 NSFPAR ID:
 10426241
 Date Published:
 Journal Name:
 Proceedings of the ACM on Measurement and Analysis of Computing Systems
 Volume:
 7
 Issue:
 2
 ISSN:
 24761249
 Page Range / eLocation ID:
 1 to 60
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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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

We consider modelfree reinforcement learning for infinitehorizon discounted Markov Decision Processes (MDPs) with a continuous state space and unknown transition kernel, when only a single sample path under an arbitrary policy of the system is available. We consider the Nearest Neighbor QLearning (NNQL) algorithm to learn the optimal Q function using nearest neighbor regression method. As the main contribution, we provide tight finite sample analysis of the convergence rate. In particular, for MDPs with a ddimensional state space and the discounted factor in (0, 1), given an arbitrary sample path with “covering time” L, we establish that the algorithm is guaranteed to output an "accurate estimate of the optimal Qfunction nearly optimal sample complexity.more » « less

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

In robust Markov decision processes (MDPs), the uncertainty in the transition kernel is addressed by finding a policy that optimizes the worstcase performance over an uncertainty set of MDPs. While much of the literature has focused on discounted MDPs, robust averagereward MDPs remain largely unexplored. In this paper, we focus on robust averagereward MDPs, where the goal is to find a policy that optimizes the worstcase average reward over an uncertainty set. We first take an approach that approximates averagereward MDPs using discounted MDPs. We prove that the robust discounted value function converges to the robust averagereward as the discount factor goes to 1, and moreover when it is large, any optimal policy of the robust discounted MDP is also an optimal policy of the robust averagereward. We further design a robust dynamic programming approach, and theoretically characterize its convergence to the optimum. Then, we investigate robust averagereward MDPs directly without using discounted MDPs as an intermediate step. We derive the robust Bellman equation for robust averagereward MDPs, prove that the optimal policy can be derived from its solution, and further design a robust relative value iteration algorithm that provably finds its solution, or equivalently, the optimal robust policy.

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