Decision Theoretic Foundations for Conformal Prediction: Optimal Uncertainty Quantification for Risk-Averse Agents
Authors: Shayan Kiyani
George J. Pappas
Aaron Roth
Hamed Hassani
Venue: ICML 2025 spotlightposter (2025)
Keywords: Decision making
Uncertainty Quantification
Prediction sets
Conformal prediction
Calibration
Risk Averse
Risk Aware
Risk sensitive
distribution free
Primary area: theory->probabilistic_methods

Abstract: A fundamental question in data-driven decision making is how to quantify the uncertainty of predictions to inform risk-sensitive downstream actions, as often required in domains such as medicine. We develop a decision-theoretic foundation linking prediction sets to risk-averse decision-making, addressing three questions: (1) What is the correct notion of uncertainty quantification for risk-averse decision makers? We prove that prediction sets are optimal for decision makers who wish to optimize their value at risk. (2) What is the optimal policy that a risk averse decision maker should use to map prediction sets to actions? We show that a simple max-min decision policy is optimal for risk-averse decision makers. Finally, (3) How can we derive prediction sets that are optimal for such decision makers? We provide an exact characterization in the population regime and a distribution free finite-sample construction. These insights leads to *Risk-Averse Calibration (RAC)*, a principled algorithm that is both *practical*—exploiting black-box predictions to enhance downstream utility—and *safe*—adhering to user-defined risk thresholds. We experimentally demonstrate RAC's advantages in medical diagnosis and recommendation systems, showing that it substantially improves the trade-off between safety and utility, delivering higher utility than existing methods while avoiding critical errors.

Decision Theoretic Foundations for Conformal Prediction: Optimal Uncertainty Quantification for Risk-Averse Agents
Shayan Kiyani George Pappas'
Aaron Roth Hamed Hassani
Abstract
high-stakes cases where incorrect treatment decision could lead to serious complications or death-it is crucial not to rely solely on a model's predictions. Instead, deci- sions must account for the uncertainty in these predictions, opting for more conservative interventions when that uncer- tainty makes the potential outcomes (e.g., complications, side effects) highly variable. Connecting uncertain pre- dictions to actionable, principled decisions is significant challenge in safety-critical domains, including medical diag- nosis, finance, robotics, and control, and requires balancing safety with utility. One extreme is to avoid any action en- tirely-sacrificingg prediction's 's practical value for absolute safety-whil the other is to aggressively exploit predictions to maximize expected utility, accepting significant downside risk at the cost of realizing poor outcomes with substantial probability. Balancing this trade-off calls for an optimal approach to risk-sensitive decision making, To this end, we focus on the following question:
A fundamental question in data-driven deci- sion making how quantify the uncertainty of predictions to inform risk-sensitive down- stream actions, as often required in domains such as medicine. We develop a decision-theoretic foundation linking prediction sets to risk-averse decision-mane making, addressing three questions: (I) What the correct notion of uncertainty quantifi- cation for risk-averse decision makers? We prove that prediction sets are optimal for decision mak- ers who wish to optimize their value at risk. (2) What is the optimal policy that a risk averse deci- sion maker should use to map prediction sets to actions? We show that simple max-min decision policy is optimal for risk-averse decision makers. Finally, (3) How can we derive prediction sets that are optimal for such decision makers? We pro- vide an exact characterization in the population regime and a distribution free finite-sample con- struction. These insights leads to Risk-Averse Cal- ibration (RAC), a principled algorithm that is both ppuctical-expliiin black-box predictions to en- hance downstream utility-and and aefe-adheringg user-defined risk thresholds. We experimentally demonstrate RAC's advantages in medical diag- nosis and recommendation systems, showing that it substantially improves the trade-off between safety and utility, delivering higher utility than existing methods while avoiding critical errors.
What is the optimal interface between prediction and action that allows for navigating the trade-off between safety and utility in high stakes applications?
The optimal design of action policy crucially depends how uncertainty is quantified. Among various methods, widely adopted saarrachhssprree by advances in con- formal prediction is to produce prediction sets rather than point estimates. But what exactly are prediction sets good for? Which decision decision-making processes make them the right language for communicating uncertainty? And, given such process, what is the optimal rule for transforming predic- tion sets into actions? To address these questions, first introduce our setting and notation. We consider feature space X' and a label set y, endowed with the distribution (x, u) ~ D. A downstream decision maker has an action set AA and a utility function u A x y + R that maps actions a and realized labels y to utilities u(a, y). which the decision maker seeks maximize. Upon observing x X' the deci- sion maker must take action E AA without observing the true label y, relying instead on predictions about y. Within this framework, we aim to answer the above questions.
1. Introduction
Predictions are frequently used inform actions. For ex- ample, in clinical medicine, patient data used to pre- dict diagnoses and outcomes when choosing treatments. In
DDeaarment of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, USA. Correspondence to: Shayan Kiyani <shayank @ seas. wesas.upenn.edx..
In seeking answers, it is instructive to reflect on what we can say about calibrated forecasts, an alternative way of quanti- fying uncertainty with well- rell-established decision- theoretic
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Decision Theoretic Foundations for Conformal Prediction
Risk Averse Calibration (RAC)
Uncertainty quantification
Decision making
Prediction
Prediction set design
Action policy design
ML model
Action
Pre- Pre-trained
Optimal predictiio Thms. 3.2 a.1
Optimal action policy Prop. 2.2, Thm. 2.3
Safety guarantee Comolary 4.2
Figure RAC pipeline, an interface between prediction and action for stakestakes applications.
foundations- has its own limitations. Suppose we are in multiclass classification setting, and we represent labels y E y' using one-hot vectors in the k-dimensional probabil-
tions to best balance the trade-off between safety and utility, The core challenge in this regard is to develop the right notion of uncertainty quantification for the predictions and optimal action policies based such uncertainty measures.
ity simplex. A forecasting rule X' -> [0, 1] is calibrated if, for every prediction P, we have EEyy f(x) D] p, meaning it is unbiased given the forecast. Then simple consequence of calibration (Foster and Vohra, I 1997; Zhao et al., 2021; Noarov et al., 2023) is that for any expecta- tion maximizing decision maker, choosing the action that would maximize expected utility as if the forecast was cor- rect is the optimal policy amongst all policies mapping forecasts to actions. Formally, if f is calibrated, then applying RR, achieves higher expected utility than any other policy map- ping forecasts to actions. In this sense, calibration is the right language for communicating uncertainty to expecta- tion maximizing-ie. risk neutral-agent,, and the right rule for such agents to ingest calibrated forecasts is to act as if they correct specifications of the label distribution.
We show that prediction sets are the right medium for com- municating uncertainty to risk-averse decision makers who seek nigh-probability guarantees on their realized utility, i.e., the quantiles of their utility distribution as formulated in (1). Specifically, we prove that optimizing action policies to maximize utility while satisfying (1) fundamentally equivalent to designing prediction sets optimally, followed by a simple max-min decision rule. This establishes pre- diction sets as a sufficient statistic for safe action policies, encapsulating all necessary information for risk-averse deci- sion making. We then derive an explicit formulation for the optimal prediction sets, which serves as the foundation for simite-sample algorithm providing distribution- safety guarantees. Put together, these results characterize the opti- mal interface between predictions and actions for risk-averse decision making as depicted in Figure 1. In more detail:
= arg max Eyy (a) [u(a, v)]
In contrast, we seek the right interface between predictions and actions for risk-averse agents. Let a(-): : X' + A be an action policy. We call v(-) : + R a utility certificate it satisfies the following safery guarantee:
1. Max-min decision rule. When given prediction sets C(z) with only a marginal coverage guarantee, risk- averse decision makers should choose their action by maximizing worst-case utility over all labels y C(z). We prove this max-min policy is minimax optimal over all data distributions satisfying the marginal coverage guarantee (Proposition 2.2).
