Multi-task regression aims at jointly solving multiple regression problems, called tasks. Compared to solving each task separately, better performances can be achieved as long as the tasks are sufficiently related. Full-conformal prediction is a framework that formulates a data-dependent prediction-region containing the unknown output-vector at any prescribed confidence level. However, explicit computation of this prediction-region is intractable in general since it requires training infinitely many predictors. The present work focuses on multi-task regression in a Reproducing Kernel Hilbert Space (RKHS) of vector-valued functions. This computational issue is addressed by designing an approximating predictionregion containing the full-conformal one. This construction is carried out in two scenarios: piq when the inter-task covariance-matrix is known, and piiq when this matrix is estimated. In terms of volume, the tightness of this approximation is assessed theoretically by means of an upper-bound in the first scenario. It is also empirically proved to improve upon the split-conformal prediction on synthetic data in both scenarios.

Knowing which training examples drive outputs is fundamental to auditing, correcting, and understanding language models, yet for modern LLMs this remains expensive, approximate, and largely post-hoc. Standard language models generate tokens through a dense network pathway, causing training data's influence to be distributed across parameters rather than organized along explicit, traceable components. We introduce a prototype language model architecture, Prototypes for Interpretable Sequence Modeling (PRISM), that forms each prediction via a sparse, non-negative mixture of learned prototypes, trained with clustering objectives that anchor each prototype to coherent neighborhoods of training examples. Across architectures from 130M to 1.6B parameters trained on up to 50B tokens, prototype language models either surpass or remain within 2.5 percentage points on average downstream accuracy of matched dense baselines. We show that sparse prototype structure localizes curvature in the loss landscape, yielding a more tractable Hessian and enabling training data attribution that is ~500x faster than post hoc baselines when consuming equivalent memory. Calibrating linear prototype controllers can improve downstream accuracy by roughly 3 points while tracing those corrections back to training neighborhoods, and targeted prototype suppression can remove model behaviors without finetuning or measurable loss in generation quality.

Scientific reasoning is an increasingly important capability of large language models, yet improving the robustness and efficiency of training such reasoning remains a key open challenge. We study this problem in instruction-based molecular optimization, where answer-only supervised fine-tuning (SFT) collapses multi-step reasoning and reinforcement learning with verifiable rewards (RLVR) suffers from sparse feedback. Reference-guided Policy Optimization (RePO) mitigates both by anchoring policy updates to dataset-provided references, but its effectiveness is tightly coupled to reference quality: weak or misaligned references impose a performance ceiling. To overcome this ceiling, we propose active reasoning, a paradigm in which the policy actively decides, on a per-instance basis, when to imitate a reference and when to reinforce its own discoveries, while continuously upgrading what it imitates. We instantiate this paradigm as Active Group Relative Policy Optimization (Active-GRPO), realized through two coupled mechanisms: active imitate-reinforce and active referencing. The former performs imitation learning when the reference still outperforms the policy's own candidates, and shifts to self-improvement via reinforcement learning once the policy has generated molecules that surpass the reference. The latter continuously upgrades the reference itself by replacing it with the best policy-generated candidate discovered so far, progressively raising the imitation target and ensuring that reference guidance remains informative--rather than restrictive--throughout training. Across TOMG-Bench MOLOPT, Active-GRPO improves average SR Sim from 0.0959 for GRPO and 0.1665 for RePO to 0.1773 under matched three-seed evaluation, with statistically significant gains on LogP, MR, and QED.

We develop a framework for analyzing the learning dynamics of $\ell_2$-adversarial training of single-index models on Gaussian mixtures in the high-dimensional limit under streaming stochastic gradient descent (SGD). We derive deterministic equivalents for a broad class of statistics of the SGD iterates, including the adversarial risk and distance to adversarial optimality, in terms of the solution to a system of ODEs. We use them to study two idealized learning rate schedules: the Polyak stepsize and exact line search. In the case of $\ell_2$-adversarial least squares with a single class, we show that, unlike noiseless standard least squares, no constant learning rate guarantees monotone descent of SGD towards a minimizer of the adversarial risk. We identify anisotropic covariance and a mismatch in ridge parameters as the main sources of suboptimality of exact line search relative to the Polyak stepsize. We also introduce a stochastic differential equation (SDE), called adversarial homogenized SGD, that captures the evolution of statistics of the iterates of SGD. For $\ell_2$-adversarial least squares, using this SDE, we show the evolution of the risk is equivalent, up to dimension-free constants, to that of SGD on standard least squares with an adaptive learning rate and adaptive $\ell_2$-regularization. When the dynamics converge, the limiting adversarial risk and SGD iterate are determined by a fixed-point equation, with the limiting iterate being equivalent to the solution of a ridge regression problem whose regularization parameter is the limiting effective regularization of SGD.

Traditional variational Kalman filtering with unknown noise statistics suffers from inconsistent process covariance estimation and slow convergence speed, limiting its practical utility. To address these issues, we introduce a surrogate variable representing the process-noise-free state, which enables explicit modeling and inference of process noise statistics. In addition, we reformulate the conventional coordinate ascent variation inference (CAVI) as a marginalized maximum a posteriori problem, followed by a single-step hyperparameter fitting. This reformulation obviates the need for multiple inner iterations inherent to CAVI and decouples the design of the covariance tracking filters. Consequently, this architecture permits the deployment of higher-order filters for covariance tracking and enables sliding-window hyperparameter estimation. Notably, when this window encompasses all historical data, the covariance tracking estimator intrinsically operates as a zero-phase filter. Numerical simulations validate the theoretical framework, demonstrating the enhanced convergence speed and superior estimation accuracy compared with existing methods.

