Gaussian Robust Classification

Abstract: Supervised learning is all about the ability to generalize knowledge. Specifically, the goal of the learning is to train a classifier using training data, in such a way that it will be capable of classifying new unseen data correctly. In order to acheive this goal, it is important to carefully design the learner, so it will not overfit the training data. The later can is done usually by adding a regularization term. The statistical learning theory explains the success of this method by claiming that it restricts the complexity of the learned model. This explanation, however, is rather abstract and does not have a geometric intuition. The generalization error of a classifier may be thought of as correlated with its robustness to perturbations of the data: a classifier that copes with disturbance is expected to generalize well. Indeed, Xu et al. [2009] have shown that the SVM formulation is equivalent to a robust optimization (RO) formulation, in which an adversary displaces the training and testing points within a ball of pre-determined radius. In this work we explore a different kind of robustness, namely changing each data point with a Gaussian cloud centered at the sample. Loss is evaluated as the expectation of an underlying loss function on the cloud. This setup fits the fact that in many applications, the data is sampled along with noise. We develop an RO framework, in which the adversary chooses the covariance of the noise. In our algorithm named GURU, the tuning parameter is a spectral bound on the noise, thus it can be estimated using physical or applicative considerations. Our experiments show that this framework performs as well as SVM and even slightly better in some cases. Generalizations for Mercer kernels and for the multiclass case are presented as well. We also show that our framework may be further generalized, using the technique of convex perspective functions.