arXiv:2510.00504v1 Announce Type: new
Abstract: When training large-scale models, the performance typically scales with the number of parameters and the dataset size according to a slow power law. A fundamental theoretical and practical question is whether comparable performance can be achieved with significantly smaller models and substantially less data. In this work, we provide a positive and constructive answer. We prove that a generic permutation-invariant function of $d$ objects can be asymptotically compressed into a function of $operatorname{polylog} d$ objects with vanishing error. This theorem yields two key implications: (Ia) a large neural network can be compressed to polylogarithmic width while preserving its learning dynamics; (Ib) a large dataset can be compressed to polylogarithmic size while leaving the loss landscape of the corresponding model unchanged. (Ia) directly establishes a proof of the textit{dynamical} lottery ticket hypothesis, which states that any ordinary network can be strongly compressed such that the learning dynamics and result remain unchanged. (Ib) shows that a neural scaling law of the form $Lsim d^{-alpha}$ can be boosted to an arbitrarily fast power law decay, and ultimately to $exp(-alpha’ sqrt[m]{d})$.