Pr[u(a(X), Y) 2 x(X) a.
(1)
In words, with probability at least 1 a, the utility of an agent following the policy a(x) is guaranteed to be at least ((x). Naturally, we aim to maximize the average value of the utility certificate / subject to satisfying the requirement in (1) 1 i.., as the risk-averse agent, we seek to maximize the average quantile of their utility, commonly referred to as the value at risk in the financial risk literature (see Sec- tion 2 for details on the problem formulation). This objec- tive yields the optimal balance between safety and utility, achieved by finding the pair (a, v) that satisfies the safety constraint while maximizing the average utility certificate.
2. Prediction-set equivalence. policy and utility certificate can be obtained by apply- ing the max-min decision rule to a suitably designed prediction set with marginal coverage (Theorem 2.3). This establishes that prediction sets are sufficient statistic for safe decision making.
The optimal pair of action
3. Optimal design of prediction sets. We formulate Risk Averse Conformal Prediction Optimization (Sec- tion 2.2) to find prediction sets that maximize the target utility quantile under the max-min policy. Using dual- ity theory, we derive an explicit, one-dimensional char-
In practice however, the true probability distribution that connects the actions to their utility values is unknown. In- stead, the decision maker must rely on (uncertain) predic-
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Decision Theoretic Foundations for Conformal Prediction
acterization of the optimal sets (Theorem 3.2), which underpins our finite-sample construction.
significantly more effective action policies.
Risk Aversion in Economics. Decision-making under risk aversion is fundamental in economics, beginning with Bernoulli's expected utility theory (Bernoulli, 1954) and formalized by Von Neumann and Morgenstern's 's axiomatic model (von Neumann and Morgenstern, 1944). Pratt (Pratt, 1964) and Arrow (Arrow, 1965) introduced precise mea- sures of risk aversion (Arrow-Pratt coefficients), while Hadar and Russell (Hadar and Russell, 1969) and Hanoch and Levy (Hanoch and Levy, 1969) developed stochastic dominance criteria. Rothschild and Stiglitz (Rothschild and Stiglitz, 1970) further refined risk comparison through mean- mean-preserve preserving spreads. Recent extensions have addressed robust criteria such as maximin and minimax- under ambiguity (Manski, 2000; 2004; Manski and Tetenov, 2007: Manski, 2011) (see also the recent survey (Royset, 2024)). Unlike these classical frameworks, our approach empha- sizes data-driven learning and distribution-free uncertainty quantification, providing risk-averse guarantees applicable to any black box pretrained model.
4. simite-sample algorithm. We propose Risk-Averse Calibration (RAC) (Section 4), which can exploit any black-box predictive model to derive action policies and utility certificates while providing a distribution- free safety guarantee (1). This guarantee holds for any given utility function.
5. Experiments. In Section 5, we compare RAC with several conformal-predicction methods (Cortes-Gomez et al., 2024; Romano et al., 2020; Sadinle et al., 2019) and best response baselines. Across multiple tasks, such as medical diagnosis, RAC achieves a superior trade-off between safety and utility, delivering higher utility at each user-specified risk threshold.
1.1. Related Work
Conformal prediction (CP), introduced by Vovk et al. (2005), provides flexible framework for constructing prediction sets with sampe- guarantees (Lei et al., 2018; Shafer and Vovk, 2008). Recent research has explored adapting CP to various decision- decision-making problems. Here, we briefly discuss the most relevant works, and provide a thorough discussion in the Appendix A.
Domain-Specific CP Methodologies. Decision making with CP has also been explored in specific domains such as robust optimization (Patel et al., 2024b; Johnstone and Cox, 2021; Yeh et al., 2024), medical tasks (Banerji et al., 2023; Vazquez and Facelli, 2022), power and energy sys- tems (Renkema et al., 2024), formal verification (Linde- mann et al., 2024), and chance-constrainedd optimization (Zhao et al., 2024). While our framework could potentially be extended these settings, each may involve additional domain- -pecifff challenges beyond the scope of this work. Additionally, recent works also explored the application of CP sets in decision making in the context of counterfactual inference (Lei and Candès, 2021; Yin et al., 2024; Jin et al., 2023). We, however, focus on risk averse decision making using prediction sets. In particular, we show that prediction sets are a sufficient statistic for risk averse agents that aim to optimize their value at risk.
Risk Control. A growing line of research extends CP be- yond coverage constraints to control more general risk mea- sures (Lindemann et al., 2023; Angelopoulos et al., 2022; 2021; Cortes-Gomez et al., 2024; Lekeufack et al., 2024). Angelopoulos et al. (2022) propose conformal risk control for risk measures over prediction sets, and Cortes-Gomez et al. (2024) extend this by constructing sets that satisfy coverage while achieving low risk. However, they do not explicitly discuss which actions their sets should inform or how to design these sets to best serve the decision maker. Lindemann et al. (2023) apply conformal prediction to safe planning, and Lekeufack et al. (2024) focus on decisions parameterized by a single scalar, calibrated to control risk. However, they restrict their action policy to predefined low-dimensional family, leaving open the question of how to jointly optimize over policy design and uncertainty quan- tification for risk-averse utility.
2. The Preliminaries of Risk-Averse Decision Making
In this section, we will formalize the central objective of a risk averse decision maker. Recall that in our stetting, upon observing
In this paper, we fill this gap by addressing three core ques- tions for a risk-averse decision maker: (1) What is the correct notion of uncertainty quantification? We prove that prediction sets are optimal for high-stakes decisions. (2) How can we design these optimal sets? We provide an ex- act pwpplationn -lvell characterization and distribution-free finite-sample construction. (3) What is the optimal policy given these sets? We show that a simple max-min rule is optimal for risk-averse utility. In Section 5, we implement Poites-Gomzzzz
X, the decision maker will have to take an action a É A. The decision maker does not observe the true label y, but its utility will depend on both the action a and label y, which is captured by given utility function u.
We focus on risk-averse decision making, where the goal is to choose actions that ensure a sufficiently high utility with high probability over the randomness of the label. That is, risk aversion prioritizes minimizing the likelihood of low- utility outcomes, even at the cost of overlooking higher but uncertain utilities. Formally, given a risk tolerance thresh-
the most recent approach in this direction, et al. (2024) and demonstrate that our framework yields
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Decision Theoretic Foundations for Conformal Prediction
old o, a decision maker facing x E X' assigns each action A a value: (a; quantile, [u(a, Y) X = s]; where Y ~ p(y z). This standard risk measure, known in financial risk literature as Value at Risk (VaR) (Duffie and Pan, 1997), represents the largest value such that, if action a is taken, the utiliity is at least Va (a; with probability 1 L-a a. Thus, the risk-averse decision maker selects the action max- imizing Va (a; z), ensuring the highest guaranteed utility.
