Edge learning refers to training machine learning models deployed on edge platforms, typically using new data accumulated onboard. The computational limitations on edge devices affect not only model optimisation, but also calculation of the predictive uncertainty of the current model on the unlabelled data, which is vital for informing model updating. In this paper, we investigate edge learning in the context of performing deep image regression on a remote sensing satellite, where a deep network is executed by an onboard computer to regress a scalar $y$ from an input image, e.g., $y$ is the percentage of pixels indicating cloud coverage or land use. We propose an uncertainty-guided edge learning (UGEL) algorithm that can accurately prioritise the data to speed up training convergence of the on-board regression model. Underpinning UGEL is the calculation of predictive uncertainty based on deep beta regression, where a deep network is used to estimate the parameters of a beta distribution for which the target $y$ for an input image has a high likelihood. Compared to established methods for uncertainty estimation that are either too costly on edge devices (e.g., require many forward passes per sample) or make strict assumptions on the predictive distribution (e.g., Gaussian), deep beta regression is computable in a single forward pass and allows more general predictive distributions. Results show that UGEL delivers faster-converging edge learning than active or semi-supervised learning. Code and models are publicly available at https://github.com/anh-vunguyen/UGEL.
Robert Hu, Siu Lun Chau, Jaime Ferrando Huertas et al. · oxford
While preference modelling is becoming one of the pillars of machine learning, the problem of preference explanation remains challenging and underexplored. In this paper, we propose \textsc{Pref-SHAP}, a Shapley value-based model explanation framework for pairwise comparison data. We derive the appropriate value functions for preference models and further extend the framework to model and explain \emph{context specific} information, such as the surface type in a tennis game. To demonstrate the utility of \textsc{Pref-SHAP}, we apply our method to a variety of synthetic and real-world datasets and show that richer and more insightful explanations can be obtained over the baseline.
We develop a framework for generalized variational inference in infinite-dimensional function spaces and use it to construct a method termed Gaussian Wasserstein inference (GWI). GWI leverages the Wasserstein distance between Gaussian measures on the Hilbert space of square-integrable functions in order to determine a variational posterior using a tractable optimisation criterion and avoids pathologies arising in standard variational function space inference. An exciting application of GWI is the ability to use deep neural networks in the variational parametrisation of GWI, combining their superior predictive performance with the principled uncertainty quantification analogous to that of Gaussian processes. The proposed method obtains state-of-the-art performance on several benchmark datasets.
Computational complexity of Bayesian learning is impeding its adoption in practical, large-scale tasks. Despite demonstrations of significant merits such as improved robustness and resilience to unseen or out-of-distribution inputs over their non- Bayesian counterparts, their practical use has faded to near insignificance. In this study, we introduce an innovative framework to mitigate the computational burden of Bayesian neural networks (BNNs). Our approach follows the principle of Bayesian techniques based on deep ensembles, but significantly reduces their cost via multiple low-rank perturbations of parameters arising from a pre-trained neural network. Both vanilla version of ensembles as well as more sophisticated schemes such as Bayesian learning with Stein Variational Gradient Descent (SVGD), previously deemed impractical for large models, can be seamlessly implemented within the proposed framework, called Bayesian Low-Rank LeArning (Bella). In a nutshell, i) Bella achieves a dramatic reduction in the number of trainable parameters required to approximate a Bayesian posterior; and ii) it not only maintains, but in some instances, surpasses the performance of conventional Bayesian learning methods and non-Bayesian baselines. Our results with large-scale tasks such as ImageNet, CAMELYON17, DomainNet, VQA with CLIP, LLaVA demonstrate the effectiveness and versatility of Bella in building highly scalable and practical Bayesian deep models for real-world applications.
Antonin Schrab, Wittawat Jitkrittum, Zoltán Szabó et al.
We discuss how MultiFIT, the Multiscale Fisher's Independence Test for Multivariate Dependence proposed by Gorsky and Ma (2022), compares to existing linear-time kernel tests based on the Hilbert-Schmidt independence criterion (HSIC). We highlight the fact that the levels of the kernel tests at any finite sample size can be controlled exactly, as it is the case with the level of MultiFIT. In our experiments, we observe some of the performance limitations of MultiFIT in terms of test power.
With the widespread application of causal inference, it is increasingly important to have tools which can test for the presence of causal effects in a diverse array of circumstances. In this vein we focus on the problem of testing for \emph{distributional} causal effects, where the treatment affects not just the mean, but also higher order moments of the distribution, as well as multidimensional or structured outcomes. We build upon a previously introduced framework, Counterfactual Mean Embeddings, for representing causal distributions within Reproducing Kernel Hilbert Spaces (RKHS) by proposing new, improved, estimators for the distributional embeddings. These improved estimators are inspired by doubly robust estimators of the causal mean, using a similar form within the kernel space. We analyse these estimators, proving they retain the doubly robust property and have improved convergence rates compared to the original estimators. This leads to new permutation based tests for distributional causal effects, using the estimators we propose as tests statistics. We experimentally and theoretically demonstrate the validity of our tests.
The problem of sequentially maximizing the expectation of a function seeks to maximize the expected value of a function of interest without having direct control on its features. Instead, the distribution of such features depends on a given context and an action taken by an agent. In contrast to Bayesian optimization, the arguments of the function are not under agent's control, but are indirectly determined by the agent's action based on a given context. If the information of the features is to be included in the maximization problem, the full conditional distribution of such features, rather than its expectation only, needs to be accounted for. Furthermore, the function is itself unknown, only counting with noisy observations of such function, and potentially requiring the use of unmatched data sets. We propose a novel algorithm for the aforementioned problem which takes into consideration the uncertainty derived from the estimation of both the conditional distribution of the features and the unknown function, by modeling the former as a Bayesian conditional mean embedding and the latter as a Gaussian process. Our algorithm empirically outperforms the current state-of-the-art algorithm in the experiments conducted.
