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.
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.
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.
We investigate the statistical properties of Temporal Difference (TD) learning with Polyak-Ruppert averaging, arguably one of the most widely used algorithms in reinforcement learning, for the task of estimating the parameters of the optimal linear approximation to the value function. Assuming independent samples, we make three theoretical contributions that improve upon the current state-of-the-art results: (i) we establish refined high-dimensional Berry-Esseen bounds over the class of convex sets, achieving faster rates than the best known results, and (ii) we propose and analyze a novel, computationally efficient online plug-in estimator of the asymptotic covariance matrix; (iii) we derive sharper high probability convergence guarantees that depend explicitly on the asymptotic variance and hold under weaker conditions than those adopted in the literature. These results enable the construction of confidence regions and simultaneous confidence intervals for the linear parameters of the value function approximation, with guaranteed finite-sample coverage. We demonstrate the applicability of our theoretical findings through numerical experiments.