Autoencoders (AEs) have emerged as powerful tools for non-linear dimensionality reduction, often surpassing traditional linear methods such as Proper Orthogonal Decomposition (POD) in scenarios characterized by slowly decaying Kolmogorov $n$-widths. In the realm of Reduced-Order Modelling (ROM), these models are increasingly utilized to learn low-dimensional representations of solution manifolds associated with parametric Partial Differential Equations (PDEs). However, the high expressivity of AEs presents a challenge: although trained networks typically minimize reconstruction error, they often struggle to capture the essential properties necessary for building accurate and robust ROMs. Recent works by arXiv:2307.15288v2 and arXiv:2506.11641v1 have tackled this challenge in fully connected AEs by proposing representation-consistent architectures, which preserve some of the properties belonging to POD. This study builds upon that concept by extending representation consistency for convolutional layers. We introduce a novel class of symmetric Convolutional AutoEncoders (CAEs) designed to embody the primary properties of manifold parametrization mappings. When integrated into a ROM framework, this architecture demonstrates significantly improved predictive capabilities. Specifically, we compared the performance of the ROMs based on classical and symmetric CAEs on three one dimensional academic test cases, namely the Linear Advection, the Viscous Burger and the Kuramoto Sivashinsky equation. Numerical results demonstrate that our proposed symmetric approach consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness.

Travel, said users should'scroll past these summaries and look at guest reviews'. Travel, said users should'scroll past these summaries and look at guest reviews'. AI-generated overview found to gloss over allegations of sexual harassment and describes hotel being sued over hygiene as'spotless' A hotel being sued for mass food poisonings was described as "spotless" and a resort where guests complained of sexual harassment by staff was praised for "friendly" service by an AI intended to summarise millions of Tripadvisor reviews. The overviews of customer feedback downplayed serious complaints, ranging from the stench of mould to a lack of mains water, according to an investigation by the consumer campaign organisation Which? The AI-generated reviews appear on the travel website's hotel webpages to help holidaymakers decide where to book.

Report suggests SpaceX is testing a handheld AI device, Musk says it's'utterly false' Report suggests SpaceX is testing a handheld AI device, Musk says it's'utterly false' The purported gadget was reported to offer a way to access xAI's models without having to use a smartphone. SpaceX, the newly-public satellite internet company that also dabbles in rockets and CSAM-generating chatbots, might be building its own AI hardware. In the minutes since our article published, Elon Musk has replied to a post on X to say that the report is utterly false. The rest of this story below covers what the original article reported. The device is slimmer than an iPhone, according to, and is powered by one of Qualcomm's Snapdragon chips.

Dad of woman, 31, killed by alligator details frantic phone call he received while she took her final breaths... as it's revealed victim noticed BUBBLES on river moments before predator struck First shark of the season spotted near one of America's busiest beaches amid Fourth of July warning Awful and avoidable reason popular girl, 19, died during boat trip to'Sex Rock': Grieving sister reveals tragic final moments and insists sibling was NOT too drunk to swim Madison Square Garden insiders leak Taylor Swift wedding details: Hour by hour schedule... the menu... and what's REALLY being built inside: 'It's so not like her' Beloved woman, 31, identified as alligator attack victim killed after predator ripped off both her arms... as nightmarish new details of her final moments are revealed America's terrifying ALS explosion: Experts reveal unexpected lifestyle habits fueling rise of deadly condition... and changes that can help REDUCE your risk after NFL icon's shock diagnosis Logan Paul and Alix Earle'kiss and make up' as Michael Rubin hosts bevy of rich and famous at his $50m Hamptons mansion ahead of exclusive White Party'Time traveler' who says he is from the year 2118 makes chilling claims about World War 3 and secret CIA inventions I tried the '1776 diet' and was stunned by the results. It banished bloating, cleared my skin and flattened my stomach in just DAYS... and you can still snack Activist says he is language policing people who use the slur'Karen' and scolds that it must only be used to describe racist white women Four housing horrors crush glittering metropolis where clued-in Americans smell the beginning of the end: 'Let them eat cake' Astonishing Sharpie trick murder victim's family used to get Florida Governor Ron DeSantis to fast-track killer's death penalty, as Sunshine State becomes execution factory Hidden face in the crowd that proves the TRUTH about Taylor Swift's MSG wedding to Travis Kelce: As singer is hit by last-minute'jitters' and can't even sleep, her private jet touches down... and absurd'bridezilla' behavior is exposed First shark of the season spotted near one of America's busiest beaches amid Fourth of July warning Americans are not the only ones heading to the beach this week. The first onshore shark was spotted lurking off the coast of Long Island on Monday, just days after New York officials told beachgoers to stay alert as sightings of the apex predators often peak around the Fourth of July. Footage captured by Joanna L Steidle shows a hammerhead shark swimming not too far from the white, sandy beach filled with people enjoying the day. ' I captured the 1st onshore shark sighting off the coast of New York for the season!' Steidle shared on X.

A cyborg cockroach survived up to three hours underwater with an experimental new oxygen tank. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. Close-up of the cyborg insect and the new diving suit, which has tubes that connect to the breathing holes of the insect and contains an oxygen generator at the back. Breakthroughs, discoveries, and DIY tips sent six days a week. By signing up, you confirm you are 16+, will receive newsletters and promotional content and agree to our Terms of Use and acknowledge the data practices in our Privacy Policy .