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Daniel Hernández-Lobato

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19 papers

2 author rows

ICML Conference 2024 Conference Paper

Variational Linearized Laplace Approximation for Bayesian Deep Learning

Luis A. Ortega Andrés
Simón Rodríguez Santana
Daniel Hernández-Lobato

The Linearized Laplace Approximation (LLA) has been recently used to perform uncertainty estimation on the predictions of pre-trained deep neural networks (DNNs). However, its widespread application is hindered by significant computational costs, particularly in scenarios with a large number of training points or DNN parameters. Consequently, additional approximations of LLA, such as Kronecker-factored or diagonal approximate GGN matrices, are utilized, potentially compromising the model’s performance. To address these challenges, we propose a new method for approximating LLA using a variational sparse Gaussian Process (GP). Our method is based on the dual RKHS formulation of GPs and retains as the predictive mean the output of the original DNN. Furthermore, it allows for efficient stochastic optimization, which results in sub-linear training time in the size of the training dataset. Specifically, its training cost is independent of the number of training points. We compare our proposed method against accelerated LLA (ELLA), which relies on the Nyström approximation, as well as other LLA variants employing the sample-then-optimize principle. Experimental results, both on regression and classification datasets, show that our method outperforms these already existing efficient variants of LLA, both in terms of the quality of the predictive distribution and in terms of total computational time.

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ICLR Conference 2023 Conference Paper

Deep Variational Implicit Processes

Luis A. Ortega Andrés
Simón Rodríguez Santana
Daniel Hernández-Lobato

Implicit processes (IPs) are a generalization of Gaussian processes (GPs). IPs may lack a closed-form expression but are easy to sample from. Examples include, among others, Bayesian neural networks or neural samplers. IPs can be used as priors over functions, resulting in flexible models with well-calibrated prediction uncertainty estimates. Methods based on IPs usually carry out function-space approximate inference, which overcomes some of the difficulties of parameter-space approximate inference. Nevertheless, the approximations employed often limit the expressiveness of the final model, resulting, e.g., in a Gaussian predictive distribution, which can be restrictive. We propose here a multi-layer generalization of IPs called the Deep Variational Implicit process (DVIP). This generalization is similar to that of deep GPs over GPs, but it is more flexible due to the use of IPs as the prior distribution over the latent functions. We describe a scalable variational inference algorithm for training DVIP and show that it outperforms previous IP-based methods and also deep GPs. We support these claims via extensive regression and classification experiments. We also evaluate DVIP on large datasets with up to several million data instances to illustrate its good scalability and performance.

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ICML Conference 2023 Conference Paper

Efficient Transformed Gaussian Processes for Non-Stationary Dependent Multi-class Classification

Juan Maroñas
Daniel Hernández-Lobato

This work introduces the Efficient Transformed Gaussian Process (ETGP), a new way of creating $C$ stochastic processes characterized by: 1) the $C$ processes are non-stationary, 2) the $C$ processes are dependent by construction without needing a mixing matrix, 3) training and making predictions is very efficient since the number of Gaussian Processes (GP) operations (e. g. inverting the inducing point’s covariance matrix) do not depend on the number of processes. This makes the ETGP particularly suited for multi-class problems with a very large number of classes, which are the problems studied in this work. ETGP exploits the recently proposed Transformed Gaussian Process (TGP), a stochastic process specified by transforming a Gaussian Process using an invertible transformation. However, unlike TGP, ETGP is constructed by transforming a single sample from a GP using $C$ invertible transformations. We derive an efficient sparse variational inference algorithm for the proposed model and demonstrate its utility in 5 classification tasks which include low/medium/large datasets and a different number of classes, ranging from just a few to hundreds. Our results show that ETGP, in general, outperforms state-of-the-art methods for multi-class classification based on GPs, and has a lower computational cost (around one order of magnitude smaller).

