A Finite Sample Analysis for Quantile Temporal Difference Learning in Distributional Reinforcement Learning

Noosaga Placements

• Reinforcement Learningsubfield95%
  The paper focuses on quantile temporal-difference learning, a core reinforcement learning algorithm, and provides a finite-sample analysis within the distributional RL framework.
  We establish a global finite-sample guarantee for synchronous quantile temporal-difference learning (QTD) in tabular distributional reinforcement learning.Distributional reinforcement learning (DRL) models the law of the random return rather than only its expectation
• Model-Based Reinforcement Learningframework90%
  The paper analyzes QTD, a model-free temporal-difference learning method, and situates it within the broader distributional RL framework, but does not directly use or extend model-based RL.
  quantile temporal-difference learning (QTD) underlies algorithms such as QR-DQN and implicit quantile networksWe study synchronous QTD with polynomially decreasing stepsizes
• Statistical Learning Theoryframework70%
  The paper provides a finite-sample analysis, which is a core topic in statistical learning theory, and uses concentration inequalities and martingale arguments typical of this framework.
  We prove a high-probability bound for the raw last iterate from an arbitrary initialization in the natural parameter domain.Freedman’s inequality therefore produces a stochastic term of order sqrt(v_m α_n / (1-γ))
• Statistical Learningsubfield60%
  The analysis heavily relies on statistical tools such as martingale concentration inequalities, variance-sensitive bounds, and quantile estimation, which are central to statistical learning theory.
  the leading last-iterate fluctuation is of order O~(T^{-a/2}/sqrt(1-γ)) and has no polynomial dependence on the number of quantilesCombining that identity with Var(ξ_{s,i} | F_t) ≲ τ_i(1-τ_i) ≲ v_m d_{s,i} telescopes the predictable variance of the propagated martingale.
• Machine Learningsubfield50%
  The paper contributes to the theoretical understanding of a machine learning algorithm (QTD) and its convergence properties, fitting within the broader machine learning theory domain.
  The finite-sample analysis of QTD is subtle even in a tabular policy-evaluation problem with a generative model.
• Policy Gradient Methodsframework30%
  The paper does not focus on policy gradient methods; it is about value-based temporal-difference learning, so this framework is only tangentially related.

Abstract

We establish a global finite-sample guarantee for synchronous quantile temporal-difference learning (QTD) in tabular distributional reinforcement learning. The proof separates two stability mechanisms. A global comparison argument, based on the order monotonicity of reward cumulative distribution functions and the $W_\infty$ contraction of the distributional Bellman operator, brings an arbitrarily initialized iterate into a local neighborhood. Inside that neighborhood, we linearize the QTD mean field. Its Jacobian is a nonsingular $M$-matrix, and the associated positive semigroup permits a variance-sensitive martingale analysis. For stepsizes $α_t=c(t+1)^{-a}$ with $a\in(1/2,1)$, the leading last-iterate fluctuation is of order $\widetilde O\bigl(T^{-a/2}/\sqrt{1-γ}\bigr)$ and has no polynomial dependence on the number of quantiles. The deterministic transient and the required burn-in can still depend on the smallest Bellman-target density, which is of order $m^{-1}$ in the worst case. The result therefore distinguishes sharply between the local stochastic fluctuation and the global sample complexity.

Paper Context