We study the squared-loss sample compression question posed by Attias, Hanneke, Kontorovich, and Sadigurschi in their ICML 2024 paper. For every nonempty class
F⊆[0,1]X and accuracy
0<α≤1 with
d=fat(F,c′α)<∞, where
c′>0 is universal, we prove the existence of a deterministic agnostic scheme of total size
O(dlog3(2/α)). Its code contains original labeled observations and auxiliary bits, and its reconstruction has empirical loss at most
α above the infimum over
F on every finite sample. This establishes the polylogarithmic formulation in Section 5 of arXiv:1810.01864v2. The construction approximates a near-optimal internal target by a fixed-step boosting ensemble and takes its equal-weight median. Average loss permits an accuracy-dependent number of rounds, so the entire ensemble can be retained without dual-dimension sparsification. Quantized target constraints and round-membership bits determine the reconstructed function exactly. We also prove that, when retained observations must come from the sample, a fixed class of fat-shattering dimension one requires total size
Ω(log(1/α)). With observations and auxiliary bits charged together, this rules out the proceedings version's log-free formulation under its original-example restriction. The framework extends to average power loss at scale
α/p.