On Gaussian approximation for entropy-regularized Q-learning with function approximation

2026-05-19 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This research paper, "On Gaussian approximation for entropy-regularized Q-learning with function approximation," by Rubtsov et al., establishes non-asymptotic Gaussian approximation bounds for Polyak–Ruppert averaged iterates in entropy-regularized asynchronous Q-learning. The algorithm uses linear function approximation and a polynomial stepsize $k^{-omega}$ where $omega \in (1/2,1)$. Assuming the observed data forms a uniformly geometrically ergodic Markov chain and suitable regularity for the projected soft Bellman equation, the authors derive a Gaussian approximation bound in convex distance with a rate of $n^{-1/4}$, up to polylogarithmic factors in $n$, where $n$ is the number of samples. This rate is determined by sample size $n$ and feature dimensionality $d$, rather than tabular cardinalities, offering a sharper convergence rate compared to existing results for vanilla Q-learning. The work also provides high-order moment bounds for the algorithm's last iterate.

Key takeaway

For AI Scientists and Research Scientists developing or analyzing reinforcement learning algorithms, this work demonstrates that entropy-regularized Q-learning with linear function approximation offers a superior non-asymptotic Gaussian approximation rate of $n^{-1/4}$ in convex distance. This finding is crucial for uncertainty quantification and statistical inference, as it provides tighter finite-sample guarantees that scale with feature dimension rather than state-action space size. You should consider adopting entropy regularization to ensure a unique optimal policy and achieve faster convergence rates in your Q-learning implementations, particularly when working with high-dimensional problems.

Key insights

Entropy-regularized Q-learning with function approximation achieves a non-asymptotic Gaussian approximation rate of $n^{-1/4}$.

Principles

• Entropy regularization ensures unique optimal policy.
• Polyak–Ruppert averaging improves statistical efficiency.
• Gaussian approximation quality depends on sample size and feature dimension.

Method

The method combines linearization of the soft Bellman recursion with Gaussian approximation for the leading martingale term, using Poisson equation framework for noise decomposition and moment bounds for iterates.

In practice

• Use $\omega=3/4$ for optimal convergence rate.
• Apply convex distance for statistical inference procedures.
• Consider entropy regularization for robust Q-learning.

Topics

• Gaussian Approximation
• Entropy-Regularized Q-learning
• Linear Function Approximation
• Central Limit Theorem
• Polyak–Ruppert Averaging

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.