On p-norm Path Following in Multiple Kernel Learning for Non-linear Feature Selection

Our objective is to develop formulations and algorithms for efficiently computing the feature selection path – i.e. the variation in classification accuracy as the fraction of selected features is varied from null to unity. Multiple Kernel Learning subject to lp≥1 regularization (lp-MKL) has been demonstrated to be one of the most effective techniques for non-linear feature selection. However, state-of-the-art lp-MKL algorithms are too computationally expensive to be invoked thousands of times to determine the entire path. We propose a novel conjecture which states that, for certain lp-MKL formulations, the number of features selected in the optimal solution monotonically decreases as p is decreased from an initial value to unity. We prove the conjecture, for a generic family of kernel target alignment based formulations, and show that the feature weights themselves decay (grow) monotonically once they are below (above) a certain threshold at optimality. This allows us to develop a path following algorithm that systematically generates optimal feature sets of decreasing size. The proposed algorithm sets certain feature weights directly to zero for potentially large intervals of p thereby reducing optimization costs while simultaneously providing approximation guarantees. We empirically demonstrate that our formulation can lead to classification accuracies which are as much as 10% higher on benchmark data sets not only as compared to other lp-MKL formulations and uniform kernel baselines but also Proceedings of the 31 st International Conference on Machine Learning, Beijing, China, 2014. JMLR: W&CP volume 32. Copyright 2014 by the author(s). leading feature selection methods. We further demonstrate that our algorithm reduces training time significantly over other path following algorithms and state-of-the-art lp-MKL optimizers such as SMO-MKL. In particular, we generate the entire feature selection path for data sets with a hundred thousand features in approximately half an hour on standard hardware. Entire path generation for such data set is well beyond the scaling capabilities of other methods.