Sample Efficient Omniprediction and Downstream Swap Regret for Non-Linear Losses
This work addresses a foundational challenge in machine learning by enabling robust prediction and omniprediction for non-linear losses, which is incremental but extends prior linear-focused methods to broader economic utility functions.
The paper tackles the problem of decision swap regret for non-linear losses in online adversarial settings, providing algorithms with polynomial sample complexity for Lipschitz loss functions, where prior bounds were limited to linear losses or had exponential scaling.
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.