{ "id": "epg35t7qaw", "slug": "minimax-optimization", "title": "Minimax optimization", "updated": "2026-09-30", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.algorithms" }, { "id": "technology.computing.ai.optimization_dp", "label": "Optimization and dynamic programming", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.optimization_dp" }, { "id": "technology.computing.ai.optimization_dp.mathematical-programming-methods", "label": "Mathematical programming methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.optimization_dp.mathematical-programming-methods" } ], "geo": [ { "id": "geo.nongeo.t1800.technology.computing", "label": "Non-geographic · 1800 to 1945: Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.technology.computing", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.technology" }, { "id": "geo.nongeo.t1800.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.technology.computing" } ] }, { "id": "geo.nongeo.t1946.technology.computing.ai.optimization_dp", "label": "Non-geographic · 1946 to 2000: Optimization and dynamic programming", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.optimization_dp", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology" }, { "id": "geo.nongeo.t1946.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing" }, { "id": "geo.nongeo.t1946.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai" }, { "id": "geo.nongeo.t1946.technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.algorithms" }, { "id": "geo.nongeo.t1946.technology.computing.ai.optimization_dp", "label": "Optimization and dynamic programming", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.optimization_dp" } ] } ], "excerpt": "Minimax optimization seeks a saddle point where one player minimizes a shared objective and an adversary maximizes it, the formal language of GANs and robust learning.", "snippet": "Minimax optimization seeks a saddle point where one player minimizes a shared objective and an adversary maximizes it, the formal language of GANs and robust learning.", "node": "technology.computing.ai.optimization_dp.mathematical-programming-methods", "markdown": "# Minimax optimization\n\nMinimax optimization computes parameters \\( x \\) that minimize a shared objective \\( f(x, y) \\) while an adversary chooses \\( y \\) to maximize it; the target is a saddle point, a pair of choices that neither player can improve by moving alone. In machine learning it is the formal language of adversarial settings: generative adversarial networks train a generator and a discriminator as a two-player game with an explicit value function[1](https://proceedings.neurips.cc/paper_files/paper/2014/file/f033ed80deb0234979a61f95710dbe25-Paper.pdf), and robust learning, distributionally robust optimization, and signal-processing design under uncertainty or jamming are all written in min-max form.[2](https://ar5iv.labs.arxiv.org/html/2006.08141) Published guarantees differ sharply by problem class, from linear convergence on bilinear games to hardness results in the fully nonconvex regime.\n\n| Fact | Statement |\n|---|---|\n| Saddle-point condition | \\( g(x_1^*, x_2) \\le g(x_1^*, x_2^*) \\le g(x_1, x_2^*) \\) for all candidate moves[3](http://proceedings.mlr.press/v132/abernethy21a/abernethy21a.pdf) |\n| GAN value function | \\( V(D,G) = \\mathbb{E}_{x \\sim p_{data}}[\\log D(x)] + \\mathbb{E}_{z \\sim p_z}[\\log(1 - D(G(z)))] \\)[1](https://proceedings.neurips.cc/paper_files/paper/2014/file/f033ed80deb0234979a61f95710dbe25-Paper.pdf) |\n| Plain GDA on bilinear games | Never converges for any step length except degenerate cases[4](https://cs.uwaterloo.ca/~y328yu/classics/extragrad.pdf) |\n| Convex-concave benchmark | Extragradient and Mirror Prox reach \\( O(1/\\epsilon) \\) duality-gap complexity, tight for first-order methods[5](https://ar5iv.labs.arxiv.org/html/2304.08389) |\n| Nonconvex-strongly-concave | GDA \\( O(\\kappa^2 \\epsilon^{-2}) \\); lower bound \\( \\Omega(\\sqrt{\\kappa}\\,\\epsilon^{-2}) \\)[6](https://proceedings.neurips.cc/paper_files/paper/2025/file/a663f8ccd0e7c87b7763bf1462aeb388-Paper-Conference.pdf) • [7](https://proceedings.neurips.cc/paper_files/paper/2021/file/0e105949d99a32ca1751703e94ece601-Paper.pdf) |\n| Nonconvex-nonconcave | A saddle point may not exist; deciding existence is NP-hard[8](https://ar5iv.labs.arxiv.org/html/2009.09623) |\n| Weak-MVI tolerance (2024) | Extended from \\( \\rho < 1/(2L) \\) to \\( \\rho < 1/L \\)[9](https://raw.githubusercontent.com/mlresearch/v235/main/assets/alacaoglu24a/alacaoglu24a.pdf) |\n\n## How it works\n\nThe problem is \\( \\min_{x \\in \\mathcal{X}} \\max_{y \\in \\mathcal{Y}} f(x, y) \\). A saddle point \\( (x^*, y^*) \\) satisfies \\( g(x_1^*, x_2) \\le g(x_1^*, x_2^*) \\le g(x_1, x_2^*) \\) for every candidate \\( x_1, x_2 \\): the minimizer cannot lower the objective by deviating, and neither can the maximizer.[3](http://proceedings.mlr.press/v132/abernethy21a/abernethy21a.pdf) Progress is measured by the duality gap, which compares each player's objective when the opponent is allowed to respond; any saddle point has gap zero.[10](https://kjtian.github.io/notes/CS%20395T%20%28Spring%202024%29/Part4_main.pdf) For continuous bilinear objectives on compact convex strategy sets, von Neumann's [minimax theorem](https://www.edgepedia.org/minimax-theorem) guarantees a saddle point with min-max equal to max-min[11](https://doi.org/10.1007/bf01448847) • [7](https://proceedings.neurips.cc/paper_files/paper/2021/file/0e105949d99a32ca1751703e94ece601-Paper.pdf), and Sion's 1958 theorem extends the guarantee to convex-[concave functions](https://www.edgepedia.org/concave-function) under regularity conditions.[12](https://doi.org/10.2140/pjm.1958.8.171) When the problem is convex in \\( x \\) and concave in \\( y \\), the associated variational inequality is monotone and classical algorithms apply; outside that class, algorithms target weaker solution concepts such as the Minty variational inequality, which asks for \\( z^* \\) with \\( \\langle \\xi, z - z^* \\rangle \\ge 0 \\) for all \\( z \\) and all \\( \\xi \\in F(z) \\)[13](https://jmlr.csail.mit.edu/papers/volume22/20-533/20-533.pdf), and optimality measures related to the Nikaidô-Isoda function.[14](https://arxiv.org/html/2210.12860v8)\n\n[Gradient descent](https://www.edgepedia.org/gradient-descent)-ascent fails for a structural reason: its update map need not be a gradient field, so the Jacobian \\( J = \\partial w / \\partial z \\) can be asymmetric and the discrete dynamics \\( z_{t+1} = w(z_t) \\) can converge to limit cycles instead of points; a fixed point is locally asymptotically stable when the spectral radius satisfies \\( \\rho(J(z^*)) < 1 \\), while the case \\( \\rho(J(z^*)) = 1 \\) requires further analysis.[28](https://doi.org/10.1155/2017/6186354) • [15](https://ar5iv.labs.arxiv.org/html/1902.00618)\n\n## How it is done\n\nGDA alternates a gradient descent step on \\( x \\) with a gradient ascent step on \\( y \\). The most common repair is two-timescale GDA with \\( \\eta_x \\ll \\eta_y \\), on the logic that the fast maximization updates in \\( y \\) track the inner maximizer while the slower minimization updates in \\( x \\) proceed.[16](https://arxiv.org/html/1906.00331v9) Practitioners also alternate updates: the standard GAN procedure runs \\( k \\) discriminator steps per generator step and replaces the generator objective \\( \\log(1 - D(G(z))) \\) with maximizing \\( \\log D(G(z)) \\), equivalently minimizing \\( -\\log D(G(z)) \\), which gives stronger early gradients.[1](https://proceedings.neurips.cc/paper_files/paper/2014/file/f033ed80deb0234979a61f95710dbe25-Paper.pdf)\n\nExtragradient takes a trial gradient step to an extrapolated point and uses the gradient at that point as the actual direction of movement[4](https://cs.uwaterloo.ca/~y328yu/classics/extragrad.pdf), at the cost of one extra gradient evaluation per iteration.