Momentum and Stochastic Momentum for Stochastic Gradient, Newton, Proximal Point and Subspace Descent Methods
| Autor: | Nicolas Loizou, Peter Richtárik |
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| Rok vydání: | 2017 |
| Předmět: |
FOS: Computer and information sciences
Control and Optimization 0211 other engineering and technologies Machine Learning (stat.ML) 010103 numerical & computational mathematics 02 engineering and technology 01 natural sciences Machine Learning (cs.LG) Statistics - Machine Learning FOS: Mathematics Applied mathematics Quadratic programming Mathematics - Numerical Analysis 0101 mathematics Mathematics - Optimization and Control Mathematics Momentum (technical analysis) 021103 operations research Applied Mathematics Linear system Computer Science - Numerical Analysis Numerical Analysis (math.NA) Computational Mathematics Computer Science - Learning Stochastic gradient descent Rate of convergence Iterated function Optimization and Control (math.OC) Convex optimization Stochastic optimization |
| DOI: | 10.48550/arxiv.1712.09677 |
| Popis: | In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these methods are studied. We choose to perform our analysis in a setting in which all of the above methods are equivalent. We prove global nonassymptotic linear convergence rates for all methods and various measures of success, including primal function values, primal iterates (in L2 sense), and dual function values. We also show that the primal iterates converge at an accelerated linear rate in the L1 sense. This is the first time a linear rate is shown for the stochastic heavy ball method (i.e., stochastic gradient descent method with momentum). Under somewhat weaker conditions, we establish a sublinear convergence rate for Cesaro averages of primal iterates. Moreover, we propose a novel concept, which we call stochastic momentum, aimed at decreasing the cost of performing the momentum step. We prove linear convergence of several stochastic methods with stochastic momentum, and show that in some sparse data regimes and for sufficiently small momentum parameters, these methods enjoy better overall complexity than methods with deterministic momentum. Finally, we perform extensive numerical testing on artificial and real datasets, including data coming from average consensus problems. Comment: 47 pages, 7 figures, 7 tables |
| Databáze: | OpenAIRE |
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