A directed acyclic graph (DAG) provides valuable prior knowledge that is often discarded in regression tasks in machine learning. We show that the independences arising from the presence of collider structures in DAGs provide meaningful inductive biases, which constrain the regression hypothesis space and improve predictive performance. We introduce collider regression, a framework to incorporate probabilistic causal knowledge from a collider in a regression problem. When the hypothesis space is a reproducing kernel Hilbert space, we prove a strictly positive generalisation benefit under mild assumptions and provide closed-form estimators of the empirical risk minimiser. Experiments on synthetic and climate model data demonstrate performance gains of the proposed methodology.
The counterfactual distribution models the effect of the treatment in the untreated group. While most of the work focuses on the expected values of the treatment effect, one may be interested in the whole counterfactual distribution or other quantities associated to it. Building on the framework of Bayesian conditional mean embeddings, we propose a Bayesian approach for modeling the counterfactual distribution, which leads to quantifying the epistemic uncertainty about the distribution. The framework naturally extends to the setting where one observes multiple treatment effects (e.g. an intermediate effect after an interim period, and an ultimate treatment effect which is of main interest) and allows for additionally modelling uncertainty about the relationship of these effects. For such goal, we present three novel Bayesian methods to estimate the expectation of the ultimate treatment effect, when only noisy samples of the dependence between intermediate and ultimate effects are provided. These methods differ on the source of uncertainty considered and allow for combining two sources of data. Moreover, we generalize these ideas to the off-policy evaluation framework, which can be seen as an extension of the counterfactual estimation problem. We empirically explore the calibration of the algorithms in two different experimental settings which require data fusion, and illustrate the value of considering the uncertainty stemming from the two sources of data.
We consider the problem of conditional independence (CI) testing and adopt a kernel-based approach. Kernel-based CI tests embed variables in reproducing kernel Hilbert spaces, regress their embeddings on the conditioning variables, and test the resulting residuals for marginal independence. This approach yields tests that are sensitive to a broad range of conditional dependencies. Existing methods, however, rely heavily on kernel ridge regression, which is computationally expensive when properly tuned and yields poorly calibrated tests when left untuned, which limits their practical usefulness. We propose the Generalised Kernel Covariance Measure (GKCM), a regression-model-agnostic kernel-based CI test that accommodates a broad class of regression estimators. Building on the Generalised Hilbertian Covariance Measure framework (Lundborg et al., 2022), we characterise conditions under which GKCM satisfies uniform asymptotic level guarantees. In simulations, GKCM paired with tree-based regression models frequently outperforms state-of-the-art CI tests across a diverse range of data-generating processes, achieving better type I error control and competitive or superior power.
We introduce squared neural Poisson point processes (SNEPPPs) by parameterising the intensity function by the squared norm of a two layer neural network. When the hidden layer is fixed and the second layer has a single neuron, our approach resembles previous uses of squared Gaussian process or kernel methods, but allowing the hidden layer to be learnt allows for additional flexibility. In many cases of interest, the integrated intensity function admits a closed form and can be computed in quadratic time in the number of hidden neurons. We enumerate a far more extensive number of such cases than has previously been discussed. Our approach is more memory and time efficient than naive implementations of squared or exponentiated kernel methods or Gaussian processes. Maximum likelihood and maximum a posteriori estimates in a reparameterisation of the final layer of the intensity function can be obtained by solving a (strongly) convex optimisation problem using projected gradient descent. We demonstrate SNEPPPs on real, and synthetic benchmarks, and provide a software implementation. https://github.com/RussellTsuchida/snefy
Label distribution learning (LDL) provides a framework wherein a distribution over categories rather than a single category is predicted, with the aim of addressing ambiguity in labeled data. Existing research on LDL mainly focuses on the task of point estimation, i.e., finding an optimal distribution in the probability simplex conditioned on the given sample. In this paper, we propose a novel label distribution learning model SNEFY-LDL, which estimates a probability distribution of all possible label distributions over the simplex, by unleashing the expressive power of the recently introduced Squared Neural Family (SNEFY), a new class of tractable probability models. As a way to summarize the fitted model, we derive the closed-form label distribution mean, variance and covariance conditioned on the given sample, which can be used to predict the ground-truth label distributions, construct label distribution confidence intervals, and measure the correlations between different labels. Moreover, more information about the label distribution prediction uncertainties can be acquired from the modeled probability density function. Extensive experiments on conformal prediction, active learning and ensemble learning are conducted, verifying SNEFY-LDL's great effectiveness in LDL uncertainty quantification. The source code of this paper is available at https://github.com/daokunzhang/SNEFY-LDL.
Causal inference grows increasingly complex as the number of confounders increases. Given treatments $X$, confounders $Z$ and outcomes $Y$, we develop a non-parametric method to test the \textit{do-null} hypothesis $H_0:\; p(y|\text{\it do}(X=x))=p(y)$ against the general alternative. Building on the Hilbert Schmidt Independence Criterion (HSIC) for marginal independence testing, we propose backdoor-HSIC (bd-HSIC) and demonstrate that it is calibrated and has power for both binary and continuous treatments under a large number of confounders. Additionally, we establish convergence properties of the estimators of covariance operators used in bd-HSIC. We investigate the advantages and disadvantages of bd-HSIC against parametric tests as well as the importance of using the do-null testing in contrast to marginal independence testing or conditional independence testing. A complete implementation can be found at \hyperlink{https://github.com/MrHuff/kgformula}{\texttt{https://github.com/MrHuff/kgformula}}.
