Gaussian process regression is a widely-applied method for function approximation and uncertainty quantification. The technique has gained popularity recently in the machine learning community due to its robustness and interpretability. The mathematical methods we discuss in this paper are an extension of the Gaussian-process framework. We are proposing advanced kernel designs that only allow for functions with certain desirable characteristics to be elements of the reproducing kernel Hilbert space (RKHS) that underlies all kernel methods and serves as the sample space for Gaussian process regression. These desirable characteristics reflect the underlying physics; two obvious examples are symmetry and periodicity constraints. In addition, non-stationary kernel designs can be defined in the same framework to yield flexible multi-task Gaussian processes. We will show the impact of advanced kernel designs on Gaussian processes using several synthetic and two scientific data sets. The results show that including domain knowledge, communicated through advanced kernel designs, has a significant impact on the accuracy and relevance of the function approximation. Gaussian processes (GPs) [14] provide a powerful mathematical framework for function approximation from data. The associated technique is generally referred to as Gaussian process regression (GPR). GPs are flexible, robust, non-parametric and naturally include uncertainty quantification.

This paper describes a general-purpose extension of max-value entropy search, a popular approach for Bayesian Optimisation (BO). A novel approximation is proposed for the information gain -- an information-theoretic quantity central to solving a range of BO problems, including noisy, multi-fidelity and batch optimisations across both continuous and highly-structured discrete spaces. Previously, these problems have been tackled separately within information-theoretic BO, each requiring a different sophisticated approximation scheme, except for batch BO, for which no computationally-lightweight information-theoretic approach has previously been proposed. GIBBON (General-purpose Information-Based Bayesian OptimisatioN) provides a single principled framework suitable for all the above, out-performing existing approaches whilst incurring substantially lower computational overheads. In addition, GIBBON does not require the problem's search space to be Euclidean and so is the first high-performance yet computationally light-weight acquisition function that supports batch BO over general highly structured input spaces like molecular search and gene design. Moreover, our principled derivation of GIBBON yields a natural interpretation of a popular batch BO heuristic based on determinantal point processes.

There has been rapidly growing interest in meta-learning as a method for increasing the flexibility and sample efficiency of reinforcement learning. One problem in this area of research, however, has been a scarcity of adequate benchmark tasks. In general, the structure underlying past benchmarks has either been too simple to be inherently interesting, or too ill-defined to support principled analysis. In the present work, we introduce a new benchmark for meta-RL research, which combines structural richness with structural transparency. Alchemy is a 3D video game, implemented in Unity, which involves a latent causal structure that is resampled procedurally from episode to episode, affording structure learning, online inference, hypothesis testing and action sequencing based on abstract domain knowledge. We evaluate a pair of powerful RL agents on Alchemy and present an in-depth analysis of one of these agents. Results clearly indicate a frank and specific failure of meta-learning, providing validation for Alchemy as a challenging benchmark for meta-RL. Concurrent with this report, we are releasing Alchemy as public resource, together with a suite of analysis tools and sample agent trajectories.

We propose a novel framework for modeling multiple multivariate point processes, each with heterogeneous event types that share an underlying space and obey the same generative mechanism. Focusing on Hawkes processes and their variants that are associated with Granger causality graphs, our model leverages an uncountable event type space and samples the graphs with different sizes from a nonparametric model called {\it graphon}. Given those graphs, we can generate the corresponding Hawkes processes and simulate event sequences. Learning this graphon-based Hawkes process model helps to 1) infer the underlying relations shared by different Hawkes processes; and 2) simulate event sequences with different event types but similar dynamics. We learn the proposed model by minimizing the hierarchical optimal transport distance between the generated event sequences and the observed ones, leading to a novel reward-augmented maximum likelihood estimation method. We analyze the properties of our model in-depth and demonstrate its rationality and effectiveness in both theory and experiments.

We introduce Invertible Dense Networks (i-DenseNets), a more parameter efficient alternative to Residual Flows. The method relies on an analysis of the Lipschitz continuity of the concatenation in DenseNets, where we enforce invertibility of the network by satisfying the Lipschitz constant. We extend this method by proposing a learnable concatenation, which not only improves the model performance but also indicates the importance of the concatenated representation. Additionally, we introduce the Concatenated LipSwish as activation function, for which we show how to enforce the Lipschitz condition and which boosts performance. The new architecture, i-DenseNet, out-performs Residual Flow and other flow-based models on density estimation evaluated in bits per dimension, where we utilize an equal parameter budget. Moreover, we show that the proposed model out-performs Residual Flows when trained as a hybrid model where the model is both a generative and a discriminative model.

