Quantile methods improve accuracy bounds in reinforcement learning returns
Near Minimax Rate for Quantile Distributional Reinforcement Learning
Machine Learning
Summary
This paper studies a way to better estimate the full range of possible future outcomes in decision-making tasks by focusing on quantiles, which are points that split data into equal parts. The authors look at how accurately these quantiles can be estimated in different settings, providing mathematical guarantees on the error. They also explore how these error bounds depend on factors like the number of quantile points used and certain properties of the reward distribution. Their analysis shows when their approach reaches the best possible accuracy and how challenging cases affect this.
What this means in practice
- •For reinforcement learning engineers: More accurately estimate full return distributions from simulation data to improve policy evaluation reliability under diverse conditions.
- •For risk management teams: Assess uncertainties in sequential decision outcomes by using quantile-based error bounds to inform safer strategy evaluations.
Authors
Zijie Cheng, Zhihua Zhang
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 $p>2$.