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ICML Conference 2022 Conference Paper

Function-space Inference with Sparse Implicit Processes

Simón Rodríguez Santana
Bryan Zaldivar
Daniel Hernández-Lobato

Implicit Processes (IPs) represent a flexible framework that can be used to describe a wide variety of models, from Bayesian neural networks, neural samplers and data generators to many others. IPs also allow for approximate inference in function-space. This change of formulation solves intrinsic degenerate problems of parameter-space approximate inference concerning the high number of parameters and their strong dependencies in large models. For this, previous works in the literature have attempted to employ IPs both to set up the prior and to approximate the resulting posterior. However, this has proven to be a challenging task. Existing methods that can tune the prior IP result in a Gaussian predictive distribution, which fails to capture important data patterns. By contrast, methods producing flexible predictive distributions by using another IP to approximate the posterior process cannot tune the prior IP to the observed data. We propose here the first method that can accomplish both goals. For this, we rely on an inducing-point representation of the prior IP, as often done in the context of sparse Gaussian processes. The result is a scalable method for approximate inference with IPs that can tune the prior IP parameters to the data, and that provides accurate non-Gaussian predictive distributions.

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ICML Conference 2022 Conference Paper

Input Dependent Sparse Gaussian Processes

Bahram Jafrasteh
Carlos Villacampa-Calvo
Daniel Hernández-Lobato

Gaussian Processes (GPs) are non-parametric models that provide accurate uncertainty estimates. Nevertheless, they have a cubic cost in the number of data instances $N$. To overcome this, sparse GP approximations are used, in which a set of $M \ll N$ inducing points is introduced. The location of the inducing points is learned by considering them parameters of an approximate posterior distribution $q$. Sparse GPs, combined with stochastic variational inference for inferring $q$ have a cost per iteration in $\mathcal{O}(M^3)$. Critically, the inducing points determine the flexibility of the model and they are often located in regions where the latent function changes. A limitation is, however, that in some tasks a large number of inducing points may be required to obtain good results. To alleviate this, we propose here to amortize the computation of the inducing points locations, as well as the parameters of $q$. For this, we use a neural network that receives a data instance as an input and outputs the corresponding inducing points locations and the parameters of $q$. We evaluate our method in several experiments, showing that it performs similar or better than other state-of-the-art sparse variational GPs. However, in our method the number of inducing points is reduced drastically since they depend on the input data. This makes our method scale to larger datasets and have faster training and prediction times.

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ICLR Conference 2021 Conference Paper

Activation-level uncertainty in deep neural networks

Pablo Morales-Alvarez
Daniel Hernández-Lobato
Rafael Molina 0001
José Miguel Hernández-Lobato

Current approaches for uncertainty estimation in deep learning often produce too confident results. Bayesian Neural Networks (BNNs) model uncertainty in the space of weights, which is usually high-dimensional and limits the quality of variational approximations. The more recent functional BNNs (fBNNs) address this only partially because, although the prior is specified in the space of functions, the posterior approximation is still defined in terms of stochastic weights. In this work we propose to move uncertainty from the weights (which are deterministic) to the activation function. Specifically, the activations are modelled with simple 1D Gaussian Processes (GP), for which a triangular kernel inspired by the ReLu non-linearity is explored. Our experiments show that activation-level stochasticity provides more reliable uncertainty estimates than BNN and fBNN, whereas it performs competitively in standard prediction tasks. We also study the connection with deep GPs, both theoretically and empirically. More precisely, we show that activation-level uncertainty requires fewer inducing points and is better suited for deep architectures.

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JMLR Journal 2021 Journal Article

Multi-class Gaussian Process Classification with Noisy Inputs

Carlos Villacampa-Calvo
Bryan Zaldívar
Eduardo C. Garrido-Merchán
Daniel Hernández-Lobato

