A Single Stepsize Suffices for Unprojected Linear TD(0): Simultaneous Robust and Fast Rates via Polyak--Ruppert Averaging

We study linear TD(0) under Markovian sampling, where data are generated along a single trajectory. We provide high-probability guarantees for a plain unprojected TD(0) algorithm with Polyak-Ruppert (PR) averaging, using a single stepsize schedule $η_t \propto \frac{1}{τ_{\mathrm{mix}}\log(t)\sqrt{t}}$ that depends on the mixing time but requires no prior knowledge of the curvature parameter $ω$. Our first result shows that such a choice of the stepsize guarantees that the TD(0) iterates are automatically and uniformly bounded with high probability, without projections and without any stability argument based on $ω$. Building on this result, we establish a simultaneous high-probability convergence guarantee for the PR average: the same stepsize yields both a robust curvature-free $\widetilde{\mathcal{O}}\!\left(\frac{τ_{\mathrm{mix}}}{\sqrt{T}}\right)$ rate and a fast curvature-dependent $\widetilde{\mathcal{O}}\!\left(\frac{τ_{\mathrm{mix}}^2}{ωT}\right)$rate, with the bound taking the minimum of the two. The core technical ingredient is a Poisson-equation toolkit for geometrically mixing Markov chains, which decomposes Markov noise into a martingale term plus a controlled remainder and enables a new self-bounding inductive argument for pathwise stability.

A Polyak-Ruppert Central Limit Theorem for SA-Adam with Momentum and Non-Convergent Adaptive Preconditioning

Adaptive optimizers combining preconditioning, momentum, and weight decay (Adam and AdamW) are, under Polyak-Ruppert averaging, candidate engines for one-pass inference. Does the averaged iterate keep the classical Polyak-Ruppert central limit theorem (CLT), with sandwich covariance $H^{-1}SH^{-1}$ (Hessian $H$, gradient covariance $S$), under momentum and non-convergent preconditioning? The preconditioner-only analysis does not carry over: with momentum the canonical decomposition collapses to a tautology. Treating the augmented state (iterate, momentum buffer) as a time-varying linear stochastic approximation (SA), we prove (under local stabilization) positive drift stability, a non-autonomous Polyak-Ruppert CLT, and a projection identity. The upshot: the iterate-marginal covariance is exactly the plain stochastic gradient descent (SGD) sandwich $H^{-1}SH^{-1}$, so the adaptivity is asymptotically invisible. This holds for SA-Adam (sub-linearly vanishing momentum gain, $γ\in(α,1)$; the sub-linear regime is essential), not constant-$β$ deployed Adam. Coupled $L_2$ weight decay yields the ridge-penalized sandwich, extending one-pass inference to regularized problems.