For four decades statistical physics has been providing a framework to analyse neural networks. A long-standing question remained on its capacity to tackle deep learning models capturing rich feature learning effects, thus going beyond the narrow networks or kernel methods analysed until now. We positively answer through the study of the supervised learning of a multi-layer perceptron. Importantly, (i) its width scales as the input dimension, making it more prone to feature learning than ultra wide networks, and more expressive than narrow ones or ones with fixed embedding layers; and (ii) we focus on the challenging interpolation regime where the number of trainable parameters and data are comparable, which forces the model to adapt to the task. We consider the matched teacher-student setting. Therefore, we provide the fundamental limits of learning random deep neural network targets and identify the sufficient statistics describing what is learnt by an optimally trained network as the data budget increases. A rich phenomenology emerges with various learning transitions. With enough data, optimal performance is attained through the model's "specialisation" towards the target, but it can be hard to reach for training algorithms which get attracted by sub-optimal solutions predicted by the theory. Specialisation occurs inhomogeneously across layers, propagating from shallow towards deep ones, but also across neurons in each layer. Furthermore, deeper targets are harder to learn. Despite its simplicity, the Bayes-optimal setting provides insights on how the depth, non-linearity and finite (proportional) width influence neural networks in the feature learning regime that are potentially relevant in much more general settings.

June Squibb sparkles opposite Cynthia Nixon in a futuristic drama, and Michelle Williams loses her way in Eugene O'Neill's Pulitzer Prize winner. Appropriately enough, Jordan Harrison's déjà-vu-inducing "Marjorie Prime" has been here before. The Off Broadway theatre Playwrights Horizons produced the poignant sci-fi play about hyperrealistic re-creations of the dead--so-called Primes, which are used as a supportive technology for the bereaved--in Anne Kauffman's spirited, delicately comic production, back in 2015. Lois Smith, then eighty-five years old, played Marjorie, a woman struggling with dementia. It's the early twenty-sixties, and so Marjorie is attended by a holographic Prime of her husband, Walter, who tells her stories from her own life.

'Hero' who wrestled gun from Bondi shooter named as Ahmed al Ahmed A hero bystander who was filmed wrestling a gun from one of the Bondi Beach attackers has been named as 43-year-old Ahmed al Ahmed. Video verified by the BBC showed Mr Ahmed run at the gunman and seize his weapon, before turning the gun round on him, forcing his retreat. Mr Ahmed, a fruit shop owner and father of two, remains in hospital, where he has undergone surgery for bullet wounds to his arm and hand, his family told 7News Australia. Eleven people were killed in the shooting on Sunday night, as more than 1,000 people attended an event to celebrate Hanukkah. The attack has since been declared by police as a terrorist incident targeting the Jewish community.

Australian police say a shooting at Bondi Beach, which killed 12 people - including one gunman - targeted the Jewish community on the first day of Hanukkah. Twenty-nine people were injured, with a second gunman in critical condition. Drone footage appears to show a gunman firing from a bridge in a nearby carpark. In the wake of a recent fatal shark attack, the BBC is off the coast in Sydney to learn how authorities are trying to protect people. The BBC's Katy Watson was in the courtroom as Erin Patterson was sentenced to life.

UN Secretary General Antonio Guterres says six peacekeepers, all Bangladesh nationals, were killed when a drone strike hit a UN logistics base in Kadugli, in Sudan's central region of Kordofan. Sudan's army says the attack came from the paramilitary Rapid Support Forces.

What is in the 28-point US plan for Ukraine? 'Ukraine is running out of men, money and time' Can the US get all sides to end the war? Why is Europe opposing Trump's peace plan? Two people were killed in a Ukrainian drone strike on the Russian city of Saratov, regional Governor Roman Busargin said in a statement on Telegram. An unspecified number of people were also injured in the attack.

He created Grand Theft Auto. Dan Houser was one of the masterminds behind revolutionary video game series Grand Theft Auto. Now, after leaving Rockstar Games and launching his own company, he's released a debut novel about a very different type of game. A Better Paradise is a dystopian vision of the near future in which an AI-led computer game goes rogue. Set in a polarised world, it finds Mark Tyburn attempting to create a virtual haven for people to find sanctuary and reconnect within themselves against an all-consuming social media hellscape.

'Tis the season to slob out on the sofa and demolish a packet of mince pies in front of a good movie, or a bad movie - or a movie that's so bad it's good. This year, as ever, a crop of new Christmas films are hoping to be part of our festive viewing - and perhaps even join the ranks of enduring classics alongside the likes of Home Alone, Elf, Love Actually and Die Hard (don't start). So, in an effort to bring you a vital public service by sorting the crackers from the turkeys, and in an attempt to get myself into the Christmas spirit, I binged as many new Christmas films as possible in a day. This is the only 2025 release on Rotten Tomatoes' list of the greatest 100 Christmas movies of all time. The Jonas Brothers find themselves stuck in the UK after wrapping up their world tour and must get home for Christmas.

At least three people have been killed and nine others wounded, when the paramilitary Rapid Support Forces (RSF) launched a drone attack on a central Sudanese city, as fighting intensifies across the vast strategic region of Kordofan that could determine the war's outcome. The strike hit a square near a police station in the Tayba neighbourhood of el-Obeid on Saturday afternoon, military sources told Al Jazeera. Several of the wounded are in critical condition, they said. Military sources reported that the Sudanese Armed Forces (SAF) had earlier struck RSF positions in the town of Um Adara in South Kordofan, while RSF forces shelled the city of Um Rawaba in the north, causing civilian casualties. An RSF drone also targeted army positions in Kosti city in White Nile state, in southeastern Sudan, destroying a military vehicle and injuring its crew, the sources added.

Russian attacks have left thousands without power in Ukraine, while a drone attack killed two people in Russia, as United States-led peace talks on ending the war, deep in its fourth year, press on. Ukrainian President Volodymyr Zelenskyy said Saturday that Russian night-time attacks damaged more than a dozen civilian facilities, disrupting power in seven regions. "They still aim to destroy our state and inflict maximum pain on our people." Kyiv and its Western allies have repeatedly said Russia is trying to cripple the Ukrainian power grid and deny civilians access to heat, light and running water for a fourth consecutive winter, in what Ukrainian officials call "weaponising" the cold. Russian attacks left parts of the Kherson region, including the regional capital, Kherson, without power, according to regional head Oleksandr Prokudin.