Missing values with mixed data types is a common problem in a large number of machine learning applications such as processing of surveys and in different medical applications. Recently, Gaussian copula models have been suggested as a means of performing imputation of missing values using a probabilistic framework. While the present Gaussian copula models have shown to yield state of the art performance, they have two limitations: they are based on an approximation that is fast but may be imprecise and they do not support unordered multinomial variables. We address the first limitation using direct and arbitrarily precise approximations both for model estimation and imputation by using randomized quasi-Monte Carlo procedures. The method we provide has lower errors for the estimated model parameters and the imputed values, compared to previously proposed methods. We also extend the previous Gaussian copula models to include unordered multinomial variables in addition to the present support of ordinal, binary, and continuous variables.

Local approximations are popular methods to scale Gaussian processes (GPs) to big data. Local approximations reduce time complexity by dividing the original dataset into subsets and training a local expert on each subset. Aggregating the experts' prediction is done assuming either conditional dependence or independence between the experts. Imposing the \emph{conditional independence assumption} (CI) between the experts renders the aggregation of different expert predictions time efficient at the cost of poor uncertainty quantification. On the other hand, modeling dependent experts can provide precise predictions and uncertainty quantification at the expense of impractically high computational costs. By eliminating weak experts via a theory-guided expert selection step, we substantially reduce the computational cost of aggregating dependent experts while ensuring calibrated uncertainty quantification. We leverage techniques from the literature on undirected graphical models, using sparse precision matrices that encode conditional dependencies between experts to select the most important experts. Moreov

Probabilistic graphical models [KF09] provide a rich and unifying framework to model structured high-dimensional distributions in terms of the local dependencies between the input variables. The problem of inference in graphical models arises in many applications across scientific disciplines, see, e.g., [WJ08]. In this work, we study the inverse problem of learning graphical models from data. Various formalizations of this general learning problem have been studied during the past five decades, see, e.g., [CL68, Das97, AKN06, WRL06, AHHK12, SW12, LW12, BMS13, BGS14, Bre15, KM17], resulting in general theory and algorithms for various settings. In this work, we focus on learning Ising models [Isi25], the prototypical family of binary undirected graphical models with applications in computer vision, computational biology, and statistical physics [Li09, JEMF06, Fel04, Cha05].

Mathematical modelling of unemployment dynamics attempts to predict the probability of a job seeker finding a job as a function of time. This is typically achieved by using information in unemployment records. These records are right censored, making survival analysis a suitable approach for parameter estimation. The proposed model uses a deep artificial neural network (ANN) as a non-linear hazard function. Through embedding, high-cardinality categorical features are analysed efficiently. The posterior distribution of the ANN parameters are estimated using a variational Bayes method. The model is evaluated on a time-to-employment data set spanning from 2011 to 2020 provided by the Slovenian public employment service. It is used to determine the employment probability over time for each individual on the record. Similar models could be applied to other questions with multi-dimensional, high-cardinality categorical data including censored records. Such data is often encountered in personal records, for example in medical records.

Federated learning (FL) allows multiple clients to collaboratively learn a globally shared model through cycles of model aggregation and local model training without the need to share data. In this paper, we comprehensively study a new problem named aggregation error (AE), arising from the model aggregation stage on a server, which is mainly induced by the heterogeneity of the client data. Due to the large discrepancies between local models, the accompanying large AE generally results in a slow convergence and an expected reduction of accuracy for FL. In order to reduce AE, we propose a novel federated learning framework from a Bayesian perspective, in which a multivariate Gaussian product mechanism is employed to aggregate the local models. It is worth noting that the product of Gaussians is still a Gaussian. This property allows us to directly aggregate local expectations and covariances in a definitely convex form, thereby greatly reducing the AE. Accordingly, on the clients, we develop a new Federated Online Laplace Approximation (FOLA) method, which can estimate the parameters of the local posterior by repeatedly accumulating priors. Specifically, in every round, the global posterior distributed from the server can be treated as the priors, and thus the local posterior can also be effectively approximated by a Gaussian using FOLA. Experimental results on benchmarks reach state-of-the-arts performance and clearly demonstrate the advantages of the proposed method.