Sarah Filippi

Benjamin Howson, Ciara Pike-Burke, Sarah Filippi

The stochastic generalised linear bandit is a well-understood model for sequential decision-making problems, with many algorithms achieving near-optimal regret guarantees under immediate feedback. However, the stringent requirement for immediate rewards is unmet in many real-world applications where the reward is almost always delayed. We study the phenomenon of delayed rewards in generalised linear bandits in a theoretical manner. We show that a natural adaptation of an optimistic algorithm to the delayed feedback achieves a regret bound where the penalty for the delays is independent of the horizon. This result significantly improves upon existing work, where the best known regret bound has the delay penalty increasing with the horizon. We verify our theoretical results through experiments on simulated data.

Guiomar Pescador-Barrios, Sarah Filippi, Mark van der Wilk

Many machine learning models require setting a parameter that controls their size before training, e.g. number of neurons in DNNs, or inducing points in GPs. Increasing capacity typically improves performance until all the information from the dataset is captured. After this point, computational cost keeps increasing, without improved performance. This leads to the question "How big is big enough?" We investigate this problem for Gaussian processes (single-layer neural networks) in continual learning. Here, data becomes available incrementally, and the final dataset size will therefore not be known before training, preventing the use of heuristics for setting a fixed model size. We develop a method to automatically adjust model size while maintaining near-optimal performance. Our experimental procedure follows the constraint that any hyperparameters must be set without seeing dataset properties, and we show that our method performs well across diverse datasets without the need to adjust its hyperparameter, showing it requires less tuning than others.

Benjamin Howson, Sarah Filippi, Ciara Pike-Burke

We study the cooperative stochastic $k$-armed bandit problem, where a network of $m$ agents collaborate to find the optimal action. In contrast to most prior work on this problem, which focuses on extending a specific algorithm to the multi-agent setting, we provide a black-box reduction that allows us to extend any single-agent bandit algorithm to the multi-agent setting. Under mild assumptions on the bandit environment, we prove that our reduction transfers the regret guarantees of the single-agent algorithm to the multi-agent setting. These guarantees are tight in subgaussian environments, in that using a near minimax optimal single-player algorithm is near minimax optimal in the multi-player setting up to an additive graph-dependent quantity. Our reduction and theoretical results are also general, and apply to many different bandit settings. By plugging in appropriate single-player algorithms, we can easily develop provably efficient algorithms for many multi-player settings such as heavy-tailed bandits, duelling bandits and bandits with local differential privacy, among others. Experimentally, our approach is competitive with or outperforms specialised multi-agent algorithms.

Michael Komodromos, Marina Evangelou, Sarah Filippi

Variational logistic regression is a popular method for approximate Bayesian inference seeing wide-spread use in many areas of machine learning including: Bayesian optimization, reinforcement learning and multi-instance learning to name a few. However, due to the intractability of the Evidence Lower Bound, authors have turned to the use of Monte Carlo, quadrature or bounds to perform inference, methods which are costly or give poor approximations to the true posterior. In this paper we introduce a new bound for the expectation of softplus function and subsequently show how this can be applied to variational logistic regression and Gaussian process classification. Unlike other bounds, our proposal does not rely on extending the variational family, or introducing additional parameters to ensure the bound is tight. In fact, we show that this bound is tighter than the state-of-the-art, and that the resulting variational posterior achieves state-of-the-art performance, whilst being significantly faster to compute than Monte-Carlo methods.

James Odgers, Ruby Sedgwick, Chrysoula Kappatou et al.

This work develops a Bayesian non-parametric approach to signal separation where the signals may vary according to latent variables. Our key contribution is to augment Gaussian Process Latent Variable Models (GPLVMs) for the case where each data point comprises the weighted sum of a known number of pure component signals, observed across several input locations. Our framework allows arbitrary non-linear variations in the signals while being able to incorporate useful priors for the linear weights, such as summing-to-one. Our contributions are particularly relevant to spectroscopy, where changing conditions may cause the underlying pure component signals to vary from sample to sample. To demonstrate the applicability to both spectroscopy and other domains, we consider several applications: a near-infrared spectroscopy dataset with varying temperatures, a simulated dataset for identifying flow configuration through a pipe, and a dataset for determining the type of rock from its reflectance.

Michael Komodromos, Eric Aboagye, Marina Evangelou et al.

Few Bayesian methods for analyzing high-dimensional sparse survival data provide scalable variable selection, effect estimation and uncertainty quantification. Such methods often either sacrifice uncertainty quantification by computing maximum a posteriori estimates, or quantify the uncertainty at high (unscalable) computational expense. We bridge this gap and develop an interpretable and scalable Bayesian proportional hazards model for prediction and variable selection, referred to as SVB. Our method, based on a mean-field variational approximation, overcomes the high computational cost of MCMC whilst retaining useful features, providing a posterior distribution for the parameters and offering a natural mechanism for variable selection via posterior inclusion probabilities. The performance of our proposed method is assessed via extensive simulations and compared against other state-of-the-art Bayesian variable selection methods, demonstrating comparable or better performance. Finally, we demonstrate how the proposed method can be used for variable selection on two transcriptomic datasets with censored survival outcomes, and how the uncertainty quantification offered by our method can be used to provide an interpretable assessment of patient risk.

