Dino Sejdinovic

CLeaR Conference 2026 Conference Paper

The Generalised Kernel Covariance Measure

• Luca Bergen
• Dino Sejdinovic
• Vanessa Didelez

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.

AAAI Conference 2025 Conference Paper

Bayesian Low-Rank Learning (Bella): A Practical Approach to Bayesian Neural Networks

• Bao Gia Doan
• Afshar Shamsi
• Xiao-Yu Guo
• Arash Mohammadi
• Hamid Alinejad-Rokny
• Dino Sejdinovic
• Damien Teney
• Damith C. Ranasinghe

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. Although, Deep ensemble methods (Seligmann et al. 2024; Lakshminarayanan, Pritzel, and Blundell 2017) have proven to be highly effective for Bayesian deep learning, their practical application is hindered by substantial computational cost. In this study, we introduce an innovative framework to mitigate the computational burden of ensemble Bayesian deep learning. We explore a more feasible alternative, inspired by the recent success of low-rank adapters, we introduce Bayesian Low-Rank LeArning (Bella). We show, 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–in accuracy and out-of-distribution generalisation–of conventional Bayesian learning methods and non-Bayesian baselines. Our extensive empirical evaluation in 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.

UAI Conference 2025 Conference Paper

Label Distribution Learning using the Squared Neural Family on the Probability Simplex

• Daokun Zhang
• Russell Tsuchida
• Dino Sejdinovic

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.

NeurIPS Conference 2025 Conference Paper

Squared families are useful conjugate priors

• Russell Tsuchida
• Jiawei Liu
• Cheng Soon Ong
• Dino Sejdinovic

Squared families of probability distributions have been studied and applied in numerous machine learning contexts. Typically, they appear as likelihoods, where their advantageous computational, geometric and statistical properties are exploited for fast estimation algorithms, representational properties and statistical guarantees. Here, we investigate the use of squared families as prior beliefs in Bayesian inference. We find that they can form helpful conjugate families, often allowing for closed-form and tractable Bayesian inference and marginal likelihoods. We apply such conjugate families to Bayesian regression in feature space using end-to-end learnable neural network features. Such a setting allows for a rich multi-modal alternative to Gaussian processes with neural network features, often called deep kernel learning. We demonstrate our method on few shot learning, outperforming existing neural methods based on Gaussian processes and normalising flows.

JMLR Journal 2024 Journal Article

A Kernel Test for Causal Association via Noise Contrastive Backdoor Adjustment

• Robert Hu
• Dino Sejdinovic
• Robin J. Evans

Causal inference grows increasingly complex as the dimension of confounders increases. Given treatments $X$, outcomes $Y$, and measured confounders $Z$, we develop a non-parametric method to test the do-null hypothesis that, after an intervention on $X$, there is no marginal dependence of $Y$ on $X$, against the general alternative. Building on the Hilbert-Schmidt Independence Criterion (HSIC) for marginal independence testing, we propose backdoor-HSIC (bd-HSIC), an importance weighted HSIC which combines density ratio estimation with kernel methods. Experiments on simulated data verify the correct size and that the estimator has power for both binary and continuous treatments under a large number of confounding variables. 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 or conditional independence testing. A complete implementation can be found at https://github.com/MrHuff/kgformula. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2024. ( edit, beta )

NeurIPS Conference 2024 Conference Paper

Bayesian Adaptive Calibration and Optimal Design

• Rafael Oliveira
• Dino Sejdinovic
• David Howard
• Edwin V. Bonilla

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 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 real data with the unknown calibration parameters. We show the benefits of our method when compared to related approaches across synthetic and real-data problems.

TMLR Journal 2024 Journal Article

Doubly Robust Kernel Statistics for Testing Distributional Treatment Effects

• Jake Fawkes
• Robert Hu
• Robin J. Evans
• Dino Sejdinovic

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, by constructing the test statistics based on the estimators we propose. We experimentally and theoretically demonstrate the validity of our tests.

