Agnostically Learning Single-Index Models using Omnipredictors

We give the first result for agnostically learning Single-Index Models (SIMs) with arbitrary monotone and Lipschitz activations. All prior work either held only in the realizable setting or required the activation to be known. Moreover, we only require the marginal to have bounded second moments, whereas all prior work required stronger distributional assumptions (such as anticoncentration or boundedness). Our algorithm is based on recent work by Gopalan et al. [2023] on Omniprediction using predictors satisfying calibrated multiaccuracy. Our analysis is simple and relies on the relationship between Bregman divergences (or matching losses) and $\ell_p$ distances. We also provide new guarantees for standard algorithms like GLMtron and logistic regression in the agnostic setting.