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C.25 (0 , g1(0)) G1∈ M exists. Then, we denote by o1the origin, ˜ o1=o1+ (0, g0(0)) G0and ¯o1=o1+ (0, g1(0)) G1(see Figure 16). Denote ¯ α=∠(o1,˜o1, x0)−π/2 and ¯β=∠(˜o1, o1,¯o1), where ∠(A, B, C ) denotes the angle between the straight lines ABandBC. Note that ¯ α≤αand ¯β≤β. Next, we bound the distance between ˜ o1and... | https://arxiv.org/abs/2503.07220v1 |
≤∥g0(0)∥ cos(¯α+¯β)( 1 + ( ∥g0(0)∥/τ+ 1) sin2(¯α+¯β) ) since cos( x)≥1−x2/2, 1/(1−x)≤1 + 2 x, sin( x)≤x, and ¯ α+¯β <1, we have ∥g1(0)∥ ≤∥g0(0)∥ cos(¯α+¯β)( 1 + ( ∥g0(0)∥/τ+ 1) sin2(¯α+¯β) ) ∥g1(0)∥ ≤∥g0(0)∥ 1−(¯α+¯β)2/2( 1 + ( ∥g0(0)∥/τ+ 1)(¯ α+¯β)2) (155) ≤ ∥g0(0)∥(1 + (¯ α+¯β)2)( 1 + ( ∥g0(0)∥/τ+ 1)(¯ α+¯β)2) (156) ... | https://arxiv.org/abs/2503.07220v1 |
2012. [9] Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. Journal of Computer and system sciences , 1999. [10] Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural computation , 15(6):1373–1396, 2003. ... | https://arxiv.org/abs/2503.07220v1 |
theory for the information age, note 1. https://www.cs.cmu.edu/ ∼venkatg/teaching/CStheory- infoage/chap1-high-dim-space.pdf , 2012. [31] Nathan Halko, Per-Gunnar Martinsson, Yoel Shkolnisky, and Mark Tygert. An algorithm for the Principal Component Analysis of large data sets. SIAM Journal on Scientific computing , 33... | https://arxiv.org/abs/2503.07220v1 |
Approximating the Riemannian met- ric from point clouds via manifold moving least squares. arXiv preprint arXiv:2007.09885 , 2020. [53] Barak Sober and David Levin. Manifold approximation by moving least-squares projection (MMLS). Constructive Approximation , pages 1–46, 2019. [54] Charles J. Stone. Optimal rates of co... | https://arxiv.org/abs/2503.07220v1 |
Is a Good Foundation Necessary for Efficient Reinforcement Learning? The Computational Role of the Base Model in Exploration Dylan J. Foster∗ Microsoft ResearchZakaria Mhammedi† Google ResearchDhruv Rohatgi‡ MIT Abstract Language model alignment (or, reinforcement learning) techniques that leverage active exploration —... | https://arxiv.org/abs/2503.07453v2 |
like PPO (Schulman et al., 2017) and Online DPO (Xu et al., 2023; Guo et al., 2024), are data-inefficient due to their reliance on passive exploration. These techniques treat the pre-trained model as a policyand iteratively update it with reinforcement learning, but since there is no explicit mechanism to promote novel... | https://arxiv.org/abs/2503.07453v2 |
et al., 2024b; Kazemnejad et al., 2024). 1.1 Background: Online Alignment from Reward-Based Feedback To motivate our computational framework, we begin by formally introducing the statistical problem of language model alignment. We adopt a contextual bandit formalism (Rafailov et al., 2023; Xiong et al., 2024a) where th... | https://arxiv.org/abs/2503.07453v2 |
algorithm reflects itsdata efficiency , i.e. how much data needs to be collected from the reward oracle to learn a good policy. Collecting high-quality reward signals can be costly or time-consuming (e.g., when human-generated, or when reward evaluation requires computationally intensive code execution or formal verifi... | https://arxiv.org/abs/2503.07453v2 |
sampling oracle queries to gather reward feedback per round, so the computational cost is no worse than the cost of gathering feedback: Tcomp=Tdata.3Unfortunately, since OnlineDPO engages in purely passive exploration, the algorithm’s data efficiency itself is unsatisfactory. We make this distinction quantitative below... | https://arxiv.org/abs/2503.07453v2 |
called XPO(see Appendix A), which augments theDPOtraining objective with a bonus designed to encourageactive exploration. This allows XPOto achieve polynomial data efficiency, irrespective of whether the base policy πrefhas favorable coverage: Tdata(ε, δ)≲d2log(δ−1) ε2. (6) Note that Ccov(π⋆ β)≫poly(d)in general, repre... | https://arxiv.org/abs/2503.07453v2 |
represented as an autoregressive policy, it is possible to achieve substantially improved runtime and oracle complexity Tcomp(replacing the coverage coefficient Ccov(π⋆ β)with antoken-level counterpart) by appealing to multi-turn exploration at the per-step (token or sub-sequence) level (Lightman et al., 2023; Qu et al... | https://arxiv.org/abs/2503.07453v2 |
on the parameter Bonly logarithmically. 6 Notice, any algorithm operating in our framework (i) must invoke the sampling oracle if it wishes to query the reward oracle with some y∼πθ(· |xt), and (ii) only has knowledge of the features ϕ(x, y)that have previously been revealed by the sampling oracle. As an example, given... | https://arxiv.org/abs/2503.07453v2 |
