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Stochastic momentum methods for non-convex learning without bounded assumptions.

Yuqing Liang1, Jinlan Liu1, Dongpo Xu1

  • 1Key Laboratory for Applied Statistics of MOE, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China.

Neural Networks : the Official Journal of the International Neural Network Society
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Summary

This study presents unified convergence rates for stochastic momentum methods on non-convex problems, relaxing prior assumptions. It offers improved stepsize schemes and validates findings with numerical experiments.

Keywords:
Last-iterate convergence rateMachine learningNon-convex optimizationPL conditionStochastic momentum methods

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Area of Science:

  • Machine Learning
  • Optimization Theory

Background:

  • Stochastic momentum methods are crucial for machine learning optimization.
  • Existing analyses often require restrictive bounded assumptions or strong stepsize conditions.
  • Non-convex optimization presents significant theoretical challenges.

Purpose of the Study:

  • To provide a unified convergence rate analysis for stochastic momentum methods.
  • To analyze methods like stochastic heavy ball (SHB) and stochastic Nesterov accelerated gradient (SNAG).
  • To relax existing theoretical assumptions, focusing on the Polyak-Łojasiewicz (PL) condition.

Main Methods:

  • Developed a theoretical framework for analyzing stochastic momentum methods.
  • Introduced the relaxed growth (RG) condition, a weaker assumption than prior work.
  • Examined convergence rates under diminishing and constant stepsize schemes.

Main Results:

  • Achieved last-iterate convergence rate for function values under the RG condition.
  • Attained sub-linear convergence for diminishing stepsizes and linear convergence for constant stepsizes (under strong growth).
  • Proposed a more flexible stepsize scheme, relaxing limitations on convergence.

Conclusions:

  • The proposed analysis offers a more unified and less restrictive theoretical foundation for stochastic momentum methods.
  • The relaxed assumptions and flexible stepsize schemes enhance the applicability of these methods.
  • Numerical experiments confirm the theoretical findings on benchmark datasets.