )$ , i.e., to solve ",
+        "bbox": [
+            171,
+            564,
+            823,
+            592
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f864310dd6c088c8f4a2d93c10743fd08c231f9f7be127ac71509ced9c96062.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathbb { A } } f ( x )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            465,
+            601,
+            532,
+            625
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Evaluating $f ( x )$ can take substantial time and computational power (Bergstra & Bengio, 2012), and may not provide gradient evaluations. Thus, machine learning practitioners have turned to Bayesian optimization for solving (1.1) (Snoek et al., 2012) because it tends to find good solutions with few function evaluations (Jones et al., 1998). ",
+        "bbox": [
+            174,
+            633,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As the computational expense of training and testing a modern deep neural network for a single set of hyperparameters has grown as long as days or weeks, it has become natural to seek ways to solve (1.1) more quickly by supplanting some evaluations of $f ( x )$ with computationally inexpensive lowfidelity approximations. Indeed, when training a neural network or most other machine learning models, we can approximate $f ( x )$ by training on less than the full training data, or using fewer training iterations. Both of these controls on fidelity can be set to achieve either better accuracy or lower computational cost across a range of values reasonably modeled as continuous. ",
+        "bbox": [
+            174,
+            695,
+            825,
+            792
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we consider optimization with evaluations of multiple fidelities and costs where the fidelity is controlled by one or more continuous parameters. We model these evaluations by a realvalued function $g ( x , s )$ where $f ( x ) : = g ( x , 1 _ { m } )$ and $s \\in [ 0 , 1 ] ^ { m }$ denotes the $m$ fidelity-control parameters. $g ( x , s )$ can be evaluated, optionally with noise, at a cost that depends on $x$ and $s$ . In the context of hyperparameter tuning, we may take $m = 2$ and let $g ( x , s _ { 1 } , s _ { 2 } )$ denote the loss on the validation set when training using hyperparameters $x$ with a fraction $s _ { 1 }$ of the training data and a fraction $s _ { 2 }$ of some maximum allowed number of training iterations. We may also set $m = 1$ and let $s$ index either training data or training iterations. We assume A is a compact connected uncountable set into which it is easy to project, such as a hyperrectangle. ",
+        "bbox": [
+            174,
+            799,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This problem setting also appears outside of hyperparameter tuning, in any application where the objective is expensive to evaluate and we may observe cheap low-fidelity approximations parameterized by a continuous vector. For example, when optimizing a system evaluated via a Monte Carlo simulator, we can evaluate a system configuration approximately by running with fewer replications. Also, when optimizing an engineering system modeled by a partial differential equation (PDE), we can evaluate a system configuration approximately by solving the PDE using a coarse grid. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            186
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Given this problem setting, we use the knowledge gradient approach (Frazier et al., 2009) to design an algorithm to adaptively select the hyperparameter configuration and fidelity to evaluate, to best support solving (1.1). By generalizing a computational technique based on the envelope theorem first developed in Wu et al. (2017), our algorithm supports parallel function evaluations, and also can take advantage of derivative observations when they are available. This algorithm chooses the point or set of points to evaluate next that maximizes the ratio of the value of information from evaluation against its cost. ",
+        "bbox": [
+            174,
+            193,
+            825,
+            290
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unlike most existing work on discrete- and continuous-fidelity Bayesian optimization, our approach considers the impact of our measurement on the future posterior distribution over the full feasible domain, while existing expected-improvement-based approaches consider its impact at only the point evaluated. One exception is the entropy-search-based method [10], which also considers the impact over the full posterior. Our approach differs from entropy search in that it chooses points to sample to directly minimize expected simple regret, while entropy search seeks to minimize the entropy of the location or value of the global optimizer, indirectly reducing simple regret. ",
+        "bbox": [
+            174,
+            296,
+            825,
+            393
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We summarize our contributions as follows. ",
+        "bbox": [
+            176,
+            401,
+            462,
+            415
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contributions of this paper: ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            421,
+            369,
+            435
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We develop a novel Bayesian Optimization algorithm, the continuous-fidelity knowledge gradient (cfKG) method, which chooses the point and fidelity to sample next that maximizes the ratio of the value of information to its cost; • After first developing this algorithm in the sequential derivative-free setting, and inspired by Wu et al. (2017), we generalize to settings where function evaluations may be performed in parallel, and where we can access possibly noisy and biased gradient information. • We show that our algorithm outperforms a number of start-of-art benchmark algorithms when optimizing common synthetic functions, tuning convolutional neural networks on CIFAR-10 and SVHN, and in a large-scale kernel learning example. ",
+        "bbox": [
+            214,
+            448,
+            825,
+            583
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The rest of the paper is organized as follows. Sect. 2 reviews related work. Sect. 3 presents the cfKG method. Sect. 4 tests cfKG on benchmarks including synthetic functions and hyperparameter tuning for deep learning and kernel learning. Sect. 5 concludes. ",
+        "bbox": [
+            178,
+            594,
+            825,
+            637
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            656,
+            341,
+            672
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Algorithms exploiting inexpensive low-fidelity approximations for hyperparameter tuning have been proposed both within and outside the field of Bayesian optimization. ",
+        "bbox": [
+            173,
+            689,
+            821,
+            717
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Outside of Bayesian optimization, Li et al. (2016) develops an early-stopping method called Hyperband that can outperform traditional Bayesian optimization when tuning hyperparameters within a random search framework by adaptively allocating a single predefined resource that can be taken to be the number of training iterations or the amount of training data. In contrast to Hyperband we allow more than one fidelity control, support derivative observations, and use one-step optimality in terms of value vs. cost to select the point or points to sample. We compare against Hyperband in our numerical experiments, and show improved performance training a convolutional neural network. ",
+        "bbox": [
+            174,
+            723,
+            825,
+            820
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Using a modified expected improvement (EI) acquisition function, Huang et al. (2006) and Lam et al. (2015) develop methods for multi-fidelity settings with a finite discrete set of low-fidelity approximations. We call this the “discrete-fidelity setting” to contrast it with the continuous-fidelity setting we consider here. We do not compare against discrete-fidelity methods in our numerical experiments as both Kandasamy et al. (2017) and Klein et al. (2017) show that continuous-fidelity methods can find good solutions in continuous-fidelity problems as much as an order of magnitude faster than discrete-fidelity ones. ",
+        "bbox": [
+            174,
+            827,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the specific context of hyperparameter tuning, Domhan et al. (2015) proposes a simple early stopping criterion combined with SMAC (an EI acquisition function with a random forest based statistical model) in which evaluation of a configuration is stopped if its performance is predicted to be worse than the current best configuration. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Kandasamy et al. (2016) generalizes the upper confidence bound (UCB) criteria to the discretefidelity setting, and further generalizes to continuous fidelities in Kandasamy et al. (2017). We compare against Kandasamy et al. (2017) in our numerical experiments, finding that cfKG provides improved performance when the budget is sufficiently large. This may be because the UCB criteria was originally designed for minimizing the cumulative regret and not simple regret, with the simple regret bounds resulting from theoretical analysis of cumulative regret being loose enough to be consistent with the empirical performance gap we observe in our experiments. It may also be because both Kandasamy et al. (2016) and Kandasamy et al. (2017) use a two-stage process in which the point to evaluate is selected without considering the fidelity, and the fidelity only selecting afterward. In contrast, cfKG selects the point and fidelity jointly, and considers the impact of fidelity choice on the best point to sample. ",
+        "bbox": [
+            174,
+            166,
+            825,
+            318
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Entropy search has also been extended to the discrete-fidelity setting by Swersky et al. (2013). Considering training iterations as a one-dimensional continuous fidelity control, Swersky et al. (2014) generalizes Swersky et al. (2013) to this specific continuous-fidelity setting. Klein et al. (2017) adapts entropy search to settings where the size of the training data is a one-dimensional continuous fidelity control. McLeod et al. (2017) extends predictive entropy search to settings with one fidelity control and provides a faster way to compute the acquisition function. ",
+        "bbox": [
+            174,
+            324,
+            823,
+            407
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Poloczek et al. (2017) develops a knowledge gradient method for the discrete-fidelity setting, but does not consider the continuous-fidelity setting. Our computational techniques are quite different from the ones developed there, as necessitated by our consideration of continuous fidelities. As an advantageous byproduct, our computational techniques also avoid the need to discrete A. ",
+        "bbox": [
+            174,
+            414,
+            823,
+            469
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This paper proposes the first knowledge gradient method for continuous-fidelity settings, and show how to generalize it to the batch and derivative-enabled settings by generalizing the computational technique developed in Wu et al. (2017). To the best of our knowledge, this is the first multi-fidelity batch Bayesian optimization algorithm. The relationship of this paper to the existing multi-fidelity BO literature is summarized in Table. 1. ",
+        "bbox": [
+            174,
+            477,
+            823,
+            545
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/abbd47b80ff52996b43e9008d934deb98e4b187290baf7ba8709de75c3bd1089.jpg",
+        "table_caption": [
+            "Table 1: Summary of the multi-fidelity BO literature. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Literature summary</td><td rowspan=1 colspan=1>EI/SMAC</td><td rowspan=1 colspan=1>Entropy Search</td><td rowspan=1 colspan=1>UCB</td><td rowspan=1 colspan=1>KnowledgeGradient</td></tr><tr><td rowspan=1 colspan=1>Discrete-fidelity</td><td rowspan=1 colspan=1>Huang et al. (2006);Lam et al. (2015)</td><td rowspan=1 colspan=1>Swersky et al. (2013)</td><td rowspan=1 colspan=1>Kandasamyet al. (2016)</td><td rowspan=1 colspan=1>Poloczeket al. (2017)</td></tr><tr><td rowspan=1 colspan=1>Continuous-fidelity</td><td rowspan=1 colspan=1>Domhan et al. (2015)</td><td rowspan=1 colspan=1>Swersky et al. (2014);Klein et al.(2017)；McLeod et al. (2017)</td><td rowspan=1 colspan=1>Kandasamyet al. (2017)</td><td rowspan=1 colspan=1>This paper</td></tr></table>",
+        "bbox": [
+            173,
+            554,
+            820,
+            655
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our method is also related to knowledge gradient methods for single-fidelity BO. In particular, the way in which we extend cfKG from the sequential (one evaluation at a time) derivative-free setting to allow batches of points and gradient observations follows work extending single-fidelity knowledge-gradient methods to the batch and derivative settings in Wu & Frazier (2016) and $\\mathrm { W u }$ et al. (2017). We also generalize the envelope-theorem based computational technique developed for the single-fidelity setting in Wu et al. (2017) to continuous fidelities. We compare with single-fidelity knowledge gradient methods in our numerical experiments, and show that cfKG is able to levarage inexpensive low-fidelity observations to provide improved performance in both derivative-free and derivative-enabled settings. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 CONTINUOUS-FIDELITY KNOWLEDGE GRADIENT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            838,
+            609,
+            853
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we propose the continuous-fidelity knowledge gradient (cfKG), a novel Bayesian optimization algorithm that exploits inexpensive low-fidelity approximations. This algorithm, like most Bayesian optimization algorithms, consists of a statistical model and an acquisition function. cfKG proceeds iteratively by fitting the statistical model (described below in Section 3.1) to all previously sampled (point, fidelty) pairs, and then finding and sampling the (point, fidelity) pair that maximizes the acquisition function. Once the budget is exhausted, at some iteration $N$ , cfKG returns as its final solution the point with the lowest estimated $g ( x , 1 _ { m } )$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To describe cfKG in detail, Sect. 3.1 first describes Gaussian process regression for modeling both $g ( x , s )$ and its cost of evaluation. This approach is standard, with the novelty arising in cfKG’s acquisition function and how we optimize it. Then, Sect. 3.2 presents the cfKG acquisition function, which values sampling a (point, fidelity) pair according to the ratio of the value of the information gained from sampling that point at that fidelity, to the cost of doing so. Sect. 3.3 generalizes an envelope-theorem based computational technique developed in $\\mathrm { W u }$ et al. (2017) to efficiently maximize this acquisition function. We discuss extensions to the derivative-enabled setting in Sect. 3.4. ",
+        "bbox": [
+            173,
+            151,
+            825,
+            250
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 GAUSSIAN PROCESSES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            268,
+            372,
+            282
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We put a Gaussian process (GP) prior (Rasmussen & Williams, 2006) on the function $g$ or its logarithm. We describe this procedure placing the prior on $g$ directly, and then discuss below when we recommend instead placing it on $( { \\bar { x } } , s ) \\mapsto \\log { \\bar { g } } ( x , s )$ . The GP prior is defined by its mean function $\\mu ^ { ( 0 ) } : \\mathbb { A } \\times [ 0 , 1 ] ^ { m } \\mapsto \\mathbb { R }$ and kernel function $K ^ { ( 0 ) } : \\{ \\mathbb { A } \\times [ 0 , 1 ] ^ { m } \\} \\times \\{ \\mathbb { A } \\times [ 0 , 1 ] ^ { m } \\} \\mapsto \\mathbb { R }$ . These mean and kernel functions have hyperparameters, whose inference we discuss below. ",
+        "bbox": [
+            174,
+            295,
+            825,
+            367
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We assume that evaluations of $g ( x , s )$ are subject to additive independent normally distributed noise with common variance $\\sigma ^ { 2 }$ . We treat the parameter $\\sigma ^ { 2 }$ as a hyperparameter of our model, and also discuss its inference below. Our assumption of normally distributed noise with constant variance is common in the BO literature (Klein et al., 2017). ",
+        "bbox": [
+            174,
+            372,
+            825,
+            429
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The posterior distribution of $g$ after $n$ function evaluations at points $\\begin{array} { r l r } { z ^ { ( 1 : n ) } } & { { } : = } & { } \\end{array}$ $\\{ ( x ^ { ( 1 ) } , s ^ { ( 1 ) } ) , ( x ^ { ( 2 ) } , s ^ { ( 2 ) } ) , \\cdot \\cdot \\cdot , ( x ^ { ( n ) } , s ^ { ( n ) } ) \\}$ with observed values $y ^ { ( 1 : n ) } : = \\{ y ^ { ( 1 ) } , y ^ { ( 2 ) } , \\cdot \\cdot \\cdot , y ^ { ( n ) } \\}$ remains a Gaussian process (Rasmussen & Williams, 2006), and $g \\mid z ^ { ( 1 : n ) } , y ^ { ( 1 : n ) } \\sim \\mathbf { G P } ( \\mu ^ { n } , K ^ { ( n ) } )$ with $\\mu ^ { n }$ and $K ^ { ( n ) }$ evaluated at a point $z = ( x , s )$ (or pair of points $z$ , $\\tilde { z } = ( \\tilde { x } , \\tilde { s } ) )$ given as follows ",
+        "bbox": [
+            174,
+            435,
+            823,
+            497
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/130e271be08460c6e5eca20407ea6c2f8027be6ceb5466c73d038662700926bd.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { \\mu ^ { ( n ) } ( z ) = \\mu ^ { ( 0 ) } \\left( z \\right) + K ^ { ( 0 ) } \\left( z , z ^ { ( 1 : n ) } \\right) \\left( K ^ { ( 0 ) } ( z ^ { ( 1 : n ) } , z ^ { ( 1 : n ) } ) + \\sigma ^ { 2 } I \\right) ^ { - 1 } \\left( y ^ { ( 1 : n ) } - \\mu ( z ^ { ( 1 : n ) } ) \\right) , } \\\\ & { } & { K ^ { ( n ) } \\left( z , \\tilde { z } \\right) = K ^ { ( 0 ) } ( z , \\tilde { z } ) - K ^ { ( 0 ) } \\left( z , z ^ { ( 1 : n ) } \\right) \\left( K ^ { ( 0 ) } ( z ^ { ( 1 : n ) } , z ^ { ( 1 : n ) } ) + \\sigma ^ { 2 } I \\right) ^ { - 1 } K ^ { ( 0 ) } \\left( z ^ { ( 1 : n ) } , \\tilde { z } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            506,
