Convergence and Regret Analysis of Q-Learning and Exploratory Policy Improvement for Controlled Diffusion Processes

Overview

This research focuses on the analytical aspects of $q$-learning and associated algorithms, specifically within the domain of controlled diffusion processes. The study addresses the convergence characteristics of these algorithms, which were initially introduced by Jia and Zhou in 2023. The scope of investigation includes exploratory policy improvement methodologies and $q$-learning.

Research Context

The algorithms under examination are those previously proposed by Jia and Zhou (J. Mach. Learn. Res., 24 (2023), 161). These algorithms are designed for application within the framework of controlled diffusion processes. The current study aims to provide a deeper understanding of their theoretical guarantees regarding convergence and performance.

Approach

The research differentiates its analytical approach for two distinct algorithmic components:

• Exploratory Policy Improvement: For this component, the study establishes conditions under which exponential convergence can be achieved. This analysis relies on specific growth and regularity assumptions concerning the model parameters relevant to the controlled diffusion processes.
• $q$-learning: For $q$-learning algorithms, the approach involves deriving quantitative bounds. These bounds characterize the error and regret associated with the algorithm's performance. The derivation of these bounds is contingent upon additional assumptions related to the function approximation techniques employed and the dynamics of the associated stochastic approximation.

Findings

• For algorithms implementing exploratory policy improvement, the research establishes exponential convergence. This finding is observed to hold under the presence of specified growth and regularity assumptions pertaining to the model parameters.
• For $q$-learning algorithms, the study derives quantitative error bounds. These bounds provide a measure of the discrepancy between the algorithm's output and the true values.
• In addition to error bounds, the research also derives quantitative regret bounds for $q$-learning. These bounds quantify the cumulative performance difference relative to an optimal policy.
• The derivation of both error and regret bounds for $q$-learning is conditional on additional assumptions. These assumptions specifically concern the characteristics of function approximation and the dynamics governing the stochastic approximation processes involved.