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The Expectation Maximisation (EM) algorithm is widely used to optimisenon-convex likelihood functions with latent variables. Many authors modifiedits simple design to fit more specific situations. For instance, theExpectation (E) step has been replaced by Monte Carlo (MC), Markov Chain MonteCarlo or tempered approximations, etc. Most of the well-studied approximationsbelong to the stochastic class. By comparison, the literature is lacking whenit comes to deterministic approximations. In this paper, we introduce atheoretical framework, with state-of-the-art convergence guarantees, for anydeterministic approximation of the E step. We analyse theoretically andempirically several approximations that fit into this framework. First, forintractable E-steps, we introduce a deterministic version of MC-EM usingRiemann sums. A straightforward method, not requiring any hyper-parameterfine-tuning, useful when the low dimensionality does not warrant a MC-EM. Then,we consider the tempered approximation, borrowed from the Simulated Annealingliterature and used to escape local extrema. We prove that the tempered EMverifies the convergence guarantees for a wider range of temperature profilesthan previously considered. We showcase empirically how new non-trivialprofiles can more successfully escape adversarial initialisations. Finally, wecombine the Riemann and tempered approximations into a method that accomplishesboth their purposes.

Deterministic Approximate EM Algorithm; Application to the Riemann Approximation EM and the Tempered EM