Sampling and Curvature in First-Exit Comparisons for Momentum SGD

Haiyi Li

First-exit prediction for momentum SGD depends on sampling information that pointwise gradient laws omit. Strongly convex finite pools with identical pointwise laws and matched iid momentum dynamics can have different reshuffled exits; even analytic pools with uniformly close pointwise laws can have separated exit probabilities. For persistent sampling in one dimension, we establish an explicit $O(h)$ mean-exit comparison with a finite-rate switching process under nonvanishing component velocities and step-size conditions; iid sampling instead averages to an ODE. Retaining realized minibatches and sampled curvature yields finite-window exit brackets beyond this regime. In 32 trained ridge-logistic runs, 29 meet prespecified targets with rounding-certified brackets; an enclosure control explains most of the width advantage. Exact-real certificates establish finite means at the original tested radii for three ResNet-18 checkpoints and give local finite-mean conditions for tanh and ViT networks; their upper bounds remain loose, and original-radius ViT mean guarantees remain unresolved despite 128 fresh continuation paths. A two-network replication gives 32/32 successful seeds in…