Fast Incremental Expectation Maximization for finite-sum optimization: nonasymptotic convergence

TL;DR

This paper introduces nonasymptotic convergence bounds for the Fast Incremental Expectation Maximization (FIEM) algorithm, improving theoretical rates and providing practical strategies for large-scale finite-sum optimization problems.

Contribution

The paper recasts FIEM within a stochastic approximation framework and derives nonasymptotic convergence bounds, achieving better rates and practical strategies for large datasets.

Findings

Achieves convergence rate scaling as  for xamples, better than previous n^{2/3} rate.

Provides two strategies for psilon-approximate stationary points with different iteration complexities.

Numerical results show improved step size choices and convergence control.

Abstract

Fast Incremental Expectation Maximization (FIEM) is a version of the EM framework for large datasets. In this paper, we first recast FIEM and other incremental EM type algorithms in the {\em Stochastic Approximation within EM} framework. Then, we provide nonasymptotic bounds for the convergence in expectation as a function of the number of examples $n$ and of the maximal number of iterations $\kmax$. We propose two strategies for achieving an $\epsilon$-approximate stationary point, respectively with $\kmax = O(n^{2/3}/\epsilon)$ and $\kmax = O(\sqrt{n}/\epsilon^{3/2})$, both strategies relying on a random termination rule before $\kmax$ and on a constant step size in the Stochastic Approximation step. Our bounds provide some improvements on the literature. First, they allow $\kmax$ to scale as $\sqrt{n}$ which is better than $n^{2/3}$ which was the best rate obtained so far; it is at the cost of a larger dependence upon the tolerance $\epsilon$, thus making this control relevant for small to medium accuracy with respect to the number of examples $n$. Second, for the $n^{2/3}$-rate, the numerical illustrations show that thanks to an optimized choice of the step size and of the bounds in terms of quantities characterizing the optimization problem at hand, our results desig a less conservative choice of the step size and provide a better control of the convergence in expectation.