min eβ2d/2, eβ−1/2, C⋆ . (10) For simplicity, consider the regime where d≥β−3. Then Theorem 2.1 shows that any algorithm needs Tcomp(ε, δ)≥Ω min eβ−1/2, Ccov(π⋆ β) to achieve non-trivial data-efficiency Tdata(ε, δ)≪ |Y|; note that the presence of eβ−1in the lower bound is fundamental, as we always have Ccov(π⋆ β... | https://arxiv.org/abs/2503.07453v2 |
y, y′) =ϕ(x, y)−ϕ(x, y′); we use these features throughout the algorithm because—per Eq.(8)—the difference in rewards r⋆(x, y)−r⋆(x, y′)is linear under Assump- tion 1.1. In the first phase, the algorithm aims to compute a spanner: a small collection Ψspanof tuples (x, y, y′)such that the second moment matrix Σspan=λId+... | https://arxiv.org/abs/2503.07453v2 |
to the spanner. Truncation allows SpannerSampling to proceed using only a weaksampling oracle (Definition 2.1): whenever the spanner phase succeeds, weareguaranteedthatπθt(y|x,y′) πref(y|x)≲Ccov(π⋆ β)for“most” pairs (x, y′)(LemmaF.6). Thismeanswecan userejection sampling (SoftmaxSampler ; Algorithm 2) at inference-time... | https://arxiv.org/abs/2503.07453v2 |
on dandRmaxwhen ε≤β(Lattimore and Szepesvári, 2020), and is independent of the coverage coefficient , reflecting active exploration. The number of prompts used by the algorithm is also independent of the coverage coefficient, though it is slightly larger than the number of reward queries. We observe that Theorem 3.1 ac... | https://arxiv.org/abs/2503.07453v2 |
Proper algorithms are closely related to the notion of training-time interventions for exploration, in the sense that any algorithm that computes exploratory policies πtby solving πt= arg min π∈ΠLt D(π) for some loss function Lt D(π)that depends on the dataset Dcollected so far will inevitably be proper in the sense of... | https://arxiv.org/abs/2503.07453v2 |
y= (a1, . . . , a H)correspond to sequences of tokens), and where πrefis explicitly represented as an autoregressive policy of the form πref(y|x) =πref(a1:H|x) =HY h=1πh,ref(ah|x, a1:h−1). In what follows, we show how to specialize SpannerSampling to this setting, then derive algorithms with improved computational effi... | https://arxiv.org/abs/2503.07453v2 |
as autoregressive linear softmax policies in general; that is, there may not exist any θ= (θ1, . . . , θ H)such that πseq θ⋆=πauto θ—see Proposition H.1.12 Applying SpannerSampling .Even though autoregressive realizability may not hold under Eq.(17), we can still apply SpannerSampling efficiently under the sequence-lev... | https://arxiv.org/abs/2503.07453v2 |
1−δ, and does so with Tdata(ε, δ)≤poly(d, H, B, ε−1,log(δ−1))reward queries and Tauto comp(ε, δ)≤poly Ccond(π⋆ β), Tdata(ε, δ) (weak) autoregressive sampling queries. As discussed above, the action-level coverage coefficient Ccond(π⋆ β)in this result can be exponentially smaller than the sequence-level coverage coeff... | https://arxiv.org/abs/2503.07453v2 |
(e.g., modifications to the DPOloss) are insufficient for computationally efficient exploration. This raises the question of whether there exist training-time interventions that induce different policy representations (e.g., based on alternative forms of regularization (Wang et al., 2024a; Huang et al., 2024b)) that mo... | https://arxiv.org/abs/2503.07453v2 |
Chi, and Bo Dai. Value-incentivized preference optimization: A unified approach to online and offline rlhf, 2024. Siu On Chan. Approximation resistance from pairwise-independent subgroups. Journal of the ACM (JACM) , 63(3):1–32, 2016. Jonathan D Chang, Wenhao Zhan, Owen Oertell, Kianté Brantley, Dipendra Misra, Jason D... | https://arxiv.org/abs/2503.07453v2 |
In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing , pages 8154–8173, 2023. Elad Hazan and Zohar Karnin. Volumetric spanners: An efficient exploration basis for learning. The Journal of Machine Learning Research , 17(1):4062–4095, 2016. 16 Audrey Huang, Adam Block, Dylan J Foster,... | https://arxiv.org/abs/2503.07453v2 |
Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. Let’s verify step by step. arXiv preprint arXiv:2305.20050 , 2023. Tianlin Liu, Shangmin Guo, Leonardo Bianco, Daniele Calandriello, Quentin Berthet, Felipe Llinares, Jessica Hoffmann, Lucas Dixon, Michal Valko, and Mathieu Blond... | https://arxiv.org/abs/2503.07453v2 |
