Measuring the Discrepancy between Conditional Distributions: Methods, Properties and Applications

We propose a simple yet powerful test statistic to quantify the discrepancy between two conditional distributions. The new statistic avoids the explicit estimation of the underlying distributions in highdimensional space and it operates on the cone of symmetric positive semidefinite (SPS) matrix using the Bregman matrix divergence. Moreover, it inherits the merits of the correntropy function to explicitly incorporate high-order statistics in the data. We present the properties of our new statistic and illustrate its connections to prior art. We finally show the applications of our new statistic on three different machine learning problems, namely the multi-task learning over graphs, the concept drift detection, and the information-theoretic feature selection, to demonstrate its utility and advantage. Code of our statistic is available at https://bit.ly/BregmanCorrentropy.Comment: manuscript accepted at IJCAI 20; added additional notes on computational complexity and auto-differentiable property; code is available at https://github.com/SJYuCNEL/Bregman-Correntropy-Conditional-Divergenc

Convex Optimization for Binary Classifier Aggregation in Multiclass Problems

Multiclass problems are often decomposed into multiple binary problems that are solved by individual binary classifiers whose results are integrated into a final answer. Various methods, including all-pairs (APs), one-versus-all (OVA), and error correcting output code (ECOC), have been studied, to decompose multiclass problems into binary problems. However, little study has been made to optimally aggregate binary problems to determine a final answer to the multiclass problem. In this paper we present a convex optimization method for an optimal aggregation of binary classifiers to estimate class membership probabilities in multiclass problems. We model the class membership probability as a softmax function which takes a conic combination of discrepancies induced by individual binary classifiers, as an input. With this model, we formulate the regularized maximum likelihood estimation as a convex optimization problem, which is solved by the primal-dual interior point method. Connections of our method to large margin classifiers are presented, showing that the large margin formulation can be considered as a limiting case of our convex formulation. Numerical experiments on synthetic and real-world data sets demonstrate that our method outperforms existing aggregation methods as well as direct methods, in terms of the classification accuracy and the quality of class membership probability estimates.Comment: Appeared in Proceedings of the 2014 SIAM International Conference on Data Mining (SDM 2014