Philipp Hennig

Max Planck Research Group Leader

Empirical Inference

Probabilistic Numerics

Office: S2.006
Max-Planck-Ring 4
72076 Tübingen
Germany

+49 7071 601 572

+49 7071 601 552

http://mml.cs.uni-tuebingen.de

Overview
Publications

I have moved!

On 1 May 2018, I was appointed as a Full Professor to the newly created Chair for the Methods of Machine Learning at the Computer Science department of the University of Tübingen. Please find my new page there. I am also keeping an adjunct position at the Max Planck Institute for Intelligent Systems.

I am interested in algorithms that estimate and express uncertainty about the result of imprecise computations. Such imprecision can arise because the computational task is not analytically tractable, because a limited computational budget only allows a partial solution, or because the description of the task is itself imprecise to begin with. Probability measures provide the formal language for the description of such uncertainty. My group and I develop computer algorithms that take in and return probability measures; we call these probabilistic numerical methods.

Research

For detailed information about our work, please see our group's homepage.
A general overview of probabilistic numerical methods can be found at http://www.probabilistic-numerics.org.
A while ago I was interviewed by Katie Gorman and Ryan Adams for Talking Machines. The interview provides a bit of a general introduction. In a bittersweet coincidence, this episode of the podcast opens with an obituary for my incredible PhD advisor, Sir David MacKay.

Teaching and Talks

Video of my talk on Bayesian Optimization at the Gaussian Process Summer School 2015
I organised the Machine Learning Summer School 2015 (videos and slides). During this MLSS, I also did a tutorial on probabilistic numerics
- part I: video and slides
- part II: video and slides
Videos of my Gaussian Process tutorial at the RNLS summer school at the ETH Zürich are online. This is the most recent and most compact video of my GP tutorial. If you want to learn about GPs in 90 minutes, I recommend watching this one instead of the older ones listed below.
I organised the Machine Learning Summer School 2013 (videos and slides)
Videos of my talks at the MLSS are up (slide animations only work in Adobe Reader):
- part I: video and slides
- part II: video and slides
- part III: video and slides
- Alongside the MLSS talks, I released a Tech Report on how to produce animations of Gaussian samples.
I spoke on GPs at the Gaussian Process Winter School 2013. Here is a video, and the slides of this talk (slide animations only work in Adobe Reader).

CV

can be found here.

If you need a bio-blurb for your event web-page or a talk introduction, here's a suggestion (sorry if this sounds like grandstanding, I've repeatedly been asked for such a text):

Philipp Hennig holds the Chair for the Methods of Machine Learning at the University of Tübingen, Germany, and an adjunct position at the Max Planck Institute for Intelligent Systems. He studied physics in Heidelberg and at Imperial College, London, and received a PhD from the University of Cambridge, UK, in 2011, under the supervision of the late Sir David JC MacKay. Since that time, he has been interested in the notion of computation as information gathering and, with collaborators, has helped re-establish the field of probabilistic numerics. Philipp primarily works in the machine learning community, where his group has made several algorithmic contributions. He has held an Emmy Noether fellowship, a Max Planck Research grant, and, in 2017, was awarded an ERC Starting Grant by the European Commission.

Inference Probability Numerical Methods

72 results (View BibTeX file of all listed publications)

2020

Integrals over Gaussians under Linear Domain Constraints

Gessner, A., Kanjilal, O., Hennig, P.

Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), 108, pages: 2764-2774, Proceedings of Machine Learning Research, (Editors: Silvia Chiappa and Roberto Calandra), PMLR, August 2020 (conference)

link (url) [BibTex]

2020

ei Gessner, A., Kanjilal, O., Hennig, P. Integrals over Gaussians under Linear Domain Constraints Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), 108, pages: 2764-2774, Proceedings of Machine Learning Research, (Editors: Silvia Chiappa and Roberto Calandra), PMLR, August 2020 (conference)

link (url) [BibTex]

Modular Block-diagonal Curvature Approximations for Feedforward Architectures

Dangel, F., Harmeling, S., Hennig, P.

Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), 108, pages: 799-808, Proceedings of Machine Learning Research, (Editors: Silvia Chiappa and Roberto Calandra), PMLR, August 2020 (conference)

link (url) [BibTex]

ei Dangel, F., Harmeling, S., Hennig, P. Modular Block-diagonal Curvature Approximations for Feedforward Architectures Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), 108, pages: 799-808, Proceedings of Machine Learning Research, (Editors: Silvia Chiappa and Roberto Calandra), PMLR, August 2020 (conference)

link (url) [BibTex]

2019

Limitations of the empirical Fisher approximation for natural gradient descent

Kunstner, F., Hennig, P., Balles, L.

Advances in Neural Information Processing Systems 32, pages: 4158-4169, (Editors: H. Wallach and H. Larochelle and A. Beygelzimer and F. d’Alché-Buc and E. Fox and R. Garnett), Curran Associates, Inc., 33rd Annual Conference on Neural Information Processing Systems, December 2019 (conference)

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link (url) [BibTex]

2019

ei pn Kunstner, F., Hennig, P., Balles, L. Limitations of the empirical Fisher approximation for natural gradient descent Advances in Neural Information Processing Systems 32, pages: 4158-4169, (Editors: H. Wallach and H. Larochelle and A. Beygelzimer and F. d’Alché-Buc and E. Fox and R. Garnett), Curran Associates, Inc., 33rd Annual Conference on Neural Information Processing Systems, December 2019 (conference)

link (url) [BibTex]

Convergence Guarantees for Adaptive Bayesian Quadrature Methods

Kanagawa, M., Hennig, P.

Advances in Neural Information Processing Systems 32, pages: 6234-6245, (Editors: H. Wallach and H. Larochelle and A. Beygelzimer and F. d’Alché-Buc and E. Fox and R. Garnett), Curran Associates, Inc., 33rd Annual Conference on Neural Information Processing Systems, December 2019 (conference)

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link (url) [BibTex]

ei pn Kanagawa, M., Hennig, P. Convergence Guarantees for Adaptive Bayesian Quadrature Methods Advances in Neural Information Processing Systems 32, pages: 6234-6245, (Editors: H. Wallach and H. Larochelle and A. Beygelzimer and F. d’Alché-Buc and E. Fox and R. Garnett), Curran Associates, Inc., 33rd Annual Conference on Neural Information Processing Systems, December 2019 (conference)

link (url) [BibTex]

DeepOBS: A Deep Learning Optimizer Benchmark Suite

Schneider, F., Balles, L., Hennig, P.

