References
- H. Abbaszadehpeivasti, E. de Klerk, and M. Zamani, Conditions for linear convergence of the gradient method for non-convex optimization, Optim. Lett. 17 (2023), pp. 1105–1125.
- K.J. Arrow, H. Azawa, L. Hurwicz, H. Uzawa, H.B. Chenery, S.M. Johnson, and S. Karlin, Studies in Linear and Non-linear Programming, Vol. 2, Stanford University Press, California, 1958.
- W. Azizian, I. Mitliagkas, S. Lacoste-Julien and G. Gidel, A tight and unified analysis of gradient-based methods for a whole spectrum of differentiable games, in International Conference on Artificial Intelligence and Statistics, PMLR, 2020, pp. 2863–2873.
- T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, SIAM, Philadelphia, PA, 1998.
- A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust Optimization, Vol. 28, Princeton University Press, Princeton, 2009.
- A. Beznosikov, B. Polyak, E. Gorbunov, D. Kovalev, and A. Gasnikov, Smooth monotone stochastic variational inequalities and saddle point problems – survey, preprint (2022). Available at arXiv:2208.13592.
- J. Bolte, T.P. Nguyen, J. Peypouquet, and B.W. Suter, From error bounds to the complexity of first-order descent methods for convex functions, Math. Program. 165 (2017), pp. 471–507.
- S. Boyd and L. Vandenberghe, Convex optimization, Cambridge University Press, Cambridge, 2004.
- E. De Klerk, F. Glineur, and A.B. Taylor, On the worst-case complexity of the gradient method with exact line search for smooth strongly convex functions, Optim. Lett. 11 (2017), pp. 1185–1199.
- Y. Drori and M. Teboulle, Performance of first-order methods for smooth convex minimization: A novel approach, Math. Program. 145 (2014), pp. 451–482.
- S.S. Du and W. Hu, Linear convergence of the primal-dual gradient method for convex–concave saddle point problems without strong convexity, in The 22nd International Conference on Artificial Intelligence and Statistics, PMLR, 2019, pp. 196–205.
- F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003.
- A. Fallah, A. Ozdaglar and S. Pattathil, An optimal multistage stochastic gradient method for minimax problems, in 2020 59th IEEE Conference on Decision and Control (CDC), IEEE, 2020, pp. 3573–3579.
- I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, Generative adversarial nets, Adv. Neural. Inf. Process. Syst. 27 (2014), pp. 1–9.
- E.Y. Hamedani and N.S. Aybat, A primal-dual algorithm with line search for general convex–concave saddle point problems, SIAM. J. Optim. 31 (2021), pp. 1299–1329.
- R. Jiang and A. Mokhtari, Generalized optimistic methods for convex–concave saddle point problems, preprint (2022). Available at arXiv:2202.09674.
- T. Liang and J. Stokes, Interaction matters: A note on non-asymptotic local convergence of generative adversarial networks, in The 22nd International Conference on Artificial Intelligence and Statistics, PMLR, 2019, pp. 907–915.
- T. Lin, C. Jin and M.I. Jordan, Near-optimal algorithms for minimax optimization, in Conference on Learning Theory, PMLR, 2020, pp. 2738–2779.
- Z.Q. Luo and P. Tseng, Error bounds and convergence analysis of feasible descent methods: A general approach, Ann. Oper. Res. 46 (1993), pp. 157–178.
- L. Mescheder, S. Nowozin, and A. Geiger, The numerics of gans, Adv. Neural. Inf. Process. Syst. 30 (2017), pp. 1–11.
- A. Mokhtari, A. Ozdaglar and S. Pattathil, A unified analysis of extra-gradient and optimistic gradient methods for saddle point problems: Proximal point approach, in International Conference on Artificial Intelligence and Statistics, PMLR, 2020, pp. 1497–1507.
- A. Mokhtari, A.E. Ozdaglar, and S. Pattathil, Convergence rate of O(1/k) for optimistic gradient and extragradient methods in smooth convex–concave saddle point problems, SIAM. J. Optim. 30 (2020), pp. 3230–3251.
- I. Necoara, Y. Nesterov, and F. Glineur, Linear convergence of first order methods for non-strongly convex optimization, Math. Program. 175 (2019), pp. 69–107.
- Y. Nesterov, Lectures on Convex Optimization, Vol. 137, Springer, Berlin, 2018.
- J. Nie, Z. Yang, and G. Zhou, The saddle point problem of polynomials, Found. Comput. Math. 22 (2022), pp. 1133–1169.
- J.W. Simpson-Porco, B.K. Poolla, N. Monshizadeh, and F. Dörfler, Input–output performance of linear–quadratic saddle-point algorithms with application to distributed resource allocation problems, IEEE. Trans. Automat. Contr. 65 (2019), pp. 2032–2045.
- A.B. Taylor, J.M. Hendrickx, and F. Glineur, Smooth strongly convex interpolation and exact worst-case performance of first-order methods, Math. Program. 161 (2017), pp. 307–345.
- Y. Wang and J. Li, Improved algorithms for convex–concave minimax optimization, Adv. Neural. Inf. Process. Syst. 33 (2020), pp. 4800–4810.
- Z. Xu, H. Zhang, Y. Xu, and G. Lan, A unified single-loop alternating gradient projection algorithm for nonconvex–concave and convex–nonconcave minimax problems, Math. Program. 201 (2023), pp. 635–706.
- M. Yang, O. Nachum, B. Dai, L. Li, and D. Schuurmans, Off-policy evaluation via the regularized Lagrangian, Adv. Neural. Inf. Process. Syst. 33 (2020), pp. 6551–6561.
- M. Zamani, H. Abbaszadehpeivasti, and E. de Klerk, The exact worst-case convergence rate of the alternating direction method of multipliers, preprint (2022). Available at arXiv:2206.09865.
- G. Zhang, X. Bao, L. Lessard, and R. Grosse, A unified analysis of first-order methods for smooth games via integral quadratic constraints, J. Mach. Learn. Res. 22 (2021), pp. 1–39.
- G. Zhang, Y. Wang, L. Lessard and R.B. Grosse, Near-optimal local convergence of alternating gradient descent–ascent for minimax optimization, in International Conference on Artificial Intelligence and Statistics, PMLR, 2022, pp. 7659–7679.
- J. Zhang, M. Hong, and S. Zhang, On lower iteration complexity bounds for the convex concave saddle point problems, Math. Program. 194 (2022), pp. 901–935.