Agnostic Learning with Unknown Utilities

Abstract

Traditional learning approaches for classification implicitly assume that each mistake has the same cost. In many real-world problems though, the utility of a decision depends on the underlying context x and decision y; for instance, misclassifying a stop sign is worse than misclassifying a road-side postbox. However, directly incorporating these utilities into the learning objective is often infeasible since these can be quite complex and difficult for humans to specify.
We formally study this as agnostic learning with unknown utilities: given a dataset S = {x_1, …, x_n} where each data point x_i ∼ 𝒟_x from some unknown distribution 𝒟_x, the objective of the learner is to output a function f in some class of decision functions ℱ with small excess risk. This risk measures the performance of the output predictor f with respect to the best predictor in the class ℱ on the unknown underlying utility u^*:𝒳×𝒴↦ [0,1]. This utility u^* is not assumed to have any specific structure and is allowed to be any bounded function. This raises an interesting question whether learning is even possible in our setup, given that obtaining a generalizable estimate of utility u^* might not be possible from finitely many samples. Surprisingly, we show that estimating the utilities of only the sampled points S suffices to learn a decision function which generalizes well. 
With this insight, we study mechanisms for eliciting information from human experts which allow a learner to estimate the utilities u^* on the set S. While humans find it difficult to directly provide utility values reliably, it is often easier for them to provide comparison feedback based on these utilities. We show that, unlike in the realizable setup, the vanilla comparison queries where humans compare a pair of decisions for a single input x are insufficient. We introduce a family of elicitation mechanisms by generalizing comparisons, called the k-comparison oracle, which enables the learner to ask for comparisons across k different inputs x at once. We show that the excess risk in our agnostic learning framework decreases at a rate of O (1/k) with such queries. This result brings out an interesting accuracy-elicitation trade-off - as the order k of the oracle increases, the comparative queries become harder to elicit from humans but allow for more accurate learning.