Sample Efficient Omniprediction and Downstream Swap Regret for Non-Linear Losses

TL;DR

This paper introduces decision swap regret for non-linear losses, providing algorithms with polynomial sample complexity bounds for omniprediction and downstream regret in adversarial settings, especially for economic utility functions.

Contribution

It generalizes swap regret to non-linear losses, offering the first polynomial sample bounds for Lipschitz functions in omniprediction and algorithms for non-linear downstream regret.

Findings

First polynomial sample complexity bounds for Lipschitz loss functions in omniprediction.

Algorithms guaranteeing swap regret bounds for non-linear loss functions over multi-dimensional outcomes.

Improved bounds for specific economic utility functions like CES, Cobb-Douglas, and Leontief.

Abstract

We define "decision swap regret" which generalizes both prediction for downstream swap regret and omniprediction, and give algorithms for obtaining it for arbitrary multi-dimensional Lipschitz loss functions in online adversarial settings. We also give sample complexity bounds in the batch setting via an online-to-batch reduction. When applied to omniprediction, our algorithm gives the first polynomial sample-complexity bounds for Lipschitz loss functions -- prior bounds either applied only to linear loss (or binary outcomes) or scaled exponentially with the error parameter even under the assumption that the loss functions were convex. When applied to prediction for downstream regret, we give the first algorithm capable of guaranteeing swap regret bounds for all downstream agents with non-linear loss functions over a multi-dimensional outcome space: prior work applied only to linear loss functions, modeling risk neutral agents. Our general bounds scale exponentially with the dimension of the outcome space, but we give improved regret and sample complexity bounds for specific families of multidimensional functions of economic interest: constant elasticity of substitution (CES), Cobb-Douglas, and Leontief utility functions.