This paper develops a unified framework, based on iterated random operator theory, to analyze the convergence of constant stepsize recursive stochastic algorithms (RSAs). RSAs use randomization to efficiently compute expectations, and so their iterates form a stochastic process. The key idea of our analysis is to lift the RSA into an appropriate higher-dimensional space and then express it as an equivalent Markov chain. Instead of determining the convergence of this Markov chain (which may not converge under constant stepsize), we study the convergence of the distribution of this Markov chain. To study this, we define a new notion of Wasserstein divergence. We show that if the distribution of the iterates in the Markov chain satisfy a contraction property with respect to the Wasserstein divergence, then the Markov chain admits an invariant distribution. We show that convergence of a large family of constant stepsize RSAs can be understood using this framework, and we provide several detailed examples.


  1. iterative random maps
  2. Wasserstein divergence
  3. stochastic gradient descent

MSC codes

  1. 93E35
  2. 60J20
  3. 68Q32

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Supplementary Material

PLEASE NOTE: These supplementary files have not been peer-reviewed.

Index of Supplementary Materials

Title of paper: Convergence of Recursive Stochastic Algorithms using Wasserstein Divergence

Authors: A. Gupta and W. Haskell

File: Supplement.pdf

Type: PDF

Contents: In this supplementary material, we derive the contraction coefficient for some algorithms that are covered in the paper.


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Information & Authors


Published In

cover image SIAM Journal on Mathematics of Data Science
SIAM Journal on Mathematics of Data Science
Pages: 1141 - 1167
ISSN (online): 2577-0187


Submitted: 4 January 2021
Accepted: 1 July 2021
Published online: 21 October 2021


  1. iterative random maps
  2. Wasserstein divergence
  3. stochastic gradient descent

MSC codes

  1. 93E35
  2. 60J20
  3. 68Q32



Funding Information

Advanced Research Projects Agency - Energy https://doi.org/10.13039/100006133

Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : ECCS 1610615

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