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Stochastic Primal-Dual Hybrid Gradient Algorithm with Adaptive Step Sizes.

Antonin Chambolle1,2, Claire Delplancke3, Matthias J Ehrhardt4

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This summary is machine-generated.

This study introduces adaptive step sizes for stochastic primal-dual hybrid gradient (SPDHG) algorithms, improving large-scale convex optimization. The new adaptive SPDHG (A-SPDHG) ensures convergence and offers practical parameter selection for enhanced performance.

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Area of Science:

  • Optimization Algorithms
  • Computational Science
  • Applied Mathematics

Background:

  • Stochastic primal-dual hybrid gradient (SPDHG) algorithms are widely used for large-scale convex optimization due to their scalability.
  • Convergence of SPDHG relies on an upper bound for the product of primal and dual step sizes.
  • Selecting optimal step size ratios for SPDHG remains a challenge in practical applications.

Purpose of the Study:

  • To develop a novel primal-dual algorithm with adaptive step sizes for convex optimization.
  • To introduce a general class of adaptive SPDHG (A-SPDHG) algorithms.
  • To provide systematic strategies for selecting step sizes in SPDHG to ensure convergence.

Main Methods:

  • Proposed a general class of adaptive SPDHG (A-SPDHG) algorithms.
  • Proved convergence properties of A-SPDHG under weak assumptions.
  • Developed concrete parameter-updating strategies for A-SPDHG.

Main Results:

  • Demonstrated the convergence of the proposed A-SPDHG algorithms.
  • Validated the effectiveness of the adaptive schemes through numerical examples.
  • Showcased successful application in computed tomography reconstruction.

Conclusions:

  • The proposed adaptive step size strategies for SPDHG algorithms enhance convergence and practical applicability.
  • A-SPDHG offers a robust solution for large-scale convex optimization problems.
  • The developed methods provide a systematic approach to step size selection, overcoming previous limitations.