Almost Sure Convergence of Linear Temporal Difference Learning with Arbitrary Features

2025-12-31 · Source: JMLR · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

Jiuqi Wang and Shangtong Zhang, in their 2026 paper published in JMLR, Volume 27, pages 1-36, establish the almost sure convergence of linear temporal difference (TD) learning without requiring linearly independent features. This finding addresses a critical limitation, as the traditional assumption of feature linear independence often does not hold in practical reinforcement learning scenarios. The authors prove that the weight iterates of linear TD converge to a bounded set, and the value estimates derived from these weights are identical almost everywhere. Their analysis also introduces a notion of local stability for the weight iterates. Crucially, this work achieves its results without modifying the standard linear TD algorithm or imposing assumptions tailored to feature dependence, relying instead on a novel characterization of bounded invariant sets of the mean Ordinary Differential Equation (ODE) of linear TD.

Key takeaway

For machine learning engineers or researchers developing reinforcement learning agents, this work validates linear TD's robustness. You can now confidently apply linear TD in environments where feature linear independence is not guaranteed, expanding its practical utility without algorithm modification. This removes a significant theoretical barrier, allowing for more flexible and reliable real-world deployments of linear TD-based systems.

Key insights

This work proves almost sure convergence for linear TD without requiring linearly independent features, addressing a key practical limitation.

Principles

• Linear TD converges almost surely even with dependent features.
• Weight iterates converge to a bounded set.
• Value estimates from converged weights are almost everywhere identical.

Method

The analysis relies on a novel characterization of bounded invariant sets of the mean Ordinary Differential Equation (ODE) of linear TD.

In practice

• This theoretical advancement supports broader application of linear TD in scenarios with dependent features.

Topics

• Reinforcement Learning
• Temporal Difference Learning
• Linear Function Approximation
• Convergence Theory
• Machine Learning Theory
• Feature Dependence

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by JMLR.