Volume 27, 2023
|Page(s)||482 - 514|
|Published online||12 April 2023|
Non-Asymptotic Analysis of Stochastic Approximation Algorithms for Streaming Data*
Sorbonne Université, Laboratoire de Probabilités, Statistique et Modélisation (LPSM), 75005 Paris, France
2 Wolfgang Pauli Institut, c/o Fakultät für Mathematik, Universität Wien, 1090 Vienna, Austria
** Corresponding author: email@example.com
Accepted: 3 March 2023
We introduce a streaming framework for analyzing stochastic approximation/optimization problems. This streaming framework is analogous to solving optimization problems using time-varying mini-batches that arrive sequentially. We provide non-asymptotic convergence rates of various gradientbased algorithms; this includes the famous Stochastic Gradient (SG) descent (a.k.a. Robbins-Monro algorithm), mini-batch SG and time-varying mini-batch SG algorithms, as well as their iterated averages (a.k.a. Polyak-Ruppert averaging). We show (i) how to accelerate convergence by choosing the learning rate according to the time-varying mini-batches, (ii) that Polyak-Ruppert averaging achieves optimal convergence in terms of attaining the Cramer-Rao lower bound, and (iii) how time-varying mini-batches together with Polyak-Ruppert averaging can provide variance reduction and accelerate convergence simultaneously, which is advantageous for many learning problems, such as online, sequential, and large-scale learning. We further demonstrate these favorable effects for various time-varying minibatches.
Mathematics Subject Classification: 62L12 / 62L20 / 68W27 / 90C25
Key words: Stochastic algorithms / stochastic optimization / machine learning / online learning / mini-batch / streaming
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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