Sampling and Curvature in First-Exit Comparisons for Momentum SGD

Polyak--Ruppert averaging gives SGD and momentum methods the same asymptotic covariance. We characterize how fast finite runs reach it and how far they can be parallelized. For strongly convex quadratics with additive noise, we prove an online--offline bridge: the empirical risk minimizer on the samples of a tail average is one Newton step from the tail-averaged query point of any online procedure. For SGD, heavy ball, and Nesterov's method with constant momentum, this step is a boundary term at the two ends of the tail and splits the gap to the offline estimator into center, covariance, and shape errors. Necessary and sufficient conditions show that, on the least curved direction, the center error sets the pace: SGD needs $m\ggκ^2$ rounds but momentum only $m\ggκ$, where $κ$ is the condition number, so at sample size $N$ the critical batch size grows from $Θ(\sqrt{N}/κ)$ to $Θ(\sqrt{N/κ})$. The covariance error separates the momentum methods: heavy ball has relative variance excess $Θ(κ^{3/2}/m)$ along the stiffest direction, whereas Nesterov has relative error $O(\sqrtκ/m)$ in every direction. The shape error is method-independent by a Berry--Esseen bound uniform in $κ$ and batch size, so confidence intervals are valid when the other two errors vanish. We further examine how the conclusions change under heavy-tailed or state-dependent noise and clipping, and establish a first-order comparison with ERM for nonlinear Nesterov averaging under explicit conditions.

Population gradient flow is a common tool for reasoning about how neural networks adapt, including after pretraining. We show that it can mispredict finite-batch stochastic gradient descent (SGD) qualitatively, and we trace the discrepancy to a specific mechanism. In a two-unit ReLU regression, a source task drives the two neurons toward positive proportionality and a target task rewards separating them. After source training for time $T$, gradient flow recovers on the target in time linear in $T$. Online SGD with batch size $b$ and step size $η$ in both phases instead fails with high probability throughout a horizon of order $e^{c/η}$ once $T \gtrsim \log(b/η)$, uniformly on an explicit set of initializations with Gaussian probability above one percent. For each fixed $T$, small-step SGD still recovers, so the failure requires the joint limit of small steps and long pretraining. At the target clone, the population instability is carried entirely by inputs on which the two ReLU gates disagree. For units at angle $δ$ these inputs form a wedge of probability $δ/π$, and weight decay shrinks the angle exponentially during pretraining. On every other input both units receive the same random linear update, which contracts their separation in conditional expectation. Bounding the cumulative probability of sampling the wedge along the exact online recursion, without a diffusion approximation, shows that recovery with fixed probability from an identical source-gradient-flow checkpoint, within $e^{c/η}$ updates, requires $Nb \gtrsim e^{λT}$ target samples and batch size $b \gtrsim ηe^{λT}$, where $N$ counts updates and $λ$ is the weight decay. In simulations, recovery is approximately a function of the disagreement budget $bδ/η$ and saturates in the horizon.

As an alternative to the standard geometric analyses, we give an exact, information-theoretic analysis of stochastic gradient descent (SGD) and its variants. We show that a preconditioned SGD step is the posterior-mean update of a Gaussian Bayes model, and that its one-step regret splits into an intrinsic-time cost and a change in comparator information. The split extends to an identity for the objective itself. Convex convergence, strict-saddle-point escape, the link between flatness and generalization, the standard learning-rate schedules, adaptive optimizers, and the noisy, momentum, heavy-tailed, and gradient-free variants of SGD each correspond to a term or a special case of this identity. We measure its terms on synthetic and real training runs. On real networks it attributes the slack of classical convergence bounds to the terms their derivations drop and separates optimizers that reach the same training loss. That separation follows the number and consistency of their steps. Its relation to which of them generalizes better differs between networks. For gradient-free SGD the identity determines how a curvature preconditioner should enter the update. The sharpness-based generalization certificate it yields, with a data-independent isotropic prior, is vacuous at network scale unless the curvature spectrum is nearly flat across all parameters.