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Near Minimax Rate for Quantile Distributional Reinforcement Learning

Zijie Cheng, Zhihua Zhang · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.29595 · cited by 0 works (OpenAlex) · open-access PDF

Russian

В этой статье мы изучаем эффективную оценку квантовой дистрибутивной политики, где распределение доходов по данной политике приближается к квантовой фиксированной точке риска Bellman $\bmη_m$. Мы исследуем статистическую точность оценки лог-лимита $\bmη_m$ по квантовым расстояниям под инстинктами Вассерштейна. Предполагая, что доступ к генерирующей модели приближается к квантовой фиксированной точке риска $\bmη_m^{(n)}$ из эмпирического процесса принятия решений Маркова и устанавливаем регулярные границы локальных ошибок под расстояниями Вассерштейна. Когда плотность вознаграждения равномерно ограничена от нуля, мы

Machine translation (Meta M2M100); the English original follows.

Abstract

In this paper, we study quantile distributional policy evaluation, where the return distribution under a given policy is approximated by the quantile Bellman fixed point $\bmη_m$. We investigate the statistical accuracy of estimating $\bmη_m$ across quantile resolutions under Wasserstein distances. Assuming access to a generative model, we construct a quantile fixed point estimator $\bmη_m^{(n)}$ from an empirical Markov decision process and establish non-asymptotic error bounds under Wasserstein distances. When reward densities are uniformly bounded away from zero, we obtain a root-$n$ high-probability bound simultaneously over all resolutions $m$ and all Wasserstein orders $1\le p\le\infty$. When reward densities vanish at the support endpoints, we show that continuity and endpoint monotonicity suffice to preserve the same resolution-independent rate for $1\le p\le2$. For $p>2$, we establish a root-$n$ bound with coefficient $m^{1/2-1/p}$, together with a slower, resolution-independent $n^{-1/p}$ bound for every finite $p>2$. We further obtain resolution-independent $W_\infty$ consistency governed by the continuity modulus of reward quantile functions. To investigate the sharpness of these bounds, we characterize the exact local asymptotic minimax risk at each fixed resolution as the expected Wasserstein-coordinate norm of the efficient Gaussian limit and prove that our estimator attains it. We then examine the resolution dependence of this optimal local risk. A regular one-state instance has resolution-independent local risk constants, whereas a single endpoint-vanishing two-state instance exhibits growth of order $m^{1/2-1/p}$, up to logarithmic factors along a resolution subsequence, for every $p>2$.

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Topics: Mathematics, Quantile, Minimax, Estimator, Wasserstein metric, Bounded function

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