Differentiating the Value Function by using Convex Duality

TL;DR

This paper introduces a convex duality-based method for differentiating value functions in parametric optimization, relaxing assumptions and providing convergence guarantees, with demonstrated effectiveness in machine learning applications including non-smooth problems.

Contribution

It presents a novel approach leveraging convex duality to compute derivatives of value functions under weaker conditions than existing methods.

Findings

Convergence rates established for derivative approximations.

Effective in non-smooth and complex parametric functions.

Outperforms some existing methods in experiments.

Abstract

We consider the differentiation of the value function for parametric optimization problems. Such problems are ubiquitous in Machine Learning applications such as structured support vector machines, matrix factorization and min-min or minimax problems in general. Existing approaches for computing the derivative rely on strong assumptions of the parametric function. Therefore, in several scenarios there is no theoretical evidence that a given algorithmic differentiation strategy computes the true gradient information of the value function. We leverage a well known result from convex duality theory to relax the conditions and to derive convergence rates of the derivative approximation for several classes of parametric optimization problems in Machine Learning. We demonstrate the versatility of our approach in several experiments, including non-smooth parametric functions. Even in settings where other approaches are applicable, our duality based strategy shows a favorable performance.