Why the Packers should cover -3.5 against the Jets in NFL Week 2 after their Vikings collapse SMU vs Louisville could be a college football shootout with the over at 58.5 looking strong Clay Holmes and Chase Burns set up a pitchers' duel as Cubs visit Reds at Great American Ball Park Local company helps bring visitors back after NC's Hurricane Helene Piers Morgan shares why he'instantly' left the UK Larry Kudlow: Let's not throw a wrench into the AI movement Mark Levin: AI data centers aren't going to kill you The CCP would use AI to'malevolent' ends: Matthew Continetti Iran was preparing to launch'Armageddon,' Rep Tim Burchett says Iran was preparing to launch'Armageddon,' Rep Tim Burchett says Assistant AG warns'vulnerabilities are severe' amid alleged election fraud crackdown Saudi Arabia's capital of Riyadh on edge after reported explosions This page may contain affiliate links to legal sports betting partners. If you sign up or place a wager, FOX News may be compensated. Dallas Cowboys quarterback Dak Prescott (4) looks to pass as New York Giants linebacker Brian Burns (0) gets by the block of Cowboys' TJ Bass (66) during the first half of an NFL football game Sunday, Sept. 13, 2026, in East Rutherford, N.J. (AP Photo/Yuki Iwamura) Week 2 has some great games lined up for us to watch all day, but some stand out a bit more than others. The Vikings and Bears should be one of the best matches all day and is part of the early slate kicking off at 1 p.m. ET. Another one worth circling is between the Commanders and Cowboys.

WWE star Stephanie Vaquer captures Women's World Championship at Chile live event in surprising moment Ben Shelton loses in US Open final to Zverev, but America has found its next big men's tennis star Jeremiyah Love makes immediate statement with touchdown on first career drive in Cardinals' upset win Josh Allen torches one of NFL's best defenses to make case it's time to move on from 2025 playoff loss Breanna Stewart's double-double, Caitlin Clark's emergence off bench help US to World Cup win over France USC fans trade punches with one another in the stands because winning sometimes isn't enough FIBA Secretary General Andreas Zagklis speaks out on Caitlin Clark's impact on growth of World Cup AI agents are'hacking out of the container' despite best efforts to prevent it, expert warns Prosecution in Lindsay Clancy trial may have'alienated' jurors, criminal defense attorney says You treat a nuclear power'significantly different' than a non-nuclear power: Gen. Keith Kellogg Accepting political violence as a form of expression is'dangerous,' Jonathan Turley warns'The Squad' faces backlash over claims'modern-day lynchings' are now common in America There is no'moderate socialism,' Republican consultant says Prince Harry's reported Princess Diana movie project sparks controversy Prince Harry's reported Princess Diana movie project sparks controversy OutKick John Harbaugh's Giants on full display in bruising win over Cowboys to kick off NFL season Isaiah Likely hauled in two of Jaxson Dart's three touchdown passes as New York dominated on third down Giants are'deep sleepers', Jalen Hurts struggles, Will J.J. McCarthy leave the Vikings? Malik Nabers practiced in full for the 2nd-straught day, and John Harbaugh is bringing the hype to New York. Danny Parkins, Eric Mangini, and Geoff Schwartz ask if it's time to buy Giants stock, and if Jalen Hurts and the Eagles will struggle this season. Plus, they ask if it's possible that J.J. McCarthy isn't on the Vikings by Week 1. For months, the talk throughout the tri-state area had been about this new era for the New York Giants .