+            816,
+            568
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This statistical approach contains several hyperparameters: the variance $\\sigma ^ { 2 }$ , and any parameters in the mean and kernel functions. We treat these hyperparameters in a Bayesian way as proposed in Snoek et al. (2012). ",
+        "bbox": [
+            174,
+            598,
+            823,
+            640
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When $g$ is the validation error in a hyperparameter optimization problem, we recommend putting a GP prior on $\\log g ( x , s )$ , rather than on $g ( x , s )$ directly, because (1) $g ( x , s )$ is nonnegative and will be allowed to be negative after log scaling, better matching the range of values assumed by the GP, and (2) because $g ( x , s )$ can climb steeply over several orders of magnitude as we move away from the optimal $x$ , making $\\log g ( x , s )$ easier to model. ",
+        "bbox": [
+            174,
+            646,
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+            717
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We analogously train a separate GP on the logarithm of the cost of evaluating $g ( x , s )$ ",
+        "bbox": [
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+            737
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 THE CFKG ACQUISITION FUNCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            756,
+            464,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "cfKG samples the point and fidelity that jointly maximize an acquisition function, which we define in this section by adopting the knowledge gradient concept (Frazier et al., 2009) in the continuousfidelity setting to value the information gained through one additional sample. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "If we were to stop sampling after $n$ samples, we would select as our solution to (1.1) a point $x$ with minimum estimated validation error $\\mu ^ { ( n ) } ( x , 1 _ { m } )$ , and this point would have a conditional expected validation error of $\\begin{array} { r } { \\operatorname* { m i n } _ { x \\in \\mathbb { A } } \\mu ^ { ( n ) } ( x , 1 _ { m } ) } \\end{array}$ under the posterior. If instead we took an additional sample at $x ^ { ( n + 1 ) }$ with the fidelity $s ^ { ( n + 1 ) }$ , then the minimum expected validation error under the resulting posterior would become minx∈A µ(n+1)(x, 1m). This quantity depends on x(n+1) and s(n+1) through the dependence of $\\boldsymbol { \\mu } ^ { ( n + 1 ) } ( x , \\boldsymbol { 1 } _ { m } )$ on the point and fidelity sampled, and is random under the posterior at iteration $n$ because $\\boldsymbol { \\mu } ^ { ( n + 1 ) } ( x , \\boldsymbol { 1 } _ { m } )$ depends on the observation $y ^ { ( n + 1 ) }$ . We discuss this dependence explicitly in Sect. 3.3. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            102,
+            823,
+            132
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The value of the information gained by sampling at $x ^ { ( n + 1 ) }$ with the fidelity $s ^ { ( n + 1 ) }$ conditioned on any particular outcome $y ^ { ( n + 1 ) }$ is thus the difference of these two expected validation errors $\\begin{array} { r } { \\operatorname* { m i n } _ { x \\in \\mathbb { A } } \\mu ^ { ( n ) } ( x , 1 _ { m } ) - \\operatorname* { m i n } _ { x \\in \\mathbb { A } } \\mu ^ { ( n + 1 ) } ( x , 1 _ { m } ) } \\end{array}$ . We then take the expectation of this difference, over the random outcome $y ^ { ( n + 1 ) }$ , to obtain the (unconditional) value of the information gained, and take the ratio of this value with the cost of obtaining it to obtain the cfKG acquistion function, ",
+        "bbox": [
+            173,
+            138,
+            825,
+            214
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d7d32e0f24d645dd7a907d6b82006c88d36b77c08c76b105972b62955be7829.jpg",
+        "text": "$$\n\\operatorname { c f K G } ( x , s ) = { \\frac { \\operatorname* { m i n } _ { x ^ { \\prime } \\in \\mathbb { A } } \\mu ^ { ( n ) } ( x ^ { \\prime } , 1 _ { m } ) - \\mathbb { E } _ { n } \\left[ \\operatorname* { m i n } _ { x ^ { \\prime } \\in \\mathbb { A } } \\mu ^ { ( n + 1 ) } ( x ^ { \\prime } , 1 _ { m } ) \\mid x ^ { ( n + 1 ) } = x , s ^ { ( n + 1 ) } = s \\right] } { \\cot ^ { ( n ) } ( x , s ) } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            171,
+            218,
+            805,
+            256
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { c o s t } ^ { ( n ) } ( x , s )$ is the estimated cost of evaluating at $x$ with the fidelity $s$ based on the observations available at iteration $n$ , according to the GP described in Sect. 3.1, and $\\mathbb { E } _ { n }$ indicates the expectation taken with respect to the posterior given $\\boldsymbol { x } ^ { ( 1 : n ) } , \\boldsymbol { s } ^ { ( 1 : n ) } , \\boldsymbol { y } ^ { ( 1 : n ) }$ . ",
+        "bbox": [
+            173,
+            261,
+            823,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The cfKG algorithm chooses to sample at the point and fidelity that jointly maximize the cfKG acquistion function ",
+        "bbox": [
+            173,
+            311,
+            820,
+            340
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/60ff86cda5c185cb919f5e519366c7e8cc6692051da654f0449cd43e6a92ba65.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { ( x , s ) \\in \\mathbb { A } \\times [ 0 , 1 ] ^ { m } } { \\mathrm { c f K G } } ( x , s ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            338,
+            586,
+            363
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Although this acquisition function considers the expected value of an improvement due to sampling, it differs from expected improvement approaches such as Lam et al. (2015) because the point at which an improvement occurs, $\\operatorname { a r g m a x } _ { x \\in \\mathbb { A } } \\mu ^ { ( n + 1 ) } ( x , 1 _ { m } )$ may differ from the point sampled. Moreover, this acquisition function allows joint valuation of both the point $x$ and the fidelity $s$ , while approaches such as Lam et al. (2015) require valuing a point $x$ assuming it will be evaluated at full fidelity and then choose the fidelity in a second stage. ",
+        "bbox": [
+            173,
+            372,
+            825,
+            458
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "cfKG generalizes naturally to batch settings where we can evaluate multiple (point, fidelity) pairs at once. We value joint evaluation of $q \\geq 1$ points $x _ { 1 : q }$ at fidelities $s _ { 1 : q }$ , where $z _ { 1 : q } =$ $\\mathsf { \\bar { ( } } ( x _ { 1 } , s _ { 1 } ) , \\ldots , ( x _ { q } , s _ { q } ) )$ , by ",
+        "bbox": [
+            173,
+            463,
+            823,
+            507
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43e8e037b27197aed398e7861356b90f751f1e1512c306c151f60917582df1a1.jpg",
+        "text": "$$\n\\mathsf { q \\mathrm { - } \\mathrm { c f K G } } ( z _ { 1 : q } ) = \\frac { \\operatorname* { m i n } _ { x ^ { \\prime } \\in \\mathbb { A } } \\mu ^ { ( n ) } ( x ^ { \\prime } , 1 _ { m } ) - \\mathbb { E } _ { n } \\left[ \\operatorname* { m i n } _ { x ^ { \\prime } \\in \\mathbb { A } } \\mu ^ { ( n + q ) } ( x ^ { \\prime } , 1 _ { m } ) \\mid z ^ { ( n + 1 : n + q ) } = z _ { 1 : q } \\right] } { \\operatorname* { m a x } _ { 1 \\leq i \\leq q } \\mathrm { c o s t } ^ { ( n ) } ( z ^ { ( n + i ) } ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            510,
+            803,
+            549
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We then modify (3.3) by sampling at the batch of points and fidelities that maximize ",
+        "bbox": [
+            174,
+            559,
+            725,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/91f9777fde39510a1180bec589ce2353128187a2d81b7c5abf838697112900fe.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { z _ { 1 : q } \\subset \\mathbb { A } \\times [ 0 , 1 ] ^ { m } } { \\bf q } \\mathrm { - c f K G } ( z _ { 1 : q } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            579,
+            573,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Although we have defined the cfKG algorithm’s sampling decision theoretically, (3.3) or (3.5) are challenging optimization problems and naive brute-force approaches are unlikely to produce highquality results with a reasonable amount of computation. Thus, in the next section, we discuss efficient computational methods for solving (3.3) and (3.5). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 ENVELOPE-THEOREM-BASED COMPUTATIONAL METHOD ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            689,
+            607,
+            703
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe computational methods for solving (3.3) and (3.5). We describe our method in the context of (3.5), and observe that (3.3) is a special case. ",
+        "bbox": [
+            171,
+            713,
+            823,
+            742
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We generalize a recently proposed envelope-theorem based computational method developed for single-fidelity optimization in Wu et al. (2017), which is used to provide unbiased estimators of both $\\mathbf { q }$ -cfKG and its gradient. We then use stochastic gradient ascent to optimize the q-cfKG acquistion function, optionally with multiple starts. ",
+        "bbox": [
+            173,
+            748,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3.1 ESTIMATING Q-CFKG ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            819,
+            382,
+            834
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To support computation, we express $\\mu ^ { ( n + q ) } ( x , 1 _ { m } )$ that results from a chosen batch of points and fidelities $z ^ { ( n + 1 : n + q ) } = z _ { 1 : q }$ as ",
+        "bbox": [
+            174,
+            842,
+            823,
+            875
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/236069233dff41e3ab7262535514d953664196caed103c70969ccda8f9dbd776.jpg",
+        "text": "$$\n\\begin{array} { l l l } { \\iota ^ { ( n + q ) } ( x , 1 _ { m } ) } & { = } & { \\mu ^ { ( n ) } ( x , 1 _ { m } ) + } \\\\ & & { K ^ { ( n ) } ( ( x , 1 _ { m } ) , z _ { 1 : q } ) \\left( K ^ { ( n ) } ( z _ { 1 : q } , z _ { 1 : q } ) + \\sigma ^ { 2 } I \\right) ^ { - 1 } \\left( y ^ { ( n + 1 : n + q ) } - \\mu ^ { ( n ) } ( z _ { 1 : q } ) \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            878,
+            826,
+            929
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Because $y ^ { ( n + 1 : n + q ) } \\ - \\ \\mu ^ { ( n ) } ( z _ { 1 : q } )$ is normally distributed with zero mean and covariance matrix $( K ^ { ( n ) } ( z _ { 1 : q } , z _ { 1 : q } ) + \\sigma ^ { 2 } I )$ with respect to the posterior after $n$ observations, we can rewrite $\\boldsymbol { \\mu } ^ { ( n + q ) } \\left( \\boldsymbol { x } , \\boldsymbol { 1 } _ { m } \\right)$ as ",
+        "bbox": [
+            173,
+            102,
+            825,
+            151
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/974460571647175ceb2e4cc718e9d87da6bc1f36ce1798120efb14615c667bcd.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\mu ^ { ( n + q ) } \\left( x , 1 _ { m } \\right) } & { = } & { \\mu ^ { ( n ) } \\left( x , 1 _ { m } \\right) + \\tilde { \\sigma } _ { n } \\left( x , z _ { 1 : q } \\right) W _ { q } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            160,
+            674,
+            183
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { q }$ is a standard $q$ -dimensional normal random vector, and ",
+        "bbox": [
+            173,
+            189,
+            606,
+            204
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7f0d3706c2b45b76f94845e781b40a40192cedf1f3f2df64f591d34efe8e7f02.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\tilde { \\sigma } _ { n } ( x , z _ { ( 1 : q ) } ) } & { = } & { K ^ { ( n ) } \\left( ( x , 1 _ { m } ) , z _ { 1 : q } \\right) \\left( D ^ { ( n ) } \\left( z _ { 1 : q } \\right) ^ { T } \\right) ^ { - 1 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            213,
+            692,
+            243
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $D ^ { ( n ) } \\left( z _ { 1 : q } \\right)$ is the Cholesky factor of the covariance matrix $K ^ { ( n ) } \\left( z _ { 1 : q } , z _ { 1 : q } \\right) + \\sigma ^ { 2 } I .$ ",
+        "bbox": [
+            171,
+            253,
+            761,
+            270
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Thus, to provide an unbiased Monte Carlo estimator of the expectation within (3.4), we may sample $W _ { q }$ , and then calculate $\\begin{array} { r } { \\operatorname* { m i n } _ { x ^ { \\prime } \\in \\mathbb { A } } \\mu ^ { ( n + q ) } ( x ^ { \\prime } , 1 _ { m } ) = \\operatorname* { m i n } _ { x ^ { \\prime } \\in \\mathbb { A } } \\mu ^ { ( n ) } \\left( x , 1 _ { m } \\right) + \\tilde { \\sigma } _ { n } \\left( x , z _ { 1 : q } \\right) W _ { q } } \\end{array}$ . To do this optimization, we use a second-order continuous optimization method, where the gradient and Hessian of $\\boldsymbol { \\mu } ^ { ( n + q ) } \\left( \\boldsymbol { x } , \\boldsymbol { 1 } _ { m } \\right)$ with respect to $x$ in (3.6) can be computed by calculating the gradient and Hessian of $\\boldsymbol { \\mu } ^ { ( n ) } \\left( \\boldsymbol { x } , \\boldsymbol { 1 } _ { m } \\right)$ and $\\tilde { \\sigma } _ { n } \\left( x , z _ { 1 : q } \\right)$ . We can then compute the $\\mathbf { q }$ -cfKG acquistion function to arbitrary accuracy by averaging many such independent Monte Carlo estimates. ",
+        "bbox": [
+            173,
+            275,
+            825,
+            364
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3.2 ESTIMATING THE GRADIENT OF Q-CFKG ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            382,
+            513,
+            397
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To solve (3.5), we generalize a recently proposed computational method based on the envelope theorem from $\\mathrm { W u }$ et al. (2017) to provide an unbiased estimator of the gradient of the $\\mathsf { q }$ -cfKG acquisition function, and then use stochastic gradient ascent. ",
+        "bbox": [
+            173,
+            406,
+            826,
+            449
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Exploiting (3.6), the q-cfKG acquisition function can be expressed as ",
+        "bbox": [
+            173,
+            455,
+            629,
+            470
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/91cca73d239e7f6a9d069117e3ad6b47617164a9a20b2b56b5f3d37e0da09c14.jpg",
+        "text": "$$\n\\mathsf { q } \\mathsf { - } \\mathsf { c f K G } ( z _ { 1 : q } ) = \\frac { \\operatorname* { m i n } _ { x \\in \\mathbb { A } } \\mu ^ { ( n ) } \\left( x , 1 _ { m } \\right) - \\mathbb { E } _ { n } \\left[ \\operatorname* { m i n } _ { x \\in \\mathbb { A } } \\left( \\mu ^ { ( n ) } \\left( x , 1 _ { m } \\right) + \\tilde { \\sigma } _ { n } \\left( x , z _ { 1 : q } \\right) W _ { q } \\right) \\right] } { \\operatorname* { m a x } _ { 1 \\leq i \\leq q } \\mathsf { c o s t } ^ { ( n ) } \\left( x _ { i } , s _ { i } \\right) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            189,
+            479,
+            774,
+            518
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { q }$ is a standard $q$ -dimensional normal random vector, $\\begin{array} { r l r } { \\tilde { \\sigma } _ { n } ( x , z _ { ( 1 : q ) } ) } & { { } = } & { } \\end{array}$ $K ^ { ( n ) } ( ( x , 1 _ { m } ) , z _ { 1 : q } ) ( D ^ { ( n ) } ( z _ { 1 : q } ) ^ { T } ) ^ { - 1 }$ , and $D ^ { ( n ) } \\left( z _ { 1 : q } \\right)$ is the Cholesky factor of the covariance matrix $K ^ { ( n ) } \\left( z _ { 1 : q } , z _ { 1 : q } \\right) ~ + ~ \\sigma ^ { 2 } I .$ $\\nabla \\mathbf { q } \\mathbf { - } \\mathbf { c f K G } ( z _ { 1 : q } )$ can be computed from $\\begin{array} { r l } {  { - \\nabla \\mathbb { E } _ { n } [ \\operatorname* { m i n } _ { x \\in \\mathbb { A } } ( \\mu ^ { ( n ) } ( x , 1 _ { m } ) + \\tilde { \\sigma } _ { n } ( x , z _ { 1 : q } ) W _ { q } ) ] } \\quad } & { { } } \\end{array}$ and $\\begin{array} { r } { \\nabla \\operatorname* { m a x } _ { 1 \\leq i \\leq q } \\operatorname { c o s t } ^ { ( n ) } \\big ( z ^ { ( n + i ) } \\big ) } \\end{array}$ , where differentiability of $\\mathrm { c o s t } ^ { ( n ) } ( \\cdot )$ implies $\\operatorname* { m a x } _ { 1 \\leq i \\leq q } \\cos t ^ { ( n ) } ( x _ { i } , s _ { i } )$ is differentiable almost everywhere. To compute the first term, under sufficient regularity conditions (L’Ecuyer, 1990) that we conjecture hold in most applications to hyperparameter tuning, one can interchange the gradient and expectation operators, ",
+        "bbox": [
+            173,
+            526,
+            826,
+            650
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/25f7c69a155695b993a502a04d5e881f4ec75da519377b70201244ad00b103b0.jpg",
+        "text": "$$\n\\nabla \\mathbb { E } _ { n } \\left[ \\operatorname* { m i n } _ { x \\in \\mathbb { A } } \\left( \\mu ^ { ( n ) } \\left( x , 1 _ { m } \\right) + \\widetilde { \\sigma } _ { n } \\left( x , z _ { 1 : q } \\right) W _ { q } \\right) \\right] = \\mathbb { E } _ { n } \\left[ \\nabla \\operatorname* { m i n } _ { x \\in \\mathbb { A } } \\left( \\mu ^ { ( n ) } ( x , 1 _ { m } ) + \\widetilde { \\sigma } _ { n } \\left( x , z _ { 1 : q } \\right) W _ { q } \\right) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            171,
+            659,
+            812,
+            694
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This technique is called infinitesimal perturbation analysis (IPA) (L’Ecuyer, 1990). ",
+        "bbox": [
+            174,
+            702,
+            717,
+            717
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Since multiplication, matrix inversion (when the inverse exists), and Cholesky factorization (Smith, 1995) preserve continuous differentiability, $( x , z _ { 1 : q } ) \\mapsto \\left( \\mu ^ { ( n ) } ( x , 1 _ { m } ) + \\tilde { \\sigma } _ { n } ( x , z _ { 1 : q } ) W _ { q } \\right)$ is continuously differentiable under mild regularity conditions. When this function is continuously differentiable and A is compact, the envelope theorem (Milgrom $\\&$ Segal, 2002, Corollary 4) implies ",
+        "bbox": [
+            173,
+            723,
+            825,
+            781
