The goal of machine learning is to find models that minimize prediction error on data that has not yet been seen. Its operational paradigm assumes access to a dataset $S$ and articulates a scheme for evaluating how well a given model performs on an arbitrary sample. The sample can be $S$ (in which case we speak of ``in-sample'' performance) or some entirely new $S'$ (in which case we speak of ``out-of-sample'' performance). Traditional analysis of generalization assumes that both in- and out-of-sample data are i.i.d.\ draws from an infinite population. However, these probabilistic assumptions cannot be verified even in principle. This paper presents an alternative view of generalization through the lens of sensitivity analysis of solutions of optimization problems to perturbations in the problem data. Under this framework, generalization bounds are obtained by purely deterministic means and take the form of variational principles that relate in-sample and out-of-sample evaluations through an error term that quantifies how close out-of-sample data are to in-sample data. Statistical assumptions can then be used \textit{ex post} to characterize the situations when this error term is small (either on average or with high probability).

Training modern neural networks often relies on large learning rates, operating at the edge of stability, where the optimization dynamics exhibit oscillatory and chaotic behavior. Empirically, this regime often yields improved generalization performance, yet the underlying mechanism remains poorly understood. In this work, we represent stochastic optimizers as random dynamical systems, which often converge to a fractal attractor set (rather than a point) with a smaller intrinsic dimension. Building on this connection and inspired by Lyapunov dimension theory, we introduce a novel notion of dimension, coined the `sharpness dimension', and prove a generalization bound based on this dimension. Our results show that generalization in the chaotic regime depends on the complete Hessian spectrum and the structure of its partial determinants, highlighting a complexity that cannot be captured by the trace or spectral norm considered in prior work. Experiments across various MLPs and transformers validate our theory while also providing new insights into the recently observed phenomenon of grokking.

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Peter Molyneux OBE is reflecting upon the future of the UK games industry in his office - and how he could soon be leaving it. The 66-year-old, who over the years has helped create iconic series such as Fable, Black & White and Theme Park, tells me Masters of Albion - his latest project as creative director of 22cans - will also be his final one. He sees it as a return to his roots - a reinvention of the god game - a genre he introduced with Populous in 1989, one where players play as a deity on high, controlling a population's inhabitants as they please. In this new iteration, players are able to build and manage settlements by day, before defending them from attacks at night, with the ability to take control of individual characters at any point. For Molyneux, once voted one of the top game creators of all time, the key idea is freedom - creating systems that respond to player curiosity rather than directing them down a fixed path.

Meta to track workers' clicks and keystrokes to train AI Meta will start tracking the way employees work, including their keystrokes and mouse clicks, to train its artificial intelligence (AI) models. The company, which owns Instagram and Facebook, told workers on Tuesday that a new tool will run on Meta's computers and internal apps, logging their activity to be used as training data for AI technology. A Meta spokesman told the BBC: If we're building agents to help people complete everyday tasks using computers, our models need real examples of how people actually use them. The data is not used for any other purpose, he said, adding that the tool has safeguards in place to protect sensitive content. But one Meta employee, who asked not to be identified, said having their smallest actions on a computer being used to train AI model as workers expect a slew of additional job cuts feels very dystopian.

Trump EXTENDS Iran ceasefire again as he backs off bombing threat amid chaos among'seriously fractured' Tehran leadership Anna Kepner's stepbrother skips court appearance as prosecutors fight to put him behind bars amid rape and murder charges New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-aging, shrinks pores, smooths wrinkles... and even banishes rosacea Truth about your Mounjaro injection site: Our expert doctors reveal exactly where you should inject yourself for the best results, what to do if your weight loss has slowed down... and the areas you should NEVER jab Driver who hit and killed jogger father-of-two sues victim's estate claiming incident left him with severe PTSD World Series winner and MLB great Garret Anderson's cause of death revealed after his sudden passing at 53 Sydney Sweeney's role is cut from The Devil Wears Prada 2 Alarm over popular new coffee chain invading the US... as experts warn of chilling secret behind its $1.99 brew Days after we got engaged, the love of my life told me he'd killed a man and buried him in a bog. I reported him to police... but then I made this irreversible mistake Ark of the Covenant's final resting place pinpointed by archaeologists as fresh search begins Wealthy realtor, 86, who'loved the finer things' disappeared into California desert after fight with daughter and grandson... then a livestreamer made horrific discovery at beauty spot Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' MORE: Death of Air Force whistleblower set to reveal UFO secrets declared'suspicious' One of the leading voices pushing for UFO disclosure has made a shocking connection between an ancient biblical text and the existence of alien life. Congresswoman Anna Paulina Luna of Florida recently posted two cryptic messages on X, one telling people to'Read the book of Enoch' and the other displaying the 15th-century painting nicknamed the ' Madonna of the UFO.' It is the latest reference the chairwoman of the House Oversight Committee's hearings on UFOs has made to the Book of Enoch while speaking about extraterrestrials and alien spacecraft. The book is an ancient Jewish religious text, written in stages between 300 and 100 BC, attributed to the biblical figure Enoch, the great-grandfather of Noah.

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Trump EXTENDS Iran ceasefire again as he backs off bombing threat amid chaos among'seriously fractured' Tehran leadership Anna Kepner's stepbrother skips court appearance as prosecutors fight to put him behind bars amid rape and murder charges New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-aging, shrinks pores, smooths wrinkles... and even banishes rosacea Truth about your Mounjaro injection site: Our expert doctors reveal exactly where you should inject yourself for the best results, what to do if your weight loss has slowed down... and the areas you should NEVER jab Driver who hit and killed jogger father-of-two sues victim's estate claiming incident left him with severe PTSD World Series winner and MLB great Garret Anderson's cause of death revealed after his sudden passing at 53 Sydney Sweeney's role is cut from The Devil Wears Prada 2 Alarm over popular new coffee chain invading the US... as experts warn of chilling secret behind its $1.99 brew Days after we got engaged, the love of my life told me he'd killed a man and buried him in a bog. I reported him to police... but then I made this irreversible mistake Ark of the Covenant's final resting place pinpointed by archaeologists as fresh search begins Wealthy realtor, 86, who'loved the finer things' disappeared into California desert after fight with daughter and grandson... then a livestreamer made horrific discovery at beauty spot Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' Space scientists spot a'sea slug'... can you see it? Space scientists have released a jaw-dropping image of a space'sea slug'. The stunning picture was taken by NASA's Hubble space telescope which is celebrating it's 36th birthday this week. Situated about 5,000 light-years from Earth, is a star-formation region called the Trifid Nebula which Hubble captured in all its cosmic wonder.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/