# Projection-Free Methods for Solving Nonconvex-Concave Saddle Point Problems

arXiv: https://arxiv.org/abs/2306.11944 · HTML: https://mlsanity.com/paper/2306.11944/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2023.

## Built on by (most cited first, 1 of 1)

- [Enhancing Stability of Physics-Informed Neural Network Training Through Saddle-Point Reformulation](https://mlsanity.com/paper/2507.16008/) (2025)

## Abstract (arXiv)

In this paper, we investigate a class of constrained saddle point (SP) problems where the objective function is nonconvex-concave and smooth. This class of problems has wide applicability in machine learning, including robust multi-class classification and dictionary learning. Several projection-based primal-dual methods have been developed for tackling this problem; however, the availability of methods with projection-free oracles remains limited. To address this gap, we propose efficient single-loop projection-free methods reliant on first-order information. In particular, using regularization and nested approximation techniques, we propose a primal-dual conditional gradient method that solely employs linear minimization oracles to handle constraints. Assuming that the constraint set in the maximization is strongly convex, our method achieves an ε-stationary solution within O(ε-6) iterations. When the projection onto the constraint set of maximization is easy to compute, we propose a one-sided projection-free method that achieves an ε-stationary solution within O(ε-4) iterations. Moreover, we present improved iteration complexities of our methods under a strong concavity assumption. To the best of our knowledge, our proposed algorithms are among the first projection-free methods with convergence guarantees for solving nonconvex-concave SP problems.