Penalty Method for Inversion-Free Deep Bilevel Optimization

TL;DR

This paper introduces a penalty-based algorithm for bilevel optimization that avoids Hessian inverse calculations, enabling efficient large-scale deep learning applications like data denoising and few-shot learning.

Contribution

The proposed method offers a simple, convergent, and scalable alternative to existing bilevel optimization techniques, handling constraints and large models effectively.

Findings

Outperforms existing methods in accuracy and speed

Effective on large-scale deep neural network problems

Converges to the exact hypergradient asymptotically

Abstract

Solving a bilevel optimization problem is at the core of several machine learning problems such as hyperparameter tuning, data denoising, meta- and few-shot learning, and training-data poisoning. Different from simultaneous or multi-objective optimization, the steepest descent direction for minimizing the upper-level cost in a bilevel problem requires the inverse of the Hessian of the lower-level cost. In this work, we propose a novel algorithm for solving bilevel optimization problems based on the classical penalty function approach. Our method avoids computing the Hessian inverse and can handle constrained bilevel problems easily. We prove the convergence of the method under mild conditions and show that the exact hypergradient is obtained asymptotically. Our method's simplicity and small space and time complexities enable us to effectively solve large-scale bilevel problems involving deep neural networks. We present results on data denoising, few-shot learning, and training-data poisoning problems in a large-scale setting. Our results show that our approach outperforms or is comparable to previously proposed methods based on automatic differentiation and approximate inversion in terms of accuracy, run-time, and convergence speed.