We study the problem of multiclass PAC learning with bandit feedback in the realizable setting. In this framework, there is an unknown data distribution over an instance space $\mathcal{X}$ and a label space $\mathcal{Y}$, as in classical multiclass PAC learning, but the learner does not observe the labels of the i.i.d. training examples. Instead, in each round, it receives an unlabeled instance, predicts its label, and receives bandit feedback indicating only whether the prediction is correct. Despite this restriction, the goal remains the same as in classical PAC learning. We provide a general characterization of the optimal sample complexity of this problem, sharp for every concept class up to logarithmic factors. Our characterization is based on a new combinatorial dimension, termed the bandit $\mathrm{DS}$ dimension, defined via generalized combinatorial structures we call pseudo-boxes. These extend the pseudo-cubes underlying the $\mathrm{DS}$ dimension by allowing a different number of neighbors in each coordinate. In contrast to the $\mathrm{DS}$ dimension, which governs the full-information setting by counting the number of coordinates in the pseudo-cube, the bandit $\mathrm{DS}$ dimension aggregates the number of neighbors across coordinates, leading to a characterization in which the sample complexity scales with the total number of neighbors. We also propose a general learning algorithm achieving the upper bound, based on an algorithmic principle called ListCascade, which connects bandit learning to list learning and may be of independent interest.

Notre Dame's Josh Yago delivers Memorial Day salute during anthem before lacrosse championship game Dak Prescott reunites with ex-fiancée Sarah Jane Ramos to celebrate daughter's first birthday Celtics guard Jaylen Brown challenges ESPN's Stephen A Smith to a debate at Harvard or MIT Wyndham Clark adds to his funky resume, TPC Craig Ranch slander and LIV Golf's pitch to new investors Unearthed fan video shows who Kyle Busch really was, NASCAR's darkest hour & Bubba Wallace's'Rowdy' story California mom speaks with compassion but brutal honesty about presence of trans athlete in daughter's sport Curt Cignetti jokes he had to'coach the hell out' of undefeated Hoosiers to be Indy 500 pace car driver A screenshot has WNBA fans asking: did a player endorse a threat toward Caitlin Clark? MLB reporter Tricia Whitaker hit with line drive during Orioles' game Defense expert argues Iran has never been'so isolated' Joey Jones calls out Dem candidate Platner for'hiding behind the Purple Hearts' of fellow vets Trump doesn't want Iran to become his Afghanistan: Mike Sarraille Any Iran deal will be judged by'how much it cost' to secure, ex-CIA station chief says Dr Rebecca Grant: Iran has'no place to go,' will have to sign a deal Pope Leo XIV calls for AI to be'disarmed' in critical warning about emerging tech Kyle Busch's family reveals NASCAR champion died from severe pneumonia that led to sepsis NEW details emerge on suspected White House gunman's prior arrests OutKick-Sports Lincoln Riley claims USC was'snaps away' from the playoff, says he's a better coach now than when at Oklahoma Lincoln Riley joins Colin Cowherd to discuss USC's schedule and their number one recruiting class, and the development of QB Jayden Maiava. Lincoln Riley's tenure as head coach of the USC Trojans hasn't been smooth sailing. When he took over ahead of the 2022 season, expectations were high that a coach with his track record would bring the Trojans back to their heyday. While with the Oklahoma Sooners, he went 55-10 and 33-7 in conference, coached in four New Year's Six games, and won 12 games three consecutive seasons.

