Daniel Hernández-Lobato

Daniel Heestermans Svendsen, Daniel Hernández-Lobato, Luca Martino et al.

Earth observation from satellites offers the possibility to monitor our planet with unprecedented accuracy. Radiative transfer models (RTMs) encode the energy transfer through the atmosphere, and are used to model and understand the Earth system, as well as to estimate the parameters that describe the status of the Earth from satellite observations by inverse modeling. However, performing inference over such simulators is a challenging problem. RTMs are nonlinear, non-differentiable and computationally costly codes, which adds a high level of difficulty in inference. In this paper, we introduce two computational techniques to infer not only point estimates of biophysical parameters but also their joint distribution. One of them is based on a variational autoencoder approach and the second one is based on a Monte Carlo Expectation Maximization (MCEM) scheme. We compare and discuss benefits and drawbacks of each approach. We also provide numerical comparisons in synthetic simulations and the real PROSAIL model, a popular RTM that combines land vegetation leaf and canopy modeling. We analyze the performance of the two approaches for modeling and inferring the distribution of three key biophysical parameters for quantifying the terrestrial biosphere.

Bahram Jafrasteh, Daniel Hernández-Lobato, Simón Pedro Lubián-López et al.

Missing values are common in many real-life datasets. However, most of the current machine learning methods can not handle missing values. This means that they should be imputed beforehand. Gaussian Processes (GPs) are non-parametric models with accurate uncertainty estimates that combined with sparse approximations and stochastic variational inference scale to large data sets. Sparse GPs can be used to compute a predictive distribution for missing data. Here, we present a hierarchical composition of sparse GPs that is used to predict missing values at each dimension using all the variables from the other dimensions. We call the approach missing GP (MGP). MGP can be trained simultaneously to impute all observed missing values. Specifically, it outputs a predictive distribution for each missing value that is then used in the imputation of other missing values. We evaluate MGP in one private clinical data set and four UCI datasets with a different percentage of missing values. We compare the performance of MGP with other state-of-the-art methods for imputing missing values, including variants based on sparse GPs and deep GPs. The results obtained show a significantly better performance of MGP.

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. ETGPs exploit the recently proposed Transformed Gaussian Process (TGP), a stochastic process specified by transforming a Gaussian Process using an invertible transformation. However, unlike TGPs, ETGPs are 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 ETGPs, in general, outperform state-of-the-art methods for multi-class classification based on GPs, and have a lower computational cost (around one order of magnitude smaller).

Luis A. Ortega, 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.

Luis A. Ortega, 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.

Simón Rodríguez Santana, Luis A. Ortega, Daniel Hernández-Lobato et al.

Model selection in machine learning (ML) is a crucial part of the Bayesian learning procedure. Model choice may impose strong biases on the resulting predictions, which can hinder the performance of methods such as Bayesian neural networks and neural samplers. On the other hand, newly proposed approaches for Bayesian ML exploit features of approximate inference in function space with implicit stochastic processes (a generalization of Gaussian processes). The approach of Sparse Implicit Processes (SIP) is particularly successful in this regard, since it is fully trainable and achieves flexible predictions. Here, we expand on the original experiments to show that SIP is capable of correcting model bias when the data generating mechanism differs strongly from the one implied by the model. We use synthetic datasets to show that SIP is capable of providing predictive distributions that reflect the data better than the exact predictions of the initial, but wrongly assumed model.

Francisco Javier Sáez-Maldonado, Juan Maroñas, Daniel Hernández-Lobato

Transformed Gaussian Processes (TGPs) are stochastic processes specified by transforming samples from the joint distribution from a prior process (typically a GP) using an invertible transformation; increasing the flexibility of the base process. Furthermore, they achieve competitive results compared with Deep Gaussian Processes (DGPs), which are another generalization constructed by a hierarchical concatenation of GPs. In this work, we propose a generalization of TGPs named Deep Transformed Gaussian Processes (DTGPs), which follows the trend of concatenating layers of stochastic processes. More precisely, we obtain a multi-layer model in which each layer is a TGP. This generalization implies an increment of flexibility with respect to both TGPs and DGPs. Exact inference in such a model is intractable. However, we show that one can use variational inference to approximate the required computations yielding a straightforward extension of the popular DSVI inference algorithm Salimbeni et al (2017). The experiments conducted evaluate the proposed novel DTGPs in multiple regression datasets, achieving good scalability and performance.