AAAI Conference 2024 Conference Paper

Exact, Fast and Expressive Poisson Point Processes via Squared Neural Families

• Russell Tsuchida
• Cheng Soon Ong
• Dino Sejdinovic

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.

ICML Conference 2024 Conference Paper

Neural-Kernel Conditional Mean Embeddings

• Eiki Shimizu
• Kenji Fukumizu
• Dino Sejdinovic

Kernel conditional mean embeddings (CMEs) offer a powerful framework for representing conditional distributions, 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.

NeurIPS Conference 2023 Conference Paper

A Rigorous Link between Deep Ensembles and (Variational) Bayesian Methods

• Veit David Wild
• Sahra Ghalebikesabi
• Dino Sejdinovic
• Jeremias Knoblauch

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.

NeurIPS Conference 2023 Conference Paper

Explaining the Uncertain: Stochastic Shapley Values for Gaussian Process Models

• Siu Lun Chau
• Krikamol Muandet
• Dino Sejdinovic

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.

TMLR Journal 2023 Journal Article

Fair Kernel Regression through Cross-Covariance Operators

• Adrian Perez-Suay
• Paula Gordaliza
• Jean-Michel Loubes
• Dino Sejdinovic
• Gustau Camps-Valls

Ensuring fairness in machine learning models is a difficult problem from both a formulation and implementation perspective. One sensible criterion for achieving fairness is Equalised Odds, which requires that subjects in protected and unprotected groups have equal true and false positive rates. However, practical implementation is challenging. This work proposes two ways to address this issue through the conditional independence operator. First, given the output values, it is used as a fairness measure of independence between model predictions and sensitive variables. Second, it is used as a regularisation term in the problem formulation, which seeks optimal models that balance performance and fairness concerning the sensitive variables. To illustrate the potential of our approach, we consider different scenarios. First, we use the Gaussian model to provide new insights into the problem formulation and numerical results on its convergence. Second, we present the formulation using the conditional cross-covariance operator. We anticipate that a closed-form solution is possible in the general problem formulation, including in the case of a kernel formulation setting. Third, we introduce a normalised criterion of the conditional independence operator. All formulations are posed under the risk minimisation principle, which leads to theoretical results on the performance. Additionally, insights are provided into using these operators under a Gaussian Process setting. Our methods are compared to state-of-the-art methods in terms of performance and fairness metrics on a representative set of real problems. The results obtained with our proposed methodology show promising performance-fairness curves. Furthermore, we discuss the usefulness of linear weights in the fair model to describe the behaviour of the features when enforcing fairness over a particular set of input features.

ICML Conference 2023 Conference Paper

Returning The Favour: When Regression Benefits From Probabilistic Causal Knowledge

• Shahine Bouabid
• Jake Fawkes
• Dino Sejdinovic

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.

NeurIPS Conference 2023 Conference Paper

Squared Neural Families: A New Class of Tractable Density Models

• Russell Tsuchida
• Cheng Soon Ong
• Dino Sejdinovic

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.

NeurIPS Conference 2022 Conference Paper

Explaining Preferences with Shapley Values

• Robert Hu
• Siu Lun Chau
• Jaime Ferrando Huertas
• Dino Sejdinovic

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.

NeurIPS Conference 2022 Conference Paper

Generalized Variational Inference in Function Spaces: Gaussian Measures meet Bayesian Deep Learning

• Veit David Wild
• Robert Hu
• Dino Sejdinovic

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 optimization criterion. It 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 parametrization 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.

NeurIPS Conference 2022 Conference Paper

Giga-scale Kernel Matrix-Vector Multiplication on GPU

• Robert Hu
• Siu Lun Chau
• Dino Sejdinovic
• Joan Glaunès

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} ($\text{F}^3$M) to address these scaling issues of KMVM for tall~($10^8\sim 10^9$) and skinny~($D\leq7$) data. Extensive experiments demonstrate that $\text{F}^3$M 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.