Dino Sejdinovic, Heiko Strathmann, Maria Lomeli Garcia et al.
A Kernel Adaptive Metropolis-Hastings algorithm is introduced, for the purpose of sampling from a target distribution with strongly nonlinear support. The algorithm embeds the trajectory of the Markov chain into a reproducing kernel Hilbert space (RKHS), such that the feature space covariance of the samples informs the choice of proposal. The procedure is computationally efficient and straightforward to implement, since the RKHS moves can be integrated out analytically: our proposal distribution in the original space is a normal distribution whose mean and covariance depend on where the current sample lies in the support of the target distribution, and adapts to its local covariance structure. Furthermore, the procedure requires neither gradients nor any other higher order information about the target, making it particularly attractive for contexts such as Pseudo-Marginal MCMC. Kernel Adaptive Metropolis-Hastings outperforms competing fixed and adaptive samplers on multivariate, highly nonlinear target distributions, arising in both real-world and synthetic examples. Code may be downloaded at https://github.com/karlnapf/kameleon-mcmc.
Siu Lun Chau, Antonin Schrab, Arthur Gretton et al. · oxford
We introduce credal two-sample testing, a new hypothesis testing framework for comparing credal sets -- convex sets of probability measures where each element captures aleatoric uncertainty and the set itself represents epistemic uncertainty that arises from the modeller's partial ignorance. Compared to classical two-sample tests, which focus on comparing precise distributions, the proposed framework provides a broader and more versatile set of hypotheses. This approach enables the direct integration of epistemic uncertainty, effectively addressing the challenges arising from partial ignorance in hypothesis testing. By generalising two-sample test to compare credal sets, our framework enables reasoning for equality, inclusion, intersection, and mutual exclusivity, each offering unique insights into the modeller's epistemic beliefs. As the first work on nonparametric hypothesis testing for comparing credal sets, we focus on finitely generated credal sets derived from i.i.d. samples from multiple distributions -- referred to as credal samples. We formalise these tests as two-sample tests with nuisance parameters and introduce the first permutation-based solution for this class of problems, significantly improving existing methods. Our approach properly incorporates the modeller's epistemic uncertainty into hypothesis testing, leading to more robust and credible conclusions, with kernel-based implementations for real-world applications.
Data valuation quantifies the intrinsic quality of individual samples to enable principled data curation, quality control, and robust learning. For time series in critical domains such as healthcare, finance, and industrial monitoring, effective valuation methods are essential yet fundamentally lacking. Existing approaches are either model-dependent, limiting their generalizability, or designed for i.i.d. data and thus fail to capture temporal dependencies, multi-scale patterns, and non-stationary dynamics inherent to sequential data. We introduce TimeLAVA, a learning-agnostic framework that values temporal segments by their marginal contribution to minimizing distributional discrepancy between evaluated and reference data. At its core is a novel Selective Wavelet-based Wasserstein discrepancy combining multi-scale wavelet transforms for temporal localization with unbalanced optimal transport for robustness to distributional shifts. Segment values are efficiently computed via sensitivity analysis without requiring model training and aggregated into point-wise scores. We provide theoretical guarantees linking valuation to model-agnostic generalization and prove bounded sensitivity to outlier contamination. Extensive experiments across anomaly detection, data pruning, and label noise detection demonstrate that TimeLAVA produces significantly more informative value scores than existing methods on diverse real-world datasets.
Kernel embeddings have emerged as a powerful tool for representing probability measures in a variety of statistical inference problems. By mapping probability measures into a reproducing kernel Hilbert space (RKHS), kernel embeddings enable flexible representations of complex relationships between variables. They serve as a mechanism for efficiently transferring the representation of a distribution downstream to other tasks, such as hypothesis testing or causal effect estimation. In the context of causal inference, the main challenges include identifying causal associations and estimating the average treatment effect from observational data, where confounding variables may obscure direct cause-and-effect relationships. Kernel embeddings provide a robust nonparametric framework for addressing these challenges. They allow for the representations of distributions of observational data and their seamless transformation into representations of interventional distributions to estimate relevant causal quantities. We overview recent research that leverages the expressiveness of kernel embeddings in tandem with causal inference.
Kernel conditional mean embeddings (CMEs) offer a powerful framework for representing conditional distribution, but they often face scalability and expressiveness challenges. In this work, we propose a new method that effectively combines the strengths of deep learning with CMEs in order to address these challenges. Specifically, our approach leverages the end-to-end neural network (NN) optimization framework using a kernel-based objective. This design circumvents the computationally expensive Gram matrix inversion required by current CME methods. To further enhance performance, we provide efficient strategies to optimize the remaining kernel hyperparameters. In conditional density estimation tasks, our NN-CME hybrid achieves competitive performance and often surpasses existing deep learning-based methods. Lastly, we showcase its remarkable versatility by seamlessly integrating it into reinforcement learning (RL) contexts. Building on Q-learning, our approach naturally leads to a new variant of distributional RL methods, which demonstrates consistent effectiveness across different environments.
