Fast Rates for Swap-Agnostic Learning of Proper Losses

Abstract

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 $O ((lo g ∣ H ∣/ m)^{2/3})$ , with a corresponding online swap-regret bound of $O (T^{1/3} (lo g ∣ H ∣)^{2/3})$ . 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 $O (T lo g ∣ H ∣)$ and $O (lo g ∣ H ∣/ m)$ , respectively. For convex, $1$ -Lipschitz proper losses, these rates improve to $O (T^{1/3} (lo g ∣ H ∣)^{2/3})$ online and $O ((lo g ∣ H ∣/ m)^{2/3})$ offline. These bounds are tight up to logarithmic factors and improve upon the $O (T^{2/3} (lo g ∣ H ∣)^{1/3})$ 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.