Sum-of-Squares Degree Barriers for the Reweighted-Hinge Method in Robust Halfspace Learning: A Christoffel-Function Characterization

arXiv:2606.17215v2 Announce Type: replace Abstract: A certificate that removes outliers sees the data only through its low-degree moments, and an adversary exploits exactly this, hiding corruption where the clean data already looks typical, in the blind spot no bounded-degree test resolves. That blind spot has an exact size: the Christoffel function of the clean marginal, the quantity data analysis thresholds to detect outliers, here read from the adversary's side as the corruption a certificate cannot remove. We turn this inversion into the organizing principle of the reweighted-hinge approach to robustly learning gamma-margin halfspaces under malicious noise (Shen 2025; Zeng-Shen 2025): the governing resource is the Sum-of-Squares degree of the certificate, and the resolution principle states that the maximal corruption mass hideable at a center c from a degree-2t certificate is exactly the Christoffel function lambda_{t+1}(c). Three consequences follow, all against the certificate method (not information-theoretic). A margin-degree tradeoff: certifying the dense pancake to error arepsilon costs SoS degree Omega(log(1/arepsilon)) or margin Omega(sqrt{log(1/arepsilon)}/sqrt{d}), so the log(1/arepsilon) margin of Shen (2025) is forced; a weighted-Chebyshev reduction makes the threshold 2t=Theta((|c|/s)^2) tight modulo one classical extremal estimate. A degree-2 outlier barrier: an explicit instance on which degree 2 is stuck at eta^{1/2} while degree 4 escapes, locating the small breakdown rate in the degree, not the analysis. A degree-2t algorithm tracing the frontier eta^{1-1/2t} (recovering Shen 2025 at t=1), with an explicit constant gain capped by the pancake density. And an information-theoretic floor of eta/(2(1-eta)), matched exactly from above; under a hard margin its two-point realizations provably require Theta(1/eta) mixture components."

Source: arXiv cs.LG | 2026-08-18