where the function a(.) X' + A is decision-policy that maps features actions such that it guarantees average util- ity according to the function vn(): X'' + R with probability at least - a, marginalized over X. Now, rather than opti- mizing over a single value for a and v for each x separately, we jointly optimize over policies a(-) and value functions vt.)² which map X' to actions and values respectively. This results in the following marginal version of the decision maker's optimization problem:
(x) = max Va ()) aεA
(2)
max quantile, [u(a, Y) X = x), Vor E X. aEA
Risk Averse Decision Policy Optimization (RA-DPO):
==
The above risk-averse utility should be contrasted with the best expected utility max, Eju(a, Y) = z². The latter leads to actions that maximize the average utility whereas the former aims to maximize the worst-case utility that can happen with probability! more risk averse at the cost of becoming more conserva- tive. It is important mention that the economic literature extensively explores various other notions of risk aversion, such as Conditional Value-at-Ree at-Risk (CVaR) (see e.g. (Royset, 2024)). However, here we only focus on the aforementioned risk measure, and the exploration of these alternative risk notions remains beyond the scope of this work.
maximize a())
Ex [x]],
subject to Pr Pr(u(a(X)) 2 ((X)) 21 21-a. a.
a. Hence the former will be
Remark 2.1. While our primary focus is on the marginal formulation of risk- averse optimization, one can also con- sider the more advanced setting of group-conditonal valid- ity (Jung et al., 2023; Gibbs et al., 2023). Specifically, for arbitrary groups 9l g1..... 99 S X, the marginal constraint in RA-DPO generalizes to: Pr [u(a(X),Y) 2 v(X) X E 9:] S 1-a, Vi E [m]. Such constraints enable finer control over risk across subpopulations crittcal in applications re- quiring group-specific guarantees. We leave the exploration of this objective to future works and believe our findings provide a principled first step toward that direction.
Marginal Version. The quantity in (2) is a point-wise or conditional quantity; i.e. to find the best action according to (2) the decision maker requires access to the conditional dis- tribution p(y(z)) In practice, such distributions are unknown, and guarantees of the form (2) are often intractable when only a finite sample of the distribution is available. An anal- ogous situation arises in conformal prediction (CP), where obtaining fully-conditional coverage guarantees known to be impossible from finite sample of data. Consequently, conformal prediction focuses on relaxed, i.c. marginal (or "group conditional", which still marginalize over part of the distribution (Bastani et al., 2022; Jung et al., 2023)) coverage guarantees which are statistically tractable.
2.1. A Prediction Set Perspective
Recall that in our setting the (feature, label) pair is gener- ated according to a distribution. The decision maker only observes the feature x based on which it will choose its action a. However, the realized utility will depend on both the action and the label y. The decision maker does not observe the label, but we assume that it has access to a predictor that provides predictions about the label given the input feature x. More specifically, we assume that the predictor will provide prediction sets of the form C(x) C V, x E X, that are guaranteed to contain the true label with high probability. We assume that the prediction sets satisfy the marginal coverage guarantee, i.e..
By analogy, we will now introduce the marginal version of (2). First we rewrite the objective. For given x E X, the value u. (z) in (2) can be equivalently written follows
Maximize V
REA,EER
subject to
Pr|u(a, Y) 2 p x=
Pr ( E 2(x))2100 1 (X
2
(4)
Let us examine the constraint in the above optimization more carefully. We are looking for action-value pairs (a, v) such that we guaranteed with probability at least 1 - a that, when taking action a, the resulting utility is at least v. Of course, to maximize utility, we should maximize over the choice of the action a and the value e which results in the above optimization. Now, the risk-averse constraint in the above optimization has the following marginal counterpart:
Given this framework, two immediate questions arise: (i) Assuming the only information that the decision maker has about the true label is through the prediction sets, how should choose its actions to maximize (risk-averse) util- ity? (ii) How should the prediction sets be designed to not
In this paper, we focus on deterministic action policies. 'Here, note that since v(x) is a utility function, its value can not be larger than the maximum achievable utility; i.c. x(x) S Umax = max, max, u(a, y) for all x E X.
Pr|u(a(X), Y) 2 x(X)) 2 1 a,
(3)
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Decision Theorctic Foundations for Conformal Prediction
only be marginally valid according (4) but also maximize the utility achieved by the decision maker?
2.2. An Equivalent Formulation via Prediction Sets
In the previous section we argued that, when deciding based prediction sets, the (minimax) optimal policy aRa and its associated value VRA are given in (5). Hence, assuming that the decision maker is playing @RA. the prediction sets C(x) should be designed to maximize the resulting utility of the decision maker while ensuring marginal coverage; Le., the following optimization:
We will proceed with answering question (i) now, and will provide an answer to question (ii) in the subsequent sections. Assuming that the decision maker can only take actions based on the prediction sets i.e. it has no other information about the label distributions then its optimal decision rule takes a simple and natural form. It will have to play the action that maximizes their utiliity u(a, y) in the worst case over labels y E C(x). We denote this optimal risk- averse (RA) decision rule by ARA : 23 + A, and the corresponding utility certificate by VRA 2 + R:
Risk Averse Conformal Prediction Optimization (RA-CPO):
Maximize Ex PRA (C(A) Ex max min u(a,y) CG)
@RA( (C(z)) = arg max min u(a,3) y): A vEC(x)
(5)
aEA BEC(X)
subject Pr[Y C(X) 2 a1-a
PRA (C(z)) max min u(a, y). aEA vEC(#)
(6)
One might expect that the result of RA-CPO, i.e. optimiz- ing the utility using prediction sets, would lead to lower utility compared to the original optimization RA-DPO. This is because: (i) The policy given in (5) is a specific policy designed to be valid even for the worst-case distribution for which the prediction sets are marginally valid (see Proposi- tion 2.2). Hence, this policy could be overly conservative; (ii) In RA-DPO the optimal action and value functions are obtained assuming full information about the data distri- bution, whereas in RA-CPO we require that information must be filtered through a (properly designed) prediction set representation. One might expect a- a-priori that passing from the actual distribution to lossy prediction set repre- sentation would discard information that is critical to finding the optimal policy. However, the following theorem shows, perhaps surprisingly, that this is not the case; the optimal action policy for any distribution can be represented as a max-min rule over a prediction set.
We will show that this decision rule is minimax optimal over the set of all distributions that are consistent with the marginal guarantee (4). Assume that the decision maker is given access to a set function C : X' + {2-).. Let us also define 52 as the set of all the data distributions that are consistent with the marginal guarantee; i.e. the set of all distributions P over (x) such that, Pr(x,y)~p] Y E C(X)]= 2 1-a. Let Letx() (-)) 2 + A be a policy that takes as input the prediction set C(r) and outputs an action. Aligned with R. RA-DPO, the value of policy π with respect to joint distribution p(z,8) y) can then be defined as:
V² (T = maximize Ex-p(s) [x(x),, v(-)
subject
Pr x-Y-p(a.y)
(M(((C()))) v()) 21-a a.
We are now interested in the policy that is minimax optimal meaning that it can perform well with respect to the worst case distribution in Ω. That is to say we want to find the policy ग" that is the answer to,
Theorem 2.3. RA-DPO and RA-CPO are equivalent. In other words, from any optimal solution of RA-DPO, de- noted by (a" (z), (xx)), we can construct an optimal solution C*(x) to RA-CPO with the same utility, i.e. Ex [RRA (C* (X))] = =xx [X)( Also, from any opti- mal solution of RA-CPO we can construct an optimal solu- tion for RA-DPO with the same utility.
Maximize Minimize v p) pelt
Proposition 2.2. Assume < 0.5 and let n² (x) be the optimal solution to (7). Then we have,
Implications. Prediction sets are a fundamental object in risk averse decision making. In particular, the optimal strat- egy of a risk averse decision maker can be formulated as playing a max min strategy over a well-designed prediction set. To fully characterize such optimal policies, the first step is to derive the optimal solution to RA-CPO.
