Gradient temporal-difference methods provide stable off-policy prediction with linear function approximation, but their practical performance is strongly affected by the geometry induced by the auxiliary-variable metric. Existing Mirror-Prox TD methods typically use the feature covariance metric, whereas hybrid TD methods suggest that behavior-policy transition information can provide a more informative update geometry. This paper proposes a behavior-induced Mirror-Prox temporal-difference method, called STHTD-MP, which replaces the covariance metric in the primal-dual saddle-point formulation with the symmetric part of the behavior-policy Bellman matrix. The method keeps a single learning rate for the primal and auxiliary variables and applies a Mirror-Prox prediction-correction step to the resulting hybrid saddle-point operator. We provide a formal convergence analysis for fixed-policy linear prediction under standard stochastic approximation assumptions: the behavior-induced metric is positive definite, the joint mean system is Hurwitz, boundedness follows from a Lyapunov argument, and the stochastic recursion converges by the ODE method. We further derive projected-oracle ergodic gap bounds and an exact mean-operator comparison with GTD2-MP based on the spectral radius of the deterministic Mirror-Prox error matrix. The analysis shows that STHTD-MP can have a smaller mean contraction factor than GTD2-MP when the behavior-induced metric improves the saddle-point geometry. Exact numerical mean-operator analysis on two-state, Random Walk, and Boyan Chain benchmarks supports this condition, while Baird's counterexample is identified as a singular boundary case where the strict assumptions fail.
Temporal-difference learning with function approximation can be unstable under off-policy sampling. TDC stabilizes off-policy TD through an auxiliary covariance correction, and TDRC further regularizes this correction in a single-timescale recursion. This paper studies a behavior-aware replacement of the auxiliary covariance geometry in the linear prediction setting, which is the standard local model for understanding the feature-space dynamics of value-function approximation. We first replace the TDC auxiliary matrix (C) by the behavior Bellman matrix (A_μ), yielding BA-TDC, and then regularize the same behavior-aware equation to obtain BA-TDRC. This two-step construction separates the contribution of behavior-aware geometry from the contribution of regularization. The linear analysis also provides a tractable model for an auxiliary-geometry design question that arises in neural-network value approximation, where feature covariances and temporal transition matrices jointly shape the last-layer correction dynamics. We give a finite-state mean-system formulation, prove fixed-point preservation and almost-sure convergence under a Hurwitz stability condition on the instantiated mean system, and compare deterministic mean rates through the spectral radius of the exact linear error recursion. Experiments on the two-state counterexample, Baird's counterexample, Random Walk, and Boyan Chain show that the behavior-aware replacement can be highly beneficial by itself on some tasks, but that regularization is necessary for robust performance across harder settings.
In this paper, we study the finite-time behavior of the TD(0) temporal-difference method with linear function approximation (LFA). We consider on-policy independent and identically distributed (i.i.d.) samples, a constant learning step, and the Polyak-Juditsky averaging method. We establish a new convergence rate, for the Mean-Square Error (MSE) on the approximated function, that is (i) fast in the sense that it admits an optimal dependency in the number of iterations k (i.e., of order 1/k), (ii) robust to ill-conditioning: it only depends on an initial error and modelindependent constants and (iii) sharp up to a multiplicative constant lower than 11. In particular, it does not depend on the smallest eigenvalue of the uncentered covariance matrix of the linear parametrization, unlike all pre-existing O(1/k) rates in the TD(0) literature. We also introduce PCTD(0), a variant of TD(0), which benefits from better convergence properties under an additional assumption of strong mixing on the Markov Chain.
Temporal difference (TD) learning with linear function approximation is a core method for policy evaluation. Its classical continuous-time description is an ordinary differential equation (ODE), which captures the asymptotic mean dynamics but neglects stochastic fluctuations determining the error floor. We introduce a stochastic differential equation (SDE) approximation for linear TD(0) under Markovian noise. The resulting model distinguishes the contraction dynamics governed by the projected Bellman operator from the influence of Markovian sampling. As a consequence, the model explains the constant-stepsize error floor through the interaction between Markovian long-run covariance and the contraction geometry of the projected Bellman operator.