We investigate the parameterized complexity of Bayesian Network Structure Learn-1 ing (BNSL), a classical problem that has received significant attention in empirical2 but also purely theoretical studies. We follow up on previous works that have3 analyzed the complexity of BNSL w.r.t. the so-called superstructure of the input.4 While known results imply that BNSL is unlikely to be fixed-parameter tractable5 even when parameterized by the size of a vertex cover in the superstructure, here we6 show that a different kind of parameterization--notably by the size of a feedback7 edge set--yields fixed-parameter tractability. We proceed by showing that this8 result can be strengthened to a localized version of the feedback edge set, and9 provide corresponding lower bounds that complement previous results to provide a10 complexity classification of BNSL w.r.t.

Simulation-based techniques such as variants of stochastic Runge-Kutta are thede facto approach for inference with stochastic differential equations (SDEs) in machine learning. These methods are general-purpose and used with parametric and non-parametric models, and neural SDEs. Stochastic Runge-Kutta relies on the use of sampling schemes that can be inefficient in high dimensions. We addressthis issue by revisiting the classical SDE literature and derive direct approximations to the (typically intractable) Fokker-Planck-Kolmogorov equation by matchingmoments. We show how this workflow is fast, scales to high-dimensional latent spaces, and is applicable to scarce-data applications, where a non-parametric SDE with a driving Gaussian process velocity field specifies the model.

Labeled data often comes at a high cost as it may require recruiting human labelers or running costly' experiments. At the same time, in many practical scenarios, one already has access to a partially labeled, potentially biased dataset that can help with the learning task at hand. Motivated by such settings, we formally initiate a study of semi-supervised active learning through the frame of linear regression.

We consider the problem where nclients transmit d-dimensional real-valued vectors using dp1 `op1qqbits each, in a manner that allows the receiver to approximately reconstruct their mean. Such compression problems naturally arise in distributed and federated learning. We provide novel mathematical results and derive computationally efficient algorithms that are more accurate than previous compression techniques. We evaluate our methods on a collection of distributed and federated learning tasks, using a variety of datasets, and show a consistent improvement over the state of the art.