#### A Stochastic Path Integral Differential EstimatoR Expectation Maximization Algorithm

Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)

#### Authors

*Gersende Fort, Eric Moulines, Hoi-To Wai*

#### Abstract

The Expectation Maximization (EM) algorithm is of key importance for inference in latent variable models including mixture of regressors and experts, missing observations. This paper introduces a novel EM algorithm, called {\tt SPIDER-EM}, for inference from a training set of size $n$, $n \gg 1$. At the core of our algorithm is an estimator of the full conditional expectation in the {\sf E}-step, adapted from the stochastic path integral differential estimator ({\tt SPIDER}) technique. We derive finite-time complexity bounds for smooth non-convex likelihood: we show that for convergence to an $\epsilon$-approximate stationary point, the complexity scales as $K_{Opt} (n,\epsilon )={\cal O}(\epsilon^{-1})$ and $K_{CE}( n,\epsilon ) = n+ \sqrt{n} {\cal O}( \epsilon^{-1} )$, where $K_{Opt}( n,\epsilon )$ and $K_{CE}(n, \epsilon )$ are respectively the number of {\sf M}-steps and the number of per-sample conditional expectations evaluations. This improves over the state-of-the-art algorithms. Numerical results support our findings.