Multicalibration is a notion of fairness for predictors that requires them to provide calibrated predictions across a large set of protected groups. Multicalibration is known to be a distinct goal than loss minimization, even for simple predictors such as linear functions. In this work, we consider the setting where the protected groups can be represented by neural networks of size $k$, and the predictors are neural networks of size $n > k$. We show that minimizing the squared loss over all neural nets of size $n$ implies multicalibration for all but a bounded number of unlucky values of $n$. We also give evidence that our bound on the number of unlucky values is tight, given our proof technique. Previously, results of the flavor that loss minimization yields multicalibration were known only for predictors that were near the ground truth, hence were rather limited in applicability. Unlike these, our results rely on the expressivity of neural nets and utilize the representation of the predictor.

翻译：多校准是一种预测器的公平性概念，要求预测器在大量受保护群体上提供校准的预测。已知即使对于线性函数等简单预测器，多校准也是一个与损失最小化不同的目标。在本工作中，我们考虑受保护群体可由大小为$k$的神经网络表示，而预测器为大小为$n > k$的神经网络的情形。我们证明，对所有大小为$n$的神经网络进行平方损失最小化，除了有界数量的不幸运$n$值外，均可实现多校准。我们还给出证据表明，根据我们的证明技术，对不幸运值数量的界限是紧的。此前，损失最小化导致多校准的结果仅适用于接近真实值的预测器，因此应用范围相当有限。与之不同，我们的结果依赖于神经网络的表达力，并利用了预测器的表示。