Educational early-warning systems increasingly operate across campuses and platforms, but combining learner records into a single repository is often restricted by privacy and governance constraints. We study whether federated learning---a training approach in which institutions keep records local and share only model updates---can serve as a practical alternative to centralised training for predicting which students are at risk of poor outcomes, without sacrificing the triage decisions the system makes. Using three public datasets spanning roughly 700 to 120,000 learners, we compare centralised training and federated training across simulated institutional clients under matched preprocessing, model family, and a controlled training schedule. We evaluate predictive accuracy, calibration, and the extent to which each approach identifies students most likely to experience the target poor outcome under limited advising capacity, and then stress-test privacy and robustness to malicious participants. On the medium and large datasets, FedAvg closely matches Central on aggregate ranking quality and shortlist precision, although individual scores and selected students can differ. Under matched privacy strength, utility remains similar, but privacy loss accumulates unevenly across institutions, and a defensive aggregation rule reduces---but does not eliminate---the effect of malicious participants. These results position federated learning as a viable collaboration pattern for learning-at-scale deployments when raw data cannot be shared, provided calibration, privacy, and integrity are evaluated explicitly rather than assumed.

In the field of parameterized complexity theory, the study of graph width measures has been intimately connected with the development of width-based model checking algorithms for combinatorial properties on graphs. In this work, we introduce a general framework to convert a large class of width-based model-checking algorithms into algorithms that can be used to test the validity of graph-theoretic conjectures on classes of graphs of bounded width. Our framework is modular and can be applied with respect to several well-studied width measures for graphs, including treewidth and cliquewidth. As a quantitative application of our framework, we prove analytically that for several long-standing graph-theoretic conjectures, there exists an algorithm that takes a number k as input and correctly determines in time double-exponential in a polynomial of k whether the conjecture is valid on all graphs of treewidth at most k. These upper bounds, which may be regarded as upper-bounds on the size of proofs/disproofs for these conjectures on the class of graphs of treewidth at most k, improve significantly on theoretical upper bounds obtained using previously available techniques.

Bounded width branching programs are a formalism that can be used to capture the notion of non-uniform constant-space computation. In this work, we study a generalized version of bounded width branching programs where instructions are defined by unitary matrices of bounded dimension. We introduce a new learning framework for these branching programs that leverages on a combination of local search techniques with gradient descent over Riemannian manifolds. We also show that gapped, read-once branching programs of bounded dimension can be learned with a polynomial number of queries in the presence of a teacher. Finally, we provide explicit near-quadratic size lower-bounds for bounded-dimension unitary branching programs, and exponential size lower-bounds for bounded-dimension read-once gapped unitary branching programs. The first lower bound is proven using a combination of Neciporuk’s lower bound technique with classic results from algebraic geometry. The second lower bound is proven within the framework of communication complexity theory.