2017. Amrith Setlur, Saurabh Garg, Xinyang Geng, Naman Garg, Virginia Smith, and Aviral Kumar. Rl on incorrect synthetic data scales the efficiency of llm math reasoning by eight-fold. arXiv preprint arXiv:2406.14532 , 2024a. Amrith Setlur, Chirag Nagpal, Adam Fisch, Xinyang Geng, Jacob Eisenstein, Rishabh Agarwal, Ale... | https://arxiv.org/abs/2503.07453v2 |
Aidan Swope, et al. Helpsteer: Multi-attribute helpfulness dataset for steerlm. arXiv preprint arXiv:2311.09528 , 2023b. Zhilin Wang, Yi Dong, Olivier Delalleau, Jiaqi Zeng, Gerald Shen, Daniel Egert, Jimmy J Zhang, Makesh Nar- simhan Sreedhar, and Oleksii Kuchaiev. Helpsteer2: Open-source dataset for training top-perf... | https://arxiv.org/abs/2503.07453v2 |
Wang. Self-exploring language models: Active preference elicitation for online alignment, 2024. Heyang Zhao, Chenlu Ye, Quanquan Gu, and Tong Zhang. Sharp analysis for kl-regularized contextual bandits and rlhf. arXiv preprint arXiv:2411.04625 , 2024. Heyang Zhao, Chenlu Ye, Wei Xiong, Quanquan Gu, and Tong Zhang. Loga... | https://arxiv.org/abs/2503.07453v2 |
. . . . . . . 42 G Proofs from Section 4 49 G.1 Overview of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 G.2 Proof of Lemma G.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 G.3 Proof of Theorem G.1 . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2503.07453v2 |
theoretical algorithms for exploration in online alignment (as well as the more abstract problem of preference-based contextual bandits and RL), but most prior algorithms are not computationally efficient when the response space Yis large (Xu et al., 2020; Novoseller et al., 2020; Pacchiano et al., 2021; Wu and Sun, 20... | https://arxiv.org/abs/2503.07453v2 |
al. (2024) achieves a similar fast rate, but requires access to an offline dataset satisfying a stringent uniform coverage assumption (and pays for the coverage coefficient statistically), while concurrent work of Zhao et al. (2025) achieves fast rates in the purely online setting, but is not computationally efficient ... | https://arxiv.org/abs/2503.07453v2 |
allowing for N > 2makes our lower bounds/impossibility results stronger, even though our algorithms themselves only use N= 2; and (ii) the absolute reward formulation—which has been used in prior work empirically (Wang et al., 2023b, 2024d,c; Xiong et al., 2024b) and in theory (Zhao et al., 2024; Wang et al., 2024b; Xi... | https://arxiv.org/abs/2503.07453v2 |
πref(y′|x)=⟨θ, ϕ(x, y)−ϕ(x, y′)⟩, buttheycannotbeusedtocompute logπθ(y|x) itself in general. We adopt this formalism because it simplifies the coverage-based lower bounds in Section 2.2; our algorithmic results only make use of the features ϕ(x, y), and hence fall into the framework of Definition 2.1. But to move beyon... | https://arxiv.org/abs/2503.07453v2 |
a sequence of Hcalls to a subroutine B2. Let Sdenote the state space of algorithm B1; the space capturing the values of all the internal variables of B1. LetSt h,−∈Sdenote the random state of B1immediately before the hth call to B2during the tth iteration; further, let St h,+∈Sdenote the random state of B1immediately a... | https://arxiv.org/abs/2503.07453v2 |
rθ⋆,y⋆(y) =⟨θ⋆, ϕθ⋆,y⋆(y)⟩andθ⋆∈B2(1) = Θ, As- sumption1.1issatisfied. Inparticular,theoptimalKL-regularizedpolicyis πθ⋆(y)∝πref(y)exp β−1 θ⋆, ϕθ⋆,y⋆(y) . It is straightforward to check that Assumption 2.1 is satisfied with Rmax=B= 1. From Eq.(4), we have Ccov(πθ⋆) =Ccov(πθ⋆)≤max y∈{0,y⋆}1 πref(y)≤1 εref≤C⋆ where th... | https://arxiv.org/abs/2503.07453v2 |
the first inequality uses the fact that θis independent of θ⋆and hence ⟨θ, θ⋆⟩is stochastically dominated by max( θ⋆ 1,0), and the final inequality uses Lemma D.2 (stated and proven in the sequel). 2.If it is a reward query y∈ Y, then the probability that the execution of Alg[q+1]deviates from Alg[q]is precisely the pr... | https://arxiv.org/abs/2503.07453v2 |
ratio to decide whether to accept each response. The only subtlety is that the density ratio πf(y|x) πref(y|x)=exp β−1f(x, y) Ey′∼πref[exp( β−1f(x, y′))], depends on the normalization constant Z(x):=Ey′∼πref exp(β−1f(x, y′)) , which is unknown. To address this, Line 3 estimates the normalization constant via sampli... | https://arxiv.org/abs/2503.07453v2 |
probability at least 1−δ, and we have P(Xi∈A|ξi= 1) = ν(A). In what follows, we omit dependence on x. Let Z:=Ey∼πref exp(β−1f(y)) denote the “true” normalization constant for πf, and observe that we have C∞=max y∈Yexp(β−1f(y)) Z. (41) We begin by giving a guarantee for the estimated normalization constant bZ. We obse... | https://arxiv.org/abs/2503.07453v2 |
