Calibrated Multiaccuracy: Deterministic and Randomized Sample Complexity

Calibrated multiaccuracy asks that a predictor be globally calibrated and multiaccurate with respect to a bounded test family $G$; that is, its residuals should be uncorrelated with every test $g(x)\in G$. This weakening of multicalibration suffices for many applications. Its sample complexity was not known to improve on that of full multicalibration, which is $\widetildeΘ\left(\frac{\log|G|}{\varepsilon^2} + \frac{1}{\varepsilon^{3}}\right)$ when $|G|$ is at least a sufficiently large power of $\log(1/\varepsilon)$. We resolve the sample complexity of $\varepsilon$-calibrated multiaccuracy, showing that it scales as $$ \widetildeΘ\left(\frac{\log|G|}{\varepsilon^2} + \frac{1}{\varepsilon^{5/2}}\right), $$ separating it from the complexity of multicalibration. The upper bound is obtained by an algorithm that outputs a randomized predictor. For deterministic predictors, we establish the tight rate $$ \widetildeΘ\left( \frac{\log|G|}{\varepsilon^2} + \frac{(\log|G|)^{1/3}}{\varepsilon^{8/3}} \right) $$ whenever $\log|G|=Ω(\log(1/\varepsilon))$. When $\log|G|\le\varepsilon^{-1+κ}$ for any fixed $κ\in(0,1)$, this rate is polynomially larger than the rate for randomized predictors. Thus, unlike for multicalibration, calibrated multiaccuracy exhibits a genuine sample-complexity separation between deterministic and randomized predictors. The matching upper and lower bounds show that the unusual cube-root dependence on $\log|G|$ is intrinsic. At $\log|G|\asymp\varepsilon^{-1}$, the minimax rates for randomized and deterministic calibrated multiaccuracy both coincide with the minimax rate for multicalibration.