Obtain sharper excess-risk bounds under additional structural assumptions

Derive tighter finite-sample excess-risk bounds for conformal risk control beyond the established worst-case order sqrt(log m / n) by imposing additional structural assumptions on the loss function beyond boundedness, Lipschitz continuity, or monotonicity, and specify the conditions under which such sharpened rates hold.

Background

The paper proves an excess risk of order sqrt(log m / n) for bounded losses over a grid and shows this rate is minimax optimal in the worst case. It also presents refined guarantees when the loss is Lipschitz (with a margin condition) and establishes exact control for monotone losses.

Further structural assumptions on the loss function may enable stronger performance guarantees than those already developed. Identifying such assumptions and quantifying the corresponding improvements remains open.