Value-Function-based Sequential Minimization for Bi-level Optimization

Gradient-based Bi-Level Optimization (BLO) methods have been widely applied to solve modern machine learning problems. However, most existing solution strategies are theoretically designed based on restrictive assumptions (e.g., convexity of the lower-level sub-problem), and computationally not applicable for high-dimensional tasks. Moreover, there are almost no gradient-based methods that can efficiently handle BLO in those challenging scenarios, such as BLO with functional constraints and pessimistic BLO. In this work, by reformulating BLO into an approximated single-level problem based on the value-function, we provide a new method, named Bi-level Value-Function-based Sequential Minimization (BVFSM), to partially address the above issues. To be specific, BVFSM constructs a series of value-function-based approximations, and thus successfully avoids the repeated calculations of recurrent gradient and Hessian inverse required by existing approaches, which are time-consuming (especially for high-dimensional tasks). We also extend BVFSM to address BLO with additional upper- and lower-level functional constraints. More importantly, we demonstrate that the algorithmic framework of BVFSM can also be used for the challenging pessimistic BLO, which has never been properly solved by existing gradient-based methods. On the theoretical side, we strictly prove the convergence of BVFSM on these types of BLO, in which the restrictive lower-level convexity assumption is completely discarded. To our best knowledge, this is the first gradient-based algorithm that can solve different kinds of BLO problems (e.g., optimistic, pessimistic and with constraints) all with solid convergence guarantees. Extensive experiments verify our theoretical investigations and demonstrate the superiority of BVFSM on various real-world applications.