We establish finite-time last-iterate guarantees for vanilla stochastic gradient descent in co-coercive games under noisy feedback. This is a broad class of games that is more general than strongly monotone games, allows for multiple Nash equilibria, and includes examples such as quadratic games with negative semidefinite interaction matrices and potential games with smooth concave potentials. Prior work in this setting has relied on relative noise models, where the noise vanishes as iterates approach equilibrium, an assumption that is often unrealistic in practice. We work instead under a substantially more general noise model in which the second moment of the noise is allowed to scale affinely with the squared norm of the iterates, an assumption natural in learning with unbounded action spaces. Under this model, we prove a last-iterate bound of order $O(\log(t)/t^{1/3})$, the first such bound for co-coercive games under non-vanishing noise. We additionally establish almost sure convergence of the iterates to the set of Nash equilibria and derive time-average convergence guarantees.

Inference-time LLM alignment methods, particularly activation steering, offer an alternative to fine-tuning by directly modifying activations during generation. Existing methods, however, often rely on non-anticipative interventions that ignore how perturbations propagate through transformer layers and lack online error feedback, resulting in suboptimal, open-loop control. To address this, we show empirically that, despite the nonlinear structure of transformer blocks, layer-wise dynamics across multiple LLM architectures and scales are well-approximated by locally-linear models. Exploiting this property, we model LLM inference as a linear time-varying dynamical system and adapt the classical linear quadratic regulator to compute feedback controllers using layer-wise Jacobians, steering activations toward desired semantic setpoints in closed-loop with minimal computational overhead and no offline training. We also derive theoretical bounds on setpoint tracking error, enabling formal guarantees on steering performance. Using a novel adaptive semantic feature setpoint signal, our method yields robust, fine-grained behavior control across models, scales, and tasks, including state-of-the-art modulation of toxicity, truthfulness, refusal, and arbitrary concepts, surpassing baseline steering methods. Our code is available at: https://github.com/trustworthyrobotics/lqr-activation-steering

We study finite-horizon continuous-time policy evaluation from discrete closed-loop trajectories under time-inhomogeneous dynamics. The target value surface solves a backward parabolic equation, but the Bellman baseline obtained from one-step recursion is only first-order in the grid width. We estimate the time-dependent generator from multi-step transitions using moment-matching coefficients that cancel lower-order truncation terms, and combine the resulting surrogate with backward regression. The main theory gives an end-to-end decomposition into generator misspecification, projection error, pooling bias, finite-sample error, and start-up error, together with a decision-frequency regime map explaining when higher-order gains should be visible. Across calibration studies, four-scale benchmarks, feature and start-up ablations, and gain-mismatch stress tests, the second-order estimator consistently improves on the Bellman baseline and remains stable in the regime where the theory predicts visible gains. These results position high-order generator regression as an interpretable continuous-time policy-evaluation method with a clear operating region.

The X and xAI owner is now working closely together with the maker of the AI coding tool. The xAI and SpaceX logos appear on a smartphone screen placed on a reflective surface onto which an abstract black and blue illustration is projected. SpaceX and AI company Cursor have struck a new partnership that could see the owner of X buy the AI company for $60 billion later this year. SpaceXAI and @cursor_ai are now working closely together to create the world's best coding and knowledge work AI, SpaceX wrote in a post on X. SpaceXAI and @cursor_ai are now working closely together to create the world's best coding and knowledge work AI. According to SpaceX, the deal allows for it to either invest $10 billion into the company known for its AI coding tool, or acquire it entirely later this year for $60 billion.

Hide-and-seek, peekaboo, and more can strengthen emotional bonds. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. Tug-of-war is the type of play that can go a long way. Breakthroughs, discoveries, and DIY tips sent six days a week. Take this as your signal to go play with your dog .

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.

Mozilla says it patched 271 Firefox vulnerabilities thanks to Anthropic's Claude Mythos Anthropic's buzzy announcement about using AI to improve cybersecurity earlier this month was met with plenty of skepticism. However, Mozilla shared some details that support use of the company's special Claude Mythos Preview model as a way to protect critical services. Using Mythos helped Mozilla's team find and patch 271 vulnerabilities in the latest release of the Firefox browser. So far we've found no category or complexity of vulnerability that humans can find that this model can't, the foundation said. The blog post from Mozilla feels like a positive sign for Anthropic's Project Glasswing.

Do the Dodgers get an unfair advantage with'bizarre' rule impacting Shohei Ohtani? Florida's Thomas Haugh ditches Draft to return to school amid swirling Todd Golden rumors Mamdani takes'Curse of the Mambino' on the chin as Mets' 11-game skid sets franchise record Cubs' co-owner pushes back on woke backlash Matt Shaw received for attending Charlie Kirk's memorial'Zig-Zag Theory': Houston Rockets will cover and even series vs. Los Angeles Lakers in Game 2 Stephen A. Smith says he believes Dianna Russini and Mike Vrabel are innocent -- for now Wisconsin teen's turkey hunt takes a wild turn when a bobcat lunges and grabs his arm on camera Matt Fitzpatrick is the king of Harbour Town, lame USA chants, and LIV Golf's telling announcement Dana Perino: Economic pressure on Iranian regime is'excruciating' IRGC's'extreme commanders' abused Iranian people for decades: Ret Lt Col Chuck DeVore Missing scientists probe was reportedly sparked after'UFO General' disappeared Iranian leaders say they don't negotiate'under the shadow of threats' VP Vance's Pakistan trip suspended as Trump weighs diplomacy vs force on Iran Former Virginia governor condemns Dem redistricting plan as'illegal power grab' California Dems warned of'blue Armageddon' in governor race This is how the US can'pressure' the Iranian regime: Former leading CENTCOM official Democratic Senate candidate criticized for remarks about JD Vance's family A stretcher crew stood by but there wasn't much left to carry off after the robot's wild self-destruction A humanoid robot, 'Lightning,' shattered the Beijing half-marathon world record this weekend, completing the race in just 50 minutes and 26 seconds, 13 miles faster than any human. Cyber expert Kurt Knutsson warns this massive leap in artificial intelligence, along with Tesla's Optimus robot, necessitates a universal'off button' and stronger guardrails to ensure safety and prevent future human replacement. I'm equally excited and terrified by it. On one hand, I'd love to have a robot around the house so it could fold my laundry and make me feel like George Jetson.

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.