By connecting convex optimization to the problem of chasing nested convex sets, the authors achieve nearly optimal oracle complexity for Lipschitz convex functions—meaning their algorithms are as efficient as theoretically possible with minimal wasted queries.
This paper develops efficient algorithms for convex optimization that achieve nearly optimal performance rates. The key innovation is reducing the problem to tracking moving convex sets, where the algorithm either finds good solutions or makes strategic cuts that force progress. The approach works for general norm constraints and achieves rates matching theoretical lower bounds.