In this work, we demonstrate the advantage of the pGMM (``powered generalized min-max'') kernel in the context of (ridge) regression. In recent prior studies, the pGMM kernel has been extensively evaluated for classification tasks, for logistic regression, support vector machines, as well as deep neural networks. In this paper, we provide an experimental study on ridge regression, to compare the pGMM kernel regression with the ordinary ridge linear regression as well as the RBF kernel ridge regression. Perhaps surprisingly, even without a tuning parameter (i.e., $p=1$ for the power parameter of the pGMM kernel), the pGMM kernel already performs well. Furthermore, by tuning the parameter $p$, this (deceptively simple) pGMM kernel even performs quite comparably to boosted trees. Boosting and boosted trees are very popular in machine learning practice. For regression tasks, typically, practitioners use $L_2$ boost, i.e., for minimizing the $L_2$ loss. Sometimes for the purpose of robustness, the $L_1$ boost might be a choice. In this study, we implement $L_p$ boost for $p\geq 1$ and include it in the package of ``Fast ABC-Boost''. Perhaps also surprisingly, the best performance (in terms of $L_2$ regression loss) is often attained at $p>2$, in some cases at $p\gg 2$. This phenomenon has already been demonstrated by Li et al (UAI 2010) in the context of k-nearest neighbor classification using $L_p$ distances. In summary, the implementation of $L_p$ boost provides practitioners the additional flexibility of tuning boosting algorithms for potentially achieving better accuracy in regression applications.

While tree methods have been popular in practice, researchers and practitioners are also looking for simple algorithms which can reach similar accuracy of trees. In 2010, (Ping Li UAI'10) developed the method of "abc-robust-logitboost" and compared it with other supervised learning methods on datasets used by the deep learning literature. In this study, we propose a series of "tunable GMM kernels" which are simple and perform largely comparably to tree methods on the same datasets. Note that "abc-robust-logitboost" substantially improved the original "GDBT" in that (a) it developed a tree-split formula based on second-order information of the derivatives of the loss function; (b) it developed a new set of derivatives for multi-class classification formulation. In the prior study in 2017, the "generalized min-max" (GMM) kernel was shown to have good performance compared to the "radial-basis function" (RBF) kernel. However, as demonstrated in this paper, the original GMM kernel is often not as competitive as tree methods on the datasets used in the deep learning literature. Since the original GMM kernel has no parameters, we propose tunable GMM kernels by adding tuning parameters in various ways. Three basic (i.e., with only one parameter) GMM kernels are the "$e$GMM kernel", "$p$GMM kernel", and "$\gamma$GMM kernel", respectively. Extensive experiments show that they are able to produce good results for a large number of classification tasks. Furthermore, the basic kernels can be combined to boost the performance.

The recently proposed "generalized min-max" (GMM) kernel can be efficiently linearized, with direct applications in large-scale statistical learning and fast near neighbor search. The linearized GMM kernel was extensively compared in with linearized radial basis function (RBF) kernel. On a large number of classification tasks, the tuning-free GMM kernel performs (surprisingly) well compared to the best-tuned RBF kernel. Nevertheless, one would naturally expect that the GMM kernel ought to be further improved if we introduce tuning parameters. In this paper, we study three simple constructions of tunable GMM kernels: (i) the exponentiated-GMM (or eGMM) kernel, (ii) the powered-GMM (or pGMM) kernel, and (iii) the exponentiated-powered-GMM (epGMM) kernel. The pGMM kernel can still be efficiently linearized by modifying the original hashing procedure for the GMM kernel. On about 60 publicly available classification datasets, we verify that the proposed tunable GMM kernels typically improve over the original GMM kernel. On some datasets, the improvements can be astonishingly significant. For example, on 11 popular datasets which were used for testing deep learning algorithms and tree methods, our experiments show that the proposed tunable GMM kernels are strong competitors to trees and deep nets. The previous studies developed tree methods including "abc-robust-logitboost" and demonstrated the excellent performance on those 11 datasets (and other datasets), by establishing the second-order tree-split formula and new derivatives for multi-class logistic loss. Compared to tree methods like "abc-robust-logitboost" (which are slow and need substantial model sizes), the tunable GMM kernels produce largely comparable results.