Two-timescale stochastic approximation (TTSA) is among the most general frameworks for iterative stochastic algorithms. This includes well-known stochastic optimization methods such as SGD variants and those designed for bilevel or minimax problems, as well as reinforcement learning like the family of gradient-based temporal difference (GTD) algorithms. In this paper, we conduct an in-depth asymptotic analysis of TTSA under controlled Markovian noise via central limit theorem (CLT), uncovering the coupled dynamics of TTSA influenced by the underlying Markov chain, which has not been addressed by previous CLT results of TTSA only with Martingale difference noise. Building upon our CLT, we expand its application horizon of efficient sampling strategies from vanilla SGD to a wider TTSA context in distributed learning, thus broadening the scope of Hu et al. (2022). In addition, we leverage our CLT result to deduce the statistical properties of GTD algorithms with nonlinear function approximation using Markovian samples and show their identical asymptotic performance, a perspective not evident from current finite-time bounds.
National Research University Higher School of Economics（俄罗斯国立高等经济研究大学）；CMAP, École Polytechnique（巴黎综合理工学院应用数学中心）；University of Bristol（布里斯托大学）；The Chinese University of Hong Kong（香港中文大学）
Linear two-timescale stochastic approximation (SA) scheme is an important class of algorithms which has become popular in reinforcement learning (RL), particularly for the policy evaluation problem. Recently, a number of works have been devoted to establishing the finite time analysis of the scheme, especially under the Markovian (non-i.i.d.) noise settings that are ubiquitous in practice. In this paper, we provide a finite-time analysis for linear two timescale SA. Our bounds show that there is no discrepancy in the convergence rate between Markovian and martingale noise, only the constants a…
Convergence of Two-Timescale Markovian Stochastic Approximations with Applications in Reinforcement Learning
Department of Computer Science, University of Virginia, Charlottesville, VA, USA（弗吉尼亚大学计算机科学系）；Data Science Lab, MIT, Cambridge, MA, USA（麻省理工学院数据科学实验室）；Mitch Daniels School of Business, Purdue University, West Lafayette, IN, USA（普渡大学米切尔丹尼尔斯商学院）；Division Office Physics, Math and Astronomy, California Institute of Technology, Pasadena, CA, USA（加州理工学院物理、数学和天文学分校）
Greedy-GQ is an off-policy two timescale algorithm for optimal control in reinforcement learning. This paper develops the first finite-sample analysis for the Greedy-GQ algorithm with linear function approximation under Markovian noise. Our finite-sample analysis provides theoretical justification for choosing stepsizes for this two timescale algorithm for faster convergence in practice, and suggests a trade-off between the convergence rate and the quality of the obtained policy. Our paper extends the finite-sample analyses of two timescale reinforcement learning algorithms from policy evaluat…
LTCI（通信与信息处理实验室）；Télécom Paris（巴黎电信学院）；Institut Polytechnique de Paris（巴黎理工学院）
Actor-critic methods integrating target networks have exhibited a stupendous empirical success in deep reinforcement learning. However, a theoretical understanding of the use of target networks in actor-critic methods is largely missing in the literature. In this paper, we reduce this gap between theory and practice by proposing the first theoretical analysis of an online target-based actor-critic algorithm with linear function approximation in the discounted reward setting. Our algorithm uses three different timescales: one for the actor and two for the critic. Instead of using the standard s…
New goodness-of-fit tests for Markovian models in time series analysis are developed which are based on the difference between a fully nonparametric estimate of the one-step transition distribution function of the observed process and that of the model class postulated under the null hypothesis. The model specification under the null allows for Markovian models, the transition mechanisms of which depend on an unknown vector of parameters and an unspecified distribution of i.i.d. innovations. Asymptotic properties of the test statistic are derived and the critical values of the test are found u…
We analyse the asymptotic properties of a continuous-time, two-timescale stochastic approximation algorithm designed for stochastic bilevel optimisation problems in continuous-time models. We obtain the weak convergence rate of this algorithm in the form of a central limit theorem. We also demonstrate how this algorithm can be applied to several continuous-time bilevel optimisation problems.
Finite-Time Bounds for Two-Time-Scale Stochastic Approximation with Arbitrary Norm Contractions and Markovian Noise
Department of Electrical Engineering, Stanford University（斯坦福大学电气工程系）；Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院工业与系统工程系）
Microsoft Research Montréal（微软研究院蒙特利尔）；MILA（蒙特利尔学习算法研究所）；McGill University（麦吉尔大学）
A central object of study in Reinforcement Learning (RL) is the Markovian policy, in which an agent's actions are chosen from a memoryless probability distribution, conditioned only on its current state. The family of Markovian policies is broad enough to be interesting, yet simple enough to be amenable to analysis. However, RL often involves more complex policies: ensembles of policies, policies over options, policies updated online, etc. Our main contribution is to prove that the occupancy measure of any non-Markovian policy, i.e., the distribution of transition samples collected with it, ca…
Adaptive and interacting Markov Chains Monte Carlo (MCMC) algorithms are a novel class of non-Markovian algorithms aimed at improving the simulation efficiency for complicated target distributions. In this paper, we study a general (non-Markovian) simulation framework covering both the adaptive and interacting MCMC algorithms. We establish a Central Limit Theorem for additive functionals of unbounded functions under a set of verifiable conditions, and identify the asymptotic variance. Our result extends all the results reported so far. An application to the interacting tempering algorithm (a s…
School of Science and Engineering（理工学院）；Shenzhen Research Institute of Big Data (SRIBD)（深圳大数据研究院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；School of Data Science（数据科学学院）；Shenzhen Institute of Artificial Intelligence and Robotics for Society (AIRS)（深圳人工智能与机器人研究院）
Decentralized Actor-Critic (AC) algorithms have been widely utilized for multi-agent reinforcement learning (MARL) and have achieved remarkable success. Apart from its empirical success, the theoretical convergence property of decentralized AC algorithms is largely unexplored. Most of the existing finite-time convergence results are derived based on either double-loop update or two-timescale step sizes rule, and this is the case even for centralized AC algorithm under a single-agent setting. In practice, the \emph{single-timescale} update is widely utilized, where actor and critic are updated …
Two timescale stochastic approximation (SA) has been widely used in value-based reinforcement learning algorithms. In the policy evaluation setting, it can model the linear and nonlinear temporal difference learning with gradient correction (TDC) algorithms as linear SA and nonlinear SA, respectively. In the policy optimization setting, two timescale nonlinear SA can also model the greedy gradient-Q (Greedy-GQ) algorithm. In previous studies, the non-asymptotic analysis of linear TDC and Greedy-GQ has been studied in the Markovian setting, with diminishing or accuracy-dependent stepsize. For t…
Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields model. Fix $n\ge 1$ and $β>0$. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate $β(n-k)/n$, where $k$ is the distance from the node to the root. Denote by $Z_n(t)$ the number of nodes with no descendants at time $t$ and let $T_n = β^{-1} n \ln(n /\ln 4) + (\ln 2)/(2 β)$. We prove that $2^{-n} Z_n(T_n + n τ)$, $τ\in\bb R$, conve…
This paper presents a finite time convergence analysis for a decentralized stochastic approximation (SA) scheme. The scheme generalizes several algorithms for decentralized machine learning and multi-agent reinforcement learning. Our proof technique involves separating the iterates into their respective consensual parts and consensus error. The consensus error is bounded in terms of the stationarity of the consensual part, while the updates of the consensual part can be analyzed as a perturbed SA scheme. Under the Markovian noise and time varying communication graph assumptions, the decentrali…