Lemma E.2. Let(y, ρ)be the random output of SoftmaxSamplerDensity . First, by Jensen’s inequality, we have that E[logρ]−E logπf(y|x) πref(y|x) ≤E logρ−logπf(y|x) πref(y|x) , and so, by letting Eacceptbe the event Eacceptin Theorem E.1, we have =E I{Eaccept} · logρ−logπf(y|x) πref(y|x) +E (1−I{Eaccept})· logρ−l... | https://arxiv.org/abs/2503.07453v2 |
that rt 1, rt 2∈[0, Rmax]almost surely, thatθ⋆∈Θ, and that Assumption 2.1 holds with parameter B. Define λ=R2 max B2. Then with probability at least 1−δ, for all t∈[T], ∥θt−θ⋆∥2 Σt+λId≤O(dR2 maxlog(BR−1 maxδ−1T)). Proof of Lemma F.3. By a standard concentration result for well-specified regression (e.g., Lemma 39 in Ji... | https://arxiv.org/abs/2503.07453v2 |
εspan(x, y′):=Py∼πθ⋆(·|x)[∥φ(x, y, y′)∥Σ−1> ν],and ε2 stat:=∥θ−θ⋆∥2 Σ. Suppose ν≤β/ε statandεspan(x, y′)≤1/2. Then for all y∈ Y, πθ(y|x, y′)≤2e2Ccov(πθ⋆)·πref(y|x). Finally Lemma F.5 and Lemma F.6 gives our main regret decomposition for truncated softmax policies. Lemma F.7 (Main regret decomposition for truncated soft... | https://arxiv.org/abs/2503.07453v2 |
ϕ(y)⟩ as the normalization constant for πθ⋆. We can write Ey∼πref exp β−1⟨θ⋆, ϕ(y)⟩ I{∥φ(y, y′)∥Σ−1≤ν} =Zπθ⋆−Ey∼πref exp β−1⟨θ⋆, ϕ(y)⟩ I{∥φ(y, y′)∥Σ−1> ν} . We can further bound Ey∼πref exp β−1⟨θ⋆, ϕ(y)⟩ I{∥φ(y, y′)∥Σ−1> ν} =Ey∼πref" exp β−1⟨θ⋆, ϕ(y)⟩ Zπθ⋆I{∥φ(y, y′)∥Σ−1> ν}# ·Zπθ⋆ =Ey∼πθ⋆[I{∥φ(y, y′)∥Σ... | https://arxiv.org/abs/2503.07453v2 |
by estimating each Jβ(bπt)and choosing bπas the minimizer over t∈[Texp](as we do in MTSS), but we omit this extra complication here. We begin by proving a number of intermediate results. We then use these results to prove Theorem 3.1 in Section F.3.2. F.3.1 Intermediate Guarantee for Spanner Construction In this sectio... | https://arxiv.org/abs/2503.07453v2 |
8 log(4 δ−1). From here, the result follows from Lemma F.8. 43 F.3.2 Proof of Theorem 3.1 Proof of Theorem 3.1. Recall that we define εstat:=c·q dR2maxlog(BR−1 maxδ−1Texp)for a sufficiently large absolute constant c >0, and use the parameter settings λ←(Rmax/B)2,ν:=β/ε stat,Mrej:= 8e2·Ccov(πθ⋆), andδrej:=T−1 exp. We wi... | https://arxiv.org/abs/2503.07453v2 |
implies that for the choice for Mrejin Line 3, we have DTV(bπt(· |x, y′), πt(· |x, y′))≤δrej (67) for all (x, y′)∈ Z good. We can further derive the following consequence. Lemma F.10. Under the event Econc, for any function f(x, y, y′)∈[0,1], Ex∼ρ,(y,y′)∼πt[f(x, y, y′)]−Ex∼ρ,(y,y′)∼bπt[f(x, y, y′)] ≤δrej+ 2Ccov(πθ⋆)·εs... | https://arxiv.org/abs/2503.07453v2 |
does not query it otherwise. Meanwhile, it queries the reward oracle twice at each round of the exploration phase. Consequently, by Lemma F.8, we have Tdata(ε, δ)≤2(|Ψspan|+Texp)≤eOd ν2+Texp ≤eO d2R2 maxlog(δ−1) β2+d2R2 maxlog2 δ−1 βε! . where we have used that ν=β/ε stat. We also observe that the number of prompts... | https://arxiv.org/abs/2503.07453v2 |
any given d∈N,β >0, and ε, δ > 0,Algsolves any d-dimensional instance with regularization parameter β, regret ε, and failure probability δ. In order to be explicit, we write Tdata(d, β, ε, δ )to denote the number of reward oracle queries used by Alg, and Tcomp(d, β, ε, δ )to denote the number of strong sampling oracle ... | https://arxiv.org/abs/2503.07453v2 |
:=δ2 16k2log(16 /δ). Then there is an algorithm Alg′for Max- k-DNF with the following guarantee: given any parameter δ >0and Max- k-DNF formula φwithdvariables and mclauses, •IfvalDNF(φ)≥δmandTdata(d, β(k, δ), ε(k, δ),1/4)≤2k, then Alg′outputs Yeswith probability at least 1/4. •IfvalDNF(φ)≤δm 16·Tdata(d,β(k,δ),ε(k,δ),1... | https://arxiv.org/abs/2503.07453v2 |
θ⋆satisfies valDNF(φ;θ⋆)clauses, and πref(i) =e−1/β/mfor all i∈[m], we get mX i=1πθ⋆(i) =Pm i=1πref(i)e1 β⟨θ⋆,ϕ(i)⟩ πref(0) +Pm i=1πref(i)e1 β⟨θ⋆,ϕ(i)⟩≥valDNF(φ;θ⋆)/m πref(0) + val DNF(φ;θ⋆)/m≥valDNF(φ;θ⋆) 2m where the first inequality uses monotonicity of z7→z πref(0)+z, and the final inequality uses that πref(0)≤1 an... | https://arxiv.org/abs/2503.07453v2 |
br(y) := 0instead of r⋆. In particular: •When Algqueries the sampling oracle with θ∈Θ,Alg′computes πθ∈∆(Y), samples y∼πθ, and passes response ytoAlg. •When Alginitiates data collection round twith exploration policy πt=πθt,Alg′computes πθt, samples y∼πθt, and passes response yand reward 0toAlg. Letqdenote the number of... | https://arxiv.org/abs/2503.07453v2 |
