Ritzy Bay Area town torn apart after teacher's daughter, 16, was behind wheel when four friends died in high-speed crash... then she posted a TikTok video that poured fuel on the flames Two CIA officers killed in Mexico when their car skidded off ravine and exploded after meeting about bust of'largest ever drug lab' Insiders claim failed AI rollout could be to blame for Tim Cook's departure from Apple - as one says'the AI era requires a different kind of leadership' Trump confronts Xi as US forces seize Chinese ship carrying mysterious'gift' to Iran New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-ageing, shrinks pores, smooths wrinkles... and even banishes rosacea Days after we got engaged, the love of my life told me he'd killed a man and buried him in a bog. I reported him to police... but then I made this irreversible mistake Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' Fury as murderer marries pen pal behind bars... as teenage victim's mom says: 'I'm serving a life sentence without my son' Kate and William join Charles and Camilla in celebrating British centenarians at Buckingham Palace as Royal Family marks the late Queen's 100th birthday US troops board second tanker as Trump accuses Iran of violating ceasefire'numerous times' - Live updates AMANDA PLATELL: Why Sarah Ferguson - with the ghost of Princess Diana at her side - is ready to sensationally blow up the Royal Family. She knows ALL their secrets... Team USA Olympics star Noah Lyles slammed for'horrible' reaction to his wife's wedding dress reveal How to lose weight when perimenopause sabotages your metabolism: I'm a trainer but when I hit 46, I piled on the pounds overnight. The new'posh' drug that's easier to order than Uber Eats - and why all my middle-class friends have ditched booze and cocaine for it: JANA HOCKING'You who have come from Europe are not going to return - I will sacrifice you': Chilling rant to tourists by Mexican pyramid gunman before he killed one and wounded seven others Autistic woman, 24, worked hard to build independent life for herself... now she's PARALYZED thanks to selfishness of stranger Even Cameron Diaz admits she's a dirty mess. I'll get hate for saying it, but we're all thinking the same thing about THAT wrinkled forehead: CAROLINE BULLOCK I tested a Marvel-style exoskeleton - so, can it really turn me into an athlete?

Technology Engineering Woman builds EpiPen cannon, because why not? It's not the most efficient way to deliver the life-saving medicine, but it's definitely the most entertaining. 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. The bolt-action device can hold up to four pens. Breakthroughs, discoveries, and DIY tips sent six days a week.

Science Archaeology'Doors to Death' reveal how Romans upgraded a stadium for bloodsport The ruins in present-day Turkey tell a grisly tale of wild animals eating prisoners. 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. The ancient city of Perge is located in present-day Turkey. Breakthroughs, discoveries, and DIY tips sent six days a week. The ancient Roman city of Perge--in present-day southern Turkey--was one of the region's most prominent urban centers.

Are corpse flowers like'Pangy' dangerous? 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. A blooming'Amorphophallus titanum,' also known as corpse flower, in Gunung Leuser National Park, North Sumatra Province, Indonesia in January 2025. Breakthroughs, discoveries, and DIY tips sent six days a week. What's big, rare, and smells like literal death?

Nancy Guthrie sheriff's appalling past revealed: Beat handcuffed suspect so badly he needed intensive care, used VILE language about woman and lied in sworn statement Vance grounded at White House as Iran peace talks in turmoil and Trump declares: 'I expect to be bombing' New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-ageing, shrinks pores, smooths wrinkles... and even banishes rosacea Days after we got engaged, the love of my life told me he'd killed a man and buried him in a bog. I reported him to police... but then I made this irreversible mistake Ark of the Covenant's final resting place pinpointed by archaeologists as fresh search begins Ritzy Bay Area town torn apart after teacher's daughter, 16, crashed car while speeding and killed four friends... then posted a TikTok video that poured fuel on the flames Jordon Hudson extends her control over Bill Belichick's empire with secret move that is set to leave his family and friends furious Two CIA officers killed in Mexico when their car skidded off ravine and exploded after meeting about bust of'largest ever drug lab' Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' AMANDA PLATELL: Why Sarah Ferguson - with the ghost of Princess Diana at her side - is ready to sensationally blow up the Royal Family. She knows ALL their secrets... Trump confronts Xi as US forces seize Chinese ship carrying mysterious'gift' to Iran Team USA Olympics star Noah Lyles slammed for'horrible' reaction to his wife's wedding dress reveal Humiliating moment runner celebrates winning marathon... only to be pipped at the line by rival in brutal finish In honour of the Queen's (purple!) reign: Kate mirrors late monarch's colourful wardrobe and wears her pearl earrings and necklace How to lose weight when perimenopause sabotages your metabolism: I'm a trainer but when I hit 46, I piled on the pounds overnight. The new'posh' drug that's easier to order than Uber Eats - and why all my middle-class friends have ditched booze and cocaine for it: JANA HOCKING Grieving mother says she went to LA school every day to complain daughter was being bullied... then tragedy struck when the lead tormentor, 12, hurled metal water bottle at victim's head Autistic woman, 24, worked hard to build independent life for herself... now she's PARALYZED thanks to selfishness of stranger Facial recognition on 21 popular devices can be easily spoofed with printed photos, tests reveal - so, is yours on the list? Facial recognition might seem like one of the safest ways to keep your phone secure, but experts say your device might be easy prey for hackers.

