Weakly-convex–concave min–max optimization: provable algorithms and applications in machine learning

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

doi:10.1080/10556788.2021.1895152

Cited by 67 publications

(95 citation statements)

References 15 publications

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“…The first result above (with nonsmooth f and finite-sum g) appears to be new and our sample complexity improves over the best known in the literature [45]. The second result is among the first in the literature to derive improved sample complexity for nonsmooth f and with g being an expectation (see also [52]).…”

Section: Contributions and Outlinementioning

confidence: 66%

“…Algorithms for solving stochastic composite optimization problems of the forms (2) and (3) have been studied recently in [6,28,34,45,53,54,57,58,60]. Since these are all stochastic or randomized algorithms, a common measure of performance is their sample complexity, i.e., the total number of samples of the component mappings g i or g ξ and their Jacobians required to output some point x such that E G( x) 2 ≤ , where is a predefined precision and G( x) is the composite gradient mapping at x (for a precise definition, see (11) in Sect.…”

Section: Related Workmentioning

confidence: 99%

“…When both f and g are smooth and g is a general expectation, the state-of-the-art sample complexity is the O( −3/2 ) obtained in [60]. When f is convex but nonsmooth and g is a finite sum of N smooth mappings, the authors of [45] applied the conjugate function of f and transformed problem (2) to a min-max saddle-point problem. The sample complexity of their method (without counting subproblem cost) is O(N −1 ).…”

Section: Related Workmentioning

confidence: 99%

“…We again apply the SARAH/Spider estimator to construct gk i and J k i . For i > 0, we use (44) and (45), where ξ is interpreted as a random index drawn from {1, . .…”

Section: Assumptionmentioning

confidence: 99%

“…Specifically, we proceed with Assumptions 1, 5 and 6. Under these assumptions, we still use the SARAH/Spider estimators in (43), (44) and (45). Since that the mean-square error bounds bounds on the estimators in Lemma 7 only depends on Assumption 5, they remain valid in this section.…”

Section: The Smooth and Expectation Casementioning

confidence: 99%

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Stochastic variance-reduced prox-linear algorithms for nonconvex composite optimization

Zhang

Xiao

2021

Math. Program.

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We consider the problem of minimizing composite functions of the form f (g(x)) + h(x), where f and h are convex functions (which can be nonsmooth) and g is a smooth vector mapping. In addition, we assume that g is the average of finite number of component mappings or the expectation over a family of random component mappings. We propose a class of stochastic variance-reduced prox-linear algorithms for solving such problems and bound their sample complexities for finding an -stationary point in terms of the total number of evaluations of the component mappings and their Jacobians. When g is a finite average of N components, we obtain sample complexity O(N + N 4/5 −1 ) for both mapping and Jacobian evaluations. When g is a general expectation, we obtain sample complexities of O( −5/2 ) and O( −3/2 ) for component mappings and their Jacobians respectively. If in addition f is smooth, then improved sample complexities of O(N + N 1/2 −1 ) and O( −3/2 ) are derived for g being a finite average and a general expectation respectively, for both component mapping and Jacobian evaluations.

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Section: Contributions and Outlinementioning

confidence: 66%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

“…We again apply the SARAH/Spider estimator to construct gk i and J k i . For i > 0, we use (44) and (45), where ξ is interpreted as a random index drawn from {1, . .…”

Section: Assumptionmentioning

confidence: 99%

Section: The Smooth and Expectation Casementioning

confidence: 99%

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Stochastic variance-reduced prox-linear algorithms for nonconvex composite optimization

Zhang

Xiao

2021

Math. Program.

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Improve robustness of machine learning via efficient optimization and conformal prediction

Yan

2024

AI Magazine

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The advance of machine learning (ML) systems in real‐world scenarios usually expects safe deployment in high‐stake applications (e.g., medical diagnosis) for critical decision‐making process. To this end, provable robustness of ML is usually required to measure and understand how reliable the deployed ML system is and how trustworthy their predictions can be. Many studies have been done to enhance the robustness in recent years from different angles, such as variance‐regularized robust objective functions and conformal prediction (CP) for uncertainty quantification on testing data. Although these tools provably improve the robustness of ML model, there is still an inevitable gap to integrate them into an end‐to‐end deployment. For example, robust objectives usually require carefully designed optimization algorithms, while CP treats ML models as black boxes. This paper is a brief introduction to our recent research focusing on filling this gap. Specifically, for learning robust objectives, we designed sample‐efficient stochastic optimization algorithms that achieves the optimal (or faster compared to existing algorithms) convergence rates. Moreover, for CP‐based uncertainty quantification, we established a framework to analyze the expected prediction set size (smaller size means more efficiency) of CP methods in both standard and adversarial settings. This paper elaborates the key challenges and our exploration towards efficient algorithms with details of background methods, notions for robustness measure, concepts of algorithmic efficiency, our proposed algorithms and results. All of them further motivate our future research on risk‐aware ML that can be critical for AI–human collaborative systems. The future work mainly targets designing conformal robust objectives and their efficient optimization algorithms.

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An implicit gradient-descent procedure for minimax problems

2022

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Weakly-convex–concave min–max optimization: provable algorithms and applications in machine learning

Cited by 67 publications

References 15 publications

Stochastic variance-reduced prox-linear algorithms for nonconvex composite optimization

Stochastic variance-reduced prox-linear algorithms for nonconvex composite optimization

Improve robustness of machine learning via efficient optimization and conformal prediction

An implicit gradient-descent procedure for minimax problems

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