d,1 k2log(16 /δ),δ2 16k2log(16 /δ),1/4 by the assumed time complexity bound for Algand the fact that for any given θ∈Θ, the distribution πθ can be explicitly computed from φin time poly( d, m). G.3 Proof of Theorem G.1 We now prove Theorem G.1 by combining Lemma G.1 with a hardness of approximation result for Max- k-D... | https://arxiv.org/abs/2503.07453v2 |
Blog(1/minx,yπref(y|x))for a universal constant c >0, and we assume that β∈(0,1). Furthermore, the time complexity of Alg′is polynomial in the time complexity of Algwith parameters (d, β, ϵ 0, δ/2)and poly d,|Y|, B, T, ϵ−1,log(δ−1),log1 minx,yπref(y|x) . Proof of Lemma G.4. For any θ∈Θ,x∈ X, and y∈ Y, observe that πθ... | https://arxiv.org/abs/2503.07453v2 |
and has time complexity at most poly(d,exp(β−1), ε−1, δ−1), under Assumption 2.1 with Rmax:= 1andB:=√ d. Without loss of generality (assuming that the output policy is represented by a circuit), the output policy is sampleable in time T:=poly(d,exp(β−1), ε−1, δ−1). Thus, by Lemma G.4 and choice of B, in the same settin... | https://arxiv.org/abs/2503.07453v2 |
a time- O(2nγ)algorithm that, given a Max- Pformula φwith at most nvariables and nclauses, has the following behavior: 1. If φis at least (1−ε)-satisfiable, then it outputs Yeswith probability at least 1/2. 2. If φis at most (2k/2k+ε)-satisfiable, then it outputs No. Then the Randomized Exponential Time Hypothesis (Con... | https://arxiv.org/abs/2503.07453v2 |
arbitrary, valDNF(φ′) =valDNF(φ)t. Finally, it’s clear that the reduction is polynomial-time in the output size. Proof of Theorem G.2. Fixk0=k0(c)∈Nsufficiently large and ε0=ε0(c)∈(0,1)sufficiently small that 17In fact, Chan (2016) shows that one can even take ε=o(1), but for our purposes a constant suffices. 57 the fo... | https://arxiv.org/abs/2503.07453v2 |
MTSS(Theorem I.1). •Appendix M: Supporting technical lemmas for the proofs of the results above. H Preliminaries for Multi-Turn Exploration In this section, we introduce formally introduce the setting and assumptions for our general multi-turn exploration results. Recall that the language model alignment problem in Sec... | https://arxiv.org/abs/2503.07453v2 |
r⋆ h= 0for all h < H, and r⋆ Hrepresents the reward for the complete response. A slightly more general formulation (e.g., Xiong et al. (2024b)) which our setup also encompasses, is where each ahrepresents a sub-sequence of tokens rather than a single token (e.g., corresponding to a portion of a proof). Online reinforce... | https://arxiv.org/abs/2503.07453v2 |
1: Q⋆ h,β(x, a) =r⋆ h(x, a) +Th,β[Q⋆ h+1,β](x, a), (76) where Th,β[f](x, a):=Eπref[Vf(xh+1)|xh=x,ah=a]and Vf(x):=βlogX a∈Aπref(a|x)·ef(x,a)/β.(77) We show in Lemma M.4 that the optimal KL-regularized policy π⋆ βsatisfies π⋆ h,β(· |x)∝πh,ref(· |x)·exp β−1Q⋆ h,β(x,·) (78) andπ⋆ h+1:H,β∈arg maxπh+1:H∈ΠQπ h,β(x, a), for ... | https://arxiv.org/abs/2503.07453v2 |
state x∈ X, action a∈ A, and layer h∈[H], and receive ϕh(x, a). We let Tcompdenote the total number of policy sampling queries used by the algorithm. For technical reasons, we require explicit query access to ϕh(x, a)in this framework, whereas in our original framework (Definition 2.1) we only required observing ϕ(x, y... | https://arxiv.org/abs/2503.07453v2 |
11: Update Σt+1 h←Σt+1 h+φh(xt h, at h)φh(xt h, at h)⊤. /* If (xt h, at h)is not too uncertain, bπtis a good candidate policy to return. */ 12:ifmax h∈[H]∥φh(xt h, at h)∥2 (Σt h)−1≤ν2/4then 13: j←t. 14:return bπ1:H=bπj 1:H. I.1 MTSSPseudocode and Overview Our main algorithm, MTSS(Algorithm 4), learns a policy in a dyna... | https://arxiv.org/abs/2503.07453v2 |
(x, a)pairs. Eventually, ∥φh(x, a)∥2 (Σt h)−1becomessmallerthan ν2for “most” state-action pairs (x, a), ensuring that πt h(and thus bπt h) approximates the optimal policy π⋆ h,β. In each iteration t∈[Tprompt ],MTSScomputes the estimates θt 1:Hin a dynamic programming fashion by fitting the difference of value functions... | https://arxiv.org/abs/2503.07453v2 |
ensure bπt=π⋆, the difference Qbπt h,β(xh, ah)−Qbπt h,β(xh,a)would be linear in φh(xh, ah)by Eq.(79), but it is not guaranteed to be linear in general; this poses challenges for deriving a regression guarantee, as the regression problem may not realizable or even approximately realizable. Fortunately, the components of... | https://arxiv.org/abs/2503.07453v2 |
L.1) implies that for sufficiently large t= Ω( d)inMTSS, the tuple (xt h, at h)returned by UncertainStateAction 66 must satisfy ∥φh(xt h, at h)∥(Σt h)−1≤ν2. Substituting this into (87) ensures that for such t, the following holds for all ℓ∈[0.. h−1]and(xℓ, aℓ)∈ Ct ℓ: Pbπt ℓ+1:hh ∥φh(xh,ah)∥2 (Σt h)−1> ν2|xℓ=xℓ,aℓ=aℓi ≤... | https://arxiv.org/abs/2503.07453v2 |
