We study the \emph{offline reinforcement learning} (offline RL) problem, where the goal is to learn a rewardmaximizing policy in an unknown \emph{Markov Decision Process} (MDP) using the data coming from a policy $\mu$. In particular, we consider the sample complexity problems of offline RL for the finite horizon MDPs. Prior works derive the informationtheoretical lower bounds based on different datacoverage assumptions and their upper bounds are expressed by the covering coefficients which lack the explicit characterization of system quantities. In this work, we analyze the \emph{Adaptive Pessimistic Value Iteration} (APVI) algorithm and derive the suboptimality upper bound that nearly matches
$
O\left(\sum_{h=1}^H\sum_{s_h,a_h}d^{\pi^\star}_h(s_h,a_h)\sqrt{\frac{\mathrm{Var}_{P_{s_h,a_h}}{(V^\star_{h+1}+r_h)}}{d^\mu_h(s_h,a_h)}}\sqrt{\frac{1}{n}}\right).
$
We also prove an informationtheoretical lower bound to show this quantity is required under the weak assumption that $d^\mu_h(s_h,a_h)>0$ if $d^{\pi^\star}_h(s_h,a_h)>0$. Here $\pi^\star$ is a optimal policy, $\mu$ is the behavior policy and $d(s_h,a_h)$ is the marginal stateaction probability. We call this adaptive bound the \emph{intrinsic offline reinforcement learning bound} since it directly implies all the existing optimal results: minimax rate under uniform datacoverage assumption, horizonfree setting, single policy concentrability, and the tight problemdependent results. Later, we extend the result to the \emph{assumptionfree} regime (where we make no assumption on $
\mu$) and obtain the assumptionfree intrinsic bound. Due to its generic form, we believe the intrinsic bound could help illuminate what makes a specific problem hard and reveal the fundamental challenges in offline RL.
more »
« less
NearOptimal Offline Reinforcement Learning via Double Variance Reduction
We consider the problem of offline reinforcement learning (RL)  a wellmotivated setting of RL that aims at policy optimization using only historical data. Despite its wide applicability, theoretical understandings of offline RL, such as its optimal sample complexity, remain largely open even in basic settings such as \emph{tabular} Markov Decision Processes (MDPs). In this paper, we propose OffPolicy Double Variance Reduction (OPDVR), a new variance reduction based algorithm for offline RL. Our main result shows that OPDVR provably identifies an ϵoptimal policy with O˜(H2/dmϵ2) episodes of offline data in the finitehorizon stationary transition setting, where H is the horizon length and dm is the minimal marginal stateaction distribution induced by the behavior policy. This improves over the best known upper bound by a factor of H. Moreover, we establish an informationtheoretic lower bound of Ω(H2/dmϵ2) which certifies that OPDVR is optimal up to logarithmic factors. Lastly, we show that OPDVR also achieves rateoptimal sample complexity under alternative settings such as the finitehorizon MDPs with nonstationary transitions and the infinite horizon MDPs with discounted rewards.
more »
« less
 NSFPAR ID:
 10346206
 Date Published:
 Journal Name:
 Advances in neural information processing systems
 Volume:
 34
 ISSN:
 10495258
 Page Range / eLocation ID:
 76777688
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


We study modelfree reinforcement learning (RL) algorithms for infinitehorizon averagereward Markov decision process (MDP), which is more appropriate for applications that involve continuing operations not divided into episodes. In contrast to episodic/discounted MDPs, theoretical understanding of modelfree RL algorithms is relatively inadequate for the averagereward setting. In this paper, we consider both the online setting and the setting with access to a simulator. We develop computationally efficient modelfree algorithms that achieve sharper guarantees on regret/sample complexity compared with existing results. In the online setting, we design an algorithm, UCBAVG, based on an optimistic variant of variancereduced Qlearning. We show that UCBAVG achieves a regret bound $\widetilde{O}(S^5A^2sp(h^*)\sqrt{T})$ after $T$ steps, where $S\times A$ is the size of stateaction space, and $sp(h^*)$ the span of the optimal bias function. Our result provides the first computationally efficient modelfree algorithm that achieves the optimal dependence in $T$ (up to log factors) for weakly communicating MDPs, which is necessary for low regret. In contrast, prior results either are suboptimal in $T$ or require strong assumptions of ergodicity or uniformly mixing of MDPs. In the simulator setting, we adapt the idea of UCBAVG to develop a modelfree algorithm that finds an $\epsilon$optimal policy with sample complexity $\widetilde{O}(SAsp^2(h^*)\epsilon^{2} + S^2Asp(h^*)\epsilon^{1}).$ This sample complexity is nearoptimal for weakly communicating MDPs, in view of the minimax lower bound $\Omega(SAsp(^*)\epsilon^{2})$. Existing work mainly focuses on ergodic MDPs and the results typically depend on $t_{mix},$ the worstcase mixing time induced by a policy. We remark that the diameter $D$ and mixing time $t_{mix}$ are both lower bounded by $sp(h^*)$, and $t_{mix}$ can be arbitrarily large for certain MDPs. On the technical side, our approach integrates two key ideas: learning an $\gamma$discounted MDP as an approximation, and leveraging referenceadvantage decomposition for variance in optimistic Qlearning. As recognized in prior work, a naive approximation by discounted MDPs results in suboptimal guarantees. A distinguishing feature of our method is maintaining estimates of valuedifference between state pairs to provide a sharper bound on the variance of reference advantage. We also crucially use a careful choice of the discounted factor $\gamma$ to balance approximation error due to discounting and the statistical learning error, and we are able to maintain a goodquality reference value function with $O(SA)$ space complexity.more » « less

