The Exact Time-Uniform Rate Frontier for Stochastic Gradient Descent on Smooth Convex Objectives
Year: 2026
Type: preprint
Abstract: We study the time-uniform convergence of the raw iterate of standard stochastic gradient descent (SGD) for unconstrained smooth convex objectives. We prove that, under standard noise assumptions, the time-uniform convergence rate gets arbitrarily close to $\sqrt{\log n / n}$ but never reaches it. More specifically, we prove that for every positive, eventually nondecreasing sequence $h$ satisfying $h(n) = o(\sqrt{n})$, a bound of order $h(n)/\sqrt{n}$, holding simultaneously for all $n$ with probability at least $1-α$ and uniformly over the problem class, is achievable if and only if \[ \sum_{j = 1}^{\infty} \frac{1}{h(2^j)^2} < \infty. \] The constructive sufficiency result follows from a dyadic horizon-free schedule together with an additive conditional-restart inequality. The necessity counterpart applies to every deterministic nonnegative schedule and holds even for a one-dimensional analytic smooth convex objective with Gaussian noise.
Source: arXiv (Cornell University)
Authors: Li, Ruijie, Chen, Kang, Wang, Tianyu
Institution: Fudan University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212025562