Gradient-flow analyses show that simplified linear transformers can learn the in-context linear-regression algorithm, but they do not explain the finite-step behavior of gradient descent at large learning rates. Motivated by empirical work on high-learning-rate transformer instabilities and by the cubic-map phase diagram for quadratic regression, we study an exactly reducible one-prompt linear-transformer training problem. After normalization, the dynamics reduce to a two-factor product map with an effective step-size parameter \(μ\). On the balanced slice, this map recovers the known scalar cubic transition from monotone convergence to catapult convergence, periodic and chaotic bounded nonconvergence, and divergence. We then analyze the full two-dimensional system and show that, for \(0<μ<2\), it has an explicit invariant Chebyshev ellipse separating forward-invariant regions; this ellipse carries off-balanced chaotic dynamics but is transversely repelling, while balanced scalar attractors can be transversely attracting. These results show that large constant learning rates can change the training attractor of the learned transformer rather than merely accelerating convergence: beyond sharp stability thresholds, finite-step training may settle into cycles, bounded chaos, or divergence instead of a single in-context linear-regression solution. We also discuss the consequences for mini-batch gradient descent based training methods.

We study contextual dynamic pricing in a semiparametric scalar-index valuation model where the latent value is $v_t=μ_\ast(\mathsf c_t)+ξ_t$, with an unknown utility map $μ_\ast$ and an unknown additive noise distribution. The key decision object is the one-dimensional oracle price map $u\mapsto p^\ast(u)$ induced by the scalar index $u=μ_\ast(\mathsf c)$ and the noise tail. Under the $β$-Hölder smoothness of the tail function for $β\geq 2$ and a revenue-geometry condition that gives a unique, stable, interior maximizer, this oracle map is itself $(β-1)$-smooth. We exploit such structure through $\mathsf{ORBIT}$, a modular coarse-to-fine policy that takes a scalar pilot index as input, localizes a benchmark price in each active bin, and learns a local polynomial approximation of the oracle map inside a trust region via bandit convex optimization. For the baseline linear utility model $μ_\ast(\mathsf c)=\mathsf c^\topθ_\ast$, an adaptive elliptical exploration scheme constructs the required scalar pilot online without distributional assumptions on the contexts. The resulting policy achieves regret $\widetilde{O}\big(T^{\frac{2β-1}{4β-3}}+\sqrt{dT}\big)$. For fixed $d$, we establish a matching lower bound in the horizon dependence, unveiling that the nonparametric oracle-map learning term is minimax sharp. The same scalar-pilot interface also yields extensions to sparse high-dimensional linear utility and nonparametric Hölder utility.

Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' READ MORE: NASA successfully changed an asteroid's orbit around the SUN An out-of-control NASA satellite has plunged back to Earth after more than 14 years in orbit. The 590-kilogram (1,300 lbs) Van Allen Probe A crashed down in the East Pacific Ocean near the Galapagos Islands at 10:37 GMT (06:37 EDT) yesterday morning. NASA says it expected most of the spacecraft to burn up in the atmosphere, but some parts may have survived re-entry and reached the surface.

Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Alexander brothers' alleged HIGH SCHOOL gang rape video: Classmates speak out on sick'taking turns' footage... as creepy unseen photos are exposed Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting NFL superstar Xavier Worthy spills all on Travis Kelce, the Chiefs' struggles... and having Taylor Swift as his No 1 fan Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Nancy Mace throws herself into Iran warzone as she goes rogue on Middle East rescue mission: 'I AM that person' One giant leap for planetary defence: NASA successfully changed an asteroid's orbit around the SUN, new study reveals Humanity has taken a'notable step forward' in its ability to deflect asteroids heading towards Earth, a new study reveals. Back in 2022, NASA deliberately smashed a spacecraft into a small asteroid'moonlet' that orbited a larger space rock. The probe, called Dart, successfully changed the path of the moonlet, called Dimorphos, around its parent asteroid, Didymos. The mission was hailed as the first-ever successful demonstration of planetary defence, proving humanity can alter an asteroid's trajectory. But now, scientists have revealed the test also knocked both asteroids off their regular orbit around the Sun.

Could AI Data Centers Be Moved to Outer Space? Massive data centers for generative AI are bad for the Earth. Data centers are being built at a frantic pace all over the world, driven by the AI boom. These facilities consume staggering amounts of electricity. By 2028, AI servers alone may use as much energy as 22 percent of US households.

Samsung Galaxy Unpacked 2026 is Feb. 25 A single collision could cause a cascading effect in orbit. Elon Musk's plan to launch millions of AI satellites could be disastrous for the planet. At the start of the month, Elon Musk announced that two of his companies -- SpaceX and xAI -- were merging, and would jointly launch a constellation of 1 million satellites to operate as orbital data centers. Musk's reputation might suggest otherwise, but according to experts, such a plan isn't a complete fantasy. However, if executed at the scale suggested, some of them believe it would have devastating effects on the environment and the sustainability of low Earth Earth orbit.

After a Falcon 9 rocket stage burned up in the atmosphere, vaporised lithium and other metals drifted over Europe. A SpaceX rocket that burned up after re-entering the atmosphere unleashed a plume of vaporised metals over Europe, a type of pollution that is expected to increase as spacecraft and satellites multiply. The upper stage of a Falcon 9, which is designed to splash down in the Pacific Ocean for possible re-use, lost control due to engine failure and fell from orbit over the north Atlantic in February 2025. We're finally solving the puzzle of how clouds will affect our climate People across Europe saw fiery debris streaking through the sky, some of which crashed behind a warehouse in Poland. Seeing the news, Robin Wing at the Leibniz Institute of Atmospheric Physics in Germany and his colleagues turned on their lidar, an instrument for atmospheric sensing.

It is a variant of shuffle coding that is many orders of magnitude faster than the original and enables'one-shot' compression of single

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Figure 5: Comparison of GenStat architecture to selected graph generative models. This proof uses two properties of LDP: composability and immunity to post-processing [2]. Figure 6 illustrates the PGM of Randomized algorithms. The GGM parameters are a function of the perturbed graph statistics as learning input. The implementation can be easily extended to directed graphs. A statistics-based GGM that takes the degree sequence as sufficient statistics [5].