Efficient sum-of-exponentials approximations for the heat kernel and their applications

In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order $O(\log(\frac{T}{\delta}) (\log(\frac{1}{\epsilon})+\log\log(\frac{T}{\delta})))$ for any x\in\bbR and $\delta \leq t \leq T$, where $\epsilon$ is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only $O(\log^2(\frac{T}{\delta}))$ terms for fixed accuracy $\epsilon$. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For $N_S$ points in the spatial discretization and $N_T$ time steps, the cost is $O(N_S N_T \log^2 \frac{T}{\delta})$ in terms of both memory and CPU time for fixed accuracy $\epsilon$. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.Comment: 23 pages, 5 figures, 3 table

Regret bounds for meta Bayesian optimization with an unknown Gaussian process prior

Bayesian optimization usually assumes that a Bayesian prior is given. However, the strong theoretical guarantees in Bayesian optimization are often regrettably compromised in practice because of unknown parameters in the prior. In this paper, we adopt a variant of empirical Bayes and show that, by estimating the Gaussian process prior from offline data sampled from the same prior and constructing unbiased estimators of the posterior, variants of both GP-UCB and probability of improvement achieve a near-zero regret bound, which decreases to a constant proportional to the observational noise as the number of offline data and the number of online evaluations increase. Empirically, we have verified our approach on challenging simulated robotic problems featuring task and motion planning.Comment: Proceedings of the Thirty-second Conference on Neural Information Processing Systems, 201

Active model learning and diverse action sampling for task and motion planning

The objective of this work is to augment the basic abilities of a robot by learning to use new sensorimotor primitives to enable the solution of complex long-horizon problems. Solving long-horizon problems in complex domains requires flexible generative planning that can combine primitive abilities in novel combinations to solve problems as they arise in the world. In order to plan to combine primitive actions, we must have models of the preconditions and effects of those actions: under what circumstances will executing this primitive achieve some particular effect in the world? We use, and develop novel improvements on, state-of-the-art methods for active learning and sampling. We use Gaussian process methods for learning the conditions of operator effectiveness from small numbers of expensive training examples collected by experimentation on a robot. We develop adaptive sampling methods for generating diverse elements of continuous sets (such as robot configurations and object poses) during planning for solving a new task, so that planning is as efficient as possible. We demonstrate these methods in an integrated system, combining newly learned models with an efficient continuous-space robot task and motion planner to learn to solve long horizon problems more efficiently than was previously possible.Comment: Proceedings of the 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Madrid, Spain. https://www.youtube.com/playlist?list=PLoWhBFPMfSzDbc8CYelsbHZa1d3uz-W_