# A Fully Parameter-Free Second-Order Algorithm for Convex-Concave Minimax Problems

arXiv: https://arxiv.org/abs/2407.03571 · HTML: https://mlsanity.com/paper/2407.03571/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 69 references that are in the map.
- First posted on arXiv 4 Jul 2024.

## Builds on (most cited first, 30 of 41)

- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach](https://mlsanity.com/paper/1901.08511/) (2019)
- [Near-Optimal Algorithms for Minimax Optimization](https://mlsanity.com/paper/2002.02417/) (2020)
- [On the convergence of single-call stochastic extra-gradient methods](https://mlsanity.com/paper/1908.08465/) (2019)
- [Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems](https://mlsanity.com/paper/1808.02901/) (2018)
- [Efficient Algorithms for Smooth Minimax Optimization](https://mlsanity.com/paper/1907.01543/) (2019)
- [Accelerated Algorithms for Smooth Convex-Concave Minimax Problems with O(1/k2) Rate on Squared Gradient Norm](https://mlsanity.com/paper/2102.07922/) (2021)
- [Weakly-Convex Concave Min-Max Optimization: Provable Algorithms and Applications in Machine Learning](https://mlsanity.com/paper/1810.02060/) (2018)
- [A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems](https://mlsanity.com/paper/2010.15768/) (2020)
- [A Unified Single-loop Alternating Gradient Projection Algorithm for Nonconvex-Concave and Convex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2006.02032/) (2020)
- [Hybrid Block Successive Approximation for One-Sided Non-Convex Min-Max Problems: Algorithms and Applications](https://mlsanity.com/paper/1902.08294/) (2019)
- [The Complexity of Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2103.15888/) (2021)
- [Accelerated Zeroth-Order and First-Order Momentum Methods from Mini to Minimax Optimization](https://mlsanity.com/paper/2008.08170/) (2020)
- [Efficient Search of First-Order Nash Equilibria in Nonconvex-Concave Smooth Min-Max Problems](https://mlsanity.com/paper/2002.07919/) (2020)
- [Simple and optimal methods for stochastic variational inequalities, I: operator extrapolation](https://mlsanity.com/paper/2011.02987/) (2020)
- [An accelerated inexact proximal point method for solving nonconvex-concave min-max problems](https://mlsanity.com/paper/1905.13433/) (2019)
- [Alternating proximal-gradient steps for (stochastic) nonconvex-concave minimax problems](https://mlsanity.com/paper/2007.13605/) (2020)
- [Zeroth-Order Algorithms for Nonconvex Minimax Problems with Improved Complexities](https://mlsanity.com/paper/2001.07819/) (2020)
- [What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?](https://mlsanity.com/paper/1902.00618/) (2019)
- [Near-Optimal Algorithms for Making the Gradient Small in Stochastic Minimax Optimization](https://mlsanity.com/paper/2208.05925/) (2022)
- [Higher-order methods for convex-concave min-max optimization and monotone variational inequalities](https://mlsanity.com/paper/2007.04528/) (2020)
- [Generalized Optimistic Methods for Convex-Concave Saddle Point Problems](https://mlsanity.com/paper/2202.09674/) (2022)
- [Towards More Efficient Stochastic Decentralized Learning: Faster Convergence and Sparse Communication](https://mlsanity.com/paper/1805.09969/) (2018)
- [Finding Second-Order Stationary Points in Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2110.04814/) (2021)
- [Zeroth-Order Alternating Gradient Descent Ascent Algorithms for a Class of Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2211.13668/) (2022)
- [Gradient-Free Methods for Saddle-Point Problem](https://mlsanity.com/paper/2005.05913/) (2020)
- [Optimal and parameter-free gradient minimization methods for convex and nonconvex optimization](https://mlsanity.com/paper/2310.12139/) (2023)
- [Adaptive and Optimal Second-order Optimistic Methods for Minimax Optimization](https://mlsanity.com/paper/2406.02016/) (2024)

## Built on by (most cited first, 3 of 3)

- [Second-Order Min-Max Optimization with Lazy Hessians](https://mlsanity.com/paper/2410.09568/) (2024)
- [Blockwise Stabilized Adaptive Cubic Regularization with Subsolvers via Recurrence](https://mlsanity.com/paper/2608.22129/) (2026)
- [Gradient Norm Regularization Second-Order Algorithms for Solving Nonconvex-Strongly Concave Minimax Problems](https://mlsanity.com/paper/2411.15769/) (2024)

## Abstract (arXiv)

In this paper, we study second-order algorithms for the convex-concave minimax problem, which has attracted much attention in many fields such as machine learning in recent years. We propose a Lipschitz-free cubic regularization (LF-CR) algorithm for solving the convex-concave minimax optimization problem without knowing the Lipschitz constant. It can be shown that the iteration complexity of the LF-CR algorithm to obtain an ε-optimal solution with respect to the restricted primal-dual gap is upper bounded by O(ρ2/3|z0-z^*|2ε-2/3) , where z0=(x0,y0) is a pair of initial points, z^*=(x^*,y^*) is a pair of optimal solutions, and ρ is the Lipschitz constant. We further propose a fully parameter-free cubic regularization (FF-CR) algorithm that does not require any parameters of the problem, including the Lipschitz constant and the upper bound of the distance from the initial point to the optimal solution. We also prove that the iteration complexity of the FF-CR algorithm to obtain an ε-optimal solution with respect to the gradient norm is upper bounded by O(ρ2/3|z0-z^*|4/3ε-2/3) . Numerical experiments show the efficiency of both algorithms. To the best of our knowledge, the proposed FF-CR algorithm is a completely parameter-free second-order algorithm, and its iteration complexity is currently the best in terms of ε under the termination criterion of the gradient norm.