Agent skills externalise reusable agent-facing behavioural knowledge and guidance as persistent artefacts that can be discovered, activated, and interpreted by LLM agents. Although a skill artefact is static at rest, its architectural responsibilities arise in use, when the artefact is selected for a run, bound to context and authority constraints, interpreted by a stochastic agent, and recorded as run evidence. We call this run-specific relation skill-in-use. This paper studies agent skill harnessing: the architectural responsibilities that govern the transition from skill artefacts to skill-in-use, bound the executable consequences associated with skill-in-use, and capture evidence for attribution, verification, repair, and evolution. This paper provides a catalogue of ten empirically grounded architectural patterns (five core, five supporting) for skill harnessing and synthesises them into a reference architecture with four responsibility layers: Supply Chain, Mediation, Execution Control, and Evidence & Feedback. We evaluate the architecture through cross-instantiation across 8 selected systems. The resulting patterns and reference architecture provide a vocabulary and diagnostic frame for analysing skill-harnessing responsibilities across agent systems.
Estimating the Conditional Average Treatment Effect (CATE) is often constrained by the high cost of obtaining outcome measurements, making active learning essential. However, conventional active learning strategies suffer from a fundamental objective mismatch. They are designed to reduce uncertainty in model parameters or in observable factual outcomes, failing to directly target the unobservable causal quantities that are the true objects of interest. To address this misalignment, we introduce the principle of causal objective alignment, which posits that acquisition functions should target unobservable causal quantities, such as the potential outcomes and the CATE, rather than indirect proxies. We operationalize this principle through the Causal-EPIG framework, which adapts the information-theoretic criterion of Expected Predictive Information Gain (EPIG) to explicitly quantify the value of a query in terms of reducing uncertainty about unobservable causal quantities. From this unified framework, we derive two distinct strategies that embody a fundamental trade-off: a comprehensive approach that robustly models the full causal mechanisms via the joint potential outcomes, and a focused approach that directly targets the CATE estimand for maximum sample efficiency. Extensive experiments demonstrate that our strategies consistently outperform standard baselines, and crucially, reveal that the optimal strategy is context-dependent, contingent on the base estimator and data complexity. Our framework thus provides a principled guide for sample-efficient CATE estimation in practice.
Russell Tsuchida, Jiawei Liu, Cheng Soon Ong et al.
We introduce squared families, which are families of probability densities obtained by squaring a linear transformation of a statistic. Squared families are singular, however their singularity can easily be handled so that they form regular models. After handling the singularity, squared families possess many convenient properties. Their Fisher information is a conformal transformation of the Hessian metric induced from a Bregman generator. The Bregman generator is the normalising constant, and yields a statistical divergence on the family. The normalising constant admits a helpful parameter-integral factorisation, meaning that only one parameter-independent integral needs to be computed for all normalising constants in the family, unlike in exponential families. Finally, the squared family kernel is the only integral that needs to be computed for the Fisher information, statistical divergence and normalising constant. We then describe how squared families are special in the broader class of $g$-families, which are obtained by applying a sufficiently regular function $g$ to a linear transformation of a statistic. After removing special singularities, positively homogeneous families and exponential families are the only $g$-families for which the Fisher information is a conformal transformation of the Hessian metric, where the generator depends on the parameter only through the normalising constant. Even-order monomial families also admit parameter-integral factorisations, unlike exponential families. We study parameter estimation and density estimation in squared families, in the well-specified and misspecified settings. We use a universal approximation property to show that squared families can learn sufficiently well-behaved target densities at a rate of $\mathcal{O}(N^{-1/2})+C n^{-1/4}$, where $N$ is the number of datapoints, $n$ is the number of parameters, and $C$ is some constant.
We propose a scalable inference algorithm for Bayes posteriors defined on a reproducing kernel Hilbert space (RKHS). Given a likelihood function and a Gaussian random element representing the prior, the corresponding Bayes posterior measure $Π_{\text{B}}$ can be obtained as the stationary distribution of an RKHS-valued Langevin diffusion. We approximate the infinite-dimensional Langevin diffusion via a projection onto the first $M$ components of the Kosambi-Karhunen-Loève expansion. Exploiting the thus obtained approximate posterior for these $M$ components, we perform inference for $Π_{\text{B}}$ by relying on the law of total probability and a sufficiency assumption. The resulting method scales as $O(M^3+JM^2)$, where $J$ is the number of samples produced from the posterior measure $Π_{\text{B}}$. Interestingly, the algorithm recovers the posterior arising from the sparse variational Gaussian process (SVGP) (see Titsias, 2009) as a special case, owed to the fact that the sufficiency assumption underlies both methods. However, whereas the SVGP is parametrically constrained to be a Gaussian process, our method is based on a non-parametric variational family $\mathcal{P}(\mathbb{R}^M)$ consisting of all probability measures on $\mathbb{R}^M$. As a result, our method is provably close to the optimal $M$-dimensional variational approximation of the Bayes posterior $Π_{\text{B}}$ in $\mathcal{P}(\mathbb{R}^M)$ for convex and Lipschitz continuous negative log likelihoods, and coincides with SVGP for the special case of a Gaussian error likelihood.
When working in a high-risk setting, having well calibrated probabilistic predictive models is a crucial requirement. However, estimators for calibration error are not always able to correctly distinguish which model is better calibrated. We propose the \emph{conditional kernel calibration error} (CKCE) which is based on the Hilbert-Schmidt norm of the difference between conditional mean operators. By working directly with the definition of strong calibration as the distance between conditional distributions, which we represent by their embeddings in reproducing kernel Hilbert spaces, the CKCE is less sensitive to the marginal distribution of predictive models. This makes it more effective for relative comparisons than previously proposed calibration metrics. Our experiments, using both synthetic and real data, show that CKCE provides a more consistent ranking of models by their calibration error and is more robust against distribution shift.