' (x) arg max min u(a,v) v). A weC(w)
(8)
To summarize, Proposition 2.2 states when the risk averse decision maker decides based on prediction set C, that con- tains the actual label with high probability, there is a simple, yet minimax optimal policy, ARA (C(x)) that guarantees the minimum utility of PRA (C(x)) with high probability. We now focus on how to design prediction sets that would be the most useful for the decision maker among all the prediction sets that provide valid marginal guarantee.
3. The Optimal Prediction Sets
We characterize the optimal solution (i.e., prediction sets) for RA-CPO given in (2.2) in terms of the conditional distri- bution p(y) x)). We begin by introducing the fundamental
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Decision Thcoretic Foundations for Conformal Prediction
proteds by
wun of godt 21
have coverage at least i.e. Pr(Y C( = = 1 the following set has the largest risk-averse utility value (RA(C)= = =aaxaeA minyeC u(a,
action 011
()
abbuii ois dil sloo sol sioua0
pictabiity
C(rr, ((cr.))=$000 { YEV: u(a(x; (), v) 2 0(0, 000.1). (11)
action 1
*Ne St2
Miso bJ allg (hb
Further, we have URA (C(a, ()) - 0(z, t).
probacility
pooto
O
The optimal sets for RA-CPO (2.2) can now be obtained based on the following re-parametrization in terms of the coverage probabilities that we assign to each E X. In order to satisfy the marginal constraint of RA-CPO, we will need to assign to each I, a coverage value t(x) such that Ex [M(X) 2 1 - a. From the above proposition, if an instance x is assigned with t units of (probability) coverage, then it can add the maximum utility amount of ((x; t) to the objective and its corresponding prediction set, which is optimal given t units of coverage assigned to x is given in (11). Hence, to find the optimal prediction sets we should find the assignment t(x) which optimally distributes the (1 - a) units of probability over the feature space X, such that the expected utility is optimized. This step is captured by the following equivalent reformulation of RA-CPO:
action Ra:
S/Pt-/007 sint Ub.
0)x,1) - maxe(G May (
This
(( = 03
((x. ang BRR (0) + St)
Figure 2. Left: Illustratoon of how the functions O and are com- puted for a given æ E X and é E [0. 1]. Here we have three actions A = {a1, (2, 0a} and four labels y = (vh, M2, Va, ya J. We also let P. := p()](z). For cach the actions ay, the value way is the (1 ()-quantile of u(a, Y). The value (zz, () corresponds to the maximum of these quantiles among the actions, and a(x, i) corresponds the maximizing action. Right: Illustration of how the function gzz, B) is obtained from (zr, f) for a given x.
notions that relate optimal utility to coverage. We define the functions 0: : X' x [0, 1] + R and a: X' x [0, 1] + ass
maximize =
Fx [x(,,((
(12)
subject to: [x[((x] 2 1 1-a.
((z, ()= max quantile [u Y) X = =r], a
(9)
(2((a, a(x, t)- = arg max max quantianllee u(a, Y) X = z/. (10)
Once the optimal solution (x) to the above re- parametrization of RA-CPO is found, then the optimal pol-
aEA
icylactions, denoted by (x(x) = a(x, ((z,(((z) (x)), are derived according to (9), and the optimal prediction set is given by:
In words, given feature x E X' and probability coverage value 1. E [0. 1]. (zr, t) is computed as follows (see also Figure 2): For each action a, we first find the (1-ti-quantile of the random variable u(a, Y) with Y being distributed according to p(ylæz). This quantile value is the largest utility achievable with probability at least & when we take action a. By maximizing such (1 -)quunitse over the choice of the action
C' (x(x) = y yey: (x)))))))) 012, (()) (13)
Let summarize what we have done so far: We proved that RA-DPO (2) and RA-CPO (2.2) are equivalent. Then, to solve the RA-CPO we used reparametrization as in (12), which we will now solve.
obtain tz), In words, for x X, the value (zr t) represents the optimal (risk-averse) utility achievable under a conditional coverage assignment f, and the maximizing action is denoted by a(x; l).
Using tools from duality theory, we can show that the op- timization problem (12) admits solution with a simple "one-dimensional" structure in terms of scalar parameter 8 R and assignment function g X x R + [0.1] defined as
Let us now explain how the function (z, t) plays a role in finding an optimal solution for RA-CPO. Fix an instance x, assume that we would like assign conditional coverage probability t to For the specific instance x, we would like to construct prediction set C(x) that with coverage at least t, i.e. Pr(Y E C(x) X xx=r) r) 2 t, where the probability is over the conditional distribution p(ylæ)). We ask: How should C(x) be designed to maximize the objective of RA- CPO? The following proposition provides the answer.
g arg max {0(I,)) + Bs}. SE [.11]
(14)
An illustration of the function (2r () is provided in Fig- ure 2. One can observe that gzz. is connected to the convex-conjugate transform of the function 0 0(x, +).
Proposition 3.1. Fix an instance 2 E X and a coverage value E [0, 1). Then, among all the sets C. C y that
Forr simplicity, we assume in this section that the maximizer of ((2, s) + Bs is unique with probability 1.
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Decision Theorctic Foundations for Conformal Prediction
Theorem 3.2. Assume that the marginal distribution of X continuous. Then, optimal solution of (12) has the form
Algorithm Risk Averse Calibration (RAC)
Input: Miscoverage level a, calibration samples [(x, YD test covariate Xn+1- For each E V: solve
t' (x) g(x,
for a value 3 E R. Consequently, the optimal prediction sets for RA-CPO are obtained using t" (x) from (13). Fur- ther, the value of 3 is a solution to the following equation a.
By arg min B SER
1 E ((x:)))
s.t.
in terms of the scalar 3: Ex (0,,,))))))))))
n+1
The main implication of the above theorem is that it provides a simple characterization of the optimal sets given access to the data distribution: (i) Find the scalar 3* that satisfies Ex (X,, (x) := (2, (*)) from (14); (iii) The optimal prediction set for x. C* (x), is then given by (13).
+ 1 E ((Xmi1:01) 21 -
Output: Compute
as (ii) For each E X' compute
CRAc(Xn+1)= =
Ey E y C(X++1: By)}
The scalar characterization via 3 is particularly useful when only approximate conditional probabilities are available. By substituting ply æ)) with an approximation in all defini- tions, we can still apply Theorem 3.2 to find a that ensures valid coverage for the corresponding prediction sets. This simple scalar calibration then yields prediction sets whose lis-averse utility are improved (and eventually becomes op- timal) as the quality of the estimated probabilities improves.
From the result of Theorem 3.2 we know that the optimal prediction sets admit "one-dimensional" structure in terms of the scalar parameter 3 E IR, and the optimal conditional coverage assignment is derived using the function g(r, 3). Hence, to simplify notation, we analogously define
(1r, B) 0a,, ((tx,)) a(x,8) = a(æ, (0x B)).
Following (13), the prediction sets take the form
4. The Main Algorithm: Risk Averse Calibration (RAC)
C( B) = { yey: u (ā(z, B). 0(z, 20,, 31
In Section 3, we derived the structure of the optimal predic- tion sets for the RA-CPO problem. These sets are defined by the functions 0 (xx t) and axx, t) given in (9), which funda- mentally relate coverage to utility and actions, as well as the assignment function g ()) introduced in (14). These quan- tities are defined based on the true conditional distribution which is often unknown in practice.
