Statistical Efficiency and Inference of Quantile Distributional Reinforcement Learning

2026-07-09 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

This paper studies quantile-based distributional reinforcement learning (RL) from a statistical efficiency perspective, focusing on distributional policy evaluation to characterize the return distribution. The authors construct an estimator, η_m^(n), based on an empirical Markov decision process for the quantile fixed point η_m. They establish a non-asymptotic error bound for η_m^(n) and η_m under the supremum W_∞ metric, showing the estimation error scales as Õ(√(m/n)). This indicates sample efficiency, achieving the optimal parametric √n convergence rate. The paper also derives the asymptotic distribution of quantile parameters √n(θ_m^(n)-θ_m) and characterizes the semiparametric efficiency bound, which the estimator attains. Furthermore, it investigates the asymptotic regime where the number of quantiles diverges, showing the limit covariance structure matches the semiparametric efficiency bound of the nonparametric model, thus maintaining asymptotic efficiency in the infinite-dimensional limit. A Berry–Esseen theorem is established for smooth functionals √n(η_m^(n)(s)-η_m(s))f, supporting statistically valid inference.

Key takeaway

For AI Scientists designing or evaluating distributional reinforcement learning algorithms, this research confirms that quantile-based methods offer statistically optimal sample efficiency, achieving √n convergence. Your choice of quantile-based estimators is validated for both fixed and diverging quantile numbers, ensuring asymptotic efficiency. This provides a strong theoretical foundation for robust inference on return distributions, informing more reliable algorithm development and performance guarantees.

Key insights

Quantile-based distributional RL estimators achieve optimal statistical efficiency and asymptotic efficiency in both fixed and infinite-dimensional settings.

Principles

• Estimation error scales as Õ(√(m/n)) for quantile-based RL.
• Quantile-based estimators attain semiparametric efficiency bounds.
• Asymptotic efficiency holds even with diverging numbers of quantiles.

Method

Construct an estimator η_m^(n) for the quantile fixed point η_m using an empirical Markov decision process, assuming a generative model.

Topics

• Distributional Reinforcement Learning
• Quantile Regression
• Statistical Efficiency
• Policy Evaluation
• Markov Decision Processes
• Asymptotic Theory

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.