Finite-Sample Analysis of Stochastic Approximation Using Smooth Convex Envelopes (2002.00874v6)

Abstract: Stochastic Approximation (SA) is a popular approach for solving fixed-point equations where the information is corrupted by noise. In this paper, we consider an SA involving a contraction mapping with respect to an arbitrary norm, and show its finite-sample error bounds while using different stepsizes. The idea is to construct a smooth Lyapunov function using the generalized Moreau envelope, and show that the iterates of SA have negative drift with respect to that Lyapunov function. Our result is applicable in Reinforcement Learning (RL). In particular, we use it to establish the first-known convergence rate of the V-trace algorithm for off-policy TD-learning. Moreover, we also use it to study TD-learning in the on-policy setting, and recover the existing state-of-the-art results for $Q$-learning. Importantly, our construction results in only a logarithmic dependence of the convergence bound on the size of the state-space.

Overview of Finite-Sample Analysis of Stochastic Approximation Using Smooth Convex Envelopes

This paper presents a detailed paper on the finite-sample behavior of Stochastic Approximation (SA) algorithms, specifically targeting cases where the underlying operator is a contraction mapping with respect to an arbitrary norm. A key methodological innovation in the paper is the use of generalized Moreau envelopes to create a smooth Lyapunov function, facilitating the derivation of finite-sample error bounds.

The authors demonstrate the general applicability of their approach across various reinforcement learning algorithms. Notably, they establish finite-sample convergence rates for the V-trace algorithm in off-policy TD-learning. Their analysis extends further to on-policy TD-learning algorithms, specifically TD$(n)$, and the classical $Q$-learning algorithm, recovering and in some cases improving upon existing convergence results.

Methodology

The core idea introduced in this work is leveraging smooth convex envelopes to manage the complexity introduced by arbitrary norms in SA. The Moreau envelope technique serves as the cornerstone, providing a way to approximate non-smooth Lyapunov functions with smooth ones. This smoothness is critical for handling the drift analyses needed while considering contraction mappings.

The SA process described is generically framed as solving a noisy fixed-point equation. The iterative algorithm, controlled through a stepsize parameter, aims to converge towards a fixed point. The efficient analysis of this convergence, specifically in finite samples, hinges on crafting a Lyapunov function that respects the contraction mapping, counteracting stochastic noise.

Key Results

The precision in convergence is quantified through finite-sample bounds. These bounds are particularly insightful as they factor in arbitrary norms, addressing a gap not typically covered by prior SA methodologies. Furthermore, the results are robust against non-uniform noise scaling. Distinctively, the dimension $d$ dependence is logarithmic, a significant insight given practical applications in high-dimensional state spaces.

Several specific results are noteworthy:

• V-trace Algorithm Performance: The paper offers the first-known finite-sample error bounds for V-trace. A unique finding is the minimal dimensional dependence, which is a logarithmic function of the state-space size. This has implications for applications like multi-agent systems where state spaces are extensive.
• TD$(n)$ Algorithm Analysis: For various choices of $n$, the convergence bounds provided offer a more granular understanding of balancing bias and variance as a function of the step size and iterative depth $n$.
• $Q$-Learning Recovery: This work recovers existing bounds in diminishing stepsize regimes, while proposing improvements for constant stepsize settings, specifically showing a reduction in dimensional dependency to logarithmic factors.

Implications and Future Directions

The implications of this research are both theoretical and practical. Theoretically, it provides a novel approach to bridging the gap where norm-specific smoothness is absent. Practically, it allows reinforcement learning applications to leverage more adaptable and efficient types of SA.

Future research could explore expanding the techniques to non-expansive operator settings without contraction, as well as implementing the approach with function approximation and larger, potentially continuous state-action spaces, which are common in modern reinforcement learning settings.

In conclusion, this paper substantially contributes to our understanding and application of SA in complex settings. Its findings extend the toolkit available to researchers working on iterative learning processes in stochastic environments and reinforce the role of innovative mathematical tools like the Moreau envelope in advancing computational methodologies.