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.

We study multigrade deep learning (MGDL) as a principled framework for structured error refinement in deep neural networks. While the approximation power of neural networks is now relatively well understood, training very deep architectures remains challenging due to highly non-convex and often ill-conditioned optimization landscapes. In contrast, for relatively shallow networks, most notably one-hidden-layer $\texttt{ReLU}$ models, training admits convex reformulations with global guarantees, motivating learning paradigms that improve stability while scaling to depth. MGDL builds upon this insight by training deep networks grade by grade: previously learned grades are frozen, and each new residual block is trained solely to reduce the remaining approximation error, yielding an interpretable and stable hierarchical refinement process. We develop an operator-theoretic foundation for MGDL and prove that, for any continuous target function, there exists a fixed-width multigrade $\texttt{ReLU}$ scheme whose residuals decrease strictly across grades and converge uniformly to zero. To the best of our knowledge, this work provides the first rigorous theoretical guarantee that grade-wise training yields provable vanishing approximation error in deep networks. Numerical experiments further illustrate the theoretical results.