Mengyan Zhang, Shahine Bouabid, Cheng Soon Ong et al.
We develop the framework of Indirect Query Bayesian Optimization (IQBO), a new class of Bayesian optimization problems where the integrated feedback is given via a conditional expectation of the unknown function $f$ to be optimized. The underlying conditional distribution can be unknown and learned from data. The goal is to find the global optimum of $f$ by adaptively querying and observing in the space transformed by the conditional distribution. This is motivated by real-world applications where one cannot access direct feedback due to privacy, hardware or computational constraints. We propose the Conditional Max-Value Entropy Search (CMES) acquisition function to address this novel setting, and propose a hierarchical search algorithm with multi-resolution feedback to improve computational efficiency. We show regret bounds for our proposed methods and demonstrate the effectiveness of our approaches on simulated optimization tasks.
Rafael Oliveira, Dino Sejdinovic, David Howard et al.
The process of calibrating computer models of natural phenomena is essential for applications in the physical sciences, where plenty of domain knowledge can be embedded into simulations and then calibrated against real observations. Current machine learning approaches, however, mostly rely on rerunning simulations over a fixed set of designs available in the observed data, potentially neglecting informative correlations across the design space and requiring a large amount of simulations. Instead, we consider the calibration process from the perspective of Bayesian adaptive experimental design and propose a data-efficient algorithm to run maximally informative simulations within a batch-sequential process. At each round, the algorithm jointly estimates the parameters of the posterior distribution and optimal designs by maximising a variational lower bound of the expected information gain. The simulator is modelled as a sample from a Gaussian process, which allows us to correlate simulations and observed data with the unknown calibration parameters. We show the benefits of our method when compared to related approaches across synthetic and real-data problems.
We present a novel approach for explaining Gaussian processes (GPs) that can utilize the full analytical covariance structure present in GPs. Our method is based on the popular solution concept of Shapley values extended to stochastic cooperative games, resulting in explanations that are random variables. The GP explanations generated using our approach satisfy similar favorable axioms to standard Shapley values and possess a tractable covariance function across features and data observations. This covariance allows for quantifying explanation uncertainties and studying the statistical dependencies between explanations. We further extend our framework to the problem of predictive explanation, and propose a Shapley prior over the explanation function to predict Shapley values for new data based on previously computed ones. Our extensive illustrations demonstrate the effectiveness of the proposed approach.
Veit David Wild, Sahra Ghalebikesabi, Dino Sejdinovic et al.
We establish the first mathematically rigorous link between Bayesian, variational Bayesian, and ensemble methods. A key step towards this it to reformulate the non-convex optimisation problem typically encountered in deep learning as a convex optimisation in the space of probability measures. On a technical level, our contribution amounts to studying generalised variational inference through the lense of Wasserstein gradient flows. The result is a unified theory of various seemingly disconnected approaches that are commonly used for uncertainty quantification in deep learning -- including deep ensembles and (variational) Bayesian methods. This offers a fresh perspective on the reasons behind the success of deep ensembles over procedures based on parameterised variational inference, and allows the derivation of new ensembling schemes with convergence guarantees. We showcase this by proposing a family of interacting deep ensembles with direct parallels to the interactions of particle systems in thermodynamics, and use our theory to prove the convergence of these algorithms to a well-defined global minimiser on the space of probability measures.
Flexible models for probability distributions are an essential ingredient in many machine learning tasks. We develop and investigate a new class of probability distributions, which we call a Squared Neural Family (SNEFY), formed by squaring the 2-norm of a neural network and normalising it with respect to a base measure. Following the reasoning similar to the well established connections between infinitely wide neural networks and Gaussian processes, we show that SNEFYs admit closed form normalising constants in many cases of interest, thereby resulting in flexible yet fully tractable density models. SNEFYs strictly generalise classical exponential families, are closed under conditioning, and have tractable marginal distributions. Their utility is illustrated on a variety of density estimation, conditional density estimation, and density estimation with missing data tasks.
In this paper we look at popular fairness methods that use causal counterfactuals. These methods capture the intuitive notion that a prediction is fair if it coincides with the prediction that would have been made if someone's race, gender or religion were counterfactually different. In order to achieve this, we must have causal models that are able to capture what someone would be like if we were to counterfactually change these traits. However, we argue that any model that can do this must lie outside the particularly well behaved class that is commonly considered in the fairness literature. This is because in fairness settings, models in this class entail a particularly strong causal assumption, normally only seen in a randomised controlled trial. We argue that in general this is unlikely to hold. Furthermore, we show in many cases it can be explicitly rejected due to the fact that samples are selected from a wider population. We show this creates difficulties for counterfactual fairness as well as for the application of more general causal fairness methods.
Kernel matrix-vector multiplication (KMVM) is a foundational operation in machine learning and scientific computing. However, as KMVM tends to scale quadratically in both memory and time, applications are often limited by these computational constraints. In this paper, we propose a novel approximation procedure coined \textit{Faster-Fast and Free Memory Method} ($\fthreem$) to address these scaling issues of KMVM for tall~($10^8\sim 10^9$) and skinny~($D\leq7$) data. Extensive experiments demonstrate that $\fthreem$ has empirical \emph{linear time and memory} complexity with a relative error of order $10^{-3}$ and can compute a full KMVM for a billion points \emph{in under a minute} on a high-end GPU, leading to a significant speed-up in comparison to existing CPU methods. We demonstrate the utility of our procedure by applying it as a drop-in for the state-of-the-art GPU-based linear solver FALKON, \emph{improving speed 1.5-5.5 times} at the cost of $<1\%$ drop in accuracy. We further demonstrate competitive results on \emph{Gaussian Process regression} coupled with significant speedups on a variety of real-world datasets.
