An interesting observation in artificial neural networks is their favorable generalization error despite typically being extremely overparameterized. It is well known that classical statistical learning methods often result in vacuous generalization errors in the case of overparameterized neural networks. Adopting the recently developed Neural Tangent (NT) kernel theory, we prove uniform generalization bounds for overparameterized neural networks in kernel regimes, when the true data generating model belongs to the reproducing kernel Hilbert space (RKHS) corresponding to the NT kernel. Importantly, our bounds capture the exact error rates depending on the differentiability of the activation functions. In order to establish these bounds, we propose the information gain of the NT kernel as a measure of complexity of the learning problem. Our analysis uses a Mercer decomposition of the NT kernel in the basis of spherical harmonics and the decay rate of the corresponding eigenvalues. As a byproduct of our results, we show the equivalence between the RKHS corresponding to the NT kernel and its counterpart corresponding to the Mat\'ern family of kernels, that induces a very general class of models. We further discuss the implications of our analysis for some recent results on the regret bounds for reinforcement learning algorithms, which use overparameterized neural networks.

Tree based ensembles such as Breiman's random forest (RF) and Gradient Boosted Trees (GBT) can be interpreted as implicit kernel generators, where the ensuing proximity matrix represents the data-driven tree ensemble kernel. Kernel perspective on the RF has been used to develop a principled framework for theoretical investigation of its statistical properties. Recently, it has been shown that the kernel interpretation is germane to other tree-based ensembles e.g. GBTs. However, practical utility of the links between kernels and the tree ensembles has not been widely explored and systematically evaluated. Focus of our work is investigation of the interplay between kernel methods and the tree based ensembles including the RF and GBT. We elucidate the performance and properties of the RF and GBT based kernels in a comprehensive simulation study comprising of continuous and binary targets. We show that for continuous targets, the RF/GBT kernels are competitive to their respective ensembles in higher dimensional scenarios, particularly in cases with larger number of noisy features. For the binary target, the RF/GBT kernels and their respective ensembles exhibit comparable performance. We provide the results from real life data sets for regression and classification to show how these insights may be leveraged in practice. Overall, our results support the tree ensemble based kernels as a valuable addition to the practitioner's toolbox. Finally, we discuss extensions of the tree ensemble based kernels for survival targets, interpretable prototype and landmarking classification and regression. We outline future line of research for kernels furnished by Bayesian counterparts of the frequentist tree ensembles.

Breiman's random forest (RF) can be interpreted as an implicit kernel generator,where the ensuing proximity matrix represents the data-driven RF kernel. Kernel perspective on the RF has been used to develop a principled framework for theoretical investigation of its statistical properties. However, practical utility of the links between kernels and the RF has not been widely explored and systematically evaluated.Focus of our work is investigation of the interplay between kernel methods and the RF. We elucidate the performance and properties of the data driven RF kernels used by regularized linear models in a comprehensive simulation study comprising of continuous, binary and survival targets. We show that for continuous and survival targets, the RF kernels are competitive to RF in higher dimensional scenarios with larger number of noisy features. For the binary target, the RF kernel and RF exhibit comparable performance. As the RF kernel asymptotically converges to the Laplace kernel, we included it in our evaluation. For most simulation setups, the RF and RFkernel outperformed the Laplace kernel. Nevertheless, in some cases the Laplace kernel was competitive, showing its potential value for applications. We also provide the results from real life data sets for the regression, classification and survival to illustrate how these insights may be leveraged in practice.Finally, we discuss further extensions of the RF kernels in the context of interpretable prototype and landmarking classification, regression and survival. We outline future line of research for kernels furnished by Bayesian counterparts of the RF.