Gaussian Limits for SGD Without Stationary Moments
Temporal dependence can separate the Gaussian approximation of stochastic gradient descent from its stationary moments. For unmodified least-squares SGD, we construct a design with standard Gaussian marginals whose stationary error has every positive moment infinite. Independent observations with the same marginals instead give finite stationary variance. Both regimes retain a Gaussian small-step limit. Our general theory establishes pathwise contraction from a finite second design moment, then uses score cancellation and localization to obtain stationary Gaussian and Ornstein--Uhlenbeck limits. Independent Gaussian regression errors yield an exact conditional Gaussian law and total-varia...
| Authors | Xiaoli Li, Wei Biao Wu |
|---|---|
| Primary category | stat.ML |
| Published | 4 October 2026 |
| Last updated at source | 4 October 2026 |
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