The twenty-first century is a data-driven era where human activities and behavior, physical phenomena, scientific discoveries, technology advancements, and almost everything that happens in the world resulting in massive generation, collection, and utilization of data. Connectivity in data is a crucial property. A straightforward example is the World Wide Web, where every webpage is connected to other web pages through hyperlinks, providing a form of directed connectivity. Combinatorial data refers to combinations of data items based on certain connectivity rules. Other forms of combinatorial data include social networks, meshes, community clusters, set systems, and molecules. This Ph.D. dissertation focuses on learning and computing with combinatorial data. We study and examine topological and connectivity features within and across connected data to improve the performance of learning and achieve high algorithmic efficiency.

Artificial intelligence commonly refers to the science and engineering of artificial systems that can carry out tasks generally associated with requiring aspects of human intelligence, such as playing games, translating languages, and driving cars. In recent years, there have been exciting advances in learning-based, data-driven approaches towards AI, and machine learning and deep learning have enabled computer systems to perceive the world in unprecedented ways. Reinforcement learning has enabled breakthroughs in complex games such as Go and challenging robotics tasks such as quadrupedal locomotion. A key aspect of intelligence is to not only make predictions, but reason about the uncertainty in these predictions, and to consider this uncertainty when making decisions. This is what this manuscript on "Probabilistic Artificial Intelligence" is about. The first part covers probabilistic approaches to machine learning. We discuss the differentiation between "epistemic" uncertainty due to lack of data and "aleatoric" uncertainty, which is irreducible and stems, e.g., from noisy observations and outcomes. We discuss concrete approaches towards probabilistic inference and modern approaches to efficient approximate inference. The second part of the manuscript is about taking uncertainty into account in sequential decision tasks. We consider active learning and Bayesian optimization -- approaches that collect data by proposing experiments that are informative for reducing the epistemic uncertainty. We then consider reinforcement learning and modern deep RL approaches that use neural network function approximation. We close by discussing modern approaches in model-based RL, which harness epistemic and aleatoric uncertainty to guide exploration, while also reasoning about safety.

Long-context large language models (LLMs) have recently shown strong performance in information retrieval and long-document QA. However, to tackle the most challenging intellectual problems, LLMs must reason effectively in long and complex contexts (e.g., frontier mathematical research). Studying how LLMs handle increasing reasoning complexity and context length is essential, yet existing benchmarks lack a solid basis for quantitative evaluation. Inspired by the abstraction of GSM-8K problems as computational graphs, and the ability to introduce noise by adding unnecessary nodes and edges, we develop a grade school math problem generator capable of producing arithmetic problems with infinite difficulty and context length under fine-grained control. Using our newly synthesized GSM-Infinite benchmark, we comprehensively evaluate existing LLMs. We find a consistent sigmoid decline in reasoning performance as complexity increases, along with a systematic inference scaling trend: exponentially increasing inference computation yields only linear performance gains. These findings underscore the fundamental limitations of current long-context LLMs and the key challenges in scaling reasoning capabilities. Our GSM-Infinite benchmark provides a scalable and controllable testbed for systematically studying and advancing LLM reasoning in long and complex contexts.

Weak-to-strong (W2S) generalization is a type of finetuning (FT) where a strong (large) student model is trained on pseudo-labels generated by a weak teacher. Surprisingly, W2S FT often outperforms the weak teacher. We seek to understand this phenomenon through the observation that FT often occurs in intrinsically low-dimensional spaces. Leveraging the low intrinsic dimensionality of FT, we analyze W2S in the ridgeless regression setting from a variance reduction perspective. For a strong student - weak teacher pair with sufficiently expressive low-dimensional feature subspaces $\mathcal{V}_s, \mathcal{V}_w$, we provide an exact characterization of the variance that dominates the generalization error of W2S. This unveils a virtue of discrepancy between the strong and weak models in W2S: the variance of the weak teacher is inherited by the strong student in $\mathcal{V}_s \cap \mathcal{V}_w$, while reduced by a factor of $\dim(\mathcal{V}_s)/N$ in the subspace of discrepancy $\mathcal{V}_w \setminus \mathcal{V}_s$ with $N$ pseudo-labels for W2S. Further, our analysis casts light on the sample complexities and the scaling of performance gap recovery in W2S. The analysis is supported with experiments on both synthetic regression problems and real vision tasks.

