Dynamic FISTA for Convex Composite Bi-Level Optimization

TL;DR

This paper introduces FBi-PG, a novel algorithm for convex bi-level optimization that achieves the fastest known convergence rate of O(1/k^2) without restrictive assumptions, with proven theoretical guarantees and numerical validation.

Contribution

The paper proposes FBi-PG, a dynamic regularized FISTA-based method for bi-level convex optimization with optimal convergence rates and improved performance under error bound conditions.

Findings

Achieves an optimal O(1/k^2) convergence rate for the inner objective.

Attains sub-linear rates for both inner and outer objectives.

Demonstrates superior performance through numerical experiments.

Abstract

In this paper, we study convex bi-level optimization problems where both the inner and outer levels are given as a composite convex minimization. We propose the Fast Bi-level Proximal Gradient (FBi-PG) algorithm, which can be interpreted as applying FISTA to a dynamic regularized composite objective function. The dynamic nature of the regularization parameters allows to achieve an optimal fast convergence rate of $O(1/k^{2})$ in terms of the inner objective function. This is the fastest known convergence rate under no additional restrictive assumptions. We also show that FBi-PG achieves sub-linear simultaneous rates in terms of both the inner and outer objective functions. Moreover, we show that under an H\"olderian type error bound assumption on the inner objective function, the FBi-PG algorithm achieves improved simultaneous rates and converges to an optimal solution of the bi-level optimization problem. Finally, we present numerical experiments demonstrating the performance of the proposed scheme.