7th International Conference on Learning Representations (ICLR), ICLR, 7th International Conference on Learning Representations (ICLR), May 2019 (conference)

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link (url) [BibTex]

ei pn Schneider, F., Balles, L., Hennig, P. DeepOBS: A Deep Learning Optimizer Benchmark Suite 7th International Conference on Learning Representations (ICLR), ICLR, 7th International Conference on Learning Representations (ICLR), May 2019 (conference)

link (url) [BibTex]

Fast and Robust Shortest Paths on Manifolds Learned from Data

Arvanitidis, G., Hauberg, S., Hennig, P., Schober, M.

Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), 89, pages: 1506-1515, (Editors: Kamalika Chaudhuri and Masashi Sugiyama), PMLR, April 2019 (conference)

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PDF link (url) [BibTex]

ei pn Arvanitidis, G., Hauberg, S., Hennig, P., Schober, M. Fast and Robust Shortest Paths on Manifolds Learned from Data Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), 89, pages: 1506-1515, (Editors: Kamalika Chaudhuri and Masashi Sugiyama), PMLR, April 2019 (conference)

PDF link (url) [BibTex]

Active Probabilistic Inference on Matrices for Pre-Conditioning in Stochastic Optimization

de Roos, F., Hennig, P.

Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), 89, pages: 1448-1457, (Editors: Kamalika Chaudhuri and Masashi Sugiyama), PMLR, April 2019 (conference)

Abstract

Pre-conditioning is a well-known concept that can significantly improve the convergence of optimization algorithms. For noise-free problems, where good pre-conditioners are not known a priori, iterative linear algebra methods offer one way to efficiently construct them. For the stochastic optimization problems that dominate contemporary machine learning, however, this approach is not readily available. We propose an iterative algorithm inspired by classic iterative linear solvers that uses a probabilistic model to actively infer a pre-conditioner in situations where Hessian-projections can only be constructed with strong Gaussian noise. The algorithm is empirically demonstrated to efficiently construct effective pre-conditioners for stochastic gradient descent and its variants. Experiments on problems of comparably low dimensionality show improved convergence. In very high-dimensional problems, such as those encountered in deep learning, the pre-conditioner effectively becomes an automatic learning-rate adaptation scheme, which we also empirically show to work well.

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PDF link (url) [BibTex]

ei pn de Roos, F., Hennig, P. Active Probabilistic Inference on Matrices for Pre-Conditioning in Stochastic Optimization Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), 89, pages: 1448-1457, (Editors: Kamalika Chaudhuri and Masashi Sugiyama), PMLR, April 2019 (conference)

PDF link (url) [BibTex]

Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective

Tronarp, F., Kersting, H., Särkkä, S. H. P.

Statistics and Computing, 29(6):1297-1315, 2019 (article)

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DOI [BibTex]

ei pn Tronarp, F., Kersting, H., Särkkä, S. H. P. Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective Statistics and Computing, 29(6):1297-1315, 2019 (article)

DOI [BibTex]

Dense connectomic reconstruction in layer 4 of the somatosensory cortex

Motta, A., Berning, M., Boergens, K. M., Staffler, B., Beining, M., Loomba, S., Hennig, P., Wissler, H., Helmstaedter, M.

Science, 366(6469):eaay3134, American Association for the Advancement of Science, 2019 (article)

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DOI [BibTex]

ei pn Motta, A., Berning, M., Boergens, K. M., Staffler, B., Beining, M., Loomba, S., Hennig, P., Wissler, H., Helmstaedter, M. Dense connectomic reconstruction in layer 4 of the somatosensory cortex Science, 366(6469):eaay3134, American Association for the Advancement of Science, 2019 (article)

DOI [BibTex]

Probabilistic Linear Solvers: A Unifying View

Bartels, S., Cockayne, J., Ipsen, I., Hennig, P.

Statistics and Computing, 29(6):1249-1263, 2019 (article)

link (url) DOI [BibTex]

pn Bartels, S., Cockayne, J., Ipsen, I., Hennig, P. Probabilistic Linear Solvers: A Unifying View Statistics and Computing, 29(6):1249-1263, 2019 (article)

link (url) DOI [BibTex]

2018

Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective

Tronarp, F., Kersting, H., Särkkä, S., Hennig, P.

ArXiv preprint 2018, arXiv:1810.03440 [stat.ME], October 2018 (article)

Abstract

We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement sequence to consists of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP---which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a Bayesian state estimation problem and all widely-used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers, which were formulated in terms of generating synthetic measurements of the vector field, come out as specific approximations. We derive novel solvers, both Gaussian and non-Gaussian, from the Bayesian state estimation problem posed in this paper and compare them with other probabilistic solvers in illustrative experiments.

link (url) Project Page [BibTex]

2018

pn Tronarp, F., Kersting, H., Särkkä, S., Hennig, P. Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective ArXiv preprint 2018, arXiv:1810.03440 [stat.ME], October 2018 (article)

link (url) Project Page [BibTex]

Convergence Rates of Gaussian ODE Filters

Kersting, H., Sullivan, T. J., Hennig, P.

arXiv preprint 2018, arXiv:1807.09737 [math.NA], July 2018 (article)

Abstract

A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution $x$ and its first $q$ derivatives a priori as a Gauss--Markov process $\boldsymbol{X}$, which is then iteratively conditioned on information about $\dot{x}$. We prove worst-case local convergence rates of order $h^{q+1}$ for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order $h^q$ in the case of $q=1$ and an integrated Brownian motion prior, and analyse how inaccurate information on $\dot{x}$ coming from approximate evaluations of $f$ affects these rates. Moreover, we present explicit formulas for the steady states and show that the posterior confidence intervals are well calibrated in all considered cases that exhibit global convergence---in the sense that they globally contract at the same rate as the truncation error.

link (url) Project Page [BibTex]

pn Kersting, H., Sullivan, T. J., Hennig, P. Convergence Rates of Gaussian ODE Filters arXiv preprint 2018, arXiv:1807.09737 [math.NA], July 2018 (article)

link (url) Project Page [BibTex]

Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences

Kanagawa, M., Hennig, P., Sejdinovic, D., Sriperumbudur, B. K.