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/de460c7f97a98998cb1387a6b48bcf823b77afa7cef199e46f3266e6ad90a82d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { n } \\left[ \\nabla \\underset { x \\in \\mathbb { A } } { \\mathrm { m i n } } \\left( \\mu ^ { ( n ) } ( x , 1 _ { m } ) + \\tilde { \\sigma } _ { n } \\left( x , z _ { 1 : q } \\right) W _ { q } \\right) \\right] } \\\\ & { = \\mathbb { E } _ { n } \\left[ \\nabla \\left( \\mu ^ { ( n ) } \\left( x ^ { * } ( W _ { q } ) , 1 _ { m } \\right) + \\tilde { \\sigma } _ { n } \\left( x ^ { * } ( W _ { q } ) , z _ { 1 : q } \\right) \\cdot W _ { q } \\right) \\right] , } \\\\ & { = \\mathbb { E } _ { n } \\left[ \\nabla \\tilde { \\sigma } _ { n } \\left( x ^ { * } ( W _ { q } ) , z _ { 1 : q } \\right) \\cdot W _ { q } \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            790,
+            696,
+            872
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { x } ^ { * } ( W _ { q } ) \\in \\arg \\operatorname* { m i n } _ { \\boldsymbol { x } \\in \\mathbb { A } } \\left( \\mu ^ { ( n ) } ( \\boldsymbol { x } , \\boldsymbol { 1 } _ { m } ) + \\tilde { \\sigma } _ { n } ( \\boldsymbol { x } , \\boldsymbol { z } _ { 1 : q } ) W _ { q } \\right)$ . We can use this unbiased gradient estimator within stochastic gradient ascent (Harold et al., 2003) to solve the optimization problem (3.5). ",
+        "bbox": [
+            176,
+            881,
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+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 EXTENSIONS TO DERIVATIVE-ENABLED SETTINGS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            104,
+            557,
+            117
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Following $\\mathbf { W } \\mathbf { u }$ et al. (2017), which developed single-fidelity BO methods for use when gradient information is available, we develop a version of the cfKG algorithm, called derivative-enabled cfKG, that can be used when gradients are available in the continuous-fidelity setting. This can be used to reduce the number of function evaluations required to find a high-quality solution to (1.1). ",
+        "bbox": [
+            174,
+            128,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "First, observe that a $G P ( \\mu ^ { n } , K ^ { n } )$ prior on $g ( x , s )$ implies a multi-output $G P ( \\tilde { \\mu } ^ { n } , \\tilde { K } ^ { n } )$ prior on $( g , \\nabla _ { x } g )$ . Then, observe that the same reasoning we used to develop the cfKG acquistion function in (3.2) can be used when when we observe gradients to motivate the acquisition function, ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/babd220d62f5824db1440b8f3c82dd3a421f1bb2e61be51b3f9d060cde29fada.jpg",
+        "text": "$$\n\\mathtt { c f - d - K G } ( x , s ) = \\frac { \\operatorname* { m i n } _ { x \\in \\mathbb { A } } \\tilde { \\mu } _ { 1 } ^ { ( n ) } ( x , 1 _ { m } ) - \\mathbb { E } _ { n } \\left[ \\operatorname* { m i n } _ { x \\in \\mathbb { A } } \\tilde { \\mu } _ { 1 } ^ { ( n + 1 ) } ( x , 1 _ { m } ) \\mid x ^ { ( n + 1 ) } = x , s ^ { ( n + 1 ) } ) = s \\right] } { \\cos \\mathfrak { c } ^ { ( n ) } ( x , s ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            242,
+            820,
+            289
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\cos \\mathbf { t } ^ { ( n ) } ( x , s )$ is now our estimate after $n$ samples of the cost of evaluating both $g$ and its gradient with respect to $x$ , and $\\tilde { \\mu } _ { 1 } ^ { ( n ) } ( x , s )$ is the posterior mean on $g ( x , s )$ in the multi-output GP. The conditional expectation is taken with respect to both the observed function value and gradient. A batch version of the derivative-enabled cfKG acquistion function can be defined analogously. We use techniques similar to those described in Sect. 3.3 to optimize these acquisition functions. ",
+        "bbox": [
+            174,
+            296,
+            825,
+            369
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 NUMERICAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            391,
+            433,
+            406
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare sequential, batch, and derivative-enabled cfKG with several benchmark algorithms on four synthetic functions, tuning convolutional neural networks on CIFAR-10 and SVHN, and on large-scale kernel learning. Our benchmarks include the traditional Bayesian optimization algorithms KG (Wu & Frazier, 2016) and EI (Jones et al., 1998), and the multi-fidelity Bayesian optimization with continuous approximation algorithm (BOCA) (Kandasamy et al., 2017). We also compare with Hyperband (Li et al., 2016) in the CIFRA-10 experiment, and with derivativeenabled KG (Wu et al., 2017) in the kernel-learning experiment. We use squared-exponential kernels with constant mean functions and integrate out the GP hyperparameters by sampling $M = 1 0$ sets of hyperparameters using the emcee package (Foreman-Mackey et al., 2013). ",
+        "bbox": [
+            173,
+            422,
+            825,
+            546
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 OPTIMIZING SYNTHETIC FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            565,
+            468,
+            579
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Inspired by numerical experiments in Kandasamy et al. (2017), we modify four standard synthetic test functions, 2-d Branin, 3-d Rosenbrock, 3-d Hartmann, and 6-d Hartmann, by adding 1 fidelity control, as described in detail in Appendix. A. We use cost $\\textstyle \\operatorname { c o s t } ( x , s ) : = \\prod _ { i = 1 } ^ { \\bar { m } } ( 0 . 0 1 + s _ { i } )$ to mimic the linear cost of training a neural network with more data and longer iterations, and $\\begin{array} { r } { \\mathrm { c o s t } ( z _ { 1 : q } ) : = \\mathrm { m a x } _ { 1 \\leq j \\leq q } \\mathrm { c o s t } ( z _ { j } ) } \\end{array}$ to mimic wall-clock time for training $q$ neural networks in parallel. Fig. 1 summarizes the results. cfKG achieves lower simple regret than competitors at the same cost. ",
+        "bbox": [
+            174,
+            590,
+            825,
+            686
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 TUNING CONVOLUTIONAL NEURAL NETWORKS ON CIFAR-10 AND SVHN ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            705,
+            732,
+            719
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We use sequential and batch cfKG to tune convolution neural networks (CNNs) on CIFAR-10 and SVHN. Our CNN consists of 3 convolutional blocks and a softmax classification layer. Each convolutional block consists of two convolutional layers with the same number of filters followed by a max-pooling layer. There is no dropout or batch-normalization layer. We split the CIFAR-10 dataset into 40000 training samples, 10000 validation samples and 10000 test samples. We split the SVHN training dataset into 67235 training samples and 6000 validation samples, and use the standard 26032 test samples. We apply standard data augmentation: horizontal and vertical shifts, and horizontal flips. We optimize 5 hyperparameters to minimize the classification error on the validation set: the learning rate, batch size, and number of filters in each convolutional block. cfKG and $\\mathbf { q }$ -cfKG use two fidelity controls: the size of the training set and the number of training iterations. Hyperband uses the size of the training set as its resource (it can use only one resource or fidelity), using a bracket size of $s _ { \\operatorname* { m a x } } = 4$ as in Li et al. (2016) and the maximum resource allowed by a single configuration set to 40000. We set the maximum number of training epochs for all algorithms to 50 for CIFAR-10 and 40 for SVHN. Fig. 2 shows the performance of cfKG relative to several benchmarks. cfKG successfully exploits the cheap approximations and find a good solution much faster than KG and Hyperband. When we train using optimized hyperparameters on the full training dataset for 200 epochs, test data classification error is $\\sim 1 2 \\%$ for CIFAR-10 and $\\sim 5 \\%$ for SVHN. ",
+        "bbox": [
+            174,
+            731,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/08281fbda24d24ce93e41224a3219369ec0988b387e4f0eb1f4f7b0c2407a9e0.jpg",
+        "image_caption": [
+            "Figure 1: Optimizing 4 synthetic functions: 2-d Branin, 3-d Rosenbrock, 3-d Hartmann, and 6-d Hartmann with 20 independent runs. Fidelity is 1-d. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            107,
+            776,
+            441
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            492,
+            821,
+            534
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 LARGE-SCALE KERNEL LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            558,
+            452,
+            570
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We use derivative-enabled cfKG (cf-d-KG) in a large-scale kernel learning example obtained by modifying the 1-d demo example for KISS-GP (Wilson & Nickisch, 2015) on the GPML website (Rasmussen & Nickisch, 2016). In this example, we optimize 3 hyperparameters (the alpha, the length scale and the variance of the noise) of a GP with an RBF kernel on 1 million training points to maximize the log marginal likelihood. We evaluate both the log marginal likelihood and its gradient using the KISS-GP framework. We use two fidelity controls: the number of training points and the number of inducing points in the KISS-GP framework. We set the maximum number of inducing points to $m = 1 0 0 0$ . We compare cf-d-KG to the derivative-enabled knowledge gradient (d-KG) (Wu et al., 2017) both with batch size $q = 8$ . Fig. 2 shows that cf-d-KG successfully utilizes the inexpensive function and gradient evaluations to find a good solution more quickly. ",
+        "bbox": [
+            173,
+            585,
+            825,
+            723
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/580470d835d17f9de2683c6601f0aeb8d69bb5c35eaa24f6276dd39aa59cf605.jpg",
+        "image_caption": [
+            "Figure 2: From left to right: Tuning convolutional neural networks on CIFAR-10 and SVHN with 6 independent runs, and kernel learning under the KISS-GP framework with 10 independent runs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            748,
+            746,
+            887
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            318,
+            118
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We propose a novel continuous-fidelity BO algorithm, cfKG, which generalizes naturally to batch and derivative settings. This algorithm can find good solutions to global optimization problems with less cost than state-of-art algorithms in applications including deep learning and kernel learning. ",
+        "bbox": [
+            174,
+            132,
+            825,
+            176
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            193,
+            285,
+            208
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "James Bergstra and Yoshua Bengio. Random search for hyper-parameter optimization. Journal of Machine Learning Research, 13(Feb): 281–305, 2012.   \nTobias Domhan, Jost Tobias Springenberg, and Frank Hutter. Speeding up automatic hyperparameter optimization of deep neural networks by extrapolation of learning curves. In IJCAI, pp. 3460–3468, 2015.   \nDaniel Foreman-Mackey, David W Hogg, Dustin Lang, and Jonathan Goodman. emcee: the mcmc hammer. Publications of the Astronomical Society of the Pacific, 125(925):306, 2013.   \nPeter Frazier, Warren Powell, and Savas Dayanik. The knowledge-gradient policy for correlated normal beliefs. INFORMS journal on Computing, 21(4):599–613, 2009.   \nJ Harold, G Kushner, and George Yin. Stochastic approximation and recursive algorithm and applications. Springer, 2003.   \nDeng Huang, TT Allen, WI Notz, and RA Miller. Sequential kriging optimization using multiple-fidelity evaluations. Structural and Multidisciplinary Optimization, 32(5):369–382, 2006.   \nDonald R Jones, Matthias Schonlau, and William J Welch. Efficient global optimization of expensive black-box functions. Journal of Global optimization, 13(4):455–492, 1998.   \nKirthevasan Kandasamy, Gautam Dasarathy, Junier B Oliva, Jeff Schneider, and Barnabas P ´ oczos. Gaussian process bandit optimisation with ´ multi-fidelity evaluations. In Advances in Neural Information Processing Systems, pp. 992–1000, 2016.   \nKirthevasan Kandasamy, Gautam Dasarathy, Jeff Schneider, and Barnabas Poczos. Multi-fidelity bayesian optimisation with continuous approximations. In ICML, 2017. Accepted for publication. ArXiv preprint 1703.06240.   \nAaron Klein, Stefan Falkner, Simon Bartels, Philipp Hennig, and Frank Hutter. Fast bayesian optimization of machine learning hyperparameters on large datasets. In Artificial Intelligence and Statistics, 2017. Accepted for publication. ArXiv preprint arXiv:1605.07079.   \nRemi Lam, Douglas Allaire, and Karen Willcox. Multifidelity optimization using statistical surrogate modeling for non-hierarchical information sources. In 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, pp. 0143, 2015.   \nPierre L’Ecuyer. A unified view of the IPA, SF, and LR gradient estimation techniques. Management Science, 36(11):1364–1383, 1990.   \nLisha Li, Kevin Jamieson, Giulia DeSalvo, Afshin Rostamizadeh, and Ameet Talwalkar. Hyperband: A novel bandit-based approach to hyperparameter optimization. arXiv preprint arXiv:1603.06560, 2016.   \nMark McLeod, Michael A Osborne, and Stephen J Roberts. Practical bayesian optimization for variable cost objectives. arXiv preprint arXiv:1703.04335, 2017.   \nPaul Milgrom and Ilya Segal. Envelope theorems for arbitrary choice sets. Econometrica, 70(2):583–601, 2002.   \nMatthias Poloczek, Jialei Wang, and Peter I Frazier. Multi-information source optimization. In Advances in Neural Information Processing Systems, 2017. Accepted for publication. ArXiv preprint 1603.00389.   \nCarl Edward Rasmussen and Hannes Nickisch. documentation for gpml matlab code version 4.0. http://www.gaussianprocess. org/gpml/code/matlab/doc/, 2016.   \nCarl Edward Rasmussen and Christopher K.˜I. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006. ISBN ISBN 0-262- 18253-X.   \nStephen P Smith. Differentiation of the cholesky algorithm. Journal of Computational and Graphical Statistics, 4(2):134 – 147, 1995.   \nJasper Snoek, Hugo Larochelle, and Ryan P Adams. Practical bayesian optimization of machine learning algorithms. In Advances in neural information processing systems, pp. 2951–2959, 2012.   \nKevin Swersky, Jasper Snoek, and Ryan P Adams. Multi-task bayesian optimization. In Advances in neural information processing systems, pp. 2004–2012, 2013.   \nKevin Swersky, Jasper Snoek, and Ryan Prescott Adams. Freeze-thaw bayesian optimization. arXiv preprint arXiv:1406.3896, 2014.   \nAndrew Gordon Wilson and Hannes Nickisch. Kernel interpolation for scalable structured gaussian processes (kiss-gp). In ICML, pp. 1775– 1784, 2015.   \nJian Wu and Peter Frazier. The parallel knowledge gradient method for batch bayesian optimization. In Advances in Neural Information Processing Systems, pp. 3126–3134, 2016.   \nJian Wu, Matthias Poloczek, Andrew Gordon Wilson, and Peter I Frazier. Bayesian optimization with gradients. In Advances in Neural Information Processing Systems, 2017. Accepted for publication. ArXiv preprint 1703.04389. ",
+        "bbox": [
+            171,
+            208,
+            828,
+            905
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "A ADDITIONAL EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            531,
+            118
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Here we define in detail the synthetic test functions on which we perform numerical experiments described in Sect. 4.1. The test functions are: ",
+        "bbox": [
+            173,
+            132,
+            825,
+            161
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e38a1d95d045b4a1aaab490a9303cc7b0fad54e01fc7c24563bc29973bf2e852.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\mathrm { a u g m e n t e d - B r a m i n } ( x , s ) } & { = } & { \\displaystyle \\left( x _ { 2 } - \\left( \\frac { 5 . 1 } { 4 \\pi ^ { 2 } } - 0 . 0 0 1 * ( 1 - s _ { 1 } ) \\right) x _ { 1 } ^ { 2 } + \\frac { 5 } { \\pi } x _ { 1 } - 6 \\right) ^ { 2 } } \\\\ & & { \\displaystyle + 1 0 * \\left( 1 - \\frac { 1 } { 8 \\pi } \\right) \\cos ( x _ { 1 } ) + 1 0 } \\\\ { \\mathrm { a g m e n t e d - H a r t m a m } ( x , s ) } & { = } & { \\displaystyle ( \\alpha _ { 1 } - 0 . 0 1 * ( 1 - s _ { 1 } ) ) \\exp \\left( - \\sum _ { j = 1 } ^ { d } A _ { i j } ( x _ { j } - P _ { \\mathrm { i j } } ) ^ { 2 } \\right) } \\\\ & & { \\displaystyle + \\sum _ { i = 2 } ^ { d } \\alpha _ { i } \\exp \\left( - \\sum _ { j = 1 } ^ { d } A _ { i j } ( x _ { j } - P _ { \\mathrm { i j } } ) ^ { 2 } \\right) } \\\\ { \\mathrm { b a r e n t e d - R o s e n b r o c k } ( x , s ) } & { = } & { \\displaystyle \\sum _ { i = 1 } ^ { 2 } \\left( 1 0 0 * ( x _ { i + 1 } - x _ { i } ^ { 2 } + 0 . 0 0 1 * ( 1 - s _ { 1 } ) ) ^ { 2 } + ( x _ { i } - 1 ) ^ { 2 } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            183,
+            166,
+            815,
+            386
+        ],
+        "page_idx": 9
+    }
+]

parse/train/eNdiU_DbM9/eNdiU_DbM9.md

@@ -0,0 +1,444 @@
+# UNCERTAINTY SETS FOR IMAGE CLASSIFIERS USING CONFORMAL PREDICTION
+
+Anastasios N. Angelopoulos∗, Stephen Bates∗, Jitendra Malik, & Michael I. Jordan
+
+Departments of Electrical Engineering and Computer Sciences and Statistics University of California, Berkeley {angelopoulos,stephenbates,malik,jordan}@cs.berkeley.edu
+
+# ABSTRACT
+
+Convolutional image classifiers can achieve high predictive accuracy, but quantifying their uncertainty remains an unresolved challenge, hindering their deployment in consequential settings. Existing uncertainty quantification techniques, such as Platt scaling, attempt to calibrate the network’s probability estimates, but they do not have formal guarantees. We present an algorithm that modifies any classifier to output a predictive set containing the true label with a user-specified probability, such as $9 0 \%$ . The algorithm is simple and fast like Platt scaling, but provides a formal finite-sample coverage guarantee for every model and dataset. Our method modifies an existing conformal prediction algorithm to give more stable predictive sets by regularizing the small scores of unlikely classes after Platt scaling. In experiments on both Imagenet and Imagenet-V2 with ResNet-152 and other classifiers, our scheme outperforms existing approaches, achieving coverage with sets that are often factors of 5 to 10 smaller than a stand-alone Platt scaling baseline.
