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An Empirical Study on The Properties of Random Bases for Kernel Methods

Neural Information Processing Systems

Kernel machines as well as neural networks possess universal function approximation properties. Nevertheless in practice their ways of choosing the appropriate function class differ. Specifically neural networks learn a representation by adapting their basis functions to the data and the task at hand, while kernel methods typically use a basis that is not adapted during training. In this work, we contrast random features of approximated kernel machines with learned features of neural networks. Our analysis reveals how these random and adaptive basis functions affect the quality of learning. Furthermore, we present basis adaptation schemes that allow for a more compact representation, while retaining the generalization properties of kernel machines.


Sparse Gaussian Processes via Parametric Families of Compactly-supported Kernels

arXiv.org Machine Learning

Gaussian processes are powerful models for probabilistic machine learning, but are limited in application by their $O(N^3)$ inference complexity. We propose a method for deriving parametric families of kernel functions with compact spatial support, which yield naturally sparse kernel matrices and enable fast Gaussian process inference via sparse linear algebra. These families generalize known compactly-supported kernel functions, such as the Wendland polynomials. The parameters of this family of kernels can be learned from data using maximum likelihood estimation. Alternatively, we can quickly compute compact approximations of a target kernel using convex optimization. We demonstrate that these approximations incur minimal error over the exact models when modeling data drawn directly from a target GP, and can out-perform the traditional GP kernels on real-world signal reconstruction tasks, while exhibiting sub-quadratic inference complexity.


Data-driven Random Fourier Features using Stein Effect

arXiv.org Machine Learning

Large-scale kernel approximation is an important problem in machine learning research. Approaches using random Fourier features have become increasingly popular [Rahimi and Recht, 2007], where kernel approximation is treated as empirical mean estimation via Monte Carlo (MC) or Quasi-Monte Carlo (QMC) integration [Yang et al., 2014]. A limitation of the current approaches is that all the features receive an equal weight summing to 1. In this paper, we propose a novel shrinkage estimator from "Stein effect", which provides a data-driven weighting strategy for random features and enjoys theoretical justifications in terms of lowering the empirical risk. We further present an efficient randomized algorithm for large-scale applications of the proposed method. Our empirical results on six benchmark data sets demonstrate the advantageous performance of this approach over representative baselines in both kernel approximation and supervised learning tasks.


Fast Prediction for Large-Scale Kernel Machines

Neural Information Processing Systems

Kernel machines such as kernel SVM and kernel ridge regression usually construct high quality models; however, their use in real-world applications remains limited due to the high prediction cost. In this paper, we present two novel insights for improving the prediction efficiency of kernel machines. First, we show that by adding “pseudo landmark points” to the classical Nystr¨om kernel approximation in an elegant way, we can significantly reduce the prediction error without much additional prediction cost. Second, we provide a new theoretical analysis on bounding the error of the solution computed by using Nystr¨om kernel approximation method, and show that the error is related to the weighted kmeans objective function where the weights are given by the model computed from the original kernel. This theoretical insight suggests a new landmark point selection technique for the situation where we have knowledge of the original model. Based on these two insights, we provide a divide-and-conquer framework for improving the prediction speed. First, we divide the whole problem into smaller local subproblems to reduce the problem size. In the second phase, we develop a kernel approximation based fast prediction approach within each subproblem. We apply our algorithm to real world large-scale classification and regression datasets, and show that the proposed algorithm is consistently and significantly better than other competitors. For example, on the Covertype classification problem, in terms of prediction time, our algorithm achieves more than 10000 times speedup over the full kernel SVM, and a two-fold speedup over the state-of-the-art LDKL approach, while obtaining much higher prediction accuracy than LDKL (95.2% vs. 89.53%).


Random Fourier Features for Kernel Ridge Regression: Approximation Bounds and Statistical Guarantees

arXiv.org Machine Learning

Random Fourier features is one of the most popular techniques for scaling up kernel methods, such as kernel ridge regression. However, despite impressive empirical results, the statistical properties of random Fourier features are still not well understood. In this paper we take steps toward filling this gap. Specifically, we approach random Fourier features from a spectral matrix approximation point of view, give tight bounds on the number of Fourier features required to achieve a spectral approximation, and show how spectral matrix approximation bounds imply statistical guarantees for kernel ridge regression. Qualitatively, our results are twofold: on the one hand, we show that random Fourier feature approximation can provably speed up kernel ridge regression under reasonable assumptions. At the same time, we show that the method is suboptimal, and sampling from a modified distribution in Fourier space, given by the leverage function of the kernel, yields provably better performance. We study this optimal sampling distribution for the Gaussian kernel, achieving a nearly complete characterization for the case of low-dimensional bounded datasets. Based on this characterization, we propose an efficient sampling scheme with guarantees superior to random Fourier features in this regime.