Gaussian processes~(Kriging) are interpolating data-driven models that are frequently applied in various disciplines. Often, Gaussian processes are trained on datasets and are subsequently embedded as surrogate models in optimization problems. These optimization problems are nonconvex and global optimization is desired. However, previous literature observed computational burdens limiting deterministic global optimization to Gaussian processes trained on few data points. We propose a reduced-space formulation for deterministic global optimization with trained Gaussian processes embedded. For optimization, the branch-and-bound solver branches only on the degrees of freedom and McCormick relaxations are propagated through explicit Gaussian process models. The approach also leads to significantly smaller and computationally cheaper subproblems for lower and upper bounding. To further accelerate convergence, we derive envelopes of common covariance functions for GPs and tight relaxations of acquisition functions used in Bayesian optimization including expected improvement, probability of improvement, and lower confidence bound. In total, we reduce computational time by orders of magnitude compared to state-of-the-art methods, thus overcoming previous computational burdens. We demonstrate the performance and scaling of the proposed method and apply it to Bayesian optimization with global optimization of the acquisition function and chance-constrained programming. The Gaussian process models, acquisition functions, and training scripts are available open-source within the "MeLOn - Machine Learning Models for Optimization" toolbox~(https://git.rwth-aachen.de/avt.svt/public/MeLOn).

We introduce a theoretical model of information acquisition under resource limitations in a noisy environment. An agent must guess the truth value of a given Boolean formula $\varphi$ after performing a bounded number of noisy tests of the truth values of variables in the formula. We observe that, in general, the problem of finding an optimal testing strategy for $\phi$ is hard, but we suggest a useful heuristic. The techniques we use also give insight into two apparently unrelated, but well-studied problems: (1) \emph{rational inattention}, that is, when it is rational to ignore pertinent information (the optimal strategy may involve hardly ever testing variables that are clearly relevant to $\phi$), and (2) what makes a formula hard to learn/remember.

Learning a Gaussian Mixture Model (GMM) is hard when the number of parameters is too large given the amount of available data. As a remedy, we propose restricting the GMM to a Gaussian Markov Random Field Mixture Model (GMRF-MM), as well as a new method for estimating the latter's sparse precision (i.e., inverse covariance) matrices. When the sparsity pattern of each matrix is known, we propose an efficient optimization method for the Maximum Likelihood Estimate (MLE) of that matrix. When it is unknown, we utilize the popular Graphical LASSO (GLASSO) to estimate that pattern. However, we show that even for a single Gaussian, when GLASSO is tuned to successfully estimate the sparsity pattern, it does so at the price of a substantial bias of the values of the nonzero entries of the matrix, and we show that this problem only worsens in a mixture setting. To overcome this, we discard the non-zero values estimated by GLASSO, keep only its pattern estimate and use it within the proposed MLE method. This yields an effective two-step procedure that removes the bias. We show that our "debiasing" approach outperforms GLASSO in both the single-GMRF and the GMRF-MM cases. We also show that when learning priors for image patches, our method outperforms GLASSO even if we merely use an educated guess about the sparsity pattern, and that our GMRF-MM outperforms the baseline GMM on real and synthetic high-dimensional datasets. Our code is available at \url{https://github.com/shahaffind/GMRF-MM}.

An influence diagram is a graphical representation of sequential decision-making under uncertainty, defining a structured decision problem by conditional probability functions and additive utility functions over discrete state and action variables. The task of finding the maximum expected utility of influence diagrams is closely related to the cost-optimal probabilistic planning, stochastic programmings, or model-based reinforcement learning. In this position paper, we address the heuristic search for solving influence diagram, where we generate admissible heuristic functions from graph decomposition schemes. Then, we demonstrate how such heuristics can guide an AND/OR branch and bound search. Finally, we briefly discuss the future directions for improving the quality of heuristic functions and search strategies.

This thesis describes work on two applications of probabilistic programming: the learning of probabilistic program code given specifications, in particular program code of one-dimensional samplers; and the facilitation of sequential Monte Carlo inference with help of data-driven proposals. The latter is presented with experimental results on a linear Gaussian model and a non-parametric dependent Dirichlet process mixture of objects model for object recognition and tracking. In Chapter 1 we provide a brief introduction to probabilistic programming. In Chapter 2 we present an approach to automatic discovery of samplers in the form of probabilistic programs. We formulate a Bayesian approach to this problem by specifying a grammar-based prior over probabilistic program code. We use an approximate Bayesian computation method to learn the programs, whose executions generate samples that statistically match observed data or analytical characteristics of distributions of interest. In our experiments we leverage different probabilistic programming systems to perform Markov chain Monte Carlo sampling over the space of programs. Experimental results have demonstrated that, using the proposed methodology, we can learn approximate and even some exact samplers. Finally, we show that our results are competitive with regard to genetic programming methods. In Chapter 3, we describe a way to facilitate sequential Monte Carlo inference in probabilistic programming using data-driven proposals. In particular, we develop a distance-based proposal for the non-parametric dependent Dirichlet process mixture of objects model. We implement this approach in the probabilistic programming system Anglican, and show that for that model data-driven proposals provide significant performance improvements. We also explore the possibility of using neural networks to improve data-driven proposals.

