## Publications

#### Structure Discovery in Nonparametric Regression through Compositional Kernel Search

David Duvenaud, James Robert Lloyd, Roger Grosse, Joshua B. Tenenbaum, Zoubin Ghahramani, June 2013. (In 30th International Conference on Machine Learning). Atlanta, Georgia, USA.

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Despite its importance, choosing the structural form of the kernel in nonparametric regression remains a black art. We define a space of kernel structures which are built compositionally by adding and multiplying a small number of base kernels. We present a method for searching over this space of structures which mirrors the scientific discovery process. The learned structures can often decompose functions into interpretable components and enable long-range extrapolation on time-series datasets. Our structure search method outperforms many widely used kernels and kernel combination methods on a variety of prediction tasks.

#### Additive Gaussian Processes

David Duvenaud, Hannes Nickisch, Carl Edward Rasmussen, 2011. (In Advances in Neural Information Processing Systems 24). Granada, Spain.

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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.

#### Avoiding pathologies in very deep networks

David Duvenaud, Oren Rippel, Ryan P. Adams, Zoubin Ghahramani, April 2014. (In 17th International Conference on Artificial Intelligence and Statistics). Reykjavik, Iceland.

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Choosing appropriate architectures and regularization strategies for deep networks is crucial to good predictive performance. To shed light on this problem, we analyze the analogous problem of constructing useful priors on compositions of functions. Specifically, we study the deep Gaussian process, a type of infinitely-wide, deep neural network. We show that in standard architectures, the representational capacity of the network tends to capture fewer degrees of freedom as the number of layers increases, retaining only a single degree of freedom in the limit. We propose an alternate network architecture which does not suffer from this pathology. We also examine deep covariance functions, obtained by composing infinitely many feature transforms. Lastly, we characterize the class of models obtained by performing dropout on Gaussian processes.

#### Optimally-Weighted Herding is Bayesian Quadrature

Ferenc Huszár, David Duvenaud, July 2012. (In 28th Conference on Uncertainty in Artificial Intelligence). Catalina Island, California.

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Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is equivalent to the posterior variance in Bayesian quadrature. We then show that sequential Bayesian quadrature can be viewed as a weighted version of kernel herding which achieves performance superior to any other weighted herding method. We demonstrate empirically a rate of convergence faster than O(1/N). Our results also imply an upper bound on the empirical error of the Bayesian quadrature estimate.

#### Warped Mixtures for Nonparametric Cluster Shapes

Tomoharu Iwata, David Duvenaud, Zoubin Ghahramani, July 2013. (In 29th Conference on Uncertainty in Artificial Intelligence). Bellevue, Washington.

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A mixture of Gaussians fit to a single curved or heavy-tailed cluster will report that the data contains many clusters. To produce more appropriate clusterings, we introduce a model which warps a latent mixture of Gaussians to produce nonparametric cluster shapes. The possibly low-dimensional latent mixture model allows us to summarize the properties of the high-dimensional clusters (or density manifolds) describing the data. The number of manifolds, as well as the shape and dimension of each manifold is automatically inferred. We derive a simple inference scheme for this model which analytically integrates out both the mixture parameters and the warping function. We show that our model is effective for density estimation, performs better than infinite Gaussian mixture models at recovering the true number of clusters, and produces interpretable summaries of high-dimensional datasets.

#### Automatic Construction and Natural-Language Description of Nonparametric Regression Models

James Robert Lloyd, David Duvenaud, Roger Grosse, Joshua B. Tenenbaum, Zoubin Ghahramani, July 2014. (In Association for the Advancement of Artificial Intelligence (AAAI)).

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This paper presents the beginnings of an automatic statistician, focusing on regression problems. Our system explores an open-ended space of statistical models to discover a good explanation of a data set, and then produces a detailed report with figures and natural-language text. Our approach treats unknown regression functions nonparametrically using Gaussian processes, which has two important consequences. First, Gaussian processes can model functions in terms of high-level properties (e.g. smoothness, trends, periodicity, changepoints). Taken together with the compositional structure of our language of models this allows us to automatically describe functions in simple terms. Second, the use of flexible nonparametric models and a rich language for composing them in an open-ended manner also results in state-of-the-art extrapolation performance evaluated over 13 real time series data sets from various domains.

#### Active Learning of Model Evidence Using Bayesian Quadrature

Michael A. Osborne, David Duvenaud, Roman Garnett, Carl Edward Rasmussen, Stephen J. Roberts, Zoubin Ghahramani, December 2012. (In Advances in Neural Information Processing Systems 25). Lake Tahoe, California, USA.

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Numerical integration is a key component of many problems in scientiﬁc computing, statistical modelling, and machine learning. Bayesian Quadrature is a model-based method for numerical integration which, relative to standard Monte Carlo methods, offers increased sample efficiency and a more robust estimate of the uncertainty in the estimated integral. We propose a novel Bayesian Quadrature approach for numerical integration when the integrand is non-negative, such as the case of computing the marginal likelihood, predictive distribution, or normalising constant of a probabilistic model. Our approach approximately marginalises the quadrature model’s hyperparameters in closed form, and introduces an active learning scheme to optimally select function evaluations, as opposed to using Monte Carlo samples. We demonstrate our method on both a number of synthetic benchmarks and a real scientiﬁc problem from astronomy.

#### Probabilistic ODE Solvers with Runge-Kutta Means

Michael Schober, David Duvenaud, Philipp Hennig, June 2014. (arXiv preprint arXiv:1406.2582).

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Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state-of-the-art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct this family such that posterior means match the outputs of the Runge-Kutta family exactly, thus inheriting their proven good properties. Remaining degrees of freedom not identified by the match to Runge-Kutta are chosen such that the posterior probability measure fits the observed structure of the ODE. Our results shed light on the structure of Runge-Kutta solvers from a new direction, provide a richer, probabilistic output, have low computational cost, and raise new research questions.