It is a common practice in the machine learning community to assume that the observed data are noise-free in the input attributes. Nevertheless, scenarios with input noise are common in real problems, as measurements are never perfectly accurate. If this input noise is not taken into account, a supervised machine learning method is expected to perform sub-optimally. In this paper, we focus on multi-class classification problems and use Gaussian processes (GPs) as the underlying classifier. Motivated by a data set coming from the astrophysics domain, we hypothesize that the observed data may contain noise in the inputs. Therefore, we devise several multi-class GP classifiers that can account for input noise. Such classifiers can be efficiently trained using variational inference to approximate the posterior distribution of the latent variables of the model. Moreover, in some situations, the amount of noise can be known before-hand. If this is the case, it can be readily introduced in the proposed methods. This prior information is expected to lead to better performance results. We have evaluated the proposed methods by carrying out several experiments, involving synthetic and real data. These include several data sets from the UCI repository, the MNIST data set and a data set coming from astrophysics. The results obtained show that, although the classification error is similar across methods, the predictive distribution of the proposed methods is better, in terms of the test log-likelihood, than the predictive distribution of a classifier based on GPs that ignores input noise. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2021. ( edit, beta )

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ICML Conference 2017 Conference Paper

Scalable Multi-Class Gaussian Process Classification using Expectation Propagation

Carlos Villacampa-Calvo
Daniel Hernández-Lobato

This paper describes an expectation propagation (EP) method for multi-class classification with Gaussian processes that scales well to very large datasets. In such a method the estimate of the log-marginal-likelihood involves a sum across the data instances. This enables efficient training using stochastic gradients and mini-batches. When this type of training is used, the computational cost does not depend on the number of data instances N. Furthermore, extra assumptions in the approximate inference process make the memory cost independent of N. The consequence is that the proposed EP method can be used on datasets with millions of instances. We compare empirically this method with alternative approaches that approximate the required computations using variational inference. The results show that it performs similar or even better than these techniques, which sometimes give significantly worse predictive distributions in terms of the test log-likelihood. Besides this, the training process of the proposed approach also seems to converge in a smaller number of iterations.

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ICML Conference 2016 Conference Paper

Black-Box Alpha Divergence Minimization

José Miguel Hernández-Lobato
Yingzhen Li
Mark Rowland 0001
Thang D. Bui
Daniel Hernández-Lobato
Richard E. Turner

Black-box alpha (BB-α) is a new approximate inference method based on the minimization of α-divergences. BB-αscales to large datasets because it can be implemented using stochastic gradient descent. BB-αcan be applied to complex probabilistic models with little effort since it only requires as input the likelihood function and its gradients. These gradients can be easily obtained using automatic differentiation. By changing the divergence parameter α, the method is able to interpolate between variational Bayes (VB) (α→0) and an algorithm similar to expectation propagation (EP) (α= 1). Experiments on probit regression and neural network regression and classification problems show that BB-αwith non-standard settings of α, such as α= 0. 5, usually produces better predictions than with α→0 (VB) or α= 1 (EP).

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ICML Conference 2016 Conference Paper

Deep Gaussian Processes for Regression using Approximate Expectation Propagation

Thang D. Bui
Daniel Hernández-Lobato
José Miguel Hernández-Lobato
Yingzhen Li
Richard E. Turner

Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian processes (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers. DGPs are nonparametric probabilistic models and as such are arguably more flexible, have a greater capacity to generalise, and provide better calibrated uncertainty estimates than alternative deep models. This paper develops a new approximate Bayesian learning scheme that enables DGPs to be applied to a range of medium to large scale regression problems for the first time. The new method uses an approximate Expectation Propagation procedure and a novel and efficient extension of the probabilistic backpropagation algorithm for learning. We evaluate the new method for non-linear regression on eleven real-world datasets, showing that it always outperforms GP regression and is almost always better than state-of-the-art deterministic and sampling-based approximate inference methods for Bayesian neural networks. As a by-product, this work provides a comprehensive analysis of six approximate Bayesian methods for training neural networks.

