Efficient Gradient Approximation Method for Constrained Bilevel Optimization

TL;DR

This paper introduces a gradient approximation method for constrained bilevel optimization problems, effectively handling large-scale, high-dimensional data with non-convex and non-differentiable objectives, and demonstrates its practical utility.

Contribution

It proposes a novel gradient-based approach that approximates descent directions in constrained bilevel problems, with proven convergence to Clarke stationary points.

Findings

Algorithm converges asymptotically to Clarke stationary points.

Effective in hyperparameter optimization tasks.

Demonstrates utility in meta-learning applications.

Abstract

Bilevel optimization has been developed for many machine learning tasks with large-scale and high-dimensional data. This paper considers a constrained bilevel optimization problem, where the lower-level optimization problem is convex with equality and inequality constraints and the upper-level optimization problem is non-convex. The overall objective function is non-convex and non-differentiable. To solve the problem, we develop a gradient-based approach, called gradient approximation method, which determines the descent direction by computing several representative gradients of the objective function inside a neighborhood of the current estimate. We show that the algorithm asymptotically converges to the set of Clarke stationary points, and demonstrate the efficacy of the algorithm by the experiments on hyperparameter optimization and meta-learning.