Spectral Reach: Understanding Neural Scaling as Progress into the Spectral Tail

Neural scaling laws describe predictable power-law relationships between model size, dataset size, compute, and performance. While these laws guide the development of modern foundation models, the mechanisms underpinning them remain poorly understood, in part due to the absence of scalable analysis tools. To close this gap, we introduce "spectral position": a scalable measure of which eigenvalues of the empirical neural tangent kernel (eNTK) currently drive loss reduction. Applying this measure to scaling experiments, we find that spectral position decreases throughout training: learning shifts from dominant eigenmodes into the spectral tail. Larger models reach further into the tail than smaller models, revealing a size-dependent capacity we call "spectral reach". This suggests why larger models achieve lower losses: they sustain learning on weak spectral signals inaccessible to smaller models. We further identify feature learning as a key enabler of spectral reach. It adaptively amplifies gradient magnitudes as learning advances, sustaining progress where frozen representations stall. This points to concrete interventions through architecture and optimizer design.

SwarmRL: Building the Future of Smart Active Systems

This work introduces SwarmRL, a Python package designed to study intelligent active particles. SwarmRL provides an easy-to-use interface for developing models to control microscopic colloids using classical control and deep reinforcement learning approaches. These models may be deployed in simulations or real-world environments under a common framework. We explain the structure of the software and its key features and demonstrate how it can be used to accelerate research. With SwarmRL, we aim to streamline research into micro-robotic control while bridging the gap between experimental and simulation-driven sciences. SwarmRL is available open-source on GitHub at https://github.com/SwarmRL/SwarmRL.