Omnipredictors for Regression and the Approximate Rank of Convex Functions

Consider the supervised learning setting where the goal is to learn to predict labels mathbf y given points mathbf x from a distribution. An textitomnipredictor for a class mathcal L of loss functions and a class mathcal C of hypotheses is a predictor whose predictions incur less expected loss than the best hypothesis in mathcal C for every loss in mathcal L. Since the work of [GKR+21] that introduced the notion, there has been a large body of work in the setting of binary labels where mathbf y ∈ 0, 1, but much less is known about the regression setting where mathbf y ∈ [0,1] can be continuous. Our main conceptual contribution is the notion of textitsufficient statistics for loss minimization over a family of loss functions: these are a set of statistics about a distribution such that knowing them allows one to take actions that minimize the expected loss for any loss in the family. The notion of sufficient statistics relates directly to the approximate rank of the family of loss functions. Our key technical contribution is a bound of O(1/varepsilon2/3) on the ε-approximate rank of convex, Lipschitz functions on the interval [0,1], which we show is tight up to a factor of polylog (1/ε). This yields improved runtimes for learning omnipredictors for the class of all convex, Lipschitz loss functions under weak learnability assumptions about the class mathcal C. We also give efficient omnipredictors when the loss families have low-degree polynomial approximations, or arise from generalized linear models (GLMs). This translation from sufficient statistics to faster omnipredictors is made possible by lifting the technique of loss outcome indistinguishability introduced by [GKH+23] for Boolean labels to the regression setting.