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

We introduce a scalable method to approximate the kernel of the Linearized Laplace Approximation (LLA). For this, we use a surrogate deep neural network (DNN) that learns a compact feature representation whose inner product replicates the Neural Tangent Kernel (NTK). This avoids the need to compute large Jacobians. Training relies solely on efficient Jacobian-vector products, allowing to compute predictive uncertainty on large-scale pre-trained DNNs. Experimental results show similar or improved uncertainty estimation and calibration compared to existing LLA approximations. Notwithstanding, biasing the learned kernel significantly enhances out-of-distribution detection. This remarks the benefits of the proposed method for finding better kernels than the NTK in the context of LLA to compute prediction uncertainty given a pre-trained DNN.

Francisco M. Castro-Macías, Pablo Morales-Álvarez, Saifuddin Syed et al.

Sampling from unnormalized multimodal distributions with limited density evaluations remains a fundamental challenge in machine learning and natural sciences. Successful approaches construct a bridge between a tractable reference and the target distribution. Parallel Tempering (PT) serves as the gold standard, while recent diffusion-based approaches offer a continuous alternative at the cost of neural training. In this work, we introduce Conditional Diffusion Sampling (CDS), a framework that combines these two paradigms. To this end, we derive Conditional Interpolants, a class of stochastic processes whose transport dynamics are governed by an exact, closed-form stochastic differential equation (SDE), requiring no neural approximation. Although these dynamics require sampling from a non-trivial initialization distribution, we show both theoretically and empirically that the cost of this initialization diminishes for sufficiently short diffusion times. CDS leverages this by a two-stage procedure: (1) PT is used to efficiently sample the initial distribution, and then (2) samples are transported via the transport SDE. This combination couples the robust global exploration of PT with efficient local transport. Experiments suggest that CDS has the potential to achieve a superior trade-off between sample quality and density evaluation cost compared to state-of-the-art samplers.

Daniel Fernández-Sánchez, Eduardo C. Garrido-Merchán, Daniel Hernández-Lobato

Bayesian optimization (BO) methods based on information theory have obtained state-of-the-art results in several tasks. These techniques heavily rely on the Kullback-Leibler (KL) divergence to compute the acquisition function. In this work, we introduce a novel information-based class of acquisition functions for BO called Alpha Entropy Search (AES). AES is based on the α-divergence, that generalizes the KL divergence. Iteratively, AES selects the next evaluation point as the one whose associated target value has the highest level of the dependency with respect to the location and associated value of the global maximum of the optimization problem. Dependency is measured in terms of the α-divergence, as an alternative to the KL divergence. Intuitively, this favors the evaluation of the objective function at the most informative points about the global maximum. The α-divergence has a free parameter α, which determines the behavior of the divergence, trading-off evaluating differences between distributions at a single mode, and evaluating differences globally. Therefore, different values of α result in different acquisition functions. AES acquisition lacks a closed-form expression. However, we propose an efficient and accurate approximation using a truncated Gaussian distribution. In practice, the value of α can be chosen by the practitioner, but here we suggest to use a combination of acquisition functions obtained by simultaneously considering a range of values of α. We provide an implementation of AES in BOTorch and we evaluate its performance in both synthetic, benchmark and real-world experiments involving the tuning of the hyper-parameters of a deep neural network. These experiments show that the performance of AES is competitive with respect to other information-based acquisition functions such as JES, MES or PES.

Seyedsaman Emami, Daniel Hernández-Lobato, Gonzalo Martínez-Muñoz

Multi-Task Learning (MTL) aims to boost predictive performance by sharing information across related tasks, yet conventional methods often suffer from negative transfer when unrelated or noisy tasks are forced to share representations. We propose Robust Multi-Task Boosting using Clustering and Local Ensembling (RMB-CLE), a principled MTL framework that integrates error-based task clustering with local ensembling. Unlike prior work that assumes fixed clusters or hand-crafted similarity metrics, RMB-CLE derives inter-task similarity directly from cross-task errors, which admit a risk decomposition into functional mismatch and irreducible noise, providing a theoretically grounded mechanism to prevent negative transfer. Tasks are grouped adaptively via agglomerative clustering, and within each cluster, a local ensemble enables robust knowledge sharing while preserving task-specific patterns. Experiments show that RMB-CLE recovers ground-truth clusters in synthetic data and consistently outperforms multi-task, single-task, and pooling-based ensemble methods across diverse real-world and synthetic benchmarks. These results demonstrate that RMB-CLE is not merely a combination of clustering and boosting but a general and scalable framework that establishes a new basis for robust multi-task learning.

