Theoretical Analysis of Bayesian Optimisation with Unknown Gaussian Process Hyper-Parameters

Bayesian optimisation has gained great popularity as a tool for optimising the parameters of machine learning algorithms and models. Somewhat ironically, setting up the hyper-parameters of Bayesian optimisation methods is notoriously hard. While reasonable practical solutions have been advanced, they can often fail to find the best optima. Surprisingly, there is little theoretical analysis of this crucial problem in the literature. To address this, we derive a cumulative regret bound for Bayesian optimisation with Gaussian processes and unknown kernel hyper-parameters in the stochastic setting. The bound, which applies to the expected improvement acquisition function and sub-Gaussian observation noise, provides us with guidelines on how to design hyper-parameter estimation methods. A simple simulation demonstrates the importance of following these guidelines.Comment: 16 pages, 1 figur

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

Learning and Designing Stochastic Processes from Logical Constraints

Stochastic processes offer a flexible mathematical formalism to model and reason about systems. Most analysis tools, however, start from the premises that models are fully specified, so that any parameters controlling the system's dynamics must be known exactly. As this is seldom the case, many methods have been devised over the last decade to infer (learn) such parameters from observations of the state of the system. In this paper, we depart from this approach by assuming that our observations are {\it qualitative} properties encoded as satisfaction of linear temporal logic formulae, as opposed to quantitative observations of the state of the system. An important feature of this approach is that it unifies naturally the system identification and the system design problems, where the properties, instead of observations, represent requirements to be satisfied. We develop a principled statistical estimation procedure based on maximising the likelihood of the system's parameters, using recent ideas from statistical machine learning. We demonstrate the efficacy and broad applicability of our method on a range of simple but non-trivial examples, including rumour spreading in social networks and hybrid models of gene regulation

Modelling transcriptional regulation with Gaussian processes

A challenging problem in systems biology is the quantitative modelling of transcriptional regulation. Transcription factors (TFs), which are the key proteins at the centre of the regulatory processes, may be subject to post-translational modification, rendering them unobservable at the mRNA level, or they may be controlled outside of the subsystem being modelled. In both cases, a mechanistic model description of the regula- tory system needs to be able to deal with latent activity profiles of the key regulators. A promising approach to deal with these difficulties is based on using Gaussian processes to define a prior distribution over the latent TF activity profiles. Inference is based on the principles of non-parametric Bayesian statistics, consistently inferring the posterior distribution of the unknown TF activities from the observed expression levels of potential target genes. The present work provides explicit solutions to the differ- ential equations needed to model the data in this manner, as well as the derivatives needed for effective optimisation. The work further explores identifiability issues not fully shown in previous work and looks at how this can cause difficulties with inference. We subsequently look at how the method works on two different TFs, including looking at how the model works with a more biologically realistic mechanistic model. Finally we analyse the effect of more biologically realistic non-Gaussian noise on the biologically realistic model showing how this can cause a reduction in the accuracy of the inference