Training Discrete Energy-Based Models with Energy Discrepancy

Training energy-based models (EBMs) on discrete spaces is challenging because sampling over such spaces can be difficult. We propose to train discrete EBMs with energy discrepancy (ED), a novel type of contrastive loss functional which only requires the evaluation of the energy function at data points and their perturbed counter parts, thus not relying on sampling strategies like Markov chain Monte Carlo (MCMC). Energy discrepancy offers theoretical guarantees for a broad class of perturbation processes of which we investigate three types: perturbations based on Bernoulli noise, based on deterministic transforms, and based on neighbourhood structures. We demonstrate their relative performance on lattice Ising models, binary synthetic data, and discrete image data sets.

Energy Discrepancies: A Score-Independent Loss for Energy-Based Models

Energy-based models are a simple yet powerful class of probabilistic models, but their widespread adoption has been limited by the computational burden of training them. We propose a novel loss function called Energy Discrepancy (ED) which does not rely on the computation of scores or expensive Markov chain Monte Carlo. We show that ED approaches the explicit score matching and negative log-likelihood loss under different limits, effectively interpolating between both. Consequently, minimum ED estimation overcomes the problem of nearsightedness encountered in score-based estimation methods, while also enjoying theoretical guarantees. Through numerical experiments, we demonstrate that ED learns low-dimensional data distributions faster and more accurately than explicit score matching or contrastive divergence. For high-dimensional image data, we describe how the manifold hypothesis puts limitations on our approach and demonstrate the effectiveness of energy discrepancy by training the energy-based model as a prior of a variational decoder model.

Using Perturbation to Improve Goodness-of-Fit Tests based on Kernelized Stein Discrepancy

Kernelized Stein discrepancy (KSD) is a score-based discrepancy widely used in goodness-of-fit tests. It can be applied even when the target distribution has an unknown normalising factor, such as in Bayesian analysis. We show theoretically and empirically that the KSD test can suffer from low power when the target and the alternative distribution have the same well-separated modes but differ in mixing proportions. We propose to perturb the observed sample via Markov transition kernels, with respect to which the target distribution is invariant. This allows us to then employ the KSD test on the perturbed sample. We provide numerical evidence that with suitably chosen kernels the proposed approach can lead to a substantially higher power than the KSD test.

A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing

We prove a convergence theorem for U-statistics of degree two, where the data dimension $d$ is allowed to scale with sample size $n$. We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite $n$ and $d$, independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with $d$ and the bandwidth.

Batch Bayesian Optimization via Particle Gradient Flows

Bayesian Optimisation (BO) methods seek to find global optima of objective functions which are only available as a black-box or are expensive to evaluate. Such methods construct a surrogate model for the objective function, quantifying the uncertainty in that surrogate through Bayesian inference. Objective evaluations are sequentially determined by maximising an acquisition function at each step. However, this ancilliary optimisation problem can be highly non-trivial to solve, due to the non-convexity of the acquisition function, particularly in the case of batch Bayesian optimisation, where multiple points are selected in every step. In this work we reformulate batch BO as an optimisation problem over the space of probability measures. We construct a new acquisition function based on multipoint expected improvement which is convex over the space of probability measures. Practical schemes for solving this `inner' optimisation problem arise naturally as gradient flows of this objective function. We demonstrate the efficacy of this new method on different benchmark functions and compare with state-of-the-art batch BO methods.

A Spectral Representation of Kernel Stein Discrepancy with Application to Goodness-of-Fit Tests for Measures on Infinite Dimensional Hilbert Spaces

Kernel Stein discrepancy (KSD) is a widely used kernel-based non-parametric measure of discrepancy between probability measures. It is often employed in the scenario where a user has a collection of samples from a candidate probability measure and wishes to compare them against a specified target probability measure. A useful property of KSD is that it may be calculated with samples from only the candidate measure and without knowledge of the normalising constant of the target measure. KSD has been employed in a range of settings including goodness-of-fit testing, parametric inference, MCMC output assessment and generative modelling. Two main issues with current KSD methodology are (i) the lack of applicability beyond the finite dimensional Euclidean setting and (ii) a lack of clarity on what influences KSD performance. This paper provides a novel spectral representation of KSD which remedies both of these, making KSD applicable to Hilbert-valued data and revealing the impact of kernel and Stein operator choice on the KSD. We demonstrate the efficacy of the proposed methodology by performing goodness-of-fit tests for various Gaussian and non-Gaussian functional models in a number of synthetic data experiments.

