A Fixed-Point of View on Gradient Methods for Big Data

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2017
Major/Subject
Mcode
Degree programme
Language
en
Pages
11
1-11
Series
Frontiers in Applied Mathematics and Statistics, Volume 3
Abstract
Interpreting gradient methods as fixed-point iterations, we provide a detailed analysis of those methods for minimizing convex objective functions. Due to their conceptual and algorithmic simplicity, gradient methods are widely used in machine learning for massive datasets (big data). In particular, stochastic gradient methods are considered the defacto standard for training deep neural networks. Studying gradient methods within the realm of fixed-point theory provides us with powerful tools to analyze their convergence properties. In particular, gradient methods using inexact or noisy gradients, such as stochastic gradient descent, can be studied conveniently using well-known results on inexact fixed-point iterations. Moreover, as we demonstrate in this paper, the fixed-point approach allows an elegant derivation of accelerations for basic gradient methods. In particular, we will show how gradient descent can be accelerated by a fixed-point preserving transformation of an operator associated with the objective function.
Description
Keywords
convex optimization, fixed point theory, big data, machine learning, contraction mapping, gradient descent, heavy balls
Citation
Jung , A 2017 , ' A Fixed-Point of View on Gradient Methods for Big Data ' , Frontiers in Applied Mathematics and Statistics , vol. 3 , pp. 1-11 . https://doi.org/10.3389/fams.2017.00018