We consider a preference learning setting where every participant chooses an ordered list of kmost preferred items among a displayed set of candidates.

Take the Portland Trail Blazers +2.5 in Game 3 Shocker! Kyle Brandt-Seth Rollins on-set spat was staged Tigers look to exploit Reds' struggles at home as Framber Valdez takes the mound in Cincinnati Watch as Eagles steal Makai Lemon with wild phone call: 'Why is Philly calling me?' Giants' draft pick has intense Jaxson Dart message: 'I'm ready to die for you' Donald Trump uses Pete Rose to justify soldier's alleged shady Maduro bet, and he's not wrong Ex-Michigan football coach Sherrone Moore's mistress reveals he got her pregnant during relationship Giants' bizarre draft decisions leave star player frustrated as true needs go unfulfilled in first round Rueben Bain's short arms and tragic car accident history contributed to his NFL Draft slide Sherrone Moore accuser Paige Shiver speaks out in new interview: he'had complete control over me' Megan Rapinoe calls on traditional WNBA media to be replaced with those who'understand queer culture' The NFL Draft continues to be one of the worst'sporting events' of the year'Fox & Friends' hosts learn country line dancing in Houston Veterans cheer Trump's order on psychedelic drugs to treat PTSD'Fox & Friends' hosts'get their Texas on' with Tecovas boots'Fox & Friends' kicks off the Fox News America 250 Tour in Houston Country artist Rich O'Toole joins'Fox & Friends' in Houston IDF finds'ambulance used by Hezbollah to conceal weapons' Hegseth shuts down reporter's EXTREME question OutKick 49ers GM John Lynch skeptical of Rams' decision to draft QB Ty Simpson with No. 13 overall pick Lynch called Simpson'a good football player' but noted the pick'surprised everybody' The San Francisco 49ers traded out of the NFL Draft's first round on Thursday, so general manager John Lynch didn't have a player to discuss when he met with reporters. No problem, because he started talking players a couple of division rivals drafted. Lynch commented on what the Arizona Cardinals and Los Angeles Rams did. San Francisco 49ers general manager John Lynch speaks at the NFL Scouting Combine at the Indiana Convention Center on Feb. 24, 2026.

Get ready for summer with discounts on robot pool cleaners from Beatbot. National Pool Opening Day is tomorrow, April 25, and summer is almost here, which means pool owners everywhere are getting ready to unveil the horrors of whatever happened during the off-season. Most of the Beatbot lineup is on sale at Amazon and Beatbot's own storefront, with prices starting at $499. Beatbot makes many of the best pool-cleaning robots we've tested, and we've highlighted our top picks below. Note that the discounts are scheduled to end on April 26, though items may sell out sooner.

But the X-class events aren't even close to the most powerful flare on record. 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. NASA's Solar Dynamics Observatory captured these images of solar flares -- seen as the bright flashes in the top right -- on April 23 and 24, 2026. The images show a subset of extreme ultraviolet light that highlights the extremely hot material in flares and which is colorized in in gold and blue on the left and teal on the right. Breakthroughs, discoveries, and DIY tips sent six days a week.

A number of gradient-based bilevel algorithms have been proposed via AIDand ITD-based hypergradient approximations. For example, AID-based hypergradient computation [4, 33, 10, 11, 19] estimates the Hessian-inverse-vector product by solving a linear system with an efficient iterative algorithm. ITD-based hypergradient computation [31, 8, 9, 6, 35, 17] involves a backpropagation over the inner-loop gradient-based optimization path. Convergence rate of AIDand ITD-based bilevel methods has been studied recently. For example, [10, 19] and [19, 17] analyzed the convergence rate and complexity of AIDand ITD-based bilevel algorithms, respectively.