+
+# 1 INTRODUCTION
+
+Imagine you are a doctor making a high-stakes medical decision based on diagnostic information from a computer vision classifier. What would you want the classifier to output in order to make the best decision? This is not a casual hypothetical; such classifiers are already used in medical settings (e.g., Razzak et al., 2018; Lundervold & Lundervold, 2019; Li et al., 2014). A maximumlikelihood diagnosis with an accompanying probability may not be the most essential piece of information. To ensure the health of the patient, you must also rule in or rule out harmful diagnoses. In other words, even if the most likely diagnosis is a stomach ache, it is equally or more important to rule out stomach cancer. Therefore, you would want the classifier to give you—in addition to an estimate of the most likely outcome—actionable uncertainty quantification, such as a set of predictions that provably covers the true diagnosis with a high probability (e.g., $9 0 \%$ ). This is called a prediction set (see Figure 1). Our paper describes a method for constructing prediction sets from any pre-trained image classifier that are formally guaranteed to contain the true class with the desired probability, relatively small, and practical to implement. Our method modifies a conformal predictor (Vovk et al., 2005) given in Romano et al. (2020) for the purpose of modern image classification in order to make it more stable in the presence of noisy small probability estimates. Just as importantly, we provide extensive evaluations and code for conformal prediction in computer vision.
+
+Formally, for a discrete response $Y \in \mathcal { Y } = \{ 1 , \dots , K \}$ and a feature vector $X \in \mathbb { R } ^ { d }$ , we desire an uncertainty set function, $\mathcal C ( X )$ , mapping a feature vector to a subset of $\{ 1 , \ldots , K \}$ such that
+
+$$
+P ( Y \in { \mathcal { C } } ( X ) ) \geq 1 - \alpha ,
+$$
+
+for a pre-specified confidence level $\alpha$ such as $10 \%$ . Conformal predictors like our method can modify any black-box classifier to output predictive sets that are rigorously guaranteed to satisfy the desired coverage property shown in Eq. (1). For evaluations, we focus on Imagenet classification using convolutional neural networks (CNNs) as the base classifiers, since this is a particularly challenging testbed. In this setting, $X$ would be the image and $Y$ would be the class label. Note that the guarantee in Eq. (1) is marginal over $X$ and $Y$ —it holds on average, not for a particular image $X$ .
+
+![](images/b83feab4a65aeb1ea9135e5026304f4f1735b7bfef87c18aee1dda3542d2f0c0.jpg)  
+Figure 1: Prediction set examples on Imagenet. We show three examples of the class fox squirrel and the $9 5 \%$ prediction sets generated by RAPS to illustrate how the size of the set changes as a function of the difficulty of a test-time image.
+
+A first approach toward this goal might be to assemble the set by including classes from highest to lowest probability (e.g., after Platt scaling and a softmax function; see Platt et al., 1999; Guo et al., 2017) until their sum just exceeds the threshold $1 - \alpha$ . We call this strategy naive and formulate it precisely in Algorithm 1. There are two problems with naive: first, the probabilities output by CNNs are known to be incorrect (Nixon et al., 2019), so the sets from naive do not achieve coverage. Second, image classification models’ tail probabilities are often badly miscalibrated, leading to large sets that do not faithfully articulate the uncertainty of the model; see Section 2.3. Moreover, smaller sets that achieve the same coverage level can be generated with other methods.
+
+The coverage problem can be solved by picking a new threshold using holdout samples. For example, with $\alpha = 1 0 \%$ , if choosing sets that contain $93 \%$ estimated probability achieves $90 \%$ coverage on the holdout set, we use the $93 \%$ cutoff instead. We refer to this algorithm, introduced in Romano et al. (2020), as Adaptive Prediction Sets (APS). The APS procedure provides coverage but still produces large sets. To fix this, we introduce a regularization technique that tempers the influence of these noisy estimates, leading to smaller, more stable sets. We describe our proposed algorithm, Regularized Adaptive Prediction Sets (RAPS), in Algorithms 2 and 3 (with APS as a special case). As we will see in Section 2, both APS and RAPS are always guaranteed to satisfy Eq. (1)—regardless of model and dataset. Furthermore, we show that RAPS is guaranteed to have better performance than choosing a fixed-size set. Both methods impose negligible computational requirements in both training and evaluation, and output useful estimates of the model’s uncertainty on a new image given, say, 1000 held-out examples.
+
+In Section 3 we conduct the most extensive evaluation of conformal prediction in deep learning to date on Imagenet and Imagenet-V2. We find that RAPS sets always have smaller average size than naive and APSsets. For example, using a ResNeXt-101, naive does not achieve coverage, while APS and RAPS achieve it almost exactly. However, APS sets have an average size of 19, while RAPS sets have an average size of 2 at $\alpha = 1 0 \%$ (Figure 2 and Table 1). We will provide an accompanying codebase that implements our method as a wrapper for any PyTorch classifier, along with code to exactly reproduce all of our experiments.
+
+# 1.1 RELATED WORK
+
+Reliably estimating predictive uncertainty for neural networks is an unsolved problem. Historically, the standard approach has been to train a Bayesian neural network to learn a distribution over network weights (Quinonero-Candela et al., 2005; MacKay, 1992; Neal, 2012; Kuleshov et al., 2018; Gal, 2016). This approach requires computational and algorithmic modifications; other approaches avoid these via ensembles (Lakshminarayanan et al., 2017; Jiang et al., 2018) or approximations of Bayesian inference (Riquelme et al., 2018; Sensoy et al., 2018). These methods also have major practical limitations; for example, ensembling requires training many copies of a neural network adversarially. Therefore, the most widely used strategy is ad-hoc traditional calibration of the softmax scores with Platt scaling (Platt et al., 1999; Guo et al., 2017; Nixon et al., 2019).
+
+This work develops a method for uncertainty quantification based on conformal prediction. Originating in the online learning literature, conformal prediction is an approach for generating predictive sets that satisfy the coverage property in Eq. (1) (Vovk et al., 1999; 2005). We use a convenient data-splitting version known as split conformal prediction that enables conformal prediction methods to be deployed for essentially any predictor (Papadopoulos et al., 2002; Lei et al., 2018). While mechanically very different from traditional calibration as discussed above, we will refer to our approach as conformal calibration to highlight that the two methodologies have overlapping but different goals.
+
+![](images/b145734d9ee59705d4626150202e0ceb926d2eef94d6631535a4e1e770ebb79e.jpg)  
+Figure 2: Coverage and average set size on Imagenet for prediction sets from three methods. All methods use a ResNet-152 as the base classifier, and results are reported for 100 random splits of Imagenet-Val, each of size 20K. See Section 3.1 for full details.
+
+Conformal prediction is a general framework, not a specific algorithm—important design decisions must be made to achieve the best performance for each context. To this end, Romano et al. (2020) and Cauchois et al. (2020) introduce techniques aimed at achieving coverage that is similar across regions of feature space, whereas Vovk et al. (2003); Hechtlinger et al. (2018) and Guan & Tibshirani (2019) introduce techniques aimed at achieving equal coverage for each class. While these methods have conceptual appeal, thus far there has been limited empirical evaluation of this general approach for state-of-the-art CNNs. Concretely, the only works that we are aware of that include some evaluation of conformal methods on ImageNet—the gold standard for benchmarking computer vision methods—are Hechtlinger et al. (2018), Park et al. (2019), Cauchois et al. (2020), and Messoudi et al. (2020), although in all four cases further experiments are needed to more fully evaluate their operating characteristics for practical deployment. At the heart of conformal prediction is the conformal score - a measure of similarity between labeled examples which is used to compare a new point to among those in a hold out set. Our theoretical contribution can be summarized as a modification of the conformal score from Romano et al. (2020) to have smaller, more stable sets. Lastly, there are alternative approaches to returning prediction sets not based on conformal prediction (Pearce et al., 2018; Zhang et al., 2018). These methods can be used as input to a conformal procedure to potentially improve performance, but they do not have finite-sample coverage guarantees when used alone.
+
+# 2 METHODS
+
+In developing uncertainty set methods to improve upon naive, we are guided by three desiderata. First and most importantly, the coverage desideratum says the sets must provide $1 - \alpha$ coverage, as discussed above. Secondly, the size desideratum says we want sets of small size, since these convey more detailed information and may be more useful in practice. Lastly, the adaptiveness desideratum says we want the sets to communicate instance-wise uncertainty: they should be smaller for easy test-time examples than for hard ones; see Figure 1 for an illustration. Coverage and size are obviously competing objectives, but size and adaptiveness are also often in tension. The size desideratum seeks small sets, while the adaptiveness desideratum seeks larger sets when the classi
+
+# Algorithm 1 Naive Prediction Sets
+
+<table><tr><td></td><td colspan="3">Input:α,sorted scores s,associated permutation of classes I,boolean rand</td></tr><tr><td></td><td colspan="3">1: procedure NAIVE(α,s,I,rand) L←1</td></tr><tr><td>2: 3:</td><td colspan="3"></td></tr><tr><td></td><td colspan="3">&gt; Stop if1-α probability exceeded L←L+1</td></tr><tr><td>4:</td><td colspan="3">if rand then</td></tr><tr><td>5:</td><td colspan="3">&gt;Break ties randomly (explained in Appendix B)</td></tr><tr><td>6:</td><td colspan="3">U ←Unif(0,1) L</td></tr><tr><td>7:</td><td colspan="3">V←(∑i=1s-(1-α))/sL</td></tr><tr><td>8:</td><td colspan="3">ifU≤Vthen</td></tr><tr><td>9:</td><td colspan="3">L←L-1</td></tr><tr><td>10:</td><td colspan="3">return {I.,.., IL}</td></tr><tr><td colspan="3">Output: The 1-α prediction set, {I1,,IL}</td><td></td></tr></table>
+
+![](images/853120ecab45b49174ade7da0ac0f77155b7a1b2e945adfe8fc36d7e61839296.jpg)  
+Figure 3: Visualizations of conformal calibration and RAPS sets. In the left panel, the $\mathbf { y }$ -axis shows the empirical coverage on the conformal calibration set, and $1 - \alpha ^ { \prime } = \lceil ( n + 1 ) ( 1 - \alpha ) \rceil / n$ . In the right panel, the printed numbers indicate the cumulative probability plus penalty mass. For the indicated value $\hat { \tau } _ { \mathrm { c c a l } }$ , the RAPS prediction set is $\{ \mathbf { c } , \mathbf { d } , \mathbf { f } , \mathbf { b } \}$ .
+
+fier is uncertain. For example, always predicting a set of size five could achieve coverage, but it is not adaptive. As noted above, both APSand RAPS achieve correct coverage, and we will show that RAPS improves upon APS according to the other two desiderata.
+
+We now turn to the specifics of our proposed method. We begin in Subsection 2.1 by describing an abstract data-splitting procedure called conformal calibration that enables the near-automatic construction of valid predictive sets (that is, sets satisfying Eq. (1)). Subsequently, in Subsection 2.2, we provide a detailed presentation of our procedure, with commentary in Section 2.3. In Subsection 2.4 we discuss the optimality of our procedure, proving that it is at least as good as the procedure that returns sets of a fixed size, unlike alternative approaches.
+
+# 2.1 CONFORMAL CALIBRATION
+
+We first review a general technique for producing valid prediction sets, following the articulation in Gupta et al. (2019). Consider a procedure that outputs a predictive set for each observation, and further suppose that this procedure has a tuning parameter $\tau$ that controls the size of the sets. (In RAPS, $\tau$ is the cumulative sum of the sorted, penalized classifier scores.) We take a small independent conformal calibration set of data, and then choose the tuning parameter $\tau$ such that the predictive sets are large enough to achieve $1 - \alpha$ coverage on this set. See Figure 3 for an illustration. This calibration step yields a choice of $\tau$ , and the resulting set is formally guaranteed to have coverage $1 - \alpha$ on a future test point from the same distribution; see Theorem 1 below.
+
+Formally, let $( X _ { i } , Y _ { i } ) _ { i = 1 , \ldots , n }$ be an independent and identically distributed (i.i.d.) set of variables that was not used for model training. Further, let $\mathcal { C } ( \boldsymbol { x } , \boldsymbol { u } , \tau ) : \mathbb { R } ^ { d } \times [ 0 , 1 ] \times \mathbb { R }  2 ^ { \mathcal { V } }$ be a setvalued function that takes a feature vector $x$ to a subset of the possible labels. The second argument $u$ is included to allow for randomized procedures; let $U _ { 1 } , \ldots , U _ { n }$ be i.i.d. uniform $[ 0 , 1 ]$ random variables that will serve as the second argument for each data point. Suppose that the sets are indexed by $\tau$ such that they are nested, meaning larger values of $\tau$ lead to larger sets:
+
+$$
+{ \mathcal { C } } ( x , u , \tau _ { 1 } ) \subseteq { \mathcal { C } } ( x , u , \tau _ { 2 } ) \quad { \mathrm { ~ i f ~ } } \quad \tau _ { 1 } \leq \tau _ { 2 } .
