Weakly-Convex Concave Min-Max Optimization: Provable Algorithms and Applications in Machine Learning

Rafique, Hassan; Liu, Mingrui; Lin, Qihang; Yang, Tianbao

doi:10.48550/arxiv.1810.02060

Cited by 44 publications

(82 citation statements)

References 16 publications

(31 reference statements)

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“…This problem has been extensively studied under different assumptions for the objective f . For instance, there are works for (strongly)-convex (strongly)-concave [46,5,44,45,74,69], non-convex (strongly)-concave [56,49,63,40,33,34,51] and non-convex non-concave [32,49,35,27,70] problems. We refer the interested readers to a recent survey about detailed developments [58].…”

Section: Minimax Problemsmentioning

confidence: 99%

“…finds applications in areas such as machine learning, game theory, signal processing, and it has been extensively studied in recent years [56,49,63,40,33,34,70]. Its specific applications include adversarial learning [42,38], reinforcement learning [12,55], AUC maximization [36], resource allocation in wireless communication [40], and Generative Adversarial Networks (GAN) [26,3,43], among others.…”

Section: Introductionmentioning

confidence: 99%

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Minimax Problems with Coupled Linear Constraints: Computational Complexity, Duality and Solution Methods

Tsaknakis

Mei

Zhang

2021

Preprint

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In this work we study a special minimax problem where there are linear constraints that couple both the minimization and maximization decision variables. The problem is a generalization of the traditional saddle point problem (which does not have the coupling constraint), and it finds applications in wireless communication, game theory, transportation, robust optimization, just to name a few. We show that the considered problem is challenging, in the sense that it violates the classical max-min inequality, and that it is NP-hard even under very strong assumptions (e.g., the objective is strongly convex-strongly concave). We then develop a duality theory for it, and analyze conditions under which the duality gap becomes zero. Leveraging the resulting duality-based relaxations, we propose a family of efficient algorithms named multiplier gradient descent (MGD), and study their convergence properties. Finally, we test the effectiveness of the proposed algorithms in an application on adversarial attacks on network flow problems.

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Section: Minimax Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Minimax Problems with Coupled Linear Constraints: Computational Complexity, Duality and Solution Methods

Tsaknakis

Mei

Zhang

2021

Preprint

View full text Add to dashboard Cite

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“…One approach is to equivalently reformulate the problem by min x {Φ(x) := max y∈Y f (x, y)}, and define an optimality notion for the local surrogate of global optimum of Φ. A series of theoretical analyses on the stationary point convergence condition of Φ with first-order algorithm were carried out to extend the convex-concave assumption to assumptions of nonconvex-strongly-concave 5 (Rafique et al, 2018;Lu et al, 2020), nonconvex-concave (Lin et al, 2020b;Nouiehed et al, 2019), and nonconvex-nonconcave 6 (Jin et al, 2020). Convergence analysis for a federated optimizer, such as FedMM that involves bounding client's drift from…”

Section: Convergence Analysismentioning

confidence: 99%

FedMM: Saddle Point Optimization for Federated Adversarial Domain Adaptation

Shen¹,

Du²,

Zhang³

et al. 2021

Preprint

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Federated adversary domain adaptation is a unique distributed minimax training task due to the prevalence of label imbalance among clients, with each client only seeing a subset of the classes of labels required to train a global model. To tackle this problem, we propose a distributed minimax optimizer referred to as FedMM, designed specifically for the federated adversary domain adaptation problem. It works well even in the extreme case where each client has different label classes and some clients only have unsupervised tasks. We prove that FedMM ensures convergence to a stationary point with domain-shifted unsupervised data. On a variety of benchmark datasets, extensive experiments show that FedMM consistently achieves either significant communication savings or significant accuracy improvements over federated optimizers based on the gradient descent ascent (GDA) algorithm. When training from scratch, for example, it outperforms other GDA based federated average methods by around 20% in accuracy over the same communication rounds; and it consistently outperforms when training from pre-trained models with an accuracy improvement from 5.4% to 9% for different networks.

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“…Convergence under the nonconvex-concave setup is less studied. Rafique et al (2018) studied a proximal version of gradient methods. Lin et al (2020a) studied gradient descent ascent with different time scales.…”

Section: Related Workmentioning

confidence: 99%

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

Li,

Tian,

Zhang

et al. 2021

Preprint

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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 Ω √ κ −2 for deterministic oracles, where defines the level of approximate stationarity and κ is the condition number. Our analysis shows that the upper bound achieved in (Lin et al., 2020b) is optimal in the and κ dependence up to logarithmic factors. For stochastic oracles, we provide a lower bound of Ω √ κ −2 + κ 1/3 −4 . It suggests that there is a significant gap between the upper bound O(κ 3 −4 ) in (Lin et al., 2020a) and our lower bound in the condition number dependence.

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Weakly-Convex Concave Min-Max Optimization: Provable Algorithms and Applications in Machine Learning

Cited by 44 publications

References 16 publications

Minimax Problems with Coupled Linear Constraints: Computational Complexity, Duality and Solution Methods

Minimax Problems with Coupled Linear Constraints: Computational Complexity, Duality and Solution Methods

FedMM: Saddle Point Optimization for Federated Adversarial Domain Adaptation

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

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