In this paper, we study norm-based regularization methods for neural networks. We compare existing penalization approaches and introduce two regularization strategies that extend classical ridge- and lasso-type penalties to neural network models. The first strategy modifies weight decay by incorporating the covariance structure of the input features into a ridge-type $\ell_2$ penalty, allowing regularization to account for feature dependence. The second combines an $\ell_1$ sparsity penalty with covariance-aware $\ell_2$ regularization, producing neural network weights that are both sparse and structurally informed. Monte Carlo simulations are used to evaluate these methods under different data-generating settings, followed by two real-data applications on building cooling-load prediction and leukemia cell-type classification from high-dimensional gene expression data. Across simulated and real-data examples, the proposed regularizers improve predictive performance on unseen data and provide more effective complexity control than standard norm-based penalties, particularly when features are correlated or high-dimensional.

Gradient Boosting Decision Trees (GBDTs) dominate tabular machine learning, with modern implementations like XGBoost, LightGBM, and CatBoost being based on Newton boosting: a second-order descent step in the space of decision trees. Despite its empirical success, the global convergence of Newton boosting is poorly understood compared to first-order boosting. In this paper, we introduce Restricted Newton Descent, which studies convex optimization with Newton's method on Hilbert spaces with inexact iterates, based on the concepts of cosine angle and weak gradient edge. Within this framework, we recover Newton boosting with GBDTs and classical finite-dimensional theory as special cases. We first prove that vanilla Newton boosting achieves a linear rate of convergence for smooth, strongly convex losses that satisfy a Hessian-dominance condition. To handle general convex losses with Lipschitz Hessians, we extend a recent gradient regularized Newton scheme to the restricted weak learner setting. This scheme minimally modifies the classical algorithm by introducing an adaptive $\ell_2$-regularization term proportional to the square root of the gradient norm at each iteration. We establish a $\mathcal{O}(\frac{1}{k^2})$ rate for this scheme, thereby obtaining a globally convergent second-order GBDT algorithm with a rate matching that of first-order boosting with Nesterov momentum. In numerical experiments, we show that our scheme converges while vanilla Newton boosting may diverge.

Decentralized learning is a learning process in which data is distributed across computational agents or collected by individual agents, and model parameters are computed as the consensus of the agents. It has gained a lot of interest for applications where agents can collaboratively learn a predictive model without sharing their own data, but sharing only their local models with their immediate neighbors to generate a global model [He et al., 2018, Hendrikx et al., 2019, Arjevani et al., 2020]. We assume there are N agents who are connected over an undirected communication network G = (V,E) where V = {1,...,N} represents the agents and E V V denotes the set of edges; i.e., if agent i and j are connected then (i,j) E implies (j,i) E. Suppose we have a collection of n independent and identically distributed (i.i.d.) data pairs zi = (ai,yi), where ai Rp is the feature vector and yi the label or response of the i-th observation. Let Z = [z1,z2,,zn] Rnp be sampled from the distribution p(Z|x) where the parameter x Rd has a common prior. The goal is to sample from the posterior distribution p(x|Z) p(Z|x)p(x) by distributing Z among N agents such that Zi = {zi1,zi2,,zini} is the subset of data exclusive to agent i.

How does live facial recognition work and how many police forces use it? Britain's biometrics watchdogs have warned that national oversight of AI-powered face scanning to catch criminals is lagging far behind the technology's rapid growth. With the Metropolitan police almost doubling the number of faces they scan in London over the past 12 months and a rising use of the technology by retailers in the UK, Prof William Webster, the biometrics commissioner for England and Wales, said the "slow pace of legislation was trying to catch up with the real world" and "the horse had gone before the cart". Dr Brian Plastow, who holds the same role in Scotland, warned the technology was "nowhere near as effective as the police claim it is" and said there was a "patchwork legal framework" throughout the UK. He said in England and Wales, police were "really just marking their own homework".

Varun Chandra advises Keir Starmer on trade negotiations including AI investment. Varun Chandra advises Keir Starmer on trade negotiations including AI investment. Exclusive: Varun Chandra's talks with Google, Meta, Apple and others raise fears of'lobbying behind closed doors' An influential government adviser close to Keir Starmer and Rachel Reeves held 16 undisclosed meetings with top US tech executives, the Guardian can reveal. The No 10 business aide Varun Chandra discussed regulatory changes, AI and Donald Trump's second administration with tech corporations during confidential meetings between October 2024 and October 2025. In one meeting he offered to help a top executive meet the prime minister directly.

OpenAI's Sam Altman, left, is a backer of Rain Neuromophics, one of the companies that received funds from the UK's Aria, the brainchild of Dominic Cummings, right OpenAI's Sam Altman, left, is a backer of Rain Neuromophics, one of the companies that received funds from the UK's Aria, the brainchild of Dominic Cummings, right Exclusive: Brainchild of Dominic Cummings, Aria is aimed at funding'crazy' scientific projects to benefit the UK Britain's "invention agency" has pledged £50m of UK taxpayer money to US tech companies and venture capital projects. Dreamed up by Dominic Cummings to fund "crazy" ideas, the Advanced Research and Invention Agency (Aria) is meant to " restore Britain's place as a scientific superpower ". But a joint investigation by the Guardian and Democracy for Sale, an investigative website, has established that more than an eighth of the agency's £400m in research and development funding over the past two years has gone to 14 US tech companies and venture capital groups, in some cases, with no clear return for the UK or Aria. One of these companies, Rain Neuromorphics, is also backed by the OpenAI chief executive, Sam Altman, and was reported to be near collapse last year, shortly after winning Aria money. It did not respond to a request for comment; two of its founders appear to have left the company.