Over the past few years, providers such as Google, Microsoft, and Amazon have started to provide customers with access to software interfaces allowing them to easily embed machine learning tasks into their applications. Overall, organizations can now use Machine Learning as a Service (MLaaS) engines to outsource complex tasks, e.g., training classifiers, performing predictions, clustering, etc. They can also let others query models trained on their data. Naturally, this approach can also be used (and is often advocated) in other contexts, including government collaborations, citizen science projects, and business-to-business partnerships. However, if malicious users were able to recover data used to train these models, the resulting information leakage would create serious issues. Likewise, if the inner parameters of the model are considered proprietary information, then access to the model should not allow an adversary to learn such parameters. In this document, we set to review privacy challenges in this space, providing a systematic review of the relevant research literature, also exploring possible countermeasures. More specifically, we provide ample background information on relevant concepts around machine learning and privacy. Then, we discuss possible adversarial models and settings, cover a wide range of attacks that relate to private and/or sensitive information leakage, and review recent results attempting to defend against such attacks. Finally, we conclude with a list of open problems that require more work, including the need for better evaluations, more targeted defenses, and the study of the relation to policy and data protection efforts.

This work considers methods for imposing sparsity in Bayesian regression with applications in nonlinear system identification. We first review automatic relevance determination (ARD) and analytically demonstrate the need to additional regularization or thresholding to achieve sparse models. We then discuss two classes of methods, regularization based and thresholding based, which build on ARD to learn parsimonious solutions to linear problems. In the case of orthogonal covariates, we analytically demonstrate favorable performance with regards to learning a small set of active terms in a linear system with a sparse solution. Several example problems are presented to compare the set of proposed methods in terms of advantages and limitations to ARD in bases with hundreds of elements. The aim of this paper is to analyze and understand the assumptions that lead to several algorithms and to provide theoretical and empirical results so that the reader may gain insight and make more informed choices regarding sparse Bayesian regression.

We consider the problem of discovering the causal process that generated a collection of datasets. We assume that all these datasets were generated by unknown sparse interventions on a structural causal model (SCM) $G$, that we want to identify. Recently, Bengio et al. (2020) argued that among all SCMs, $G$ is the fastest to adapt from one dataset to another, and proposed a meta-learning criterion to identify the causal direction in a two-variable SCM. While the experiments were promising, the theoretical justification was incomplete. Our contribution is a theoretical investigation of the adaptation speed of simple two-variable SCMs. We use convergence rates from stochastic optimization to justify that a relevant proxy for adaptation speed is distance in parameter space after intervention. Using this proxy, we show that the SCM with the correct causal direction is advantaged for categorical and normal cause-effect datasets when the intervention is on the cause variable. When the intervention is on the effect variable, we provide a more nuanced picture which highlights that the fastest-to-adapt heuristic is not always valid. Code to reproduce experiments is available at https://github.com/remilepriol/causal-adaptation-speed

Traceability is a fundamental component of the modern software development process that helps to ensure properly functioning, secure programs. Due to the high cost of manually establishing trace links, researchers have developed automated approaches that draw relationships between pairs of textual software artifacts using similarity measures. However, the effectiveness of such techniques are often limited as they only utilize a single measure of artifact similarity and cannot simultaneously model (implicit and explicit) relationships across groups of diverse development artifacts. In this paper, we illustrate how these limitations can be overcome through the use of a tailored probabilistic model. To this end, we design and implement a HierarchiCal PrObabilistic Model for SoftwarE Traceability (Comet) that is able to infer candidate trace links. Comet is capable of modeling relationships between artifacts by combining the complementary observational prowess of multiple measures of textual similarity. Additionally, our model can holistically incorporate information from a diverse set of sources, including developer feedback and transitive (often implicit) relationships among groups of software artifacts, to improve inference accuracy. We conduct a comprehensive empirical evaluation of Comet that illustrates an improvement over a set of optimally configured baselines of $\approx$14% in the best case and $\approx$5% across all subjects in terms of average precision. The comparative effectiveness of Comet in practice, where optimal configuration is typically not possible, is likely to be higher. Finally, we illustrate Comets potential for practical applicability in a survey with developers from Cisco Systems who used a prototype Comet Jenkins plugin.

In this paper we propose an efficient stochastic optimization algorithm to search for Bayesian experimental designs such that the expected information gain is maximized. The gradient of the expected information gain with respect to experimental design parameters is given by a nested expectation, for which the standard Monte Carlo method using a fixed number of inner samples yields a biased estimator. In this paper, applying the idea of randomized multilevel Monte Carlo methods, we introduce an unbiased Monte Carlo estimator for the gradient of the expected information gain with finite expected squared $\ell_2$-norm and finite expected computational cost per sample. Our unbiased estimator can be combined well with stochastic gradient descent algorithms, which results in our proposal of an optimization algorithm to search for an optimal Bayesian experimental design. Numerical experiments confirm that our proposed algorithm works well not only for a simple test problem but also for a more realistic pharmacokinetic problem.