Fast Learning Beyond Finite Variance: Fractional Bernstein Geometry under Heavy-Tailed Losses

Classical fast-rate theory typically relies on second-moment or related Bernstein-type control. We ask whether fast rates remain possible when variance does not exist. For finite classes, we introduce a fractional Bernstein condition $\mathbb{E}|Z_f{|}^q\le B{\Delta }_f^{\gamma },q\in (1,2],$ for centered relative losses. An observable median-of-means pairwise selector achieves $\Delta (\hat{f})\lesssim B^{1/(q-\gamma )}{\left(\frac{\log (M/\delta )}{n}\right)}^{\frac{q-1}{q-\gamma }},$ and a matching minimax lower bound shows that this interpolation law is sharp up to constants. We derive the fractional Bernstein geometry from primitive $L^q$ increment regularity and polynomial risk growth; in particular, Lipschitz increments and quadratic growth yield $\gamma =q/2$and the exponent $2(q-1)/q$. We then show that this regime arises in vector quantization genuinely below finite variance. Relative squared losses require only a first moment, and therefore remain well defined in regimes where every absolute population distortion is infinite. A weighted Voronoi margin condition yields local quadratic relative-risk growth without requiring a second moment. For a symmetric two-component Student-\(t\) family, we prove the explicit separation law $A_{\mathrm{V}}(a)=O_{\nu ,d}\left((s/a{)}^{\nu }\right).$ Hence, sufficient separation yields local finite-dictionary relative quantization at rate ${\left(\frac{\log (M/\delta )}{n}\right)}^{\frac{2(q-1)}{q}},1<q<\nu \le 2,$ despite $\mathbb{E}\Vert X{\Vert }^2=\infty$.