Statistical guarantees for continuous-time policy evaluation: blessing of ellipticity and new tradeoffs

2025-02-06

AI Summary

This paper addresses the non-asymptotic estimation of policy value functions for continuous-time Markov diffusion processes, based on a single discretely observed ergodic trajectory. We propose a least-squares temporal difference (LSTD) estimator and establish the first non-asymptotic statistical guarantee under the first-order Sobolev norm. Key contributions include: (i) revealing the decisive role of diffusion ellipticity in ensuring robustness over infinite time horizons; (ii) characterizing, for the first time, the fundamental trade-off between approximation error and Markov/martingale statistical errors; and (iii) breaking the conventional finite-horizon assumption to achieve an $O(1/sqrt{T})$ convergence rate, where the trajectory length $T$ scales only linearly with the mixing time and the dimension of the basis functions. Our results provide a rigorous theoretical foundation for diffusion-based modeling in continuous control and reinforcement learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We study the estimation of the value function for continuous-time Markov diffusion processes using a single, discretely observed ergodic trajectory. Our work provides non-asymptotic statistical guarantees for the least-squares temporal-difference (LSTD) method, with performance measured in the first-order Sobolev norm. Specifically, the estimator attains an $O(1 / sqrt{T})$ convergence rate when using a trajectory of length $T$; notably, this rate is achieved as long as $T$ scales nearly linearly with both the mixing time of the diffusion and the number of basis functions employed. A key insight of our approach is that the ellipticity inherent in the diffusion process ensures robust performance even as the effective horizon diverges to infinity. Moreover, we demonstrate that the Markovian component of the statistical error can be controlled by the approximation error, while the martingale component grows at a slower rate relative to the number of basis functions. By carefully balancing these two sources of error, our analysis reveals novel trade-offs between approximation and statistical errors.

Problem

• Estimates value function for continuous-time Markov processes
• Provides non-asymptotic guarantees for LSTD method
• Balances approximation and statistical errors in estimation

Innovation

• Continuous-time Markov diffusion processes
• Least-squares temporal-difference method
• Ellipticity ensures robust performance