Benjamin Howson, Ciara Pike-Burke, Sarah Filippi

There are many algorithms for regret minimisation in episodic reinforcement learning. This problem is well-understood from a theoretical perspective, providing that the sequences of states, actions and rewards associated with each episode are available to the algorithm updating the policy immediately after every interaction with the environment. However, feedback is almost always delayed in practice. In this paper, we study the impact of delayed feedback in episodic reinforcement learning from a theoretical perspective and propose two general-purpose approaches to handling the delays. The first involves updating as soon as new information becomes available, whereas the second waits before using newly observed information to update the policy. For the class of optimistic algorithms and either approach, we show that the regret increases by an additive term involving the number of states, actions, episode length, the expected delay and an algorithm-dependent constant. We empirically investigate the impact of various delay distributions on the regret of optimistic algorithms to validate our theoretical results.

Stamatina Lamprinakou, Mauricio Barahona, Seth Flaxman et al.

The effectiveness of Bayesian Additive Regression Trees (BART) has been demonstrated in a variety of contexts including non-parametric regression and classification. A BART scheme for estimating the intensity of inhomogeneous Poisson processes is introduced. Poisson intensity estimation is a vital task in various applications including medical imaging, astrophysics and network traffic analysis. The new approach enables full posterior inference of the intensity in a non-parametric regression setting. The performance of the novel scheme is demonstrated through simulation studies on synthetic and real datasets up to five dimensions, and the new scheme is compared with alternative approaches.

Onur Teymur, Sarah Filippi

This article introduces a Bayesian nonparametric method for quantifying the relative evidence in a dataset in favour of the dependence or independence of two variables conditional on a third. The approach uses Polya tree priors on spaces of conditional probability densities, accounting for uncertainty in the form of the underlying distributions in a nonparametric way. The Bayesian perspective provides an inherently symmetric probability measure of conditional dependence or independence, a feature particularly advantageous in causal discovery and not employed in existing procedures of this type.

Jonathan Ish-Horowicz, Dana Udwin, Seth Flaxman et al.

While the success of deep neural networks (DNNs) is well-established across a variety of domains, our ability to explain and interpret these methods is limited. Unlike previously proposed local methods which try to explain particular classification decisions, we focus on global interpretability and ask a universally applicable question: given a trained model, which features are the most important? In the context of neural networks, a feature is rarely important on its own, so our strategy is specifically designed to leverage partial covariance structures and incorporate variable dependence into feature ranking. Our methodological contributions in this paper are two-fold. First, we propose an effect size analogue for DNNs that is appropriate for applications with highly collinear predictors (ubiquitous in computer vision). Second, we extend the recently proposed "RelATive cEntrality" (RATE) measure (Crawford et al., 2019) to the Bayesian deep learning setting. RATE applies an information theoretic criterion to the posterior distribution of effect sizes to assess feature significance. We apply our framework to three broad application areas: computer vision, natural language processing, and social science.

Qinyi Zhang, Sarah Filippi, Arthur Gretton et al.

Representations of probability measures in reproducing kernel Hilbert spaces provide a flexible framework for fully nonparametric hypothesis tests of independence, which can capture any type of departure from independence, including nonlinear associations and multivariate interactions. However, these approaches come with an at least quadratic computational cost in the number of observations, which can be prohibitive in many applications. Arguably, it is exactly in such large-scale datasets that capturing any type of dependence is of interest, so striking a favourable tradeoff between computational efficiency and test performance for kernel independence tests would have a direct impact on their applicability in practice. In this contribution, we provide an extensive study of the use of large-scale kernel approximations in the context of independence testing, contrasting block-based, Nystrom and random Fourier feature approaches. Through a variety of synthetic data experiments, it is demonstrated that our novel large scale methods give comparable performance with existing methods whilst using significantly less computation time and memory.

Seth Flaxman, Dino Sejdinovic, John P. Cunningham et al.

Kernel methods are one of the mainstays of machine learning, but the problem of kernel learning remains challenging, with only a few heuristics and very little theory. This is of particular importance in methods based on estimation of kernel mean embeddings of probability measures. For characteristic kernels, which include most commonly used ones, the kernel mean embedding uniquely determines its probability measure, so it can be used to design a powerful statistical testing framework, which includes nonparametric two-sample and independence tests. In practice, however, the performance of these tests can be very sensitive to the choice of kernel and its lengthscale parameters. To address this central issue, we propose a new probabilistic model for kernel mean embeddings, the Bayesian Kernel Embedding model, combining a Gaussian process prior over the Reproducing Kernel Hilbert Space containing the mean embedding with a conjugate likelihood function, thus yielding a closed form posterior over the mean embedding. The posterior mean of our model is closely related to recently proposed shrinkage estimators for kernel mean embeddings, while the posterior uncertainty is a new, interesting feature with various possible applications. Critically for the purposes of kernel learning, our model gives a simple, closed form marginal pseudolikelihood of the observed data given the kernel hyperparameters. This marginal pseudolikelihood can either be optimized to inform the hyperparameter choice or fully Bayesian inference can be used.