Strong error analysis for stochastic gradient descent optimization algorithms

TL;DR

This paper provides a rigorous strong error analysis for stochastic gradient descent algorithms, proving convergence in the strong L^p sense with near-optimal order under standard assumptions, using Lyapunov functions.

Contribution

It introduces a novel convergence framework for SGD based on Lyapunov functions, achieving strong L^p convergence rates for arbitrary p and small epsilon.

Findings

Proves strong L^p convergence of SGD with order 1/2 - epsilon.

Develops a general convergence machinery using Lyapunov functions.

Achieves convergence results under relaxed moment conditions.

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 $\varepsilon \in (0,\infty)$ and every arbitrarily large $p\in (0,\infty)$ that the considered SGD optimization algorithm converges in the strong $L^p$-sense with order $\frac{1}{2}-\varepsilon$ 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 Lyapunov-type functions in order to achieve for every arbitrarily large $ p \in (0,\infty) $ strong $ L^p $-convergence rates. This article also contains an extensive review of results on SGD optimization algorithms in the scientific literature.