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Global convergence of Langevin dynamics based algorithms for nonconvex optimization

Publication ,  Conference
Xu, P; Zou, D; Chen, J; Gu, Q
Published in: Advances in Neural Information Processing Systems
January 1, 2018

We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with n component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical approximations to Langevin dynamics, which leads to faster convergence rates. Specifically, we show that gradient Langevin dynamics (GLD) and stochastic gradient Langevin dynamics (SGLD) converge to the almost minimizer2 within Õe(nd/(λε)) and Õe(d7/(λ5ε5)) stochastic gradient evaluations respectively3, where d is the problem dimension, and λ is the spectral gap of the Markov chain generated by GLD. Both results improve upon the best known gradient complexity4 results [45]. Furthermore, for the first time we prove the global convergence guarantee for variance reduced stochastic gradient Langevin dynamics (SVRG-LD) to the almost minimizer within Õe(pnd5/(λ4ε5/2)) stochastic gradient evaluations, which outperforms the gradient complexities of GLD and SGLD in a wide regime. Our theoretical analyses shed some light on using Langevin dynamics based algorithms for nonconvex optimization with provable guarantees.

Duke Scholars

Published In

Advances in Neural Information Processing Systems

ISSN

1049-5258

Publication Date

January 1, 2018

Volume

2018-December

Start / End Page

3122 / 3133

Related Subject Headings

  • 4611 Machine learning
  • 1702 Cognitive Sciences
  • 1701 Psychology
 

Citation

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ICMJE
MLA
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Xu, P., Zou, D., Chen, J., & Gu, Q. (2018). Global convergence of Langevin dynamics based algorithms for nonconvex optimization. In Advances in Neural Information Processing Systems (Vol. 2018-December, pp. 3122–3133).
Xu, P., D. Zou, J. Chen, and Q. Gu. “Global convergence of Langevin dynamics based algorithms for nonconvex optimization.” In Advances in Neural Information Processing Systems, 2018-December:3122–33, 2018.
Xu P, Zou D, Chen J, Gu Q. Global convergence of Langevin dynamics based algorithms for nonconvex optimization. In: Advances in Neural Information Processing Systems. 2018. p. 3122–33.
Xu, P., et al. “Global convergence of Langevin dynamics based algorithms for nonconvex optimization.” Advances in Neural Information Processing Systems, vol. 2018-December, 2018, pp. 3122–33.
Xu P, Zou D, Chen J, Gu Q. Global convergence of Langevin dynamics based algorithms for nonconvex optimization. Advances in Neural Information Processing Systems. 2018. p. 3122–3133.

Published In

Advances in Neural Information Processing Systems

ISSN

1049-5258

Publication Date

January 1, 2018

Volume

2018-December

Start / End Page

3122 / 3133

Related Subject Headings

  • 4611 Machine learning
  • 1702 Cognitive Sciences
  • 1701 Psychology