Extend quadratic behavior beyond current density regularity assumptions

Identify relaxed regularity conditions on the density p(q) under which quadratic population-regret behavior persists, quantify how the reference point s controls the approximation bias of the regularized curvature matrix \(\mathsf K_{\lambda,s}\), and develop practical guidelines for selecting the regularization parameters \(\lambda\) and s.

Background

The paper establishes a local quadratic expansion of population regret for contextual linear optimization over a polyhedral feasible set under Lipschitz regularity assumptions on the conditional-cost density p(q) and perturbation field h(q). It approximates the singular population-curvature measure supported on normal-fan walls using the regularized curvature matrix $\mathsf K_{\lambda,s}$, which depends on a regularization parameter $\lambda$ and a reference point s in the relative interior of the feasible set.

The authors leave unresolved how far the quadratic-regret result extends under weaker assumptions on p(q), how the choice of s affects the approximation bias of $\mathsf K_{\lambda,s}$, and how practitioners should select $\lambda$ and s. Resolving these issues would support broader theoretical applicability and more reliable implementation of the curvature-based approximation.