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1. A Black-box Approach for Non-stationary Multi-agent Reinforcement Learning

   Jiang, Haozhe; Cui, Qiwen; Xiong, Zhihan; Fazel, Maryam; Du, Simon S (January 2024, Proceedings of the International Conference on Learning Representations)

   We investigate learning the equilibria in non-stationary multi-agent systems and address the challenges that differentiate multi-agent learning from single-agent learning. Specifically, we focus on games with bandit feedback, where testing an equilibrium can result in substantial regret even when the gap to be tested is small, and the existence of multiple optimal solutions (equilibria) in stationary games poses extra challenges. To overcome these obstacles, we propose a versatile black-box approach applicable to a broad spectrum of problems, such as general-sum games, potential games, and Markov games, when equipped with appropriate learning and testing oracles for stationary environments. Our algorithms can achieve O(∆^1/4 T^3/4) regret when the degree of nonstationarity, as measured by total variation ∆, is known, and O(∆^1/5 T^4/5) regret when ∆ is unknown, where T is the number of rounds. Meanwhile, our algorithm inherits the favorable dependence on number of agents from the oracles. As a side contribution that may be independent of interest, we show how to test for various types of equilibria by a black-box reduction to single-agent learning, which includes Nash equilibria, correlated equilibria, and coarse correlated equilibria. 

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2. 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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3. 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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4. Adaptive Online Estimation of Piecewise Polynomial Trends

   Baby, Dheeraj; Wang, Yu-Xiang (December 2020, NeurIPS 2020)

   null (Ed.)

   We consider the framework of non-stationary stochastic optimization (Besbes et al., 2015) with squared error losses and noisy gradient feedback where the dynamic regret of an online learner against a time varying comparator sequence is studied. Motivated from the theory of non-parametric regression, we introduce a new variational constraint that enforces the comparator sequence to belong to a discrete k^{th} order Total Variation ball of radius C_n. This variational constraint models comparators that have piece-wise polynomial structure which has many relevant practical applications (Tibshirani, 2014). By establishing connections to the theory of wavelet based non-parametric regression, we design a polynomial time algorithm that achieves the nearly optimal dynamic regret of ~O(n^{1/(2k+3)} C_n^{2/(2k+3)}). The proposed policy is adaptive to the unknown radius C_n. Further, we show that the same policy is minimax optimal for several other non-parametric families of interest. 

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5. On Regret with Multiple Best Arms

   Zhu, Yinglun; Nowak, Robert (November 2022, Advances in Neural Information Processing Systems)

   We study a regret minimization problem with the existence of multiple best/near-optimal arms in the multi-armed bandit setting. We consider the case when the number of arms/actions is comparable or much larger than the time horizon, and make no assumptions about the structure of the bandit instance. Our goal is to design algorithms that can automatically adapt to the unknown hardness of the problem, i.e., the number of best arms. Our setting captures many modern applications of bandit algorithms where the action space is enormous and the information about the underlying instance/structure is unavailable. We first propose an adaptive algorithm that is agnostic to the hardness level and theoretically derive its regret bound. We then prove a lower bound for our problem setting, which indicates: (1) no algorithm can be minimax optimal simultaneously over all hardness levels; and (2) our algorithm achieves a rate function that is Pareto optimal. With additional knowledge of the expected reward of the best arm, we propose another adaptive algorithm that is minimax optimal, up to polylog factors, over all hardness levels. Experimental results confirm our theoretical guarantees and show advantages of our algorithms over the previous state-of-the-art. 

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