NeurIPS Conference 2022 Conference Paper

RKHS-SHAP: Shapley Values for Kernel Methods

• Siu Lun Chau
• Robert Hu
• Javier González
• Dino Sejdinovic

Feature attribution for kernel methods is often heuristic and not individualised for each prediction. To address this, we turn to the concept of Shapley values (SV), a coalition game theoretical framework that has previously been applied to different machine learning model interpretation tasks, such as linear models, tree ensembles and deep networks. By analysing SVs from a functional perspective, we propose RKHS-SHAP, an attribution method for kernel machines that can efficiently compute both Interventional and Observational Shapley values using kernel mean embeddings of distributions. We show theoretically that our method is robust with respect to local perturbations - a key yet often overlooked desideratum for consistent model interpretation. Further, we propose Shapley regulariser, applicable to a general empirical risk minimisation framework, allowing learning while controlling the level of specific feature's contributions to the model. We demonstrate that the Shapley regulariser enables learning which is robust to covariate shift of a given feature and fair learning which controls the SVs of sensitive features.

CLeaR Conference 2022 Conference Paper

Selection, Ignorability and Challenges With Causal Fairness

• Jake Fawkes
• Robin Evans
• Dino Sejdinovic

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.

NeurIPS Conference 2021 Conference Paper

BayesIMP: Uncertainty Quantification for Causal Data Fusion

• Siu Lun Chau
• Jean-Francois Ton
• Javier González
• Yee Teh
• Dino Sejdinovic

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 data arising from 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 sample size, principled uncertainty quantification becomes essential. To that end, we introduce \emph{Bayesian Causal Mean Processes}, the 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 informativeness of our uncertainty estimation, we apply our method to the Causal Bayesian Optimisation task and show improvements over state-of-the-art methods.

NeurIPS Conference 2021 Conference Paper

Deconditional Downscaling with Gaussian Processes

• Siu Lun Chau
• Shahine Bouabid
• Dino Sejdinovic

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.

AAAI Conference 2021 Conference Paper

Meta Learning for Causal Direction

• Jean-François 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 (task), 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.

JMLR Journal 2021 Journal Article

Towards a Unified Analysis of Random Fourier Features

• Zhu Li
• Jean-Francois Ton
• Dino Oglic
• Dino Sejdinovic

Random Fourier features is a widely used, simple, and effective technique for scaling up kernel methods. The existing theoretical analysis of the approach, however, remains focused on specific learning tasks and typically gives pessimistic bounds which are at odds with the empirical results. We tackle these problems and provide the first unified risk analysis of learning with random Fourier features using the squared error and Lipschitz continuous loss functions. In our bounds, the trade-off between the computational cost and the learning risk convergence rate is problem specific and expressed in terms of the regularization parameter and the number of effective degrees of freedom. We study both the standard random Fourier features method for which we improve the existing bounds on the number of features required to guarantee the corresponding minimax risk convergence rate of kernel ridge regression, as well as a data-dependent modification which samples features proportional to ridge leverage scores and further reduces the required number of features. As ridge leverage scores are expensive to compute, we devise a simple approximation scheme which provably reduces the computational cost without loss of statistical efficiency. Our empirical results illustrate the effectiveness of the proposed scheme relative to the standard random Fourier features method. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

UAI Conference 2021 Conference Paper

Variational inference with continuously-indexed normalizing flows

• Anthony L. Caterini
• Robert Cornish
• Dino Sejdinovic
• Arnaud Doucet

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.

ICML Conference 2020 Conference Paper

Inter-domain Deep Gaussian Processes

• Tim G. J. Rudner
• Dino Sejdinovic
• Yarin Gal

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.

