Multi-task Causal Learning with Gaussian Processes

This paper studies the problem of learning the correlation structure of a set of intervention functions defined on the directed acyclic graph (DAG) of a causal model. This is useful when we are interested in jointly learning the causal effects of interventions on different subsets of variables in a DAG, which is common in field such as healthcare or operations research. We propose the first multi-task causal Gaussian process (GP) model, which we call DAG-GP, that allows for information sharing across continuous interventions and across experiments on different variables. DAG-GP accommodates different assumptions in terms of data availability and captures the correlation between functions lying in input spaces of different dimensionality via a well-defined integral operator. We give theoretical results detailing when and how the DAG-GP model can be formulated depending on the DAG. We test both the quality of its predictions and its calibrated uncertainties. Compared to single-task models, DAG-GP achieves the best fitting performance in a variety of real and synthetic settings. In addition, it helps to select optimal interventions faster than competing approaches when used within sequential decision making frameworks, like active learning or Bayesian optimization.

Exoplanet Validation with Machine Learning: 50 new validated Kepler planets

Over 30% of the ~4000 known exoplanets to date have been discovered using 'validation', where the statistical likelihood of a transit arising from a false positive (FP), non-planetary scenario is calculated. For the large majority of these validated planets calculations were performed using the vespa algorithm (Morton et al. 2016). Regardless of the strengths and weaknesses of vespa, it is highly desirable for the catalogue of known planets not to be dependent on a single method. We demonstrate the use of machine learning algorithms, specifically a gaussian process classifier (GPC) reinforced by other models, to perform probabilistic planet validation incorporating prior probabilities for possible FP scenarios. The GPC can attain a mean log-loss per sample of 0.54 when separating confirmed planets from FPs in the Kepler threshold crossing event (TCE) catalogue. Our models can validate thousands of unseen candidates in seconds once applicable vetting metrics are calculated, and can be adapted to work with the active TESS mission, where the large number of observed targets necessitates the use of automated algorithms. We discuss the limitations and caveats of this methodology, and after accounting for possible failure modes newly validate 50 Kepler candidates as planets, sanity checking the validations by confirming them with vespa using up to date stellar information. Concerning discrepancies with vespa arise for many other candidates, which typically resolve in favour of our models. Given such issues, we caution against using single-method planet validation with either method until the discrepancies are fully understood.

Variational Autoencoding of PDE Inverse Problems

Specifying a governing physical model in the presence of missing physics and recovering its parameters are two intertwined and fundamental problems in science. Modern machine learning allows one to circumvent these, via emulators and surrogates, but in doing so disregards prior knowledge and physical laws that are especially important for small data regimes, interpretability, and decision making. In this work we fold the mechanistic model into a flexible data-driven surrogate to arrive at a physically structured decoder network. This provides accelerated inference for the Bayesian inverse problem, and can act as a drop-in regulariser that encodes a-priori physical information. We employ the variational form of the PDE problem and introduce stochastic local approximations as a form of model based data augmentation. We demonstrate both the accuracy and increased computational efficiency of the framework on real world settings and structured spatial processes.

Distribution Regression for Continuous-Time Processes via the Expected Signature

We introduce a learning framework to infer macroscopic properties of an evolving system from longitudinal trajectories of its components. By considering probability measures on continuous paths we view this problem as a distribution regression task for continuous-time processes and propose two distinct solutions leveraging the recently established properties of the expected signature. Firstly, we embed the measures in a Hilbert space, enabling the application of an existing kernel-based technique. Secondly, we recast the complex task of learning a non-linear regression function on probability measures to a simpler functional linear regression on the signature of a single vector-valued path. We provide theoretical results on the universality of both approaches, and demonstrate empirically their robustness to densely and irregularly sampled multivariate time-series, outperforming existing methods adapted to this task on both synthetic and real-world examples from thermodynamics, mathematical finance and agricultural science.

