Multi-component datasets with intricate dependencies, like industrial assemblies or multi-modal imaging, challenge current generative modeling techniques. Existing Multi-component Variational AutoEncoders typically rely on simplified aggregation strategies, neglecting critical nuances and consequently compromising structural coherence across generated components. To explicitly address this gap, we introduce the Gaussian Markov Random Field Multi-Component Variational AutoEncoder , a novel generative framework embedding Gaussian Markov Random Fields into both prior and posterior distributions. This design choice explicitly models cross-component relationships, enabling richer representation and faithful reproduction of complex interactions. Empirically, our GMRF MCVAE achieves state-of-the-art performance on a synthetic Copula dataset specifically constructed to evaluate intricate component relationships, demonstrates competitive results on the PolyMNIST benchmark, and significantly enhances structural coherence on the real-world BIKED dataset. Our results indicate that the GMRF MCVAE is especially suited for practical applications demanding robust and realistic modeling of multi-component coherence

In this PhD thesis, we propose a novel framework for uncertainty quantification in machine learning, which is based on proper scores. Uncertainty quantification is an important cornerstone for trustworthy and reliable machine learning applications in practice. Usually, approaches to uncertainty quantification are problem-specific, and solutions and insights cannot be readily transferred from one task to another. Proper scores are loss functions minimized by predicting the target distribution. Due to their very general definition, proper scores apply to regression, classification, or even generative modeling tasks. We contribute several theoretical results, that connect epistemic uncertainty, aleatoric uncertainty, and model calibration with proper scores, resulting in a general and widely applicable framework. We achieve this by introducing a general bias-variance decomposition for strictly proper scores via functional Bregman divergences. Specifically, we use the kernel score, a kernel-based proper score, for evaluating sample-based generative models in various domains, like image, audio, and natural language generation. This includes a novel approach for uncertainty estimation of large language models, which outperforms state-of-the-art baselines. Further, we generalize the calibration-sharpness decomposition beyond classification, which motivates the definition of proper calibration errors. We then introduce a novel estimator for proper calibration errors in classification, and a novel risk-based approach to compare different estimators for squared calibration errors. Last, we offer a decomposition of the kernel spherical score, another kernel-based proper score, allowing a more fine-grained and interpretable evaluation of generative image models.

Accurate quantification of uncertainty in neural network predictions remains a central challenge for scientific applications involving high-dimensional, correlated data. While existing methods capture either aleatoric or epistemic uncertainty, few offer closed-form, multidimensional distributions that preserve spatial correlation while remaining computationally tractable. In this work, we present a framework for training neural networks with a multidimensional Gaussian loss, generating closed-form predictive distributions over outputs with non-identically distributed and heteroscedastic structure. Our approach captures aleatoric uncertainty by iteratively estimating the means and covariance matrices, and is demonstrated on a super-resolution example. We leverage a Fourier representation of the covariance matrix to stabilize network training and preserve spatial correlation. We introduce a novel regularization strategy -- referred to as information sharing -- that interpolates between image-specific and global covariance estimates, enabling convergence of the super-resolution downscaling network trained on image-specific distributional loss functions. This framework allows for efficient sampling, explicit correlation modeling, and extensions to more complex distribution families all without disrupting prediction performance. We demonstrate the method on a surface wind speed downscaling task and discuss its broader applicability to uncertainty-aware prediction in scientific models.

Standard techniques for aligning large language models (LLMs) utilize human-produced data, which could limit the capability of any aligned LLM to human level. Label refinement and weak training have emerged as promising strategies to address this superalignment problem. In this work, we adopt probabilistic assumptions commonly used to study label refinement and analyze whether refinement can be outperformed by alternative approaches, including computationally intractable oracle methods. We show that both weak training and label refinement suffer from irreducible error, leaving a performance gap between label refinement and the oracle. These results motivate future research into developing alternative methods for weak to strong generalization that synthesize the practicality of label refinement or weak training and the optimality of the oracle procedure.

This paper investigates a perceptron, a simple neural network model, with ReLU activation and a ridge-regularized mean squared error (RR-MSE). Our approach leverages the fact that the RR-MSE for ReLU perceptron is piecewise polynomial, enabling a systematic analysis using tools from computational algebra. In particular, we develop a Divide-Enumerate-Merge strategy that exhaustively enumerates all local minima of the RR-MSE. By virtue of the algebraic formulation, our approach can identify not only the typical zero-dimensional minima (i.e., isolated points) obtained by numerical optimization, but also higher-dimensional minima (i.e., connected sets such as curves, surfaces, or hypersurfaces). Although computational algebraic methods are computationally very intensive for perceptrons of practical size, as a proof of concept, we apply the proposed approach in practice to minimal perceptrons with a few hidden units.

