102 research outputs found
Bayesian Quadrature for Multiple Related Integrals
Bayesian probabilistic numerical methods are a set of tools providing
posterior distributions on the output of numerical methods. The use of these
methods is usually motivated by the fact that they can represent our
uncertainty due to incomplete/finite information about the continuous
mathematical problem being approximated. In this paper, we demonstrate that
this paradigm can provide additional advantages, such as the possibility of
transferring information between several numerical methods. This allows users
to represent uncertainty in a more faithful manner and, as a by-product,
provide increased numerical efficiency. We propose the first such numerical
method by extending the well-known Bayesian quadrature algorithm to the case
where we are interested in computing the integral of several related functions.
We then prove convergence rates for the method in the well-specified and
misspecified cases, and demonstrate its efficiency in the context of
multi-fidelity models for complex engineering systems and a problem of global
illumination in computer graphics.Comment: Proceedings of the 35th International Conference on Machine Learning
(ICML), PMLR 80:5369-5378, 201
Convergence Guarantees for Gaussian Process Means with Misspecified Likelihoods and Smoothness
Gaussian processes are ubiquitous in machine learning, statistics, and
applied mathematics. They provide a flexible modelling framework for
approximating functions, whilst simultaneously quantifying uncertainty.
However, this is only true when the model is well-specified, which is often not
the case in practice. In this paper, we study the properties of Gaussian
process means when the smoothness of the model and the likelihood function are
misspecified. In this setting, an important theoretical question of practial
relevance is how accurate the Gaussian process approximations will be given the
difficulty of the problem, our model and the extent of the misspecification.
The answer to this problem is particularly useful since it can inform our
choice of model and experimental design. In particular, we describe how the
experimental design and choice of kernel and kernel hyperparameters can be
adapted to alleviate model misspecification
Deep Bayesian Quadrature Policy Optimization
We study the problem of obtaining accurate policy gradient estimates using a
finite number of samples. Monte-Carlo methods have been the default choice for
policy gradient estimation, despite suffering from high variance in the
gradient estimates. On the other hand, more sample efficient alternatives like
Bayesian quadrature methods have received little attention due to their high
computational complexity. In this work, we propose deep Bayesian quadrature
policy gradient (DBQPG), a computationally efficient high-dimensional
generalization of Bayesian quadrature, for policy gradient estimation. We show
that DBQPG can substitute Monte-Carlo estimation in policy gradient methods,
and demonstrate its effectiveness on a set of continuous control benchmarks. In
comparison to Monte-Carlo estimation, DBQPG provides (i) more accurate gradient
estimates with a significantly lower variance, (ii) a consistent improvement in
the sample complexity and average return for several deep policy gradient
algorithms, and, (iii) the uncertainty in gradient estimation that can be
incorporated to further improve the performance.Comment: Conference paper: AAAI-21. Code available at
https://github.com/Akella17/Deep-Bayesian-Quadrature-Policy-Optimizatio
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