Convex Bi-Level Optimization Problems with Non-smooth Outer Objective Function

TL;DR

This paper introduces the Bi-Sub-Gradient method, a simple first-order approach for convex bi-level optimization with non-smooth outer objectives, achieving sub-linear to linear convergence rates.

Contribution

It generalizes the sub-gradient method to bi-level problems, providing convergence guarantees and handling non-smooth outer functions with minimal computational complexity.

Findings

Achieves sub-linear convergence rates for general convex bi-level problems.

Improves to linear convergence when the outer objective is strongly convex.

Proves convergence of the solution sequence to the optimal set.

Abstract

In this paper, we propose the Bi-Sub-Gradient (Bi-SG) method, which is a generalization of the classical sub-gradient method to the setting of convex bi-level optimization problems. This is a first-order method that is very easy to implement in the sense that it requires only a computation of the associated proximal mapping or a sub-gradient of the outer non-smooth objective function, in addition to a proximal gradient step on the inner optimization problem. We show, under very mild assumptions, that Bi-SG tackles bi-level optimization problems and achieves sub-linear rates both in terms of the inner and outer objective functions. Moreover, if the outer objective function is additionally strongly convex (still could be non-smooth), the outer rate can be improved to a linear rate. Last, we prove that the distance of the generated sequence to the set of optimal solutions of the bi-level problem converges to zero.