Penalized Bregman divergence for large-dimensional regression and classification.

Regularization methods are characterized by loss functions measuring data fits and penalty terms constraining model parameters. The commonly used quadratic loss is not suitable for classification with binary responses, whereas the loglikelihood function is not readily applicable to models where the exact distribution of observations is unknown or not fully specified. We introduce the penalized Bregman divergence by replacing the negative loglikelihood in the conventional penalized likelihood with Bregman divergence, which encompasses many commonly used loss functions in the regression analysis, classification procedures and machine learning literature. We investigate new statistical properties of the resulting class of estimators with the number p(n) of parameters either diverging with the sample size n or even nearly comparable with n, and develop statistical inference tools. It is shown that the resulting penalized estimator, combined with appropriate penalties, achieves the same oracle property as the penalized likelihood estimator, but asymptotically does not rely on the complete specification of the underlying distribution. Furthermore, the choice of loss function in the penalized classifiers has an asymptotically relatively negligible impact on classification performance. We illustrate the proposed method for quasilikelihood regression and binary classification with simulation evaluation and real-data application.