Learning Heteroscedastic Models by Convex Programming under Group Sparsity

Abstract

Popular sparse estimation methods based on ${\ell }_1$-relaxation, such as the Lasso and the Dantzig selector, require the knowledge of the variance of the noise in order to properly tune the regularization parameter. This constitutes a major obstacle in applying these methods in several frameworks---such as time series, random fields, inverse problems---for which the noise is rarely homoscedastic and its level is hard to know in advance. In this paper, we propose a new approach to the joint estimation of the conditional mean and the conditional variance in a high-dimensional (auto-) regression setting. An attractive feature of the proposed estimator is that it is efficiently computable even for very large scale problems by solving a second-order cone program (SOCP). We present theoretical analysis and numerical results assessing the performance of the proposed procedure.

Proceedings of the 30 th International Conference on Machine Learning (2013) http://icml.cc/2013/?page_id=43

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• Composite Self-Concordant Minimization
• Adaptive Lasso and group-Lasso for functional Poisson regression
• On estimation of the diagonal elements of a sparse precision matrix
• Heteroscedasticity-aware residuals-based contextual stochastic optimization
• Mean and variance estimation in high-dimensional heteroscedastic models with non-convex penalties