Ensemble Inference Methods for Models With Noisy and Expensive Likelihoods

The increasing availability of data presents an opportunity to calibrate unknown parameters which appear in complex models of phenomena in the biomedical, physical and social sciences. However, model complexity often leads to parameter-to-data maps which are expensive to evaluate and are only available through noisy approximations. This paper is concerned with the use of interacting particle systems for the solution of the resulting inverse problems for parameters. Of particular interest is the case where the available forward model evaluations are subject to rapid fluctuations, in parameter space, superimposed on the smoothly varying large scale parametric structure of interest. Multiscale analysis is used to study the behaviour of interacting particle system algorithms when such rapid fluctuations, which we refer to as noise, pollute the large scale parametric dependence of the parameter-to-data map. Ensemble Kalman methods (which are derivative-free) and Langevin-based methods (which use the derivative of the parameter-to-data map) are compared in this light. The ensemble Kalman methods are shown to behave favourably in the presence of noise in the parameter-to-data map, whereas Langevin methods are adversely affected. On the other hand, Langevin methods have the correct equilibrium distribution in the setting of noise-free forward models, whilst ensemble Kalman methods only provide an uncontrolled approximation, except in the linear case. Therefore a new class of algorithms, ensemble Gaussian process samplers, which combine the benefits of both ensemble Kalman and Langevin methods, are introduced and shown to perform favourably.

Data Driven Density Functional Theory: A case for Physics Informed Learning

We propose a novel data-driven approach to solving a classical statistical mechanics problem: given data on collective motion of particles, characterise the set of free energies associated with the system of particles. We demonstrate empirically that the particle data contains all the information necessary to infer a free energy. While traditional physical modelling seeks to construct analytically tractable approximations, the proposed approach leverages modern Bayesian computational capabilities to accomplish this in a purely data-driven fashion. The Bayesian paradigm permits us to combine underpinning physical principles with simulation data to obtain uncertainty-quantified predictions of the free energy, in the form of a probability distribution over the family of free energies consistent with the observed particle data. In the present work we focus on classical statistical mechanical systems with excluded volume interactions. Using standard coarse-graining methods, our results can be made applicable to systems with realistic attractive-repulsive interactions. We validate our method on a paradigmatic and computationally cheap case of a one-dimensional fluid. With the appropriate particle data, it is possible to learn canonical and grand-canonical representations of the underlying physical system. Extensions to higher-dimensional systems are conceptually straightforward.

Probabilistic Gradients for Fast Calibration of Differential Equation Models

Calibration of large-scale differential equation models to observational or experimental data is a widespread challenge throughout applied sciences and engineering. A crucial bottleneck in state-of-the art calibration methods is the calculation of local sensitivities, i.e. derivatives of the loss function with respect to the estimated parameters, which often necessitates several numerical solves of the underlying system of partial or ordinary differential equations. In this paper we present a new probabilistic approach to computing local sensitivities. The proposed method has several advantages over classical methods. Firstly, it operates within a constrained computational budget and provides a probabilistic quantification of uncertainty incurred in the sensitivities from this constraint. Secondly, information from previous sensitivity estimates can be recycled in subsequent computations, reducing the overall computational effort for iterative gradient-based calibration methods. The methodology presented is applied to two challenging test problems and compared against classical methods.

A Kernel Two-Sample Test for Functional Data

We propose a nonparametric two-sample test procedure based on Maximum Mean Discrepancy (MMD) for testing the hypothesis that two samples of functions have the same underlying distribution, using a kernel defined on a function space. This construction is motivated by a scaling analysis of the efficiency of MMD-based tests for datasets of increasing dimension. Theoretical properties of kernels on function spaces and their associated MMD are established and employed to ascertain the efficacy of the newly proposed test, as well as to assess the effects of using functional reconstructions based on discretised function samples. The theoretical results are demonstrated over a range of synthetic and real world datasets.