key challenge is to show that misspecification errors propagate favorably across the layers h∈[H], avoiding the dreaded error amplification phenomenon (Wang et al., 2021) in which misspecification errors compound exponentially. For this, our main insight is that the regularization parameter βallows for benign error pro... | https://arxiv.org/abs/2503.07453v2 |
all ℓ∈[0.. h−1]and(xℓ, aℓ)∈ Cℓ, Pbπℓ+1:hh ∥φh(xh,ah)∥2 Σ−1 h>2 ζ∨ ∥φh(ˆxh,ˆah)∥2 Σ−1 h |xℓ=xℓ,aℓ=aℓi ≤max h∈[H]4 log 16H|Ch| λδ′ζ N,(90) where φh(·,·):=ϕh(·,·)−ϕh(·,a). •Furthermore, there exists Xh,span⊆ Xsuch that for all ℓ∈[0.. h−1]and(xℓ, aℓ)∈ Cℓ,Pbπℓ+1:h−1[xh∈ Xh,span|xℓ=xℓ,aℓ=aℓ]≥1−max h∈[H]4 Nlog32HN|Ch| λδ′... | https://arxiv.org/abs/2503.07453v2 |
≤2 NX (x,a)∈Dℓ(xℓ,aℓ)I Pπrefh ∥φh(xh,ah)∥2 Σ−1 h> γ|xh=xi >4 log 32HN|Cℓ| λδ′ζ N +4 log 16H|Cℓ| λδ′ζ N. (96) Substituting γℓ(xℓ, aℓ)forγin (96) and using (95), we get that under E′∩ E′′, for all ℓ∈[0.. h−1]and (xℓ, aℓ)∈ Cℓ: Pbπℓ+1:h−1 Pπrefh ∥φh(xh,ah)∥2 Σ−1 h>2 ζ∨ ∥φh(ˆxh,ˆah)∥2 Σ−1 h |xhi >4 log 3... | https://arxiv.org/abs/2503.07453v2 |
(xh,ah)∈ChHX ℓ=h+1Ebπθ Bℓ,θ(xℓ)2|xh=xh,ah=ah , (100) where C1:= 6400 B2H2dlog(3N/δ′) + 3840 H2B2|Ch|, C2:= 19200 |Ch| · 12R2 max+ 48β2log(4Mlog(4 ˜δ−1))2+ 85H2B2 + 3072 |Ch|H2Bβlog(4Mlog(4 ˜δ−1)), C3:= 16H· 8R2 maxlog(4Mlog(4 ˜δ−1))2+ 200 H2B2 + 3072 HB2, and for x∈ X: Bℓ,θ(x):= min 1, max π∈{πℓ,θ,π⋆ ℓ,β,π⋆ ℓ,β}C... | https://arxiv.org/abs/2503.07453v2 |
|||bf−f⋆|||2≤8X (xh,ah)∈ChX zh∈D(xh,ah)ξ(xh, ah, zh)·(bf(xh, ah)−f⋆(xh, ah))−2|||bf−f⋆|||2 | {z } I + 16X (xh,ah)∈ChX zh∈D(xh,ah)b(xh, ah)2 | {z } II. (105) Bounding Term I .We first bound Term I, which reflects the (nearly) mean-zero noise in the regression targets. Concretely, by Lemma K.2 (stated and proven in the s... | https://arxiv.org/abs/2503.07453v2 |
< 16Ccond(πℓ,θ|xℓ)2|xh=xh,ah=ah . (112) It remains to bound the absolute value term on the right-hand side of (110). As a starting point, note that for all ℓ∈[h+ 1.. H], Q⋆ ℓ,β(·,·)−β·logπℓ,θ(· | ·) πℓ,ref(· | ·) ≤HB+ 2Hβ sup (x,a)∈X×A logπℓ,θ(a|x) πℓ,ref(a|x) ≤5HB, since e−2B/β≤πℓ,θ′(·|·) πℓ,ref(·|·)≤e2B/βfor all θ′ ... | https://arxiv.org/abs/2503.07453v2 |
Let (x1,a1,ρ1,r1), . . . , (xH,aH,ρH,rh)be the sequence of random variables generated via the process (ah,ρh) =SoftmaxSamplerDensityβ,M, ˜δ(⟨φh(xh,·), θh⟩;xh, πref),rh∼ r⋆ h(xh,ah),xh+1∼Ph(· |xh,ah), initialized from x1∼P0(· |∅)(we use xH+1to denote a terminal state with zero reward). We write Pbπθ[·]andEbπθ[·]to denot... | https://arxiv.org/abs/2503.07453v2 |
7680 HR2 maxX (xh,ah)∈ChR tHX ℓ=h+1Ebπt Bt ℓ(xℓ)2|xh=xh,ah=ah , (123) wherebπtis as in Algorithm 4; πt h(· |x)∝πh,ref(· |x)·eφt h(x,·)⊤θt h/β; (124) πt,⋆ h,β(· |x)∝πh,ref(· |x)·eφt h(x,·)⊤θ⋆ h,β/β; (125) φt h(x,·):=φh(x,·)·In ∥φh(x,·)∥2 (Σt h)−1≤ν2o ; φh(·,·):=ϕh(·,·)−ϕh(·,a), (126) with aas in Algorithm 4; and C1:= ... | https://arxiv.org/abs/2503.07453v2 |
Lemma J.2 (and the conditioning on Espan) and the definition of Jspan, we have that for all h∈[H],ℓ∈[0.. h−1], and (xℓ, aℓ)∈ Cj ℓ: Pbπj ℓ+1:hh ∥φh(xh,ah)∥2 (Σj h)−1> ν2|xℓ=xℓ,aℓ=aℓi 82 =Pbπj ℓ+1:hh ∥φh(xh,ah)∥2 (Σj h)−1> ν2∨ 2∥uj h∥2 (λI+Uj h)−1 |xℓ=xℓ,aℓ=aℓi , =Pbπj ℓ+1:hh ∥φh(xh,ah)∥2 (Σj h)−1> ν2∨ 2∥φh(xj h, aj h... | https://arxiv.org/abs/2503.07453v2 |
all (xh, ah)∈ Cj handℓ∈[h+ 1.. H]: Ebπj DKL πj ℓ(· |xℓ)∥πj,⋆ ℓ,β(· |xℓ)2 |xh=xh,ah=ah ≤Ebπj β−1X a∈Aπj ℓ(a|xℓ)· ∥φj ℓ(xℓ, a)∥2 (Σj ℓ)−1!2 |xh=xh,ah=ah ∥θ⋆ ℓ,β−θj ℓ∥4 Σj ℓ,(134) where φjis as in Lemma K.4. Now, by Lemma E.3, we have that for any x∈ Xandℓ∈[h+ 1.. H]: X a∈Aπj ℓ(a|x)· ∥φj ℓ(x, a)∥2 (Σj ℓ)−1−X a∈Ab... | https://arxiv.org/abs/2503.07453v2 |
H]and(xh, ah)∈ Cj h: Pbπj[Mrej<4Ccond(πj ℓ|xℓ)|xh=xh,ah=ah]≤εspan, (143) 85 and Pbπj Mrej<16Ccond(πj ℓ|xℓ)2|xh=xh,ah=ah ≤εspan. (144) Finally, we have that for all ℓ∈[h+ 1.. H]and(xh, ah)∈ Cj h: Ebπj" min 1, Ccond(πj ℓ|xℓ)·s 2 Mrej! |xh=xh,ah=ah# ≤Ebπj" I{xℓ∈ Xj ℓ,span} ·min 1, Ccond(πj ℓ|xℓ)·s 2 Mrej! |xh=xh,ah=ah# ... | https://arxiv.org/abs/2503.07453v2 |
E.3, we have that for all ℓ∈[H]: Ebπj"X a∈Abπj ℓ(a|xℓ)· Q⋆ ℓ,β(xℓ, a)−β·logπj ℓ(a|xℓ) πℓ,ref(a|xℓ)# 87 −Ebπj"X a∈Aπj ℓ(a|xℓ)· Q⋆ ℓ,β(xℓ, a)−β·logπj ℓ(a|xℓ) πℓ,ref(a|xℓ)# ≤5BHδ rej+ 5BH·Pbπj[Mrej<4Ccond(πj ℓ|x)]. (151) Combining this with (149) and (150), we get that Jβ(π⋆ β)−Jβ(bπj 1:H) ≤HX h=1Ebπj"X a∈Aπ⋆ h,β(a|xh... | https://arxiv.org/abs/2503.07453v2 |
