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 texttt{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-integrated 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 $K_{operatorname{Opt}} (n,epsilon )={cal O}(epsilon^{-1})$ and $K_{operatorname{CE}}( n,epsilon ) = n+ sqrt{n} {cal O}(epsilon^{-1} )$, where $K_{operatorname{Opt}}( n,epsilon )$ and $K_{operatorname{CE}}(n, epsilon )$ 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.
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