12,799 research outputs found
-MLE: A fast algorithm for learning statistical mixture models
We describe -MLE, a fast and efficient local search algorithm for learning
finite statistical mixtures of exponential families such as Gaussian mixture
models. Mixture models are traditionally learned using the
expectation-maximization (EM) soft clustering technique that monotonically
increases the incomplete (expected complete) likelihood. Given prescribed
mixture weights, the hard clustering -MLE algorithm iteratively assigns data
to the most likely weighted component and update the component models using
Maximum Likelihood Estimators (MLEs). Using the duality between exponential
families and Bregman divergences, we prove that the local convergence of the
complete likelihood of -MLE follows directly from the convergence of a dual
additively weighted Bregman hard clustering. The inner loop of -MLE can be
implemented using any -means heuristic like the celebrated Lloyd's batched
or Hartigan's greedy swap updates. We then show how to update the mixture
weights by minimizing a cross-entropy criterion that implies to update weights
by taking the relative proportion of cluster points, and reiterate the mixture
parameter update and mixture weight update processes until convergence. Hard EM
is interpreted as a special case of -MLE when both the component update and
the weight update are performed successively in the inner loop. To initialize
-MLE, we propose -MLE++, a careful initialization of -MLE guaranteeing
probabilistically a global bound on the best possible complete likelihood.Comment: 31 pages, Extend preliminary paper presented at IEEE ICASSP 201
Differentially Private Empirical Risk Minimization with Sparsity-Inducing Norms
Differential privacy is concerned about the prediction quality while
measuring the privacy impact on individuals whose information is contained in
the data. We consider differentially private risk minimization problems with
regularizers that induce structured sparsity. These regularizers are known to
be convex but they are often non-differentiable. We analyze the standard
differentially private algorithms, such as output perturbation, Frank-Wolfe and
objective perturbation. Output perturbation is a differentially private
algorithm that is known to perform well for minimizing risks that are strongly
convex. Previous works have derived excess risk bounds that are independent of
the dimensionality. In this paper, we assume a particular class of convex but
non-smooth regularizers that induce structured sparsity and loss functions for
generalized linear models. We also consider differentially private Frank-Wolfe
algorithms to optimize the dual of the risk minimization problem. We derive
excess risk bounds for both these algorithms. Both the bounds depend on the
Gaussian width of the unit ball of the dual norm. We also show that objective
perturbation of the risk minimization problems is equivalent to the output
perturbation of a dual optimization problem. This is the first work that
analyzes the dual optimization problems of risk minimization problems in the
context of differential privacy
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