WNBA superstar Caitlin Clark will be the grand marshal of this year's Indianapolis 500 Victor Wembanyama's historic game one performance was personal, Spurs star reveals in postgame interview Dana White says gnats at Trump's White House Rose Garden dinner raised concerns for outdoor UFC events High school athlete slams CIF's shared podium rule as humiliating response that fails female competitors Kuwaiti Muslim jiu-jitsu champion refuses Israeli athlete's handshake: 'We do not respect them at all' Caitlin Clark's fiery Fever teammate tells WNBA haters to relax with perfect three-word response Red Sox legend Jason Varitek's wife appears to take massive swipe at team after ugly ouster Taiwan warns US about China's regional ambitions as Trump weighs arms deal Nate Bargatze takes clean comedy to big screen with'The Breadwinner' Retired vice admiral on Iran standoff: Trump has'time on his hands' Jury dismisses Elon Musk's lawsuit against OpenAI and Sam Altman Strikes must resume if Iran fails to negotiate'in good faith': Brig Gen John Teichert Trace Gallagher: What does liberal America want? 'Rededicate 250' faith event draws thousands to DC OutKick-Sports Brendan Sorsby's lawsuit against NCAA could set a dangerous precedent in college sports moving forward Chris Fallica weighs in on the Brendan Sorby sports betting incident. Fallica is skeptical on if Sorsby will even play college sports again after checking himself into a betting rehab. Brendan Sorsby's college football career should likely be over, according to rules put in place, after the gambling revelations detailed this week in a lawsuit filed against the NCAA by his own attorneys. At a time when athletes are suing the NCAA over nearly every restriction tied to earning opportunities, this case feels far more straightforward.

PK Subban makes good on commitment, donates $10M to Montreal Children's Hospital Golfer Mackenzine Hughes sticks a tee shot on the green after it bounces off a cameraman's head Is Avalanche star Nathan MacKinnon a psychopath for wearing flippers and goggles in the hotel pool? Kevin Durant's media company says Caitlin Clark is the third most marketable WNBA player Rally erupts at California girls' track meet amid trans feud between White House and Newsom's office Tyreek Hill's court battle with OnlyFans model who accused him of breaking her leg ends with shocking twist Best bet for Caitlin Clark's points prop in Dallas Wings at Indiana Fever 2026 WNBA season opener Fernando Tatis Jr. ripped for embarrassing fielding blunder that led to little league grand slam Women's tennis legend speaks out on California trans athlete controversy as Newsom faces criticism Dale Earnhardt Jr. buries mouthy NASCAR fan who attempted to insult his intelligence Some viewers absolutely slammed Bad Bunny's Super Bowl halftime show and complained to the FCC I don't buy that Iran has a'divided government,' US Navy captain says Democratic congressman blames Trump for disruption of world's oil supply Putin is'really worried' about Ukrainian drone strikes: National security expert OH, DEER!: Nursing home receives unexpected visitor Does the U.S. Still Need NATO? What Will Trump Do in China? OutKick Caitlin Clark says'I honestly could have probably got a couple more calls' in Fever's season opener Clark's coach and teammates have previously called out WNBA refs for not protecting the league's star player Indiana Fever star Lexie Hull opens up on dealing with hostile attention amid team's rise in popularity Indiana Fever shooting guard Lexie Hull spoke to Fox News Digital about the challenges of the team's surge in attention since 2024. Caitlin Clark made her return from injury to the WNBA on Saturday, but had to settle for a loss.

Why is Israel still in southern Lebanon? A war to shape Lebanon's future Video captured massive explosions in southern Lebanon in what the Israeli military called strikes on a Hezbollah tunnel. Other attacks happened nearby, as Israeli Defence Minister Israel Katz vowed that southern Lebanon's fate will be like Gaza's. Ukrainian drones strike Russia's Tuapse refinery for third time Qatar says using Hormuz Strait as political weapon is'unacceptable' Australia's top diplomat visits China to talk energy security

Diego Pavia, the former Vanderbilt quarterback and Heisman runner-up, accepted an invitation to the Baltimore Ravens' rookie minicamp on a tryout basis after going undrafted.

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.

The show exists in a strange world where men repeatedly confess their love for each other. Does it make them better people? Much ink has been spilled, and countless TikToks recorded, in an effort to explain the female fervor unleashed by the series " Heated Rivalry ." I, a thirty-eight-year-old woman who owns a T-shirt that bears the logo of Shane Hollander's Montreal Metros and another that celebrates Ilya Rozanov's Boston Raiders (Valentine's Day gifts, it should be said, from my indulgent husband), don't find its appeal so mystifying. Two gorgeous young men, as elegantly muscled as Myron's discus thrower, have ecstatically unbridled, mutually satisfying sex to a soundtrack designed to tickle elder millennials' nostalgia-pleasure centers, all while falling in the kind of soul-sustaining love that most of us can only dream of.