Musk's SpaceX in merger talks with xAI ahead of planned IPO, source says SpaceX combining with xAI would bring Elon Musk's rockets, Starlink satellites, the X social media platform and Grok AI chatbot under one roof. NEW YORK - Elon Musk's SpaceX and xAI are in discussions to merge ahead of a blockbuster public offering planned for later this year. The combination would bring Musk's rockets, Starlink satellites, the X social media platform and Grok AI chatbot under one roof, according to a person briefed on the matter and two recent company filings. The plan, which Reuters is reporting exclusively, would give fresh momentum to SpaceX's effort to launch data centers into orbit as Musk battles for supremacy in the rapidly escalating artificial intelligence race against tech giants like Google, Meta and OpenAI. Musk, the world's richest man, is the CEO of both the private space company SpaceX and the artificial intelligence company xAI, which controls his social media platform X.

Score-based diffusion models have recently been extended to infinite-dimensional function spaces, with uses such as inverse problems arising from partial differential equations. In the Bayesian formulation of inverse problems, the aim is to sample from a posterior distribution over functions obtained by conditioning a prior on noisy observations. While diffusion models provide expressive priors in function space, the theory of conditioning them to sample from the posterior remains open. We address this, assuming that either the prior lies in the Cameron-Martin space, or is absolutely continuous with respect to a Gaussian measure. We prove that the models can be conditioned using an infinite-dimensional extension of Doob's $h$-transform, and that the conditional score decomposes into an unconditional score and a guidance term. As the guidance term is intractable, we propose a simulation-free score matching objective (called Supervised Guidance Training) enabling efficient and stable posterior sampling. We illustrate the theory with numerical examples on Bayesian inverse problems in function spaces. In summary, our work offers the first function-space method for fine-tuning trained diffusion models to accurately sample from a posterior.

Generative Flow Networks (GFlowNets; GFNs) are a class of generative models that learn to sample compositional objects proportionally to their a priori unknown value, their reward. We focus on the case where the reward has a specified, actionable structure, namely that it is submodular. We show submodularity can be harnessed to retrieve upper bounds on the reward of compositional objects that have not yet been observed. We provide in-depth analyses of the probability of such bounds occurring, as well as how many unobserved compositional objects can be covered by a bound. Following the Optimism in the Face of Uncertainty principle, we then introduce SUBo-GFN, which uses the submodular upper bounds to train a GFN. We show that SUBo-GFN generates orders of magnitude more training data than classical GFNs for the same number of queries to the reward function. We demonstrate the effectiveness of SUBo-GFN in terms of distribution matching and high-quality candidate generation on synthetic and real-world submodular tasks.

Conformal prediction (CP) has become a cornerstone of distribution-free uncertainty quantification, conventionally evaluated by its coverage and interval length. This work critically examines the sufficiency of these standard metrics. We demonstrate that the interval length might be deceptively improved through a counter-intuitive approach termed Prejudicial Trick (PT), while the coverage remains valid. Specifically, for any given test sample, PT probabilistically returns an interval, which is either null or constructed using an adjusted confidence level, thereby preserving marginal coverage. While PT potentially yields a deceptively lower interval length, it introduces practical vulnerabilities: the same input can yield completely different prediction intervals across repeated runs of the algorithm. We formally derive the conditions under which PT achieves these misleading improvements and provides extensive empirical evidence across various regression and classification tasks. Furthermore, we introduce a new metric interval stability which helps detect whether a new CP method implicitly improves the length based on such PT-like techniques.

We provide a complete theory of optimal universal rates for binary classification in the agnostic setting. This extends the realizable-case theory of Bousquet, Hanneke, Moran, van Handel, and Yehudayoff (2021) by removing the realizability assumption on the distribution. We identify a fundamental tetrachotomy of optimal rates: for every concept class, the optimal universal rate of convergence of the excess error rate is one of $e^{-n}$, $e^{-o(n)}$, $o(n^{-1/2})$, or arbitrarily slow. We further identify simple combinatorial structures which determine which of these categories any given concept class falls into.

Such failure can be caused by distribution shift (also known as dataset shift) between the training and test datasets. For this reason, distribution shift and domain adaptation (a notion comprising techniques for tackling distribution shift) has been a major research topic in machine learning for some time. This paper takes the perspective of Kouw and Loog (2021) and studies the case where feature observations from the test dataset are available for analysis but observations of labels are missing. Under these circumstances, without any assumptions on the nature of the distribution shift between the training and test datasets meaningful prediction of the labels in the test dataset or of their distribution is not feasible. See Kouw and Loog (2021) for a survey of approaches to domain adaptation and their related assumptions. Arguably, covariate shift (also known as population drift) and label shift (also known as prior probability shift or target shift) are the most popular specific distribution shift assumptions, both for their intuiveness as well as their computational manageability. However, exclusive covariate and label shift assumptions have been criticised for being insufficient for common domain adaptation tasks (e.g.

