John Fetterman's own WIFE turns on him over ICE as Senator comes under fire for his silence on shooting of Alex Pretti What happened to Alex Pretti's key witness? Trump voices disapproval over Alex Pretti shooting and gives Minneapolis immigrant ultimatum: 'I don't like it' Shocking history of handgun Alex Pretti was carrying when he was shot by Border Patrol is revealed... as judge bans Trump administration from'destroying evidence' from scene Sinister truth about Celine Dion's song All By Myself: Singer's producer reveals bombshell secrets of her 26-year age gap marriage... that he swore not to tell until her husband René died The nastiest clique in Hollywood have had their dirty secret outed... there's no coming back from this: MAUREEN CALLAHAN Sandra Bullock's Blind Side costar Quinton Aaron is'fighting for his life' in hospital after falling at home China's top general accused of leaking nuclear secrets to the US amid President Xi's military purge Daredevil Alex Honnold reveals'embarrassing' fee he was paid to climb 101-story Taiwan skyscraper live on Netflix without a harness Ariana Grande and Cynthia Erivo'creeped a lot of people out' says anonymous Oscar voter amid Wicked snubs Forensic video analysis of Alex Pretti's final 30 seconds exposes'John Wayne gun' question that can't be ignored Taylor Swift's mom Andrea and Donna Kelce make rare public appearance together ahead of pop star's wedding to Travis Kelce Seedy underbelly of America's exclusive golf clubs... as cart girls expose ultra-rich world of sex scandals and drunken debauchery My son nearly died from the flu and had to relearn how to walk... as new'super' strain causes child hospitalizations to rise Real estate mogul is sensationally found GUILTY of murdering football coach's son outside mall Are your memories merely illusions? Whether it is a beloved childhood recollection or the simple knowledge of what you had for breakfast, our memories often seem like the only things we can be certain of. But a group of leading physicists now argue that all of your memories could merely be an illusion. According to this bold theory, all of your recollections are more likely to arise from random fluctuations in space than from real past events.

Federal immigration enforcement faces challenges as Minnesota ice agents deal with violent protests and uncooperative local officials nationwide.

A British AI startup that makes realistic video avatars has almost doubled its valuation to $4bn (£3bn), in a boost for the UK technology sector. Synthesia was valued at $2.1bn last year and moved into new offices in central London, marking the moment with a ceremony attended by the Sadiq Khan, the city's mayor, and Peter Kyle, then technology secretary. On Monday, it announced its latest funding round, led by an existing investor, Google Ventures, had raised $200m and valued the British company at $4bn. Google Ventures is the search firm's venture capital arm. Synthesia uses human actors to generate digital avatars of people and also offers employers the ability to create replicas of their staff.

British businesses reported an average 11.5% increase in productivity thanks to AI, the study found. British businesses reported an average 11.5% increase in productivity thanks to AI, the study found. The UK is losing more jobs than it is creating because of artificial intelligence and is being hit harder than rival large economies, new research suggests. British companies reported that AI had resulted in net job losses over the past 12 months, down 8% - the highest rate among other leading economies including the US, Japan, Germany and Australia, according to a study by the investment bank Morgan Stanley. The research, which was shared with Bloomberg, surveyed companies using AI for at least a year across five industries: consumer staples and retail, real estate, transport, healthcare equipment and cars.

At least three people have been killed and two others are still missing after a fire broke out at a food factory near the central Greek city of Trikala, officials say. The blaze began in the early hours of Monday at a Violanta biscuit factory, where 13 workers were on site, according to local media. Eight people managed to escape, while firefighters later recovered three bodies from the building. Drone footage showed thick smoke billowing from the fire. A powerful explosion was reportedly heard before it broke out but an investigation into the cause of the blaze is ongoing.