Generalized Bayesian Filtering via Sequential Monte Carlo

We introduce a framework for inference in general state-space hidden Markov models (HMMs) under likelihood misspecification. In particular, we leverage the loss-theoretic perspective of generalized Bayesian inference (GBI) to define generalized filtering recursions in HMMs, that can tackle the problem of inference under model misspecification. In doing so, we arrive at principled procedures for robust inference against observation contamination through the $\beta$-divergence. Operationalizing the proposed framework is made possible via sequential Monte Carlo methods (SMC). The standard particle methods, and their associated convergence results, are readily generalized to the new setting. We demonstrate our approach to object tracking and Gaussian process regression problems, and observe improved performance over standard filtering algorithms.

Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimization

Within the context of empirical risk minimization, see Raginsky, Rakhlin, and Telgarsky (2017), we are concerned with a non-asymptotic analysis of sampling algorithms used in optimization. In particular, we obtain non-asymptotic error bounds for a popular class of algorithms called Stochastic Gradient Langevin Dynamics (SGLD). These results are derived in appropriate Wasserstein distances in the absence of the log-concavity of the target distribution. More precisely, the local Lipschitzness of the stochastic gradient $H(\theta, x)$ is assumed, and furthermore, the dissipativity and convexity at infinity condition are relaxed by removing the uniform dependence in $x$.

Probabilistic sequential matrix factorization

We introduce the probabilistic sequential matrix factorization (PSMF) method for factorizing time-varying and non-stationary datasets consisting of high-dimensional time-series. In particular, we consider nonlinear-Gaussian state-space models in which sequential approximate inference results in the factorization of a data matrix into a dictionary and time-varying coefficients with (possibly nonlinear) Markovian dependencies. The assumed Markovian structure on the coefficients enables us to encode temporal dependencies into a low-dimensional feature space. The proposed inference method is solely based on an approximate extended Kalman filtering scheme which makes the resulting method particularly efficient. The PSMF can account for temporal nonlinearities and, more importantly, can be used to calibrate and estimate generic differentiable nonlinear subspace models. We show that the PSMF can be used in multiple contexts: modelling time series with a periodic subspace, robustifying changepoint detection methods, and imputing missing-data in high-dimensional time-series of air pollutants measured across London.

Multi-resolution Multi-task Gaussian Processes

We consider evidence integration from potentially dependent observation processes under varying spatio-temporal sampling resolutions and noise levels. We develop a multi-resolution multi-task (MRGP) framework while allowing for both inter-task and intra-task multi-resolution and multi-fidelity. We develop shallow Gaussian Process (GP) mixtures that approximate the difficult to estimate joint likelihood with a composite one and deep GP constructions that naturally handle biases in the mean. By doing so, we generalize and outperform state of the art GP compositions and offer information-theoretic corrections and efficient variational approximations. We demonstrate the competitiveness of MRGPs on synthetic settings and on the challenging problem of hyper-local estimation of air pollution levels across London from multiple sensing modalities operating at disparate spatio-temporal resolutions.

On the Constrained Least-cost Tour Problem

We introduce the Constrained Least-cost Tour (CLT) problem: given an undirected graph with weight and cost functions on the edges, minimise the total cost of a tour rooted at a start vertex such that the total weight lies within a given range. CLT is related to the family of Travelling Salesman Problems with Profits, but differs by defining the weight function on edges instead of vertices, and by requiring the total weight to be within a range instead of being at least some quota. We prove CLT is $\mathcal{NP}$-hard, even in the simple case when the input graph is a path. We derive an informative lower bound by relaxing the integrality of edges and propose a heuristic motivated by this relaxation. For the case that requires the tour to be a simple cycle, we develop two heuristics which exploit Suurballe's algorithm to find low-cost, weight-feasible cycles. We demonstrate our algorithms by addressing a real-world problem that affects urban populations: finding routes that minimise air pollution exposure for walking, running and cycling in the city of London.