Convergence of the Chambolle-Pock Algorithm in the Absence of Monotonicity

• URL: http://arxiv.org/abs/2312.06540v2
• Date: Tue, 9 Jul 2024 14:51:36 GMT
• Title: Convergence of the Chambolle-Pock Algorithm in the Absence of Monotonicity
• Authors: Brecht Evens, Puya Latafat, Panagiotis Patrinos,
• Abstract summary: The Chambolle-Pock algorithm (CPA) has gained popularity over the last decade due to its success in solving large-scale convex structured problems. This work extends its convergence analysis for problems with varying degrees of (non)monotonicity, quantified through a so-called weak Minty condition on the associated primal-dual operator.
• Score: 4.307128674848627
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Chambolle-Pock algorithm (CPA), also known as the primal-dual hybrid gradient method, has gained popularity over the last decade due to its success in solving large-scale convex structured problems. This work extends its convergence analysis for problems with varying degrees of (non)monotonicity, quantified through a so-called oblique weak Minty condition on the associated primal-dual operator. Our results reveal novel stepsize and relaxation parameter ranges which do not only depend on the norm of the linear mapping, but also on its other singular values. In particular, in nonmonotone settings, in addition to the classical stepsize conditions, extra bounds on the stepsizes and relaxation parameters are required. On the other hand, in the strongly monotone setting, the relaxation parameter is allowed to exceed the classical upper bound of two. Moreover, we build upon the recently introduced class of semimonotone operators, providing sufficient convergence conditions for CPA when the individual operators are semimonotone. Since this class of operators encompasses traditional operator classes including (hypo)- and co(hypo)-monotone operators, this analysis recovers and extends existing results for CPA. Tightness of the proposed stepsize ranges is demonstrated through several examples.

Related papers

• On the Hypomonotone Class of Variational Inequalities [4.204990010424083]
  We study the behavior of the extragradient algorithm when applied to hypomonotone operators. We provide a characterization theorem that identifies the conditions under which the extragradient algorithm fails to converge.
  arXiv  Detail & Related papers  (2024-10-11T18:35:48Z)
• Variance Reduced Halpern Iteration for Finite-Sum Monotone Inclusions [18.086061048484616]
  We study finite-sum monotone inclusion problems, which model broad classes of equilibrium problems. Our main contributions are variants of the classical Halpern iteration that employ variance reduction to obtain improved complexity guarantees. We argue that, up to poly-logarithmic factors, this complexity is unimprovable in the monotone Lipschitz setting.
  arXiv  Detail & Related papers  (2023-10-04T17:24:45Z)
• Single-Call Stochastic Extragradient Methods for Structured Non-monotone Variational Inequalities: Improved Analysis under Weaker Conditions [21.681130192954583]
  Single-call extragradient methods are one of the most efficient algorithms for solving large-scale min-max optimization problems. We provide convergence guarantees for two classes of VIPs: (i) quasi-strongly monotone problems (a generalization of strongly monotone problems) and (ii) weak Minty variational inequalities (a generalization of monotone and Minty VIPs) Our convergence analysis holds under the arbitrary sampling paradigm, which includes importance sampling and various mini-batching strategies as special cases.
  arXiv  Detail & Related papers  (2023-02-27T18:53:28Z)
• High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance [59.211456992422136]
  We propose algorithms with high-probability convergence results under less restrictive assumptions. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes in optimization.
  arXiv  Detail & Related papers  (2023-02-02T10:37:23Z)
• A Primal-Dual Approach to Solving Variational Inequalities with General Constraints [54.62996442406718]
  Yang et al. (2023) recently showed how to use first-order gradient methods to solve general variational inequalities. We prove the convergence of this method and show that the gap function of the last iterate of the method decreases at a rate of $O(frac1sqrtK)$ when the operator is $L$-Lipschitz and monotone.
  arXiv  Detail & Related papers  (2022-10-27T17:59:09Z)
• Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noise [64.85879194013407]
  We prove the first high-probability results with logarithmic dependence on the confidence level for methods for solving monotone and structured non-monotone VIPs. Our results match the best-known ones in the light-tails case and are novel for structured non-monotone problems. In addition, we numerically validate that the gradient noise of many practical formulations is heavy-tailed and show that clipping improves the performance of SEG/SGDA.
  arXiv  Detail & Related papers  (2022-06-02T15:21:55Z)
• Optimal variance-reduced stochastic approximation in Banach spaces [114.8734960258221]
  We study the problem of estimating the fixed point of a contractive operator defined on a separable Banach space. We establish non-asymptotic bounds for both the operator defect and the estimation error.
  arXiv  Detail & Related papers  (2022-01-21T02:46:57Z)
• Stochastic Gradient Descent-Ascent and Consensus Optimization for Smooth Games: Convergence Analysis under Expected Co-coercivity [49.66890309455787]
  We introduce the expected co-coercivity condition, explain its benefits, and provide the first last-iterate convergence guarantees of SGDA and SCO. We prove linear convergence of both methods to a neighborhood of the solution when they use constant step-size. Our convergence guarantees hold under the arbitrary sampling paradigm, and we give insights into the complexity of minibatching.
  arXiv  Detail & Related papers  (2021-06-30T18:32:46Z)
• Lipschitz Bounded Equilibrium Networks [3.2872586139884623]
  This paper introduces new parameterizations of equilibrium neural networks, i.e. networks defined by implicit equations. The new parameterization admits a Lipschitz bound during training via unconstrained optimization. In image classification experiments we show that the Lipschitz bounds are very accurate and improve robustness to adversarial attacks.
  arXiv  Detail & Related papers  (2020-10-05T01:00:40Z)
• Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
  We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
  arXiv  Detail & Related papers  (2020-06-03T17:54:43Z)
• On operator growth and emergent Poincar\'e symmetries [0.0]
  We consider operator growth for generic large-N gauge theories at finite temperature. The algebra of these modes allows for a simple analysis of the operators with whom the initial operator mixes over time. We show all these approaches have a natural formulation in terms of the Gelfand-Naimark-Segal (GNS) construction.
  arXiv  Detail & Related papers  (2020-02-10T15:29:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.