A constraint-based approach to function interpolation, with application to performance estimation for weakly convex optimisation

TL;DR

This paper introduces a novel algebraic approach to derive interpolation constraints for function classes, enabling tighter performance guarantees in optimization, exemplified by weakly convex functions.

Contribution

It presents a new algebraic method for deriving interpolation constraints, applicable without relying on analytic properties, and demonstrates improved performance bounds for weakly convex functions.

Findings

Derived interpolation constraints for weakly convex functions

Outperformed existing bounds on subgradient method performance

Provided a general algebraic framework for function class analysis

Abstract

We consider the problem of obtaining interpolation constraints for function classes, i.e., necessary and sufficient constraints that a set of points, function values and (sub)gradients must satisfy to ensure the existence of a global function of the class considered, consistent with this set. The derivation of such constraints is crucial, e.g., in the performance analysis of optimization methods, since obtaining a priori tight performance guarantees requires using a tight description of function classes of interest. We propose an approach that allows setting aside all analytic properties of the function class to work only at an algebraic level, and to obtain counterexamples when a condition characterizing a function class cannot serve as an interpolation constraint. As an illustration, we provide interpolation constraints for the class of weakly convex functions with bounded subgradients, and rely on these constraints to outperform state-of-the-art bounds on the performance of the subgradient method on this class.