arXiv subjects

Dino Sejdinovic

Publications and source records attributed to Dino Sejdinovic.

At least 73 records · Page 4Linked to original sources

arXiv2021

Towards A Unified Analysis of Random Fourier Features

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 \emph{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 \emph{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.

stat.ML↗

arXiv2021

Benign Overfitting and Noisy Features

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.

stat.ML↗

arXiv2021

Bayesian Approaches to Distribution Regression

Distribution regression has recently attracted much interest as a generic solution to the problem of supervised learning where labels are available at the group level, rather than at the individual level. Current approaches, however, do not propagate the uncertainty in observations due to sampling variability in the groups. This effectively assumes that small and large groups are estimated equally well, and should have equal weight in the final regression. We account for this uncertainty with a Bayesian distribution regression formalism, improving the robustness and performance of the model when group sizes vary. We frame our models in a neural network style, allowing for simple MAP inference using backpropagation to learn the parameters, as well as MCMC-based inference which can fully propagate uncertainty. We demonstrate our approach on illustrative toy datasets, as well as on a challenging problem of predicting age from images.

stat.ML↗

arXiv2020

A kernel- and optimal transport- based test of independence between covariates and right-censored lifetimes

We propose a nonparametric test of independence, termed optHSIC, between a covariate and a right-censored lifetime. Because the presence of censoring creates a challenge in applying the standard permutation-based testing approaches, we use optimal transport to transform the censored dataset into an uncensored one, while preserving the relevant dependencies. We then apply a permutation test using the kernel-based dependence measure as a statistic to the transformed dataset. The type 1 error is proven to be correct in the case where censoring is independent of the covariate. Experiments indicate that optHSIC has power against a much wider class of alternatives than Cox proportional hazards regression and that it has the correct type 1 control even in the challenging cases where censoring strongly depends on the covariate.

math.ST↗

arXiv2020

Inter-domain Deep Gaussian Processes

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.

stat.ML↗

arXiv2020

Kernel-based Graph Learning from Smooth Signals: A Functional Viewpoint

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.

stat.ML↗

arXiv2020

A Perspective on Gaussian Processes for Earth Observation

Earth observation (EO) by airborne and satellite remote sensing and in-situ observations play a fundamental role in monitoring our planet. In the last decade, machine learning and Gaussian processes (GPs) in particular has attained outstanding results in the estimation of bio-geo-physical variables from the acquired images at local and global scales in a time-resolved manner. GPs provide not only accurate estimates but also principled uncertainty estimates for the predictions, can easily accommodate multimodal data coming from different sensors and from multitemporal acquisitions, allow the introduction of physical knowledge, and a formal treatment of uncertainty quantification and error propagation. Despite great advances in forward and inverse modelling, GP models still have to face important challenges that are revised in this perspective paper. GP models should evolve towards data-driven physics-aware models that respect signal characteristics, be consistent with elementary laws of physics, and move from pure regression to observational causal inference.

cs.LG↗

arXiv2020

Consistency of permutation tests for HSIC and dHSIC

The Hilbert--Schmidt Independence Criterion (HSIC) is a popular measure of the dependency between two random variables. The statistic dHSIC is an extension of HSIC that can be used to test joint independence of $d$ random variables. Such hypothesis testing for (joint) independence is often done using a permutation test, which compares the observed data with randomly permuted datasets. The main contribution of this work is proving that the power of such independence tests converges to 1 as the sample size converges to infinity. This answers a question that was asked in (Pfister, 2018) Additionally this work proves correct type 1 error rate of HSIC and dHSIC permutation tests and provides guidance on how to select the number of permutations one uses in practice. While correct type 1 error rate was already proved in (Pfister, 2018), we provide a modified proof following (Berrett, 2019), which extends to the case of non-continuous data. The number of permutations to use was studied e.g. by (Marozzi, 2004) but not in the context of HSIC and with a slight difference in the estimate of the $p$-value and for permutations rather than vectors of permutations. While the last two points have limited novelty we include these to give a complete overview of permutation testing in the context of HSIC and dHSIC.

math.ST↗

arXiv2020

Large Scale Tensor Regression using Kernels and Variational Inference

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.

stat.ML↗

arXiv2020

Discussion of "Functional Models for Time-Varying Random Objects'' by Dubey and Müller

