How well do SEM algorithms imitate EM algorithms? A non-asymptotic analysis for mixture models

In this paper, we present a theoretical and an experimental comparison of EM and SEM algorithms for different mixture models. The SEM algorithm is a stochastic variant of the EM algorithm. The qualitative intuition behind the SEM algorithm is simple: If the number of observations is large enough, then we expect that an update step of the stochastic SEM algorithm is similar to the corresponding update step of the deterministic EM algorithm. In this paper, we quantify this intuition. We show that with high probability the update equations of any EM-like algorithm and its stochastic variant are similar, given that the input set satisfies certain properties. For instance, this result applies to the well-known EM and SEM algorithm for Gaussian mixture models and EM-like and SEM-like heuristics for multivariate power exponential distributions. Our experiments confirm that our theoretical results also hold for a large number of successive update steps. Thereby we complement the known asymptotic results for the SEM algorithm. We also show that, for multivariate Gaussian and multivariate Laplacian mixture models, an update step of SEM runs nearly twice as fast as an EM update set.