choice of νin Algorithm 4). Therefore, we have that for all ℓ∈[H],x∈ Xj ℓ,span, and π∈ {πj ℓ,πj,⋆ ℓ,β}: Ccond(π|x)≤2eCcond(π⋆ ℓ,β). (159) Now, by (158) and the fact that Mrej≥8eCcond(π⋆ ℓ,β)(seeSection I.1.4 ), we have that for all ℓ∈[H]: x∈ Xj ℓ,spanonly if Mrej≥4Ccond(πj ℓ|x)and so by (139), we have for all ℓ∈[H]: 1−... | https://arxiv.org/abs/2503.07453v2 |
we have Ea∼πh,ref(·|x) exp(β−1⟨φh(x,a)−φh(x, a), θh⟩) ≥Ea∼πh,ref(·|x)h exp(β−1⟨φh(x,a)−φh(x, a), θh⟩)·In ∥φh(x,a)∥2 Σ−1 h≤ν2oi , and so by Hölder’s inequality, we have ≥Ea∼πh,ref(·|x) e−2 β∥θh−θ⋆ h,β∥Σh·∥φh(x,a)−∥φh(x,a)∥Σ−1 heβ−1⟨φh(x,a)−φh(x,a),θ⋆ h,β⟩·In ∥φh(x,a)∥2 Σ−1 h≤ν2o , ≥Ea∼πh,ref(·|x)" e−2 β∥θh−θ⋆ h,β∥Σh... | https://arxiv.org/abs/2503.07453v2 |
h,β(a|x) πh,ref(a|x) ≥max π Eπ φh(xh,ah)⊤θ⋆ h,β|xh=x −βDKL(π(· |x)∥πh,ref(· |x)) 93 −Rmax·Pπh ∥φh(xh,ah)∥2 Σ−1 h> ν2|xh=xio , ≥Eπ⋆ h,β φh(xh,ah)⊤θ⋆ h,β|xh=x −βDKL π⋆ h,β(· |x)∥πh,ref(· |x) −Rmax·Pπ⋆ h,βh ∥φh(xh,ah)∥2 Σ−1 h> ν2|xh=xi , ≥Eπ⋆ h,β φh(xh,ah)⊤θ⋆ h,β|xh=x −βDKL π⋆ h,β(· |x)∥πh,ref(· |x) −Rmax·min... | https://arxiv.org/abs/2503.07453v2 |
ℓ=hr⋆ ℓ(xℓ,aℓ)−βHX ℓ=hlogπ′ ℓ(aℓ|xℓ) πℓ,ref(aℓ|xℓ)|xh=x# =HX ℓ=hEπ′"X a∈Aπℓ(a|xℓ)· Qπ ℓ,β(xℓ, a)−β·logπℓ(a|xℓ) πℓ,ref(a|xℓ) |xh=x# −HX ℓ=hEπ′"X a∈Aπ′ ℓ(a|xℓ)· Qπ ℓ,β(xℓ, a)−β·logπ′ ℓ(a|xℓ) πℓ,ref(a|xℓ) |xh=x# . Proof of Lemma M.6. First, for any π1:H⊂ {π:X → ∆(A)},(x, a)∈ X × A ,h∈[H], define rπ h(x, a):=r⋆ h(x, a)... | https://arxiv.org/abs/2503.07453v2 |
Concentration via Metastable Mixing, with Applications to the Supercritical Exponential Random Graph Model Vilas Winstein University of California, Berkeley vilas@berkeley.edu March 10, 2025 Abstract It is a folklore belief that metastable wells in low-temperature statistical mechanics models exhibit high-temperature b... | https://arxiv.org/abs/2503.07571v1 |
. . . . . . . . . . . . . 8 2.2 Coupling the ERGM and the Erd˝ os-R´ enyi model . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Approximating edge counts by normal distributions . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Review of the exponential random graph model 12 3.1 Warm-up: the Curie-Weiss mode... | https://arxiv.org/abs/2503.07571v1 |
. . . . 38 7 Alternative approaches 41 7.1 A strategy that worked for the Curie-Weiss model . . . . . . . . . . . . . . . . . . . . . . . . 42 7.2 Initially Gaussian tail for large Lipschitz observables . . . . . . . . . . . . . . . . . . . . . . . 43 1 Introduction Exponential random graph models (ERGMs) are exponenti... | https://arxiv.org/abs/2503.07571v1 |
via a large deviations principle which characterizes the system to first order. Large deviations principles, however, leave open the question of the microscopic behavior, i.e. the fluctuations of various observables of the system at finite size, in the same way that the standard law of large numbers nX i=1Yi=n·E[Y1] +o... | https://arxiv.org/abs/2503.07571v1 |
supercritical orlow-temperature regime if Lβ has at least two local maximizers p∗∈[0,1] with L′′ β(p∗)<0 (see Figure 2, bottom row). Values of βoutside of these two regimes are said to be critical . The large deviations principle thus states that in the subcritical regime, the ERGM is close in cut distance toG(n, p∗) f... | https://arxiv.org/abs/2503.07571v1 |
of the technique of [Cha05], necessitating a deeper study of the energy landscape of the ERGM. Such analysis of the energy landscape of low-temperature models has garnered much recent interest, and has been carried out in settings such as the lattice Ising model as well as partially in some disordered 4 models such as ... | https://arxiv.org/abs/2503.07571v1 |
set of the complete graph Kn, namely all size-2 subsets of [ n]. For a function f:{0,1}([n] 2)→Rand a vector v∈R([n] 2) ≥0, we say that fisv-Lipschitz if |f(x)−f(y)| ≤ve whenever xandydiffer only at index e∈ [n] 2 . The main theorem of this work is the following concentration inequality for v-Lipschitz observables und... | https://arxiv.org/abs/2503.07571v1 |
our proof strategy, which is adapted from that of [GN24]. In that work, a key input to the corresponding result was the FKG inequality, which simply does not hold for the conditioned measure µ. This is an issue especially in the phase coexistence regime, when |Mβ|>1. As such, in the present work we introduce a new tech... | https://arxiv.org/abs/2503.07571v1 |
in the next few paragraphs. Suppose we have some function fof a random variable X, and there is a Markov chain ( Xt) on the state space which allows one to obtain approximate samples of X. If the Markov chain mixes rapidly, and the value of f(Xt) does not change by much from one time step to the next, then it is intuit... | https://arxiv.org/abs/2503.07571v1 |
