Abstract
Classical stochastic-approximation analyses treat the covariance of stochastic gradients as an exogenous modeling input. We show that under exchangeable mini-batch sampling this covariance is identified by the sampling mechanism itself: to leading order it is the projected covariance of per-sample gradients. In well-specified likelihood problems this reduces locally to projected Fisher information; for general M-estimation losses the same object is the projected gradient covariance G*(theta), which together with the Hessian induces sandwich/Godambe geometry. This identification -- not the subsequent diffusion or Lyapunov machinery, which is classical once the noise matrix is given -- is the paper's main contribution. It endogenizes the diffusion coefficient (with effective temperature tau = eta/b), determines the stationary covariance via a Lyapunov equation whose inputs are now structurally fixed, and selects the identified statistical geometry as the natural metric for convergence analysis. We prove matching upper and lower bounds of order Theta(1/N) for risk in this metric under an oracle budget N; the lower bound is established first via a van Trees argument in the parametric Fisher setting and then extended to adaptive oracle transcripts under a predictable-information condition and mild conditional likelihood regularity. Translating these bounds into oracle complexity yields epsilon-stationarity guarantees in the Fisher dual norm that depend on an intrinsic effective dimension d_eff and a statistical condition number kappa_F, rather than ambient dimension or Euclidean conditioning. Numerical experiments confirm the Lyapunov predictions at both continuous-time and discrete-time levels and show that scalar temperature matching cannot reproduce directional noise structure.