Arxiv e-prints, arXiv:1805.08845v1 [stat.ML], 2018 (article)

Abstract

This paper is an attempt to bridge the conceptual gaps between researchers working on the two widely used approaches based on positive definite kernels: Bayesian learning or inference using Gaussian processes on the one side, and frequentist kernel methods based on reproducing kernel Hilbert spaces on the other. It is widely known in machine learning that these two formalisms are closely related; for instance, the estimator of kernel ridge regression is identical to the posterior mean of Gaussian process regression. However, they have been studied and developed almost independently by two essentially separate communities, and this makes it difficult to seamlessly transfer results between them. Our aim is to overcome this potential difficulty. To this end, we review several old and new results and concepts from either side, and juxtapose algorithmic quantities from each framework to highlight close similarities. We also provide discussions on subtle philosophical and theoretical differences between the two approaches.

pn ei

arXiv [BibTex]

pn ei Kanagawa, M., Hennig, P., Sejdinovic, D., Sriperumbudur, B. K. Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences Arxiv e-prints, arXiv:1805.08845v1 [stat.ML], 2018 (article)

arXiv [BibTex]

Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients

Balles, L., Hennig, P.

In Proceedings of the 35th International Conference on Machine Learning (ICML), 2018 (inproceedings) Accepted

Abstract

The ADAM optimizer is exceedingly popular in the deep learning community. Often it works very well, sometimes it doesn't. Why? We interpret ADAM as a combination of two aspects: for each weight, the update direction is determined by the sign of stochastic gradients, whereas the update magnitude is determined by an estimate of their relative variance. We disentangle these two aspects and analyze them in isolation, gaining insight into the mechanisms underlying ADAM. This analysis also extends recent results on adverse effects of ADAM on generalization, isolating the sign aspect as the problematic one. Transferring the variance adaptation to SGD gives rise to a novel method, completing the practitioner's toolbox for problems where ADAM fails.

link (url) Project Page [BibTex]

pn Balles, L., Hennig, P. Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients In Proceedings of the 35th International Conference on Machine Learning (ICML), 2018 (inproceedings) Accepted

link (url) Project Page [BibTex]

A probabilistic model for the numerical solution of initial value problems

Schober, M., Särkkä, S., Philipp Hennig,

Statistics and Computing, Springer US, 2018 (article)

Abstract

We study connections between ordinary differential equation (ODE) solvers and probabilistic regression methods in statistics. We provide a new view of probabilistic ODE solvers as active inference agents operating on stochastic differential equation models that estimate the unknown initial value problem (IVP) solution from approximate observations of the solution derivative, as provided by the ODE dynamics. Adding to this picture, we show that several multistep methods of Nordsieck form can be recast as Kalman filtering on q-times integrated Wiener processes. Doing so provides a family of IVP solvers that return a Gaussian posterior measure, rather than a point estimate. We show that some such methods have low computational overhead, nontrivial convergence order, and that the posterior has a calibrated concentration rate. Additionally, we suggest a step size adaptation algorithm which completes the proposed method to a practically useful implementation, which we experimentally evaluate using a representative set of standard codes in the DETEST benchmark set.

PDF Code DOI Project Page [BibTex]

pn Schober, M., Särkkä, S., Philipp Hennig, A probabilistic model for the numerical solution of initial value problems Statistics and Computing, Springer US, 2018 (article)

PDF Code DOI Project Page [BibTex]

Analytical incorporation of fractionation effects in probabilistic treatment planning for intensity-modulated proton therapy

Wahl, N., Hennig, P., Wieser, H., Bangert, M.

Medical Physics, 2018 (article)

[BibTex]

pn Wahl, N., Hennig, P., Wieser, H., Bangert, M. Analytical incorporation of fractionation effects in probabilistic treatment planning for intensity-modulated proton therapy Medical Physics, 2018 (article)

[BibTex]

2017

On the Design of LQR Kernels for Efficient Controller Learning

Marco, A., Hennig, P., Schaal, S., Trimpe, S.

Proceedings of the 56th IEEE Annual Conference on Decision and Control (CDC), pages: 5193-5200, IEEE, IEEE Conference on Decision and Control, December 2017 (conference)

Abstract

Finding optimal feedback controllers for nonlinear dynamic systems from data is hard. Recently, Bayesian optimization (BO) has been proposed as a powerful framework for direct controller tuning from experimental trials. For selecting the next query point and finding the global optimum, BO relies on a probabilistic description of the latent objective function, typically a Gaussian process (GP). As is shown herein, GPs with a common kernel choice can, however, lead to poor learning outcomes on standard quadratic control problems. For a first-order system, we construct two kernels that specifically leverage the structure of the well-known Linear Quadratic Regulator (LQR), yet retain the flexibility of Bayesian nonparametric learning. Simulations of uncertain linear and nonlinear systems demonstrate that the LQR kernels yield superior learning performance.

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arXiv PDF On the Design of LQR Kernels for Efficient Controller Learning - CDC presentation DOI Project Page [BibTex]

2017

am ics pn Marco, A., Hennig, P., Schaal, S., Trimpe, S. On the Design of LQR Kernels for Efficient Controller Learning Proceedings of the 56th IEEE Annual Conference on Decision and Control (CDC), pages: 5193-5200, IEEE, IEEE Conference on Decision and Control, December 2017 (conference)

arXiv PDF On the Design of LQR Kernels for Efficient Controller Learning - CDC presentation DOI Project Page [BibTex]

Probabilistic Line Searches for Stochastic Optimization

Mahsereci, M., Hennig, P.

Journal of Machine Learning Research, 18(119):1-59, November 2017 (article)

link (url) Project Page [BibTex]

pn Mahsereci, M., Hennig, P. Probabilistic Line Searches for Stochastic Optimization Journal of Machine Learning Research, 18(119):1-59, November 2017 (article)

link (url) Project Page [BibTex]

Coupling Adaptive Batch Sizes with Learning Rates

Balles, L., Romero, J., Hennig, P.

