Fast Rates for Swap-Agnostic Learning of Proper Losses

Swap-agnostic learning strengthens classical agnostic learning by allowing the comparator to select a different hypothesis on each level set of the learner's predictions. This benchmark captures prediction-dependent postprocessing, but appears to require solving a separate agnostic-learning problem for every possible prediction value. We show that, for proper losses, these prediction-level comparisons can instead be controlled jointly. Our main result is an offline swap-agnostic learner for any fixed proper loss. For a finite hypothesis class H and any fixed smooth proper loss, the excess risk from m i.i.d. samples is widetildeO((log |H|/m)2/3), with a corresponding online swap-regret bound of widetildeO(T1/3(log |H|)2/3). We also give algorithms whose predictions are simultaneously swap-agnostic for entire families of losses. For all proper losses bounded in [-1,1], we obtain online and offline rates of widetildeO(√Tlog |H|) and widetildeO(√log |H|/m), respectively. For convex, 1-Lipschitz proper losses, these rates improve to widetildeO(T1/3(log |H|)2/3) online and widetildeO((log |H|/m)2/3) offline. These bounds are tight up to logarithmic factors and improve upon the widetildeO(T2/3(log |H|)1/3) rate implied by the swap-omniprediction guarantee of Luo et al. (2025). Our main technical contribution is a reduction from swap-agnostic learning to a second-order form of multicalibration, obtained via Blackwell approachability with a Bernstein-style variance correction.