Pedro Jiménez, Luis A. Ortega, Pablo Morales-Álvarez et al.

Deep neural networks (DNNs) often produce overconfident out-of-distribution predictions, motivating Bayesian uncertainty quantification. The Linearized Laplace Approximation (LLA) achieves this by linearizing the DNN and applying Laplace inference to the resulting model. Importantly, the linear model is also used for prediction. We argue this linearization in the posterior may degrade fidelity to the true Laplace approximation. To alleviate this problem, without increasing significantly the computational cost, we propose the Quadratic Laplace Approximation (QLA). QLA approximates each second order factor in the approximate Laplace log-posterior using a rank-one factor obtained via efficient power iterations. QLA is expected to yield a posterior precision closer to that of the full Laplace without forming the full Hessian, which is typically intractable. For prediction, QLA also uses the linearized model. Empirically, QLA yields modest yet consistent uncertainty estimation improvements over LLA on five regression datasets.

Seyedsaman Emami, Gonzalo Martínez-Muñoz, Daniel Hernández-Lobato

Multi-task learning (MTL) has shown effectiveness in exploiting shared information across tasks to improve generalization. MTL assumes tasks share similarities that can improve performance. In addition, boosting algorithms have demonstrated exceptional performance across diverse learning problems, primarily due to their ability to focus on hard-to-learn instances and iteratively reduce residual errors. This makes them a promising approach for learning multi-task problems. However, real-world MTL scenarios often involve tasks that are not well-aligned (known as outlier or adversarial tasks), which do not share beneficial similarities with others and can, in fact, deteriorate the performance of the overall model. To overcome this challenge, we propose Robust-Multi-Task Gradient Boosting (R-MTGB), a novel boosting framework that explicitly models and adapts to task heterogeneity during training. R-MTGB structures the learning process into three sequential blocks: (1) learning shared patterns, (2) partitioning tasks into outliers and non-outliers with regularized parameters, and (3) fine-tuning task-specific predictors. This architecture enables R-MTGB to automatically detect and penalize outlier tasks while promoting effective knowledge transfer among related tasks. Our method integrates these mechanisms seamlessly within gradient boosting, allowing robust handling of noisy or adversarial tasks without sacrificing accuracy. Extensive experiments on both synthetic benchmarks and real-world datasets demonstrate that our approach successfully isolates outliers, transfers knowledge, and consistently reduces prediction errors for each task individually, and achieves overall performance gains across all tasks. These results highlight robustness, adaptability, and reliable convergence of R-MTGB in challenging MTL environments.

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

Recently, there has been an increasing interest in performing post-hoc uncertainty estimation about the predictions of pre-trained deep neural networks (DNNs). Given a pre-trained DNN via back-propagation, these methods enhance the original network by adding output confidence measures, such as error bars, without compromising its initial accuracy. In this context, we introduce a novel family of sparse variational Gaussian processes (GPs), where the posterior mean is fixed to any continuous function when using a universal kernel. Specifically, we fix the mean of this GP to the output of the pre-trained DNN, allowing our approach to effectively fit the GP's predictive variances to estimate the DNN prediction uncertainty. Our approach leverages variational inference (VI) for efficient stochastic optimization, with training costs that remain independent of the number of training points, scaling efficiently to large datasets such as ImageNet. The proposed method, called fixed mean GP (FMGP), is architecture-agnostic, relying solely on the pre-trained model's outputs to adjust the predictive variances. Experimental results demonstrate that FMGP improves both uncertainty estimation and computational efficiency when compared to state-of-the-art methods.

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.

