The Exact Time-Uniform Rate Frontier for Stochastic Gradient Descent on Smooth Convex Objectives

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 unbiased norm-sub-Gaussian gradient noise, 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 $1-\alpha$ 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.