Exact Tail Laws for Nonlinear SGD with Infinite-Variance Gradients

With infinite-variance gradient noise, SGD can have infinite expected excess objective even when most iterates are close to the optimum. We characterize its accuracy through exact first-order tail probabilities for the last iterate of nonlinear, constant-stepsize SGD. For regularly varying innovations of index $t\in(1,2)$, the leading measure transports one large gradient shock through the local Hessian flow. This is the same measure that governs the central stable Ornstein--Uhlenbeck limit; we prove its relative accuracy at shrinking tolerances above the central scale, with state-dependent…

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