Bahram Jafrasteh, Carlos Villacampa-Calvo, Daniel Hernández-Lobato

Gaussian Processes (GPs) are Bayesian models that provide uncertainty estimates associated to the predictions made. They are also very flexible due to their non-parametric nature. Nevertheless, GPs suffer from poor scalability as the number of training instances N increases. More precisely, they have a cubic cost with respect to $N$. To overcome this problem, sparse GP approximations are often used, where a set of $M \ll N$ inducing points is introduced during training. The location of the inducing points is learned by considering them as parameters of an approximate posterior distribution $q$. Sparse GPs, combined with variational inference for inferring $q$, reduce the training cost of GPs to $\mathcal{O}(M^3)$. Critically, the inducing points determine the flexibility of the model and they are often located in regions of the input space where the latent function changes. A limitation is, however, that for some learning tasks a large number of inducing points may be required to obtain a good prediction performance. To address this limitation, we propose here to amortize the computation of the inducing points locations, as well as the parameters of the variational posterior approximation q. For this, we use a neural network that receives the observed data as an input and outputs the 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 GP approaches. However, with our method the number of inducing points is reduced drastically due to their dependency on the input data. This makes our method scale to larger datasets and have faster training and prediction times.

Daniel Fernández-Sánchez, Eduardo C. Garrido-Merchán, Daniel Hernández-Lobato

We present MESMOC+, an improved version of Max-value Entropy search for Multi-Objective Bayesian optimization with Constraints (MESMOC). MESMOC+ can be used to solve constrained multi-objective problems when the objectives and the constraints are expensive to evaluate. MESMOC+ works by minimizing the entropy of the solution of the optimization problem in function space, i.e., the Pareto frontier, to guide the search for the optimum. The cost of MESMOC+ is linear in the number of objectives and constraints. Furthermore, it is often significantly smaller than the cost of alternative methods based on minimizing the entropy of the Pareto set. The reason for this is that it is easier to approximate the required computations in MESMOC+. Moreover, MESMOC+'s acquisition function is expressed as the sum of one acquisition per each black-box (objective or constraint). Thus, it can be used in a decoupled evaluation setting in which one chooses not only the next input location to evaluate, but also which black-box to evaluate there. We compare MESMOC+ with related methods in synthetic and real optimization problems. These experiments show that the entropy estimation provided by MESMOC+ is more accurate than that of previous methods. This leads to better optimization results. MESMOC+ is also competitive with other information-based methods for constrained multi-objective Bayesian optimization, but it is significantly faster.

Eduardo C. Garrido-Merchán, Daniel Hernández-Lobato

Real-world problems often involve the optimization of several objectives under multiple constraints. An example is the hyper-parameter tuning problem of machine learning algorithms. In particular, the minimization of the estimation of the generalization error of a deep neural network and at the same time the minimization of its prediction time. We may also consider as a constraint that the deep neural network must be implemented in a chip with an area below some size. Here, both the objectives and the constraint are black boxes, i.e., functions whose analytical expressions are unknown and are expensive to evaluate. Bayesian optimization (BO) methodologies have given state-of-the-art results for the optimization of black-boxes. Nevertheless, most BO methods are sequential and evaluate the objectives and the constraints at just one input location, iteratively. Sometimes, however, we may have resources to evaluate several configurations in parallel. Notwithstanding, no parallel BO method has been proposed to deal with the optimization of multiple objectives under several constraints. If the expensive evaluations can be carried out in parallel (as when a cluster of computers is available), sequential evaluations result in a waste of resources. This article introduces PPESMOC, Parallel Predictive Entropy Search for Multi-objective Bayesian Optimization with Constraints, an information-based batch method for the simultaneous optimization of multiple expensive-to-evaluate black-box functions under the presence of several constraints. Iteratively, PPESMOC selects a batch of input locations at which to evaluate the black-boxes so as to maximally reduce the entropy of the Pareto set of the optimization problem. We present empirical evidence in the form of synthetic, benchmark and real-world experiments that illustrate the effectiveness of PPESMOC.

Carlos Villacampa-Calvo, Bryan Zaldivar, Eduardo C. Garrido-Merchán et al.

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.

