Bilevel optimization problems are at the center of several important 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 requires the inverse of the Hessian of the lower-level cost. In this paper, we propose a new method for solving bilevel optimization problems using the classical penalty function approach which avoids computing the inverse and can also handle additional constraints easily. We prove the convergence of the method under mild conditions and show that the exact hypergradient is obtained asymptotically. Our methods 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 and show that our method outperforms or is comparable to previously proposed methods based on automatic differentiation and approximate inversion in terms of accuracy, run-time and convergence speed.
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