NeurIPS Conference 2019 Conference Paper

Hyperparameter Learning via Distributional Transfer

• Ho Chung Law
• Peilin Zhao
• Leung Sing Chan
• Junzhou Huang
• Dino Sejdinovic

Bayesian optimisation is a popular technique for hyperparameter learning but typically requires initial exploration even in cases where similar prior tasks have been solved. We propose to transfer information across tasks using learnt representations of training datasets used in those tasks. This results in a joint Gaussian process model on hyperparameters and data representations. Representations make use of the framework of distribution embeddings into reproducing kernel Hilbert spaces. The developed method has a faster convergence compared to existing baselines, in some cases requiring only a few evaluations of the target objective.

ICML Conference 2019 Conference Paper

Towards a Unified Analysis of Random Fourier Features

• Zhu Li
• Jean-Francois Ton
• Dino Oglic
• Dino Sejdinovic

Random Fourier features is a widely used, simple, and effective technique for scaling up kernel methods. The existing theoretical analysis of the approach, however, remains focused on specific learning tasks and typically gives pessimistic bounds which are at odds with the empirical results. We tackle these problems and provide the first unified risk analysis of learning with random Fourier features using the squared error and Lipschitz continuous loss functions. In our bounds, the trade-off between the computational cost and the expected risk convergence rate is problem specific and expressed in terms of the regularization parameter and the number of effective degrees of freedom. We study both the standard random Fourier features method for which we improve the existing bounds on the number of features required to guarantee the corresponding minimax risk convergence rate of kernel ridge regression, as well as a data-dependent modification which samples features proportional to ridge leverage scores and further reduces the required number of features. As ridge leverage scores are expensive to compute, we devise a simple approximation scheme which provably reduces the computational cost without loss of statistical efficiency.

NeurIPS Conference 2018 Conference Paper

Causal Inference via Kernel Deviance Measures

• Jovana Mitrovic
• Dino Sejdinovic
• Yee Whye Teh

Discovering the causal structure among a set of variables is a fundamental problem in many areas of science. In this paper, we propose Kernel Conditional Deviance for Causal Inference (KCDC) a fully nonparametric causal discovery method based on purely observational data. From a novel interpretation of the notion of asymmetry between cause and effect, we derive a corresponding asymmetry measure using the framework of reproducing kernel Hilbert spaces. Based on this, we propose three decision rules for causal discovery. We demonstrate the wide applicability and robustness of our method across a range of diverse synthetic datasets. Furthermore, we test our method on real-world time series data and the real-world benchmark dataset Tübingen Cause-Effect Pairs where we outperform state-of-the-art approaches.

NeurIPS Conference 2018 Conference Paper

Hamiltonian Variational Auto-Encoder

• Anthony Caterini
• Arnaud Doucet
• Dino Sejdinovic

Variational Auto-Encoders (VAE) have become very popular techniques to perform inference and learning in latent variable models as they allow us to leverage the rich representational power of neural networks to obtain flexible approximations of the posterior of latent variables as well as tight evidence lower bounds (ELBO). Com- bined with stochastic variational inference, this provides a methodology scaling to large datasets. However, for this methodology to be practically efficient, it is neces- sary to obtain low-variance unbiased estimators of the ELBO and its gradients with respect to the parameters of interest. While the use of Markov chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo (HMC) has been previously suggested to achieve this [23, 26], the proposed methods require specifying reverse kernels which have a large impact on performance. Additionally, the resulting unbiased estimator of the ELBO for most MCMC kernels is typically not amenable to the reparameterization trick. We show here how to optimally select reverse kernels in this setting and, by building upon Hamiltonian Importance Sampling (HIS) [17], we obtain a scheme that provides low-variance unbiased estimators of the ELBO and its gradients using the reparameterization trick. This allows us to develop a Hamiltonian Variational Auto-Encoder (HVAE). This method can be re-interpreted as a target-informed normalizing flow [20] which, within our context, only requires a few evaluations of the gradient of the sampled likelihood and trivial Jacobian calculations at each iteration.

