Gradient Descent Ascent for Minimax Problems on Riemannian Manifolds

Huang, Feihu; Gao, Shangqian

doi:10.1109/tpami.2023.3234160

Cited by 13 publications

(10 citation statements)

References 19 publications

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“…Moreover, the lower complexity bounds have also been studied for centralized minimax algorithms in (Zhang, Hong, and Zhang 2019;Zhang et al 2021a;. Additionally, more methods employing variance reduction have been considered to improve the performance of the stochastic minimax algorithms, e.g., see (Xu et al 2020;Huang, Wu, and Huang 2021;Luo et al 2020;Zhang, Aybat, and Gurbuzbalaban 2022). In this paper, to control the noise accumulation, we propose DGDA-VR, a decentralized method employing the SPIDER variance reduction technique (Fang et al 2018), a variant of SARAH (Nguyen et al 2017a).…”

Section: Related Workmentioning

confidence: 99%

Jointly Improving the Sample and Communication Complexities in Decentralized Stochastic Minimax Optimization

Zhang,

Mancino-Ball,

Aybat

et al. 2024

AAAI

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We propose a novel single-loop decentralized algorithm, DGDA-VR, for solving the stochastic nonconvex strongly-concave minimax problems over a connected network of agents, which are equipped with stochastic first-order oracles to estimate their local gradients. DGDA-VR, incorporating variance reduction, achieves O(ε^−3) oracle complexity and O(ε^−2) communication complexity without resorting to multi-communication rounds – both are optimal, i.e., matching the lower bounds for this class of problems. Since DGDA-VR does not require multiple communication rounds, it is applicable to a broader range of decentralized computational environments. To the best of our knowledge, this is the first distributed method using a single communication round in each iteration to jointly optimize the oracle and communication complexities for the problem considered here.

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Section: Related Workmentioning

confidence: 99%

Jointly Improving the Sample and Communication Complexities in Decentralized Stochastic Minimax Optimization

Zhang,

Mancino-Ball,

Aybat

et al. 2024

AAAI

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“…Subsequently, accelerated gradient descent ascent methods (Luo et al 2020;Xu et al 2020) emerged, leveraging variance reduction techniques to tackle stochastic minimax problems based on the variance reduction techniques. Additionally, research efforts have been made to explore non-smooth nonconvex-strongly-concave minimax optimization (Huang, Gao, and Huang 2020;Chen et al 2020). Moreover, (Huang, Gao, and Huang 2020) proposed the Riemannian stochastic gradient descent ascent method and some variants for the Riemannian minimax optimization problem.…”

Section: Related Workmentioning

confidence: 99%

“…Approaches such as Stochastic Gradient Descent Ascent (SGDA) have been proposed (Lin, Jin, and Jordan 2020), accompanied by innovations like variance-reduced SGDA (Luo et al 2020;Xu et al 2020) that aim to expedite convergence rates. Moreover, the application of Riemannian manifold-based optimization has been explored (Huang, Gao, and Huang 2020) across different minimax scenarios, showcasing the breadth of methodologies available. However, all of these methods are only designed for the non-compositional problem.…”

Section: Introductionmentioning

confidence: 99%

“…The theoretical result shows that our proposed NSTORM method can achieve the optimal sample complexity of O(κ 3 /ϵ 3 ). To the best of our knowledge, NSTORM is the first method to match the best sample complexity in existing minimax optimization studies (Huang, Wu, and Hu 2023;Luo et al 2020) without requiring a large batch size. We also demonstrate that NSTORM can achieve the same sample complexity under the Polyak-Łojasiewicz (PL)-condition, which indicates an insightful extension of NSTORM.…”

Section: Introductionmentioning

confidence: 99%

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Faster Stochastic Variance Reduction Methods for Compositional MiniMax Optimization

Liu,

Pan,

Duan

et al. 2024

AAAI

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This paper delves into the realm of stochastic optimization for compositional minimax optimization—a pivotal challenge across various machine learning domains, including deep AUC and reinforcement learning policy evaluation. Despite its significance, the problem of compositional minimax optimization is still under-explored. Adding to the complexity, current methods of compositional minimax optimization are plagued by sub-optimal complexities or heavy reliance on sizable batch sizes. To respond to these constraints, this paper introduces a novel method, called Nested STOchastic Recursive Momentum (NSTORM), which can achieve the optimal sample complexity and obtain the nearly accuracy solution, matching the existing minimax methods. We also demonstrate that NSTORM can achieve the same sample complexity under the Polyak-Lojasiewicz (PL)-condition—an insightful extension of its capabilities. Yet, NSTORM encounters an issue with its requirement for low learning rates, potentially constraining its real-world applicability in machine learning. To overcome this hurdle, we present ADAptive NSTORM (ADA-NSTORM) with adaptive learning rates. We demonstrate that ADA-NSTORM can achieve the same sample complexity but the experimental results show its more effectiveness. All the proposed complexities indicate that our proposed methods can match lower bounds to existing minimax optimizations, without requiring a large batch size in each iteration. Extensive experiments support the efficiency of our proposed methods.

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“…More recently, Huang et al [2023], Junchi et al [2022], presented some efficient adaptive stochastic GDA methods for NC-SC minimax optimization. Meanwhile, Lu et al [2020], Chen et al [2021], Huang et al [2021b] proposed the effective and efficient two-timescale proximal GDA methods to solve NC-SC minimax problems with nonsmooth regularization. In general, the two-timescale GDA methods for minimax optimization can be more easily implemented than multi-step GDA methods, and perform better than the multi-step GDA methods in practice.…”

Section: Algorithmmentioning

confidence: 99%

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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Gradient Descent Ascent for Minimax Problems on Riemannian Manifolds

Cited by 13 publications

References 19 publications

Jointly Improving the Sample and Communication Complexities in Decentralized Stochastic Minimax Optimization

Jointly Improving the Sample and Communication Complexities in Decentralized Stochastic Minimax Optimization

Faster Stochastic Variance Reduction Methods for Compositional MiniMax Optimization

Smellscapes of Nanjing Road

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