Extend CRC analysis to continuous parameter spaces

Extend the finite-sample expectation analysis of conformal risk control for non-monotonic bounded losses from a finite discrete parameter grid to a continuous parameter space by deriving conditions and a selection rule that ensure control of the expected loss at a target level α without assuming monotonicity of the loss function.

Background

The paper establishes finite-sample guarantees for conformal risk control (CRC) with bounded, potentially non-monotonic losses when the tuning parameter is selected from a finite grid of size m. The main bound shows an excess risk of order sqrt(log m / n), and a matching lower bound demonstrates minimax optimality in this discretized setting.

While these results cover many practical scenarios where thresholds or decision rules are discretized, many applications involve tuning parameters that vary over a continuous domain. Extending the analysis beyond finite grids would require addressing complexity measures appropriate to continuous spaces while maintaining distribution-free (or robust) control of expected loss without assuming monotonicity.