Jonas Schuff, Dominic T. Lennon, Simon Geyer et al.
Pauli spin blockade (PSB) can be employed as a great resource for spin qubit initialisation and readout even at elevated temperatures but it can be difficult to identify. We present a machine learning algorithm capable of automatically identifying PSB using charge transport measurements. The scarcity of PSB data is circumvented by training the algorithm with simulated data and by using cross-device validation. We demonstrate our approach on a silicon field-effect transistor device and report an accuracy of 96% on different test devices, giving evidence that the approach is robust to device variability. The approach is expected to be employable across all types of quantum dot devices.
The discrepancies between reality and simulation impede the optimisation and scalability of solid-state quantum devices. Disorder induced by the unpredictable distribution of material defects is one of the major contributions to the reality gap. We bridge this gap using physics-aware machine learning, in particular, using an approach combining a physical model, deep learning, Gaussian random field, and Bayesian inference. This approach has enabled us to infer the disorder potential of a nanoscale electronic device from electron transport data. This inference is validated by verifying the algorithm's predictions about the gate voltage values required for a laterally-defined quantum dot device in AlGaAs/GaAs to produce current features corresponding to a double quantum dot regime.
The potential of Si and SiGe-based devices for the scaling of quantum circuits is tainted by device variability. Each device needs to be tuned to operation conditions. We give a key step towards tackling this variability with an algorithm that, without modification, is capable of tuning a 4-gate Si FinFET, a 5-gate GeSi nanowire and a 7-gate SiGe heterostructure double quantum dot device from scratch. We achieve tuning times of 30, 10, and 92 minutes, respectively. The algorithm also provides insight into the parameter space landscape for each of these devices. These results show that overarching solutions for the tuning of quantum devices are enabled by machine learning.
Siu Lun Chau, Jean-François Ton, Javier González et al.
While causal models are becoming one of the mainstays of machine learning, the problem of uncertainty quantification in causal inference remains challenging. In this paper, we study the causal data fusion problem, where datasets pertaining to multiple causal graphs are combined to estimate the average treatment effect of a target variable. As data arises from multiple sources and can vary in quality and quantity, principled uncertainty quantification becomes essential. To that end, we introduce Bayesian Interventional Mean Processes, a framework which combines ideas from probabilistic integration and kernel mean embeddings to represent interventional distributions in the reproducing kernel Hilbert space, while taking into account the uncertainty within each causal graph. To demonstrate the utility of our uncertainty estimation, we apply our method to the Causal Bayesian Optimisation task and show improvements over state-of-the-art methods.
We investigate the connections between sparse approximation methods for making kernel methods and Gaussian processes (GPs) scalable to large-scale data, focusing on the Nyström method and the Sparse Variational Gaussian Processes (SVGP). While sparse approximation methods for GPs and kernel methods share some algebraic similarities, the literature lacks a deep understanding of how and why they are related. This may pose an obstacle to the communications between the GP and kernel communities, making it difficult to transfer results from one side to the other. Our motivation is to remove this obstacle, by clarifying the connections between the sparse approximations for GPs and kernel methods. In this work, we study the two popular approaches, the Nyström and SVGP approximations, in the context of a regression problem, and establish various connections and equivalences between them. In particular, we provide an RKHS interpretation of the SVGP approximation, and show that the Evidence Lower Bound of the SVGP contains the objective function of the Nyström approximation, revealing the origin of the algebraic equivalence between the two approaches. We also study recently established convergence results for the SVGP and how they are related to the approximation quality of the Nyström method.
Refining low-resolution (LR) spatial fields with high-resolution (HR) information, often known as statistical downscaling, is challenging as the diversity of spatial datasets often prevents direct matching of observations. Yet, when LR samples are modeled as aggregate conditional means of HR samples with respect to a mediating variable that is globally observed, the recovery of the underlying fine-grained field can be framed as taking an "inverse" of the conditional expectation, namely a deconditioning problem. In this work, we propose a Bayesian formulation of deconditioning which naturally recovers the initial reproducing kernel Hilbert space formulation from Hsu and Ramos (2019). We extend deconditioning to a downscaling setup and devise efficient conditional mean embedding estimator for multiresolution data. By treating conditional expectations as inter-domain features of the underlying field, a posterior for the latent field can be established as a solution to the deconditioning problem. Furthermore, we show that this solution can be viewed as a two-staged vector-valued kernel ridge regressor and show that it has a minimax optimal convergence rate under mild assumptions. Lastly, we demonstrate its proficiency in a synthetic and a real-world atmospheric field downscaling problem, showing substantial improvements over existing methods.
