Logistic regression with weight grouping priors

A generalization of the commonly used Maximum Likelihood based learning algorithm for the logistic regression model is considered. It is well known that using the Laplace prior (L1 penalty) on model coefficients leads to a variable selection effect, when most of the coefficients vanish. It is argued that variable selection is not always desirable; it is often better to group correlated variables together and assign equal weights to them. Two new kinds of a-priori distributions over weights are investigated: Gaussian Extremal Mixture (GEM) and Laplacian Extremal Mixture (LEM) which enforce grouping of model coefficients in a manner analogous to L1 and L2 regularization. An efficient learning algorithm is presented, which simultaneously finds model weights and the hyperparameters of those priors. Examples are shown in the experimental part where the proposed a-priori distributions outperform Gauss and Laplace priors as well as other methods which take coefficient grouping into account, such as the elastic net. Theoretical results on parameter shrinkage and sample complexity are also included.