Automatic Differentiation of Optimization Algorithms with Time-Varying Updates

TL;DR

This paper develops a method to differentiate through time-varying optimization algorithms, providing convergence guarantees and demonstrating that the derivative iterates reflect the original algorithm's convergence rate.

Contribution

It introduces a framework for automatic differentiation of optimization algorithms with changing parameters, with theoretical convergence guarantees and practical applications.

Findings

Convergence rates of derivative iterates match those of the original algorithms.

Validated the approach on regularized linear and logistic regression problems.

Demonstrated the method's effectiveness through numerical experiments.

Abstract

Numerous Optimization Algorithms have a time-varying update rule thanks to, for instance, a changing step size, momentum parameter or, Hessian approximation. In this paper, we apply unrolled or automatic differentiation to a time-varying iterative process and provide convergence (rate) guarantees for the resulting derivative iterates. We adapt these convergence results and apply them to proximal gradient descent with variable step size and FISTA when solving partly smooth problems. We confirm our findings numerically by solving $\ell_1$ and $\ell_2$-regularized linear and logisitc regression respectively. Our theoretical and numerical results show that the convergence rate of the algorithm is reflected in its derivative iterates.