  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.