NeurIPS Conference 2018 Conference Paper

Variational Learning on Aggregate Outputs with Gaussian Processes

• Ho Chung Law
• Dino Sejdinovic
• Ewan Cameron
• Tim Lucas
• Seth Flaxman
• Katherine Battle
• Kenji Fukumizu

While a typical supervised learning framework assumes that the inputs and the outputs are measured at the same levels of granularity, many applications, including global mapping of disease, only have access to outputs at a much coarser level than that of the inputs. Aggregation of outputs makes generalization to new inputs much more difficult. We consider an approach to this problem based on variational learning with a model of output aggregation and Gaussian processes, where aggregation leads to intractability of the standard evidence lower bounds. We propose new bounds and tractable approximations, leading to improved prediction accuracy and scalability to large datasets, while explicitly taking uncertainty into account. We develop a framework which extends to several types of likelihoods, including the Poisson model for aggregated count data. We apply our framework to a challenging and important problem, the fine-scale spatial modelling of malaria incidence, with over 1 million observations.

UAI Conference 2017 Conference Paper

Feature-to-Feature Regression for a Two-Step Conditional Independence Test

• Qinyi Zhang 0002
• Sarah Filippi
• Seth R. Flaxman
• Dino Sejdinovic

The algorithms for causal discovery and more broadly for learning the structure of graphical models require well calibrated and consistent conditional independence (CI) tests. We revisit the CI tests which are based on two-step procedures and involve regression with subsequent (unconditional) independence test (RESIT) on regression residuals and investigate the assumptions under which these tests operate. In particular, we demonstrate that when going beyond simple functional relationships with additive noise, such tests can lead to an inflated number of false discoveries. We study the relationship of these tests with those based on dependence measures using reproducing kernel Hilbert spaces (RKHS) and propose an extension of RESIT which uses RKHS-valued regression. The resulting test inherits the simple two-step testing procedure of RESIT, while giving correct Type I control and competitive power. When used as a component of the PC algorithm, the proposed test is more robust to the case where hidden variables induce a switching behaviour in the associations present in the data.

NeurIPS Conference 2017 Conference Paper

Testing and Learning on Distributions with Symmetric Noise Invariance

• Ho Chung Law
• Christopher Yau
• Dino Sejdinovic

Kernel embeddings of distributions and the Maximum Mean Discrepancy (MMD), the resulting distance between distributions, are useful tools for fully nonparametric two-sample testing and learning on distributions. However, it is rarely that all possible differences between samples are of interest -- discovered differences can be due to different types of measurement noise, data collection artefacts or other irrelevant sources of variability. We propose distances between distributions which encode invariance to additive symmetric noise, aimed at testing whether the assumed true underlying processes differ. Moreover, we construct invariant features of distributions, leading to learning algorithms robust to the impairment of the input distributions with symmetric additive noise.

UAI Conference 2016 Conference Paper

Bayesian Learning of Kernel Embeddings

• Seth R. Flaxman
• Dino Sejdinovic
• John P. Cunningham
• Sarah Filippi

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.

ICML Conference 2016 Conference Paper

DR-ABC: Approximate Bayesian Computation with Kernel-Based Distribution Regression

• Jovana Mitrovic
• Dino Sejdinovic
• Yee Whye Teh

Performing exact posterior inference in complex generative models is often difficult or impossible due to an expensive to evaluate or intractable likelihood function. Approximate Bayesian computation (ABC) is an inference framework that constructs an approximation to the true likelihood based on the similarity between the observed and simulated data as measured by a predefined set of summary statistics. Although the choice of informative problem-specific summary statistics crucially influences the quality of the likelihood approximation and hence also the quality of the posterior sample in ABC, there are only few principled general-purpose approaches to the selection or construction of such summary statistics. In this paper, we develop a novel framework for solving this problem. We model the functional relationship between the data and the optimal choice (with respect to a loss function) of summary statistics using kernel-based distribution regression. Furthermore, we extend our approach to incorporate kernel-based regression from conditional distributions, thus appropriately taking into account the specific structure of the posited generative model. We show that our approach can be implemented in a computationally and statistically efficient way using the random Fourier features framework for large-scale kernel learning. In addition to that, our framework outperforms related methods by a large margin on toy and real-world data, including hierarchical and time series models.

