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.
Modern deep learning optimization features heterogeneous parameter structures, noisy gradients, and highly nonconvex landscapes, posing significant challenges for both algorithm design and theoretical analysis. Motivated by the limitations of SGD and the success of adaptive optimizers, we propose {\it Schattor}, a family of adaptive first-order methods based on Schatten norms. Schattor unifies SGD and the recently proposed matrix-variate adaptive optimizer Muon within a single Schatten-norm-based framework. We establish dimension-free stationarity guarantees for methods in the Schattor family for stochastic matrix optimization problems via a novel matrix martingale moment bound. We also develop multi-block extensions that adaptively balance block-wise optimization progress and prove dimension-free stationarity guarantees in this more general setting.