Detail from Holbein's sketch of an unidentified woman, which it is claimed may depict Anne Boleyn. Detail from Holbein's sketch of an unidentified woman, which it is claimed may depict Anne Boleyn. They are two small sketches by the Renaissance master Hans Holbein: one has long been considered to be a portrait of Henry VIII's doomed second wife, Anne Boleyn, and the other is of an unknown woman whose name was lost to time. Now researchers using AI have discovered that the unnamed woman might be the tragic queen after all, while the other figure could in fact be Boleyn's mother. The works, which belong to the royal collection and are known as the Windsor sketch and the Unidentified Woman respectively, were analysed by a team at the University of Bradford, who found that they might have been incorrectly inscribed in the 1700s, leading to a misunderstanding that has lasted centuries.

Trump reveals'absolutely pathetic' first words he says Bill Maher uttered at fabled White House visit Charles had late-stage ALS and couldn't speak or move. Bigamist pastor's'marriage scam': Five-times-wed author told women God wanted them to be together for twisted ulterior motive, wives say The rat'apocalypse' forcing residents in a northwestern state to catch vermin with their bare hands Kylie Jenner's BFF Stassie reveals dramatic butt reduction in skimpy bikini after cosmetic surgery'regrets' I had agonising acid reflux every day - but then overnight it stopped thanks to something you can buy in any supermarket. With 39 bedrooms, 59 bathrooms and its own X-ray machine, America's most expensive home hits the market for $400million - but will anyone afford to buy it? 'Dog Whisperer' Cesar Millan reveals price of the world's'safest' collar - and it TALKS to your pet Moment'disgruntled former employee' smashed car full of explosives into Portland athletics club caught on camera'She's spiralling badly': How Meghan and Harry have burned ALL their bridges as insiders reveal spectacular fallout with Anna Wintour and Kardashians, money woes - and'problems' that are worse than anyone realises Former FBI deputy director Dan Bongino'living in fear' as he issues astonishing warning after mysteriously leaving intel agency Whether it's a celebration or a bottomless brunch, nothing hits the spot quite like a glass of fizz. But it turns out you've probably been drinking prosecco wrong this entire time.

The Iran war has strengthened Ukraine in surprising ways. Could a ceasefire with Russia be closer? When Ukrainian President Volodymyr Zelensky, serious-faced and clad in black, strolled down a lilac carpet in Saudi Arabia in March, it marked a moment in the US-Israeli war in Iran. In a post on X, he said his visit was to strengthen the protection of lives. Zelensky, who carries the weight of Ukraine's own war with Russia on his shoulders, has been seizing the moment, flying to the Gulf to publicly showcase the international value and marketability of Kyiv's learned-on-the-battlefield military nous in drone warfare. Ukraine says it has now signed deals with Saudi Arabia, the UAE and Qatar - all hit by Iranian missiles and drones in recent weeks - to share drone expertise and technology, tightening alliances and benefitting from business - and it hopes defence deals - with wealthy US-allied countries.

TPUSA issues blistering response to Hollywood nepo baby who called Erika Kirk a'sociopath' and urged Trump to'kill' organization Who's The Boss? star Judith Light, 77, has fans concerned with strange poses on red carpet Shock as Home Depot rival closes all 15 of its stores and declares bankruptcy thanks to consumers' reluctance to spend ROBERT HARDMAN: What Trump told me about the King and William. Men everywhere secretly have the same complaint about their sex lives. It's NOT about looks or frequency... Spirit Airlines prepares to shut down as Trump's rescue deal falls apart I'm the REAL Emily from Devil Wears Prada: Anna Wintour's assistant played by Emily Blunt reveals herself... and cutthroat behind-scenes details that the movie did NOT include The Devil Wears Prada 2 review: Searingly silly, ridiculous sequel is a complete disgrace to fashion... and guilty of the biggest sin of all: JANE TIPPETT The ultimate Ozempic survival kit: Experts reveal cheap drugstore remedies and one miracle food every GLP-1 user needs to ease side effects... meaning you can take a HIGHER dose and lose MORE weight Mom stunned to discover she is pregnant with twins just WEEKS after giving birth: 'I was in denial' Alleged JPMorgan sex slave unmasked as crisis sparks drama at America's biggest bank: 'Everyone's wondering what Jamie thinks' Time machines may seem better suited to science fiction than the physics lab, but experts say this futuristic technology could become a reality. Researchers have revealed how time travel could really work by using the laws of quantum physics. While their method won't let you hop back to the time of the dinosaurs, scientists say it could be possible to send messages into the past.