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Register a new userRegularized Maximum Likelihood Estimation and Feature Selection in Mixtures-of-Experts Models
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Mixture of Experts (MoE) are successful models for modeling heterogeneous data in many statistical learning problems including regression, clustering and classification. Generally fitted by maximum likelihood estimation via the well-known EM algorithm, their application to high-dimensional problems is still therefore challenging. We consider the problem of fitting and feature selection in MoE models, and propose a regularized maximum likelihood estimation approach that encourages sparse solutions for heterogeneous regression data models with potentially high-dimensional predictors. Unlike state-of-the art regularized MLE for MoE, the proposed modelings do not require an approximate of the penalty function. We develop two hybrid EM algorithms: an Expectation-Majorization-Maximization (EM/MM) algorithm, and an EM algorithm with coordinate ascent algorithm. The proposed algorithms allow to automatically obtaining sparse solutions without thresholding, and avoid matrix inversion by allowing univariate parameter updates. An experimental study shows the good performance of the algorithms in terms of recovering the actual sparse solutions, parameter estimation, and clustering of heterogeneous regression data.
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Mixtures-of-Experts models and their maximum likelihood estimation (MLE) via the EM algorithm have been thoroughly studied in the statistics and machine learning literature. They are subject of a growing investigation in the context of modeling with high-dimensional predictors with regularized MLE. We examine MoE with Gaussian gating network, for clustering and regression, and propose an $ell_1$-regularized MLE to encourage sparse models and deal with the high-dimensional setting. We develop an EM-Lasso algorithm to perform parameter estimation and utilize a BIC-like criterion to select the model parameters, including the sparsity tuning hyperparameters. Experiments conducted on simulated data show the good performance of the proposed regularized MLE compared to the standard MLE with the EM algorithm.
Mixtures-of-Experts (MoE) are conditional mixture models that have shown their performance in modeling heterogeneity in data in many statistical learning approaches for prediction, including regression and classification, as well as for clustering. Their estimation in high-dimensional problems is still however challenging. We consider the problem of parameter estimation and feature selection in MoE models with different generalized linear experts models, and propose a regularized maximum likelihood estimation that efficiently encourages sparse solutions for heterogeneous data with high-dimensional predictors. The developed proximal-Newton EM algorithm includes proximal Newton-type procedures to update the model parameter by monotonically maximizing the objective function and allows to perform efficient estimation and feature selection. An experimental study shows the good performance of the algorithms in terms of recovering the actual sparse solutions, parameter estimation, and clustering of heterogeneous regression data, compared to the main state-of-the art competitors.
This article investigates the origin of numerical issues in maximum likelihood parameter estimation for Gaussian process (GP) interpolation and investigates simple but effective strategies for improving commonly used open-source software implementations. This work targets a basic problem but a host of studies, particularly in the literature of Bayesian optimization, rely on off-the-shelf GP implementations. For the conclusions of these studies to be reliable and reproducible, robust GP implementations are critical.
Marginal-likelihood based model-selection, even though promising, is rarely used in deep learning due to estimation difficulties. Instead, most approaches rely on validation data, which may not be readily available. In this work, we present a scalable marginal-likelihood estimation method to select both hyperparameters and network architectures, based on the training data alone. Some hyperparameters can be estimated online during training, simplifying the procedure. Our marginal-likelihood estimate is based on Laplaces method and Gauss-Newton approximations to the Hessian, and it outperforms cross-validation and manual-tuning on standard regression and image classification datasets, especially in terms of calibration and out-of-distribution detection. Our work shows that marginal likelihoods can improve generalization and be useful when validation data is unavailable (e.g., in nonstationary settings).
In unsupervised classification, Hidden Markov Models (HMM) are used to account for a neighborhood structure between observations. The emission distributions are often supposed to belong to some parametric family. In this paper, a semiparametric modeling where the emission distributions are a mixture of parametric distributions is proposed to get a higher flexibility. We show that the classical EM algorithm can be adapted to infer the model parameters. For the initialisation step, starting from a large number of components, a hierarchical method to combine them into the hidden states is proposed. Three likelihood-based criteria to select the components to be combined are discussed. To estimate the number of hidden states, BIC-like criteria are derived. A simulation study is carried out both to determine the best combination between the merging criteria and the model selection criteria and to evaluate the accuracy of classification. The proposed method is also illustrated using a biological dataset from the model plant Arabidopsis thaliana. A R package HMMmix is freely available on the CRAN.
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