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1. 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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2. 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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3. On the Sample Complexity of Vanilla Model-Based Offline Reinforcement Learning with Dependent Samples

   https://doi.org/10.1609/aaai.v37i7.25989  

   Karabag, Mustafa O; Topcu, Ufuk (June 2023, Proceedings of the AAAI Conference on Artificial Intelligence)

   Offline reinforcement learning (offline RL) considers problems where learning is performed using only previously collected samples and is helpful for the settings in which collecting new data is costly or risky. In model-based offline RL, the learner performs estimation (or optimization) using a model constructed according to the empirical transition frequencies. We analyze the sample complexity of vanilla model-based offline RL with dependent samples in the infinite-horizon discounted-reward setting. In our setting, the samples obey the dynamics of the Markov decision process and, consequently, may have interdependencies. Under no assumption of independent samples, we provide a high-probability, polynomial sample complexity bound for vanilla model-based off-policy evaluation that requires partial or uniform coverage. We extend this result to the off-policy optimization under uniform coverage. As a comparison to the model-based approach, we analyze the sample complexity of off-policy evaluation with vanilla importance sampling in the infinite-horizon setting. Finally, we provide an estimator that outperforms the sample-mean estimator for almost deterministic dynamics that are prevalent in reinforcement learning. 

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4. 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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5. Pessimistic Q-Learning for Offline Reinforcement Learning: Towards Optimal Sample Complexity

   Shi, Laixi; Li, Gen; Wei, Yuting; Chen, Yuxin; Chi, Yuejie (January 2022, Proceedings of the 39th International Conference on Machine Learning)

   Offline or batch reinforcement learning seeks to learn a near-optimal 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 model-based algorithms (e.g., value iteration with lower confidence bounds) have been theoretically investigated, their model-free 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 Q-learning in the context of finite-horizon Markov decision processes, and characterize its sample complexity under the single-policy concentrability assumption which does not require the full coverage of the state-action space. In addition, a variance-reduced pessimistic Q-learning algorithm is proposed to achieve near-optimal sample complexity. Altogether, this work highlights the efficiency of model-free algorithms in offline RL when used in conjunction with pessimism and variance reduction. 

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