An official website of the United States government
Motivated by applications in large-scale and multi-agent reinforcement learning, we study the non-asymptotic performance of stochastic approximation (SA) schemes with delayed updates under Markovian sampling. While the effect of delays has been extensively studied for optimization, the manner in which they interact with the underlying Markov process to shape the finite-time performance of SA remains poorly understood. In this context, our first main contribution is to show that under time-varying bounded delays, the delayed SA update rule guarantees exponentially fast convergence of the\emph {last iterate} to a ball around the SA operator’s fixed point. Notably, our bound is\emph {tight} in its dependence on both the maximum delay , and the mixing time . To achieve this tight bound, we develop a novel inductive proof technique that, unlike various existing delayed-optimization analyses, relies on establishing uniform boundedness of the iterates. As such, our proof may be of independent interest. Next, to mitigate the impact of the maximum delay on the convergence rate, we provide the first finite-time analysis of a delay-adaptive SA scheme under Markovian sampling. In particular, we show that the exponent of convergence of this scheme gets scaled down by , as opposed to for the vanilla delayed SA rule; here, denotes the average delay across all iterations. Moreover, the adaptive scheme requires no prior knowledge of the delay sequence for step-size tuning. Our theoretical findings shed light on the finite-time effects of delays for a broad class of algorithms, including TD learning, Q-learning, and stochastic gradient.
more »
« less
The Collusion of Memory and Nonlinearity in Stochastic Approximation With Constant Stepsize
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.
more »
« less
- PAR ID:
- 10573136
- Publisher / Repository:
- 38th Conference on Neural Information Processing Systems (NeurIPS 2024)
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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.more » « less
-
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.more » « less
-
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.more » « less
-
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.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Conference Paper:
- The DOI is not currently available.
