Penalty Method for Inversion-Free Deep Bilevel Optimization

Bilevel optimizations are at the center of several important machine learning problems such as hyperparameter tuning, data denoising, few-shot learning, data poisoning. Different from simultaneous or multi-objective optimization, obtaining the exact descent direction for continuous bilevel optimization requires computing the inverse of the hessian of the lower-level cost function, even for first order methods. In this paper, we propose a new method for solving bilevel optimization, using the penalty function, which avoids computing the inverse of the hessian. We prove convergence of the method under mild conditions and show that it computes the exact hypergradient asymptotically. Small space and time complexity of our method allows us to solve large-scale bilevel optimization problems involving deep neural networks with up to 3.8M upper-level and 1.4M lower-level variables. We present results of our method for data denoising on MNIST/CIFAR10/SVHN datasets, for few-shot learning on Omniglot/Mini-Imagenet datasets and for training-data poisoning on MNIST/Imagenet datasets. In all experiments, our method outperforms or is comparable to previously proposed methods both in terms of accuracy and run-time.