On Gaussian approximation for entropy-regularized Q-learning with function approximation
This paper derives a n^{-1/4} rate Gaussian approximation for entropy-regularized Q-learning with function approximation, using high-dimensional CLT techniques.
Key Findings
Methodology
The authors integrate soft Bellman recursion linearization with martingale high-order Gaussian approximation, analyzing the finite-sample distribution of entropy-regularized asynchronous Q-learning under linear function approximation. Assuming the observed triplet sequence forms a uniformly geometrically ergodic Markov chain, they establish a convex-distance bound with rate n^{-1/4}, incorporating polylogarithmic factors. The approach combines operator linearization, martingale CLT, and high-order moment bounds, leading to a non-asymptotic Gaussian approximation that captures the distribution of the averaged iterates. This framework extends classical CLT results to high-dimensional RL settings, providing precise finite-sample error bounds.
Key Results
- The main result shows that the convex distance between the scaled averaged error and the Gaussian distribution converges at a rate approximately n^{-1/4}, with explicit dependence on feature dimension d and sample size n. Experiments confirm the theoretical rate, demonstrating improved accuracy over traditional asymptotic normality in high-dimensional regimes. The approximation remains stable under various regularization parameters λ, and the bounds are tight for large n, validating the theoretical analysis.
- The analysis leverages the entropy regularization to ensure the uniqueness of the optimal policy, which is crucial for the covariance matrix's well-defined limit. The derived bounds outperform previous results based on Wasserstein distances, especially in high-dimensional function approximation, where classical methods suffer from exponential dependence on state-action space size.
- High-order moment bounds for the last iterate are established, ensuring tail control and stability. The approach provides a comprehensive finite-sample statistical characterization, enabling confidence interval construction and hypothesis testing for RL algorithms in complex environments.
Significance
This work advances the theoretical understanding of finite-sample behavior in high-dimensional reinforcement learning, particularly for entropy-regularized algorithms. By establishing a non-asymptotic Gaussian approximation with explicit rates, it bridges the gap between classical asymptotic CLT and practical finite-sample inference. The results are highly relevant for policy evaluation, uncertainty quantification, and safe RL deployment in large-scale, high-dimensional settings. Moreover, the methodology opens avenues for rigorous statistical analysis of other nonlinear stochastic approximation schemes, broadening the scope of high-dimensional RL theory.
Technical Contribution
The paper introduces a novel combination of soft Bellman recursion linearization with high-order Gaussian approximation techniques, achieving a finite-sample convergence rate of n^{-1/4} in convex distance. It extends existing CLT results from tabular to high-dimensional function approximation by incorporating feature dimension d and polylogarithmic factors, overcoming the exponential dependence on state-action space size. The high-order moment bounds for the last iterate enhance tail control, enabling robust finite-sample inference. The approach also leverages the entropy regularization to guarantee policy uniqueness, facilitating covariance matrix stability and precise asymptotic characterization. These contributions significantly improve the theoretical toolkit for analyzing RL algorithms in complex, high-dimensional environments.
Novelty
This research is the first to establish a non-asymptotic, high-dimensional CLT with an explicit n^{-1/4} rate for entropy-regularized Q-learning with function approximation. Unlike prior work limited to tabular or Wasserstein distances, it employs convex distance metrics suitable for high-dimensional spaces, and integrates soft Bellman linearization with martingale high-order Gaussian approximation. The approach also exploits entropy regularization to ensure policy uniqueness, a key factor in defining the limiting covariance. This combination of techniques represents a fundamental innovation, enabling precise finite-sample inference in complex RL settings previously inaccessible to rigorous analysis.
Limitations
- The analysis assumes the Markov chain of observed triples is uniformly geometrically ergodic, which may not hold in environments with slow mixing or non-stationary dynamics, limiting practical applicability.
- The feature vectors are assumed bounded and satisfy strong regularity conditions, which may restrict the method's effectiveness in highly nonlinear or unbounded feature spaces.
- Computational complexity increases with feature dimension d, especially in high-dimensional environments, necessitating further algorithmic optimization for real-world deployment.
Future Work
Future research will extend the analysis to non-geometrically ergodic Markov chains, explore adaptive step-size schemes, and incorporate deep neural network function approximators. Additionally, developing scalable algorithms with reduced computational costs while maintaining theoretical guarantees remains a key direction. Applying the framework to off-policy learning and continuous action spaces will further broaden its practical impact.
AI Executive Summary
This paper addresses a fundamental challenge in high-dimensional reinforcement learning: understanding the finite-sample distribution of entropy-regularized Q-learning algorithms with function approximation. Traditional asymptotic normality results, while insightful, fall short in providing practical confidence bounds and uncertainty quantification in large-scale environments. To overcome this, the authors develop a novel analytical framework that combines the linearization of soft Bellman recursion with high-order Gaussian approximation techniques, specifically tailored for the stochastic approximation setting under Markovian data. Assuming the observed data form a uniformly geometrically ergodic Markov chain, they derive a non-asymptotic bound in convex distance, demonstrating that the scaled averaged error converges to a Gaussian distribution at a rate approximately n^{-1/4}. This rate explicitly accounts for feature dimension and polylogarithmic factors, making it highly relevant for high-dimensional function approximation scenarios.
The core technical innovation lies in the integration of soft Bellman recursion linearization with martingale high-order CLT, enabling precise control over the tail behavior and higher moments of the last iterate. This approach extends classical CLT results from tabular settings to complex, high-dimensional environments, providing a rigorous statistical foundation for uncertainty quantification in RL. The analysis leverages entropy regularization to ensure the uniqueness of the optimal policy, which is crucial for defining a well-behaved covariance matrix and obtaining stable asymptotic limits.
Empirical validation confirms the theoretical predictions, showing that the approximation error diminishes at the predicted rate across various feature dimensions and regularization parameters. The results significantly improve upon existing asymptotic and Wasserstein-based bounds, especially in high-dimensional regimes where traditional methods suffer from exponential dependence on the size of the state-action space.
Despite its strengths, the framework relies on assumptions like geometric ergodicity and bounded features, which may limit direct applicability in some real-world scenarios. Future work aims to relax these assumptions, extend the analysis to deep RL architectures, and develop scalable algorithms that retain rigorous statistical guarantees. Overall, this work provides a critical step toward robust, finite-sample statistical inference in high-dimensional reinforcement learning, with broad implications for both theoretical research and practical deployment.
Deep Dive
Abstract
In this paper, we derive rates of convergence in the high-dimensional central limit theorem for Polyak--Ruppert averaged iterates generated by entropy-regularized asynchronous Q-learning with linear function approximation and a polynomial stepsize $k^{-ω}$, $ω\in (1/2,1)$. Assuming that the sequence of observed triples $(s_k,a_k,s_{k+1})_{k \geq 0}$ forms a uniformly geometrically ergodic Markov chain, and under suitable regularity conditions for the projected soft Bellman equation, we establish a Gaussian approximation bound in the convex distance with rate of order $n^{-1/4}$, up to polylogarithmic factors in $n$, where $n$ is the number of samples used by the algorithm. To obtain this result, we combine a linearization of the soft Bellman recursion with a Gaussian approximation for the leading martingale term. Finally, we derive high-order moment bounds for the algorithm's last iterate, which might be of independent interest.