Finite-grid conformal risk control

Input a deterministic ordered grid $\mathrm{\Lambda }=\{{\lambda }_1<\cdots <{\lambda }_M\}$ with $M\ge 1$, a calibration size $n\ge 1$, a bound $B>0$, a target $\alpha \in [0,B)$, and calibration losses $L_i({\lambda }_j)\in [0,B]$ for $1\le i\le n$ and $1\le j\le M$. Also supply the prediction rule $d_{{\lambda }_j}$ associated with each grid point. For $j=1,\dots ,M$ in order, compute $T_j=\frac{{\sum }_{i=1}^n{L}_i({\lambda }_j)+B}{n+1}\mathrm{.}$ Return $({\lambda }_j,d_{{\lambda }_j},\text{pass})$ at the first index for which $T_j\le \alpha$. If no index passes, return $({\lambda }_M,d_{{\lambda }_M},\text{fallback})$. Monotonicity and the endpoint-safety assumption from the finite nested decision family are hypotheses for the risk theorem, not checks of the unobserved test loss.

Finite nested decision families and bounded risk: the grid, bounded loss array, loss-nesting convention, endpoint-safety condition, and marginal expected test risk.

Angelopoulos et al., Conformal Risk Control, §1.1, equation (4)