Masayoshi Son, chairman and chief executive officer of SoftBank Group, speaks during the Future Investment Initiative (FII) Institute Priority Asia conference in Tokyo in December. SoftBank Group has halted talks about an acquisition of U.S. data center operator Switch, a setback to founder Masayoshi Son's ambition to roll out Stargate artificial intelligence infrastructure, according to people familiar with the matter. For months, Son pursued a deal of around $50 billion for Switch, convinced that direct control of the latter's network of energy-efficient data centers would help the $500 billion Stargate push to generate computing power for partner OpenAI. But earlier this month, Son conceded that a full acquisition was off the table and scrapped a planned January announcement, the people said, asking not to be named because the matter is private. The two sides remain in active discussions about a partial investment or a partnership, they said. In a time of both misinformation and too much information, quality journalism is more crucial than ever.

Could Ukraine hold a presidential election right now? Will Europe use frozen Russian assets to fund war? How can Ukraine rebuild China ties? 'Ukraine is running out of men, money and time' More than 1,300 apartment buildings in the Ukrainian capital, Kyiv, were still without heating following Russia's missile and drone attacks on Saturday, according to Mayor Vitalii Klitschko. Over the past week alone, Russia launched more than 1,700 attack drones, at least 1,380 guided aerial bombs, and 69 missiles on Ukraine, mainly targeting the energy sector, critical infrastructure, and residential buildings, according to Ukrainian President Volodymyr Zelenskyy.

Iranian drone swarms pose serious threat to U.S. military assets as reports emerge that Iran's Supreme Leader has gone underground, according to military drone expert Cameron Chell.

Understanding the curvature evolution of the loss landscape is fundamental to analyzing the training dynamics of neural networks. The most commonly studied measure, Hessian sharpness ($λ_{\max}^H$) -- the largest eigenvalue of the loss Hessian -- determines local training stability and interacts with the learning rate throughout training. Despite its significance in analyzing training dynamics, direct measurement of Hessian sharpness remains prohibitive for Large Language Models (LLMs) due to high computational cost. We analyze $\textit{critical sharpness}$ ($λ_c$), a computationally efficient measure requiring fewer than $10$ forward passes given the update direction $Δ\mathbfθ$. Critically, this measure captures well-documented Hessian sharpness phenomena, including progressive sharpening and Edge of Stability. Using this measure, we provide the first demonstration of these sharpness phenomena at scale, up to $7$B parameters, spanning both pre-training and mid-training of OLMo-2 models. We further introduce $\textit{relative critical sharpness}$ ($λ_c^{1\to 2}$), which quantifies the curvature of one loss landscape while optimizing another, to analyze the transition from pre-training to fine-tuning and guide data mixing strategies. Critical sharpness provides practitioners with a practical tool for diagnosing curvature dynamics and informing data composition choices at scale. More broadly, our work shows that scalable curvature measures can provide actionable insights for large-scale training.

We introduce FedSGM, a unified framework for federated constrained optimization that addresses four major challenges in federated learning (FL): functional constraints, communication bottlenecks, local updates, and partial client participation. Building on the switching gradient method, FedSGM provides projection-free, primal-only updates, avoiding expensive dual-variable tuning or inner solvers. To handle communication limits, FedSGM incorporates bi-directional error feedback, correcting the bias introduced by compression while explicitly understanding the interaction between compression noise and multi-step local updates. We derive convergence guarantees showing that the averaged iterate achieves the canonical $\boldsymbol{\mathcal{O}}(1/\sqrt{T})$ rate, with additional high-probability bounds that decouple optimization progress from sampling noise due to partial participation. Additionally, we introduce a soft switching version of FedSGM to stabilize updates near the feasibility boundary. To our knowledge, FedSGM is the first framework to unify functional constraints, compression, multiple local updates, and partial client participation, establishing a theoretically grounded foundation for constrained federated learning. Finally, we validate the theoretical guarantees of FedSGM via experimentation on Neyman-Pearson classification and constrained Markov decision process (CMDP) tasks.