+$$
+
+To find a function that will achieve $1 - \alpha$ coverage on test data, we select the smallest $\tau$ that gives at least $1 - \alpha$ coverage on the conformal calibration set, with a slight correction to account for the finite sample size:
+
+$$
+\hat { \tau } _ { \mathrm { c c a l } } = \operatorname* { i n f } \left\{ \tau : \frac { | \{ i : Y _ { i } \in \mathcal { C } ( X _ { i } , U _ { i } , \tau ) \} | } { n } \geq \frac { \lceil ( n + 1 ) ( 1 - \alpha ) \rceil } { n } \right\} .
+$$
+
+The set function $\mathcal { C } ( x , u , \tau )$ with this data-driven choice of $\tau$ is guaranteed to have correct finitesample coverage on a fresh test observation, as stated formally next.
+
+Theorem 1 (Conformal calibration coverage guarantee). Suppose $( X _ { i } , Y _ { i } , U _ { i } ) _ { i = 1 , \dots , n }$ and $( X _ { n + 1 } , Y _ { n + 1 } , U _ { n + 1 } )$ are i.i.d. and let $\mathcal { C } ( x , u , \tau )$ be a set-valued function satisfying the nesting property in Eq. (2). Suppose further that the sets $\mathcal { C } ( x , u , \tau )$ grow to include all labels for large enough $\tau$ : for all $x \in \mathbb { R } ^ { d }$ , $\boldsymbol { \mathcal { C } } ( \boldsymbol { x } , \boldsymbol { u } , \tau ) = \boldsymbol { \mathcal { y } }$ for some $\tau$ . Then for $\hat { \tau } _ { \mathrm { c c a l } }$ defined as in Eq. (3), we have the following coverage guarantee:
+
+$$
+\begin{array} { r } { P \Big ( Y _ { n + 1 } \in \mathcal { C } ( X _ { n + 1 } , U _ { n + 1 } , \hat { \tau } _ { \mathrm { c c a l } } ) \Big ) \geq 1 - \alpha . } \end{array}
+$$
+
+This is the same coverage property as Eq. (1) in the introduction, written in a more explicit manner. The result is not new—a special case of this result leveraging sample-splitting first appears in the regression setting in Papadopoulos et al. (2002), and the core idea of conformal prediction was introduced even earlier; see (Vovk et al., 2005).
+
+As a technical remark, the theorem also holds if the observations to satisfy the weaker condition of exchangeability; see Vovk et al. (2005). In addition, for most families of set-valued functions $\mathcal { C } ( x , u , \tau )$ there is a matching upper bound:
+
+$$
+P \Big ( Y _ { n + 1 } \in \mathcal { C } ( X _ { n + 1 } , U _ { n + 1 } , \hat { \tau } _ { \mathrm { c c a l } } ) \Big ) \leq 1 - \alpha + \frac { 1 } { n + 1 } .
+$$
+
+Roughly speaking, this will hold whenever the sets grow smoothly in $\tau$ . See Lei et al. (2018) for a formal statement of the required conditions.
+
+# 2.2 OUR METHOD
+
+Conformal calibration is a powerful general idea, allowing one to achieve the coverage desideratum for any choice of sets $\mathcal { C } ( x , u , \tau )$ . Nonetheless, this is not yet a full solution, since the quality of the resulting prediction sets can vary dramatically depending on the design of $\mathcal { C } ( x , u , \tau )$ . In particular, we recall the size and adaptiveness desiderata from Section 1—we want our uncertainty sets to be as small as possible while faithfully articulating the instance-wise uncertainty of each test point. In this section, we explicitly give our algorithm, which can be viewed as a special case of conformal calibration with the uncertainty sets $\mathcal { C }$ designed to extract information from CNNs.
+
+Our algorithm has three main ingredients. First, for a feature vector $x$ , the base model computes class probabilities $\hat { \pi } _ { x } \in \mathbb { R } ^ { k }$ , and we order the classes from most probable to least probable. Then, we add a regularization term to promote small predictive sets. Finally, we conformally calibrate the penalized prediction sets to guarantee coverage on future test points.
+
+Formally, let $\begin{array} { r } { \rho _ { x } ( y ) = \sum _ { y ^ { \prime } = 1 } ^ { K } \hat { \pi } _ { x } ( y ^ { \prime } ) \mathbb { I } _ { \{ \hat { \pi } _ { x } ( y ^ { \prime } ) > \hat { \pi } _ { x } ( y ) \} } } \end{array}$ be the total probability mass of the set of labels that are more likely than $y$ . These are all the labels that will be included before $y$ is included. In addition, let $o _ { x } ( y ) = | \{ y ^ { \prime } \in \mathcal { y } : \hat { \pi } _ { x } ( y ^ { \prime } ) \geq \hat { \pi } _ { x } ( y ) \} |$ be the ranking of $y$ among the label based on the probabilities $\hat { \pi }$ . For example, if $y$ is the third most likely label, then $o _ { x } ( y ) = 3$ .1 We take
+
+$$
+\mathcal { C } ^ { * } ( x , u , \tau ) : = \Big \{ y : \rho _ { x } ( y ) + \hat { \pi } _ { x } ( y ) \cdot u + \underbrace { \lambda \cdot ( o _ { x } ( y ) - k _ { r e g } ) ^ { + } } _ { \mathrm { r e g u l a r i z a t i o n } } \leq \tau \Big \} ,
+$$
+
+where $( z ) ^ { + }$ denotes the positive part of $z$ and $\lambda , k _ { r e g } \ge 0$ are regularization hyperparameters that are introduced to encourage small set sizes. See Figure 3 for a visualization of a RAPS predictive set and Appendix $\mathrm { E }$ for a discussion of how to select $k _ { r e g }$ and $\lambda$ .
+
+Since this is the heart of our proposal, we carefully parse each term. First, the $\rho _ { x } ( y )$ term increases as $y$ ranges from the most probable to least probable label, so our sets will prefer to include the $y$ that are predicted to be the most probable. The second term, $\hat { \pi } _ { x } ( y ) \cdot u$ , is a randomized term to handle the fact that the value will jump discretely with the inclusion of each new $y$ . The randomization term can never impact more than one value of $y$ : there is at most one value of $y$ such that $y \in \mathcal { C } ( x , 0 , \tau )$ but $y \notin \mathcal { C } ( x , 1 , \tau )$ . These first two terms can be viewed as the CDF transform after arranging the classes from most likely to least likely, randomized in the usual way to result in a continuous uniform random variable (cf. Romano et al., 2020). We discuss randomization further in Appendix B.
+
+Lastly, the regularization promotes small set sizes: for values of $y$ that occur farther down the ordered list of classes, the term $\bar { \lambda ^ { \cdot } } ( o _ { x } ( y ) - k _ { r e g } ) ^ { + }$ makes that value of $y$ require a higher value of $\tau$ before it is included in the predictive set. For example, if $k _ { r e g } = 5$ , then the sixth most likely value of $y$ has an extra penalty of size $\lambda$ , so it will never be included until $\tau$ exceeds $\rho _ { x } ( y ) + \hat { \pi } _ { x } ( y ) \cdot u + \lambda .$ , whereas it enters when $\tau$ exceeds $\rho _ { x } ( y ) + { \hat { \pi } } _ { x } ( y ) \cdot u$ in the nonregularized version. Our method has the following coverage property:
+
+Proposition 1 (RAPS coverage guarantee). Suppose $( X _ { i } , Y _ { i } , U _ { i } ) _ { i = 1 , \dots , n }$ and $( X _ { n + 1 } , Y _ { n + 1 } , U _ { n + 1 } )$ are i.i.d. and let $\mathcal { C } ^ { * } ( x , u , \tau )$ be defined as in Eq. (4). Suppose further that $\hat { \pi } _ { x } ( y ) > 0$ for all $x$ and $y$ . Then for $\hat { \tau } _ { \mathrm { c c a l } }$ defined as in Eq. (3), we have the following coverage guarantee:
+
+$$
+1 - \alpha \le P \Big ( Y _ { n + 1 } \in \mathcal { C } ^ { * } ( X _ { n + 1 } , U _ { n + 1 } , \widehat { \tau } _ { \mathrm { c c a l } } ) \Big ) \le 1 - \alpha + \frac { 1 } { n + 1 } .
+$$
+
+For ease of notation, we assume distinct probabilities. Else, label-ordering ties should be broken randomly.
+
+Input: $\alpha ; s \in [ 0 , 1 ] ^ { n \times K }$ , $I \in \{ 1 , . . . , K \} ^ { n \times K }$ , and one-hot $y \in \{ 0 , 1 \} ^ { K }$ corresponding respectively to the sorted scores, the associated permutation of indexes, and labels for each of $n$ examples in the calibration set; $k _ { r e g }$ ; $\lambda$ ; boolean rand   
+1: procedure RAPSC $( \alpha , \bar { s } , I , y , \lambda )$   
+2: for $i \in \{ 1 , \cdots , n \}$ do   
+3: $L _ { i } \gets \{ \ : j : \ : \dot { I _ { i , j } } = y _ { i } \ : \}$   
+4: $E _ { i } \gets \Sigma _ { j = 0 } ^ { L _ { i } } s _ { i , j } + \lambda ( L _ { i } - k _ { r e g } + 1 ) ^ { + }$   
+5: if rand then   
+6: $\begin{array} { l } { { U \sim \mathrm { U n i f } ( 0 , 1 ) } } \\ { { E _ { i }  E _ { i } - s _ { i , L _ { i } } + U \ast s _ { i , L _ { i } } } } \end{array}$   
+7:   
+8: $\hat { \tau } _ { c c a l } \gets$ the $\lceil ( 1 - \alpha ) ( 1 + n ) \rceil$ largest value in $\{ E _ { i } \} _ { i = 1 } ^ { n }$   
+9: return τˆccal
+
+![](images/0d21abcc0b71c07dfbe6e87c3bc9ddce258700b8d7443f22b06e401fe31a5451.jpg)
+
+Output: The generalized quantile, $\hat { \tau } _ { c c a l }$ . The value in Eq. (3)
+
+# Algorithm 3 RAPS Prediction Sets
+
+Input: $\alpha$ , sorted scores $s$ and the associated permutation of classes $I$ for a test-time example, $\hat { \tau } _ { c c a l }$ from Algorithm 2, $k _ { r e g }$ , $\lambda$ , boolean rand   
+1: procedure $\mathrm { R A P S } ( \alpha , s , \bar { I } , \hat { \tau } _ { c c a l } , k _ { r e g } , \lambda , r a n d )$   
+2: $L \gets | \ j \in \mathcal { V } : \ \Sigma _ { i = 0 } ^ { j } s _ { i } + \lambda ( L - k _ { r e g } ) ^ { + } \leq \hat { \tau } _ { c c a l } \ | + 1$   
+3: $V  ( \hat { \tau } _ { c c a l } - \Sigma _ { i = 0 } ^ { L - 1 } s _ { i } - \lambda ( L - k _ { r e g } ) ^ { + } + s _ { L - 1 } ) / s _ { L - 1 }$   
+4: if rand & $V \leq U \sim \operatorname { U n i f } ( 0 , 1 )$ then   
+5: L ← L − 1   
+6: return $\mathcal { C } = \left\{ I _ { 1 } , . . . I _ { L } \right\}$ $\triangleright$ The $L$ most likely classes   
+Output: The $1 - \alpha$ confidence set, $\mathcal { C }$ . The set in Eq. (4)
+
+Note that the first inequality is a corollary of Theorem 1, and the second inequality is a special case of the remark in Section 2.1. The restriction that $\hat { \pi } _ { x } ( y ) > 0$ is not necessary for the first inequality.
+
+# 2.3 WHY REGULARIZE?
+
+In our experiments, the sets from APS are larger than necessary, because APS is sensitive to the noisy probability estimates far down the list of classes. This noise leads to a permutation problem of unlikely classes, where ordering of the classes with small probability estimates is determined mostly by random chance. If $5 \%$ of the true classes from the calibration set are deep in the tail due to the permutation problem, APS will choose large $9 5 \%$ predictive sets; see Figure 2. The inclusion of the RAPS regularization causes the algorithm to avoid using the unreliable probabilities in the tail; see Figure 4. We discuss how RAPS improves the adaptiveness of APS in Section 4 and Appendix E.
+
+# 2.4 OPTIMALITY CONSIDERATIONS
+
+To complement these experimental results, we now formally prove that RAPS with the correct regularization parameters will always dominate the simple procedure that returns a fixed set size. (Section 3.5 shows the parameters are easy to select and RAPS is not sensitive to their values). For a feature vector $x$ , let ${ \hat { y } } _ { ( j ) } ( x )$ be the label with the $j$ th highest predicted probability. We define the top- $k$ predictive sets to be $\{ \hat { y } _ { ( 1 ) } ( x ) , \ldots , \hat { y } _ { ( k ) } ( x ) \}$ .
+
+Proposition 2 (RAPS dominates top- $\mathbf { \nabla } \cdot \mathbf { k }$ sets). Suppose $( X _ { i } , Y _ { i } , U _ { i } ) _ { i = 1 , \dots , n }$ and $( X _ { n + 1 } , Y _ { n + 1 } , U _ { n + 1 } )$ are i.i.d. draws. Let $k ^ { * }$ be the smallest $k$ such that the top- $k$ predictive sets have coverage at least $\lceil ( n + 1 ) ( 1 - \alpha ) \rceil / n$ on the conformal calibration points $( X _ { i } , Y _ { i } ) _ { i = 1 , \dots , n }$ . Take $\mathcal { C } ^ { * } ( x , u , \tau )$ as in Eq. (4) with any $k _ { r e g } \leq k ^ { * }$ and $\lambda = 1$ . Then with $\hat { \tau } _ { \mathrm { c c a l } }$ chosen as in Eq. (3), we have
+
+$$
+\mathcal { C } ^ { * } ( X _ { n + 1 } , U _ { n + 1 } , \hat { \tau } _ { \mathrm { c c a l } } ) \subseteq \{ \hat { y } _ { ( 1 ) } ( x ) , \dots , \hat { y } _ { ( k ^ { * } ) } ( x ) \} .
+$$
+
+In words, the RAPS procedure with heavy regularization will be at least as good as the top- $k$ procedure in the sense that it has smaller or same average set size while maintaining the desired coverage level. This is not true of either the naive baseline or the APS procedure; Table 2 shows that these two procedures usually return predictive sets with size much larger than $k ^ { * }$ .
+
+# 3 EXPERIMENTS
+
+Published as a conference paper at ICLR 2021   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Accuracy</td><td colspan="4">Coverage</td><td colspan="4">Size</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top K</td><td>Naive</td><td>APS</td><td>RAPS</td><td>Top K</td><td>Naive</td><td>APS</td><td>RAPS</td></tr><tr><td>ResNeXt101</td><td>0.793</td><td>0.945</td><td>0.900</td><td>0.889</td><td>0.900</td><td>0.900</td><td>2.42</td><td>17.1</td><td>19.7</td><td>2.00</td></tr><tr><td>ResNet152</td><td>0.783</td><td>0.94</td><td>0.900</td><td>0.895</td><td>0.900</td><td>0.900</td><td>2.63</td><td>9.78</td><td>10.4</td><td>2.11</td></tr><tr><td>ResNet101</td><td>0.774</td><td>0.936</td><td>0.900</td><td>0.896</td><td>0.900</td><td>0.900</td><td>2.83</td><td>10.3</td><td>10.7</td><td>2.25</td></tr><tr><td>ResNet50</td><td>0.761</td><td>0.929</td><td>0.900</td><td>0.896</td><td>0.900</td><td>0.900</td><td>3.14</td><td>11.8</td><td>12.3</td><td>2.57</td></tr><tr><td>ResNet18</td><td>0.698</td><td>0.891</td><td>0.900</td><td>0.895</td><td>0.900</td><td>0.900</td><td>5.72</td><td>15.5</td><td>16.2</td><td>4.43</td></tr><tr><td>DenseNet161</td><td>0.771</td><td>0.936</td><td>0.900</td><td>0.894</td><td>0.900</td><td>0.900</td><td>2.84</td><td>11.2</td><td>12.1</td><td>2.29</td></tr><tr><td>VGG16</td><td>0.716</td><td>0.904</td><td>0.900</td><td>0.895</td><td>0.901</td><td>0.900</td><td>4.75</td><td>13.4</td><td>14.1</td><td>3.54</td></tr><tr><td>Inception</td><td>0.695</td><td>0.887</td><td>0.900</td><td>0.885</td><td>0.900</td><td>0.901</td><td>6.30</td><td>75.4</td><td>89.1</td><td>5.32</td></tr><tr><td>ShuffleNet</td><td>0.694</td><td>0.883</td><td>0.900</td><td>0.891</td><td>0.900</td><td>0.900</td><td>6.46</td><td>28.9</td><td>31.9</td><td>5.05</td></tr></table>
+
+Table 1: Results on Imagenet-Val. We report coverage and size of the optimal, randomized fixed sets, naive, APS, and RAPS sets for nine different Imagenet classifiers. The median-of-means for each column is reported over 100 different trials. See Section 3.1 for full details.
