Quivering Karmelo Anthony is convicted of murdering Austin Metcalf, 17... but now prosecutors have granted him Hail Mary that could see him jailed for as little as TWO YEARS Trump's $70B immigration crackdown passes the House as sneaky loophole allows $1.8B weaponization'slush fund' to survive I watched footage of the race crime that split America. She's always by Trump's side, trusted with the White House's biggest secrets... and she influences millions Trump ERUPTS behind closed doors as top Republican pleads with him to axe Tulsi Gabbard's spy-chief replacement Leaked transcript of UNAIRED 60 Minutes interview exposes REAL reason'callous' CBS star Scott Pelley'deserved to be fired' Epstein's massage fixer looks PETRIFIED as she's dragged into explosive congressional grilling - and reveals jaw-dropping'blackmail' theory Caitlyn Jenner biographer and Robin Riker's ex William Hasley found dead on hiking trail at 78 Eva Longoria reunites with ex Tony Parker 15 years after cheating scandal split... as shocked fans react Shamed ex mayor Misty Roberts is sentenced to 90 DAYS as she's branded a'predator with hair extensions' by enraged mother of 17-year-old sex assault victim My compulsive bathroom habit that so many are guilty of left me in excruciating pain. DR STUART reveals early signs... cures that work in days... and when to worry Inside Travis Kelce's plan to become'the Shaq of the NFL' after wedding Taylor Swift Moment Real Housewives star Lenny Hochstein's sexual assault accuser'dances' as she leaves Star Island mansion - before filing $100k civil lawsuit Madonna's wild sex claim about JFK Jr now draws surprising response from his outspoken nephew Jack Schlossberg Zodiac killer case takes bombshell turn as unsolved cipher is CRACKED... and America's top codebreakers say evidence is all pointing to one man Kennedy heir Jack Schlossberg concedes Trump is a'genius' as aspiring congressman reveals what he'deeply respects' about president'Great' mom, 32, tried to gas herself and her three young kids to death after inviting them to'popcorn sleepover' in car, prosecutors allege Want to lose up to a stone in six weeks, plus boost your mood and energy levels? Fermented foods from kefir to kombucha are having a moment, hailed for their gut health benefits. But experts say we could be overlooking one of the healthiest ferments out there: sauerkraut.

How Much of Data-Center Activism Is Really AI Slop? Anti-AI sentiment is genuine, but its online expression looks stranger and stranger. Americans are wary of AI in general, and they are especially suspicious of the AI data centers that are popping up across the country like enormous mushrooms. A majority do not want a new data center built in their town. Across the country, community groups have organized to protest individual projects, and activists have successfully lobbied local and state politicians to place moratoriums on the facilities' construction.

Stochastic gradient Langevin dynamics combined with Gibbs updates (SGLD--Gibbs) provides a highly scalable approach to approximate Bayesian inference in latent variable models. However, it remains unclear how to tune the algorithm's hyperparameters in a principled manner to ensure the uncertainty estimates are statistically meaningful. In this work, we address this gap in tuning guidance by developing a statistical scaling limit theory for SGLD--Gibbs. We derive a joint asymptotic limit for the global parameters and latent variables under appropriate space-time rescaling. We show that global parameters converge to a diffusion-type limit, while each latent variable converges to a jump process, reflecting the use of intermittent Gibbs updates. This joint jump-diffusion structure reveals how latent-variable randomness contributes to the stationary distribution of the global parameters. We leverage our results to propose explicit guidance on hyperparameter tuning for SGLD--Gibbs that ensures meaningful uncertainty quantification. Numerical experiments show that SGLD--Gibbs with our tuning guidance leads to better parameter estimates, uncertainty quantification, and predictive performance than stochastic variational inference.

The Bayesian paradigm offers principled tools for sequential decision-making under uncertainty, but its reliance on a probabilistic model for all parameters can hinder the incorporation of complex structural constraints. We introduce a minimalist Bayesian framework that places a prior only on the location of the optimum, while eliminating nuisance parameters through profile likelihood. This yields a generalized posterior that naturally accommodates structural constraints. As a direct instantiation, we develop MINimalist Thompson Sampling (MINTS). For multi-armed bandits with mean constraints, we establish near-optimal non-asymptotic regret guarantees and sharp almost-sure asymptotic regret characterizations. In particular, MINTS attains the classical Lai--Robbins constant in the unstructured setting and automatically adapts to unimodal structure, achieving the sharp constant determined only by the immediate neighbors of the optimal arm.

We study quantile regression in an extrapolation regime where the covariate takes unusually large values. Under regular variation assumptions, extreme observations can be effectively characterized through their angular components, enabling learning strategies that focus on the angle of the most extreme observations. This approach is formalized through the minimization of an asymptotic conditional risk that localizes learning in the tail of the covariate distribution. We propose a novel Support Vector Machine (SVM) framework for extreme quantile regression, leveraging reproducing kernel Hilbert spaces to handle high-dimensional and nonlinear settings. Our method also accommodates unbounded response variables and avoids restrictive transformations. We establish finite-sample learning guarantees under mild regularity assumptions. The proposed framework unifies ideas from statistical learning and multivariate extremes, providing a tractable and theoretically grounded approach to extrapolation. We complement our theoretical findings with an empirical study on river flow data from the Danube, demonstrating the practical relevance of our methods.

