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A line of recent work has characterized the behavior of the EM algorithm in favorable settings in which the population likelihood is locally strongly concave around its maximizing argument. Examples include suitably separated Gaussian mixture models and mixtures of linear regressions. We consider instead over-fitted settings in which the likelihood need not be strongly concave, or, equivalently, when the Fisher information matrix might be singular. In such settings, it is known that a global maximum of the MLE based on n samples can have a non-standard n −1/4 rate of convergence. How does the EM algorithm behave in such settings? Focusing on the simple setting of a two-component mixture fit to a multivariate Gaussian distribution, we study the behavior of the EM algorithm both when the mixture weights are different (unbalanced case), and are equal (balanced case). Our analysis reveals a sharp distinction between these cases: in the former, the EM algorithm converges geometrically to a point at Euclidean distance O((d/n) 1/2 ) from the true parameter, whereas in the latter case, the convergence rate is exponentially slower, and the fixed point has a much lower O((d/n) 1/4 ) accuracy. The slower convergence in the balanced over-fitted case arises from the singularity of the Fisher information matrix. Analysis of this singular case requires the introduction of some novel analysis techniques, in particular we make use of a careful form of localization in the associated empirical process, and develop a recursive argument to progressively sharpen the statistical rate.

Hamiltonian Monte Carlo (HMC) is a state-of-the-art Markov chain Monte Carlo sampling algorithm for drawing samples from smooth probability densities over continuous spaces. We study the variant most widely used in practice, Metropolized HMC with the Störmer-Verlet or leapfrog integrator, and make two primary contributions. First, we provide a non-asymptotic upper bound on the mixing time of the Metropolized HMC with explicit choices of stepsize and number of leapfrog steps. This bound gives a precise quantification of the faster convergence of Metropolized HMC relative to simpler MCMC algorithms such as the Metropolized random walk, or Metropolized Langevin algorithm. Second, we provide a general framework for sharpening mixing time bounds Markov chains initialized at a substantial distance from the target distribution over continuous spaces. We apply this sharpening device to the Metropolized random walk and Langevin algorithms, thereby obtaining improved mixing time bounds from a non-warm initial distribution.

Consider an infinite sequence of independent, uniformly chosen points from r0, 1s d . After looking at each point in the sequence, an overseer is allowed to either keep it or reject it, and this choice may depend on the locations of all previously kept points. However, the overseer must keep at least one of every two consecutive points. We call a sequence generated in this fashion a two-thinning sequence. Here, the purpose of the overseer is to control the discrepancy of the empirical distribution of points, that is, after selecting n points, to reduce the maximal deviation of the number of points inside any axis-parallel hyper-rectangle of volume A from nA. Our main result is an explicit low complexity two-thinning strategy which guarantees discrepancy of Oplog 2d`1 nq for all n with high probability (compare with Θp ? n log log nq without thinning). The case d " 1 of this result answers a question of Benjamini.We also extend the construction to achieve the same asymptotic bound for (1`β)-thinning, a set-up in which rejecting is only allowed with probability β independently for each point. In addition, we suggest an improved and simplified strategy which we conjecture to guarantee discrepancy of Oplog d`1 nq (compare with θplog d nq, the best known construction of a low discrepancy sequence). Finally, we provide theoretical and empirical evidence for our conjecture, and provide simulations supporting the viability of our construction for applications.