12 results (BibTeX)

2017


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On the Design of LQR Kernels for Efficient Controller Learning

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

Proceedings of the 56th IEEE Conference on Decision and Control, December 2017 (conference) Accepted

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.

arXiv PDF Project Page [BibTex]

2017

arXiv PDF Project Page [BibTex]


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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) [BibTex]


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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]

link (url) [BibTex]


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Follow the Signs for Robust Stochastic Optimization

Balles, L., Hennig, P.

arXiv preprint arXiv:1705.07774, 2017 (article)

Abstract
Stochastic noise on gradients is now a common feature in machine learning. It complicates the design of optimization algorithms, and its effect can be unintuitive: We show that in some settings, particularly those of low signal-to-noise ratio, it can be helpful to discard all but the signs of stochastic gradient elements. In fact, we argue that three popular existing methods already approximate this very paradigm. We devise novel stochastic optimization algorithms that explicitly follow stochastic sign estimates while appropriately accounting for their uncertainty. These methods favorably compare to the state of the art on a number of benchmark problems.

link (url) Project Page [BibTex]


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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.

link (url) Project Page [BibTex]


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Dynamic Time-of-Flight

Schober, M., Adam, A., Yair, O., Mazor, S., Nowozin, S.

The IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2017), 2017 (conference) Accepted

[BibTex]

[BibTex]


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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) [BibTex]

pdf link (url) [BibTex]


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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 International Conference on Robotics and Automation, May 2017 (inproceedings)

PDF arXiv ICRA 2017 Spotlight presentation DOI Project Page [BibTex]

PDF arXiv ICRA 2017 Spotlight presentation DOI Project Page [BibTex]


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Coupling Adaptive Batch Sizes with Learning Rates

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

In Proceedings of the Thirty-Third Conference on Uncertainty in Artificial Intelligence (UAI), pages: 410-419, (Editors: Gal Elidan and Kristian Kersting), 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.

Code link (url) Project Page [BibTex]

Code link (url) Project Page [BibTex]