+
+In this section we report on experiments that study the performance of the predictive sets from naive, APS, and RAPS, evaluating each based on the three desiderata above. We begin with a brief preview of the experiments. In Experiment 1, we evaluate naive, APS, and RAPS on ImagenetVal. Both APS and RAPS provided almost exact coverage, while naive sets had coverage slightly below the specified level. APS has larger sets on average than naive and RAPS. RAPS has a much smaller average set size than APS and naive. In Experiment 2, we repeat Experiment 1 on Imagenet-V2, and the conclusions still hold. In Experiment 3, we produce histograms of set sizes for naive, APS, and RAPS for several different values of $\lambda$ , illustrating a simple tradeoff between set size and adaptiveness. In Experiment 4, we compute histograms of RAPS sets stratified by image difficulty, showing that RAPS sets are smaller for easier images than for difficult ones. In Experiment 5, we report the performance of RAPS with many values of the tuning parameters.
+
+In our experiments, we use nine standard, pretrained Imagenet classifiers from the torchvision repository (Paszke et al., 2019) with standard normalization, resize, and crop parameters. Before applying naive, APS, or RAPS, we calibrated the classifiers using the standard temperature scaling/Platt scaling procedure as in Guo et al. (2017) on the calibration set. Thereafter, naive, APS, and RAPS were applied, with RAPS using a data-driven choice of parameters described in Appendix E. We use the randomized versions of these algorithms—see Appendix B for a discussion.
+
+# 3.1 EXPERIMENT 1: COVERAGE VS SET SIZE ON IMAGENET
+
+In this experiment, we calculated the coverage and mean set size of each procedure for two different choices of $\alpha$ . Over 100 trials, we randomly sampled two subsets of Imagenet-Val: one conformal calibration subset of size 20K and one evaluation subset of size 20K. The median-of-means over trials for both coverage and set size are reported in Table 1. Figure 2 illustrates the performances of naive, APS, and RAPS; RAPS has much smaller sets than both naive and APS, while achieving coverage. We also report results from a conformalized fixed- $\mathbf { \nabla \cdot k }$ procedure, which finds the smallest fixed set size achieving coverage on the holdout set, $k ^ { * }$ , then predicts sets of size $k ^ { * } - 1$ or $k ^ { * }$ on new examples in order to achieve exact coverage; see Algorithm 4 in Appendix E.
+
+# 3.2 EXPERIMENT 2: COVERAGE VS SET SIZE ON IMAGENET-V2
+
+The same procedure as Experiment 1 was repeated on Imagenet-V2, with exactly the same normalization, resize, and crop parameters. The size of the calibration and evaluation sets was 5K, since Imagenet-V2 is a smaller dataset. The result shows that our method can still provide coverage even for models trained on different distributions, as long as the conformal calibration set comes from the new distribution. The variance of the coverage is higher due to having less data.
+
+3.3 EXPERIMENT 3: SET SIZES OF N A I V E, APS, AND RAPS ON IMAGENET
+
+We investigate the effect of regularization in more detail. For three values of $\lambda$ , we collected the set sizes produced by each of naive, APS, and RAPS and report their histograms in Figure 4.
+
+# 3.4 EXPERIMENT 4: ADAPTIVENESS OF RAPS ON IMAGENET
+
+We now show that RAPS sets are smaller for easy images than hard ones, addressing the adaptiveness desideratum. Table 4 reports the size-stratified coverages of RAPS at the $90 \%$ level with $k _ { r e g } = 5$ and different choices of $\lambda$ . When $\lambda$ is small, RAPS allows sets to be large. But when $\lambda = 1$ , RAPS clips sets to be a maximum of size 5. Table 7 (in the Appendix) stratifies by image difficulty, showing that RAPS sets are small for easy examples and large for hard ones. Experiments 3 and 4 together illustrate the tradeoff between adaptiveness and size: as the average set size decreases, the RAPS procedure truncates sets larger than the smallest fixed set that provides coverage, taming the heavy tail of the APS procedure. Since RAPS with large $\lambda$ undercovers hard examples, it must compensate by taking larger sets for easy examples to ensure the $1 - \alpha$ marginal coverage guarantee. However, the size only increases slightly since easy images are more common than hard ones, and the total probability mass can often exceed $\hat { \tau } _ { c c a l }$ by including only one more class. If this behavior is not desired, we can instead automatically pick $\lambda$ to optimize the adaptiveness of RAPS; see Section 4.
+
+Published as a conference paper at ICLR 2021   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Accuracy</td><td colspan="4">Coverage</td><td colspan="4">Size</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top K</td><td>Naive</td><td>APS</td><td>RAPS</td><td>Top K</td><td>Naive</td><td>APS</td><td>RAPS</td></tr><tr><td>ResNeXt101</td><td>0.678</td><td>0.874</td><td>0.900</td><td>0.888</td><td>0.899</td><td>0.899</td><td>7.48</td><td>43.0</td><td>50.8</td><td>6.18</td></tr><tr><td>ResNet152</td><td>0.67</td><td>0.876</td><td>0.899</td><td>0.896</td><td>0.900</td><td>0.900</td><td>7.18</td><td>25.8</td><td>27.2</td><td>5.69</td></tr><tr><td>ResNet101</td><td>0.657</td><td>0.859</td><td>0.901</td><td>0.894</td><td>0.900</td><td>0.898</td><td>9.21</td><td>28.7</td><td>30.7</td><td>6.93</td></tr><tr><td>ResNet50</td><td>0.634</td><td>0.847</td><td>0.898</td><td>0.894</td><td>0.899</td><td>0.900</td><td>10.3</td><td>30.3</td><td>32.3</td><td>7.80</td></tr><tr><td>ResNet18</td><td>0.572</td><td>0.802</td><td>0.902</td><td>0.895</td><td>0.900</td><td>0.900</td><td>17.5</td><td>35.3</td><td>37.4</td><td>13.3</td></tr><tr><td>DenseNet161</td><td>0.653</td><td>0.862</td><td>0.902</td><td>0.895</td><td>0.901</td><td>0.901</td><td>8.6</td><td>29.9</td><td>32.4</td><td>6.93</td></tr><tr><td>VGG16</td><td>0.588</td><td>0.817</td><td>0.902</td><td>0.897</td><td>0.900</td><td>0.899</td><td>15.1</td><td>31.9</td><td>32.8</td><td>11.2</td></tr><tr><td>Inception</td><td>0.573</td><td>0.797</td><td>0.900</td><td>0.893</td><td>0.900</td><td>0.899</td><td>21.8</td><td>145.0</td><td>155.0</td><td>20.5</td></tr><tr><td>ShuffleNet</td><td>0.559</td><td>0.781</td><td>0.899</td><td>0.892</td><td>0.900</td><td>0.899</td><td>26.0</td><td>66.2</td><td>71.7</td><td>22.5</td></tr></table>
+
+Table 2: Results on Imagenet-V2. We report coverage and size of the optimal, randomized fixed sets, naive, APS, and RAPS sets for nine different Imagenet classifiers. The median-of-means for each column is reported over 100 different trials at the $10 \%$ level. See Section 3.2 for full details.
+
+![](images/a536df49415615fe82ae947fd6220877a37f4fe1d2ee3aa82499fe99157eb273.jpg)  
+Figure 4: Set sizes produced with ResNet-152. See Section 3.3 for details.
+
+# 3.5 EXPERIMENT 5: CHOICE OF TUNING PARAMETERS
+
+While any value of the tuning parameters $\lambda$ and $k _ { r e g }$ lead to coverage (Proposition 1), some values will lead to smaller sets. In Experiments 1 and 2, we chose $k _ { r e g }$ and $\lambda$ adaptively from data (see Appendix E), achieving strong results for all models and choices of the coverage level. Table 3 gives the performance of RAPS with many choices of $k _ { r e g }$ and $\lambda$ for ResNet-152.
+
+# 4 ADAPTIVENESS AND CONDITIONAL COVERAGE
+
+In this section, we point to a definition of adaptiveness that is more natural for the image classification setting than the existing notion of conditional coverage. We show that APS does not satisfy conditional coverage, and that RAPS with small $\lambda$ outperforms it in terms of adaptiveness.
+
+We say that a set-valued predictor $\mathcal { C } : \mathbb { R } ^ { d } \to 2 ^ { \mathcal { V } }$ satisfies exact conditional coverage if $P ( Y \in$ ${ \mathcal { C } } ( X ) { \\mathrm { ~ } } { \mathrm { ~ } } | { \mathrm { ~ } } X = x ) = 1 - \alpha$ for each $x$ . Distribution-free guarantees on conditional coverage are impossible (Vovk, 2012; Lei & Wasserman, 2014), but many algorithms try to satisfy it approximately (Romano et al., 2019; 2020; Cauchois et al., 2020). In a similar spirit, Tibshirani et al. (2019) suggest a notion of local conditional coverage, where one asks for coverage in a neighborhood of each point, weighted according to a chosen kernel. Cauchois et al. (2020) introduce the worst-case slab metric for measuring violations of the conditional coverage property. We present a different way of measuring violations of conditional coverage.
+
+Proposition 3. Suppose $P ( Y \in { \mathcal { C } } ( X ) \mid X = x ) = 1 - \alpha$ for each $x \in \mathbb { R } ^ { d }$ . Then, $P ( Y \in { \mathcal { C } } ( X ) \mid \{ | C ( X ) | \in { \mathcal { A } } \} ) = 1 - \alpha$ for any $\mathcal { A } \subset \{ 0 , 1 , 2 , \dotsc \}$ .
+
+In words, if conditional coverage holds, then coverage holds after stratifying by set size. Based on this result, In Appendix E, we introduce the size-stratified coverage violation criterion, a simple and pragmatic way of quantifying adaptiveness. Then, we automatically tune $\lambda$ on this metric so RAPS markedly outperforms the adaptiveness of APS (see Table 8).
+
+<table><tr><td>kreg|入</td><td>0</td><td>1e-4</td><td>1e-3</td><td>0.01</td><td>0.02</td><td>0.05</td><td>0.2</td><td>0.5</td><td>0.7</td><td>1.0</td></tr><tr><td>1</td><td>11.2</td><td>10.2</td><td>7.0</td><td>3.6</td><td>2.9</td><td>2.3</td><td>2.1</td><td>2.3</td><td>2.2</td><td>2.2</td></tr><tr><td>2</td><td>11.2</td><td>10.2</td><td>7.1</td><td>3.7</td><td>3.0</td><td>2.4</td><td>2.1</td><td>2.3</td><td>2.2</td><td>2.2</td></tr><tr><td>5</td><td>11.2</td><td>10.2</td><td>7.2</td><td>3.9</td><td>3.4</td><td>2.9</td><td>2.6</td><td>2.5</td><td>2.5</td><td>2.5</td></tr><tr><td>10</td><td>11.2</td><td>10.2</td><td>7.4</td><td>4.5</td><td>4.0</td><td>3.6</td><td>3.4</td><td>3.4</td><td>3.4</td><td>3.4</td></tr><tr><td>50</td><td>11.2</td><td>10.6</td><td>8.7</td><td>7.2</td><td>7.0</td><td>6.9</td><td>6.9</td><td>6.9</td><td>6.9</td><td>6.9</td></tr></table>
+
+Table 3: Set sizes of RAPS with parameters $k _ { r e g }$ and $\lambda$ , a ResNet-152, and coverage level $90 \%$   
+
+<table><tr><td></td><td colspan="2">入=0</td><td colspan="2">λ= 0.001</td><td colspan="2">入=0.01</td><td colspan="2">入=0.1</td><td colspan="2">入=1</td></tr><tr><td>size</td><td>cnt</td><td>cvg</td><td>cnt</td><td>cvg</td><td>cnt</td><td>cvg</td><td>cnt</td><td>cvg</td><td>cnt</td><td>cvg</td></tr><tr><td>0 to1</td><td>11627</td><td>0.88</td><td>11539</td><td>0.88</td><td>11225</td><td>0.89</td><td>10476</td><td>0.92</td><td>10027</td><td>0.93</td></tr><tr><td>2 to3</td><td>3687</td><td>0.91</td><td>3702</td><td>0.91</td><td>3741</td><td>0.92</td><td>3845</td><td>0.93</td><td>3922</td><td>0.94</td></tr><tr><td>4 to 6</td><td>1239</td><td>0.91</td><td>1290</td><td>0.91</td><td>1706</td><td>0.92</td><td>4221</td><td>0.89</td><td>6051</td><td>0.83</td></tr><tr><td>7 to 10</td><td>688</td><td>0.93</td><td>765</td><td>0.93</td><td>1314</td><td>0.91</td><td>1436</td><td>0.71</td><td>0</td><td></td></tr><tr><td>11 to 100</td><td>2207</td><td>0.94</td><td>2604</td><td>0.93</td><td>2014</td><td>0.86</td><td>22</td><td>0.59</td><td>0</td><td></td></tr><tr><td>101 to 1000</td><td>552</td><td>0.97</td><td>100</td><td>0.90</td><td>0</td><td></td><td>0</td><td></td><td>0</td><td></td></tr></table>
+
+Table 4: Coverage conditional on set size. We report average coverage of images stratified by the size of the set output by RAPS using a ResNet-152 for varying $\lambda$ . The marginal coverage rate is $90 \%$ .
+
+In Table 4, we report on the coverage of APS and RAPS, stratified by the size of the prediction set. Turning our attention to the $\lambda = 0$ column, we see that when APS outputs a set of size $1 0 1 - 1 0 0 0$ , APS has coverage $9 7 \%$ , substantially higher than $90 \%$ nominal rate. By Proposition 3, we conclude that APS is not achieving exact conditional coverage, because the scores are far from the oracle probabilities. The APS procedure still achieves marginal coverage by overcovering hard examples and undercovering easy ones, an undesirable behavior. Alternatively, RAPS can be used to regularize the set sizes—for $\lambda = . 0 0 1$ to $\lambda = . 0 1$ the coverage stratified by set size is more balanced. In summary, even purely based on the adaptiveness desideratum, RAPS with light regularization is preferable to APS. Note that as the size of the training data increases, as long as $\hat { \pi }$ is consistent, naive and APS will become more stable, and so we expect less regularization will be needed.
+
+Lastly, we argue that conditional coverage is a poor notion of adaptiveness when the best possible model (i.e., one fit on infinite data) has high accuracy. Given such a model, the oracle procedure from Romano et al. (2020) would return the correct label with probability $1 - \alpha$ and the empty set with probability $\alpha$ . That is, having correct conditional coverage for high-signal problems where $Y$ is perfectly determined by $X$ requires a perfect classifier. In our experiments on ImageNet, APS does not approximate this behavior. Therefore, conditional coverage isn’t the right goal for prediction sets with realistic sample sizes. Proposition 3 suggests a relaxation. We could require that we have the right coverage, no matter the size of the prediction set: $P ( Y \in { \mathcal { C } } ( X ) \mid \{ | C ( x ) | \in A \} ) \geq 1 - \alpha$ for any $\mathcal { A } \subset \{ 0 , 1 , 2 , . . . \}$ ; Appendix E.2 develops this idea. We view this as a promising way to reason about adaptiveness in high-signal problems such as image classification.
