Fast incremental expectation maximization for finite-sum optimization: nonasymptotic convergence

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 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 $K_\mathrm {max}$. We propose two strategies for achieving an $\epsilon$-approximate stationary point, respectively with $K_\mathrm {max}= O(n^{2/3}/\epsilon )$ and $K_\mathrm {max}= O(\sqrt{n}/\epsilon ^{3/2})$, both strategies relying on a random termination rule before $K_\mathrm {max}$ and on a constant step size in the Stochastic Approximation step. Our bounds provide some improvements on the literature. First, they allow $K_\mathrm {max}$ 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 design a less conservative choice of the step size and provide a better control of the convergence in expectation.