A Statistical Learning Approach to Modal Regression

This paper studies the nonparametric modal regression problem systematically from a statistical learning view. Originally motivated by pursuing a theoretical understanding of the maximum correntropy criterion based regression (MCCR), our study reveals that MCCR with a tending-to-zero scale parameter is essentially modal regression. We show that nonparametric modal regression problem can be approached via the classical empirical risk minimization. Some efforts are then made to develop a framework for analyzing and implementing modal regression. For instance, the modal regression function is described, the modal regression risk is defined explicitly and its \textit{Bayes} rule is characterized; for the sake of computational tractability, the surrogate modal regression risk, which is termed as the generalization risk in our study, is introduced. On the theoretical side, the excess modal regression risk, the excess generalization risk, the function estimation error, and the relations among the above three quantities are studied rigorously. It turns out that under mild conditions, function estimation consistency and convergence may be pursued in modal regression as in vanilla regression protocols, such as mean regression, median regression, and quantile regression. However, it outperforms these regression models in terms of robustness as shown in our study from a re-descending M-estimation view. This coincides with and in return explains the merits of MCCR on robustness. On the practical side, the implementation issues of modal regression including the computational algorithm and the tuning parameters selection are discussed. Numerical assessments on modal regression are also conducted to verify our findings empirically.