Penalty Method for Inversion-Free Deep Bilevel Optimization
This addresses a bottleneck in machine learning applications such as hyperparameter tuning and meta-learning, offering a more efficient solution for large-scale deep learning problems.
The paper tackles the computational challenge of solving bilevel optimization problems, which require Hessian inverses, by proposing a penalty-based method that avoids this inversion and handles constraints, achieving competitive or superior performance in accuracy, run-time, and convergence speed on tasks like data denoising and few-shot learning.
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.