We consider frequently used scoring rules for right-censored survival regression models such as time-dependent concordance, survival-CRPS, integrated Brier score and integrated binomial log-likelihood, and prove that neither of them is a proper scoring rule. This means that the true survival distribution may be scored worse than incorrect distributions, leading to inaccurate estimation. We prove that, in contrast to these scores, the right-censored log-likelihood is a proper scoring rule, i.e., the highest expected score is achieved by the true distribution. Despite this, modern feed-forward neural-network-based survival regression models are unable to train and validate directly on the right-censored log-likelihood, due to its intractability, and resort to the aforementioned alternatives, i.e., non-proper scoring rules. We therefore propose a simple novel survival regression method capable of directly optimizing log-likelihood using a monotonic restriction on the time-dependent weights, coined SurvivalMonotonic-net (SuMo-net). SuMo-net achieves state-of-the-art log-likelihood scores across several datasets with 20--100$\times$ computational speedup on inference over existing state-of-the-art neural methods, and is readily applicable to datasets with several million observations.
Inter-domain Gaussian processes (GPs) allow for high flexibility and low computational cost when performing approximate inference in GP models. They are particularly suitable for modeling data exhibiting global structure but are limited to stationary covariance functions and thus fail to model non-stationary data effectively. We propose Inter-domain Deep Gaussian Processes, an extension of inter-domain shallow GPs that combines the advantages of inter-domain and deep Gaussian processes (DGPs), and demonstrate how to leverage existing approximate inference methods to perform simple and scalable approximate inference using inter-domain features in DGPs. We assess the performance of our method on a range of regression tasks and demonstrate that it outperforms inter-domain shallow GPs and conventional DGPs on challenging large-scale real-world datasets exhibiting both global structure as well as a high-degree of non-stationarity.
The problem of graph learning concerns the construction of an explicit topological structure revealing the relationship between nodes representing data entities, which plays an increasingly important role in the success of many graph-based representations and algorithms in the field of machine learning and graph signal processing. In this paper, we propose a novel graph learning framework that incorporates the node-side and observation-side information, and in particular the covariates that help to explain the dependency structures in graph signals. To this end, we consider graph signals as functions in the reproducing kernel Hilbert space associated with a Kronecker product kernel, and integrate functional learning with smoothness-promoting graph learning to learn a graph representing the relationship between nodes. The functional learning increases the robustness of graph learning against missing and incomplete information in the graph signals. In addition, we develop a novel graph-based regularisation method which, when combined with the Kronecker product kernel, enables our model to capture both the dependency explained by the graph and the dependency due to graph signals observed under different but related circumstances, e.g. different points in time. The latter means the graph signals are free from the i.i.d. assumptions required by the classical graph learning models. Experiments on both synthetic and real-world data show that our methods outperform the state-of-the-art models in learning a meaningful graph topology from graph signals, in particular under heavy noise, missing values, and multiple dependency.
Modern machine learning often operates in the regime where the number of parameters is much higher than the number of data points, with zero training loss and yet good generalization, thereby contradicting the classical bias-variance trade-off. This \textit{benign overfitting} phenomenon has recently been characterized using so called \textit{double descent} curves where the risk undergoes another descent (in addition to the classical U-shaped learning curve when the number of parameters is small) as we increase the number of parameters beyond a certain threshold. In this paper, we examine the conditions under which \textit{Benign Overfitting} occurs in the random feature (RF) models, i.e. in a two-layer neural network with fixed first layer weights. We adopt a new view of random feature and show that \textit{benign overfitting} arises due to the noise which resides in such features (the noise may already be present in the data and propagate to the features or it may be added by the user to the features directly) and plays an important implicit regularization role in the phenomenon.
Anthony Caterini, Rob Cornish, Dino Sejdinovic et al.
Continuously-indexed flows (CIFs) have recently achieved improvements over baseline normalizing flows on a variety of density estimation tasks. CIFs do not possess a closed-form marginal density, and so, unlike standard flows, cannot be plugged in directly to a variational inference (VI) scheme in order to produce a more expressive family of approximate posteriors. However, we show here how CIFs can be used as part of an auxiliary VI scheme to formulate and train expressive posterior approximations in a natural way. We exploit the conditional independence structure of multi-layer CIFs to build the required auxiliary inference models, which we show empirically yield low-variance estimators of the model evidence. We then demonstrate the advantages of CIFs over baseline flows in VI problems when the posterior distribution of interest possesses a complicated topology, obtaining improved results in both the Bayesian inference and surrogate maximum likelihood settings.
Jean-Francois Ton, Dino Sejdinovic, Kenji Fukumizu
The inaccessibility of controlled randomized trials due to inherent constraints in many fields of science has been a fundamental issue in causal inference. In this paper, we focus on distinguishing the cause from effect in the bivariate setting under limited observational data. Based on recent developments in meta learning as well as in causal inference, we introduce a novel generative model that allows distinguishing cause and effect in the small data setting. Using a learnt task variable that contains distributional information of each dataset, we propose an end-to-end algorithm that makes use of similar training datasets at test time. We demonstrate our method on various synthetic as well as real-world data and show that it is able to maintain high accuracy in detecting directions across varying dataset sizes.
We revisit widely used preferential Gaussian processes by Chu et al.(2005) and challenge their modelling assumption that imposes rankability of data items via latent utility function values. We propose a generalisation of pgp which can capture more expressive latent preferential structures in the data and thus be used to model inconsistent preferences, i.e. where transitivity is violated, or to discover clusters of comparable items via spectral decomposition of the learned preference functions. We also consider the properties of associated covariance kernel functions and its reproducing kernel Hilbert Space (RKHS), giving a simple construction that satisfies universality in the space of preference functions. Finally, we provide an extensive set of numerical experiments on simulated and real-world datasets showcasing the competitiveness of our proposed method with state-of-the-art. Our experimental findings support the conjecture that violations of rankability are ubiquitous in real-world preferential data.