In Proceedings Conference on Uncertainty in Artificial Intelligence (UAI) 2017, pages: 410-419, (Editors: Gal Elidan and Kristian Kersting), Association for Uncertainty in Artificial Intelligence (AUAI), Conference on Uncertainty in Artificial Intelligence (UAI), August 2017 (inproceedings)

Abstract

Mini-batch stochastic gradient descent and variants thereof have become standard for large-scale empirical risk minimization like the training of neural networks. These methods are usually used with a constant batch size chosen by simple empirical inspection. The batch size significantly influences the behavior of the stochastic optimization algorithm, though, since it determines the variance of the gradient estimates. This variance also changes over the optimization process; when using a constant batch size, stability and convergence is thus often enforced by means of a (manually tuned) decreasing learning rate schedule. We propose a practical method for dynamic batch size adaptation. It estimates the variance of the stochastic gradients and adapts the batch size to decrease the variance proportionally to the value of the objective function, removing the need for the aforementioned learning rate decrease. In contrast to recent related work, our algorithm couples the batch size to the learning rate, directly reflecting the known relationship between the two. On three image classification benchmarks, our batch size adaptation yields faster optimization convergence, while simultaneously simplifying learning rate tuning. A TensorFlow implementation is available.

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Code link (url) Project Page [BibTex]

ps pn Balles, L., Romero, J., Hennig, P. Coupling Adaptive Batch Sizes with Learning Rates In Proceedings Conference on Uncertainty in Artificial Intelligence (UAI) 2017, pages: 410-419, (Editors: Gal Elidan and Kristian Kersting), Association for Uncertainty in Artificial Intelligence (AUAI), Conference on Uncertainty in Artificial Intelligence (UAI), August 2017 (inproceedings)

Code link (url) Project Page [BibTex]

Virtual vs. Real: Trading Off Simulations and Physical Experiments in Reinforcement Learning with Bayesian Optimization

Marco, A., Berkenkamp, F., Hennig, P., Schoellig, A. P., Krause, A., Schaal, S., Trimpe, S.

In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pages: 1557-1563, IEEE, Piscataway, NJ, USA, IEEE International Conference on Robotics and Automation (ICRA), May 2017 (inproceedings)

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PDF arXiv ICRA 2017 Spotlight presentation Virtual vs. Real - Video explanation DOI Project Page [BibTex]

am ics pn Marco, A., Berkenkamp, F., Hennig, P., Schoellig, A. P., Krause, A., Schaal, S., Trimpe, S. Virtual vs. Real: Trading Off Simulations and Physical Experiments in Reinforcement Learning with Bayesian Optimization In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pages: 1557-1563, IEEE, Piscataway, NJ, USA, IEEE International Conference on Robotics and Automation (ICRA), May 2017 (inproceedings)

PDF arXiv ICRA 2017 Spotlight presentation Virtual vs. Real - Video explanation DOI Project Page [BibTex]

Fast Bayesian Optimization of Machine Learning Hyperparameters on Large Datasets

Klein, A., Falkner, S., Bartels, S., Hennig, P., Hutter, F.

Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS 2017), 54, pages: 528-536, Proceedings of Machine Learning Research, (Editors: Sign, Aarti and Zhu, Jerry), PMLR, April 2017 (conference)

pdf link (url) Project Page [BibTex]

pn Klein, A., Falkner, S., Bartels, S., Hennig, P., Hutter, F. Fast Bayesian Optimization of Machine Learning Hyperparameters on Large Datasets Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS 2017), 54, pages: 528-536, Proceedings of Machine Learning Research, (Editors: Sign, Aarti and Zhu, Jerry), PMLR, April 2017 (conference)

pdf link (url) Project Page [BibTex]

Computing with Uncertainty

Hennig, P.

2017 (mpi_year_book)

Abstract

Machine learning requires computer hardware to reliable and efficiently compute estimations for ever more complex and fundamentally incomputable quantities. A research team at MPI for Intelligent Systems in Tübingen develops new algorithms which purposely lower the precision of computations and return an explicit measure of uncertainty over the correct result alongside the estimate. Doing so allows for more flexible management of resources, and increases the reliability of intelligent systems.

link (url) DOI [BibTex]

Hennig, P. Computing with Uncertainty 2017 (mpi_year_book)

link (url) DOI [BibTex]

Early Stopping Without a Validation Set

Mahsereci, M., Balles, L., Lassner, C., Hennig, P.

arXiv preprint arXiv:1703.09580, 2017 (article)

Abstract

Early stopping is a widely used technique to prevent poor generalization performance when training an over-expressive model by means of gradient-based optimization. To find a good point to halt the optimizer, a common practice is to split the dataset into a training and a smaller validation set to obtain an ongoing estimate of the generalization performance. In this paper we propose a novel early stopping criterion which is based on fast-to-compute, local statistics of the computed gradients and entirely removes the need for a held-out validation set. Our experiments show that this is a viable approach in the setting of least-squares and logistic regression as well as neural networks.

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link (url) Project Page Project Page [BibTex]

ps pn Mahsereci, M., Balles, L., Lassner, C., Hennig, P. Early Stopping Without a Validation Set arXiv preprint arXiv:1703.09580, 2017 (article)

link (url) Project Page Project Page [BibTex]

Krylov Subspace Recycling for Fast Iterative Least-Squares in Machine Learning

Roos, F. D., Hennig, P.

arXiv preprint arXiv:1706.00241, 2017 (article)

Abstract

Solving symmetric positive definite linear problems is a fundamental computational task in machine learning. The exact solution, famously, is cubicly expensive in the size of the matrix. To alleviate this problem, several linear-time approximations, such as spectral and inducing-point methods, have been suggested and are now in wide use. These are low-rank approximations that choose the low-rank space a priori and do not refine it over time. While this allows linear cost in the data-set size, it also causes a finite, uncorrected approximation error. Authors from numerical linear algebra have explored ways to iteratively refine such low-rank approximations, at a cost of a small number of matrix-vector multiplications. This idea is particularly interesting in the many situations in machine learning where one has to solve a sequence of related symmetric positive definite linear problems. From the machine learning perspective, such deflation methods can be interpreted as transfer learning of a low-rank approximation across a time-series of numerical tasks. We study the use of such methods for our field. Our empirical results show that, on regression and classification problems of intermediate size, this approach can interpolate between low computational cost and numerical precision.

link (url) Project Page [BibTex]

pn Roos, F. D., Hennig, P. Krylov Subspace Recycling for Fast Iterative Least-Squares in Machine Learning arXiv preprint arXiv:1706.00241, 2017 (article)

link (url) Project Page [BibTex]

Fast Bayesian hyperparameter optimization on large datasets

Klein, A., Falkner, S., Bartels, S., Hennig, P., Hutter, F.

