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1. More Benefits of Being Distributional: Second-Order Bounds for Reinforcement Learning

   Wang, Kaiwen; Oertell, Owen; Agarwal, Alekh; Kallus, Nathan; Sun, Wen (July 2024, Proceedings of the 41st International Conference on Machine Learning)

   In this paper, we prove that Distributional Re- inforcement Learning (DistRL), which learns the return distribution, can obtain second-order bounds in both online and offline RL in general settings with function approximation. Second- order bounds are instance-dependent bounds that scale with the variance of return, which we prove are tighter than the previously known small-loss bounds of distributional RL. To the best of our knowledge, our results are the first second-order bounds for low-rank MDPs and for offline RL. When specializing to contextual bandits (one-step RL problem), we show that a distributional learn- ing based optimism algorithm achieves a second- order worst-case regret bound, and a second-order gap dependent bound, simultaneously. We also empirically demonstrate the benefit of DistRL in contextual bandits on real-world datasets. We highlight that our analysis with DistRL is rela- tively simple, follows the general framework of optimism in the face of uncertainty and does not require weighted regression. Our results suggest that DistRL is a promising framework for obtain- ing second-order bounds in general RL settings, thus further reinforcing the benefits of DistRL. 

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2. The Benefits of Being Distributional: Small-Loss Bounds for Reinforcement Learning

   Wang, Kaiwen; Zhou, Kevin; Wu, Runzhe; Kallus, Nathan; Sun, Wen (December 2023, 37th Conference on Neural Information Processing Systems (NeurIPS 2023))

   While distributional reinforcement learning (DistRL) has been empirically effective, the question of when and why it is better than vanilla, non-distributional RL has remained unanswered. This paper explains the benefits of DistRL through the lens of small-loss bounds, which are instance-dependent bounds that scale with optimal achievable cost. Particularly, our bounds converge much faster than those from non-distributional approaches if the optimal cost is small. As warmup, we propose a distributional contextual bandit (DistCB) algorithm, which we show enjoys small-loss regret bounds and empirically outperforms the state-of-the-art on three real-world tasks. In online RL, we propose a DistRL algorithm that constructs confidence sets using maximum likelihood estimation. We prove that our algorithm enjoys novel small-loss PAC bounds in low-rank MDPs. As part of our analysis, we introduce the l1 distributional eluder dimension which may be of independent interest. Then, in offline RL, we show that pessimistic DistRL enjoys small-loss PAC bounds that are novel to the offline setting and are more robust to bad single-policy coverage. 

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3. AdaLinUCB: Opportunistic Learning for Contextual Bandits

   https://doi.org/10.24963/ijcai.2019/336  

   Guo, Xueying; Wang, Xiaoxiao; Liu, Xin (August 2019, Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence)

   In this paper, we propose and study opportunistic contextual bandits - a special case of contextual bandits where the exploration cost varies under different environmental conditions, such as network load or return variation in recommendations. When the exploration cost is low, so is the actual regret of pulling a sub-optimal arm (e.g., trying a suboptimal recommendation). Therefore, intuitively, we could explore more when the exploration cost is relatively low and exploit more when the exploration cost is relatively high. Inspired by this intuition, for opportunistic contextual bandits with Linear payoffs, we propose an Adaptive Upper-Confidence-Bound algorithm (AdaLinUCB) to adaptively balance the exploration-exploitation trade-off for opportunistic learning. We prove that AdaLinUCB achieves O((log T)^2) problem-dependent regret upper bound, which has a smaller coefficient than that of the traditional LinUCB algorithm. Moreover, based on both synthetic and real-world dataset, we show that AdaLinUCB significantly outperforms other contextual bandit algorithms, under large exploration cost fluctuations. 

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4. Optimal Batched Linear Bandits

   Ren, Xuanfei; Jin, Tianyuan; Xu, Pan (May 2024, Proceedings of The 41st International Conference on Machine Learning (ICML))

   We introduce the E$^4$ algorithm for the batched linear bandit problem, incorporating an Explore-Estimate-Eliminate-Exploit framework. With a proper choice of exploration rate, we prove E$^4$ achieves the finite-time minimax optimal regret with only $$O(\log\log T)$$ batches, and the asymptotically optimal regret with only $$3$$ batches as $$T\rightarrow\infty$$, where $$T$$ is the time horizon. We further prove a lower bound on the batch complexity of linear contextual bandits showing that any asymptotically optimal algorithm must require at least $$3$$ batches in expectation as $$T\rightarrow\infty$$, which indicates E$^4$ achieves the asymptotic optimality in regret and batch complexity simultaneously. To the best of our knowledge, E$^4$ is the first algorithm for linear bandits that simultaneously achieves the minimax and asymptotic optimality in regret with the corresponding optimal batch complexities. In addition, we show that with another choice of exploration rate E$^4$ achieves an instance-dependent regret bound requiring at most $$O(\log T)$$ batches, and maintains the minimax optimality and asymptotic optimality. We conduct thorough experiments to evaluate our algorithm on randomly generated instances and the challenging \textit{End of Optimism} instances \citep{lattimore2017end} which were shown to be hard to learn for optimism based algorithms. Empirical results show that E$^4$ consistently outperforms baseline algorithms with respect to regret minimization, batch complexity, and computational efficiency. 

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5. Optimal Batched Linear Bandits

   Ren, Xuanfei; Jin, Tianyuan; Xu, Pan (May 2024, Proceedings of the 41st International Conference on Machine Learning)

   We introduce the E$^4$ algorithm for the batched linear bandit problem, incorporating an Explore-Estimate-Eliminate-Exploit framework. With a proper choice of exploration rate, we prove E$^4$ achieves the finite-time minimax optimal regret with only $$O(\log\log T)$$ batches, and the asymptotically optimal regret with only $$3$$ batches as $$T\rightarrow\infty$$, where $$T$$ is the time horizon. We further prove a lower bound on the batch complexity of linear contextual bandits showing that any asymptotically optimal algorithm must require at least $$3$$ batches in expectation as $$T\rightarrow\infty$$, which indicates E$^4$ achieves the asymptotic optimality in regret and batch complexity simultaneously. To the best of our knowledge, E$^4$ is the first algorithm for linear bandits that simultaneously achieves the minimax and asymptotic optimality in regret with the corresponding optimal batch complexities. In addition, we show that with another choice of exploration rate E$^4$ achieves an instance-dependent regret bound requiring at most $$O(\log T)$$ batches, and maintains the minimax optimality and asymptotic optimality. We conduct thorough experiments to evaluate our algorithm on randomly generated instances and the challenging \textit{End of Optimism} instances \citep{lattimore2017end} which were shown to be hard to learn for optimism based algorithms. Empirical results show that E$^4$ consistently outperforms baseline algorithms with respect to regret minimization, batch complexity, and computational efficiency. 

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