This work studies the statistical limits of uniform convergence for offline policy evaluation (OPE) problems with modelbased methods (for episodic MDP) and provides a unified framework towards optimal learning for several wellmotivated offline tasks. Uniform OPE supΠQπ−Q̂ π<ϵ is a stronger measure than the pointwise OPE and ensures offline learning when Π contains all policies (the global class). In this paper, we establish an Ω(H2S/dmϵ2) lower bound (over modelbased family) for the global uniform OPE and our main result establishes an upper bound of Õ (H2/dmϵ2) for the \emph{local} uniform convergence that applies to all \emph{nearempirically optimal} policies for the MDPs with \emph{stationary} transition. Here dm is the minimal marginal stateaction probability. Critically, the highlight in achieving the optimal rate Õ (H2/dmϵ2) is our design of \emph{singleton absorbing MDP}, which is a new sharp analysis tool that works with the modelbased approach. We generalize such a modelbased framework to the new settings: offline taskagnostic and the offline rewardfree with optimal complexity Õ (H2log(K)/dmϵ2) (K is the number of tasks) and Õ (H2S/dmϵ2) respectively. These results provide a unified solution for simultaneously solving different offline RL problems.more » « less

Offline or batch reinforcement learning seeks to learn a nearoptimal policy using history data without active exploration of the environment. To counter the insufficient coverage and sample scarcity of many offline datasets, the principle of pessimism has been recently introduced to mitigate high bias of the estimated values. While pessimistic variants of modelbased algorithms (e.g., value iteration with lower confidence bounds) have been theoretically investigated, their modelfree counterparts — which do not require explicit model estimation — have not been adequately studied, especially in terms of sample efficiency. To address this inadequacy, we study a pessimistic variant of Qlearning in the context of finitehorizon Markov decision processes, and characterize its sample complexity under the singlepolicy concentrability assumption which does not require the full coverage of the stateaction space. In addition, a variancereduced pessimistic Qlearning algorithm is proposed to achieve nearoptimal sample complexity. Altogether, this work highlights the efficiency of modelfree algorithms in offline RL when used in conjunction with pessimism and variance reduction.more » « less

Abstract Achieving sample efficiency in online episodic reinforcement learning (RL) requires optimally balancing exploration and exploitation. When it comes to a finitehorizon episodic Markov decision process with $S$ states, $A$ actions and horizon length $H$, substantial progress has been achieved toward characterizing the minimaxoptimal regret, which scales on the order of $\sqrt{H^2SAT}$ (modulo log factors) with $T$ the total number of samples. While several competing solution paradigms have been proposed to minimize regret, they are either memoryinefficient, or fall short of optimality unless the sample size exceeds an enormous threshold (e.g. $S^6A^4 \,\mathrm{poly}(H)$ for existing modelfree methods). To overcome such a large sample size barrier to efficient RL, we design a novel modelfree algorithm, with space complexity $O(SAH)$, that achieves nearoptimal regret as soon as the sample size exceeds the order of $SA\,\mathrm{poly}(H)$. In terms of this sample size requirement (also referred to the initial burnin cost), our method improves—by at least a factor of $S^5A^3$—upon any prior memoryefficient algorithm that is asymptotically regretoptimal. Leveraging the recently introduced variance reduction strategy (also called referenceadvantage decomposition), the proposed algorithm employs an earlysettled reference update rule, with the aid of two Qlearning sequences with upper and lower confidence bounds. The design principle of our earlysettled variance reduction method might be of independent interest to other RL settings that involve intricate exploration–exploitation tradeoffs.more » « less