Strong error analysis for stochastic gradient descent optimization algorithms
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Date
2018-01-31Type
- Report
ETH Bibliography
yes
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Abstract
Stochastic gradient descent (SGD) optimization algorithms are key ingredients in a series of machine learning applications. In this article we perform a rigorous strong error analysis for SGD optimization algorithms. In particular, we prove for every arbitrarily small ε ∈ (0,∞) and every arbitrarily large p ∈ (0,∞) that the considered SGD optimization algorithm converges in the strong Lp-sense with order 1/2 − ε to the global minimum of the objective function of the considered stochastic approximation problem under standard convexity-type assumptions on the objective function and relaxed assumptions on the moments of the stochastic errors appearing in the employed SGD optimization algorithm. The key ideas in our convergence proof are, first, to employ techniques from the theory of Lyapunov-type functions for dynamical systems to develop a general convergence machinery for SGD optimization algorithms based on such functions, then, to apply this general machinery to concrete Lyapunov-type functions with polynomial structures, and, thereafter, to perform an induction argument along the powers appearing in the Lyapunovtype functions in order to achieve for every arbitrarily large p ∈ (0,∞) strong Lp-convergence rates. This article also contains an extensive review of results on SGD optimization algorithms in the scientific literature. Show more
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publishedExternal links
Journal / series
SAM Research ReportVolume
Publisher
Seminar for Applied Mathematics, ETH ZurichOrganisational unit
03951 - Jentzen, Arnulf (ehemalig) / Jentzen, Arnulf (former)
02204 - RiskLab / RiskLab
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ETH Bibliography
yes
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