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Life sciences · Preprint
arXiv · September 4, 2026
Posted before peer review. The findings may change or fail to hold.
This is a theoretical preprint establishing convergence rate bounds for stochastic gradient descent with random reshuffling under various smoothness and convexity assumptions. The work derives matching upper and lower bounds for optimization error but contains no empirical validation, clinical data, or peer review.
Preprint.
Last-epoch convergence rate Õ(T^{-2} + n²T^{-3}) with constant component stepsize under strong convexity and Lipschitz Hessian Hölder-continuous average Hessian with ν ≥ 1/2 adds Õ(n^{1+ν}T^{-2-2ν}) while preserving quadratic rate Composite proximal extension yields error bound Õ(β_*²/K² + T^{-2} + n²T^{-3} + n^{1+ν}T^{-2-2ν}) with matching lower bound for ν ≥ 1/2
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This is a theoretical mathematics paper on optimization algorithms that has not been peer reviewed; it presents convergence rate proofs for SGD with random reshuffling but carries no clinical, experimental, or empirical validation.
Quoted from the source exactly as published.
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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 is required. More generally, a $ν$-Hölder-continuous average Hessian adds only $\widetilde O(n^{1+ν}T^{-2-2ν})$, so every $ν\ge 1/2$ preserves the quadratic rate. Under convex components, a decreasing-stepsize result removes the large-epoch requirement and recovers the same two-term scale once $nK$ exceeds the condition-number scale. We also analyze epoch-wise ProxRR for $\mathcal P=F+ψ$. Writing $x^\dagger$ for the composite minimizer and $β_\star=\|\nabla F(x^\dagger)\|$, we prove \[ \mathbb E\|y_K-x^\dagger\|^2 =\widetilde O\!\left( \frac{β_\star^2}{K^2} +T^{-2}+n^2T^{-3} +n^{1+ν}T^{-2-2ν} \right). \] For $ν\ge 1/2$, we show that the $β_\star^2/K^2$ splitting term is unavoidable and obtain a matching lower bound up to logarithms in the stated constant-stepsize regime.
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