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JMLR Journal 2016 Journal Article

Non-linear Causal Inference using Gaussianity Measures

Daniel Hernández-Lobato
Pablo Morales-Mombiela
David Lopez-Paz
Alberto Suárez

We provide theoretical and empirical evidence for a type of asymmetry between causes and effects that is present when these are related via linear models contaminated with additive non- Gaussian noise. Assuming that the causes and the effects have the same distribution, we show that the distribution of the residuals of a linear fit in the anti-causal direction is closer to a Gaussian than the distribution of the residuals in the causal direction. This Gaussianization effect is characterized by reduction of the magnitude of the high-order cumulants and by an increment of the differential entropy of the residuals. The problem of non-linear causal inference is addressed by performing an embedding in an expanded feature space, in which the relation between causes and effects can be assumed to be linear. The effectiveness of a method to discriminate between causes and effects based on this type of asymmetry is illustrated in a variety of experiments using different measures of Gaussianity. The proposed method is shown to be competitive with state-of-the-art techniques for causal inference. [abs] [ pdf ][ bib ] &copy JMLR 2016. ( edit, beta )

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ICML Conference 2016 Conference Paper

Predictive Entropy Search for Multi-objective Bayesian Optimization

Daniel Hernández-Lobato
José Miguel Hernández-Lobato
Amar Shah 0001
Ryan P. Adams

We present \small PESMO, a Bayesian method for identifying the Pareto set of multi-objective optimization problems, when the functions are expensive to evaluate. \small PESMO chooses the evaluation points to maximally reduce the entropy of the posterior distribution over the Pareto set. The \small PESMO acquisition function is decomposed as a sum of objective-specific acquisition functions, which makes it possible to use the algorithm in \emphdecoupled scenarios in which the objectives can be evaluated separately and perhaps with different costs. This decoupling capability is useful to identify difficult objectives that require more evaluations. \small PESMO also offers gains in efficiency, as its cost scales linearly with the number of objectives, in comparison to the exponential cost of other methods. We compare \small PESMO with other methods on synthetic and real-world problems. The results show that \small PESMO produces better recommendations with a smaller number of evaluations, and that a decoupled evaluation can lead to improvements in performance, particularly when the number of objectives is large.

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ICML Conference 2015 Conference Paper

A Probabilistic Model for Dirty Multi-task Feature Selection

Daniel Hernández-Lobato
José Miguel Hernández-Lobato
Zoubin Ghahramani

Multi-task feature selection methods often make the hypothesis that learning tasks share relevant and irrelevant features. However, this hypothesis may be too restrictive in practice. For example, there may be a few tasks with specific relevant and irrelevant features (outlier tasks). Similarly, a few of the features may be relevant for only some of the tasks (outlier features). To account for this, we propose a model for multi-task feature selection based on a robust prior distribution that introduces a set of binary latent variables to identify outlier tasks and outlier features. Expectation propagation can be used for efficient approximate inference under the proposed prior. Several experiments show that a model based on the new robust prior provides better predictive performance than other benchmark methods.

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NeurIPS Conference 2014 Conference Paper

Mind the Nuisance: Gaussian Process Classification using Privileged Noise

Daniel Hernández-Lobato
Viktoriia Sharmanska
Kristian Kersting
Christoph Lampert
Novi Quadrianto

The learning with privileged information setting has recently attracted a lot of attention within the machine learning community, as it allows the integration of additional knowledge into the training process of a classifier, even when this comes in the form of a data modality that is not available at test time. Here, we show that privileged information can naturally be treated as noise in the latent function of a Gaussian process classifier (GPC). That is, in contrast to the standard GPC setting, the latent function is not just a nuisance but a feature: it becomes a natural measure of confidence about the training data by modulating the slope of the GPC probit likelihood function. Extensive experiments on public datasets show that the proposed GPC method using privileged noise, called GPC+, improves over a standard GPC without privileged knowledge, and also over the current state-of-the-art SVM-based method, SVM+. Moreover, we show that advanced neural networks and deep learning methods can be compressed as privileged information.

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NeurIPS Conference 2013 Conference Paper

Gaussian Process Conditional Copulas with Applications to Financial Time Series

José Miguel Hernández-Lobato
James Lloyd
Daniel Hernández-Lobato

The estimation of dependencies between multiple variables is a central problem in the analysis of financial time series. A common approach is to express these dependencies in terms of a copula function. Typically the copula function is assumed to be constant but this may be innacurate when there are covariates that could have a large influence on the dependence structure of the data. To account for this, a Bayesian framework for the estimation of conditional copulas is proposed. In this framework the parameters of a copula are non-linearly related to some arbitrary conditioning variables. We evaluate the ability of our method to predict time-varying dependencies on several equities and currencies and observe consistent performance gains compared to static copula models and other time-varying copula methods.