Effective and reliable evaluation is essential for advancing empirical machine learning. However, the increasing accessibility of generalist models and the progress towards ever more complex, high-level tasks make systematic evaluation more challenging. Benchmarks are plagued by various biases, artifacts, or leakage, while models may behave unreliably due to poorly explored failure modes. Haphazard treatments and inconsistent formulations of such "monsters" can contribute to a duplication of efforts, a lack of trust in results, and unsupported inferences. In this position paper, we argue causality offers an ideal framework to systematically address these challenges. By making causal assumptions in an approach explicit, we can faithfully model phenomena, formulate testable hypotheses with explanatory power, and leverage principled tools for analysis. To make causal model design more accessible, we identify several useful Common Abstract Topologies (CATs) in causal graphs which help gain insight into the reasoning abilities in large language models. Through a series of case studies, we demonstrate how the precise yet pragmatic language of causality clarifies the strengths and limitations of a method and inspires new approaches for systematic progress.

Federated learning (FL) is a popular paradigm for collaborative training which avoids direct data exposure between clients. However, data privacy issues still remain: FL-trained large language models are capable of memorizing and completing phrases and sentences contained in training data when given with their prefixes. Thus, it is possible for adversarial and honest-but-curious clients to recover training data of other participants simply through targeted prompting. In this work, we demonstrate that a popular and simple fine-tuning strategy, low-rank adaptation (LoRA), reduces memorization during FL up to a factor of 10. We study this effect by performing a medical question-answering fine-tuning task and injecting multiple replicas of out-of-distribution sensitive sequences drawn from an external clinical dataset. We observe a reduction in memorization for a wide variety of Llama 2 and 3 models, and find that LoRA can reduce memorization in centralized learning as well. Furthermore, we show that LoRA can be combined with other privacy-preserving techniques such as gradient clipping and Gaussian noising, secure aggregation, and Goldfish loss to further improve record-level privacy while maintaining performance.

Flexible and scalable decentralized learning solutions are fundamentally important in the application of multi-agent systems. While several recent approaches introduce (ensembles of) kernel machines in the distributed setting, Bayesian solutions are much more limited. We introduce a fully decentralized, asymptotically exact solution to computing the random feature approximation of Gaussian processes. We further address the choice of hyperparameters by introducing an ensembling scheme for Bayesian multiple kernel learning based on online Bayesian model averaging. The resulting algorithm is tested against Bayesian and frequentist methods on simulated and real-world datasets.

Data-driven models of robot motion constructed using principles from Geometric Mechanics have been shown to produce useful predictions of robot motion for a variety of robots. For robots with a useful number of DoF, these geometric mechanics models can only be constructed in the neighborhood of a gait. Here we show how Gaussian Mixture Models (GMM) can be used as a form of manifold learning that learns the structure of the Geometric Mechanics "motility map" and demonstrate: [i] a sizable improvement in prediction quality when compared to the previously published methods; [ii] a method that can be applied to any motion dataset and not only periodic gait data; [iii] a way to pre-process the data-set to facilitate extrapolation in places where the motility map is known to be linear. Our results can be applied anywhere a data-driven geometric motion model might be useful.

Explaining the hidden representations of Large Language Models (LLMs) is a perspective to understand LLMs' underlying inference logic and improve their reliability in application scenarios. However, previous methods introduce external ''black-box'' modules to explain ''black-box'' LLMs, increasing the potential uncertainty and failing to provide faithful explanations. In this paper, we propose a self-explaining method SEER, enhancing LLMs' explainability by aggregating the same concept and disentangling the different concepts in the representation space. In this way, SEER provides faithful explanations carried by representations synchronously with the LLMs' output. Additionally, we showcase the applications of SEER on trustworthiness-related tasks (e.g., the safety risks classification and detoxification tasks), where self-explained LLMs achieve consistent improvement in explainability and performance. More crucially, we theoretically analyze the improvement of SEER on LLMs' generalization ability through optimal transport theory.

As foundational models reshape scientific discovery, a bottleneck persists in dynamical system reconstruction (DSR): the ability to learn across system hierarchies. Many meta-learning approaches have been applied successfully to single systems, but falter when confronted with sparse, loosely related datasets requiring multiple hierarchies to be learned. Mixture of Experts (MoE) offers a natural paradigm to address these challenges. Despite their potential, we demonstrate that naive MoEs are inadequate for the nuanced demands of hierarchical DSR, largely due to their gradient descent-based gating update mechanism which leads to slow updates and conflicted routing during training. To overcome this limitation, we introduce MixER: Mixture of Expert Reconstructors, a novel sparse top-1 MoE layer employing a custom gating update algorithm based on $K$-means and least squares. Extensive experiments validate MixER's capabilities, demonstrating efficient training and scalability to systems of up to ten parametric ordinary differential equations. However, our layer underperforms state-of-the-art meta-learners in high-data regimes, particularly when each expert is constrained to process only a fraction of a dataset composed of highly related data points. Further analysis with synthetic and neuroscientific time series suggests that the quality of the contextual representations generated by MixER is closely linked to the presence of hierarchical structure in the data.