+
+# 5 DISCUSSION
+
+For classification tasks with many possible labels, our method enables a researcher to take any base classifier and return predictive sets guaranteed to achieve a pre-specified error level, such as $90 \%$ , while retaining small average size. It is simple to deploy, so it is an attractive, automatic way to quantify the uncertainty of image classifiers—an essential task in such settings as medical diagnostics, self-driving vehicles, and flagging dangerous internet content. Predictive sets in computer vision (from RAPS and other conformal methods) have many further uses, since they systematically identify hard test-time examples. Finding such examples is useful in active learning where one only has resources to label a small number of points. In a different direction, one can improve efficiency of a classifier by using a cheap classifier outputting a prediction set first, and an expensive one only when the cheap classifier outputs a large set (a cascade; see, e.g., Li et al. 2015), and Fisch et al. (2021) for an implementation of conformal prediction in this setting. One can also use predictive sets during model development to identify failure cases and outliers and suggest strategies for improving its performance. Prediction sets are most useful for problems with many classes; returning to our initial medical motivation, we envision RAPS could be used by a doctor to automatically screen for a large number of diseases (e.g. via a blood sample) and refer the patient to relevant specialists.
+
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+
+# A PROOFS
+
+Theorem $^ { l }$ . Let $\begin{array} { r } { s ( x , u , y ) = \operatorname* { i n f } _ { \tau } \{ y \in \mathcal { C } ( x , u , \tau ) \} } \end{array}$ , and let $s _ { i } = s ( X _ { i } , U _ { i } , Y _ { i } )$ for $i = 1 , \ldots , n$ . Then
+
+$$
+\{ y : s ( x , u , y ) \leq \tau \} = \{ y : y \in \mathcal { C } ( x , u , \tau ) \}
+$$
+
+because $\mathcal { C } ( x , u , \tau )$ is a finite set growing in $\tau$ by the assumption in Eq. (2). Thus,
+
+$$
+\tau : | \{ i : s _ { i } \leq \tau \} | \geq \lceil ( 1 - \alpha ) ( n + 1 ) \rceil \} = \left\{ \tau : \frac { | \{ i : Y _ { i } \in \mathcal { C } ( X _ { i } , U _ { i } , \tau ) \} | } { n } \geq \frac { \lceil ( n + 1 ) ( 1 - \alpha ) \rceil } { n } \right\} .
+$$
+
+Considering the left expression, the infimum over $\tau$ of the set on the left hand side is the $\lceil ( 1 ~ - ~ \alpha ) ( n ~ + ~ 1 ) \rceil$ smallest value of the $s _ { i }$ , so this is the value of $\hat { \tau } _ { \mathrm { c c a l } }$ . Since $s _ { 1 } , \ldots , s _ { n } , s ( X _ { n + 1 } , U _ { n + 1 } , Y _ { n + 1 } )$ are exchangeable random variables, $\begin{array} { r l } { | \{ i } ^ { \cdot } \cdot  & { { } : } \end{array}$ $s ( X _ { n + 1 } , U _ { n + 1 } , Y _ { n + 1 } ) \ > \ s _ { i } \} |$ is stochastically dominated by the discrete uniform distribution on $\{ 0 , 1 , \ldots , n \}$ . We thus have that
+
+$$
+\begin{array} { r l } & { \mathsf { P } \left( Y _ { n + 1 } \notin \mathcal { C } ( X _ { n + 1 } , U _ { n + 1 } , \hat { \tau } _ { \mathrm { c e a l } } ) \right) = P \left( s ( X _ { n + 1 } , U _ { n + 1 } , Y _ { n + 1 } ) > \hat { \tau } _ { \mathrm { c e a l } } \right) } \\ & { \qquad = P \left( | \{ i : s ( X _ { n + 1 } , U _ { n + 1 } , Y _ { n + 1 } ) > s _ { i } \} | \ge \lceil ( n + 1 ) ( 1 - \alpha ) \rceil \right) } \\ & { \qquad = P \left( \frac { \left[ \{ i : s ( X _ { n + 1 } , U _ { n + 1 } , Y _ { n + 1 } ) > s _ { i } \} \right] } { n + 1 } \ge \frac { \lceil ( n + 1 ) ( 1 - \alpha ) \rceil } { n + 1 } \right) } \\ & { \qquad \le \alpha . } \end{array}
+$$
+
+Proposition $^ { l }$ . The lower bound follows from Theorem 1. To prove the upper bound, using the result from Theorem 2.2 of Lei et al. (2018) it suffices to show that the variables $s ( X _ { i } , U _ { i } , Y _ { i } ) =$ $\operatorname* { i n f } \{ \tau : Y _ { i } \in \mathcal { C } ( X _ { i } , U _ { i } , \tau ) \}$ are almost surely distinct. To this end, note that that
+
+$$
+s ( X _ { i } , U _ { i } , Y _ { i } ) = \rho _ { X _ { i } } ( Y _ { i } ) + \hat { \pi } _ { X _ { i } } ( Y _ { i } ) \cdot U _ { i } + \lambda ( o _ { X _ { i } } ( Y _ { i } ) - k _ { r e g } ) ^ { + } ,
+$$
+
+and due to the middle term of the sum, these values are distinct almost surely provided $\hat { \pi } _ { X _ { i } } ( Y _ { i } ) >$ 0. □
+
+Proposition 2. We first show that $\hat { \tau } _ { \mathrm { c c a l } } \leq 1 + k ^ { * } - k _ { r e g }$ . Note that since at least $\lceil ( 1 - \alpha ) ( n + 1 ) \rceil$ of the conformal calibration points are covered by a set of size $k ^ { * }$ , at least $\lceil ( 1 - \alpha ) ( n + 1 ) \rceil$ of the $E _ { i }$ in Algorithm 2 are less than or equal to $1 + k ^ { * } - k _ { r e g }$ . Thus, by the definition of $\hat { \tau } _ { \mathrm { c c a l } }$ , we have that it is less than or equal to $1 + k ^ { * } - k _ { r e g }$ . Then, note that by the definition of $\mathcal { C } ^ { * }$ in Eq. (4), we have that
+
+$$
+| { \mathcal { C } } ^ { * } ( X _ { n + 1 } , U _ { n + 1 } , { \hat { \tau } } _ { \mathrm { c c a l } } ) | \leq k ^ { * } .
+$$
+
+as long as $\hat { \tau } _ { \mathrm { c c a l } } \leq 1 + k ^ { * } - k _ { r e q }$ , since for the $k ^ { * } + 1$ most likely class, the sum in Eq. (4) will exceed $\lambda \cdot ( 1 + k ^ { * } - k _ { r e g } ) = ( 1 + k ^ { * } - k _ { r e g } ) \geq \hat { \tau } _ { \mathrm { c c a l } } .$ , and so the $k ^ { * } + 1$ class will not be in the set.
+
+Proposition 3. Suppose $P ( Y \in { \mathcal { C } } ( X ) \mid X = x ) = 1 - \alpha$ for each $x \in \mathbb { R } ^ { d }$ . Then,
+
+$$
+\begin{array} { r l }  P ( Y \in \mathcal { C } ( X ) \mid | C ( X ) | \in \mathcal { A } ) = \displaystyle \frac { \int _ { x } P ( Y \in \mathcal { C } ( x ) \mid X = x \} { P ( | \mathcal { C } ( X ) | \in \mathcal { A } ) } \mathbb { I } _ { \{ | \mathcal { C } ( x ) | \in \mathcal { A } \} } d P ( x ) } & { } \\ { = \displaystyle \frac { \int _ { x } ( 1 - \alpha ) \mathbb { I } _ { \{ | \mathcal { C } ( x ) | \in \mathcal { A } \} } d P ( x ) } { P ( | \mathcal { C } ( X ) | \in \mathcal { A } ) } } & { } \\ { = 1 - \alpha . } \end{array}
+$$
+
+Published as a conference paper at ICLR 2021   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Accuracy</td><td colspan="4">Coverage</td><td colspan="4">Size</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top K</td><td>Naive</td><td>APS</td><td>RAPS</td><td>Top K</td><td>Naive</td><td>APS</td><td>RAPS</td></tr><tr><td>ResNeXt101</td><td>0.794</td><td>0.945</td><td>0.905</td><td>0.938</td><td>0.950</td><td>0.950</td><td>5.64</td><td>36.4</td><td>46.3</td><td>4.21</td></tr><tr><td>ResNet152</td><td>0.783</td><td>0.940</td><td>0.950</td><td>0.943</td><td>0.950</td><td>0.950</td><td>6.36</td><td>19.6</td><td>22.5</td><td>4.40</td></tr><tr><td>ResNet101</td><td>0.774</td><td>0.936</td><td>0.950</td><td>0.944</td><td>0.950</td><td>0.950</td><td>6.79</td><td>20.6</td><td>23.2</td><td>4.79</td></tr><tr><td>ResNet50</td><td>0.762</td><td>0.929</td><td>0.951</td><td>0.943</td><td>0.950</td><td>0.950</td><td>8.12</td><td>22.9</td><td>26.2</td><td>5.57</td></tr><tr><td>ResNet18</td><td>0.698</td><td>0.891</td><td>0.950</td><td>0.943</td><td>0.950</td><td>0.950</td><td>16.0</td><td>28.9</td><td>33.2</td><td>11.7</td></tr><tr><td>DenseNet161</td><td>0.772</td><td>0.936</td><td>0.950</td><td>0.942</td><td>0.950</td><td>0.950</td><td>6.84</td><td>23.4</td><td>28.0</td><td>5.09</td></tr><tr><td>VGG16</td><td>0.716</td><td>0.904</td><td>0.950</td><td>0.943</td><td>0.950</td><td>0.950</td><td>12.9</td><td>24.6</td><td>27.8</td><td>8.98</td></tr><tr><td>Inception</td><td>0.695</td><td>0.887</td><td>0.950</td><td>0.937</td><td>0.950</td><td>0.950</td><td>20.3</td><td>142.0</td><td>168.0</td><td>18.5</td></tr><tr><td>ShuffleNet</td><td>0.694</td><td>0.883</td><td>0.950</td><td>0.940</td><td>0.950</td><td>0.950</td><td>19.3</td><td>58.7</td><td>71.6</td><td>16.3</td></tr></table>
+
+Table 5: Results on Imagenet-Val. We report coverage and size of the optimal, randomized fixed sets, naive, APS, and RAPS sets for nine different Imagenet classifiers. The median-of-means for each column is reported over 100 different trials. See Section 3.1 for full details.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Accuracy</td><td colspan="4">Coverage</td><td colspan="4">Size</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top K</td><td>Naive</td><td>APS</td><td>RAPS</td><td>Top K</td><td>Naive</td><td>APS</td><td>RAPS</td></tr><tr><td>ResNeXt101</td><td>0.678</td><td>0.875</td><td>0.950</td><td>0.937</td><td>0.950</td><td>0.950</td><td>21.7</td><td>86.2</td><td>107.0</td><td>18.5</td></tr><tr><td>ResNet152</td><td>0.670</td><td>0.876</td><td>0.951</td><td>0.944</td><td>0.950</td><td>0.950</td><td>21.3</td><td>51.0</td><td>56.6</td><td>16.2</td></tr><tr><td>ResNet101</td><td>0.656</td><td>0.86</td><td>0.95</td><td>0.944</td><td>0.950</td><td>0.949</td><td>25.7</td><td>55.8</td><td>63.1</td><td>19.1</td></tr><tr><td>ResNet50</td><td>0.634</td><td>0.847</td><td>0.949</td><td>0.944</td><td>0.949</td><td>0.950</td><td>29.5</td><td>58.6</td><td>65.9</td><td>21.5</td></tr><tr><td>ResNet18</td><td>0.572</td><td>0.802</td><td>0.950</td><td>0.942</td><td>0.950</td><td>0.949</td><td>48.3</td><td>65.0</td><td>74.0</td><td>35.3</td></tr><tr><td>DenseNet161</td><td>0.653</td><td>0.861</td><td>0.951</td><td>0.941</td><td>0.950</td><td>0.949</td><td>25.9</td><td>60.0</td><td>72.7</td><td>20.4</td></tr><tr><td>VGG16</td><td>0.588</td><td>0.816</td><td>0.950</td><td>0.943</td><td>0.950</td><td>0.949</td><td>38.5</td><td>57.8</td><td>63.9</td><td>26.4</td></tr><tr><td>Inception</td><td>0.573</td><td>0.797</td><td>0.950</td><td>0.943</td><td>0.949</td><td>0.950</td><td>73.1</td><td>253.0</td><td>275.0</td><td>70.2</td></tr><tr><td>ShuffleNet</td><td>0.560</td><td>0.781</td><td>0.950</td><td>0.941</td><td>0.949</td><td>0.949</td><td>80.0</td><td>125.0</td><td>140.0</td><td>67.4</td></tr></table>
+
+Table 6: Results on Imagenet-V2. We report coverage and size of the optimal, randomized fixed sets, naive, APS, and RAPS sets for nine different Imagenet classifiers. The median-of-means for each column is reported over 100 different trials at the $5 \%$ level. See Section 3.2 for full details.
+
+# B RANDOMIZED PREDICTORS
+
+The reader may wonder why we choose to use a randomized procedure. The randomization is needed to achieve $1 - \alpha$ coverage exactly, which we will explain via an example. Note that the randomization is of little practical importance, since the predictive set output by the randomized procedure will differ from the that of the non-randomized procedure by at most one element.
+
+Turning to an example, assume for a particular input image we expect a set of size $k$ to have $9 1 \%$ coverage, and a set of size $k - 1$ to have $8 9 \%$ coverage. In order to achieve our desired coverage of $9 0 \%$ , we randomly choose size $k$ or $k - 1$ with equal probability. In general, the probabilities will not be equal, but rather chosen so the weighted average of the two coverages is exactly $9 0 \%$ . If a user of our method desires deterministic sets, it is easy to turn off this randomization with a single flag, resulting in slightly conservative sets.
+
+# C IMAGENET AND IMAGENETV2 RESULTS FOR $\alpha = 5 \%$
+
+We repeated Experiments 1 and 2 with $\alpha = 5 \%$ . See the results in Tables 5 and 6.
+
+# D COVERAGE AND SIZE CONDITIONAL ON IMAGE DIFFICULTY
+
+In order to probe the adaptiveness properties of APS and RAPS we stratified coverage and size by image difficulty (the position of the true label in the list of most likely to least likely classes, based on the classifier predictions) in Table 7. With increasing $\lambda$ , coverage decreases for more difficult images and increases for easier ones. In the most difficult regime, even though APS can output large sets, those sets still rarely contain the true class. This suggests regularization is a sensible way to stabilize the sets. As a final word on Table 7, notice that as $\lambda$ increases, coverage improves for the more common medium-difficulty examples, although not for very rare and difficult ones.
