Quantitative Weak Convergence for Discrete Stochastic Processes

Xiang Cheng; Peter L. Bartlett; Michael I. Jordan

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arXiv · 1902.00832

Quantitative Weak Convergence for Discrete Stochastic Processes

Xiang Cheng·Peter L. Bartlett·Michael I. Jordan

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Abstract

In this paper, we quantitative convergence in $W_2$ for a family of Langevin-like stochastic processes that includes stochastic gradient descent and related gradient-based algorithms. Under certain regularity assumptions, we show that the iterates of these stochastic processes converge to an invariant distribution at a rate of $\tilde{O}\lrp{1/\sqrt{k}}$ where $k$ is the number of steps; this rate is provably tight up to log factors. Our result reduces to a quantitative form of the classical Central Limit Theorem in the special case when the potential is quadratic.

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BibTeX RIS

Xiang Cheng, Peter L. Bartlett, Michael I. Jordan. 2019-02-03. Quantitative Weak Convergence for Discrete Stochastic Processes. https://arxiv.org/abs/1902.00832

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