The distance from calibration, introduced by Błasiok, Gopalan, Hu, and Nakkiran (STOC 2023), has recently emerged as a central measure of miscalibration for probabilistic predictors. We study the fundamental problems of computing and estimating this quantity, given either an exact description of the data distribution or only sample access to it. We give an efficient algorithm that exactly computes the calibration distance when the distribution has a uniform marginal and noiseless labels, which improves the $O(1/\sqrt{|\mathcal{X}|})$ additive approximation of Qiao and Zheng (COLT 2024) for this special case. Perhaps surprisingly, the problem becomes $\mathsf{NP}$-hard when either of the two assumptions is removed. We extend our algorithm to a polynomial-time approximation scheme for the general case. For the estimation problem, we show that $Θ(1/ε^3)$ samples are sufficient and necessary for the empirical calibration distance to be upper bounded by the true distance plus $ε$. In contrast, a polynomial dependence on the domain size -- incurred by the learning-based baseline -- is unavoidable for two-sided estimation. Our positive results are based on simple sparsifications of both the distribution and the target predictor, which significantly reduce the search space for computation and lead to stronger concentration for the estimation problem. To prove the hardness results, we introduce new techniques for certifying lower bounds on the calibration distance -- a problem that is hard in general due to its $\textsf{co-NP}$-completeness.

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We call this distinction thediscrimination risk. We prove that a higher discrimination risk can amplify the unfairness of a machine learning model applied totheimputed data.

In practical situations, the clustering result must be stable against points missing in the input data so that we can make trustworthy andconsistentdecisions.

If A is (",0)- di erentiallyprivate, thenA is " Yis (", )- di B : Y anyrandomB Ais (", )- di erA is thenthealgoB Ais - z CDP. Proof.Algorithm1 isthedesiredsamplerAPo.Itis (", ) - di erentiallyprivatesinceA is (", ) - di erentially private.

The publicly verifiable record of the randomization is available athttps://www.

Werecallthisresult from [4] (see Theorem 7 thereinformoredetails).

Theorem B.1 (Performance difference bound for Model-based RL). Mi denote the inconsistency between the learned dynamics PMi and the true dynamics, i.e. ϵ For L1 L3, with the performance gap approximation of M1 and π1, we apply Lemma C.2, and Here, dπMi denotes the distribution of state-action pair induced by policy π under the dynamical modelMi. Theorem B.3 (Refined bound with constraints). Let µ and v be two probability distributions on the configuration space X, according to LemmaC.1,thenwehaveDTV(µ Under these definitions, we can yield the following intermediate outcome by applying the results from B.2and B.1 Here, we take the time-varying linear quadratic regulator as an instance for illustrating the rationality of our assumption on α.

In the first model, we are givenn i.i.d.

In the first model, we are givenn i.i.d.