We study the learning dynamics of a multi-pass, mini-batch Stochastic Gradient Descent (SGD) procedure for empirical risk minimization in high-dimensional multi-index models with isotropic random data. In an asymptotic regime where the sample size $n$ and data dimension $d$ increase proportionally, for any sub-linear batch size $κ\asymp n^α$ where $α\in [0,1)$, and for a commensurate ``critical'' scaling of the learning rate, we provide an asymptotically exact characterization of the coordinate-wise dynamics of SGD. This characterization takes the form of a system of dynamical mean-field equations, driven by a scalar Poisson jump process that represents the asymptotic limit of SGD sampling noise. We develop an analogous characterization of the Stochastic Modified Equation (SME) which provides a Gaussian diffusion approximation to SGD. Our analyses imply that the limiting dynamics for SGD are the same for any batch size scaling $α\in [0,1)$, and that under a commensurate scaling of the learning rate, dynamics of SGD, SME, and gradient flow are mutually distinct, with those of SGD and SME coinciding in the special case of a linear model. We recover a known dynamical mean-field characterization of gradient flow in a limit of small learning rate, and of one-pass/online SGD in a limit of increasing sample size $n/d \to \infty$.

Pathophysiolpgical modelling of brain systems from microscale to macroscale remains difficult in group comparisons partly because of the infeasibility of modelling the interactions of thousands of neurons at the scales involved. Here, to address the challenge, we present a novel approach to construct differential causal networks directly from electroencephalogram (EEG) data. The proposed network is based on conditionally coupled neuronal circuits which describe the average behaviour of interacting neuron populations that contribute to observed EEG data. In the network, each node represents a parameterised local neural system while directed edges stand for node-wise connections with transmission parameters. The network is hierarchically structured in the sense that node and edge parameters are varying in subjects but follow a mixed-effects model. A novel evolutionary optimisation algorithm for parameter inference in the proposed method is developed using a loss function derived from Chen-Fliess expansions of stochastic differential equations. The method is demonstrated by application to the fitting of coupled Jansen-Rit local models. The performance of the proposed method is evaluated on both synthetic and real EEG data. In the real EEG data analysis, we track changes in the parameters that characterise dynamic causality within brains that demonstrate epileptic activity. We show evidence of network functional disruptions, due to imbalance of excitatory-inhibitory interneurons and altered epileptic brain connectivity, before and during seizure periods.

Exhaustively evaluating many large language models (LLMs) on a large suite of benchmarks is expensive. We cast benchmarking as finite-population inference and, under a fixed query budget, seek tight confidence intervals (CIs) for model accuracy with valid frequentist coverage. We propose Factorized Active Querying (FAQ), which (a) leverages historical information through a Bayesian factor model; (b) adaptively selects questions using a hybrid variance-reduction/active-learning sampling policy; and (c) maintains validity through Proactive Active Inference -- a finite-population extension of active inference (Zrnic & Candès, 2024) that enables direct question selection while preserving coverage. With negligible overhead cost, FAQ delivers up to $5\times$ effective sample size gains over strong baselines on two benchmark suites, across varying historical-data missingness levels: this means that it matches the CI width of uniform sampling while using up to $5\times$ fewer queries. We release our source code and our curated datasets to support reproducible evaluation and future research.

Emergent phenomena -- onset of epileptic seizures, sudden customer churn, or pandemic outbreaks -- often arise from hidden causal interactions in complex systems. We propose a machine learning method for their early detection that addresses a core challenge: unveiling and harnessing a system's latent causal structure despite the data-generating process being unknown and partially observed. The method learns an optimal feature representation from a one-parameter family of estimators -- powers of the empirical covariance or precision matrix -- offering a principled way to tune in to the underlying structure driving the emergence of critical events. A supervised learning module then classifies the learned representation. We prove structural consistency of the family and demonstrate the empirical soundness of our approach on seizure detection and churn prediction, attaining competitive results in both. Beyond prediction, and toward explainability, we ascertain that the optimal covariance power exhibits evidence of good identifiability while capturing structural signatures, thus reconciling predictive performance with interpretable statistical structure.