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

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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 $\varepsilon$ costs SoS degree $\Omega(\log(1/\varepsilon))$ or margin $\Omega(\sqrt{\log(1/\varepsilon)}/\sqrt{d})$, so the $\log(1/\varepsilon)$ 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 …

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