Primal-dual subgradient methods for minimizing uniformly convex functions

TL;DR

This paper introduces non-Euclidean primal-dual subgradient algorithms for optimizing uniformly convex functions, providing adaptive methods with near-optimal accuracy bounds for both deterministic and stochastic settings.

Contribution

It develops adaptive primal-dual subgradient algorithms tailored for uniformly convex functions, achieving near-optimal accuracy bounds with respect to iteration count.

Findings

Algorithms are adaptive to strong or uniform convexity parameters.

Accuracy bounds match optimal algorithms up to a logarithmic factor.

Applicable to both deterministic and stochastic optimization problems.

Abstract

We discuss non-Euclidean deterministic and stochastic algorithms for optimization problems with strongly and uniformly convex objectives. We provide accuracy bounds for the performance of these algorithms and design methods which are adaptive with respect to the parameters of strong or uniform convexity of the objective: in the case when the total number of iterations $N$ is fixed, their accuracy coincides, up to a logarithmic in $N$ factor with the accuracy of optimal algorithms.