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Centered Permutation Prefixes for SGD with Random Reshuffling: Sharp Rates, Hölder Geometry, and Composite Proximal Extensions

arXiv · AI, language, vision and robotics · article · Sep 4, 2026 · UTC

We study stochastic gradient descent with random reshuffling for finite sums \[ F(x)=\frac1n\sum_{i=1}^n f_i(x). \] For fresh reshuffling with a constant component stepsize, if each $f_i$ has an $L$-Lipschitz gradient and the average $F$ is $μ$-strongly convex with a Lipschitz-continuous Hessian, we prove the last-epoch rate \[ \mathbb E[F(y_K)-F(x_\star)] =\widetilde O\!\left(T^{-2}+n^2T^{-3}\right), \qquad T=nK, \] matching the known quadratic lower bound in its $(n,K)$-dependence. The components may be nonconvex, and no componentwise Hessian continuity or separate bounded-iterate assumption

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