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1. Prelimit Coupling and Steady-State Convergence of Constant-stepsize Nonsmooth Contractive SA

   https://doi.org/10.1145/3673660.3655076  

   Zhang, Yixuan; Huo, Dongyan Lucy; Chen, Yudong; Xie, Qiaomin (June 2024, ACM SIGMETRICS Performance Evaluation Review)

   Motivated by Q-learning, we study nonsmooth contractive stochastic approximation (SA) with constant stepsize. We focus on two important classes of dynamics: 1) nonsmooth contractive SA with additive noise, and 2) synchronous and asynchronous Q-learning, which features both additive and multiplicative noise. For both dynamics, we establish weak convergence of the iterates to a stationary limit distribution in Wasserstein distance. Furthermore, we propose a prelimit coupling technique for establishing steady-state convergence and characterize the limit of the stationary distribution as the stepsize goes to zero. Using this result, we derive that the asymptotic bias of nonsmooth SA is proportional to the square root of the stepsize, which stands in sharp contrast to smooth SA. This bias characterization allows for the use of Richardson-Romberg extrapolation for bias reduction in nonsmooth SA. 

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2. Constant Stepsize Q-learning: Distributional Convergence, Bias and Extrapolation

   Zhang, Yixuan; Xie, Qiaomin (August 2024, Reinforcement Learning Journal)

   Stochastic Approximation (SA) is a widely used algorithmic approach in various fields, including optimization and reinforcement learning (RL). Among RL algorithms, Q-learning is particularly popular due to its empirical success. In this paper, we study asynchronous Q-learning with constant stepsize, which is commonly used in practice for its fast convergence. By connecting the constant stepsize Q-learning to a time-homogeneous Markov chain, we show the distributional convergence of the iterates in Wasserstein distance and establish its exponential convergence rate. We also establish a Central Limit Theory for Q-learning iterates, demonstrating the asymptotic normality of the averaged iterates. Moreover, we provide an explicit expansion of the asymptotic bias of the averaged iterate in stepsize. Specifically, the bias is proportional to the stepsize up to higher-order terms and we provide an explicit expression for the linear coefficient. This precise characterization of the bias allows the application of Richardson-Romberg (RR) extrapolation technique to construct a new estimate that is provably closer to the optimal Q function. Numerical results corroborate our theoretical finding on the improvement of the RR extrapolation method. 

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3. The Collusion of Memory and Nonlinearity in Stochastic Approximation With Constant Stepsize

   Huo, Dongyan; Zhang, Yixuan; Chen, Yudong; Xie, Qiaomin (September 2024, 38th Conference on Neural Information Processing Systems (NeurIPS 2024))

   In this work, we investigate stochastic approximation (SA) with Markovian data and nonlinear updates under constant stepsize. Existing work has primarily focused on either i.i.d. data or linear update rules. We take a new perspective and carefully examine the simultaneous presence of Markovian dependency of data and nonlinear update rules, delineating how the interplay between these two structures leads to complications that are not captured by prior techniques. By leveraging the smoothness and recurrence properties of the SA updates, we develop a fine-grained analysis of the correlation between the SA iterates and Markovian data. This enables us to overcome the obstacles in existing analysis and establish for the first time the weak convergence of the joint process. Furthermore, we present a precise characterization of the asymptotic bias of the SA iterates. As a by-product of our analysis, we derive finite-time bounds on higher moment and present non-asymptotic geometric convergence rates for the iterates, along with a Central Limit Theorem. 

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4. A Lyapunov Theory for Finite-Sample Guarantees of Markovian Stochastic Approximation

   https://doi.org/10.1287/opre.2022.0249  

   Chen, Zaiwei; Maguluri, Siva T; Shakkottai, Sanjay; Shanmugam, Karthikeyan (October 2023, Operations Research)

   This paper develops a unified Lyapunov framework for finite-sample analysis of a Markovian stochastic approximation (SA) algorithm under a contraction operator with respect to an arbitrary norm. The main novelty lies in the construction of a valid Lyapunov function called the generalized Moreau envelope. The smoothness and an approximation property of the generalized Moreau envelope enable us to derive a one-step Lyapunov drift inequality, which is the key to establishing the finite-sample bounds. Our SA result has wide applications, especially in the context of reinforcement learning (RL). Specifically, we show that a large class of value-based RL algorithms can be modeled in the exact form of our Markovian SA algorithm. Therefore, our SA results immediately imply finite-sample guarantees for popular RL algorithms such as n-step temporal difference (TD) learning, TD(𝜆), off-policy V-trace, and Q-learning. As byproducts, by analyzing the convergence bounds of n-step TD and TD(𝜆), we provide theoretical insight into the problem about the efficiency of bootstrapping. Moreover, our finite-sample bounds of off-policy V-trace explicitly capture the tradeoff between the variance of the stochastic iterates and the bias in the limit. 

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5. Effectiveness of Constant Stepsize in Markovian LSA and Statistical Inference

   https://doi.org/10.1609/aaai.v38i18.30028  

   Huo, Dongyan Lucy; Chen, Yudong; Xie, Qiaomin (March 2024, Proceedings of the AAAI Conference on Artificial Intelligence)

   In this paper, we study the effectiveness of using a constant stepsize in statistical inference via linear stochastic approximation (LSA) algorithms with Markovian data. After establishing a Central Limit Theorem (CLT), we outline an inference procedure that uses averaged LSA iterates to construct confidence intervals (CIs). Our procedure leverages the fast mixing property of constant-stepsize LSA for better covariance estimation and employs Richardson-Romberg (RR) extrapolation to reduce the bias induced by constant stepsize and Markovian data. We develop theoretical results for guiding stepsize selection in RR extrapolation, and identify several important settings where the bias provably vanishes even without extrapolation. We conduct extensive numerical experiments and compare against classical inference approaches. Our results show that using a constant stepsize enjoys easy hyperparameter tuning, fast convergence, and consistently better CI coverage, especially when data is limited. 

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