I am attempting to solve a mathematical conundrum that has stumped many of humanity's greatest thinkers. I have zero mathematical training, apart from a distant undergraduate physics degree, which should put my odds of success at slim to none. But I also have a trick up my sleeve - a kind of mathematical genie that can conjure arcane secrets seemingly out of thin air. I make a short request concerning an esoteric conjecture in number theory, then cross my fingers. Perhaps "genie" is a bit too strong - I'm simply using GPT 5.5 Pro, the latest iteration of OpenAI's flagship model. But for mathematicians, modern AI models appear to have a spark of magic.

Alan Turing proposed a test for machine intelligence: could a computer convince a human it was human? Recently, a friend told me over coffee about some disheartening feedback she had received. "They said it was good," she said, "but that it read like it was written by AI." Knowing her, I understood immediately what had happened. Her credibility was being questioned not because her work was poor, but because it was too good - too clear, too fluent, too polished. The rapid acceleration of artificial intelligence tools is changing how we think about good writing.

Voters deliver verdict on embattled'womanizer' and Nazi-tattooed candidate in crucial Maine race that could determine Senate power balance I watched footage of the race crime that split America. My compulsive bathroom habit that so many are guilty of left me in excruciating pain. DR STUART reveals early signs... cures that work in days... and when to worry Nancy Mace is OUSTED from politics after Trump extracts Epstein'revenge' in South Carolina governor's race Leaked transcript of UNAIRED 60 Minutes interview exposes REAL reason'callous' CBS star Scott Pelley'deserved to be fired' She's always by Trump's side, trusted with the White House's biggest secrets... and she influences millions Woke Canadian lawmakers fly into hilarious rage after conservative asks country's top scientist to define a woman Austin Metcalf's heartbroken father tells court how son's death destroyed him: 'We were robbed' Eva Longoria reunites with ex Tony Parker 15 years after cheating scandal split... as shocked fans react Inside Travis Kelce's plan to become'the Shaq of the NFL' after wedding Taylor Swift Zodiac killer case takes bombshell turn as unsolved cipher is CRACKED... and America's top codebreakers say evidence is all pointing to one man Caitlyn Jenner biographer and Robin Riker's ex William Hasley found dead on hiking trail at 78 Trump ERUPTS behind closed doors as top Republican pleads with him to axe Tulsi Gabbard's spy-chief replacement Are you over 45 and looking for a new job? If AI is to be believed, you might be too old. Scientists from the University of Melbourne asked ChatGPT for help finding candidates for fictional roles, and found a clear bias towards younger applicants.

We study true self-avoiding walk (TSAW) as a mechanism for improving empirical integral estimation via Markov chain Monte Carlo (MCMC). We consider finite-state adaptive sampling dynamics associated with an irreducible Markov kernel $P$ on a finite set, with stationary distribution $π$, in which the transition probabilities are penalized according to empirical overuse. Our main result is that the empirical occupation counts $L_t(i)$ and transition counts $N_t(i,j)$ of the resulting TSAW-based walk satisfy \[ L_t(i)-tπ_i = O(\sqrt{\log t}) \quad\text{and}\quad N_t(i,j)-tπ_iP_{ij}=O(\sqrt{\log t}) \qquad\text{almost surely} \] for every state $i$ and every edge $(i,j)$ with $P_{ij}>0$. Consequently, for every bounded function $f:V\to\mathbb R$, the error of our integral estimator converges as \[ \left|\frac1t\sum_{s=0}^{t-1} f(X_s)-\sum_{i\in V}π_i f(i)\right| = O\left(\frac{\sqrt{\log t}}{t}\right) \qquad\text{almost surely}. \] These results show that, in contrast with the usual $t^{-1/2}$ error scaling for empirical averages under standard random-walk-based methods, TSAW-based estimator yields empirical integral errors of order $O(\sqrt{\log t}/t)$ almost surely, thereby achieving a substantially sharper dependence on the sample size $t$.

We study stochastic linear bandits under a natural combination of batching and communication constraints: the time horizon is partitioned into batches of equal size $B$, and during each batch the learner sends $B$ requested arm pulls to an agent, who then observes the corresponding $B$ rewards and responds with a single bit of feedback to the learner. For each batch, the learner specifies the 1-bit quantization rule the agent uses, which may depend on all previously received bits but not on any past rewards directly. This setting addresses a significant yet unexplored ``middle ground'' between previous models having per-round quantization only or total bit budgets only. We establish a minimax lower bound showing that $Ω(B\min\{d,\log\lvert \mathcal{A} \rvert\})$ regret is unavoidable due to the 1-bit communication bottleneck, even in the absence of noise. Combined with standard statistical limits, this yields a general lower bound of $\widetildeΩ(B\min\{d,\log\lvert \mathcal{A} \rvert\} + \sqrt{dT \min\{d,\log\lvert \mathcal{A} \rvert\}})$. We develop two phased-elimination algorithms based on $G$-optimal designs and 1-bit mean estimation. The first achieves $\widetilde{O}(dB + d\sqrt{T})$ regret, matching the lower bound up to logarithmic factors when $\lvert \mathcal{A} \rvert = \exp(Ω(d))$, and the second incorporates a safe-arm identification and warm-start procedure to obtain $\widetilde{O}(B\log\lvert \mathcal{A} \rvert + d^{3/2}\sqrt{B} + \sqrt{dT\log\lvert \mathcal{A} \rvert})$ regret, which is near-optimal in broad scaling regimes of $(\lvert \mathcal{A} \rvert, B, d, T)$. Together, our results demonstrate that a single bit of feedback per batch suffices to nearly match the minimax regret of unconstrained linear bandits in broad scaling regimes, even for batch sizes as large as $Θ(\sqrt{T})$.