# Local Saddle Point Optimization: A Curvature Exploitation Approach

arXiv: https://arxiv.org/abs/1805.05751 · HTML: https://mlsanity.com/paper/1805.05751/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2018.

## Built on by (most cited first, 28 of 28)

- [Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity](https://mlsanity.com/paper/2112.05604/) (2021)
- [Proximal Gradient Descent-Ascent: Variable Convergence under KL Geometry](https://mlsanity.com/paper/2102.04653/) (2021)
- [On the Convergence of Stochastic Extragradient for Bilinear Games using Restarted Iteration Averaging](https://mlsanity.com/paper/2107.00464/) (2021)
- [Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2408.11974/) (2024)
- [Understanding Overparameterization in Generative Adversarial Networks](https://mlsanity.com/paper/2104.05605/) (2021)
- [Riemannian Hamiltonian methods for min-max optimization on manifolds](https://mlsanity.com/paper/2204.11418/) (2022)
- [Universal Gradient Descent Ascent Method for Nonconvex-Nonconcave Minimax Optimization](https://mlsanity.com/paper/2212.12978/) (2022)
- [Learning in Matrix Games can be Arbitrarily Complex](https://mlsanity.com/paper/2103.03405/) (2021)
- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [Solving Min-Max Optimization with Hidden Structure via Gradient Descent Ascent](https://mlsanity.com/paper/2101.05248/) (2021)
- [Minimax Optimization with Smooth Algorithmic Adversaries](https://mlsanity.com/paper/2106.01488/) (2021)
- [Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equations](https://mlsanity.com/paper/2112.13826/) (2021)
- [Direct-Search for a Class of Stochastic Min-Max Problems](https://mlsanity.com/paper/2102.11386/) (2021)
- [Exploring Energy Landscapes for Minimal Counterfactual Explanations: Applications in Cybersecurity and Beyond](https://mlsanity.com/paper/2503.18185/) (2025)
- [A Cubic Regularization Approach for Finding Local Minimax Points in Nonconvex Minimax Optimization](https://mlsanity.com/paper/2110.07098/) (2021)
- [Accelerated Proximal Alternating Gradient-Descent-Ascent for Nonconvex Minimax Machine Learning](https://mlsanity.com/paper/2112.11663/) (2021)
- [STay-ON-the-Ridge: Guaranteed Convergence to Local Minimax Equilibrium in Nonconvex-Nonconcave Games](https://mlsanity.com/paper/2210.09769/) (2022)
- [On Characterizing GAN Convergence Through Proximal Duality Gap](https://mlsanity.com/paper/2105.04801/) (2021)
- [Connected Superlevel Set in (Deep) Reinforcement Learning and its Application to Minimax Theorems](https://mlsanity.com/paper/2303.12981/) (2023)
- [PAPAL: A Provable PArticle-based Primal-Dual ALgorithm for Mixed Nash Equilibrium](https://mlsanity.com/paper/2303.00970/) (2023)
- [Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectives](https://mlsanity.com/paper/2501.17122/) (2025)
- [Continuous-time Analysis for Variational Inequalities: An Overview and Desiderata](https://mlsanity.com/paper/2207.07105/) (2022)
- [A Framework for Finding Local Saddle Points in Two-Player Zero-Sum Black-Box Games](https://mlsanity.com/paper/2503.18224/) (2025)
- [Enhancing Stability of Physics-Informed Neural Network Training Through Saddle-Point Reformulation](https://mlsanity.com/paper/2507.16008/) (2025)
- [GDA-AM: On the effectiveness of solving minimax optimization via Anderson Acceleration](https://mlsanity.com/paper/2110.02457/) (2021)
- [Large deviations for interacting particle dynamics for finding mixed equilibria in zero-sum games](https://mlsanity.com/paper/2206.15177/) (2022)
- [Adversarial Reinforcement Learning Framework for ESP Cheater Simulation](https://mlsanity.com/paper/2509.24274/) (2025)

## Abstract (arXiv)

Gradient-based optimization methods are the most popular choice for finding local optima for classical minimization and saddle point problems. Here, we highlight a systemic issue of gradient dynamics that arise for saddle point problems, namely the presence of undesired stable stationary points that are no local optima. We propose a novel optimization approach that exploits curvature information in order to escape from these undesired stationary points. We prove that different optimization methods, including gradient method and Adagrad, equipped with curvature exploitation can escape non-optimal stationary points. We also provide empirical results on common saddle point problems which confirm the advantage of using curvature exploitation.