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Hoang Phuc Hau Luu1, Hoai Minh Le1, Hoai An Le Thi2

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Summary

This study introduces a novel algorithm for complex optimization problems with uncertain data, using Markov chains instead of independent samples. The method, Markov chain stochastic DCA, shows promise in deep learning applications.

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Area of Science:

  • Optimization Theory
  • Machine Learning
  • Stochastic Processes

Background:

  • Nonsmooth nonconvex stochastic difference-of-convex (DC) programs often involve endogenous uncertainty.
  • Standard methods assume independent and identically distributed (i.i.d.) samples, which are not always available.
  • Markovian noise is prevalent in Bayesian inference, reinforcement learning, and high-dimensional optimization.

Purpose of the Study:

  • To develop a stochastic algorithm for DC programs with Markovian noise.
  • To analyze the convergence properties of the proposed algorithm.
  • To apply the algorithm to deep learning problems using partial differential equations (PDEs) regularization.

Main Methods:

  • A novel algorithm, Markov chain stochastic DCA (MCSDCA), is designed based on the DC algorithm (DCA).
  • Convergence is analyzed in both asymptotic and nonasymptotic senses.
  • Two variants, MCSDCA-odLD and MCSDCA-udLD, are developed using overdamped and underdamped Langevin dynamics for PDE regularization in deep learning.

Main Results:

  • The MCSDCA algorithm is established for a class of nonsmooth nonconvex stochastic DC programs with Markovian noise.
  • Theoretical convergence guarantees are provided for the proposed method.
  • Numerical experiments demonstrate the effectiveness of MCSDCA variants on time series prediction and image classification tasks.

Conclusions:

  • The proposed MCSDCA algorithm effectively handles nonsmooth nonconvex stochastic DC programs with Markovian noise.
  • The application to deep learning via PDE regularization shows promising results in empirical evaluations.
  • The study contributes a robust optimization framework for scenarios lacking i.i.d. data.