Simón Rodríguez Santana, Daniel Hernández-Lobato

Neural networks are popular state-of-the-art models for many different tasks.They are often trained via back-propagation to find a value of the weights that correctly predicts the observed data. Although back-propagation has shown good performance in many applications, it cannot easily output an estimate of the uncertainty in the predictions made. Estimating the uncertainty in the predictions is a critical aspect with important applications, and one method to obtain this information is following a Bayesian approach to estimate a posterior distribution on the model parameters. This posterior distribution summarizes which parameter values are compatible with the data, but is usually intractable and has to be approximated. Several mechanisms have been considered for solving this problem. We propose here a general method for approximate Bayesian inference that is based on minimizingα-divergences and that allows for flexible approximate distributions. The method is evaluated in the context of Bayesian neural networks on extensive experiments. The results show that, in regression problems, it often gives better performance in terms of the test log-likelihoodand sometimes in terms of the squared error. In classification problems, however, it gives competitive results.

Irene Córdoba, Eduardo C. Garrido-Merchán, Daniel Hernández-Lobato et al.

The PC algorithm is a popular method for learning the structure of Gaussian Bayesian networks. It carries out statistical tests to determine absent edges in the network. It is hence governed by two parameters: (i) The type of test, and (ii) its significance level. These parameters are usually set to values recommended by an expert. Nevertheless, such an approach can suffer from human bias, leading to suboptimal reconstruction results. In this paper we consider a more principled approach for choosing these parameters in an automatic way. For this we optimize a reconstruction score evaluated on a set of different Gaussian Bayesian networks. This objective is expensive to evaluate and lacks a closed-form expression, which means that Bayesian optimization (BO) is a natural choice. BO methods use a model to guide the search and are hence able to exploit smoothness properties of the objective surface. We show that the parameters found by a BO method outperform those found by a random search strategy and the expert recommendation. Importantly, we have found that an often overlooked statistical test provides the best over-all reconstruction results.

Eduardo C. Garrido-Merchán, Daniel Hernández-Lobato

Bayesian Optimization (BO) methods are useful for optimizing functions that are expen- sive to evaluate, lack an analytical expression and whose evaluations can be contaminated by noise. These methods rely on a probabilistic model of the objective function, typically a Gaussian process (GP), upon which an acquisition function is built. The acquisition function guides the optimization process and measures the expected utility of performing an evaluation of the objective at a new point. GPs assume continous input variables. When this is not the case, for example when some of the input variables take categorical or integer values, one has to introduce extra approximations. Consider a suggested input location taking values in the real line. Before doing the evaluation of the objective, a common approach is to use a one hot encoding approximation for categorical variables, or to round to the closest integer, in the case of integer-valued variables. We show that this can lead to problems in the optimization process and describe a more principled approach to account for input variables that are categorical or integer-valued. We illustrate in both synthetic and a real experiments the utility of our approach, which significantly improves the results of standard BO methods using Gaussian processes on problems with categorical or integer-valued variables.

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.

Eduardo C. Garrido-Merchán, Daniel Hernández-Lobato

Bayesian optimization (BO) methods are useful for optimizing functions that are expensive to evaluate, lack an analytical expression and whose evaluations can be contaminated by noise. These methods rely on a probabilistic model of the objective function, typically a Gaussian process (GP), upon which an acquisition function is built. This function guides the optimization process and measures the expected utility of performing an evaluation of the objective at a new point. GPs assume continous input variables. When this is not the case, such as when some of the input variables take integer values, one has to introduce extra approximations. A common approach is to round the suggested variable value to the closest integer before doing the evaluation of the objective. We show that this can lead to problems in the optimization process and describe a more principled approach to account for input variables that are integer-valued. We illustrate in both synthetic and a real experiments the utility of our approach, which significantly improves the results of standard BO methods on problems involving integer-valued variables.

Eduardo C. Garrido-Merchán, Daniel Hernández-Lobato

This work presents PESMOC, Predictive Entropy Search for Multi-objective Bayesian Optimization with Constraints, an information-based strategy for the simultaneous optimization of multiple expensive-to-evaluate black-box functions under the presence of several constraints. PESMOC can hence be used to solve a wide range of optimization problems. Iteratively, PESMOC chooses an input location on which to evaluate the objective functions and the constraints so as to maximally reduce the entropy of the Pareto set of the corresponding optimization problem. The constraints considered in PESMOC are assumed to have similar properties to those of the objective functions in typical Bayesian optimization problems. That is, they do not have a known expression (which prevents gradient computation), their evaluation is considered to be very expensive, and the resulting observations may be corrupted by noise. These constraints arise in a plethora of expensive black-box optimization problems. We carry out synthetic experiments to illustrate the effectiveness of PESMOC, where we sample both the objectives and the constraints from a Gaussian process prior. The results obtained show that PESMOC is able to provide better recommendations with a smaller number of evaluations than a strategy based on random search.

