[Submitted on 21 Oct 2026 (v1), last revised 3 Nov 2026 (this version, v2)]
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 $bmeta_m$. We investigate the statistical accuracy of estimating $bmeta_m$ across quantile resolutions under Wasserstein distances. Assuming access to a generative model, we construct a quantile fixed point estimator $bmeta_m^{(n)}$ from an empirical Markov decision process of $n$ observations per state-action pair 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 $1le pleinfty$. When reward densities vanish at the support endpoints, we show that continuity and endpoint monotonicity suffice to preserve the same resolution-independent rate for $1le ple2$. 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 $p>2$.
Submission history
From: Zijie Cheng [view email]
[v1] Wed, 21 Oct 2026 11:35:36 UTC (70 KB)
[v2] Tue, 3 Nov 2026 12:46:34 UTC (53 KB)
