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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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.

We consider the problem of Robust PCA in the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem. From a statistical standpoint this problem has been recently well-studied, and conditions on when recovery is possible (how many observations do we need, how many corruptions can we tolerate) via polynomial-time algorithms is by now understood. This paper presents and analyzes a non-convex optimization approach that greatly reduces the computational complexity of the above problems, compared to the best available algorithms. In particular, in the fully observed case, with r denoting rank and d dimension, we reduce the complexity from O(r2d2 log(1/ε)) to O(rd2 log(1/ε)) - a big savings when the rank is big. For the partially observed case, we show the complexity of our algorithm is no more than O(r4dlog dlog(1/ε)). Not only is this the best-known run-time for a provable algorithm under partial observation, but in the setting where r is small compared to d, it also allows for near-linear-in-drun-time that can be exploited in the fully-observed case as well, by simply running our algorithm on a subset of the observations.

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

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/

Driver who hit and killed jogger father-of-two sues victim's estate claiming incident left him with severe PTSD Sydney Sweeney's role is cut from The Devil Wears Prada 2 New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-aging, shrinks pores, smooths wrinkles... and even banishes rosacea Alarm over popular new coffee chain invading the US... as experts warn of chilling secret behind its $1.99 brew Ark of the Covenant's final resting place pinpointed by archaeologists as fresh search begins 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 America's scariest hospital where patient and multiple staffers have been attacked in weeks since pepper spray and batons were banned Jordon Hudson extends her control over Bill Belichick's empire with secret move that is set to leave his family and friends furious Beloved Republican mayor of small Great Plains town could be deported over'mistake' he insists was an innocent one Humiliating moment runner celebrates winning marathon... only to be pipped at the line by rival in brutal finish Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' A parade of US Air Force jets has been detected flying near the Middle East on Tuesday, as President Donald Trump's ceasefire with Iran hangs in the balance. Flight tracking websites show at least five C-17 Globemaster III aircraft moving toward Saudi Arabia after leaving a key Nato installation in Germany . Trump announced a 14-day ceasefire with Iran on April 7, with the pause in hostilities expected to expire on Wednesday evening, April 22 . On Tuesday, Trump signaled that US forces are prepared to resume military operations if negotiations fail to produce an agreement. Trump told reporters the military is'raring to go,' adding that he expects bombing could resume if no deal is reached before the deadline, while expressing confidence that talks could still result in what he described as a'great deal.'

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