Thang D. Bui, Daniel Hernández-Lobato, Yingzhen Li et al.

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.

Daniel Hernández-Lobato, José Miguel Hernández-Lobato, Amar Shah et al.

We present PESMO, a Bayesian method for identifying the Pareto set of multi-objective optimization problems, when the functions are expensive to evaluate. The central idea of PESMO is to choose evaluation points so as to maximally reduce the entropy of the posterior distribution over the Pareto set. Critically, the PESMO multi-objective acquisition function can be decomposed as a sum of objective-specific acquisition functions, which enables the algorithm to be used in \emph{decoupled} scenarios in which the objectives can be evaluated separately and perhaps with different costs. This decoupling capability also makes it possible to identify difficult objectives that require more evaluations. 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 PESMO with other related methods for multi-objective Bayesian optimization on synthetic and real-world problems. The results show that PESMO produces better recommendations with a smaller number of evaluations of the objectives, and that a decoupled evaluation can lead to improvements in performance, particularly when the number of objectives is large.

Thang D. Bui, José Miguel Hernández-Lobato, Yingzhen Li et al.

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 probabilistic and non-parametric and as such are arguably more flexible, have a greater capacity to generalise, and provide better calibrated uncertainty estimates than alternative deep models. The focus of this paper is scalable approximate Bayesian learning of these networks. The paper develops a novel and efficient extension of probabilistic backpropagation, a state-of-the-art method for training Bayesian neural networks, that can be used to train DGPs. The new method leverages a recently proposed method for scaling Expectation Propagation, called stochastic Expectation Propagation. The method is able to automatically discover useful input warping, expansion or compression, and it is therefore is a flexible form of Bayesian kernel design. We demonstrate the success of the new method for supervised learning on several real-world datasets, showing that it typically outperforms GP regression and is never much worse.

Daniel Hernández-Lobato, José Miguel Hernández-Lobato, Yingzhen Li et al.

A method for large scale Gaussian process classification has been recently proposed based on expectation propagation (EP). Such a method allows Gaussian process classifiers to be trained on very large datasets that were out of the reach of previous deployments of EP and has been shown to be competitive with related techniques based on stochastic variational inference. Nevertheless, the memory resources required scale linearly with the dataset size, unlike in variational methods. This is a severe limitation when the number of instances is very large. Here we show that this problem is avoided when stochastic EP is used to train the model.

José Miguel Hernández-Lobato, Yingzhen Li, Mark Rowland et al.

Black-box alpha (BB-$α$) is a new approximate inference method based on the minimization of $α$-divergences. BB-$α$ scales to large datasets because it can be implemented using stochastic gradient descent. BB-$α$ can 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) ($α\rightarrow 0$) and an algorithm similar to expectation propagation (EP) ($α= 1$). Experiments on probit regression and neural network regression and classification problems show that BB-$α$ with non-standard settings of $α$, such as $α= 0.5$, usually produces better predictions than with $α\rightarrow 0$ (VB) or $α= 1$ (EP).

Daniel Hernández-Lobato, José Miguel Hernández-Lobato

Variational methods have been recently considered for scaling the training process of Gaussian process classifiers to large datasets. As an alternative, we describe here how to train these classifiers efficiently using expectation propagation. The proposed method allows for handling datasets with millions of data instances. More precisely, it can be used for (i) training in a distributed fashion where the data instances are sent to different nodes in which the required computations are carried out, and for (ii) maximizing an estimate of the marginal likelihood using a stochastic approximation of the gradient. Several experiments indicate that the method described is competitive with the variational approach.

Daniel Hernández-Lobato, Pablo Morales-Mombiela, David Lopez-Paz et al.

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.

Daniel Hernández-Lobato, Viktoriia Sharmanska, Kristian Kersting et al.

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 sigmoid 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.

José Miguel Hernández-Lobato, James Robert 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 inaccurate 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.