Unifying lower bounds on prediction dimension of convex surrogates

Jessica Finocchiaro, Rafael Frongillo, Bo Waggoner

Keywords: Property elicitation, surrogate loss functions, consistency, prediction dimension

TL;DR: We present a framework for deriving lower bounds on prediction dimension that strengthens and broadens previous bounds in the literature.

Abstract: The convex consistency dimension of a supervised learning task is the lowest prediction dimension $d$ such that there exists a convex surrogate $L : \mathbb{R}^d \times \mathcal Y \to \mathbb R$ that is consistent for the given task. We present a new tool based on property elicitation, $d$-flats, for lower-bounding convex consistency dimension. This tool unifies approaches from a variety of domains, including continuous and discrete prediction problems. We use $d$-flats to obtain a new lower bound on the convex consistency dimension of risk measures, resolving an open question due to Frongillo and Kash (NeurIPS 2015). In discrete prediction settings, we show that the $d$-flats approach recovers and even tightens previous lower bounds using feasible subspace dimension.

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