We study the sample complexity of policy gradient for log-growth control -- the problem of learning, from observed state transitions, a feedback gain that optimally stabilizes a scalar linear system driven through a multiplicative-noise actuation channel. The objective $J(K) = \mathbb{E}[\log|1+BK|]$ is the top Lyapunov exponent of the closed loop. This problem carries a structural difficulty we call the cusp obstruction: the optimal gain $K^*$ always places the noise singularity $b_{\rm sing}(K) = -1/K$ in the interior of the support. At this singular optimum the policy gradient exists only as a Cauchy principal value, not as a Lebesgue integral, and the natural single-sample gradient estimator has infinite variance. Standard first-order stochastic-optimization analysis is thus inapplicable at the optimum, and merely smoothing the objective does not resolve the difficulty. The obstruction, however, has an exploitable symmetry: the Cauchy kernel is an odd function of the displacement from the moving pole, so pairing each observation with its reflection through the pole cancels the divergent part. This one cancellation simultaneously controls the population curvature, the gradient-estimator variance, and the bias incurred when the noise density is estimated. Combining these bounds with a closed-form single-transition gradient oracle, we prove that projected mini-batch policy gradient, initialized in any compact subset of the stabilizing region, attains total sample complexity $\tilde{O}(1/η)$ when the noise density is known and $\tilde{O}(η^{-(2s+1)/(2s)})$ when it must be estimated, for $C^s$ noise densities with $s \geq 2$.

Reinforcement learning (RL) often has a hierarchical structure, where an upper-level (UL) learner selects model parameters and a lower-level (LL) decision-making process responds, naturally leading to a bilevel optimization problem. Most existing bilevel RL methods assume a single-policy LL Markov decision process (MDP), and therefore fail to capture competitive structures arising in applications such as incentive design, where multiple policies interact. We study bilevel optimization problems in which the LL problem is a regularized min-max zero-sum Markov game and the UL objective is optimized through the saddle-point equilibrium induced by the LL game. In this work, we propose penalty-augmented Nikaido-Isoda descent-ascent (PANDA), a penalty-based first-order policy-gradient method based on the Nikaido-Isoda function. By exploiting the min-max game structure, PANDA avoids computing UL hypergradients and does not require second-order information. We prove that PANDA converges to stationary points without convexity assumptions on either the UL or LL objectives. Moreover, PANDA reaches an $ε$-stationary point in $\tilde{\mathcal{O}}(ε^{-1})$ iterations with sample complexity $\tilde{\mathcal{O}}(ε^{-3})$, matching the best-known rates for bilevel RL with single-policy LL MDPs. Experiments demonstrate the superior performance of PANDA over closely related baselines.

Russia'relentlessly targeting' critical infrastructure and democracy, GCHQ says The UK is at a moment of consequence as Russia is relentlessly targeting critical infrastructure, the UK's largest spy agency will warn. GCHQ Director Anne Keast-Butler will set out threats facing the UK and the measures she believes need to be taken to confront them when she makes her inaugural public speech on Wednesday. Russia has been blamed for a string of espionage plots on British soil and, more recently, waging an undeclared'hybrid war' against the UK and other Nato countries. The Kremlin has denied the allegations. Keast-Butler says GCHQ is working tirelessly to fend off cyber attacks and counter what she calls reckless sabotage and assassination attempts.

India's communists once ruled millions. For the first time since 1957, India no longer has a single communist-led state government. The defeat of the Communist Party of India (Marxist)-led Left Democratic Front (LDF) in Kerala this month, after a decade in power, marked the end - at least for now - of one of the world's most enduring experiments in democratic communism. At their peak, India's communist parties ruled states stretching from West Bengal to Kerala and Tripura. They impacted the lives of more than 100 million people through trade unions, peasant organisations, student wings and disciplined cadre networks.

Nasa has released details of robotic landers, hopping drones and vehicles it aims to send to the Moon as part of US plans to build a lunar base. Amazon founder Jeff Bezos's space company Blue Origin is one of several companies picked to build the machines. The US wants to land Americans back on the Moon before President Donald Trump leaves office in 2028. But Nasa is competing with China to return humans to the lunar surface, meaning the space agency is under pressure to appear to be winning the new space race. China is forging ahead with its own plans to land humans on the Moon by 2030.

An undergraduate student first spotted the translucent species off the coast of Taiwan. 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. Breakthroughs, discoveries, and DIY tips sent six days a week. These are some of the ingredients that come together to make, a newly identified species of sea slug, or nudibranch, found swimming in Taiwan. "Taiwanese divers call it'sesame' in Chinese and it is also small like a sesame seed, hence the name," researchers explain in a statement .

