Kernel Methods

In this paper we consider the fundamental problem of semi-supervised kernel function learning. We propose a general regularized framework for learning a kernel matrix, and then demonstrate an equivalence between our proposed kernel matrix learning framework and a general linear transformation learning problem. Our result shows that the learned kernel matrices parameterize a linear transformation kernel function and can be applied inductively to new data points. Furthermore, our result gives a constructive method for kernelizing most existing Mahalanobis metric learning formulations. To make our results practical for large-scale data, we modify our framework to limit the number of parameters in the optimization process.

We introduce a new family of positive-definite kernel functions that mimic the computation in large, multilayer neural nets. These kernel functions can be used in shallow architectures, such as support vector machines (SVMs), or in deep kernel-based architectures that we call multilayer kernel machines (MKMs). We evaluate SVMs and MKMs with these kernel functions on problems designed to illustrate the advantages of deep architectures. On several problems, we obtain better results than previous, leading benchmarks from both SVMs with Gaussian kernels as well as deep belief nets. Papers published at the Neural Information Processing Systems Conference.

We propose a novel kernel approach to dimension reduction for supervised learning: feature extraction and variable selection; the former constructs a small number of features from predictors, and the latter finds a subset of predictors. First, a method of linear feature extraction is proposed using the gradient of regression function, based on the recent development of the kernel method. In comparison with other existing methods, the proposed one has wide applicability without strong assumptions on the regressor or type of variables, and uses computationally simple eigendecomposition, thus applicable to large data sets. Second, in combination of a sparse penalty, the method is extended to variable selection, following the approach by Chen et al. (2010). Experimental results show that the proposed methods successfully find effective features and variables without parametric models.

We consider the problem of improving kernel approximation via randomized feature maps. These maps arise as Monte Carlo approximation to integral representations of kernel functions and scale up kernel methods for larger datasets. Based on an efficient numerical integration technique, we propose a unifying approach that reinterprets the previous random features methods and extends to better estimates of the kernel approximation. We derive the convergence behavior and conduct an extensive empirical study that supports our hypothesis. Papers published at the Neural Information Processing Systems Conference.

Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a low-dimensional Euclidean embedding of the data set that reflects the given similarity triplets, we aim at defining kernel functions that correspond to high-dimensional embeddings. These kernel functions can subsequently be used to apply any kernel method to the data set. Papers published at the Neural Information Processing Systems Conference.

Empirical risk minimization (ERM) is ubiquitous in machine learning and underlies most supervised learning methods. While there is a large body of work on algorithms for various ERM problems, the exact computational complexity of ERM is still not understood. We address this issue for multiple popular ERM problems including kernel SVMs, kernel ridge regression, and training the final layer of a neural network. In particular, we give conditional hardness results for these problems based on complexity-theoretic assumptions such as the Strong Exponential Time Hypothesis. Under these assumptions, we show that there are no algorithms that solve the aforementioned ERM problems to high accuracy in sub-quadratic time.

Kernel methods provide a principled way to perform non linear, nonparametric learning. They rely on solid functional analytic foundations and enjoy optimal statistical properties. However, at least in their basic form, they have limited applicability in large scale scenarios because of stringent computational requirements in terms of time and especially memory. In this paper, we take a substantial step in scaling up kernel methods, proposing FALKON, a novel algorithm that allows to efficiently process millions of points. FALKON is derived combining several algorithmic principles, namely stochastic subsampling, iterative solvers and preconditioning.

The general perception is that kernel methods are not scalable, so neural nets become the choice for large-scale nonlinear learning problems. Have we tried hard enough for kernel methods? In this paper, we propose an approach that scales up kernel methods using a novel concept called doubly stochastic functional gradients''. Based on the fact that many kernel methods can be expressed as convex optimization problems, our approach solves the optimization problems by making two unbiased stochastic approximations to the functional gradient---one using random training points and another using random features associated with the kernel---and performing descent steps with this noisy functional gradient. Our algorithm is simple, need no commit to a preset number of random features, and allows the flexibility of the function class to grow as we see more incoming data in the streaming setting.

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.

By interpreting the algorithm as Randomized Coordinate Descent in the infinite-dimensional space, we show the proposed approach converges to a solution comparable within $\eps$-precision to exact kernel method by drawing $O(1/\eps)$ number of random features, contrasted to the $O(1/\eps 2)$-type convergence achieved by Monte-Carlo analysis in current Random Feature literature. In our experiments, the Sparse Random Feature algorithm obtains sparse solution that requires less memory and prediction time while maintains comparable performance on tasks of regression and classification. Papers published at the Neural Information Processing Systems Conference.