[17](https://arxiv.org/pdf/1901.08511) Optimistic GDA and extragradient can both be read as approximations of the proximal point method, which explains why they converge on bilinear problems where GDA diverges.[17](https://arxiv.org/pdf/1901.08511) Hyperparameters matter: correction-term ratios above 0.5 make generalized OGDA diverge and become unstable.[18](https://papers.neurips.cc/paper_files/paper/2022/file/ca4f6e86453e4b117dd3263792053cf5-Paper-Conference.pdf) Symplectic gradient adjustment targets stable fixed points in potential and Hamiltonian games.[16](https://arxiv.org/html/1906.00331v9)\n\n## Origin\n\nVon Neumann stated the minimax theorem in \"Zur Theorie der Gesellschaftsspiele\" (Mathematische Annalen, 1928)[11](https://doi.org/10.1007/bf01448847); the bilinear minimax problem together with this theorem was a cornerstone in the development of game theory.[19](https://www.jmlr.org/papers/volume26/22-0863/22-0863.pdf) Sion's \"On general minimax theorems\" (Pacific Journal of Mathematics, 1958) generalized the result from bilinear to general convex-concave games.[12](https://doi.org/10.2140/pjm.1958.8.171) A discrete-time algorithmic line then grew in the variational-inequality literature, where extragradient's linear convergence for smooth strongly convex-strongly concave and bilinear functions was established.[17](https://arxiv.org/pdf/1901.08511) Minimax entered mainstream machine learning through adversarial generative modeling and adversarial learning, which motivated the nonconvex-nonconcave literature.[19](https://www.jmlr.org/papers/volume26/22-0863/22-0863.pdf) Lin, Jin and Jordan (2019, arXiv) gave the first systematic complexity analysis of two-timescale GDA for structured nonconvex minimax problems[20](https://doi.org/10.48550/arxiv.1906.00331), reporting the \\( \\tilde{O}(\\sqrt{\\kappa}\\,\\epsilon^{-2}) \\) GDA upper bound in the nonconvex-strongly-concave setting.[7](https://proceedings.neurips.cc/paper_files/paper/2021/file/0e105949d99a32ca1751703e94ece601-Paper.pdf)\n\n## Variants\n\n**Convex-concave.** Extragradient and Mirror Prox converge in duality gap at \\( O(1/\\epsilon) \\), tight for first-order methods[5](https://ar5iv.labs.arxiv.org/html/2304.08389); on squared gradient norm the first-order lower bound is \\( \\Omega(\\epsilon^{-1/2}) \\).[14](https://arxiv.org/html/2210.12860v8)\n\n**Bilinear and strongly convex-strongly concave.** State-of-the-art first-order methods find an approximate [Nash equilibrium](https://www.edgepedia.org/nash-equilibrium) in \\( \\tilde{O}(\\kappa_x + \\kappa_y) \\) gradient evaluations, where \\( \\kappa_x, \\kappa_y \\) are condition numbers[21](https://proceedings.mlr.press/v125/lin20a.html), and extragradient converges linearly in \\( O(\\kappa \\log(1/\\epsilon)) \\) iterations.[17](https://arxiv.org/pdf/1901.08511)\n\n**Nonconvex.** In the nonconvex-strongly-concave setting, two-timescale GDA needs \\( O(\\kappa^2 \\epsilon^{-2}) \\) gradient evaluations and stochastic GDA \\( O(\\kappa^3 \\epsilon^{-4}) \\)[16](https://arxiv.org/html/1906.00331v9), against a deterministic lower bound of \\( \\Omega(\\sqrt{\\kappa}\\,\\epsilon^{-2}) \\)[7](https://proceedings.neurips.cc/paper_files/paper/2021/file/0e105949d99a32ca1751703e94ece601-Paper.pdf); proximal point methods with acceleration improve the rate to \\( \\tilde{O}(\\sqrt{\\kappa}\\,\\epsilon^{-2}) \\).[6](https://proceedings.neurips.cc/paper_files/paper/2025/file/a663f8ccd0e7c87b7763bf1462aeb388-Paper-Conference.pdf) For nonconvex-concave problems, published rates differ in scope: GDA and SGDA analyses give \\( O(\\epsilon^{-6}) \\) and \\( O(\\epsilon^{-8}) \\)[16](https://arxiv.org/html/1906.00331v9), while other work describes typical rates as \\( O(\\epsilon^{-4}) \\) once strong concavity is absent.