Year: 2026
Type: preprint
Abstract: We study the last iterate of the projected subGradient Method with a constant stepsize for convex $L$-Lipschitz objectives in $\mathbb{R}^d$, minimized over closed convex sets. Assuming that the initial iterate $x_1$ has distance at most $R$ from a minimizer, we establish matching upper and lower bounds on the minimax optimization error at $x_n$, where the infimum is taken over constant stepsizes. Specifically, this error is of order $L\cdot R\cdot\sqrt{\min\{d,\log n\}/n}$. Notably, when the dimension is fixed, the minimax error decays as $1/\sqrt{n}$, without the extra logarithmic factor intrinsic to dimension-free analyses. Our analysis exploits a combinatorial argument based on linear algebra and novel geometric constructions.
Source: arXiv (Cornell University)
Institutions: Aarhus University, University of Ottawa, University of Bristol
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7169798721