New Proof Sharpens Bounds for Asynchronous Temporal-Difference Learning in Reinforcement Learning

A new theoretical paper posted to arXiv establishes near-minimax-optimal convergence rates for importance-weighted temporal-difference (TD) learning when the algorithm learns from a single trajectory of Markovian data. The work addresses a long-standing gap between the rates achievable in synchronous, i.i.d. settings and the slower rates often observed in practice when data arrives sequentially from a Markov decision process.

The paper, submitted by researchers affiliated with the arXiv cs.LG category, studies the last iterate of standard importance-weighted TD learning. The setting is a finite Markov decision process where a behavior policy generates a single trajectory, and a target policy is evaluated using importance sampling ratios bounded by a parameter ρ. The analysis covers irreducible, possibly periodic state chains and provides a finite-time, high-probability bound on the sup-norm error.

Key facts

The main result shows that the last iterate achieves sup-norm error at most ε with high probability after Õ(ρ H³ / (μ_min ε²) + h_tar) transitions, for ε between 0 and 1. The statistical term retains the cubic effective-horizon dependence of synchronous TD, while the initialization cost is additive in h_tar, not multiplicative. Setting the behavior policy equal to the target policy recovers the on-policy rate.

What the new bound means for practitioners

The result directly affects anyone deploying TD learning for policy evaluation in reinforcement learning, especially when off-policy data is used. The cubic dependence on H, the effective horizon, confirms that longer-horizon problems require proportionally more data. The additive initialization cost means that the time to reach a good estimate does not compound with the horizon, which is a practical improvement over prior bounds that mixed these terms multiplicatively.

For aperiodic behavior chains, the paper relates h_tar to the mixing time of the chain, providing a concrete bound that connects the theory to common practice in Markov chain Monte Carlo and reinforcement learning.

Practical state-count rule

A notable contribution is an empirical state-count rule that achieves the same rate without requiring prior knowledge of model parameters. Given the importance sampling ratios and their bound, a practitioner can set step sizes based on how many times each state has been visited. This removes the need to estimate mixing times or stationary probabilities ahead of time, bringing the theoretical guarantee closer to a usable algorithm.

Coverage beyond standard TD learning

The proof framework is general enough to cover nonnegative affine importance-weighted prediction updates. This includes successor features, transition-dependent discounting, and coupled finite-mode predictions. A perturbation estimate also yields a time-uniform guarantee for online return-variance estimation, which is relevant for risk-aware reinforcement learning applications.

Matching lower bound

The paper does not stop at an upper bound. A three-state, two-action construction provides matching minimax lower bounds for both the statistical term and the initialization cost, up to logarithmic factors. This means the rates cannot be substantially improved without changing the problem class, establishing the result as near-optimal.

What the result does not cover

The analysis assumes a finite state space and known importance sampling ratios. Infinite state spaces, function approximation, or settings where the behavior policy is unknown are not addressed. The practical state-count rule still requires the importance ratio bound to be known or estimated.

Source: arXiv cs.LG, “Near-Minimax-Optimal Rates for Asynchronous TD Learning with Markovian Data,” https://arxiv.org/abs/2609.38880