+
+<table><tr><td></td><td></td><td colspan="2">入=0</td><td colspan="2">入=0.001</td><td colspan="2">入= 0.01</td><td colspan="2">入=0.1</td><td colspan="2">入=1</td></tr><tr><td>difficulty</td><td>count</td><td>cvg</td><td>SZ</td><td>cvg</td><td>SZ</td><td>cvg</td><td>SZ</td><td>cvg</td><td>SZ</td><td>cvg</td><td>SZ</td></tr><tr><td>1</td><td>15668</td><td>0.95</td><td>5.2</td><td>0.95</td><td>3.8</td><td>0.96</td><td>2.5</td><td>0.97</td><td>2.0</td><td>0.98</td><td>2.0</td></tr><tr><td>2 to3</td><td>2578</td><td>0.78</td><td>15.7</td><td>0.78</td><td>10.5</td><td>0.80</td><td>6.0</td><td>0.84</td><td>3.9</td><td>0.86</td><td>3.6</td></tr><tr><td>4 to6</td><td>717</td><td>0.68</td><td>31.7</td><td>0.68</td><td>19.7</td><td>0.70</td><td>9.7</td><td>0.71</td><td>5.3</td><td>0.64</td><td>4.4</td></tr><tr><td>7 to 10</td><td>334</td><td>0.63</td><td>41.0</td><td>0.63</td><td>24.9</td><td>0.60</td><td>11.6</td><td>0.22</td><td>5.7</td><td>0.00</td><td>4.5</td></tr><tr><td>11 to 100</td><td>622</td><td>0.55</td><td>57.8</td><td>0.51</td><td>34.1</td><td>0.26</td><td>14.7</td><td>0.00</td><td>6.4</td><td>0.00</td><td>4.6</td></tr><tr><td>101 to 1000</td><td>81</td><td>0.23</td><td>96.7</td><td>0.00</td><td>51.6</td><td>0.00</td><td>19.1</td><td>0.00</td><td>7.1</td><td>0.00</td><td>4.7</td></tr></table>
+
+Table 7: Coverage and size conditional on difficulty. We report coverage and size of RAPS sets using ResNet152 with $k _ { r e g } = 5$ and varying $\lambda$ (recall that $\lambda = 0$ is the APS procedure). The desired coverage level is $90 \%$ The ‘difficulty’ is the ranking of the true class’s estimated probability.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Violation at α = 10%</td><td colspan="2">Violationatα= 5%</td></tr><tr><td>APS</td><td>RAPS</td><td>APS</td><td>RAPS</td></tr><tr><td>ResNeXt101</td><td>0.090</td><td>0.049</td><td>0.048</td><td>0.021</td></tr><tr><td>ResNet152</td><td>0.069</td><td>0.038</td><td>0.037</td><td>0.017</td></tr><tr><td>ResNet101</td><td>0.073</td><td>0.041</td><td>0.038</td><td>0.017</td></tr><tr><td>ResNet50</td><td>0.069</td><td>0.037</td><td>0.037</td><td>0.016</td></tr><tr><td>ResNet18</td><td>0.046</td><td>0.025</td><td>0.032</td><td>0.019</td></tr><tr><td>DenseNet161</td><td>0.080</td><td>0.047</td><td>0.040</td><td>0.016</td></tr><tr><td>VGG16</td><td>0.046</td><td>0.022</td><td>0.030</td><td>0.022</td></tr><tr><td>Inception</td><td>0.085</td><td>0.045</td><td>0.043</td><td>0.023</td></tr><tr><td>ShuffleNet</td><td>0.061</td><td>0.033</td><td>0.035</td><td>0.020</td></tr></table>
+
+Table 8: Adaptiveness results after automatically tuning λ. We report the median size-stratified coverage violations of APS and RAPS over 10 trials. See Appendix E.2 for experimental details.
+
+# E CHOOSING $k _ { r e g }$ AND $\lambda$ TO OPTIMIZE SET SIZE AND ADAPTIVENESS
+
+This section describes two procedures for picking $k _ { r e g }$ and $\lambda$ that optimize for set size or adaptiveness, outperforming APS in both cases.
+
+# E.1 OPTIMIZING SET SIZE WITH RAPS
+
+# Algorithm 4 Adaptive Fixed-K
+
+Input: $\alpha$ $x ; I \in \{ 1 , . . . , K \} ^ { n \times K }$ , and one-hot $y \in \{ 0 , 1 \} ^ { K }$ corresponding respectively to the classes from highest to lowest estimated probability mass, and labels for each of $n$ examples in the dataset   
+1: procedure GET-KSTAR $( \alpha , I , y )$   
+2: for $i \in \{ 1 , \cdots , n \}$ do   
+3: $L _ { i } \gets \{ \ : j : I _ { i , j } = y _ { i } \ : \}$   
+4: $\hat { k } ^ { * } \gets \mathrm { t h e } \left\lceil ( 1 - \alpha ) ( 1 + n ) \right\rceil$ largest value in $\{ L _ { i } \} _ { i = 1 } ^ { n }$   
+5: return $\hat { k } ^ { * }$
+
+Output: The estimate of the smallest fixed size set that achieves coverage, $\hat { k } ^ { * }$
+
+To produce Tables 1, 5, 2, and 6, we chose $k _ { r e g }$ and $\lambda$ adaptively. This required an extra data splitting step, where a small amount of tuning data $\left\{ x _ { i } , y _ { i } \right\} _ { i = 1 } ^ { m }$ were used to estimate $k ^ { * }$ , and then $k _ { r e g }$ is set to $k ^ { * }$ . Taking $m \approx 1 0 0 0$ was sufficient, since the algorithm is fairly insensitive to $k _ { r e g }$ (see Table 3). Then, $\hat { k } ^ { * }$ was calculated with Algorithm 4. We produced the Imagenet V2 tables with $m = 1 0 0 0$ and the Imagenet tables with $m = 1 0 0 0 0$ .
+
+After choosing $\hat { k } ^ { * }$ , we chose $\lambda$ to have small set size. We used the same tuning data to pick $\hat { k } ^ { * }$ and $\lambda$ for simplicity (this does not invalidate our coverage guarantee since conformal calibration still uses fresh data). A coarse grid search on $\lambda$ sufficed, since small parameter variations have little impact on RAPS. For example, we chose the $\lambda \in \{ 0 . 0 0 1 , 0 . 0 1 , 0 . 1 , 0 . 2 , 0 . 5 \}$ that achieved the smallest size on the $m$ holdout samples in order to produce Tables 1, 5, 2, and 6. We include a subroutine that automatically chooses $\hat { k } ^ { * }$ and $\lambda$ to optimize size in our GitHub codebase.
+
+# E.2 OPTIMIZING ADAPTIVENESS WITH RAPS
+
+In this appendix, we show empirically that RAPS with an automatically chosen set of $k _ { r e g }$ and $\lambda$ improves the adaptiveness of APS. Recall our discussion in Section 4 and Proposition 3, wherein we propose size-stratified coverage as a useful definition of adaptiveness in image classification. After picking $k _ { r e g }$ as in Appendix E, we can choose $\lambda$ using the same tuning data to optimize this notion of adaptiveness.
+
+We now describe a particular manifestation of our adaptiveness criterion that we will use to optimize $\lambda$ . Consider disjoint set-size strata $\{ S _ { i } \} _ { i = 1 } ^ { i = s }$ , where $\bigcup _ { j = 1 } ^ { j = s } S _ { i } = \{ 1 , \ldots , | \mathcal { V } | \}$ . Then define the indexes of examples stratified by the prediction set size of each example from algorithm $\mathcal { C }$ as ${ \mathcal { I } } _ { j } = \{ i :$ $| { \mathcal { C } } ( X _ { i } , Y _ { i } , U _ { i } ) | \in S _ { j } \}$ . Then we can define the size-stratified coverage violation of an algorithm $\mathcal { C }$ on strata $\{ S \} _ { i = 1 } ^ { i = s }$ as
+
+$$
+\mathrm { S S C V } ( \mathscr C , \{ \mathrm S \} _ { \mathrm { j = 1 } } ^ { \mathrm { j = s } } ) = \operatorname* { s u p } _ { \mathrm { j } } \bigg | \frac { | \{ \mathrm { i } : \mathrm { Y } _ { \mathrm { i } } \in \mathscr C ( \mathrm { X } _ { \mathrm { i } } , \mathrm Y _ { \mathrm { i } } , \mathrm U _ { \mathrm { i } } ) , \mathrm { i } \in \mathcal J _ { \mathrm { j } } \} | } { | \mathcal J _ { \mathrm { j } } | } - ( 1 - \alpha ) \bigg | .
+$$
+
+In words, Eq. (5) is the worst-case deviation of $\mathcal { C }$ from exact coverage when it outputs sets of a certain size. Computing the size-stratified coverage violation thus only requires post-stratifying the results of $\mathcal { C }$ on a set of labeled examples. If conditional coverage held, the worst stratum coverage violation would be 0 by Proposition 3.
+
+To maximize adaptiveness, we’d like to choose $\lambda$ to minimize the size-stratified coverage violation of RAPS. Write $\mathcal { C } _ { \lambda }$ to mean the RAPS procedure for a fixed choice of $k _ { r e g }$ and $\lambda$ . Then we would like to pick
+
+$$
+\lambda = \underset { \lambda ^ { \prime } } { \arg \operatorname* { m i n } } \mathrm { S S C V } ( \mathcal C _ { \lambda ^ { \prime } } , \{ \mathrm { S } \} _ { \mathrm { j = 1 } } ^ { \mathrm { j = s } } ) .
+$$
+
+In our experiments, we choose a relatively coarse partitioning of the possible set sizes: 0-1, 2-3, 4- 10, 11-100, and 101-1000. Then, we chose the $\lambda \in \{ 0 . 0 0 0 0 1 , 0 . 0 0 0 1 , 0 . 0 0 0 8 , 0 . 0 0 1 , 0 . 0 0 1 5 , 0 . 0 0 2 \}$ which minimized the size-stratified coverage violation on the tuning set. The results in Table 8 show RAPS always outperforms the adaptiveness of APS on the test set, even with this coarse, automated choice of parameters. The table reports the median size-stratified coverage violation over 10 independent trials of APS and RAPS with automated parameter tuning.
+
+# F COMPARISON WITH LEAST AMBIGUOUS SET-VALUED CLASSIFIERS
+
+In this section, we compare RAPS to the Least Ambiguous Set-valued Classifier (LAC) method introduced in Sadinle et al. (2019), an alternative conformal procedure that is designed to have small sets. The LAC method provable gives the smallest possible average set size in the case where the input probabilities are correct, with the idea that these sets should be small even when the estimated probabilities are only approximately correct. In the notation of this paper, the LAC method considers nested sets of the following form:
+
+$$
+\begin{array} { r } { \mathcal { C } ^ { \mathtt { L A C } } ( x , \tau ) : = \{ y : \hat { \pi } _ { x } ( y ) \geq 1 - \tau \} , } \end{array}
+$$
+
+which can be calibrated using as before in using $\hat { \tau } _ { \mathrm { c c a l } }$ from Eq. (3).
+
+We first compare naive, APS, RAPS, and LAC in terms of power and coverage in Table 9. In this experiment, we tuned RAPS to have small set size as described in Appendix E.1. We see that LAC also achieves correct coverage, as expected since it is a conformal method and satisfies the guarantee from Theorem 1. We further see that it has systematically smaller sets that RAPS, although the difference is slight compared to the gap between APS and RAPS or APS and LAC.
+
+We next compare RAPS to LAC in terms of adaptiveness, tuning RAPS as in Section E.2. First, in Table 10, we report on the coverage of LAC for images of different difficulties, and see that LAC has dramatically worse coverage for hard images than for easy ones. Comparing this to RAPS in Table 7, we see that RAPS also has worse coverage for more difficult images, although the gap is much smaller for RAPS. Next, in Table 11, we report on the SSCV metric for of adaptiveness (and conditional coverage) for APS, RAPS, and LAC. We find that APS and RAPS have much better adaptiveness than LAC, with RAPS being the overall winner. The results of all of these comparisons are expected: LAC is not targetting adpativeness and instead trying to achieve the smallest possible set size. It succeeds at its goal, sacrificing adaptiveness to do so.
+
+<table><tr><td rowspan="2">Model</td><td colspan="2"> Accuracy</td><td colspan="6">Coverage</td><td colspan="5">Size</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top K</td><td>Naive</td><td>APS</td><td>RAPS</td><td></td><td>LAC</td><td>Top K</td><td>Naive</td><td>APS</td><td>RAPS</td><td>LAC</td></tr><tr><td>ResNeXt101</td><td>0.793</td><td>0.945</td><td>0.900</td><td>0.889</td><td>0.900</td><td>0.900</td><td>0.900</td><td>2.42</td><td>17.2</td><td>19.9</td><td></td><td>2.01</td><td>1.65</td></tr><tr><td>ResNet152</td><td>0.783</td><td>0.941</td><td>0.900</td><td>0.894</td><td>0.900</td><td></td><td>0.900</td><td>0.900</td><td>2.64</td><td>9.68</td><td>10.4</td><td>2.09</td><td>1.76</td></tr><tr><td>ResNet101</td><td>0.774</td><td>0.936</td><td>0.900</td><td>0.895</td><td>0.900</td><td>0.900</td><td>0.900</td><td></td><td>2.83</td><td>10.0</td><td>10.8</td><td>2.25</td><td>1.87</td></tr><tr><td>ResNet50</td><td>0.761</td><td>0.929</td><td>0.899</td><td>0.896</td><td>0.900</td><td>0.900</td><td>0.900</td><td></td><td>3.13</td><td>11.7</td><td>12.3</td><td>2.55</td><td>2.05</td></tr><tr><td>ResNet18</td><td>0.698</td><td>0.891</td><td>0.900</td><td>0.895</td><td>0.900</td><td>0.900</td><td></td><td>0.900</td><td>5.74</td><td>15.3</td><td>16.1</td><td>4.38</td><td>3.64</td></tr><tr><td>DenseNet161</td><td>0.771</td><td>0.936</td><td>0.900</td><td>0.894</td><td>0.900</td><td>0.900</td><td>0.900</td><td></td><td>2.84</td><td>11.2</td><td>12.0</td><td>2.29</td><td>1.90</td></tr><tr><td>VGG16</td><td>0.716</td><td>0.904</td><td>0.900</td><td>0.896</td><td>0.901</td><td>0.900</td><td>0.900</td><td></td><td>4.75</td><td>13.4</td><td>14.1</td><td>3.54</td><td>3.00</td></tr><tr><td>Inception</td><td>0.695</td><td>0.886</td><td>0.899</td><td>0.884</td><td>0.900</td><td>0.899</td><td>0.900</td><td></td><td>6.27</td><td>74.8</td><td>88.8</td><td>5.24</td><td>4.06</td></tr><tr><td>ShuffleNet</td><td>0.694</td><td>0.883</td><td>0.900</td><td>0.892</td><td>0.900</td><td>0.899</td><td>0.900</td><td></td><td>6.45</td><td>28.8</td><td>32.1</td><td>5.01</td><td>4.14</td></tr></table>
+
+Results on Imagenet-Val. We report coverage and size of the optimal, randomized fixed sets, n a i ve , AP S , RAP S , and the LAC sets for nine different Imagenet classifiers . The median-of-means for each column is reported over 1OO different trials at the $10 \%$ 3 . 1 for full details .
+
+<table><tr><td>difficulty</td><td>count</td><td>cvg</td><td>SZ</td></tr><tr><td>1</td><td>15668</td><td>1.00</td><td>1.5</td></tr><tr><td>2 to3</td><td>2578</td><td>0.81</td><td>2.6</td></tr><tr><td>4 to 6</td><td>717</td><td>0.23</td><td>3.0</td></tr><tr><td>7 to 10</td><td>334</td><td>0.00</td><td>2.9</td></tr><tr><td>11 to 100</td><td>622</td><td>0.00</td><td>2.7</td></tr><tr><td>101 to 1000</td><td>81</td><td>0.00</td><td>2.4</td></tr></table>
+
+Table 10: Coverage and size conditional on difficulty. We report coverage and size of the LAC sets for ResNet-152.
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">Violation at α= 10%</td><td colspan="3">Violation at α = 5%</td></tr><tr><td>APS</td><td>RAPS</td><td>LAC</td><td>APS</td><td>RAPS</td><td>LAC</td></tr><tr><td>ResNeXt101</td><td>0.086</td><td>0.047</td><td>0.217</td><td>0.047</td><td>0.022</td><td>0.127</td></tr><tr><td>ResNet152</td><td>0.067</td><td>0.039</td><td>0.156</td><td>0.04</td><td>0.021</td><td>0.141</td></tr><tr><td>ResNet101</td><td>0.075</td><td>0.060</td><td>0.152</td><td>0.039</td><td>0.015</td><td>0.120</td></tr><tr><td>ResNet50</td><td>0.071</td><td>0.042</td><td>0.131</td><td>0.037</td><td>0.014</td><td>0.109</td></tr><tr><td>ResNet18</td><td>0.050</td><td>0.024</td><td>0.196</td><td>0.031</td><td>0.021</td><td>0.059</td></tr><tr><td>DenseNet161</td><td>0.076</td><td>0.055</td><td>0.140</td><td>0.039</td><td>0.016</td><td>0.110</td></tr><tr><td>VGG16</td><td>0.051</td><td>0.023</td><td>0.165</td><td>0.029</td><td>0.019</td><td>0.070</td></tr><tr><td>Inception</td><td>0.086</td><td>0.042</td><td>0.181</td><td>0.043</td><td>0.023</td><td>0.135</td></tr><tr><td>ShuffleNet</td><td>0.058</td><td>0.030</td><td>0.192</td><td>0.033</td><td>0.019</td><td>0.045</td></tr></table>
+
+Table 11: Adaptiveness results after automatically tuning λ. We report the median SSCV of APS RAPS and LAC over 10 trials. See Appendix E.2 for experimental details.