Electronic Journal of Statistics, 11, 2017 (article)

[BibTex]

pn Klein, A., Falkner, S., Bartels, S., Hennig, P., Hutter, F. Fast Bayesian hyperparameter optimization on large datasets Electronic Journal of Statistics, 11, 2017 (article)

[BibTex]

Probabilistic Active Learning of Functions in Structural Causal Models

Rubenstein, P. K., Tolstikhin, I., Hennig, P., Schölkopf, B.

2017 (misc)

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Arxiv [BibTex]

ei pn Rubenstein, P. K., Tolstikhin, I., Hennig, P., Schölkopf, B. Probabilistic Active Learning of Functions in Structural Causal Models 2017 (misc)

Arxiv [BibTex]

New Directions for Learning with Kernels and Gaussian Processes (Dagstuhl Seminar 16481)

Gretton, A., Hennig, P., Rasmussen, C., Schölkopf, B.

Dagstuhl Reports, 6(11):142-167, 2017 (book)

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DOI [BibTex]

ei pn Gretton, A., Hennig, P., Rasmussen, C., Schölkopf, B. New Directions for Learning with Kernels and Gaussian Processes (Dagstuhl Seminar 16481) Dagstuhl Reports, 6(11):142-167, 2017 (book)

DOI [BibTex]

Efficiency of analytical and sampling-based uncertainty propagation in intensity-modulated proton therapy

Wahl, N., Hennig, P., Wieser, H. P., Bangert, M.

Physics in Medicine & Biology, 62(14):5790-5807, 2017 (article)

Abstract

The sensitivity of intensity-modulated proton therapy (IMPT) treatment plans to uncertainties can be quantified and mitigated with robust/min-max and stochastic/probabilistic treatment analysis and optimization techniques. Those methods usually rely on sparse random, importance, or worst-case sampling. Inevitably, this imposes a trade-off between computational speed and accuracy of the uncertainty propagation. Here, we investigate analytical probabilistic modeling (APM) as an alternative for uncertainty propagation and minimization in IMPT that does not rely on scenario sampling. APM propagates probability distributions over range and setup uncertainties via a Gaussian pencil-beam approximation into moments of the probability distributions over the resulting dose in closed form. It supports arbitrary correlation models and allows for efficient incorporation of fractionation effects regarding random and systematic errors. We evaluate the trade-off between run-time and accuracy of APM uncertainty computations on three patient datasets. Results are compared against reference computations facilitating importance and random sampling. Two approximation techniques to accelerate uncertainty propagation and minimization based on probabilistic treatment plan optimization are presented. Runtimes are measured on CPU and GPU platforms, dosimetric accuracy is quantified in comparison to a sampling-based benchmark (5000 random samples). APM accurately propagates range and setup uncertainties into dose uncertainties at competitive run-times (GPU ##IMG## [http://ej.iop.org/images/0031-9155/62/14/5790/pmbaa6ec5ieqn001.gif] {$\leqslant {5}$} min). The resulting standard deviation (expectation value) of dose show average global ##IMG## [http://ej.iop.org/images/0031-9155/62/14/5790/pmbaa6ec5ieqn002.gif] {$\gamma_{{3}\% / {3}~{\rm mm}}$} pass rates between 94.2% and 99.9% (98.4% and 100.0%). All investigated importance sampling strategies provided less accuracy at higher run-times considering only a single fraction. Considering fractionation, APM uncertainty propagation and treatment plan optimization was proven to be possible at constant time complexity, while run-times of sampling-based computations are linear in the number of fractions. Using sum sampling within APM, uncertainty propagation can only be accelerated at the cost of reduced accuracy in variance calculations. For probabilistic plan optimization, we were able to approximate the necessary pre-computations within seconds, yielding treatment plans of similar quality as gained from exact uncertainty propagation. APM is suited to enhance the trade-off between speed and accuracy in uncertainty propagation and probabilistic treatment plan optimization, especially in the context of fractionation. This brings fully-fledged APM computations within reach of clinical application.

link (url) [BibTex]

pn Wahl, N., Hennig, P., Wieser, H. P., Bangert, M. Efficiency of analytical and sampling-based uncertainty propagation in intensity-modulated proton therapy Physics in Medicine & Biology, 62(14):5790-5807, 2017 (article)

link (url) [BibTex]

Analytical probabilistic modeling of RBE-weighted dose for ion therapy

Wieser, H., Hennig, P., Wahl, N., Bangert, M.

Physics in Medicine and Biology (PMB), 62(23):8959-8982, 2017 (article)

link (url) [BibTex]

pn Wieser, H., Hennig, P., Wahl, N., Bangert, M. Analytical probabilistic modeling of RBE-weighted dose for ion therapy Physics in Medicine and Biology (PMB), 62(23):8959-8982, 2017 (article)

link (url) [BibTex]

2016

Approximate dual control maintaining the value of information with an application to building control

Klenske, E. D., Hennig, P., Schölkopf, B., Zeilinger, M. N.

In European Control Conference (ECC), pages: 800-806, June 2016 (inproceedings)

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PDF DOI [BibTex]

2016

ei pn Klenske, E. D., Hennig, P., Schölkopf, B., Zeilinger, M. N. Approximate dual control maintaining the value of information with an application to building control In European Control Conference (ECC), pages: 800-806, June 2016 (inproceedings)

PDF DOI [BibTex]

Active Uncertainty Calibration in Bayesian ODE Solvers

Kersting, H., Hennig, P.

Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence (UAI), pages: 309-318, (Editors: Ihler, A. and Janzing, D.), AUAI Press, June 2016 (conference)

Abstract

There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al.~introduced a sampling-based class of methods that are `well-calibrated' in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al.~pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the trade-off between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations.

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link (url) Project Page Project Page [BibTex]

ei pn Kersting, H., Hennig, P. Active Uncertainty Calibration in Bayesian ODE Solvers Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence (UAI), pages: 309-318, (Editors: Ihler, A. and Janzing, D.), AUAI Press, June 2016 (conference)

link (url) Project Page Project Page [BibTex]

Automatic LQR Tuning Based on Gaussian Process Global Optimization

Marco, A., Hennig, P., Bohg, J., Schaal, S., Trimpe, S.

In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pages: 270-277, IEEE, IEEE International Conference on Robotics and Automation, May 2016 (inproceedings)

Abstract

This paper proposes an automatic controller tuning framework based on linear optimal control combined with Bayesian optimization. With this framework, an initial set of controller gains is automatically improved according to a pre-defined performance objective evaluated from experimental data. The underlying Bayesian optimization algorithm is Entropy Search, which represents the latent objective as a Gaussian process and constructs an explicit belief over the location of the objective minimum. This is used to maximize the information gain from each experimental evaluation. Thus, this framework shall yield improved controllers with fewer evaluations compared to alternative approaches. A seven-degree- of-freedom robot arm balancing an inverted pole is used as the experimental demonstrator. Results of a two- and four- dimensional tuning problems highlight the method’s potential for automatic controller tuning on robotic platforms.

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Video - Automatic LQR Tuning Based on Gaussian Process Global Optimization - ICRA 2016 Video - Automatic Controller Tuning on a Two-legged Robot PDF DOI Project Page [BibTex]

am ics pn Marco, A., Hennig, P., Bohg, J., Schaal, S., Trimpe, S. Automatic LQR Tuning Based on Gaussian Process Global Optimization In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pages: 270-277, IEEE, IEEE International Conference on Robotics and Automation, May 2016 (inproceedings)

Video - Automatic LQR Tuning Based on Gaussian Process Global Optimization - ICRA 2016 Video - Automatic Controller Tuning on a Two-legged Robot PDF DOI Project Page [BibTex]

Batch Bayesian Optimization via Local Penalization

González, J., Dai, Z., Hennig, P., Lawrence, N.

Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS), 51, pages: 648-657, JMLR Workshop and Conference Proceedings, (Editors: Gretton, A. and Robert, C. C.), May 2016 (conference)

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link (url) Project Page [BibTex]

ei pn González, J., Dai, Z., Hennig, P., Lawrence, N. Batch Bayesian Optimization via Local Penalization Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS), 51, pages: 648-657, JMLR Workshop and Conference Proceedings, (Editors: Gretton, A. and Robert, C. C.), May 2016 (conference)

link (url) Project Page [BibTex]

Probabilistic Approximate Least-Squares

Bartels, S., Hennig, P.

Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS), 51, pages: 676-684, JMLR Workshop and Conference Proceedings, (Editors: Gretton, A. and Robert, C. C. ), May 2016 (conference)

Abstract

Least-squares and kernel-ridge / Gaussian process regression are among the foundational algorithms of statistics and machine learning. Famously, the worst-case cost of exact nonparametric regression grows cubically with the data-set size; but a growing number of approximations have been developed that estimate good solutions at lower cost. These algorithms typically return point estimators, without measures of uncertainty. Leveraging recent results casting elementary linear algebra operations as probabilistic inference, we propose a new approximate method for nonparametric least-squares that affords a probabilistic uncertainty estimate over the error between the approximate and exact least-squares solution (this is not the same as the posterior variance of the associated Gaussian process regressor). This allows estimating the error of the least-squares solution on a subset of the data relative to the full-data solution. The uncertainty can be used to control the computational effort invested in the approximation. Our algorithm has linear cost in the data-set size, and a simple formal form, so that it can be implemented with a few lines of code in programming languages with linear algebra functionality.

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link (url) Project Page Project Page [BibTex]

ei pn Bartels, S., Hennig, P. Probabilistic Approximate Least-Squares Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS), 51, pages: 676-684, JMLR Workshop and Conference Proceedings, (Editors: Gretton, A. and Robert, C. C. ), May 2016 (conference)

link (url) Project Page Project Page [BibTex]

Gaussian Process-Based Predictive Control for Periodic Error Correction

Klenske, E. D., Zeilinger, M., Schölkopf, B., Hennig, P.

IEEE Transactions on Control Systems Technology , 24(1):110-121, 2016 (article)

ei pn

PDF DOI [BibTex]

ei pn Klenske, E. D., Zeilinger, M., Schölkopf, B., Hennig, P. Gaussian Process-Based Predictive Control for Periodic Error Correction IEEE Transactions on Control Systems Technology , 24(1):110-121, 2016 (article)

PDF DOI [BibTex]

Dual Control for Approximate Bayesian Reinforcement Learning

Klenske, E. D., Hennig, P.

Journal of Machine Learning Research, 17(127):1-30, 2016 (article)

ei pn

PDF link (url) [BibTex]

ei pn Klenske, E. D., Hennig, P. Dual Control for Approximate Bayesian Reinforcement Learning Journal of Machine Learning Research, 17(127):1-30, 2016 (article)

PDF link (url) [BibTex]

2015

Automatic LQR Tuning Based on Gaussian Process Optimization: Early Experimental Results

Marco, A., Hennig, P., Bohg, J., Schaal, S., Trimpe, S.

