Title: Why Ask One When You Can Ask $k$? Learning-to-Defer to the Top-$k$ Experts
Primary area: learning theory
Keywords: learning to defer, selective prediction, routing, machine learning

Abstract: Existing _Learning-to-Defer_ (L2D) frameworks are limited to _single-expert deferral_, forcing each query to rely on only one expert and preventing the use of collective expertise. We introduce the first framework for _Top-$k$ Learning-to-Defer_, which allocates queries to the $k$ most cost-effective entities. Our formulation unifies and strictly generalizes prior approaches, including the _one-stage_ and _two-stage_ regimes, _selective prediction_, and classical cascades. In particular, it recovers the usual Top-1 deferral rule as a special case while enabling principled collaboration with multiple experts when $k>1$. We further propose _Top-$k(x)$ Learning-to-Defer_, an adaptive variant that learns the optimal number of experts per query based on input difficulty, expert quality, and consultation cost. To enable practical learning, we develop a novel surrogate loss that is Bayes-consistent, $\mathcal{H}_h$-consistent in the one-stage setting, and $(\mathcal{H}_r,\mathcal{H}_g)$-consistent in the two-stage setting. Crucially, this surrogate is independent of $k$, allowing a single policy to be learned once and deployed flexibly across $k$. Experiments across both regimes show that Top-$k$ and Top-$k(x)$ deliver superior accuracy–cost trade-offs, opening a new direction for multi-expert deferral in L2D.

TL;DR: Generalizing Learning-to-Defer: From Single-Expert Deferral to Top-$k$ Expert Selection