[6](https://proceedings.neurips.cc/paper_files/paper/2025/file/a663f8ccd0e7c87b7763bf1462aeb388-Paper-Conference.pdf) Weakly-convex-weakly-concave problems are solved through sequences of strongly monotone variational inequalities with \\( O(1/\\epsilon^6) \\) complexity.[13](https://jmlr.csail.mit.edu/papers/volume22/20-533/20-533.pdf) The weak-MVI parameter range for first-order methods was extended from \\( \\rho < 1/(2L) \\) to \\( \\rho < 1/L \\) using conic nonexpansiveness of operators.[9](https://raw.githubusercontent.com/mlresearch/v235/main/assets/alacaoglu24a/alacaoglu24a.pdf)\n\n## Applications\n\nIn GAN training, at the discriminator optimum the generator's criterion is \\( C(G) = -\\log 4 + 2 \\cdot \\mathrm{JSD}(p_{data} \\,\\|\\, p_g) \\), minimized if and only if \\( p_g = p_{data} \\).[1](https://proceedings.neurips.cc/paper_files/paper/2014/file/f033ed80deb0234979a61f95710dbe25-Paper.pdf) The Wasserstein GAN is written \\( \\min_v \\max_w \\mathbb{E}_{x \\sim p_{data}}[D_w(x)] - \\mathbb{E}_{z \\sim p_z}[D_w(G_v(z))] \\) with a 1-Lipschitz critic, weakly-convex-weakly-concave under smoothness assumptions.[22](https://pure-oai.bham.ac.uk/ws/portalfiles/portal/143292814/lei21b.pdf) [Distributionally robust optimization](https://www.edgepedia.org/distributionally-robust-optimization), learning with non-decomposable losses, reinforcement learning, and AUC maximization are also formulated as minimax problems.[23](https://arxiv.org/pdf/2002.05309) • [22](https://pure-oai.bham.ac.uk/ws/portalfiles/portal/143292814/lei21b.pdf) In signal processing and robust learning, any design problem with model uncertainty or an adversary, including fair beamforming, robust transceiver design, and communication in the presence of jammers, takes min-max form.[2](https://ar5iv.labs.arxiv.org/html/2006.08141)\n\n## Limitations and alternatives\n\nCycling is structural, not an artifact: on simple bilinear games SGDA produces recurrent orbits containing no critical point of the objective, and constructed almost-bilinear problems admit spurious limit cycles, one of which is unstable and repels trajectories approaching the true solution.[24](https://ar5iv.labs.arxiv.org/html/2006.09065) A saddle point need not exist without convex-concave structure; \\( \\min_{x \\in [0,1]} \\max_{y \\in [0,1]} (x - y)^2 \\) has none.[8](https://ar5iv.labs.arxiv.org/html/2009.09623) Stable limit points of GDA are not necessarily Nash equilibria.[15](https://ar5iv.labs.arxiv.org/html/1902.00618) Computationally, deciding whether an approximate min-max point exists is NP-hard and finding an approximate local min-max point is PPAD-complete[8](https://ar5iv.labs.arxiv.org/html/2009.09623), while approximately finding a stationary point in the general constrained setting is FNP-complete.[5](https://ar5iv.labs.arxiv.org/html/2304.08389)\n\n**Alternatives** change the problem rather than the step size. Playing against smooth algorithmic adversaries, such as multi-step stochastic gradient ascent, gives monotonic progress with no limit cycles and \\( O(\\epsilon^{-4}) \\) complexity (\\( O(\\epsilon^{-3}) \\) for a proximal variant).[25](https://ar5iv.labs.arxiv.org/html/2106.01488) K-beam \\( \\epsilon \\)-subgradient methods track \\( K \\) candidate inner solutions when the argmax is non-unique or discontinuous in the outer variable.[26](https://arxiv.org/pdf/1805.11640v2.pdf) A simulated-annealing method converges to a local min-max equilibrium from any starting point and avoids mode collapse in GAN training at per-iteration cost similar to GDA.