Distributionally robust PAC learning under Cressie–Read balls has polynomial sample complexity for every fixed order above one. The Kullback–Leibler endpoint was left open because it amplifies rare events on a different scale. We determine that endpoint. Let $\rho>0$ be fixed. For binary classification with zero-one loss, both realizable and agnostic KL-robust learning have sample complexity $\Theta_{\rho} \left( \frac{e^{\rho/\epsilon}}{\epsilon} \bigl(d+\log(1/\delta)\bigr)\right)$ samples, where $d$ is the VC dimension. The realizable statement is sharp without logarithmic factors. It follows from an exact reduction to ordinary PAC learning at the nominal accuracy $A_{\rho}(\epsilon)\sim e^{-1}\epsilon e^{-\rho/\epsilon}$. The agnostic result requires a different argument. Relative-VC control of ERM loses a factor of order $1/\epsilon$ at this singular endpoint. We combine a recent small-error improper learner with ERM. A new inverse-geometry dichotomy shows that multiplicative comparator slack is harmless exactly where the ERM logarithm is costly, and conversely. Validation between the two candidates gives the same sharp rate as in the realizable case. We also resolve the singular transition from Cressie–Read to KL. If the order is $k=1+\lambda\epsilon/\rho$, then the inverse nominal scale has exponent $(\rho/\epsilon)\log(1+\lambda)/\lambda$. This boundary layer interpolates continuously between KL and every near-KL polynomial regime, and shows that the limits $k\downarrow1$ and $\epsilon\downarrow0$ do not commute.
Keywords: distributionally robust learning, Kullback-Leibler divergence, PAC learning, Cressie-Read divergences, sample complexity, robustness