Trump's secret NATO ultimatum sparks panic as US'pulls jets, bombers and EVERY submarine from Europe' Iraq war widow left speechless at Trump cabinet's actions after she made humble plea for someone to visit husband's grave on Memorial Day Condo bloodbath hits US hotspots as values plunge to lowest in decades and terrified investors issue doom-laden warning: 'Not just a price correction' Scandals plague the'horse girls' of America's'spoiled brat capital': Insiders lift lid on VIP world hit by vile claims and furious backlash Lisa Rinna gets political as she SLAMS Spencer Pratt's run for LA mayor while taking a jab at Donald Trump Kyle Busch's bitter NASCAR rival reveals heartbreaking sign he'wasn't well' in final meeting before he died I'm a doctor, and treat men with premature ejaculation. Furious followers demand REAL story from heiress Belle Burden as she's accused of lying about her finances in divorce memoir Spencer Pratt fires back at The Price Is Right host Drew Carey with Epstein jab after he called LA mayor hopeful a'serial scammer' Ever wish your dog could speak to you? AI collar can translate your pet's barks with 95% accuracy, experts claim The half-price Hamptons: Insiders reveal America's new sanctuary, where the beaches are untouched and a'quiet luxury' charm endures So many of my female friends are resorting to a risky new sex taboo to spice up their marriages. You'll know women secretly doing it too... but we simply can't let this become normal: JANA HOCKING Donald Trump fires back at Joe Rogan's criticism of UFC White House event... amid podcaster's slating of president he endorsed Danielle Fishel, 45, was everyone's favorite girlfriend in Boy Meets World, see her now in rare appearance I got addicted to the stimulant that Trump insiders are secretly using... it can obliterate your sexual performance and ruined my life When Alex suffered a mortifying accident in bed with her new partner, she put it down to an embarrassing one-off. Little did she know she had a condition which is silently affecting thousands of women in their 50s and 60s... Ever wish your dog could speak to you? AI collar can translate your pet's barks with 95% accuracy, experts claim If you've ever wondered what your dog's barks really mean, a new ' AI collar' claims to translate their noises with remarkable accuracy. Chinese startup Meng Xiaoyi has launched a device that it alleges can translate animal sounds into human language.

Israel has launched an intensive wave of strikes across swathes of southern and eastern Lebanon, after vowing to step up its military action against Hezbollah. The Israeli military said it hit more than 100 Hezbollah infrastructure sites and fighters during what was one of the heaviest nights of bombardment since a US-brokered ceasefire began in mid-April. Strikes in the Bekaa Valley village of Mashghara killed 11 people, including two children, Lebanon's health ministry said. The military said it hit sites where terrorist activity was identified. It came after Israel's Prime Minister Benjamin Netanyahu said he had given the instruction to press the pedal even harder in targeting Hezbollah.

The Tokyo District Court on Monday found two former executives of artificial intelligence developer Alt guilty of window dressing in violation of the financial instruments and exchange law. The Tokyo District Court on Monday found two former executives of Japanese artificial intelligence developer Alt guilty of window dressing in violation of the financial instruments and exchange law. Former executive officer Katsuya Asai, 46, and former treasury and accounting division chief Takayuki Ariizumi, 53, were both sentenced to three years in prison, suspended for five years. The Tokyo-based company was fined ¥300 million ($1.89 million). Noting that fictitious sales at the firm reached about ¥11 billion in total, Judge Shoji Miyata said, "The window-dressing rate was extremely high, and the company achieved a stock listing that should not have been approved."

Greg Sankey makes it clear that SEC didn't start the 16-team CFP format discussion, that's on the Big Ten Emmanuel Acho says it was'pretty stupid' for Jaxson Dart to introduce President Trump Lincoln Riley claims USC was'snaps away' from the playoff, says he's a better coach now than when at Oklahoma 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 Brit Hume: A Trump endorsement'repeatedly' gives candidates a leg up Democrats' 2028 presidential hopefuls face scrutiny over elitism, political attacks'The Five' reveals what fans always wanted to know about them 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'Fox News @ Night' panelists evaluate Spencer Pratt's Los Angeles mayoral campaign. Milwaukee Bucks forward Kyle Kuzma is betting big that LA will change its ways. Kuzma added some intrigue to next week's nonpartisan primary, placing a $36,000 bet that former The Hills reality star Spencer Pratt will pull off an upset victory and become the next mayor of Los Angeles. With the June 2 vote just days away, Kuzma, who won a championship with the Lakers in 2020, is backing Pratt's campaign.