Optimal Decision-Making Based on Prediction Sets

TL;DR

This work addresses how to make high-stakes decisions using prediction sets that come with a distribution-free coverage guarantee $\Pr(Y\in C(X))\ge 1-\alpha$. It develops a minimax decision framework that yields a closed-form worst-case risk $L_S(a;\alpha)=\ell^{\mathrm{in}}_S(a)+\alpha(\ell^{\mathrm{out}}_S(a)-\ell^{\mathrm{in}}_S(a))_+$ and a per-instance optimal policy that minimizes this risk, bridging prediction-set design with downstream actions. The authors then solve the population design problem, show a one-dimensional dual structure with a scalar $\beta^*$ that determines the optimal coverage allocation $t^*(x)$, and construct risk-optimal prediction sets $C^*(x)$ together with actions $a^*(x)$. To deploy this in practice, they introduce Risk-Optimal Conformal Prediction (ROCP), which estimates necessary quantities from data and guarantees finite-sample marginal coverage via conformal calibration. Empirical results in medical diagnosis and autonomous-driving-like settings demonstrate ROCP's advantage in reducing worst-case risk and critical mistakes, especially when out-of-set errors are costly. The approach provides a principled, decision-aware extension of conformal prediction with practical applicability to safety-critical systems.

Abstract

Prediction sets can wrap around any ML model to cover unknown test outcomes with a guaranteed probability. Yet, it remains unclear how to use them optimally for downstream decision-making. Here, we propose a decision-theoretic framework that seeks to minimize the expected loss (risk) against a worst-case distribution consistent with the prediction set's coverage guarantee. We first characterize the minimax optimal policy for a fixed prediction set, showing that it balances the worst-case loss inside the set with a penalty for potential losses outside the set. Building on this, we derive the optimal prediction set construction that minimizes the resulting robust risk subject to a coverage constraint. Finally, we introduce Risk-Optimal Conformal Prediction (ROCP), a practical algorithm that targets these risk-minimizing sets while maintaining finite-sample distribution-free marginal coverage. Empirical evaluations on medical diagnosis and safety-critical decision-making tasks demonstrate that ROCP reduces critical mistakes compared to baselines, particularly when out-of-set errors are costly.

Figures (3)

• Figure 1: The pipeline of Risk-Optimal Conformal Prediction (ROCP).
• Figure 2: Medical diagnosis experiments. Results under two treatment-loss specifications: the baseline loss matrix from kiyani2025decision (left) and our safety-critical variant. Each panel reports, as a function of miscoverage level $\alpha$: (a) average realized worst-case risk certificate; (b) average realized loss; (c) empirical miscoverage; (d) critical mistake rate for critical labels, defined as the fraction of test points with true label $y_c$ for which the chosen action attains $\arg\max_{a\in\mathcal{A}}\ell(a,y_c)$. All results are averaged over 20 random train/calibration/test splits; error bars show $\pm 1$ standard error.
• Figure 3: Toy autonomous driving experiment. (a) average realized worst-case risk certificate; (b) average realized loss; (c) empirical miscoverage; and (d) a critical mistake is defined as selecting an action that incurs the collision penalty (i.e., loss at least $M$) in that state. In (d), the x-axis labels 001, 010, $\ldots$, 111 denote the 3-bit hazard state $y=(y_a,y_\ell,y_r)$. Results are averaged over 20 random splits; error bars show $\pm 1$ standard error.

Theorems & Definitions (13)

• Lemma 2.1: Worst-case expected loss under miscoverage level of $\alpha$
• Theorem 2.2: Optimal policy and risk
• Proposition 3.1
• Remark 3.2: Edge cases
• Theorem 3.3
• Proposition 4.1: Distribution-free marginal coverage
• Remark B.1: Measurable selection for \ref{['pi-star']}
• Definition B.2: Normal integrands rockafellar1998variational
• Theorem B.3
• proof : Proof of Theorem \ref{['thm:pop-opt-general']}