We present a new Gaussian process (GP) regression model whose covariance is parameterized by the the locations of M pseudo-input points, which we learn by a gradient based optimization. We take M ≪ N, where N is the number of real data points, and hence obtain a sparse regression method which has O(M 2 N) training cost and O(M 2) prediction cost per test case. We also find hyperparameters of the covariance function in the same joint optimization. The method can be viewed as a Bayesian regression model with particular input dependent noise. The method turns out to be closely related to several other sparse GP approaches, and we discuss the relation in detail. We finally demonstrate its performance on some large data sets, and make a direct comparison to other sparse GP methods. We show that our method can match full GP performance with small M, i.e. very sparse solutions, and it significantly outperforms other approaches in this regime. 1

We present a method for the sparse greedy approximation of Bayesian Gaussian process regression, featuring a novel heuristic for very fast forward selection. Our method is essentially as fast as an equivalent one which selects the "support" patterns at random, yet it can outperform random selection on hard curve fitting tasks. More importantly, it leads to a suciently stable approximation of the log marginal likelihood of the training data, which can be optimised to adjust a large number of hyperparameters automatically.

Gaussian processes (GPs) are natural generalisations of multivariate Gaussian random variables to in nite (countably or continuous) index sets. GPs have been applied in a large number of elds to a diverse range of ends, and very many deep theoretical analyses of various properties are available. This paper gives an introduction to Gaussian processes on a fairly elementary level with special emphasis on characteristics relevant in machine learning. It draws explicit connections to branches such as spline smoothing models and support vector machines in which similar ideas have been investigated.

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These lecture notes are based on the work of Neal (1996), Williams and

We present a variational Bayesian method for model selection over families of kernels classifiers like Support Vector machines or Gaussian processes. The algorithm needs no user interaction and is able to adapt a large number of kernel parameters to given data without having to sacrifice training cases for validation. This opens the possibility to use sophisticated families of kernels in situations where the small "standard kernel" classes are clearly inappropriate. We relate the method to other work done on Gaussian processes and clarify the relation between Support Vector machines and certain Gaussian process models.

Relational reinforcement learning is a Q-learning technique for relational state-action spaces. It aims to enable agents to learn how to act in an environment that has no natural representation as a tuple of constants. In this case, the learning algorithm used to approximate the mapping between state-action pairs and their so called Q(uality)-value has to be not only very reliable, but it also has to be able to handle the relational representation of state-action pairs. In this paper we investigate...

We show that the Gaussian random fields and harmonic energy minimizing function framework for semi-supervised learning can be viewed in terms of Gaussian processes, with covariance matrices derived from the graph Laplacian. We derive hyperparameter learning with evidence maximization, and give an empirical study of various ways to parameterize the graph weights.

Gaussian processes are a promising non-linear interpolation tool (Williams 1995; Williams and Rasmussen 1996), but it is not straightforward to solve classification problems with them. In this paper the variational methods of Jaakkola and Jordan (1996) are applied to Gaussian processes to produce an efficient Bayesian binary classifier. 1 Introduction Assume that we have some data D which consists of inputs fx n g N n=1 in some space, real or discrete, and corresponding targets t n which are binary categorical variables. We shall model this data using a Bayesian conditional classifier which predicts t conditional on x. We assume the existence of a function a(x) which models the `logit' log P (t=1jx) P (t=0jx) as a function of x. Thus P (t = 1jx; a(x)) = 1 1 + exp(\Gammaa(x)) (1) To complete the model we place a prior distribution over the unknown function a(x). There are two approaches to this. In the standard parametric approach, a(x) is a parameterized function a(x; w) where the...

We present an overview of evolutionary algorithms that use empirical models of the fitness function to accelerate convergence, distinguishing between Evolution Control and the Surrogate Approach. We describe the Gaussian process model and propose using it as an inexpensive fitness function surrogate. Implementation issues such as efficient and numerically stable computation, exploration vs. exploitation, local modeling, multiple objectives and constraints, and failed evaluations are addressed. Our resulting Gaussian Process Optimization Procedure (GPOP) clearly outperforms other evolutionary strategies on standard test functions as well as on a real-world problem: the optimization of stationary gas turbine compressor profiles.