We introduce a Gaussian process model of functions which are additive. An additive function is one which decomposes into a sum of low-dimensional functions, each depending on only a subset of the input variables. Additive GPs generalize both Generalized Additive Models, and the standard GP models which use squared-exponential kernels. Hyperparameter learning in this model can be seen as Bayesian Hierarchical Kernel Learning (HKL). We introduce an expressive but tractable parameterization of the kernel function, which allows efficient evaluation of all input interaction terms, whose number is exponential in the input dimension. The additional structure discoverable by this model results in increased interpretability, as well as state-of-the-art predictive power in regression tasks.
@cite_9 previously showed how mixtures of kernels can be learnt by gradient descent in the Gaussian process framework. They call this . However, their approach learns a mixture over a small, fixed set of kernels, while our method learns a mixture over all possible products of those kernels.
@cite_9
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We build up the mathematical connection between the “Expectation-Maximization” (EM) algorithm and gradient-based approaches for maximum likelihood learning of finite gaussian mixtures. We show that the EM step in parameter space is obtained from the gradient via a projection matrix P, and we provide an explicit expression for the matrix. We then analyze the convergence of EM in terms of special properties of P and provide new results analyzing the effect that P has on the likelihood surface. Based on these mathematical results, we present a comparative discussion of the advantages and disadvantages of EM and other algorithms for the learning of gaussian mixture models. In this paper, we present a new framework for large scale online kernel learning, making kernel methods efficient and scalable for large-scale online learning applications. Unlike the regular budget online kernel learning scheme that usually uses some budget maintenance strategies to bound the number of support vectors, our framework explores a completely different approach of kernel functional approximation techniques to make the subsequent online learning task efficient and scalable. Specifically, we present two different online kernel machine learning algorithms: (i) Fourier Online Gradient Descent (FOGD) algorithm that applies the random Fourier features for approximating kernel functions; and (ii) Nystrom Online Gradient Descent (NOGD) algorithm that applies the Nystrom method to approximate large kernel matrices. We explore these two approaches to tackle three online learning tasks: binary classification, multi-class classification, and regression. The encouraging results of our experiments on large-scale datasets validate the effectiveness and efficiency of the proposed algorithms, making them potentially more practical than the family of existing budget online kernel learning approaches. We introduce a kernel approximation strategy that enables computation of the Gaussian process log marginal likelihood and all hyperparameter derivatives in @math time. Our GRIEF kernel consists of @math eigenfunctions found using a Nystrom approximation from a dense Cartesian product grid of inducing points. By exploiting algebraic properties of Kronecker and Khatri-Rao tensor products, computational complexity of the training procedure can be practically independent of the number of inducing points. This allows us to use arbitrarily many inducing points to achieve a globally accurate kernel approximation, even in high-dimensional problems. The fast likelihood evaluation enables type-I or II Bayesian inference on large-scale datasets. We benchmark our algorithms on real-world problems with up to two-million training points and @math inducing points. The general perception is that kernel methods are not scalable, and neural nets are the methods of choice for nonlinear learning problems. Or have we simply not tried hard enough for kernel methods? Here we propose an approach that scales up kernel methods using a novel concept called "doubly stochastic functional gradients". Our approach relies on the fact that many kernel methods can be expressed as convex optimization problems, and we solve the problems by making two unbiased stochastic approximations to the functional gradient, one using random training points and another using random functions associated with the kernel, and then descending using this noisy functional gradient. We show that a function produced by this procedure after @math iterations converges to the optimal function in the reproducing kernel Hilbert space in rate @math , and achieves a generalization performance of @math . This doubly stochasticity also allows us to avoid keeping the support vectors and to implement the algorithm in a small memory footprint, which is linear in number of iterations and independent of data dimension. Our approach can readily scale kernel methods up to the regimes which are dominated by neural nets. We show that our method can achieve competitive performance to neural nets in datasets such as 8 million handwritten digits from MNIST, 2.3 million energy materials from MolecularSpace, and 1 million photos from ImageNet.