[27](http://arxiv.org/pdf/2006.12376v2)\n\n## References\n\n1. [Generative Adversarial Nets (Goodfellow et al., NeurIPS 2014)](https://proceedings.neurips.cc/paper_files/paper/2014/file/f033ed80deb0234979a61f95710dbe25-Paper.pdf)\n2. [Non-convex Min-Max Optimization: Applications, Challenges, and Recent Theoretical Advances (IEEE Signal Processing Magazine survey)](https://ar5iv.labs.arxiv.org/html/2006.08141)\n3. [Last-Iterate Convergence Rates for Min-Max Optimization: Convergence of Hamiltonian Gradient Descent and Consensus Optimization (ICML 2021)](http://proceedings.mlr.press/v132/abernethy21a/abernethy21a.pdf)\n4. [The Extragradient Method for Finding Saddle Points and Other Problems (Korpelevich, scanned original)](https://cs.uwaterloo.ca/~y328yu/classics/extragrad.pdf)\n5. [Beyond first-order methods for non-convex non-concave min-max optimization](https://ar5iv.labs.arxiv.org/html/2304.08389)\n6. [Semi-infinite Nonconvex Constrained Min-Max Optimization (NeurIPS 2025)](https://proceedings.neurips.cc/paper_files/paper/2025/file/a663f8ccd0e7c87b7763bf1462aeb388-Paper-Conference.pdf)\n7. [Complexity Lower Bounds for Nonconvex-Strongly-Concave Min-Max Optimization (NeurIPS 2021)](https://proceedings.neurips.cc/paper_files/paper/2021/file/0e105949d99a32ca1751703e94ece601-Paper.pdf)\n8. [The Complexity of Constrained Min-Max Optimization](https://ar5iv.labs.arxiv.org/html/2009.09623)\n9. [Revisiting Inexact Fixed-Point Iterations for Min-Max Problems: Stochasticity and Structured Nonconvexity (ICML 2024)](https://raw.githubusercontent.com/mlresearch/v235/main/assets/alacaoglu24a/alacaoglu24a.pdf)\n10. [Part4 main (kjtian.github.io)](https://kjtian.github.io/notes/CS%20395T%20%28Spring%202024%29/Part4_main.pdf)\n11. [J. v. Neumann (1928). Zur Theorie der Gesellschaftsspiele. Mathematische Annalen.](https://doi.org/10.1007/bf01448847)\n12. [Maurice Sion (1958). On general minimax theorems. Pacific Journal of Mathematics.](https://doi.org/10.2140/pjm.1958.8.171)\n13. [First-order Convergence Theory for Weakly-Convex-Weakly-Concave Min-max Problems (JMLR)](https://jmlr.csail.mit.edu/papers/volume22/20-533/20-533.pdf)\n14. [Second-order min-max optimization methods (Newton-MinMax)](https://arxiv.org/html/2210.12860v8)\n15. [What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization? (Jin, Netrapalli, Jordan; ICLR 2020)](https://ar5iv.labs.arxiv.org/html/1902.00618)\n16. [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems (Lin, Jin, Jordan)](https://arxiv.org/html/1906.00331v9)\n17. [A Unified Framework for HGDA and OGDA (Mokhtari, Ozdaglar, Pattathil)](https://arxiv.org/pdf/1901.08511)\n18. [Tight Analysis of Extra-gradient and Optimistic Gradient Methods For Nonconvex Minimax Problems (NeurIPS 2022)](https://papers.neurips.cc/paper_files/paper/2022/file/ca4f6e86453e4b117dd3263792053cf5-Paper-Conference.pdf)\n19. [Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization (JMLR vol. 26)](https://www.jmlr.org/papers/volume26/22-0863/22-0863.pdf)\n20. [Lin, Tianyi, Jin, Chi, Jordan, Michael I. (2019). On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1906.00331)\n21. [Near-Optimal Algorithms for Minimax Optimization (COLT 2020)](https://proceedings.mlr.press/v125/lin20a.html)\n22. [Generalization Analysis of Stochastic Gradient Methods for Minimax Problems (University of Birmingham repository)](https://pure-oai.bham.ac.uk/ws/portalfiles/portal/143292814/lei21b.pdf)\n23. [Optimal Epoch Stochastic Gradient Descent Ascent Methods for Min-Max Optimization (Epoch-GDA)](https://arxiv.org/pdf/2002.05309)\n24. [The Limits of Min-Max Optimization Algorithms: Convergence to Spurious Non-Critical Sets](https://ar5iv.labs.arxiv.org/html/2006.09065)\n25. [Minimax Optimization with Smooth Algorithmic Adversaries](https://ar5iv.labs.arxiv.org/html/2106.01488)\n26. [K-Beam Minimax: Efficient Optimization for Deep Adversarial Learning (NeurIPS 2018)](https://arxiv.org/pdf/1805.11640v2.pdf)\n27. [A Convergent Algorithm for Nonconvex-Nonconcave Min-Max Optimization (simulated-annealing based)](http://arxiv.org/pdf/2006.12376v2)\n28. 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