# Complexity Lower Bounds for Nonconvex-Strongly-Concave Min-Max Optimization

arXiv: https://arxiv.org/abs/2104.08708 · HTML: https://mlsanity.com/paper/2104.08708/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 56 references that are in the map.
- First posted on arXiv 18 Apr 2021.

## Builds on (most cited first, 30 of 32)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [A Variational Inequality Perspective on Generative Adversarial Networks](https://mlsanity.com/paper/1802.10551/) (2018)
- [Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimization](https://mlsanity.com/paper/1308.6594/) (2013)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [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)
- [Independent Policy Gradient Methods for Competitive Reinforcement Learning](https://mlsanity.com/paper/2101.04233/) (2021)
- [Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization](https://mlsanity.com/paper/2011.00364/) (2020)
- [On the convergence of single-call stochastic extra-gradient methods](https://mlsanity.com/paper/1908.08465/) (2019)
- [The Complexity of Constrained Min-Max Optimization](https://mlsanity.com/paper/2009.09623/) (2020)
- [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)
- [Last-iterate Convergence of Decentralized Optimistic Gradient Descent/Ascent in Infinite-horizon Competitive Markov Games](https://mlsanity.com/paper/2102.04540/) (2021)
- [Last Iterate is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems](https://mlsanity.com/paper/2002.00057/) (2020)
- [Lower Bounds for Finding Stationary Points II: First-Order Methods](https://mlsanity.com/paper/1711.00841/) (2017)
- [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)
- [Efficient Search of First-Order Nash Equilibria in Nonconvex-Concave Smooth Min-Max Problems](https://mlsanity.com/paper/2002.07919/) (2020)
- [Global Convergence and Variance-Reduced Optimization for a Class of Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2002.09621/) (2020)
- [On Solving Minimax Optimization Locally: A Follow-the-Ridge Approach](https://mlsanity.com/paper/1910.07512/) (2019)
- [An accelerated inexact proximal point method for solving nonconvex-concave min-max problems](https://mlsanity.com/paper/1905.13433/) (2019)
- [Stabilizing Adversarial Nets With Prediction Methods](https://mlsanity.com/paper/1705.07364/) (2017)
- [On the Iteration Complexity of Oblivious First-Order Optimization Algorithms](https://mlsanity.com/paper/1605.03529/) (2016)

## Built on by (most cited first, 30 of 36)

- [On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation](https://mlsanity.com/paper/2309.01753/) (2023)
- [Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity](https://mlsanity.com/paper/2112.05604/) (2021)
- [Federated Minimax Optimization: Improved Convergence Analyses and Algorithms](https://mlsanity.com/paper/2203.04850/) (2022)
- [Universal Approximation Under Constraints is Possible with Transformers](https://mlsanity.com/paper/2110.03303/) (2021)
- [Nest Your Adaptive Algorithm for Parameter-Agnostic Nonconvex Minimax Optimization](https://mlsanity.com/paper/2206.00743/) (2022)
- [Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2408.11974/) (2024)
- [Unified Convergence Analysis for Adaptive Optimization with Moving Average Estimator](https://mlsanity.com/paper/2104.14840/) (2021)
- [AdaGDA: Faster Adaptive Gradient Descent Ascent Methods for Minimax Optimization](https://mlsanity.com/paper/2106.16101/) (2021)
- [Nonsmooth Nonconvex-Nonconcave Minimax Optimization: Primal-Dual Balancing and Iteration Complexity Analysis](https://mlsanity.com/paper/2209.10825/) (2022)
- [On Convergence of Gradient Descent Ascent: A Tight Local Analysis](https://mlsanity.com/paper/2207.00957/) (2022)
- [Minimax Optimization with Smooth Algorithmic Adversaries](https://mlsanity.com/paper/2106.01488/) (2021)
- [Nonconvex-Nonconcave Min-Max Optimization with a Small Maximization Domain](https://mlsanity.com/paper/2110.03950/) (2021)
- [TiAda: A Time-scale Adaptive Algorithm for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2210.17478/) (2022)
- [Federated Minimax Optimization with Client Heterogeneity](https://mlsanity.com/paper/2302.04249/) (2023)
- [Generalization Bounds of Nonconvex-(Strongly)-Concave Stochastic Minimax Optimization](https://mlsanity.com/paper/2205.14278/) (2022)
- [Tight Analysis of Extra-gradient and Optimistic Gradient Methods For Nonconvex Minimax Problems](https://mlsanity.com/paper/2210.09382/) (2022)
- [Worst-case generation via minimax optimization in Wasserstein space](https://mlsanity.com/paper/2512.08176/) (2025)
- [Universal Regular Conditional Distributions](https://mlsanity.com/paper/2105.07743/) (2021)
- [Stochastic Bilevel Optimization with Heavy-Tailed Noise](https://mlsanity.com/paper/2509.14952/) (2025)
- [A Survey of Decision Making in Adversarial Games](https://mlsanity.com/paper/2207.07971/) (2022)
- [Optimal Guarantees for Algorithmic Reproducibility and Gradient Complexity in Convex Optimization](https://mlsanity.com/paper/2310.17759/) (2023)
- [Lower Complexity Bounds for Nonconvex-Strongly-Convex Bilevel Optimization with First-Order Oracles](https://mlsanity.com/paper/2511.19656/) (2025)
- [On the Condition Number Dependency in Bilevel Optimization](https://mlsanity.com/paper/2511.22331/) (2025)
- [Two trust region type algorithms for solving nonconvex-strongly concave minimax problems](https://mlsanity.com/paper/2402.09807/) (2024)
- [Accelerated Stochastic Min-Max Optimization Based on Bias-corrected Momentum](https://mlsanity.com/paper/2406.13041/) (2024)
- [Achieving Near-Optimal Convergence for Distributed Minimax Optimization with Adaptive Stepsizes](https://mlsanity.com/paper/2406.02939/) (2024)
- [Stochastic Compositional Minimax Optimization with Provable Convergence Guarantees](https://mlsanity.com/paper/2408.12505/) (2024)
- [Tight Stochastic Condition-Number Dependence in Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2609.30877/) (2026)
- [Iterative Minimax Games with Coupled Linear Constraints](https://mlsanity.com/paper/2212.04672/) (2022)
- [First-order Methods for Affinely Constrained Composite Non-convex Non-smooth Problems: Lower Complexity Bound and Near-optimal Methods](https://mlsanity.com/paper/2307.07605/) (2023)

## Abstract (arXiv)

We provide a first-order oracle complexity lower bound for finding stationary points of min-max optimization problems where the objective function is smooth, nonconvex in the minimization variable, and strongly concave in the maximization variable. We establish a lower bound of Omegaleft(√kappaε-2right) for deterministic oracles, where ε defines the level of approximate stationarity and kappa is the condition number. Our analysis shows that the upper bound achieved in (Lin et al., 2020b) is optimal in the ε and kappa dependence up to logarithmic factors. For stochastic oracles, we provide a lower bound of Omegaleft(√kappaε-2 + kappa1/3ε-4right). It suggests that there is a significant gap between the upper bound O(kappa3 ε-4) in (Lin et al., 2020a) and our lower bound in the condition number dependence.