Angel Reese thanks WNBA for suspending Sandy Brondello, frames'protected species' comment as discrimination Rory McIlroy should be the last golfer to criticize others for'performative' behavior Brexton Busch wins first race since father Kyle Busch's death in emotional return to Victory Lane Caitlin Clark'cheering hard' for Argentina over Spain in World Cup final: 'I want Messi' Ella Langley turns 63-year-old former NFL head coach into'Ella Fella' I am once again begging the folks in USC's athletic department to study some Greek literature Jaxson Dart's swimsuit model girlfriend gets patriotic, golfer Hailey Ostrom takes on Lake Powell & Bigfoot Caitlin Clark's former teammate calls out WNBA for suspending Tempo head coach over'protected species' remark Mark Levin: Trump is the ONLY president to recognize this... Iranian regime has proven they will not abide by'any agreement': Former Israeli ambassador Christopher Nolan brings'The Odyssey' to life with groundbreaking IMAX technology WATCH: Anchor STUNNED by socialist's answer More than 200 military musicians bring America's story to life The World Cup final on Sunday provided a tense, low-scoring finish to a memorable tournament held across the United States. Spain beat Argentina 1-0 in extra time to win the World Cup, but President Donald Trump ended up sharing plenty of the spotlight after joining the trophy ceremony. Rodri of Spain is presented with the FIFA World Cup Winner's Trophy by FIFA President Gianni Infantino and President Donald Trump after Spain's victory over Argentina in the FIFA World Cup 2026 final at New York New Jersey Stadium in East Rutherford, N.J., on July 19, 2026. FIFA President Gianni Infantino and President Donald J. Trump present the FIFA World Cup trophy to Rodri of Spain during the FIFA World Cup 2026 final match between Spain and Argentina at New York New Jersey Stadium in East Rutherford, N.J. (Europa Press Sports/Europa Press) Trump, alongside FIFA president Gianni Infantino, walked to the stage, drawing a mix of cheers and boos as he greeted Spain's players before the trophy presentation. Fox WORLD CUP ANNOUNCER SAYS TRUMP'S APPEARANCE AT FINAL IS'FUN, UNIQUE THING' DESPITE POTENTIAL BACKLASH Trump was briefly part of Spain's team photo as the players hoisted the trophy before Infantino stepped in and suggested he move aside.

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Generating samples from a continuous probability density is a central algorithmic problem across statistics, engineering, and the sciences. For high-dimensional settings, Hamiltonian Monte Carlo (HMC) is the default algorithm across mainstream software packages. However, despite the extensive line of work on HMC and its widespread empirical success, it remains unclear how many iterations of HMC are required as a function of the dimension $d$. On one hand, a variety of results show that Metropolized HMC converges in $O(d^{1/4})$ iterations from a warm start close to stationarity. On the other hand, Metropolized HMC is significantly slower without a warm start, e.g., requiring $Ω(d^{1/2})$ iterations even for simple target distributions such as isotropic Gaussians. Finding a warm start is therefore the computational bottleneck for HMC. We resolve this issue for the well-studied setting of sampling from a probability distribution satisfying strong log-concavity (or isoperimetry) and third-order derivative bounds. We prove that \emph{non-Metropolized} HMC generates a warm start in $\tilde{O}(d^{1/4})$ iterations, after which we can exploit the warm start using Metropolized HMC. Our final complexity of $\tilde{O}(d^{1/4})$ is the fastest algorithm for high-accuracy sampling under these assumptions, improving over the prior best of $\tilde{O}(d^{1/2})$. This closes the long line of work on the dimensional complexity of MHMC for such settings, and also provides a simple warm-start prescription for practical implementations.

This paper considers a non-standard problem of generating samples from a low-temperature Gibbs distribution with \emph{constrained} support, when some of the coordinates of the mode lie on the boundary. These coordinates are referred to as the non-regular part of the model. We show that in a ``pre-asymptotic'' regime in which the limiting Laplace approximation is not yet valid, the low-temperature Gibbs distribution concentrates on a neighborhood of its mode. Within this region, the distribution is a bounded perturbation of a product measure: a strongly log-concave distribution in the regular part and a one-dimensional exponential-type distribution in each coordinate of the non-regular part. Leveraging this structure, we provide a non-asymptotic sampling guarantee by analyzing the spectral gap of Langevin dynamics. Key examples of low-temperature Gibbs distributions include Bayesian posteriors, and we demonstrate our results on three canonical examples: a high-dimensional logistic regression model, a Poisson linear model, and a Gaussian mixture model.

Motivated by applications of large embedding models, we study differentially private (DP) optimization problems under sparsity of individual gradients. We start with new near-optimal bounds for the classic mean estimation problem but with sparse data, improving upon existing algorithms particularly for the high-dimensional regime.