# Matching Multi-Loop Complexities with a Single Loop: Optimal Optimization Stationarity and Best-Known Game Stationarity in Nonconvex--Concave Minimax Optimization

arXiv: https://arxiv.org/abs/2609.17973 · HTML: https://mlsanity.com/paper/2609.17973/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 14 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization](https://mlsanity.com/paper/1911.08731/) (2019)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Lower Bounds for Finding Stationary Points I](https://mlsanity.com/paper/1710.11606/) (2017)
- [Near-Optimal Algorithms for Minimax Optimization](https://mlsanity.com/paper/2002.02417/) (2020)
- [Efficient Algorithms for Smooth Minimax Optimization](https://mlsanity.com/paper/1907.01543/) (2019)
- [Weakly-Convex Concave Min-Max Optimization: Provable Algorithms and Applications in Machine Learning](https://mlsanity.com/paper/1810.02060/) (2018)
- [A Unified Single-loop Alternating Gradient Projection Algorithm for Nonconvex-Concave and Convex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2006.02032/) (2020)
- [Learning with Average Top-k Loss](https://mlsanity.com/paper/1705.08826/) (2017)
- [Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity](https://mlsanity.com/paper/2112.05604/) (2021)
- [The Complexity of Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2103.15888/) (2021)
- [Complexity Lower Bounds for Nonconvex-Strongly-Concave Min-Max Optimization](https://mlsanity.com/paper/2104.08708/) (2021)
- [Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2408.11974/) (2024)
- [Smoothing Meets Perturbation: Unified and Tight Analysis for Nonconvex-Concave Minimax Optimization](https://mlsanity.com/paper/2602.14185/) (2026)

## Abstract (arXiv)

We introduce a new single-loop algorithmic framework for smooth nonconvex--concave minimax optimization. The resulting projected damped extragradient method combines projected extragradient updates, dual momentum, and a moving proximal center. Under both the optimization-stationarity and game-stationarity criteria, our method achieves the best-known complexity among single-loop first-order methods. For optimization stationarity, our method achieves a gradient complexity of O(L2DYbarDelta0varepsilon-3), where L is the gradient Lipschitz constant, DY bounds the diameter of the dual feasible set, and barDelta0 is an initialization quantity involving the value-function gap and the initial gradients. Moreover, by incorporating a fixed-center warm-up phase, the complexity can be improved to O(L2DYDeltaφvarepsilon-3), up to an additive lower-order cost, where Deltaφ:=φ(x0)-infxφ(x). We further establish a lower bound of Omega(L2DYDeltaφvarepsilon-3) for optimization stationarity over projected zero-respecting first-order methods. This lower bound proves that the warm-started version of our algorithm is optimal up to a constant factor for optimization stationarity within this oracle class. For game stationarity, our method achieves O!(L3/2DY1/2Deltaφvarepsilon-5/2) gradient complexity. This matches the best-known complexity of multi-loop first-order methods, thereby establishing the same complexity with a single-loop algorithmic structure. Under dual strong concavity, the proposed framework achieves O!(√kappa,LDeltaφvarepsilon-2) leading complexity for both stationarity criteria, where kappa=L/μ is the dual condition number, up to an additive initialization cost. The varepsilon-2 accuracy dependence is optimal under fixed regularity and initialization bounds.