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1. Near-optimal local convergence of alternating gradient descent-ascent for minimax optimization

   Zhang, Guodong ; Wang, Yuanhao ; Lessard, Laurent ; Grosse, Roger B. ( March 2022 , Proceedings of Machine Learning Research)

   Smooth minimax games often proceed by simultaneous or alternating gradient updates. Although algorithms with alternating updates are commonly used in practice, the majority of existing theoretical analyses focus on simultaneous algorithms for convenience of analysis. In this paper, we study alternating gradient descent-ascent (Alt-GDA) in minimax games and show that Alt-GDA is superior to its simultaneous counterpart (Sim-GDA) in many settings. We prove that Alt-GDA achieves a near-optimal local convergence rate for strongly convex-strongly concave (SCSC) problems while Sim-GDA converges at a much slower rate. To our knowledge, this is the first result of any setting showing that Alt-GDA converges faster than Sim-GDA by more than a constant. We further adapt the theory of integral quadratic constraints (IQC) and show that Alt-GDA attains the same rate globally for a subclass of SCSC minimax problems. Empirically, we demonstrate that alternating updates speed up GAN training significantly and the use of optimism only helps for simultaneous algorithms. 

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2. Stochastic Smoothed Gradient Descent Ascent for Federated Minimax Optimization

   Shen, W ; Huang, M ; Zhang, J ; Shen, C ( July 2024 , Proceedings of The 27th International Conference on Artificial Intelligence and Statistics)

   In recent years, federated minimax optimization has attracted growing interest due to its extensive applications in various machine learning tasks. While Smoothed Alternative Gradient Descent Ascent (Smoothed-AGDA) has proved successful in centralized nonconvex minimax optimization, how and whether smoothing techniques could be helpful in a federated setting remains unexplored. In this paper, we propose a new algorithm termed Federated Stochastic Smoothed Gradient Descent Ascent (FESS-GDA), which utilizes the smoothing technique for federated minimax optimization. We prove that FESS-GDA can be uniformly applied to solve several classes of federated minimax problems and prove new or better analytical convergence results for these settings. We showcase the practical efficiency of FESS-GDA in practical federated learning tasks of training generative adversarial networks (GANs) and fair classification. 

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3. A Diffusion Approximation Theory of Momentum Stochastic Gradient Descent in Nonconvex Optimization

   https://doi.org/10.1287/stsy.2021.0083  

   Liu, Tianyi ; Chen, Zhehui ; Zhou, Enlu ; Zhao, Tuo ( December 2021 , Stochastic Systems)

   Momentum stochastic gradient descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning (e.g., training deep neural networks, variational Bayesian inference, etc.). Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points but hurts the convergence within the neighborhood of optima (if without the step size annealing or momentum annealing). Our theoretical discovery partially corroborates the empirical success of MSGD in training deep neural networks. 

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4. Federated Minimax Optimization with Client Heterogeneity

   Sharma, Pranay ; Panda, Rohan ; Joshi, Gauri ( December 2023 , Transactions on machine learning research)

   Minimax optimization has seen a surge in interest with the advent of modern applications such as GANs, and it is inherently more challenging than simple minimization. The difficulty is exacerbated by the training data residing at multiple edge devices or clients, especially when these clients can have heterogeneous datasets and heterogeneous local computation capabilities. We propose a general federated minimax optimization framework that subsumes such settings and several existing methods like Local SGDA. We show that naive aggregation of model updates made by clients running unequal number of local steps can result in optimizing a mismatched objective function – a phenomenon previously observed in standard federated minimization. To fix this problem, we propose normalizing the client updates by the number of local steps. We analyze the convergence of the proposed algorithm for classes of nonconvex-concave and nonconvex-nonconcave functions and characterize the impact of heterogeneous client data, partial client participation, and heterogeneous local computations. For all the function classes considered, we significantly improve the existing computation and communication complexity results. Experimental results support our theoretical claims. 

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5. The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization

   Daskalakis, Constantinos ; Panageas, Ioannis ( January 2018 , 32nd Annual Conference on Neural Information Processing Systems)

   Motivated by applications in Optimization, Game Theory, and the training of Generative Adversarial Networks, the convergence properties of first order methods in min-max problems have received extensive study. It has been recognized that they may cycle, and there is no good understanding of their limit points when they do not. When they converge, do they converge to local min-max solutions? We characterize the limit points of two basic first order methods, namely Gradient Descent/Ascent (GDA) and Optimistic Gradient Descent Ascent (OGDA). We show that both dynamics avoid unstable critical points for almost all initializations. Moreover, for small step sizes and under mild assumptions, the set of \{OGDA\}-stable critical points is a superset of \{GDA\}-stable critical points, which is a superset of local min-max solutions (strict in some cases). The connecting thread is that the behavior of these dynamics can be studied from a dynamical systems perspective. 

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