A Constrained Optimization Approach to Bilevel Optimization with Multiple Inner Minima

Bilevel optimization has found extensive applications in modern machine learning problems such as hyperparameter optimization, neural architecture search, meta-learning, etc. While bilevel problems with a unique inner minimal point (e.g., where the inner function is strongly convex) are well understood, bilevel problems with multiple inner minimal points remains to be a challenging and open problem. Existing algorithms designed for such a problem were applicable to restricted situations and do not come with the full guarantee of convergence. In this paper, we propose a new approach, which convert the bilevel problem to an equivalent constrained optimization, and then the primal-dual algorithm can be used to solve the problem. Such an approach enjoys a few advantages including (a) addresses the multiple inner minima challenge; (b) features fully first-order efficiency without involving second-order Hessian and Jacobian computations, as opposed to most existing gradient-based bilevel algorithms; (c) admits the convergence guarantee via constrained nonconvex optimization. Our experiments further demonstrate the desired performance of the proposed approach.