Exact minimax optimal FO-LMO algorithms and matching hard instances

Determine the exactly minimax optimal deterministic first-order linear minimization oracle (FO-LMO) algorithms and the corresponding worst-case problem instances—including the precise constant factors—for minimizing L-smooth, strongly convex objectives over α-strongly convex constraint sets, thereby closing the constant-factor gap between existing upper and lower bounds.

Background

The paper proves a universal lower bound on the iteration complexity of deterministic FO-LMO methods over strongly convex sets, matching the accelerated Frank–Wolfe upper bound in order but not in constants. Specifically, the lower bound’s constant is 1/528, while the known upper bound has the constant 9/2.

This mismatch raises a natural minimax question: identify the exact worst-case instance and the exactly optimal algorithm under the FO-LMO model for strongly convex constrained problems, akin to results obtained via Performance Estimation Problem (PEP) techniques in unconstrained settings.