for our applications. 1.5 Acknowledgements I was partially supported by the NSF Graduate Research Fellowship grant DGE 2146752. I would like to thank my advisor, Shirshendu Ganguly, for suggesting this project and for helpful discussions. I would also like to thank Sourav Chatterjee, Persi Diaconis, Sumit Mukherjee, an... | https://arxiv.org/abs/2503.07571v1 |
which match the data closely. No attempt was made to find the true curves of best fit, we simply aim to demonstrate the power laws with reasonable constants. Let us discuss how the fluctuation bounds guaranteed by Theorem 1.1 compare with the observed fluctu- ations. For this, we need to calculate the norms of the vari... | https://arxiv.org/abs/2503.07571v1 |
noting that the removal of an edge typically does not destroy the maximal number, n−2, of triangles. This is because the density of the graph is typically close to p∗, meaning that removing an edge will instead typically destroy close to ( p∗)2ntriangles. Moreover, only around ( p∗)2 n 2 edges participate in a potenti... | https://arxiv.org/abs/2503.07571v1 |
is p∗+O(n−1/2). A slightly weaker version of this fact which follows from our results is stated as Proposition 6.5 in Section 6.2 below. However, the middle plot of Figure 6 suggests that the average signed discrepancy is actually much smaller than n3/2, meaning that roughly the same number of edges are added and remov... | https://arxiv.org/abs/2503.07571v1 |
bounded away from 0, or at least of order much larger than n−1/2. 2.3 Approximating edge counts by normal distributions Finally we turn to Theorem 1.3, and to the subject of central limit theorems for edge counts more broadly. To get a more accurate picture of the distribution of the edge count, we generated many more ... | https://arxiv.org/abs/2503.07571v1 |
the supercritical regime, with a bound of order n−1on the Kolmogorov-Smirnov statistic for the full edge count. 11 3 Review of the exponential random graph model In this section we provide a more detailed review of past work on the exponential random graph model (ERGM) which is relevant for the current article. 3.1 War... | https://arxiv.org/abs/2503.07571v1 |
that we resample the spins in Xvia Glauber dynamics. This is a discrete-time Markov chain where at each step we choose an index iuniformly at random and resample the spin Xi, according to its distribution conditional on the rest of the configuration. Given that we have resampled the spin Xi, the probability that it bec... | https://arxiv.org/abs/2503.07571v1 |
perhaps an exponentially long time before eventually arriving at m∗. This phenomenon is known as metastability . Note that the large deviations principle (1) does not automatically give detailed information about the fluctuations of the magnetization m(X) under the Gibbs measure. Nonetheless, it was proved in [EN78] th... | https://arxiv.org/abs/2503.07571v1 |
n ×j−1 n,j n . Notice however that this interpretation is not injective, since a graphon has no well-defined number of vertices, and, for example, subdividing the squares leads to a different adjacency matrix for a graph with twice as many vertices which nonetheless has the same graphon representation. Just as we co... | https://arxiv.org/abs/2503.07571v1 |
more dynamical perspective, resampling edges in Xvia Glauber dynamics. At each time step, a uniformly random edge e∈ [n] 2 is chosen to be resampled, and starting from Xthe subsequent state is either X+eorX−e, i.e. the same configuration but with Xefixed to be 1 or 0 respectively. On the event that an edge eis selecte... | https://arxiv.org/abs/2503.07571v1 |
mixing result combined with a method of [Cha05] which allows one to translate certain markers of rapid mixing into concentration inequalities. We remind the reader that for v∈R([n] 2), we say f: Ω→Risv-Lipschitz if |f(X+e)−f(X−e)| ≤vefor every e∈ [n] 2 . The main result of [GN24] is the following concentration inequal... | https://arxiv.org/abs/2503.07571v1 |
We remark that the result of this section is general, and may be used beyond the setting of the ERGM. Heuristically, if one has a fast sampling algorithm for Xsuch that at each step, the value of an observable does not change too much, then one would expect that observable to exhibit some form of concentration, since m... | https://arxiv.org/abs/2503.07571v1 |
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