Multiplicative gating is widely used in neural architectures and has recently been applied to attention layers to improve performance and training stability in large language models. Despite the success of gated attention, the mathematical implications of gated attention mechanisms remain poorly understood. We study attention through the geometry of its representations by modeling outputs as mean parameters of Gaussian distributions and analyzing the induced Fisher--Rao geometry. We show that ungated attention operator is restricted to intrinsically flat statistical manifolds due to its affine structure, while multiplicative gating enables non-flat geometries, including positively curved manifolds that are unattainable in the ungated setting. These results establish a geometric expressivity gap between ungated and gated attention. Empirically, we show that gated models exhibit higher representation curvature and improved performance on tasks requiring nonlinear decision boundaries whereas they provide no consistent advantage on tasks with linear decision boundaries. Furthermore, we identify a structured regime in which curvature accumulates under composition, yielding a systematic depth amplification effect.

Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across a range of deep learning training problems. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue.

In online clustering problems, there is often a large amount of uncertainty over possible cluster assignments that cannot be resolved until more data are observed. This difficulty is compounded when clusters follow complex distributions, as is the case with text data. Sequential Monte Carlo (SMC) methods give a natural way of representing and updating this uncertainty over time, but have prohibitive memory requirements for large-scale problems. We propose a novel SMC algorithm that decomposes clustering problems into approximately independent subproblems, allowing a more compact representation of the algorithm state. Our approach is motivated by the knowledge base construction problem, and we show that our method is able to accurately and efficiently solve clustering problems in this setting and others where traditional SMC struggles.

Data-driven operations management often relies on parameters estimated from costly human-generated labels. Recent advances in large language models (LLMs) and other AI systems offer inexpensive auxiliary data, but introduce a new challenge: AI outputs are not direct observations of the target outcomes, but could involve high-dimensional representations with complex and unknown relationships to human labels. Conventional methods leverage AI predictions as direct proxies for true labels, which can be inefficient or unreliable when this relationship is weak or misspecified. We propose Generative Augmented Inference (GAI), a general framework that incorporates AI-generated outputs as informative features for estimating models of human-labeled outcomes. GAI uses an orthogonal moment construction that enables consistent estimation and valid inference with flexible, nonparametric relationship between LLM-generated outputs and human labels. We establish asymptotic normality and show a "safe default" property: relative to human-data-only estimators, GAI weakly improves estimation efficiency under arbitrary auxiliary signals and yields strict gains whenever the auxiliary information is predictive. Empirically, GAI outperforms benchmarks across diverse settings. In conjoint analysis with weak auxiliary signals, GAI reduces estimation error by about 50% and lowers human labeling requirements by over 75%. In retail pricing, where all methods access the same auxiliary inputs, GAI consistently outperforms alternative estimators, highlighting the value of its construction rather than differences in information. In health insurance choice, it cuts labeling requirements by over 90% while maintaining decision accuracy. Across applications, GAI improves confidence interval coverage without inflating width. Overall, GAI provides a principled and scalable approach to integrating AI-generated information.

We study downlink beam and rate adaptation in a multi-user mmWave MISO system where multiple base stations (BSs), each using analog beamforming from finite codebooks, serve multiple single-antenna user equipments (UEs) with a unique beam per UE and discrete data transmission rates. BSs learn about transmission success based on ACK/NACK feedback. To encode service goals, we introduce a satisficing throughput threshold $τ_r$ and cast joint beam and rate adaptation as a combinatorial semi-bandit over beam-rate tuples. Within this framework, we propose SAT-CTS, a lightweight, threshold-aware policy that blends conservative confidence estimates with posterior sampling, steering learning toward meeting $τ_r$ rather than merely maximizing. Our main theoretical contribution provides the first finite-time regret bounds for combinatorial semi-bandits with satisficing objective: when $τ_r$ is realizable, we upper bound the cumulative satisficing regret to the target with a time-independent constant, and when $τ_r$ is non-realizable, we show that SAT-CTS incurs only a finite expected transient outside committed CTS rounds, after which its regret is governed by the sum of the regret contributions of restarted CTS rounds, yielding an $O((\log T)^2)$ standard regret bound. On the practical side, we evaluate the performance via cumulative satisficing regret to $τ_r$ alongside standard regret and fairness. Experiments with time-varying sparse multipath channels show that SAT-CTS consistently reduces satisficing regret and maintains competitive standard regret, while achieving favorable average throughput and fairness across users, indicating that feedback-efficient learning can equitably allocate beams and rates to meet QoS targets without channel state knowledge.