UAI Conference 2016 Conference Paper

Super-Sampling with a Reservoir

• Brooks Paige
• Dino Sejdinovic
• Frank Wood

We introduce an alternative to reservoir sampling, a classic and popular algorithm for drawing a fixed-size subsample from streaming data in a single pass. Rather than draw a random sample, our approach performs an online optimization which aims to select the subset that provides the best overall approximation to the full data set, as judged using a kernel two-sample test. This produces subsets which minimize the worst-case relative error when computing expectations of functions in a specified function class, using just the samples from the subset. Kernel functions are approximated using random Fourier features, and the subset of samples itself is stored in a random projection tree. The resulting algorithm runs in a single pass through the whole data set, and has a per-iteration computational complexity logarithmic in the size of the subset. These “supersamples” subsampled from the full data provide a concise summary, as demonstrated empirically on mixture models and the MNIST dataset.

NeurIPS Conference 2015 Conference Paper

Fast Two-Sample Testing with Analytic Representations of Probability Measures

• Kacper Chwialkowski
• Aaditya Ramdas
• Dino Sejdinovic
• Arthur Gretton

We propose a class of nonparametric two-sample tests with a cost linear in the sample size. Two tests are given, both based on an ensemble of distances between analytic functions representing each of the distributions. The first test uses smoothed empirical characteristic functions to represent the distributions, the second uses distribution embeddings in a reproducing kernel Hilbert space. Analyticity implies that differences in the distributions may be detected almost surely at a finite number of randomly chosen locations/frequencies. The new tests are consistent against a larger class of alternatives than the previous linear-time tests based on the (non-smoothed) empirical characteristic functions, while being much faster than the current state-of-the-art quadratic-time kernel-based or energy distance-based tests. Experiments on artificial benchmarks and on challenging real-world testing problems demonstrate that our tests give a better power/time tradeoff than competing approaches, and in some cases, better outright power than even the most expensive quadratic-time tests. This performance advantage is retained even in high dimensions, and in cases where the difference in distributions is not observable with low order statistics.

NeurIPS Conference 2015 Conference Paper

Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families

• Heiko Strathmann
• Dino Sejdinovic
• Samuel Livingstone
• Zoltan Szabo
• Arthur Gretton

We propose Kernel Hamiltonian Monte Carlo (KMC), a gradient-free adaptive MCMC algorithm based on Hamiltonian Monte Carlo (HMC). On target densities where classical HMC is not an option due to intractable gradients, KMC adaptively learns the target's gradient structure by fitting an exponential family model in a Reproducing Kernel Hilbert Space. Computational costs are reduced by two novel efficient approximations to this gradient. While being asymptotically exact, KMC mimics HMC in terms of sampling efficiency, and offers substantial mixing improvements over state-of-the-art gradient free samplers. We support our claims with experimental studies on both toy and real-world applications, including Approximate Bayesian Computation and exact-approximate MCMC.

UAI Conference 2015 Conference Paper

Kernel-Based Just-In-Time Learning for Passing Expectation Propagation Messages

• Wittawat Jitkrittum
• Arthur Gretton
• Nicolas Heess
• S. M. Ali Eslami
• Balaji Lakshminarayanan
• Dino Sejdinovic
• Zoltán Szabó 0001

We propose an efficient nonparametric strategy for learning a message operator in expectation propagation (EP), which takes as input the set of incoming messages to a factor node, and produces an outgoing message as output. This learned operator replaces the multivariate integral required in classical EP, which may not have an analytic expression. We use kernel-based regression, which is trained on a set of probability distributions representing the incoming messages, and the associated outgoing messages. The kernel approach has two main advantages: first, it is fast, as it is implemented using a novel two-layer random feature representation of the input message distributions; second, it has principled uncertainty estimates, and can be cheaply updated online, meaning it can request and incorporate new training data when it encounters inputs on which it is uncertain. In experiments, our approach is able to solve learning problems where a single message operator is required for multiple, substantially different data sets (logistic regression for a variety of classification problems), where it is essential to accurately assess uncertainty and to efficiently and robustly update the message operator.