The utilization of Machine Learning (ML) in contemporary software systems is extensive and continually expanding. However, its usage is energy-intensive, contributing to increased carbon emissions and demanding significant resources. While numerous studies examine the performance and accuracy of ML, only a limited few focus on its environmental aspects, particularly energy consumption. In addition, despite emerging efforts to compare energy consumption across various programming languages for specific algorithms and tasks, there remains a gap specifically in comparing these languages for ML-based tasks. This paper aims to raise awareness of the energy costs associated with employing different programming languages for ML model training and inference. Through this empirical study, we measure and compare the energy consumption along with run-time performance of five regression and five classification tasks implemented in Python and R, the two most popular programming languages in this context. Our study results reveal a statistically significant difference in costs between the two languages in 95% of the cases examined. Furthermore, our analysis demonstrates that the choice of programming language can influence energy efficiency significantly, up to 99.16% during model training and up to 99.8% during inferences, for a given ML task.

Markov Decision Problems (MDPs) provide a founda-tional framework for modelling sequential decision-making across diverse domains, guided by optimality criteria such as discounted and average rewards. However, these criteria have inherent limitations: discounted optimality may overly prioritise short-term rewards, while average optimality relies on strong structural assumptions. Blackwell optimality addresses these challenges, offering a robust and comprehensive criterion that ensures optimality under both discounted and average reward frameworks. Despite its theoretical appeal, existing algorithms for computing Blackwell Optimal (BO) policies are computationally expensive or hard to implement. In this paper we describe procedures for computing BO policies using an ordering of rational functions in the vicinity of 1 . We adapt state-of-the-art algorithms for deterministic and general MDPs, replacing numerical evaluations with symbolic operations on rational functions to derive bounds independent of bit complexity. For deterministic MDPs, we give the first strongly polynomial-time algorithms for computing BO policies, and for general MDPs we obtain the first subexponential-time algorithm. We further generalise several policy iteration algorithms, extending the best known upper bounds from the discounted to the Blackwell criterion.

These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads us to the framework of well-posedness and first considerations regarding reconstruction and inversion approaches. The second chapter then first deals with classical regularization theory of inverse problems in Hilbert spaces. After introducing the pseudo-inverse, we review the concept of convergent regularization. Within this chapter we then proceed to ask the question of how to realize practical reconstruction algorithms. Here, we mainly focus on Tikhonov and sparsity promoting regularization in finite dimensional spaces. In the third chapter, we dive into modern deep-learning methods, which allow solving inverse problems in a data-dependent approach. The intersection between inverse problems and machine learning is a rapidly growing field and our exposition here restricts itself to a very limited selection of topics. Among them are learned regularization, fully-learned Bayesian estimation, post-processing strategies and plug-n-play methods.

Controlling high-dimensional and nonlinear musculoskeletal models of the human body is a foundational scientific challenge. Recent machine learning breakthroughs have heralded policies that master individual skills like reaching, object manipulation and locomotion in musculoskeletal systems with many degrees of freedom. However, these agents are merely "specialists", achieving high performance for a single skill. In this work, we develop Arnold, a generalist policy that masters multiple tasks and embodiments. Arnold combines behavior cloning and fine-tuning with PPO to achieve expert or super-expert performance in 14 challenging control tasks from dexterous object manipulation to locomotion. A key innovation is Arnold's sensorimotor vocabulary, a compositional representation of the semantics of heterogeneous sensory modalities, objectives, and actuators. Arnold leverages this vocabulary via a transformer architecture to deal with the variable observation and action spaces of each task. This framework supports efficient multi-task, multi-embodiment learning and facilitates rapid adaptation to novel tasks. Finally, we analyze Arnold to provide insights into biological motor control, corroborating recent findings on the limited transferability of muscle synergies across tasks.

To further enhance the sampling process towards distribution regions with desirable metric values, we propose a joint preference alignment scheme applicable to both flow matching and Markov bridge frameworks. Furthermore, we extend our model to also explore the conformational landscape of the protein by jointly sampling side chain angles and molecules. Small molecules are the predominant class of FDA-approved drugs with a share of 85%, and more than 95% of known drugs target human or pathogen proteins (Santos et al., 2017). At the same time, the cost and duration of the development of new drugs are skyrocketing (Simoens & Huys, 2021). This sparks increasing interest in the computational design of small molecular compounds that bind specifically to disease-associated proteins and thus reduce the amount of costly experimental testing. In recent years, the machine learning community has contributed a plethora of generative tools addressing drug design from various angles (Du et al., 2024). However, these methods typically require careful tuning of the objective function to avoid exploiting imperfect computational oracles and overly maximizing one desired property (e.g. Additionally, one often aims to design a suitable 3D binding pose along with the chemical structure of the molecule, which substantially increases the degrees of freedom. Many optimization algorithms struggle to efficiently navigate such vast design spaces. Following a different approach, probabilistic generative models learn to generate drug-like molecules directly from data (Hoogeboom et al., 2022; Vignac et al., 2022). Here, the design objectives are implicitly encoded in the training data set. While these methods may not outperform direct optimization on isolated metrics, they are well suited for the multifaceted nature of drug design as they learn "what a drug looks like" in a more general way. Once trained on sufficient high-quality data, these models can capture a more holistic picture of the molecular space compared to models optimized for a limited set of target metrics. The strength of generative modeling lies in its ability to reproduce patterns seen in the training data.