Without any assumptions about data generation, multiple causal models may explain our observations equally well. To avoid selecting a single arbitrary model that could result in unsafe decisions if it does not match reality, it is therefore essential to maintain a notion of epistemic uncertainty about our possible candidates. This thesis studies the problem of structure learning from a Bayesian perspective, approximating the posterior distribution over the structure of a causal model, represented as a directed acyclic graph (DAG), given data. It introduces Generative Flow Networks (GFlowNets), a novel class of probabilistic models designed for modeling distributions over discrete and compositional objects such as graphs. They treat generation as a sequential decision making problem, constructing samples of a target distribution defined up to a normalization constant piece by piece. In the first part of this thesis, we present the mathematical foundations of GFlowNets, their connections to existing domains of machine learning and statistics such as variational inference and reinforcement learning, and their extensions beyond discrete problems. In the second part of this thesis, we show how GFlowNets can approximate the posterior distribution over DAG structures of causal Bayesian Networks, along with the parameters of its causal mechanisms, given observational and experimental data.

Context. Developing secure and reliable software remains a key challenge in software engineering (SE). The ever-evolving technological landscape offers both opportunities and threats, creating a dynamic space where chaos and order compete. Secure software engineering (SSE) must continuously address vulnerabilities that endanger software systems and carry broader socio-economic risks, such as compromising critical national infrastructure and causing significant financial losses. Researchers and practitioners have explored methodologies like Static Application Security Testing Tools (SASTTs) and artificial intelligence (AI) approaches, including machine learning (ML) and large language models (LLMs), to detect and mitigate these vulnerabilities. Each method has unique strengths and limitations. Aim. This thesis seeks to bring order to the chaos in SSE by addressing domain-specific differences that impact AI accuracy. Methodology. The research employs a mix of empirical strategies, such as evaluating effort-aware metrics, analyzing SASTTs, conducting method-level analysis, and leveraging evidence-based techniques like systematic dataset reviews. These approaches help characterize vulnerability prediction datasets. Results. Key findings include limitations in static analysis tools for identifying vulnerabilities, gaps in SASTT coverage of vulnerability types, weak relationships among vulnerability severity scores, improved defect prediction accuracy using just-in-time modeling, and threats posed by untouched methods. Conclusions. This thesis highlights the complexity of SSE and the importance of contextual knowledge in improving AI-driven vulnerability and defect prediction. The comprehensive analysis advances effective prediction models, benefiting both researchers and practitioners.

Current methods commonly used for uncertainty quantification (UQ) in deep learning (DL) models utilize Bayesian methods which are computationally expensive and time-consuming. In this paper, we provide a detailed study of UQ based on evidential deep learning (EDL) for deep neural network models designed to identify jets in high energy proton-proton collisions at the Large Hadron Collider and explore its utility in anomaly detection. EDL is a DL approach that treats learning as an evidence acquisition process designed to provide confidence (or epistemic uncertainty) about test data. Using publicly available datasets for jet classification benchmarking, we explore hyperparameter optimizations for EDL applied to the challenge of UQ for jet identification. We also investigate how the uncertainty is distributed for each jet class, how this method can be implemented for the detection of anomalies, how the uncertainty compares with Bayesian ensemble methods, and how the uncertainty maps onto latent spaces for the models. Our studies uncover some pitfalls of EDL applied to anomaly detection and a more effective way to quantify uncertainty from EDL as compared with the foundational EDL setup. These studies illustrate a methodological approach to interpreting EDL in jet classification models, providing new insights on how EDL quantifies uncertainty and detects out-of-distribution data which may lead to improved EDL methods for DL models applied to classification tasks.

Error accumulation is effective for gradient sparsification in distributed settings: initially-unselected gradient entries are eventually selected as their accumulated error exceeds a certain level. The accumulation essentially behaves as a scaling of the learning rate for the selected entries. Although this property prevents the slow-down of lateral movements in distributed gradient descent, it can deteriorate convergence in some settings. This work proposes a novel sparsification scheme that controls the learning rate scaling of error accumulation. The development of this scheme follows two major steps: first, gradient sparsification is formulated as an inverse probability (inference) problem, and the Bayesian optimal sparsification mask is derived as a maximum-a-posteriori estimator. Using the prior distribution inherited from Top-$k$, we derive a new sparsification algorithm which can be interpreted as a regularized form of Top-$k$. We call this algorithm regularized Top-$k$ (RegTop-$k$). It utilizes past aggregated gradients to evaluate posterior statistics of the next aggregation. It then prioritizes the local accumulated gradient entries based on these posterior statistics. We validate our derivation through numerical experiments. In distributed linear regression, it is observed that while Top-$k$ remains at a fixed distance from the global optimum, RegTop-$k$ converges to the global optimum at significantly higher compression ratios. We further demonstrate the generalization of this observation by employing RegTop-$k$ in distributed training of ResNet-18 on CIFAR-10, where it noticeably outperforms Top-$k$.