We can now present main algorithm.
Theorem 4.1. Assuming that the calibration data {(,,, Yi) and (Xn+1, are exchangeable, we have
Pr [Ym+1= E CRAC(Xn+H))
2 1-a
Put differently, Theorem 4.1 states that the prediction sets constructed by RAC have the so-called property of distribution-fiit coverage guarantee. Recalling the detini- tions (5), we can now state the following corollary.
In this section, we consider the finite-sample setting. We assume access to calibration samples {(X,, ?) and a predictive model f X + Ay. which assigns a | dimensional probability vector to each æ E X. The output fz represents approximate label probabilities, such as those from pre-trained model's softmax layer. We denote by fy)) the probability assigned to label for input x.
Corollary 4.2. Assuming that the calibration data ((X,, Y and (Xn+1; Yn+1) are exchangeuble, we huve
Pr [u(aRA (CRAc(Xx+11). Yn+1) 2 PRA(CRAC(A.)
Using the model f. we will estimate the functions 0, a, and g. defined in (9) and (14), by substituting the true con- ditional probabilities with their estimated counterparts ob- tained via f. Concretely,
2 21aa.
Putting the pieces together, Corollary 4.2 ensures that a simple max-min decision policy over RAC-constructed pre- diction sets provides a pair of action policy and utility certifi- cate, namely aRA a (CRac(Xees) and VRA (CrAc (Xtest) )), providing a distribution free safety guarantee according to (1). Moreover, Theorem 3.2 highlights RAC's practical rele- vance in terms of exploiting the predictive model. Specifi- cally, RAC's utility performance depends on the quality of (17) the predictive model f: if f closely estimates the true con- ditional probabilities, then the model-band definitions in
(xx t) = max quantile Y ~ fa]. a
(15)
á(z,t) arg max quantile Y) Y ~ fu]. (16)
A
and
g(2, B) arg max {0(z,)) + 35]
0.

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Decision Theorctic Foundations for Conformal Prediction
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Medical Diagnosis Experiment
MovieLens Recommendation Experiment
Figure 3. Results from two experiments. (a) Average realized max-min value as function of a. (b) Fraction of wrong critical decisions: medical diagnosis, severe omission appropriate care (e-g., failing to act on pneumonia or COVID-19 cases); in MovieLens, the percentage of movies rated 1 or that were incorrectly recommended. (c) Average realized utility. (d) Realized miscoverage.
(15) and (17) approximate their true counterparts in (9) and (14), ensuring that RAC-informed decisions align closely with the optimal ones, as guaranteed by Theorem 3.2.
trading off average utility against the avoidance of catas- trophic errors. We compare in terms of safety and utility using the following metrics: (a) Average realized max-min value: The mean of the worst-case utility across the pre- diction sets (i.e., the average of VRA (5)). (b) Fraction of critical mistakes: For samples with a critical ground- truth label, we report the fraction of cases in which each method chooses the worst action. (c) Average realized util- ity: The empirical mean of the realized utilities across all test samples. (d) Realized miscoverage: The fraction of test samples for which the true label is not in the prediction set.
5. Experiments
In this section, given a pre-trained model which assigns probability (x) to input-label pair (x., y), we compare RAC with two groups of baselines:
Calibration +
We calibrate the model on the calibration data using a strengthened version of decision calibration (Zhao et al., 2021), specifically the variant from (Noarov et al., 2023), which provides swap regret bounds. We then apply the best-response policy: best-response(z)
5.1. Medical Diagnosis
arg max.e.A (x) [u(a, v)]- While this method may achieve higher average utility, it fully trusts the model and is prone critical errors.
In this experiment, we explore decision making in medi- cal diagnosis and treatment risk-sensitive application. We use the COVID-19 Radiography Database (Chowdhury et al., 2020; Rahman et al., 2021), containing chest X-ray images of four classes: Normal, Pneumonia, COVID-19, and Lung Opacity. The data randomly split into training (70%), calibration (10%), and test (20%) sets. We then fine-tune Incept inceptioovv on_v3 model (Szegedy et al., 2015; 2016) (pretrained by google on ImageNet) by retraining the higher layers, while preserving the early-layer features.
Conformal Prediction + Max-Min. We construct (1 a) -v-valid prediction sets using split conformal prediction with three different scoring rules. The decision policy then applies the max-min rule from Section 2: aRA (C(x)) = arg max minyeC(s) u(a, y), which we proved is the optimal strategy when deciding based prediction sets:
score-1 (Sadinle et al., 2019): 1 fa(y).
capture clinical priorities, we employ the utility matrix in Table 1, which maps each true condition (row) to set of actions (column). Although we use the specific matrix below, setup accommodate any alternative design. (Further details on the Al-assisted construction appear in the Appendix C) All the baselines then will be calibrated to connect model's predictions to these four actions.
score-2 (Romano al., 2020): fo)
y J. [v())()()))
score-3 (Cortes-Gomez et al., 2024): greedy scoring rule tailored the max-min policy.
By varying a, we can control the degree of conservativeness,
8

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Decision Theoretic Foundations for Conformal Prediction
True Label
No Action Antibioties Quarantine Testing
Compatison of My Pots
(deg villt 68)
Mivsing en some of the lighest ukily actions os result of reducing the
Pneumonia
Normal
10
0
10
2
3
2
4
7
0.-
BAC 2.00 Jilly 7.522
RAC Jtllty 7.46
7.291
COVID-19
0
3
10
8
Lung Opacity
4
4
10
Reserting conservative actions
Table 1. Utility matrix for the four-class chest X-ray task.
Reducing the critica errors
10
After training, we vary the nominal miscoverage parameter a during calibration to study its impact on performance. As shown in Figure 3()), our method achieves the best trade- off curve among baselines, providing higher worst-case utilities for every nominal Equivalently, it offers stronger utility certificates at each high-pni croh-brbbbbblitty threshold. In Figure 3(c), it also consistently outperforms other prediction set-based methods in terms of average utility.
Usiby
Figure 4. Comparison of the full utility distribution for different variants of RAC and the best-response method.
Table 2: if a movie with true rating y is recommended, the utility is y - 3, while not recommending yields O. We vary
While the best-response method attains the highest overall average utility, Figure 3bb highlights its susceptibility to critical mistakes. For example, in COVID-19 cases, best- response chooses no action over 60% of the time, recom- mending wrong treatment on large fraction of patients with 19. Our rvsk-everse policy (RAC) drive this error rate below 10% (at a = 0.02), incurring only a mod- est (under 5%) drop in average utility. Finally, Figure 3(d) baselines achieve their target miscoverage levels, ensuring the associated high- probability utility guarantees remain statistically valid. Ad- ditionally, in Figure 4, we also report the full distribution of the utility of the actions made by RAC for different values of alpha. There, it is even more clear that as we increase 1 - a, RAC avoids the extremely low utility actions at the cost of missing on some of the highest utility ones, by resorting to conservative decisions.
Action
1
2
3
4
5
Not Rec
0
0
0
0
0
Rec
-2
-1
0
+1
+2
Table 2. Utility matrix for the Moviel.nns recommendation task.
the nominal miscoverage a during calibration and measure performance test data. As shown in Figure 3(a), our method achieves the best trade-off among baselines, offering stronger utility certificates (worst-case utility) at all a levels. Figure 3(c) also shows that our approach outperforms other Ca-based methods in average utility.
confirms that all prediction-set-baee
A Although the best-response method achieves the highest overall average utility, Figure 3(b) reveals its vulnerability to "critical mistakes" frequently frequell" recommending movies rated 1 or 2. Such failures can undermine user trust and harm companies policy in keeping their customers. In contrast, RAC (a = 0.05) cuts these critical errors by 75%, while incurring only a modest (15%) reduction in average utility.
