Convex Synthesis of Accelerated Gradient Algorithms

TL;DR

This paper introduces a convex framework for designing accelerated gradient algorithms for strongly convex functions, leveraging control theory tools to optimize convergence rates.

Contribution

It provides a novel convex synthesis method using integral quadratic constraints and Youla parameterization for accelerated gradient algorithms.

Findings

Explicit formulas for optimal convergence rates

Convex semi-definite programming formulation

Extension to extremum control problems

Abstract

We present a convex solution for the design of generalized accelerated gradient algorithms for strongly convex objective functions with Lipschitz continuous gradients. We utilize integral quadratic constraints and the Youla parameterization from robust control theory to formulate a solution of the algorithm design problem as a convex semi-definite program. We establish explicit formulas for the optimal convergence rates and extend the proposed synthesis solution to extremum control problems.