Moreover, we show that the system exhibits Poincaré recurrence, meaning that almost all orbits return arbitrarily close to their initial conditions infinitely often.

Large Language Models (LLMs) have demonstrated remarkable capabilities in many real-world applications. Nonetheless, LLMs are often criticized for their tendencytoproduce hallucinations, wherein themodels fabricate incorrectstatements on tasks beyond their knowledge and perception.

Atthesame time, thedevelopment ofequivariant methods, which preservetheir output under the action of rotation or some other transformation, has proven to be important for geometry problems across these domains. In this work, we observethat Adam -- when treated as afunction that maps initial conditions to optimized results --isnotrotation equivariant forvector-valued parameters due to per-coordinate moment updates. This leads to significant artifacts and biases in practice.

We show with confidence bound based bidding, and'Average Pricing' there