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Adaptive Restart of the Optimized Gradient Method for Convex Optimization.

Donghwan Kim1, Jeffrey A Fessler1

  • 1University of Michigan, Ann Arbor, MI.

Journal of Optimization Theory and Applications
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PubMed
Summary
This summary is machine-generated.

Adaptive restart accelerates the optimized gradient method, a first-order optimization algorithm. This technique improves convergence for convex optimization problems, even for nonsmooth composite functions.

Keywords:
80M5090C0690C25Accelerated gradient methodConvex optimizationFirst-order algorithmsOptimized gradient methodRestarting

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Area of Science:

  • Optimization Algorithms
  • Convex Optimization
  • Numerical Analysis

Background:

  • First-order methods like Nesterov's fast gradient method are effective for convex problems but can oscillate.
  • Adaptive restarting enhances convergence for fast gradient methods, especially with unknown parameters or ill-conditioned regions.
  • The optimized gradient method offers improved convergence bounds over fast gradient methods for large-dimensional smooth convex problems.

Purpose of the Study:

  • To investigate adaptive restart as a heuristic acceleration technique for the optimized gradient method.
  • To derive new step coefficients for the optimized gradient method in strongly convex quadratic problems.
  • To demonstrate the effectiveness of adaptive restart for both smooth and nonsmooth composite convex functions.

Main Methods:

  • Derivation of new step coefficients for the optimized gradient method.
  • Heuristic analysis of adaptive restart's impact on the optimized gradient method.
  • Numerical experiments comparing accelerated and non-accelerated optimized gradient methods.

Main Results:

  • New step coefficients yield faster convergence for the optimized gradient method compared to its standard version.
  • Adaptive restart significantly accelerates the convergence of the optimized gradient method.
  • Adaptive restart proves beneficial for a proximal version of the optimized gradient method applied to nonsmooth composite functions.

Conclusions:

  • Adaptive restart is a viable and effective heuristic for accelerating the optimized gradient method.
  • The optimized gradient method, when combined with adaptive restart, offers superior convergence properties.
  • This acceleration strategy shows promise for a broader range of convex optimization problems, including nonsmooth ones.