The exact information-based complexity of smooth convex minimization

TL;DR

This paper establishes a tight lower bound on the complexity of smooth convex minimization, matching the performance of the Optimized Gradient Method, and introduces a new technique for constructing such functions.

Contribution

It provides a new lower bound that exactly matches the performance of an existing optimal method, demonstrating the bound's tightness and practical relevance.

Findings

Lower bound matches the performance of the Optimized Gradient Method

New construction technique for smooth convex functions

Bound is proven to be tight and achievable

Abstract

We obtain a new lower bound on the information-based complexity of first-order minimization of smooth and convex functions. We show that the bound matches the worst-case performance of the recently introduced Optimized Gradient Method, thereby establishing that the bound is tight and can be realized by an efficient algorithm. The proof is based on a novel construction technique of smooth and convex functions.