Machine Learning in Planning and Control of Robot Motion Workshop at the IEEE/RSJ International Conference on Intelligent Robots and Systems (iROS), pages: , , Machine Learning in Planning and Control of Robot Motion Workshop, October 2015 (conference)

Abstract

This paper proposes an automatic controller tuning framework based on linear optimal control combined with Bayesian optimization. With this framework, an initial set of controller gains is automatically improved according to a pre-defined performance objective evaluated from experimental data. The underlying Bayesian optimization algorithm is Entropy Search, which represents the latent objective as a Gaussian process and constructs an explicit belief over the location of the objective minimum. This is used to maximize the information gain from each experimental evaluation. Thus, this framework shall yield improved controllers with fewer evaluations compared to alternative approaches. A seven-degree-of-freedom robot arm balancing an inverted pole is used as the experimental demonstrator. Preliminary results of a low-dimensional tuning problem highlight the method’s potential for automatic controller tuning on robotic platforms.

am ei ics pn

PDF DOI Project Page [BibTex]

2015

am ei ics pn Marco, A., Hennig, P., Bohg, J., Schaal, S., Trimpe, S. Automatic LQR Tuning Based on Gaussian Process Optimization: Early Experimental Results Machine Learning in Planning and Control of Robot Motion Workshop at the IEEE/RSJ International Conference on Intelligent Robots and Systems (iROS), pages: , , Machine Learning in Planning and Control of Robot Motion Workshop, October 2015 (conference)

PDF DOI Project Page [BibTex]

Probabilistic Interpretation of Linear Solvers

Hennig, P.

SIAM Journal on Optimization, 25(1):234-260, 2015 (article)

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Web PDF link (url) DOI [BibTex]

ei pn Hennig, P. Probabilistic Interpretation of Linear Solvers SIAM Journal on Optimization, 25(1):234-260, 2015 (article)

Web PDF link (url) DOI [BibTex]

Inference of Cause and Effect with Unsupervised Inverse Regression

Sgouritsa, E., Janzing, D., Hennig, P., Schölkopf, B.

In Proceedings of the 18th International Conference on Artificial Intelligence and Statistics, 38, pages: 847-855, JMLR Workshop and Conference Proceedings, (Editors: Lebanon, G. and Vishwanathan, S.V.N.), JMLR.org, AISTATS, 2015 (inproceedings)

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Web PDF [BibTex]

ei pn Sgouritsa, E., Janzing, D., Hennig, P., Schölkopf, B. Inference of Cause and Effect with Unsupervised Inverse Regression In Proceedings of the 18th International Conference on Artificial Intelligence and Statistics, 38, pages: 847-855, JMLR Workshop and Conference Proceedings, (Editors: Lebanon, G. and Vishwanathan, S.V.N.), JMLR.org, AISTATS, 2015 (inproceedings)

Web PDF [BibTex]

Probabilistic Line Searches for Stochastic Optimization

Mahsereci, M., Hennig, P.

In Advances in Neural Information Processing Systems 28, pages: 181-189, (Editors: C. Cortes, N.D. Lawrence, D.D. Lee, M. Sugiyama and R. Garnett), Curran Associates, Inc., 29th Annual Conference on Neural Information Processing Systems (NIPS), 2015 (inproceedings)

Abstract

In deterministic optimization, line searches are a standard tool ensuring stability and efficiency. Where only stochastic gradients are available, no direct equivalent has so far been formulated, because uncertain gradients do not allow for a strict sequence of decisions collapsing the search space. We construct a probabilistic line search by combining the structure of existing deterministic methods with notions from Bayesian optimization. Our method retains a Gaussian process surrogate of the univariate optimization objective, and uses a probabilistic belief over the Wolfe conditions to monitor the descent. The algorithm has very low computational cost, and no user-controlled parameters. Experiments show that it effectively removes the need to define a learning rate for stochastic gradient descent. [You can find the matlab research code under `attachments' below. The zip-file contains a minimal working example. The docstring in probLineSearch.m contains additional information. A more polished implementation in C++ will be published here at a later point. For comments and questions about the code please write to mmahsereci@tue.mpg.de.]

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Matlab research code link (url) [BibTex]

ei pn Mahsereci, M., Hennig, P. Probabilistic Line Searches for Stochastic Optimization In Advances in Neural Information Processing Systems 28, pages: 181-189, (Editors: C. Cortes, N.D. Lawrence, D.D. Lee, M. Sugiyama and R. Garnett), Curran Associates, Inc., 29th Annual Conference on Neural Information Processing Systems (NIPS), 2015 (inproceedings)

Matlab research code link (url) [BibTex]

A Random Riemannian Metric for Probabilistic Shortest-Path Tractography

Hauberg, S., Schober, M., Liptrot, M., Hennig, P., Feragen, A.

In 18th International Conference on Medical Image Computing and Computer Assisted Intervention, 9349, pages: 597-604, Lecture Notes in Computer Science, MICCAI, 2015 (inproceedings)

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PDF DOI [BibTex]

ei pn Hauberg, S., Schober, M., Liptrot, M., Hennig, P., Feragen, A. A Random Riemannian Metric for Probabilistic Shortest-Path Tractography In 18th International Conference on Medical Image Computing and Computer Assisted Intervention, 9349, pages: 597-604, Lecture Notes in Computer Science, MICCAI, 2015 (inproceedings)

PDF DOI [BibTex]

Probabilistic numerics and uncertainty in computations

Hennig, P., Osborne, M. A., Girolami, M.

Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471(2179), 2015 (article)

Abstract

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data have led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimizers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.

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PDF DOI [BibTex]

ei pn Hennig, P., Osborne, M. A., Girolami, M. Probabilistic numerics and uncertainty in computations Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471(2179), 2015 (article)

PDF DOI [BibTex]

2014

Probabilistic Progress Bars

Kiefel, M., Schuler, C., Hennig, P.

In Conference on Pattern Recognition (GCPR), 8753, pages: 331-341, Lecture Notes in Computer Science, (Editors: Jiang, X., Hornegger, J., and Koch, R.), Springer, GCPR, September 2014 (inproceedings)

Abstract

Predicting the time at which the integral over a stochastic process reaches a target level is a value of interest in many applications. Often, such computations have to be made at low cost, in real time. As an intuitive example that captures many features of this problem class, we choose progress bars, a ubiquitous element of computer user interfaces. These predictors are usually based on simple point estimators, with no error modelling. This leads to fluctuating behaviour confusing to the user. It also does not provide a distribution prediction (risk values), which are crucial for many other application areas. We construct and empirically evaluate a fast, constant cost algorithm using a Gauss-Markov process model which provides more information to the user.

