mlsanityData to 7 Oct 2026

Local Saddle Point Optimization: A Curvature Exploitation Approach

arXiv 1805.05751 ↗First posted 15 May 2018cs.LG math.OC stat.MLcited by 28

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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.

Key facts. Cited by 28 papers in the map (data through 7 Oct 2026). First posted on arXiv 15 May 2018.

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Citing papers per year: 2021: 13; 2022: 7; 2023: 2; 2024: 1; 2025: 5.

28 papers.

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  1. 2021Faster Single-loop Algorithms for Minimax Optimization without Strong Concavitycited by 47
  2. 2021On the Convergence of Stochastic Extragradient for Bilinear Games using Restarted Iteration Averagingcited by 18
  3. 2021Proximal Gradient Descent-Ascent: Variable Convergence under KL Geometrycited by 17
  4. 2024Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimizationcited by 15
  5. 2021Understanding Overparameterization in Generative Adversarial Networkscited by 14
  6. 2022Riemannian Hamiltonian methods for min-max optimization on manifoldscited by 12
  7. 2021Learning in Matrix Games can be Arbitrarily Complexcited by 11
  8. 2022Universal Gradient Descent Ascent Method for Nonconvex-Nonconcave Minimax Optimizationcited by 11
  9. 2022First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spacescited by 10
  10. 2022Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithmcited by 9
  11. 2021Solving Min-Max Optimization with Hidden Structure via Gradient Descent Ascentcited by 8
  12. 2021Minimax Optimization with Smooth Algorithmic Adversariescited by 8
16 more
  1. 2021Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equationscited by 8
  2. 2021Direct-Search for a Class of Stochastic Min-Max Problemscited by 7
  3. 2025Exploring Energy Landscapes for Minimal Counterfactual Explanations: Applications in Cybersecurity and Beyondcited by 6
  4. 2021A Cubic Regularization Approach for Finding Local Minimax Points in Nonconvex Minimax Optimizationcited by 5
  5. 2021Accelerated Proximal Alternating Gradient-Descent-Ascent for Nonconvex Minimax Machine Learningcited by 4
  6. 2022STay-ON-the-Ridge: Guaranteed Convergence to Local Minimax Equilibrium in Nonconvex-Nonconcave Gamescited by 4
  7. 2021On Characterizing GAN Convergence Through Proximal Duality Gapcited by 3
  8. 2023Connected Superlevel Set in (Deep) Reinforcement Learning and its Application to Minimax Theoremscited by 3
  9. 2023PAPAL: A Provable PArticle-based Primal-Dual ALgorithm for Mixed Nash Equilibriumcited by 2
  10. 2025Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectivescited by 2
  11. 2022Continuous-time Analysis for Variational Inequalities: An Overview and Desideratacited by 1
  12. 2025A Framework for Finding Local Saddle Points in Two-Player Zero-Sum Black-Box Gamescited by 1
  13. 2025Enhancing Stability of Physics-Informed Neural Network Training Through Saddle-Point Reformulationcited by 1
  14. 2021GDA-AM: On the effectiveness of solving minimax optimization via Anderson Accelerationcited by 0
  15. 2022Large deviations for interacting particle dynamics for finding mixed equilibria in zero-sum gamescited by 0
  16. 2025Adversarial Reinforcement Learning Framework for ESP Cheater Simulationcited by 0