Ordinary Nonconvex SGD under Distance-Dependent Moments: Finite-Horizon Stationarity and Nagaev Bounds

Abstract

Uniform noise-moment bounds exclude stochastic gradients whose variability increases with the iterate. We study ordinary, single-sample stochastic gradient descent for smooth, lower-bounded, possibly nonconvex objectives under distance-dependent conditional moments. Under second moments alone, a direct descent--displacement argument yields $T^{-1/3}$ expected average squared-gradient stationarity with a horizon-dependent stepsize. An explicit oracle-complexity corollary matches the known smooth Blum--Gladyshev (BG-0) lower bound, including the $Lb_2\Delta^3\varepsilon^{-6}$ and $L\Delta\sigma^2\varepsilon^{-4}$ stochastic terms, where $\Delta$ is the initial objective gap and $\sigma^2+b_2\|x-x_1\|^2$ bounds the variance. Thus unchanged SGD attains the minimax stochastic complexity in this second-moment class. For $p>2$, predictable localization and a Hilbert-space Fuk--Nagaev inequality yield a high-probability bound separating logarithmic variance and polynomial rare-shock contributions. The localization radius is derived from the recursion: no bounded-iterate assumption, clipping, normalization, momentum, or increasing batch size is needed. We also give increasing-confidence rates, an objective-gap-growth refinement recovering root-$T$ stationarity, and stochastic $L^p$-Lipschitz examples. The broad BG-0 optimality statement is distinguished from the smaller mean-square-smooth class, in which additional oracle structure permits faster algorithms.