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website+code pdf DOI [BibTex]

2014

ei ps pn Kiefel, M., Schuler, C., Hennig, P. Probabilistic Progress Bars In Conference on Pattern Recognition (GCPR), 8753, pages: 331-341, Lecture Notes in Computer Science, (Editors: Jiang, X., Hornegger, J., and Koch, R.), Springer, GCPR, September 2014 (inproceedings)

website+code pdf DOI [BibTex]

Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics

Hennig, P., Hauberg, S.

In Proceedings of the 17th International Conference on Artificial Intelligence and Statistics, 33, pages: 347-355, JMLR: Workshop and Conference Proceedings, (Editors: S Kaski and J Corander), Microtome Publishing, Brookline, MA, AISTATS, April 2014 (inproceedings)

Abstract

We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian manifolds, where non-analytic ordinary differential equations are involved in virtually all computations. The probabilistic formulation permits marginalising the uncertainty of the numerical solution such that statistics are less sensitive to inaccuracies. This leads to new Riemannian algorithms for mean value computations and principal geodesic analysis. Marginalisation also means results can be less precise than point estimates, enabling a noticeable speed-up over the state of the art. Our approach is an argument for a wider point that uncertainty caused by numerical calculations should be tracked throughout the pipeline of machine learning algorithms.

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pdf Youtube Supplements Project page link (url) [BibTex]

ei ps pn Hennig, P., Hauberg, S. Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics In Proceedings of the 17th International Conference on Artificial Intelligence and Statistics, 33, pages: 347-355, JMLR: Workshop and Conference Proceedings, (Editors: S Kaski and J Corander), Microtome Publishing, Brookline, MA, AISTATS, April 2014 (inproceedings)

pdf Youtube Supplements Project page link (url) [BibTex]

Local Gaussian Regression

Meier, F., Hennig, P., Schaal, S.

arXiv preprint, March 2014, clmc (misc)

Abstract

Abstract: Locally weighted regression was created as a nonparametric learning method that is computationally efficient, can learn from very large amounts of data and add data incrementally. An interesting feature of locally weighted regression is that it can work with ...

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Web link (url) [BibTex]

am pn Meier, F., Hennig, P., Schaal, S. Local Gaussian Regression arXiv preprint, March 2014, clmc (misc)

Web link (url) [BibTex]

Probabilistic ODE Solvers with Runge-Kutta Means

Schober, M., Duvenaud, D., Hennig, P.

In Advances in Neural Information Processing Systems 27, pages: 739-747, (Editors: Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence and K.Q. Weinberger), Curran Associates, Inc., 28th Annual Conference on Neural Information Processing Systems (NIPS), 2014 (inproceedings)

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Web link (url) [BibTex]

ei pn Schober, M., Duvenaud, D., Hennig, P. Probabilistic ODE Solvers with Runge-Kutta Means In Advances in Neural Information Processing Systems 27, pages: 739-747, (Editors: Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence and K.Q. Weinberger), Curran Associates, Inc., 28th Annual Conference on Neural Information Processing Systems (NIPS), 2014 (inproceedings)

Web link (url) [BibTex]

Active Learning of Linear Embeddings for Gaussian Processes

Garnett, R., Osborne, M., Hennig, P.

In Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence, pages: 230-239, (Editors: NL Zhang and J Tian), AUAI Press , Corvallis, Oregon, UAI2014, 2014, another link: http://arxiv.org/abs/1310.6740 (inproceedings)

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PDF Web [BibTex]

ei pn Garnett, R., Osborne, M., Hennig, P. Active Learning of Linear Embeddings for Gaussian Processes In Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence, pages: 230-239, (Editors: NL Zhang and J Tian), AUAI Press , Corvallis, Oregon, UAI2014, 2014, another link: http://arxiv.org/abs/1310.6740 (inproceedings)

PDF Web [BibTex]

Probabilistic Shortest Path Tractography in DTI Using Gaussian Process ODE Solvers

Schober, M., Kasenburg, N., Feragen, A., Hennig, P., Hauberg, S.

In Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014, Lecture Notes in Computer Science Vol. 8675, pages: 265-272, (Editors: P. Golland, N. Hata, C. Barillot, J. Hornegger and R. Howe), Springer, Heidelberg, MICCAI, 2014 (inproceedings)

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DOI [BibTex]

ei pn Schober, M., Kasenburg, N., Feragen, A., Hennig, P., Hauberg, S. Probabilistic Shortest Path Tractography in DTI Using Gaussian Process ODE Solvers In Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014, Lecture Notes in Computer Science Vol. 8675, pages: 265-272, (Editors: P. Golland, N. Hata, C. Barillot, J. Hornegger and R. Howe), Springer, Heidelberg, MICCAI, 2014 (inproceedings)

DOI [BibTex]

Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature

Gunter, T., Osborne, M., Garnett, R., Hennig, P., Roberts, S.

In Advances in Neural Information Processing Systems 27, pages: 2789-2797, (Editors: Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence and K.Q. Weinberger), Curran Associates, Inc., 28th Annual Conference on Neural Information Processing Systems (NIPS), 2014 (inproceedings)

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Web link (url) [BibTex]

ei pn Gunter, T., Osborne, M., Garnett, R., Hennig, P., Roberts, S. Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature In Advances in Neural Information Processing Systems 27, pages: 2789-2797, (Editors: Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence and K.Q. Weinberger), Curran Associates, Inc., 28th Annual Conference on Neural Information Processing Systems (NIPS), 2014 (inproceedings)

Web link (url) [BibTex]

Incremental Local Gaussian Regression

Meier, F., Hennig, P., Schaal, S.

In Advances in Neural Information Processing Systems 27, pages: 972-980, (Editors: Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence and K.Q. Weinberger), 28th Annual Conference on Neural Information Processing Systems (NIPS), 2014, clmc (inproceedings)

am ei pn

PDF link (url) [BibTex]

am ei pn Meier, F., Hennig, P., Schaal, S. Incremental Local Gaussian Regression In Advances in Neural Information Processing Systems 27, pages: 972-980, (Editors: Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence and K.Q. Weinberger), 28th Annual Conference on Neural Information Processing Systems (NIPS), 2014, clmc (inproceedings)

PDF link (url) [BibTex]

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