In this paper, we propose to use the concept of local fairness for auditing and ranking redistricting plans. Given a redistricting plan, a deviating group is a population-balanced contiguous region in which a majority of individuals are of the same interest and in the minority of their respective districts; such a set of individuals have a justified complaint with how the redistricting plan was drawn. A redistricting plan with no deviating groups is called locally fair. We show that the problem of auditing a given plan for local fairness is NP-complete. We present an MCMC approach for auditing as well as ranking redistricting plans. We also present a dynamic programming based algorithm for the auditing problem that we use to demonstrate the efficacy of our MCMC approach. Using these tools, we test local fairness on real-world election data, showing that it is indeed possible to find plans that are almost or exactly locally fair. Further, we show that such plans can be generated while sacrificing very little in terms of compactness and existing fairness measures such as competitiveness of the districts or seat shares of the plans.

A routine training flight over Arizona in January 2023 took an unusual turn when a U.S. Air Force F-16D was struck by what was initially reported as an unidentified object, but now U.S. defense officials say was a small drone. Fox News confirmed that the incident, which occurred near Gila Bend, Arizona, on Jan. 19, 2023, was a routine training mission and was witnessed by the instructor pilot seated in the rear of the two-seat aircraft. According to a U.S. defense official, the pilot observed a "mostly white and orange object" collide with the left side of the aircraft canopy, the transparent covering over the cockpit. Initially, the object was thought to be a bird, a common hazard for aircraft. But after conducting checks during the flight and a detailed inspection upon landing at Tucson International Airport, the crew found "zero evidence" of a bird strike.

CNN host Jake Tapper told Joe Scarborough during a Wednesday conversation on "Morning Joe" that former President Biden made an effort to convince the MSNBC host that he was fit to run for re-election. The Washington Post editorial board called for more oversight of the Oval Office on Wednesday to ensure a cover-up of the president's health doesn't happen again following revelations in a bombshell book alleging the White House hid former President Joe Biden's decline from the public. "It now seems that, for a considerable time, Biden might have lacked the stamina and cognitive capacity the job demands -- and that his family and closest aides concealed this from the public," the paper's editorial board wrote. "Their apparent decision to put personal loyalties ahead of their duty to the country must be reckoned with. A legal mechanism should be considered to ensure that this doesn't happen again," the board proposed.

Although the Cannes Film Festival is the world's most prestigious movie showcase, its spotlight rarely falls on nonfiction film. Years go by without a single documentary competing for its biggest honor, the Palme d'Or, and there is no separate documentary prize. Juliette Binoche, the president of this year's jury, devoted part of her opening-night remarks to Fatma Hassona, the Palestinian photojournalist who was killed in an Israeli airstrike the day after it was announced that her documentary Put Your Soul on Your Hand and Walk would be premiering at Cannes. But the film itself was slotted into a low-profile sidebar devoted to independent productions. The festival did, however, roll out the red carpet for The Six Billion Dollar Man, Eugene Jarecki's portrait of WikiLeaks founder Julian Assange, which premiered out of competition on Wednesday evening.

President Donald Trump evoked Elon Musk during his Oval Office meeting with South Africa's president on Wednesday, during talks about the ongoing attacks white farmers in the country are facing. Trump went back and forth with President Cyril Ramaphosa over whether what is occurring in South Africa is indeed a "genocide" against white farmers. At one point, during the conversation, a reporter asked Trump how the United States and South Africa might be able to improve their relations. The president said that relations with South Africa are an important matter to him, noting he has several personal friends who are from there, including professional golfers Ernie Els and Retief Goosen, who were present at Tuesday's meeting, and Elon Musk. President Donald Trump and Elon Musk attend a UFC 309 at Madison Square Garden last November. Unprompted, Trump added that while Musk may be a South African native, he doesn't want to "get [him] involved" in the ongoing foreign diplomacy matters that played out during Tuesday's meeting.

US chip exports controls have been a "failure", the head of Nvidia, Jensen Huang, told a tech forum on Wednesday, as the Chinese government separately slammed US warnings to other countries against using Chinese tech. Successive US administrations have imposed restrictions on the sale of hi-tech AI chips to China, in an effort to curb China's military advancement and protect US dominance of the AI industry. But Huang told the Computex tech forum in Taipei that the controls had instead spurred on Chinese developers. "The local companies are very, very talented and very determined, and the export control gave them the spirit, the energy and the government support to accelerate their development," Huang told media the Computex tech show in Taipei. "I think, all in all, the export control was a failure."

Each of our estimators is based on a simple, general approach to designing differentially private mechanisms, but with novel technical steps to make the estimator private and sample-efficient. Our first estimator samples a point with approximately maximum Tukey depth using the exponential mechanism, but restricted to the set of points of large Tukey depth. Proving that this mechanism is private requires a novel analysis. Our second estimator perturbs the empirical mean of the data set with noise calibrated to the empirical covariance, without releasing the covariance itself. Its sample complexity guarantees hold more generally for subgaussian distributions, albeit with a slightly worse dependence on the privacy parameter. For both estimators, careful preprocessing of the data is required to satisfy differential privacy.

Linear systems are the bedrock of virtually all numerical computation. Machine learning poses specific challenges for the solution of such systems due to their scale, characteristic structure, stochasticity and the central role of uncertainty in the field. Unifying earlier work we propose a class of probabilistic linear solvers which jointly infer the matrix, its inverse and the solution from matrix-vector product observations. This class emerges from a fundamental set of desiderata which constrains the space of possible algorithms and recovers the method of conjugate gradients under certain conditions. We demonstrate how to incorporate prior spectral information in order to calibrate uncertainty and experimentally showcase the potential of such solvers for machine learning.

Former United States President Joe Biden was not diagnosed with prostate cancer before last week, and received his "last known" blood test for the disease more than a decade ago, his office has said. The Biden camp's statement on Tuesday came as critics, including current President Donald Trump, stoked scepticism over the timing of the diagnosis, which has reanimated questions about whether the former president misled the public about his health while in office. "President Biden's last known PSA was in 2014," Biden's office said in the brief statement, referring to the prostate-specific antigen test used to detect prostate cancer. "Prior to Friday, President Biden had never been diagnosed with prostate cancer." On Monday, Trump said he was "surprised" that the public had not been notified about Biden's diagnosis "a long time ago".

Google has injected AI features into practically all of its products, now including Chrome. At Google I/O, the tech giant's annual event, Google announced that Gemini is coming to Chrome as it transitions into the generative AI era, antitrust issues be damned. Gemini's integration with the browser means users can ask questions about information on sites, or even navigate to those sites, while browsing the web. Gemini on Chrome will be available to Chrome users on Windows and macOS, but only for paying subscribers to Google AI Pro and AI Ultra plans, which cost 20 and 250 a month, respectively. Meanwhile, Google is in the remedial phase of its antitrust case, which the U.S. Department of Justice is prosecuting. Google has been ruled a monopoly for leveraging its Chrome browser in anti-competitive ways.