Andy Neely

Tim Pearce, Russell Tsuchida, Mohamed Zaki et al.

A simple, flexible approach to creating expressive priors in Gaussian process (GP) models makes new kernels from a combination of basic kernels, e.g. summing a periodic and linear kernel can capture seasonal variation with a long term trend. Despite a well-studied link between GPs and Bayesian neural networks (BNNs), the BNN analogue of this has not yet been explored. This paper derives BNN architectures mirroring such kernel combinations. Furthermore, it shows how BNNs can produce periodic kernels, which are often useful in this context. These ideas provide a principled approach to designing BNNs that incorporate prior knowledge about a function. We showcase the practical value of these ideas with illustrative experiments in supervised and reinforcement learning settings.

Tim Pearce, Mohamed Zaki, Andy Neely

Ensembles of neural networks (NNs) have long been used to estimate predictive uncertainty; a small number of NNs are trained from different initialisations and sometimes on differing versions of the dataset. The variance of the ensemble's predictions is interpreted as its epistemic uncertainty. The appeal of ensembling stems from being a collection of regular NNs - this makes them both scalable and easily implementable. They have achieved strong empirical results in recent years, often presented as a practical alternative to more costly Bayesian NNs (BNNs). The departure from Bayesian methodology is of concern since the Bayesian framework provides a principled, widely-accepted approach to handling uncertainty. In this extended abstract we derive and implement a modified NN ensembling scheme, which provides a consistent estimator of the Bayesian posterior in wide NNs - regularising parameters about values drawn from a prior distribution.

Tim Pearce, Felix Leibfried, Alexandra Brintrup et al.

Understanding the uncertainty of a neural network's (NN) predictions is essential for many purposes. The Bayesian framework provides a principled approach to this, however applying it to NNs is challenging due to large numbers of parameters and data. Ensembling NNs provides an easily implementable, scalable method for uncertainty quantification, however, it has been criticised for not being Bayesian. This work proposes one modification to the usual process that we argue does result in approximate Bayesian inference; regularising parameters about values drawn from a distribution which can be set equal to the prior. A theoretical analysis of the procedure in a simplified setting suggests the recovered posterior is centred correctly but tends to have an underestimated marginal variance, and overestimated correlation. However, two conditions can lead to exact recovery. We argue that these conditions are partially present in NNs. Empirical evaluations demonstrate it has an advantage over standard ensembling, and is competitive with variational methods.

Tim Pearce, Nicolas Anastassacos, Mohamed Zaki et al.

The use of ensembles of neural networks (NNs) for the quantification of predictive uncertainty is widespread. However, the current justification is intuitive rather than analytical. This work proposes one minor modification to the normal ensembling methodology, which we prove allows the ensemble to perform Bayesian inference, hence converging to the corresponding Gaussian Process as both the total number of NNs, and the size of each, tend to infinity. This working paper provides early-stage results in a reinforcement learning setting, analysing the practicality of the technique for an ensemble of small, finite number. Using the uncertainty estimates produced by anchored ensembles to govern the exploration-exploitation process results in steadier, more stable learning.

Tim Pearce, Mohamed Zaki, Alexandra Brintrup et al.

This paper considers the generation of prediction intervals (PIs) by neural networks for quantifying uncertainty in regression tasks. It is axiomatic that high-quality PIs should be as narrow as possible, whilst capturing a specified portion of data. We derive a loss function directly from this axiom that requires no distributional assumption. We show how its form derives from a likelihood principle, that it can be used with gradient descent, and that model uncertainty is accounted for in ensembled form. Benchmark experiments show the method outperforms current state-of-the-art uncertainty quantification methods, reducing average PI width by over 10%.