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JMLR Journal 2013 Journal Article

Generalized Spike-and-Slab Priors for Bayesian Group Feature Selection Using Expectation Propagation

Daniel Hernández-Lobato
José Miguel Hernández-Lobato
Pierre Dupont

We describe a Bayesian method for group feature selection in linear regression problems. The method is based on a generalized version of the standard spike-and-slab prior distribution which is often used for individual feature selection. Exact Bayesian inference under the prior considered is infeasible for typical regression problems. However, approximate inference can be carried out efficiently using Expectation Propagation (EP). A detailed analysis of the generalized spike-and-slab prior shows that it is well suited for regression problems that are sparse at the group level. Furthermore, this prior can be used to introduce prior knowledge about specific groups of features that are a priori believed to be more relevant. An experimental evaluation compares the performance of the proposed method with those of group LASSO, Bayesian group LASSO, automatic relevance determination and additional variants used for group feature selection. The results of these experiments show that a model based on the generalized spike-and-slab prior and the EP algorithm has state-of-the-art prediction performance in the problems analyzed. Furthermore, this model is also very useful to carry out sequential experimental design (also known as active learning), where the data instances that are most informative are iteratively included in the training set, reducing the number of instances needed to obtain a particular level of prediction accuracy. [abs] [ pdf ][ bib ] &copy JMLR 2013. ( edit, beta )

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NeurIPS Conference 2013 Conference Paper

Learning Feature Selection Dependencies in Multi-task Learning

Daniel Hernández-Lobato
José Miguel Hernández-Lobato

A probabilistic model based on the horseshoe prior is proposed for learning dependencies in the process of identifying relevant features for prediction. Exact inference is intractable in this model. However, expectation propagation offers an approximate alternative. Because the process of estimating feature selection dependencies may suffer from over-fitting in the model proposed, additional data from a multi-task learning scenario are considered for induction. The same model can be used in this setting with few modifications. Furthermore, the assumptions made are less restrictive than in other multi-task methods: The different tasks must share feature selection dependencies, but can have different relevant features and model coefficients. Experiments with real and synthetic data show that this model performs better than other multi-task alternatives from the literature. The experiments also show that the model is able to induce suitable feature selection dependencies for the problems considered, only from the training data.

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IJCAI Conference 2013 Conference Paper

Statistical Tests for the Detection of the Arrow of Time in Vector Autoregressive Models

Pablo Morales-Mombiela
Daniel Hernández-Lobato
Alberto Suárez

The problem of detecting the direction of time in vector Autoregressive (VAR) processes using statistical techniques is considered. By analogy to causal AR(1) processes with non-Gaussian noise, we conjecture that the distribution of the time reversed residuals of a linear VAR model is closer to a Gaussian than the distribution of actual residuals in the forward direction. Experiments with simulated data illustrate the validity of the conjecture. Based on these results, we design a decision rule for detecting the direction of VAR processes. The correct direction in time (forward) is the one in which the residuals of the time series are less Gaussian. A series of experiments illustrate the superior results of the proposed rule when compared with other methods based on independence tests.

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NeurIPS Conference 2011 Conference Paper

Robust Multi-Class Gaussian Process Classification

Daniel Hernández-Lobato
Jose Hernández-lobato
Pierre Dupont

Multi-class Gaussian Process Classifiers (MGPCs) are often affected by over-fitting problems when labeling errors occur far from the decision boundaries. To prevent this, we investigate a robust MGPC (RMGPC) which considers labeling errors independently of their distance to the decision boundaries. Expectation propagation is used for approximate inference. Experiments with several datasets in which noise is injected in the class labels illustrate the benefits of RMGPC. This method performs better than other Gaussian process alternatives based on considering latent Gaussian noise or heavy-tailed processes. When no noise is injected in the labels, RMGPC still performs equal or better than the other methods. Finally, we show how RMGPC can be used for successfully identifying data instances which are difficult to classify accurately in practice.

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