Solving Nonconvex-Nonconcave Min-Max Problems exhibiting Weak Minty Solutions

Böhm, Axel

doi:10.48550/arxiv.2201.12247

Cited by 3 publications

(5 citation statements)

References 30 publications

(54 reference statements)

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“…Clearly, these updates of ν k and θ k are exactly shown in (8). Now, under the choice of parameters as in (12), (11) reduces to…”

Section: Convergence Analysismentioning

confidence: 85%

“…Note that we can choose different η such that Lη ≤ 1, but not necessary to be η = γ + 2ρ as in (8). The update rule (8) in Theorem 3.1 only provides one choice of η and γ and it makes our analysis simpler.…”

Section: Convergence Analysismentioning

confidence: 99%

“…Classical methods include gradient or forward, extragradient, past-extragradient, optimistic gradient, proximal-point, forward-backward splitting, forward-backward-forward splitting, Douglas-Rachford splitting, forward-reflected-backward splitting, golden ratio, projective splitting methods, and their variants, see, e.g., [5,16,19,21,38,24,43,45,56,65]. However, extensions to non-monotone settings remain limited, see, e.g., [8,13,23,33,39,54,53] for a few examples.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Extragradient-Type Methods with $\mathcal{O}(1/k)$ Convergence Rates for Co-Hypomonotone Inclusions

Tran-Dinh¹

2023

Preprint

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In this paper, we develop two "Nesterov's accelerated" variants of the well-known extragradient method to approximate a solution of a co-hypomonotone inclusion constituted by the sum of two operators, where one is Lipschitz continuous and the other is possibly multivalued. The first scheme can be viewed as an accelerated variant of Tseng's forwardbackward-forward splitting method, while the second one is a variant of the reflected forward-backward splitting method, which requires only one evaluation of the Lipschitz operator, and one resolvent of the multivalued operator. Under a proper choice of the algorithmic parameters and appropriate conditions on the co-hypomonotone parameter, we theoretically prove that both algorithms achieve O (1/k) convergence rates on the norm of the residual, where k is the iteration counter. Our results can be viewed as alternatives of a recent class of Halpern-type schemes for root-finding problems.

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“…Clearly, these updates of ν k and θ k are exactly shown in (8). Now, under the choice of parameters as in (12), (11) reduces to…”

Section: Convergence Analysismentioning

confidence: 85%

Section: Convergence Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Extragradient-Type Methods with $\mathcal{O}(1/k)$ Convergence Rates for Co-Hypomonotone Inclusions

Tran-Dinh¹

2023

Preprint

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“…There is a noticeable growing interest of the community in studying the theoretical convergence guarantees of deterministic methods for solving VIP with non-monotone operators F (x) having a certain structure, e.g., negative comonotonicty [Diakonikolas et al, 2021, Lee and Kim, 2021, Böhm, 2022, quasi-strong monotonicity [Song et al, 2020, Mertikopoulos andZhou, 2019] and/or star-cocoercivity [Loizou et al, 2021, Gorbunov et al, 2022b,a, Beznosikov et al, 2022. In the context of stochastic VIPs, SEG (with different extrapolation and update stepsizes) is analyzed under negative comonotonicity by Diakonikolas et al [2021] and under quasi-strong monotonicity by Gorbunov et al [2022a], while SGDA is studied under quasi-strong monotonicity and/or star-cocoercivity by [Loizou et al, 2021, Beznosikov et al, 2022.…”

Section: Structured Non-monotonicitymentioning

confidence: 99%

Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noise

Gorbunov¹,

Danilova²,

Dobre³

et al. 2022

Preprint

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Stochastic first-order methods such as Stochastic Extragradient (SEG) or Stochastic Gradient Descent-Ascent (SGDA) for solving smooth minimax problems and, more generally, variational inequality problems (VIP) have been gaining a lot of attention in recent years due to the growing popularity of adversarial formulations in machine learning. However, while high-probability convergence bounds are known to reflect the actual behavior of stochastic methods more accurately, most convergence results are provided in expectation. Moreover, the only known high-probability complexity results have been derived under restrictive sub-Gaussian (light-tailed) noise and bounded domain Assump. [Juditsky et al., 2011a]. In this work, we prove the first high-probability complexity results with logarithmic dependence on the confidence level for stochastic methods for solving monotone and structured non-monotone VIPs with non-sub-Gaussian (heavy-tailed) noise and unbounded domains. In the monotone case, our results match the best-known ones in the light-tails case [Juditsky et al., 2011a], and are novel for structured non-monotone problems such as negative comonotone, quasi-strongly monotone, and/or star-cocoercive ones. We achieve these results by studying SEG and SGDA with clipping. In addition, we numerically validate that the gradient noise of many practical GAN formulations is heavy-tailed and show that clipping improves the performance of SEG/SGDA.

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“…The development of efficient algorithms with provable convergence has recently been the focus of interest in machine learning, particularly in the unconstrained setting [e.g., Tseng, 1995, Daskalakis et al, 2018, Mokhtari et al, 2019, 2020, Golowich et al, 2020b, Azizian et al, 2020, Chavdarova et al, 2021a, Gorbunov et al, 2022, Bot et al, 2022, Bohm, 2022. Our focus is the constrained setting.…”

Section: Introductionmentioning

confidence: 99%

Solving Constrained Variational Inequalities via a First-order Interior Point-based Method

Yang¹,

Jordan²,

Chavdarova³

2022

Preprint

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We develop an interior-point approach to solve constrained variational inequality (cVI) problems. Inspired by the efficacy of the alternating direction method of multipliers (ADMM) method in the single-objective context, we generalize ADMM to derive a first-order method for cVIs, that we refer to as ADMM-based interior point method for constrained VIs (ACVI). We provide convergence guarantees for ACVI in two general classes of problems: (i) when the operator is ξ-monotone, and (ii) when it is monotone, the constraints are active and the game is not purely rotational. When the operator is in addition L-Lipschitz for the latter case, we match known lower bounds on rates for the gap function of O(1/ √ K) and O(1/K) for the last and average iterate, respectively. To the best of our knowledge, this is the first presentation of a first-order interior-point method for the general cVI problem that has a global convergence guarantee. Moreover, unlike previous work in this setting, ACVI provides a means to solve cVIs when the constraints are nontrivial. Empirical analyses demonstrate clear advantages of ACVI over common first-order methods. In particular, (i) cyclical behavior is notably reduced as our methods approach the solution from the analytic center, and (ii) unlike projection-based methods that oscillate when near a constraint, ACVI efficiently handles the constraints.

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Solving Nonconvex-Nonconcave Min-Max Problems exhibiting Weak Minty Solutions

Cited by 3 publications

References 30 publications

Extragradient-Type Methods with $\mathcal{O}(1/k)$ Convergence Rates for Co-Hypomonotone Inclusions

Extragradient-Type Methods with $\mathcal{O}(1/k)$ Convergence Rates for Co-Hypomonotone Inclusions

Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noise

Solving Constrained Variational Inequalities via a First-order Interior Point-based Method

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