On the Convergence of Semi-Gradient TD: A Mean Field Perspective

Relative temporal-difference (TD) learning was introduced to mitigate the slow convergence of TD methods when the discount factor approaches one by subtracting a baseline from the temporal-difference update. While this idea has been studied in the tabular setting, stability guarantees with function approximation remain poorly understood. This paper analyzes relative TD learning with linear function approximation. We establish stability conditions for the algorithm and show that the choice of baseline distribution plays a central role. In particular, when the baseline is chosen as the empirical distribution of the state-action process, the algorithm is stable for any non-negative baseline weight and any discount factor. We also provide a sensitivity analysis of the resulting parameter estimates, characterizing both asymptotic bias and covariance. The asymptotic covariance and asymptotic bias are shown to remain uniformly bounded as the discount factor approaches one.

Emphatic temporal-difference learning (ETD) stabilizes the expected off-policy TD update and changes its projection geometry, but neither property determines constant-stepsize sampled dynamics. We construct an ergodic two-state counterexample in which the ETD mean map contracts while the sampled product has a positive top Lyapunov exponent. Regenerative-cycle analysis separates this sign from the infinite variance of the follow-on trace. We introduce regularized emphatic TD (RETD), a normalized first-order post-shock repair that leaves the trace and importance ratios unchanged, stores the emphatic TD signal in a leaky scalar state, and releases a delayed correction. RETD's raw equilibrium is an affine shift of the ETD equilibrium; single- and two-regularization readouts recover the ETD fixed point exactly. We prove almost-sure convergence for harmonic diminishing stepsizes and a conditional constant-stepsize moment-contraction result from a Markovian random-product bound. RETD has certified negative exponents on the two-state construction and one Baird point, whereas the positive Baird ETD sign remains numerical. Paired 10,000-run experiments validate both separations, fixed-point recovery, a nonmonotone stability region, and task dependence. RETD changes post-shock dynamics; it does not reduce the shared follow-on-trace variance.

Temporal difference (TD) learning is a fundamental algorithm for estimating value functions in reinforcement learning. Recent finite-time analyses of TD with linear function approximation quantify its theoretical convergence rate. However, they often require setting the algorithm parameters using problem-dependent quantities that are difficult to estimate in practice -- such as the minimum eigenvalue of the feature covariance ($ω$) or the mixing time of the underlying Markov chain ($τ_{\text{mix}}$). In addition, some analyses rely on nonstandard and impractical modifications, exacerbating the gap between theory and practice. To address these limitations, we use an exponential step-size schedule with the standard TD(0) algorithm. We analyze the resulting method under two sampling regimes: independent and identically distributed (i.i.d.) sampling from the stationary distribution, and the more practical Markovian sampling along a single trajectory. In the i.i.d.\ setting, the proposed algorithm does not require knowledge of problem-dependent quantities such as $ω$, and attains the optimal bias-variance trade-off for the last iterate. In the Markovian setting, we propose a regularized TD(0) algorithm with an exponential step-size schedule. The resulting algorithm achieves a comparable convergence rate to prior works, without requiring projections, iterate averaging, or knowledge of $τ_{\text{mix}}$ or $ω$.