The plots reported in this experiment can serve broader pur- pose beyond evaluation: they provide a practical interface for choosing the right level of risk aversion in real-world deployments. By inspecting trade-offs across different a values, e.g. by looking at plots similar to Figures and 4, practitioners can tune the system to their needs-for in- stance, favoring safety over utility in high-stakes settings like medicine, or vice versa in lower-risk applications.
6. Discussion and Future Work
In this paper, established the pacision-thhoretii founda- tions of conformal prediction, showing that valid prediction sets act as sufficient statistics for risk-averse agents opti- mizing their value at risk. We developed an algorithmic interface linking predictions from any black-box model to actions with marginal, distribution-fee safety guarantees.
5.2. Recommender Systems
We next consider a risk-sensitive recommendation scenario using the MovieLens dataset. Each data point is a user- movie pair (zz (user features, movie features), y). where the label y E {1, 3, 4, 5) is the user's rating. We split the data into training (80%), calibration (10%), and test (10%), and train neural network classifier f (details in the Appendix) to estimate the probability distribution (z).
Although our focus has been primarily on marginal safety guarantees, we acknowledge the practical importance of stronger conditional guarantees. These include group- conditional (based on covariate characteristics), label- conditional (based on true labels), and action- conditiion safety (based on chosen actions). Extending our results systematically to these more nuanced scenarios presents promising directions for future research.
At test time, the policy must decide whether to recommend or not recommend a movie. We use the utility function in

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Decision Thcoretic Foundations for Conformal Prediction
Acknowledgments
M. E. Chowdhury, T. Rahman, A. Khandakar, R. Mazhar, M. Kadir, Z. B. Mahbub, K. R. Islam, M. S. Khan, A. lqbal, N. Al Emadi, et al. Can ai help in screening viral and covid-19 pneumonia? leee Access, 8:132655-132676 2020.
This work was supported by the NSF Institute for CORE Emerging Methods in Data Science (EnCORE) and NSF grant FAI-2147212. The authors wish thank John Cherian, Natalie Collina, and Bruce D. Lee for helpful discussions.
S. Cortes- Comte, C. Patiñi,, Y. Byun, S. Wu, E. Horvitz, and
B. Wilder. Decision-focusee uncertainty quantification. arXiv preprint arXiv:2410.01761 767, 2024.
1. Demirel, V. Nguyen, and A. Krause. Escada: Efficient safety and context aware dose allocation for precision medicine. In Advancey in Neural Information Processing Systems, vol- ume 35, pages 1-12, 2022. URL https:/ pestps://arxiis /arxiv /
Impact Statement
We developed a decision theoretic framework for conformal prediction. We do not anticipate negative societal impact.
abs/2111, 13415.
D. Duffie and J. Pan. An overview of value at risk. Journal of derivatives, 4(3):7-49, 1997.
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Decision Theoretic Foundations for Conformal Prediction
A. Extended related works
The foundational idea of prediction sets can be traced back to early studies by Wilks (1941); Wald (1943); Scheffe and Tukey (1945); Tukey (1947). The initial concepts of conformal prediction (CP) were introduced in Saunders et al. (1999); Vovk et al. (1999; 2005). With the advancement of machine learning, conformal prediction has become a widely adopted framework for constructing prediction sets (Vovk, 2013; Papadopoulos et al., 2002; Lei et al., 2018; Romano et al., 2020; 2019; Park et al., 2022; Angelopoulos et al., 2020). There has been growing body of work aiming to adapt conformal prediction methods for a range of decision decision-making problems, In the following, we will discuss the ones relevant to the present work.
Risk Control. A growing line of research extends CP beyond coverage constraints to control more general risk measures (Lindemann et al., 2023; Angelopoulos et al., 2022; 2021; Cortes-Gomer et al., 2024; Lekeufack et al., 2024; Zecchin and Simeone, 2024a; Blot et al., 2024; Zecchin and Simeone, 2024b). In particular, Angelopoulos al. (2022) propose conformal risk control for monotone risk measures over prediction sets, and constructing sets that satisfy coverage while achieving low risk. However, these works do not explicitly discuss which actions their sets should inform or how to design these sets to best serve the decision maker. Moreover, Lindemann et al. (2023) applies conformal prediction to safe planning, and Lekeufack et al. (2024) focuses on decisions parameterized by a single scalar, calibrated to control risk. However, these works restrict their action policy to a predefined low-dimensional family, leaving open the question of how to jointly optimize over policy design and uncertainty quantification for risk- lisereeerse utility.
et al. (2024) extend this by
In this paper, we fill this gap by addressing three core questions for risk-averse decision maker: (1) What is the correct notion of uncertainty quantification? We prove that prediction sets are optimal for high-stakes decisions. (2) How can we design these optimal sets? We provide an exact population-level characterization and distribution-fred finite-sample construction. (3) What is the optimal policy given these sets? We show that a simple max-min rule is optimal for risk-averse utility. In Section 5, we implement the most recent approach in this direction (Cortes-Gomez et al., 2024), and demonstrate that our framework yields significantly more effective action policies.
On top of the fundamental differences mentioned, there are also technical differences. After proving the equivalence of the risk-averse objective defined in Section 2 to the prediction set optimization called RA-CPO in Section 2.2, one might think we can define a risk function of the form (CC) = max minyeC(a) u(a, y). and then apply risk controlling methods to control this risk. However, controlling this risk alone is meaningless, as it is always possible to control the risk by outputting trivial sets. Hence, the risk should be controlled combined with coverage guarantees. The only risk controlling framework that additionally allows for a coverage constraint is the work of Cortes-Gomez et al. (2024), which we compare our performance with in Section 5, and show our superior performance in handling the safety utility trade-off. Furthermore, the defined loss function 7 for generic utility function u, lacks any (approximate) separability property or sub- uub-oouuarrity, which are essential for algorithmic development of Cortes-Gomez et al. (2024). We, however, work directly with the max-min objective and do not rely on any assumptions. For readers familiar with nested conformal prediction (Gupta et al., 2022), perhaps another way to elaborate on this important technical difference is to look at Section ??, where in Theorem 3.2, we characterize the optimal prediction sets over the population. It is clear then that the optimal sets do not necessarily form nested sequence of sets as we sweep the miscoverage threshold a. This is in contrast to when we want to find optimal sets corresponding to minimum average prediction set size (or any other separable objective). There, the optimal characterization is of the form p<y]=) 2 q (or more generally of the form s(x, y) S q for some score function s), where q is tuned to satisfy the marginal coverage constraint (Lei et al., 2013; Sadinle et al., 2019; Kiyani et al., 2024b). This distinction hints to the sub-optimality of the algorithms that rely on monotonicity properties of the risk, e.g. thresholding a score function, in obtaining the best risk averse action policies and safety guarantee.