NeurIPS Conference 2014 Conference Paper

A Wild Bootstrap for Degenerate Kernel Tests

• Kacper Chwialkowski
• Dino Sejdinovic
• Arthur Gretton

A wild bootstrap method for nonparametric hypothesis tests based on kernel distribution embeddings is proposed. This bootstrap method is used to construct provably consistent tests that apply to random processes, for which the naive permutation-based bootstrap fails. It applies to a large group of kernel tests based on V-statistics, which are degenerate under the null hypothesis, and non-degenerate elsewhere. To illustrate this approach, we construct a two-sample test, an instantaneous independence test and a multiple lag independence test for time series. In experiments, the wild bootstrap gives strong performance on synthetic examples, on audio data, and in performance benchmarking for the Gibbs sampler. The code is available at https: //github. com/kacperChwialkowski/wildBootstrap.

ICML Conference 2014 Conference Paper

Kernel Adaptive Metropolis-Hastings

• Dino Sejdinovic
• Heiko Strathmann
• Maria Lomeli Garcia
• Christophe Andrieu
• Arthur Gretton

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.

NeurIPS Conference 2013 Conference Paper

A Kernel Test for Three-Variable Interactions

• Dino Sejdinovic
• Arthur Gretton
• Wicher Bergsma

We introduce kernel nonparametric tests for Lancaster three-variable interaction and for total independence, using embeddings of signed measures into a reproducing kernel Hilbert space. The resulting test statistics are straightforward to compute, and are used in powerful three-variable interaction tests, which are consistent against all alternatives for a large family of reproducing kernels. We show the Lancaster test to be sensitive to cases where two independent causes individually have weak influence on a third dependent variable, but their combined effect has a strong influence. This makes the Lancaster test especially suited to finding structure in directed graphical models, where it outperforms competing nonparametric tests in detecting such V-structures.

ICML Conference 2012 Conference Paper

Hypothesis testing using pairwise distances and associated kernels

• Dino Sejdinovic
• Arthur Gretton
• Bharath K. Sriperumbudur
• Kenji Fukumizu

NeurIPS Conference 2012 Conference Paper

Optimal kernel choice for large-scale two-sample tests

• Arthur Gretton
• Dino Sejdinovic
• Heiko Strathmann
• Sivaraman Balakrishnan
• Massimiliano Pontil
• Kenji Fukumizu
• Bharath Sriperumbudur

Abstract Given samples from distributions $p$ and $q$, a two-sample test determines whether to reject the null hypothesis that $p=q$, based on the value of a test statistic measuring the distance between the samples. One choice of test statistic is the maximum mean discrepancy (MMD), which is a distance between embeddings of the probability distributions in a reproducing kernel Hilbert space. The kernel used in obtaining these embeddings is thus critical in ensuring the test has high power, and correctly distinguishes unlike distributions with high probability. A means of parameter selection for the two-sample test based on the MMD is proposed. For a given test level (an upper bound on the probability of making a Type I error), the kernel is chosen so as to maximize the test power, and minimize the probability of making a Type II error. The test statistic, test threshold, and optimization over the kernel parameters are obtained with cost linear in the sample size. These properties make the kernel selection and test procedures suited to data streams, where the observations cannot all be stored in memory. In experiments, the new kernel selection approach yields a more powerful test than earlier kernel selection heuristics.