Characterization of surrogates yielding fast rates and tight convergence neighborhoods

Characterize the class of surrogate functions h:R^d→R for which the surrogate-loss framework ψ(x)=1/2 h^2(x) yields fast convergence rates and tight neighbourhoods of convergence.

Background

The surrogate-based view connects Polyak-type updates to minimizing ψ(x)=1/2 h^2(x), leading to analyses that often guarantee convergence to a neighbourhood rather than exact minima when h remains positive. The paper’s negative results show such neighbourhoods can be unavoidable in general.

The authors highlight that identifying which surrogate constructions deliver both fast rates and tight (small) convergence neighbourhoods is not yet understood, leaving open the task of systematically classifying surrogates that balance speed and accuracy within this framework.