However, there is a noticeable gap in analysis for multiclass classification, with only a handful of results which themselves are restricted to the cross-entropy loss.

We study the problem of learning an optimal regression function subject to a fairness constraint. It requires that, conditionally on the sensitive feature, the distribution of the function output remains the same. This constraint naturally extends the notion of demographic parity, often used in classification, to the regression setting. We tackle this problem by leveraging on a proxy-discretized version, for which we derive an explicit expression of the optimal fair predictor. This result naturally suggests a two stage approach, in which we first estimate the (unconstrained) regression function from a set of labeled data and then we recalibrate it with another set of unlabeled data.

Proposition 5 (equivalence between transferability and transfer measures).

Based oninvariant features, a high-performing classifier on source domains could hopefully behave equally well on a target domain. In other words, we hope the invariant features to be transferable. However, in practice, there are no perfectly transferable features, andsomealgorithmsseemtolearn"moretransferable"featuresthanothers.

A key object in sequential simulation is the sequence of distributions, called the policy, fromwhich togenerate therandom variables, called particles, usedtoapproximate theintegralsof interest.

Since utilities can serveas target values to learn the scoring function through square loss regression, the optimality ofsorting byexpected utilities isequivalent tothe consistencyofregression.

This paper addresses the question of which of these tasks are asymptotically solved by sorting by decreasing order of expected utility, for some suitable notion of utility, or, equivalently, when is square loss regression consistent for ranking via score-and-sort?