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

Frontiers in Applied Mathematics and Statistics

Jung, Alexander¹ 

[1] Department of Computer Science, Aalto University, Espoo, Finland

关键词: convex optimization;    Gradient methods;    Fixed-point iterations;    big data;    machine learning;    heavy balls;    First order methods;   

DOI  :  10.3389/fams.2017.00018

学科分类：数学（综合）

来源: Frontiers

【 摘 要 】

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.

【 授权许可】

CC BY