Bilevel optimization problems are at the center of several important machine learning problems such as hyperparameter tuning, data denoising, meta- and few-shot learning, data poisoning. Different from simultaneous or multi-objective optimization, bilevel optimization requires computing the inverse of the Hessian of the lower-level cost function to obtain the exact descent direction for the upper-level cost. In this paper, we propose a new method for solving deep bilevel optimization problems using the penalty function which avoids computing the inverse. We prove convergence of our method under mild conditions and show that it computes the exact hypergradient asymptotically. Small space and time complexity of our method enables us to solve large-scale bilevel problems involving deep neural networks with several million parameters. 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.
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