We consider spectral approaches to the problem of ranking n players given their incomplete and noisy pairwise comparisons, but revisit this classical problem in light of player covariate information. We propose three spectral ranking methods that incorporate player covariates and are based on seriation, low-rank structure assumption and canonical correlation, respectively. Extensive numerical simulations on both synthetic and real-world data sets demonstrated that our proposed methods compare favorably to existing state-of-the-art covariate-based ranking algorithms.
We outline an inherent weakness of tensor factorization models when latent factors are expressed as a function of side information and propose a novel method to mitigate this weakness. We coin our method \textit{Kernel Fried Tensor}(KFT) and present it as a large scale forecasting tool for high dimensional data. Our results show superior performance against \textit{LightGBM} and \textit{Field Aware Factorization Machines}(FFM), two algorithms with proven track records widely used in industrial forecasting. We also develop a variational inference framework for KFT and associate our forecasts with calibrated uncertainty estimates on three large scale datasets. Furthermore, KFT is empirically shown to be robust against uninformative side information in terms of constants and Gaussian noise.
Quantum devices with a large number of gate electrodes allow for precise control of device parameters. This capability is hard to fully exploit due to the complex dependence of these parameters on applied gate voltages. We experimentally demonstrate an algorithm capable of fine-tuning several device parameters at once. The algorithm acquires a measurement and assigns it a score using a variational auto-encoder. Gate voltage settings are set to optimise this score in real-time in an unsupervised fashion. We report fine-tuning times of a double quantum dot device within approximately 40 min.
Device variability is a bottleneck for the scalability of semiconductor quantum devices. Increasing device control comes at the cost of a large parameter space that has to be explored in order to find the optimal operating conditions. We demonstrate a statistical tuning algorithm that navigates this entire parameter space, using just a few modelling assumptions, in the search for specific electron transport features. We focused on gate-defined quantum dot devices, demonstrating fully automated tuning of two different devices to double quantum dot regimes in an up to eight-dimensional gate voltage space. We considered a parameter space defined by the maximum range of each gate voltage in these devices, demonstrating expected tuning in under 70 minutes. This performance exceeded a human benchmark, although we recognise that there is room for improvement in the performance of both humans and machines. Our approach is approximately 180 times faster than a pure random search of the parameter space, and it is readily applicable to different material systems and device architectures. With an efficient navigation of the gate voltage space we are able to give a quantitative measurement of device variability, from one device to another and after a thermal cycle of a device. This is a key demonstration of the use of machine learning techniques to explore and optimise the parameter space of quantum devices and overcome the challenge of device variability.
Tamara Fernandez, Arthur Gretton, David Rindt et al.
We introduce a general non-parametric independence test between right-censored survival times and covariates, which may be multivariate. Our test statistic has a dual interpretation, first in terms of the supremum of a potentially infinite collection of weight-indexed log-rank tests, with weight functions belonging to a reproducing kernel Hilbert space (RKHS) of functions; and second, as the norm of the difference of embeddings of certain finite measures into the RKHS, similar to the Hilbert-Schmidt Independence Criterion (HSIC) test-statistic. We study the asymptotic properties of the test, finding sufficient conditions to ensure our test correctly rejects the null hypothesis under any alternative. The test statistic can be computed straightforwardly, and the rejection threshold is obtained via an asymptotically consistent Wild Bootstrap procedure. Extensive investigations on both simulated and real data suggest that our testing procedure generally performs better than competing approaches in detecting complex non-linear dependence.
Duncan Watson-Parris, Samuel Sutherland, Matthew Christensen et al.
One of the most pressing questions in climate science is that of the effect of anthropogenic aerosol on the Earth's energy balance. Aerosols provide the `seeds' on which cloud droplets form, and changes in the amount of aerosol available to a cloud can change its brightness and other physical properties such as optical thickness and spatial extent. Clouds play a critical role in moderating global temperatures and small perturbations can lead to significant amounts of cooling or warming. Uncertainty in this effect is so large it is not currently known if it is negligible, or provides a large enough cooling to largely negate present-day warming by CO2. This work uses deep convolutional neural networks to look for two particular perturbations in clouds due to anthropogenic aerosol and assess their properties and prevalence, providing valuable insights into their climatic effects.
Zhu Li, Adrian Perez-Suay, Gustau Camps-Valls et al.
Current adoption of machine learning in industrial, societal and economical activities has raised concerns about the fairness, equity and ethics of automated decisions. Predictive models are often developed using biased datasets and thus retain or even exacerbate biases in their decisions and recommendations. Removing the sensitive covariates, such as gender or race, is insufficient to remedy this issue since the biases may be retained due to other related covariates. We present a regularization approach to this problem that trades off predictive accuracy of the learned models (with respect to biased labels) for the fairness in terms of statistical parity, i.e. independence of the decisions from the sensitive covariates. In particular, we consider a general framework of regularized empirical risk minimization over reproducing kernel Hilbert spaces and impose an additional regularizer of dependence between predictors and sensitive covariates using kernel-based measures of dependence, namely the Hilbert-Schmidt Independence Criterion (HSIC) and its normalized version. This approach leads to a closed-form solution in the case of squared loss, i.e. ridge regression. Moreover, we show that the dependence regularizer has an interpretation as modifying the corresponding Gaussian process (GP) prior. As a consequence, a GP model with a prior that encourages fairness to sensitive variables can be derived, allowing principled hyperparameter selection and studying of the relative relevance of covariates under fairness constraints. Experimental results in synthetic examples and in real problems of income and crime prediction illustrate the potential of the approach to improve fairness of automated decisions.