Single-loop Projection-free and Projected Gradient-based Algorithms for Nonconvex-concave Saddle Point Problems with Bilevel Structure

In this paper, we explore a class of constrained saddle point problems with a bilevel structure, where the upper-level objective function is nonconvex-concave and smooth subject to a strongly convex lower-level objective function. This class of problems finds wide applicability in machine learning, including robust multi-task learning, adversarial learning, and robust meta-learning. While some studies have focused on simpler formulations, e.g., when the upper-level objective function is linear in the maximization component, there remains a significant gap in developing efficient projection-free and projection-based algorithms with theoretical guarantees for more general settings. To bridge this gap, we propose efficient single-loop one-sided projection-free, and fully projection-based primal-dual methods. By leveraging regularization and nested approximation techniques, we initially devise a bilevel primal-dual one-sided projection-free algorithm, requiring $\mathcal{O}(\epsilon^{-4})$ iterations to attain an $\epsilon$-stationary point. Subsequently, we develop a bilevel primal-dual fully projected algorithm, capable of achieving an $\epsilon$-stationary solution within $\mathcal{O}(\epsilon^{-5})$ iterations. To the best of our knowledge, our proposed algorithms represent among the first methods for solving general non-bilinear saddle point problems with a bilevel structure.