A unified single-loop alternating gradient projection algorithm for nonconvex–concave and convex–nonconcave minimax problems

Xu, Zi; Zhang, Huiling; Xu, Yang; Lan, Guanghui

doi:10.1007/s10107-022-01919-z

Cited by 16 publications

(6 citation statements)

References 46 publications

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“…It is demonstrated in Appendix E that h I is µ-weakly convex in lots of cases. Following (Xie, Koyejo, and Gupta 2019;Davis and Drusvyatskiy 2019), the definition and first-order condition of µ-weakly convex function are given as follows.…”

Section: Refining Hyper-polyhedral Approximationmentioning

confidence: 99%

Provably Convergent Federated Trilevel Learning

Jiao,

Yang,

et al. 2024

AAAI

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Trilevel learning, also called trilevel optimization (TLO), has been recognized as a powerful modelling tool for hierarchical decision process and widely applied in many machine learning applications, such as robust neural architecture search, hyperparameter optimization, and domain adaptation. Tackling TLO problems has presented a great challenge due to their nested decision-making structure. In addition, existing works on TLO face the following key challenges: 1) they all focus on the non-distributed setting, which may lead to privacy breach; 2) they do not offer any non-asymptotic convergence analysis which characterizes how fast an algorithm converges. To address the aforementioned challenges, this paper proposes an asynchronous federated trilevel optimization method to solve TLO problems. The proposed method utilizes u-cuts to construct a hyper-polyhedral approximation for the TLO problem and solve it in an asynchronous manner. We demonstrate that the proposed u-cuts are applicable to not only convex functions but also a wide range of non-convex functions that meet the u-weakly convex assumption. Furthermore, we theoretically analyze the non-asymptotic convergence rate for the proposed method by showing its iteration complexity to obtain ϵ-stationary point is upper bounded by O(1/ϵ²). Extensive experiments on real-world datasets have been conducted to elucidate the superiority of the proposed method, e.g., it has a faster convergence rate with a maximum acceleration of approximately 80%.

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Section: Refining Hyper-polyhedral Approximationmentioning

confidence: 99%

Provably Convergent Federated Trilevel Learning

Jiao,

Yang,

et al. 2024

AAAI

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“…The minimax problem has broad applications in different areas including game theory [40], training generative adversarial networks (GANs) [41,42], adversarial and robust machine learning [43,44], resource allocation over networks [45], and distributed optimization [46,47]; to name just a few. For solving this problem, (alternating) gradient descent-ascent is the most popular method used in the existing literature; see for example [48][49][50][51][52][53][54] and the references therein. This method iteratively updates the estimates x k , y k of the desired solution as…”

Section: Minimax Optimizationmentioning

confidence: 99%

Nonlinear Two-Time-Scale Stochastic Approximation: Convergence and Finite-Time Performance

Doan

2023

IEEE Trans. Automat. Contr.

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This paper proposes to develop a new variant of the two-time-scale stochastic approximation to find the roots of two coupled nonlinear operators, assuming only noisy samples of these operators can be observed. Our key idea is to leverage the classic Ruppert-Polyak averaging technique to dynamically estimate the operators through their samples. The estimated values of these averaging steps will then be used in the two-time-scale stochastic approximation updates to find the desired solution. Our main theoretical result is to show that under the strongly monotone condition of the underlying nonlinear operators the mean-squared errors of the iterates generated by the proposed method converge to zero at an optimal rate O(1/k), where k is the number of iterations. Our result significantly improves the existing result of two-time-scale stochastic approximation, where the best known finite-time convergence rate is O(1/k 2/3 ). We illustrate this result by applying the proposed method to develop new reinforcement learning algorithms with improved performance.

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“…Reference Assumption f (x, •) Complexity batch size ALR ZO-VRAGDA Xu et al [2022] PL O(ǫ −3 ) O(ǫ −2 ) × Smoothed-AGDA Yang et al [2022] PL…”

Section: Algorithmmentioning

confidence: 99%

“…More recently, Chen et al [2022] propose a class of faster stochastic GDA (i.e.,SPIDER-GDA and AccSPIDER-GDA) methods based on variance-reduced technique for finite-sum minimax optimization under PL condition that includes PL-PL and Nonconvex-PL minimax optimizations. Meanwhile, Xu et al [2022] proposed a class of variance-reduced zeroth-order methods for stochastic Nonconvex-PL minimax problems.…”

Section: Algorithmmentioning

confidence: 99%

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Smellscapes of Nanjing Road

Huang¹

2022

Sensing China

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Bilevel optimization is a popular two-level hierarchical optimization, which has been widely applied to many machine learning tasks such as hyperparameter learning, meta learning and continual learning. Although many bilevel optimization methods recently have been developed, the bilevel methods are not well studied when the lower-level problem is nonconvex. To fill this gap, in the paper, we study a class of nonconvex bilevel optimization problems, which both upper-level and lower-level problems are nonconvex, and the lower-level problem satisfies Polyak-Lojasiewicz (PL) condition. We propose an efficient momentum-based gradient bilevel method (MGBiO) to solve these deterministic problems. Meanwhile, we propose a class of efficient momentum-based stochastic gradient bilevel methods (MSGBiO and VR-MSGBiO) to solve these stochastic problems. Moreover, we provide a useful convergence analysis framework for our methods. Specifically, under some mild conditions, we prove that our MGBiO method has a sample (or gradient) complexity of O(ǫ −2 ) for finding an ǫ-stationary solution of the deterministic bilevel problems (i.e., ∇F (x) ≤ ǫ), which improves the existing best results by a factor of O(ǫ −1 ). Meanwhile, we prove that our MSGBiO and VR-MSGBiO methods have sample complexities of Õ(ǫ −4 ) and Õ(ǫ −3 ), respectively, in finding an ǫ-stationary solution of the stochastic bilevel problems (i.e., E ∇F (x) ≤ ǫ), which improves the existing best results by a factor of O(ǫ −3 ). This manuscript commemorates the mathematician Boris Polyak (1935 -2023).

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A unified single-loop alternating gradient projection algorithm for nonconvex–concave and convex–nonconcave minimax problems

Cited by 16 publications

References 46 publications

Provably Convergent Federated Trilevel Learning

Provably Convergent Federated Trilevel Learning

Nonlinear Two-Time-Scale Stochastic Approximation: Convergence and Finite-Time Performance

Smellscapes of Nanjing Road

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