Robust Optimization. The max- min policy that we will discuss in Section 2.1 also naturally arises at the intersection of uncertainty quantification and robust optimization (Patel et al., 2024b; Johnstone and Cox, 2021; Chenreddy and Delage, 2024; Li et al., 2025; Yeh et al., 2024: Cao, 2024; Wang et al., 2023; Lou al., 2024: Patel et al., 2024a; Elmachtoub al., 2023; Lin et al., 2024; Chan et al., 2024; 2023; Chan and Kaw, 2020). In robust optimization, decision-making under uncertainty is typically formulated as minimax problem, where optimal decision is sought against worst-case realizations within an uncertainty set. Despite a structural resemblance of these works to our framework in that they involve optimization over an uncertainty set, their scope and objectives have some fundamental differences from ours. We fix any black-bor box predictive model and any utility function, and in contrast to existing approaches, we jointly characterize the optimal notions of uncertainty quantification and action policy. Specifically, we ask: (1) What is the appropriate uncertainty quantification
13

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Decision Theoretic Foundations for Conformal Prediction
for risk-averse decision makers? We answer that prediction sets are optimal for achieving high-probability utility guarantees. (2) How should these prediction sets be optimally constructed? We provide distribution-free, finite-sample construction that characterizes the optimal sets, (3) What is the optimal decision policy given these sets? We prove that the max-min rule is provably optimal for risk averse agents. In doing so, our Risk-Averse Calibration (RAC) method offers a principled alternative to uncertainty sets based on heuristic conformity score designs, thereby contributing to the growing intersection of conformal prediction and robust optimization. Additionally,, on a more technical note, in Section ??, we show that the optimal prediction sets that lead to optimal safe action policies when used tandem with the max-min rule do not necessarily take the form of thresholding a score function (i.e., s(r, y) S 9 for some score function s). There, we characterize an alternative form that, in fact, captures the optimal prediction sets in the context of risk-averse decision-making That is to say, our results hint to principled alternative to conventional score-based prediction sets in the pipeline of robust optimization to avoid suboptimality.
Risk Aversion in Economics. Decision-making under risk aversion is a foundational topic in economics, shaped by seminal contributions. Bernoulli (Bernoulli, 1954) introduced expected utility theory, explaining risk aversion via diminishing marginal utility. Von Neumann and Morgenstern (von Neumann and Morgenstern, 1944) formalized this with an axiomatic model of rational choice under uncertainty. Pratt (Pratt, 1964) and Arrow (Arrow, 1965) developed the Arrow-Pratt coefficients, providing precise measures of risk aversion and distinguishing between increasing and decreasing risk sensitivity. Hadar and Russell (Hadar and Russell, 1969), along with Hanoch and Levy (Hanoch and Levy, 1969), introduced stochastic dominance to rank risky alternatives for risk-averse agents. Rothschild and Stiglitz. (Rothschild and Stiglitz, 1970) deepened this framework by defining mean-preserving spreads, a formal notion of increased risk. More recent extensions introduced robust decision-making making criteria, such as maximin and minimax-regred applicable under ambiguous or unknown probabilities (Manski, 2000; 2004; Manski and Tetenov, 2007; Manski, 2011). Collectively, these works established the theoretical underpinnings of risk aversion that continue to influence modern economic theory (for a recent survey look at (Royset, 2024)). In contrast these works, our work focuses on data-driven learning and uncertainty quantification aspects of the risk averse decision making. We develop distribution-fii methods capable of leveraging any black-box pretrained model, accompanied with risk aversion guarantees.
Further Related Work. The potential connection of CP ideas to decision making has also been explored in Vovk and Bendtsen (2018), from the point of view of conformal predictive distributions. Conformal predictive distributions produce calibrated distributions rather than prediction sets see e.g. (Vovk et al., 2017; 2018; 2020). Therefore, they are best to be compared with calibrated forecasts as the methodologies developed in Vovk and Bendtsen (2018) are also targeting expectation maximizer-i.e. risk neutral- agents. Additionally, recent works also explored the application of CP sets in decision making in the context of counterfactual inference (Lei and Candès, 2021; Yin et al., 2024; Jin et al., 2023). We, however, focus on risk averse decision making using prediction sets. In particular, we show that prediction sets are a sufficient statistic for risk averse agents that aim to optimize their value at risk.
Alternatively, Bayesian methods for risk-averse decision-making often employ Gaussian Processes (GPs) to optimize measures like Value-at- at- Ri-R and Conditional Value-at Ris-Riik; e.g. look at (Sui et al., 2015; Nguyen et al., 2021; Demirel et al., 2022: Cakmak et al., 2020; Lin al., 2022; Baudry et al., 2021; Jia al., 2024). These approaches rely on accurate Bayesian posterior distributions, thus implicitly assuming well-specified probabilistic models. Our conformal approach complements rather than competes with Bayesian methods: our theoretical results (up to Section ??) can be directly employed even in Bayesian settings. In fact, when Bayesian approximations are reliable, one can take advantage of our optimal prediction sets derivation in population, and then calibrate the prediction sets with finite sample under Bayesian models, without employing the finite-sample calibration of Section 4. Alternatively, even when Bayesian assumptions' precision is uncertain, one can still start from Bayesian posteriors and further calibrate prediction sets using our approach, ensuring robust safety guarantees.
Although our primary aim is to develop general framework to construct prediction sets for high-stakes decision-making, we note that conformal prediction sets have been explored in wide range of specific applications and domains of high-stakes nature, For instance, CP methods have been adapted and used in medical tasks (Banerji et al., 2023), power and energy systems (Renkema et al., 2024), formal verification and control (Lindemann et al., 2024), laance-constrained optimization (Zhao et al., 2024), and more generally Sun et al. (2024); Ramalingam et al. (2024); Kiyani et al. (2024a); Straitouri et al. (2023); Vishwakarma et al. (2024); Kiyani et al. (2024b); van der Laan and Alaa (2024); Noorani et al. (2024). Our framework could potentially be extended to these domains, yet each may present additional, domain-specific challenges that lie beyond the scope of this work.
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Decision Thcorctic Foundations for Conformal Prediction
B. Proofs
B.1. Proof Proposition 2.2
We prove that the risk-averse decision rule
@RA (C(x:) = arg max min u(a,y) y) aeA yEC(a)
solves the minimax problem in (7).
Part I: Upper bound for any arbitrary policy. satisfying
Let (() 2 - A be any policy, and let C(-) be fixed set function
Pr Y C(X) 2 aa. (x-Y)-p
We construct "worst-case" distribution in Ω for Pick any z E 2' for which C(r) # 00. Define distribution p" (z, y) by
{
for some y E arg sec)) (TT((()))
(x = = 1, pi(Y = y x = =) =
otherwise.
Under p", we have Y C(X) almost surely (since ((x) is nonempty and we place all mass on a label in C(x)). Hence p" Ω because the marginal coverage constraint
Pr (X,Y)ap'
(Y C(X) > 1
is satisfied. But under this distribution, the utility of π (C(x)) is forced to be
min vEC(a)
(=)(C())).
since Y chosen (with probability 1) to be the worst-case label within C(x). Thus, for this specific æ, no matter how we choose æ, its achievable value is at most minyec(() u (T((C)))) v). Also,
min u (=((C(r)) v) I max min u(a, y), μеC(w)
aEA gEC(2)
Because x was arbitrary (among those with C(x) + 0), repeating the same argument for each such æ yields
inf v" (T, p)
inf max min u(a,)) v).
pEn
C( aEA yEC(2)
In other words, any policy π cannot achieve value larger than the above infimum for the inner minimization in (7).
Part 2: Achievability by the max min policy. Next, we show that the policy
T" (C(x)) = arg max min u(a, y)
a gEC(x)
matches the upper bound from Part and thus minimax optimal. Consider any p E Ω.
