Heterogeneous Multi-output Gaussian Process Prediction

arXiv.org Machine Learning

We present a novel extension of multi-output Gaussian processes for handling heterogeneous outputs. We assume that each output has its own likelihood function and use a vector-valued Gaussian process prior to jointly model the parameters in all likelihoods as latent functions. Our multi-output Gaussian process uses a covariance function with a linear model of coregionalisation form. Assuming conditional independence across the underlying latent functions together with an inducing variable framework, we are able to obtain tractable variational bounds amenable to stochastic variational inference. We illustrate the performance of the model on synthetic data and two real datasets: a human behavioral study and a demographic high-dimensional dataset.


Heterogeneous Multi-output Gaussian Process Prediction

Neural Information Processing Systems

We present a novel extension of multi-output Gaussian processes for handling heterogeneous outputs. We assume that each output has its own likelihood function and use a vector-valued Gaussian process prior to jointly model the parameters in all likelihoods as latent functions. Our multi-output Gaussian process uses a covariance function with a linear model of coregionalisation form. Assuming conditional independence across the underlying latent functions together with an inducing variable framework, we are able to obtain tractable variational bounds amenable to stochastic variational inference. We illustrate the performance of the model on synthetic data and two real datasets: a human behavioral study and a demographic high-dimensional dataset.


Continual Multi-task Gaussian Processes

arXiv.org Machine Learning

We address the problem of continual learning in multi-task Gaussian process (GP) models for handling sequential input-output observations. Our approach extends the existing prior-posterior recursion of online Bayesian inference, i.e.\ past posterior discoveries become future prior beliefs, to the infinite functional space setting of GP. For a reason of scalability, we introduce variational inference together with an sparse approximation based on inducing inputs. As a consequence, we obtain tractable continual lower-bounds where two novel Kullback-Leibler (KL) divergences intervene in a natural way. The key technical property of our method is the recursive reconstruction of conditional GP priors conditioned on the variational parameters learned so far. To achieve this goal, we introduce a novel factorization of past variational distributions, where the predictive GP equation propagates the posterior uncertainty forward. We then demonstrate that it is possible to derive GP models over many types of sequential observations, either discrete or continuous and amenable to stochastic optimization. The continual inference approach is also applicable to scenarios where potential multi-channel or heterogeneous observations might appear. Extensive experiments demonstrate that the method is fully scalable, shows a reliable performance and is robust to uncertainty error propagation over a plenty of synthetic and real-world datasets.


Multi-task Learning for Aggregated Data using Gaussian Processes

arXiv.org Machine Learning

Aggregated data is commonplace in areas such as epidemiology and demography. For example, census data for a population is usually given as averages defined over time periods or spatial resolutions (city, region or countries). In this paper, we present a novel multi-task learning model based on Gaussian processes for joint learning of variables that have been aggregated at different input scales. Our model represents each task as the linear combination of the realizations of latent processes that are integrated at a different scale per task. We are then able to compute the cross-covariance between the different tasks either analytically or numerically. We also allow each task to have a potentially different likelihood model and provide a variational lower bound that can be optimised in a stochastic fashion making our model suitable for larger datasets. We show examples of the model in a synthetic example, a fertility dataset and an air pollution prediction application.


A Framework for Interdomain and Multioutput Gaussian Processes

arXiv.org Machine Learning

One obstacle to the use of Gaussian processes (GPs) in large-scale problems, and as a component in deep learning system, is the need for bespoke derivations and implementations for small variations in the model or inference. In order to improve the utility of GPs we need a modular system that allows rapid implementation and testing, as seen in the neural network community. We present a mathematical and software framework for scalable approximate inference in GPs, which combines interdomain approximations and multiple outputs. Our framework, implemented in GPflow, provides a unified interface for many existing multioutput models, as well as more recent convolutional structures. This simplifies the creation of deep models with GPs, and we hope that this work will encourage more interest in this approach.