Fast and Robust Convergence Rate for TD(0) with Linear Function Approximation, Universal Learning Steps and I.I.D. Samples

2026-06-04 · Source: Takara TLDR - Daily AI Papers · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, medium

Summary

A new analysis of the TD(0) temporal-difference method with linear function approximation (LFA) establishes a fast and robust convergence rate for the Mean-Square Error (MSE) on the approximated function. This study, conducted under on-policy independent and identically distributed (i.i.d.) samples, a constant learning step, and Polyak-Juditsky averaging, demonstrates a convergence rate of order 1/k, which is optimal in its dependency on the number of iterations k. A key finding is the rate's robustness to ill-conditioning, as it relies solely on an initial error and model-independent constants, notably avoiding dependency on the smallest eigenvalue of the uncentered covariance matrix—a common factor in prior O(1/k) TD(0) rates. The established rate is also sharp, up to a multiplicative constant lower than 11. Additionally, the paper introduces PCTD(0), a variant of TD(0) designed for improved convergence properties under a strong mixing assumption on the Markov Chain.

Key takeaway

For Machine Learning Engineers optimizing reinforcement learning agents with TD(0) and linear function approximation, this research suggests you can achieve optimal O(1/k) convergence rates that are robust to ill-conditioning. You should consider implementing Polyak-Juditsky averaging and constant learning steps, as these contribute to a convergence rate independent of the feature covariance matrix's smallest eigenvalue. This removes a significant practical hurdle, allowing for more stable and predictable performance without needing to estimate complex problem-dependent quantities.

Key insights

A new TD(0) convergence rate with LFA is fast, robust to ill-conditioning, and independent of the covariance matrix's smallest eigenvalue.

Principles

• Optimal O(1/k) convergence can be achieved without ill-conditioning dependency.
• Polyak-Juditsky averaging aids robust convergence in TD(0).
• Strong mixing assumptions can enable better TD(0) variant convergence.

Method

The paper analyzes TD(0) with LFA using on-policy i.i.d. samples, a constant learning step, and Polyak-Juditsky averaging to derive a new MSE convergence rate. It also introduces PCTD(0).

In practice

• Consider TD(0) variants like PCTD(0) for strong mixing environments.
• Apply Polyak-Juditsky averaging for robust TD(0) convergence.

Topics

• Temporal Difference Learning
• Linear Function Approximation
• Convergence Rate Analysis
• Reinforcement Learning
• Polyak-Juditsky Averaging
• Ill-conditioning Robustness

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

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Editorial summary, takeaway, and curation by AIssential. Original article published by Takara TLDR - Daily AI Papers.