We study the implicit bias of momentum-based optimizers on homogeneous models. We first extend existing results on the implicit bias of steepest descent in homogeneous models to normalized steepest descent with an optional learning rate schedule. We then show that for smooth homogeneous models, momentum steepest descent algorithms like Muon (spectral norm), MomentumGD ($\ell_2$ norm), and Signum ($\ell_\infty$ norm) are approximate steepest descent trajectories under a decaying learning rate schedule, proving that these algorithms too have a bias towards KKT points of the corresponding margin maximization problem. We extend the analysis to Adam (without the stability constant), which maximizes the $\ell_\infty$ margin, and to Muon-Signum and Muon-Adam, which maximize a hybrid norm. Our experiments corroborate the theory and show that the identity of the margin maximized depends on the choice of optimizer. Overall, our results extend earlier lines of work on steepest descent in homogeneous models and momentum-based optimizers in linear models.

Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Alexander brothers' alleged HIGH SCHOOL gang rape video: Classmates speak out on sick'taking turns' footage... as creepy unseen photos are exposed Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Live Nation executives mocked'stupid' concert-goers in emails where they bragged about how to best rip them off: '$60 for closer grass' NFL superstar Xavier Worthy spills all on Travis Kelce, the Chiefs' struggles... and having Taylor Swift as his No 1 fan Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Nancy Mace throws herself into Iran warzone as she goes rogue on Middle East rescue mission: 'I AM that person' Hidden toxins in kids' treats EXPOSED: Health guru Jillian Michaels' sit-down with Casey DeSantis reveals dangers lurking in popular foods Interactive map reveals your nearest nuclear shelter and states that are MOST exposed... amid fears of US attack: Make an emergency plan now The fear of a nuclear apocalypse has reached levels not seen in decades as the US and Israel launch a deadly new conflict with Iran, raising alarms across capitals and prompting emergency diplomatic efforts to prevent a wider war. For Americans, the pressing question may soon shift from geopolitics to personal preparedness, including where the nearest fallout shelter is located and how to protect themselves if tensions escalate further. There is currently no public list of active shelters available for everyday Americans, since most are defunct or privately owned. But survival expert and Air Force veteran Sean Gold has built his own fallout shelter map, revealing that the vast majority of these radiation bunkers are scattered throughout America's largest cities. The map can be found on his survival guide website, TruePrepper .

Environment Conservation World's largest acidic geyser erupts for first time since 2020 Echinus Geyser in Yellowstone National Park is in one of the park's hottest and most dynamic regions. Breakthroughs, discoveries, and DIY tips sent six days a week. The world's largest acidic geyser is erupting for the first time in six years. Yellowstone National Park's Echinus Geyser is part of the very active Norris Geyser Basin in Wyoming. In early February, the geyser began spewing out acid and water up to 30 feet into the air.

ROTC students at Old Dominion subdued and killed ISIS-linked gunman who left one dead, two wounded after shouting'Allahu Akbar' and opened fire Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Face it, your cat doesn't care about you: Felines'rarely help' their owners - while dogs engage with humans like toddlers, study finds They're renowned for being aloof - and now it turns out your pet cat really won't lift a paw if you need help. A new study has revealed that while felines are happy to sit and watch you struggle, a dog will attempt to provide some assistance. Researchers from Eötvös Loránd University in Hungary investigated how untrained pet dogs, pet cats and 16-24-month-old toddlers responded when a familiar person searched for a hidden object.

ROTC students at Old Dominion subdued and killed ISIS-linked gunman who left one dead, two wounded after shouting'Allahu Akbar' and opened fire Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Do YOU know your Scouse from your Geordie? Study reveals the UK's hardest accents to understand - with the Essex drawl at the top of the list READ MORE: New York City's iconic accent is dying out, study finds From the approachable Geordie twang to the instantly recognisable Edinburgh lilt, the UK is home to some of the most distinctive accents in the world. Now, experts have revealed which accents are the most difficult to understand.

The US and Iran are trading blows in the Gulf with a simple drone that costs as little as $50,000 to make. But why is a slow, cheap and relatively primitive drone seeing use in 2026 alongside hypersonic missiles and stealth jets? Iran invented the relatively simple Shahed 136 attack drone, but is now fending off US copies launched against it in combat. Why, when the US military has expensive, cutting-edge and hi-tech weapons, is it making flimsy drones powered by a motorbike engine? Iranian company Shahed Aviation Industries originally designed the 136.

Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Scientists claim hair salons are'hubs of trust' where people feel comfortable discussing global warming Scientists are calling on an unexpected warrior in the fight against climate change - the humble hairdresser. Experts say the UK's hair stylists are emerging as'powerful, under-recognised influencers' in tackling the climate crisis. New research shows that hairdressers are seen as confidantes or even counsellors, and are uniquely placed to hold influential conversations about global warming.

Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' 'Pollen bomb' warning as levels soar across the UK: Hay fever sufferers say their'eyes are burning' as tree pollen spikes amid the balmy weather After more than 50 days of consecutive rain, the UK is finally enjoying some sunshine, with temperatures set to hit highs of 19 C this week. But it seems every silver lining has a cloud, as the warm weather has brought an aggressive'pollen bomb' with it. While grass pollen is low, tree pollen levels across the UK have soared amid the warm weather.

Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Alexander brothers' alleged HIGH SCHOOL gang rape video: Classmates speak out on sick'taking turns' footage... as creepy unseen photos are exposed Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Live Nation executives mocked'stupid' concert-goers in emails where they bragged about how to best rip them off: '$60 for closer grass' NFL superstar Xavier Worthy spills all on Travis Kelce, the Chiefs' struggles... and having Taylor Swift as his No 1 fan Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Nancy Mace throws herself into Iran warzone as she goes rogue on Middle East rescue mission: 'I AM that person' Hidden toxins in kids' treats EXPOSED: Health guru Jillian Michaels' sit-down with Casey DeSantis reveals dangers lurking in popular foods I hacked NASA's systems and finally found the truth about UFOs...the cover-up is astonishing READ MORE: Mystery as UFO vault with 3.8 million files is wiped clean hours after Trump demands alien docs released A British computer hacker has stepped out from the shadows to retell his discovery of an alleged UFO photo while accessing NASA's systems. Gary McKinnon was accused of perpetrating the'biggest military computer hack of all time' after breaching 97 government sites in 2002. Though he has long insisted he was not searching for aliens, McKinnon said he was looking for evidence of suppressed energy or propulsion technology when he allegedly infiltrated NASA's Johnson Space Center. That is when, he claimed, he found a high-resolution image of a cigar-shaped craft hovering above Earth. 'Then there's like slowly a hemisphere started appearing, and I'm thinking that's a planet.

Drones aren't yet licensed to carry passengers, but some may already be airlifting wounded personnel off the battlefield and could be employed for smuggling people Still from a promotional video for Skysurfer, a US company that sells "ultralight aircraft" for personal, recreational use The first passenger-carrying drones may already be in use. These aren't sophisticated urban air taxis, but crudely modified cargo drones transporting combat casualties and criminals. Heavy-lift drones are essentially scaled-up versions of the familiar quadcopters. Hair-raising videos of hobbyists carried by home-made drones show that the basic technology is simple enough. But meeting aircraft safety requirements for passenger transport takes years, and drone-makers, including Volocopter, EHang and Eve Air Mobility, are all aiming to get vehicles certified this year or next.