Assistive mobile robots are a transformative technology that helps persons with disabilities regain the ability to move freely. Although autonomous wheelchairs significantly reduce user effort, they still require human input to allow users to maintain control and adapt to changing environments. Brain Computer Interface (BCI) stands out as a highly user-friendly option that does not require physical movement. Current BCI systems can understand whether users want to accelerate or decelerate, but they implement these changes in discrete speed steps rather than allowing for smooth, continuous velocity adjustments. This limitation prevents the systems from mimicking the natural, fluid speed changes seen in human self-paced motion. The authors aim to address this limitation by redesigning the perception-action cycle in a BCI controlled robotic system: improving how the robotic agent interprets the user's motion intentions (world state) and implementing these actions in a way that better reflects natural physical properties of motion, such as inertia and damping. The scope of this paper focuses on the perception aspect. We asked and answered a normative question "what computation should the robotic agent carry out to optimally perceive incomplete or noisy sensory observations?" Empirical EEG data were collected, and probabilistic representation that served as world state distributions were learned and evaluated in a Generative Adversarial Network framework. The ROS framework was established that connected with a Gazebo environment containing a digital twin of an indoor space and a virtual model of a robotic wheelchair. Signal processing and statistical analyses were implemented to identity the most discriminative features in the spatial-spectral-temporal dimensions, which are then used to construct the world model for the robotic agent to interpret user motion intentions as a Bayesian observer.

Monotone learning refers to learning processes in which expected performance consistently improves as more training data is introduced. Non-monotone behavior of machine learning has been the topic of a series of recent works, with various proposals that ensure monotonicity by applying transformations or wrappers on learning algorithms. In this work, from a different perspective, we tackle the topic of monotone learning within the framework of Probably Approximately Correct (PAC) learning theory. Following the mechanism that estimates sample complexity of a PAC-learnable problem, we derive a performance lower bound for that problem, and prove the monotonicity of that bound as the sample sizes increase. By calculating the lower bound distribution, we are able to prove that given a PAC-learnable problem with a hypothesis space that is either of finite size or of finite VC dimension, any learning algorithm based on Empirical Risk Minimization (ERM) is monotone if training samples are independent and identically distributed (i.i.d.). We further carry out an experiment on two concrete machine learning problems, one of which has a finite hypothesis set, and the other of finite VC dimension, and compared the experimental data for the empirical risk distributions with the estimated theoretical bound. The results of the comparison have confirmed the monotonicity of learning for the two PAC-learnable problems.

An event sequence generated by a temporal point process is often associated with a hidden and structured event branching process that captures the triggering relations between its historical and current events. In this study, we design a new plug-and-play module based on the Bregman ADMM (BADMM) algorithm, which infers event branches associated with event sequences in the maximum likelihood estimation framework of temporal point processes (TPPs). Specifically, we formulate the inference of event branches as an optimization problem for the event transition matrix under sparse and low-rank constraints, which is embedded in existing TPP models or their learning paradigms. We can implement this optimization problem based on subspace clustering and sparse group-lasso, respectively, and solve it using the Bregman ADMM algorithm, whose unrolling leads to the proposed BADMM module. When learning a classic TPP (e.g., Hawkes process) by the expectation-maximization algorithm, the BADMM module helps derive structured responsibility matrices in the E-step. Similarly, the BADMM module helps derive low-rank and sparse attention maps for the neural TPPs with self-attention layers. The structured responsibility matrices and attention maps, which work as learned event transition matrices, indicate event branches, e.g., inferring isolated events and those key events triggering many subsequent events. Experiments on both synthetic and real-world data show that plugging our BADMM module into existing TPP models and learning paradigms can improve model performance and provide us with interpretable structured event branches.

Expanding on neural operators, we propose a novel framework for stochastic process learning across arbitrary domains. In particular, we develop operator flow matching (OFM) for learning stochastic process priors on function spaces. OFM provides the probability density of the values of any collection of points and enables mathematically tractable functional regression at new points with mean and density estimation. Our method outperforms state-of-the-art models in stochastic process learning, functional regression, and prior learning.

We introduce a new multimodal optimization approach called Natural Variational Annealing (NVA) that combines the strengths of three foundational concepts to simultaneously search for multiple global and local modes of black-box nonconvex objectives. First, it implements a simultaneous search by using variational posteriors, such as, mixtures of Gaussians. Second, it applies annealing to gradually trade off exploration for exploitation. Finally, it learns the variational search distribution using natural-gradient learning where updates resemble well-known and easy-to-implement algorithms. The three concepts come together in NVA giving rise to new algorithms and also allowing us to incorporate "fitness shaping", a core concept from evolutionary algorithms. We assess the quality of search on simulations and compare them to methods using gradient descent and evolution strategies. We also provide an application to a real-world inverse problem in planetary science.

Multi-modal generative AI systems, such as those combining vision and language, rely on contrastive pre-training to learn representations across different modalities. While their practical benefits are widely acknowledged, a rigorous theoretical understanding of the contrastive pre-training framework remains limited. This paper develops a theoretical framework to explain the success of contrastive pre-training in downstream tasks, such as zero-shot classification, conditional diffusion models, and vision-language models. We introduce the concept of approximate sufficient statistics, a generalization of the classical sufficient statistics, and show that near-minimizers of the contrastive pre-training loss are approximately sufficient, making them adaptable to diverse downstream tasks. We further propose the Joint Generative Hierarchical Model for the joint distribution of images and text, showing that transformers can efficiently approximate relevant functions within this model via belief propagation. Building on this framework, we derive sample complexity guarantees for multi-modal learning based on contrastive pre-trained representations. Numerical simulations validate these theoretical findings, demonstrating the strong generalization performance of contrastively pre-trained transformers in various multi-modal tasks.