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1. Optimal Uniform OPE and Model-based Offline Reinforcement Learning in Time-Homogeneous, Reward-Free and Task-Agnostic Settings

   Yin, Ming ; Wang, Yu-Xiang ( October 2021 , Advances in neural information processing systems)

   This work studies the statistical limits of uniform convergence for offline policy evaluation (OPE) problems with model-based methods (for episodic MDP) and provides a unified framework towards optimal learning for several well-motivated offline tasks. Uniform OPE supΠ|Qπ−Q̂ π|<ϵ is a stronger measure than the point-wise 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 model-based 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{near-empirically optimal} policies for the MDPs with \emph{stationary} transition. Here dm is the minimal marginal state-action 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 model-based approach. We generalize such a model-based framework to the new settings: offline task-agnostic and the offline reward-free 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. 

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2. Towards instance-optimal offline reinforcement learning with pessimism

   Yin, Ming ; Wang, Yu-Xiang ( December 2021 , Advances in neural information processing systems)

   We study the \emph{offline reinforcement learning} (offline RL) problem, where the goal is to learn a reward-maximizing 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 information-theoretical lower bounds based on different data-coverage 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 information-theoretical 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 state-action 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 data-coverage assumption, horizon-free setting, single policy concentrability, and the tight problem-dependent results. Later, we extend the result to the \emph{assumption-free} regime (where we make no assumption on $ \mu$) and obtain the assumption-free 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. 

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3. Near-Optimal Offline Reinforcement Learning via Double Variance Reduction

   Yin, Ming ; Bai, Yu ; Wang, Yu-Xiang ( December 2021 , Advances in neural information processing systems)

   We consider the problem of offline reinforcement learning (RL) -- a well-motivated 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 Off-Policy Double Variance Reduction (OPDVR), a new variance reduction based algorithm for offline RL. Our main result shows that OPDVR provably identifies an ϵ-optimal policy with O˜(H2/dmϵ2) episodes of offline data in the finite-horizon stationary transition setting, where H is the horizon length and dm is the minimal marginal state-action distribution induced by the behavior policy. This improves over the best known upper bound by a factor of H. Moreover, we establish an information-theoretic lower bound of Ω(H2/dmϵ2) which certifies that OPDVR is optimal up to logarithmic factors. Lastly, we show that OPDVR also achieves rate-optimal sample complexity under alternative settings such as the finite-horizon MDPs with non-stationary transitions and the infinite horizon MDPs with discounted rewards. 

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4. Sharper Model-free Reinforcement Learning for Average-reward Markov Decision Processes

   Zhang, Zihan ; Xie, Qiaomin ( July 2023 , Conference on Learning Theory)

   We study model-free reinforcement learning (RL) algorithms for infinite-horizon average-reward 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 model-free RL algorithms is relatively inadequate for the average-reward setting. In this paper, we consider both the online setting and the setting with access to a simulator. We develop computationally efficient model-free algorithms that achieve sharper guarantees on regret/sample complexity compared with existing results. In the online setting, we design an algorithm, UCB-AVG, based on an optimistic variant of variance-reduced Q-learning. We show that UCB-AVG achieves a regret bound $\widetilde{O}(S^5A^2sp(h^*)\sqrt{T})$ after $T$ steps, where $S\times A$ is the size of state-action space, and $sp(h^*)$ the span of the optimal bias function. Our result provides the first computationally efficient model-free 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 UCB-AVG to develop a model-free 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 near-optimal 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 worst-case 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 reference-advantage decomposition for variance in optimistic Q-learning. 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 value-difference 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 good-quality reference value function with $O(SA)$ space complexity. 

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5. Overcoming the Long Horizon Barrier for Sample-Efficient Reinforcement Learning with Latent Low-Rank Structure

   https://doi.org/10.1145/3589973  

   Sam, Tyler ; Chen, Yudong ; Yu, Christina Lee ( May 2023 , Proceedings of the ACM on Measurement and Analysis of Computing Systems)

   The practicality of reinforcement learning algorithms has been limited due to poor scaling with respect to the problem size, as the sample complexity of learning an ε-optimal policy is Ω(|S||A|H/ ε2) over worst case instances of an MDP with state space S, action space A, and horizon H. We consider a class of MDPs for which the associated optimal Q* function is low rank, where the latent features are unknown. While one would hope to achieve linear sample complexity in |S| and |A| due to the low rank structure, we show that without imposing further assumptions beyond low rank of Q*, if one is constrained to estimate the Q function using only observations from a subset of entries, there is a worst case instance in which one must incur a sample complexity exponential in the horizon H to learn a near optimal policy. We subsequently show that under stronger low rank structural assumptions, given access to a generative model, Low Rank Monte Carlo Policy Iteration (LR-MCPI) and Low Rank Empirical Value Iteration (LR-EVI) achieve the desired sample complexity of Õ((|S|+|A|)poly (d,H)/ε2) for a rank d setting, which is minimax optimal with respect to the scaling of |S|, |A|, and ε. In contrast to literature on linear and low-rank MDPs, we do not require a known feature mapping, our algorithm is computationally simple, and our results hold for long time horizons. Our results provide insights on the minimal low-rank structural assumptions required on the MDP with respect to the transition kernel versus the optimal action-value function. 

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