# A Finite-Sample Analysis of Quantile Temporal-Difference Learning

arXiv: https://arxiv.org/abs/2608.27313 · HTML: https://mlsanity.com/paper/2608.27313/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 33 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [A Distributional Perspective on Reinforcement Learning](https://mlsanity.com/paper/1707.06887/) (2017)
- [Distributional Reinforcement Learning with Quantile Regression](https://mlsanity.com/paper/1710.10044/) (2017)
- [Implicit Quantile Networks for Distributional Reinforcement Learning](https://mlsanity.com/paper/1806.06923/) (2018)
- [Distributed Distributional Deterministic Policy Gradients](https://mlsanity.com/paper/1804.08617/) (2018)
- [Tight Analyses for Non-Smooth Stochastic Gradient Descent](https://mlsanity.com/paper/1812.05217/) (2018)
- [Fully Parameterized Quantile Function for Distributional Reinforcement Learning](https://mlsanity.com/paper/1911.02140/) (2019)
- [An Analysis of Categorical Distributional Reinforcement Learning](https://mlsanity.com/paper/1802.08163/) (2018)
- [Statistics and Samples in Distributional Reinforcement Learning](https://mlsanity.com/paper/1902.08102/) (2019)
- [An Analysis of Quantile Temporal-Difference Learning](https://mlsanity.com/paper/2301.04462/) (2023)
- [Quantile QT-Opt for Risk-Aware Vision-Based Robotic Grasping](https://mlsanity.com/paper/1910.02787/) (2019)
- [The Statistical Benefits of Quantile Temporal-Difference Learning for Value Estimation](https://mlsanity.com/paper/2305.18388/) (2023)
- [Statistical Efficiency of Distributional Temporal Difference Learning and Freedman's Inequality in Hilbert Spaces](https://mlsanity.com/paper/2403.05811/) (2024)
- [A Finite Sample Analysis of Distributional TD Learning with Linear Function Approximation](https://mlsanity.com/paper/2502.14172/) (2025)
- [Statistical Efficiency and Inference of Quantile Distributional Reinforcement Learning](https://mlsanity.com/paper/2607.08444/) (2026)
- [Online Inference for Quantile Temporal Difference Learning in Distributional Reinforcement Learning](https://mlsanity.com/paper/2608.12973/) (2026)

## Abstract (arXiv)

Quantile temporal-difference learning (QTD) is an effective method for learning return distributions through quantile approximation, yet its finite-time behavior remains poorly understood. Its update is nonlinear and nonsmooth, and the stability needed for a sharp convergence rate holds only near the target. We establish a global high-probability last-iterate guarantee for synchronous tabular QTD under general positive, nonincreasing step-size sequences and arbitrary initialization in the natural parameter range. For polynomially decaying step sizes with exponent a∈(0,1), the last iterate converges to the target at rate T-a/2 in the infinity norm, up to logarithmic and lower-order terms. A suitably tuned harmonic schedule recovers the T-1/2 statistical rate up to logarithmic factors. For the m-quantile representation, its ∞-Wasserstein error scales as √m/T up to logarithmic factors, matching the leading polynomial dependence on the quantile resolution and sample size of the corresponding model-based estimator. The proof uses a two-stage global-to-local argument. From arbitrary initialization, Bellman contraction and CDF monotonicity first bring the iterate close to the target, after which, a novel variance--drift matching argument sharpens the control of accumulated noise and local contraction reduces the remaining errors, yielding the sharp rate. Simulations verify the predicted polynomial decay and assess the finite-time entrance bound.