The discussion focuses on metric covariance, a new association measure between paired random objects in a metric space, developed by Dubey and Müller, and on its relationship with other similar concepts which have previously appeared in the literature, including distance covariance by Székely et al, as well as its generalisations which rely on the formalism of reproducing kernel Hilbert spaces (RKHS).

stat.ME↗

arXiv2019

Detecting anthropogenic cloud perturbations with deep learning

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.

physics.ao-ph↗

arXiv2019

Kernel Dependence Regularizers and Gaussian Processes with Applications to Algorithmic Fairness

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.

stat.ML↗

arXiv2019

Hyperparameter Learning via Distributional Transfer

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.

stat.ML↗

arXiv2018

Rejoinder for "Probabilistic Integration: A Role in Statistical Computation?"

This article is the rejoinder for the paper "Probabilistic Integration: A Role in Statistical Computation?" to appear in Statistical Science with discussion. We would first like to thank the reviewers and many of our colleagues who helped shape this paper, the editor for selecting our paper for discussion, and of course all of the discussants for their thoughtful, insightful and constructive comments. In this rejoinder, we respond to some of the points raised by the discussants and comment further on the fundamental questions underlying the paper: (i) Should Bayesian ideas be used in numerical analysis?, and (ii) If so, what role should such approaches have in statistical computation?

stat.CO↗

arXiv2018

Hamiltonian Variational Auto-Encoder

Variational Auto-Encoders (VAEs) 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 (ELBOs). Combined with stochastic variational inference, this provides a methodology scaling to large datasets. However, for this methodology to be practically efficient, it is necessary 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 reinterpreted 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.

cs.LG↗

arXiv2018

A Differentially Private Kernel Two-Sample Test

Kernel two-sample testing is a useful statistical tool in determining whether data samples arise from different distributions without imposing any parametric assumptions on those distributions. However, raw data samples can expose sensitive information about individuals who participate in scientific studies, which makes the current tests vulnerable to privacy breaches. Hence, we design a new framework for kernel two-sample testing conforming to differential privacy constraints, in order to guarantee the privacy of subjects in the data. Unlike existing differentially private parametric tests that simply add noise to data, kernel-based testing imposes a challenge due to a complex dependence of test statistics on the raw data, as these statistics correspond to estimators of distances between representations of probability measures in Hilbert spaces. Our approach considers finite dimensional approximations to those representations. As a result, a simple chi-squared test is obtained, where a test statistic depends on a mean and covariance of empirical differences between the samples, which we perturb for a privacy guarantee. We investigate the utility of our framework in two realistic settings and conclude that our method requires only a relatively modest increase in sample size to achieve a similar level of power to the non-private tests in both settings.

stat.ML↗

arXiv2018

Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences

This paper is an attempt to bridge the conceptual gaps between researchers working on the two widely used approaches based on positive definite kernels: Bayesian learning or inference using Gaussian processes on the one side, and frequentist kernel methods based on reproducing kernel Hilbert spaces on the other. It is widely known in machine learning that these two formalisms are closely related; for instance, the estimator of kernel ridge regression is identical to the posterior mean of Gaussian process regression. However, they have been studied and developed almost independently by two essentially separate communities, and this makes it difficult to seamlessly transfer results between them. Our aim is to overcome this potential difficulty. To this end, we review several old and new results and concepts from either side, and juxtapose algorithmic quantities from each framework to highlight close similarities. We also provide discussions on subtle philosophical and theoretical differences between the two approaches.

stat.ML↗

arXiv2018

Detecting causal associations in large nonlinear time series datasets

Identifying causal relationships from observational time series data is a key problem in disciplines such as climate science or neuroscience, where experiments are often not possible. Data-driven causal inference is challenging since datasets are often high-dimensional and nonlinear with limited sample sizes. Here we introduce a novel method that flexibly combines linear or nonlinear conditional independence tests with a causal discovery algorithm that allows to reconstruct causal networks from large-scale time series datasets. We validate the method on a well-established climatic teleconnection connecting the tropical Pacific with extra-tropical temperatures and using large-scale synthetic datasets mimicking the typical properties of real data. The experiments demonstrate that our method outperforms alternative techniques in detection power from small to large-scale datasets and opens up entirely new possibilities to discover causal networks from time series across a range of research fields.

stat.ME↗