Define
v(x) = max min u(a, v).
a vEC(z)
For those & E X' such that C(x) is empty put v(x) = max,5.4 maxyey u(a, y). We claim that with probability least
1 a, the policy ARA (C()) achieves utility at least v(x). Indeed, on the event (Y C(X)) (which has probability
least 1 by assumption), it holds that
((nn(C(())) r)
min a ((nA(C(X)) = v(X).
2
yCC(X)
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Decision Thcorctic Foundations for Conformal Prediction
Thus, setting the target utility each x to v(x) satísfies
Pr (((xn(((()),))) 2 v(x) 2 a-a.
(x
By definition of v² (s,), this implies
1/ PR
E Ex-p [(X]
Ex Ex-p
max
min SEO(X)
u(a.y)]
inf max min u(a.y).
2. C(a) aEA yEC(a)
Since p E Q was arbitrary, we have shown
inf v' (P
max min u(a.z). ((a) d sEC(a)
inf
PEn
Comparing with the upper bound in Part 1 establishes that aRa attains the best possible (minimax) value. Hence
(x) (RA(C((F) arg max min u(a,y) aEA yEC(a)
solves the minimax problem (7).
B.2. Proof Theorem 2.3
We give a constructive proof by showing how from each solution of RA-DPO we can construct a feasible solution of RA-CPO without losing any utility, and vice versa. By applying this to the optimal solutions both problems, we obtain the result of the theorem.
(I) From RA-DPO to RA-CPO. Suppose we have an feasible solution (a(-), to the RA-DPO problem. Consider pair (())),())) such that a: X' + A and X' + 10, tmax]. Here, we have Umax maza maxy u(a, y). and mentioned in Section 2, since v is utility certificate its value at any x should be less than Umax- Since (a, v) is feasible solution of RA-DPO, it satisfies the following:
Prx,y [u(a(X),Y)
a.
Define prediction set
C(x) {ur u(((x)))) 2 v(x))
(18)
In words, C(x) is the set of labels y for which the utility u u(a(z)))) 3) is at least v(x). By definition, have
Pr[Y E C(x) =| = Pr(u(a(X), 2 ((x) =
As result, have
Pr[y'e(X)] = Ex [Pr[Y E C(X) |x]
= Exx [Pr[u(a(X),, V) 2 v(X) x]]
Prr(u(a(X), Y) 2 v(x)
2 l
Hence, C(-) satisfies the marginal coverage constraint of RA-CPO.
Next, we will improve the prediction sets C to new prediction sets Ō which satisfy the marginal guarantee but can potentially have larger value under the objective of RA-CPO. The basic idea is to consider points x E X' such that C(x) is empty and augment an additional element to those empty sets. Recall that we defined Umax := max,e maxyey u(a, y). Hence, there exists least one (action, label) pair, which we call (amaxx 3max) such that umax = u maamaxx Bmax). Now, let define
Xempty {xr E X C(x) = =0},
where 0 denotes the empty set. We now update C(.) to Õ(-) as follows:
if x E Xempty
Č(x) = {ymaa )
if æ & Xompey
Ö(x) C(x).
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Decision Theoretic Foundations for Conformal Prediction
Note that have for any X' that C(x) C(r)... Hence, C() satísfies the marginal coverage guarantee as C(.) is marginally valid.
Next, we show that the RA-CPO objective under C() is at least equal to the RA-DPO objective under (a(-), Recall that the RA-CPO objective evaluated at Č(-) is
Ex
max
min u(a, v) aEA vEČ(X)
To bound this objective value, we consider two cases based on whether of not r belongs to Xempty.
Consider first the case & Kempty. By definition of (x)) from (18), we have minyec(s) u(a(x). 2 v(x). Hence, for $ Kempty, by noting that C(x) A 0, we have
max min u(a,y) 2 EtA vEC(x)
min u(a(x),)) 2 v(x).
yEC(m)
Therefore, for r & Xempty. by noting that C(x) = C(x), we have
max min u(a,y) y) aEA vEC(#)
max min u(a, v) 2 v(x).
atA VEC(2)
Now, let's consider the other case where E Kempty. For this case, we not that as Cxx) {ymax and from the fact that for any E X' we have vxx) Umax, we simply derive
max min u(a,v) v) = Umax 2 v(x). aEA =EČ(m)
Therefore, putting the two cases above together, we have proven
Ex max min u(a, u(a,w) w) 2 Ex[e(X)
aEA vEC(X)
Hence, have constructed a feasible solution RA-CPO, namely Ö(-), that achieves an objective value for RA-CPO which is at least as big the value of RA-DPO achieved by (())),(()) Thus, starting from solution of RA-DPO, have constructed solution RA-CPO with at least the same objective value.
(II) From RA-CPO to RA-DPO. Conversely, suppose we have feasible solution C(-) to RA-CPO, which is marginally valid, i.e.
Pr[Y E C(X) 2 a
Define the action policy a(-) and utility certificate v(-) as follows:
a(x) arg
min u(a,v), and aEA yEC(a)
v(x) = max min u(a, v). aeA yEC(Z)
is now easy to see that
Pr Pr(u(a(X),)) 2 2x(x) Pr C(X) 2 a
Moreover, by definition of v(x), we can easily deduce
Ex[u(X)] = Ex imax min u(a, v)]. a yEC(X)
Thus, from a feasible solution of RA-CPO, we constructed a feasible solution to RA-DPO that attains the same objective value, proving the equivalence in the other direction.
B.3. Proof of Proposition 3.1
Proof of Proposition 3. Fix any instance x E X' and coverage value / E [0, 1). Recall from (9) that
0xx,t) t) max ptrantile-,Iuucc Y) x/, a(x,t) arg max quantile, [u(a, Y) x x aEA
A
We show that among all sets C' with Pr(Y E C X x| 2 t, the set
C(r,t) = {v E y u(a(zz (z(zz t) 2 0(z,t)}
maximizes the risk-avers utility VRA(C) = max.e.4 mingec u(a, y): and the maximum value (z,, t).
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Decision Thcorctic Foundations for Conformal Prediction
Step 1: Any set C with coverage has risk-averse utility at most (zr, l).
Take arbitrary set C' y satisfying
Pr(y E C X = x]
Then for any action E A.
minu(a, v) yEC
quantile Lu(a,, Y) Y) -
(The reason that with probability at least I, Y lies C, and so the (1 - -quantile of u(a, Y) cannot be smaller than the smallest utility on this event.) Taking the maximum over a yields
max min u(a, y) aEA YEC
<
max quantile [u[a, Y) X = ((r,t).
aEA
Hence no set with coverage at least i can achieve risk-averse utility larger than (
Step 2: The set C(x,t) t) attains coverage i and achieves (ar, t). Consider C(x,, t) = { u(a(a,() t). ()) By
definition of the (1 --=-)nlunttlle, we have
Pr[u(a(z,(),) () 2 0 1) X = x] 2 4
which implies Pr[Y E C(r, X x) 2 l. Moreover, for every E C(r, 1), by construction
((a(x,t),)) 2 0(r,t), t),
so
u(a(,,t),u) 2 0(x,t). vEC(s.))
min
Thus
PRA (C(x,t)
max min u(a, v) 2 min ul (((xxtt),)) 2 ((x,t). a vEC(z.))
vEC(..t)
Combining both steps shows that Cax,, t) is an optimal choice among sets with coverage at least t, and its risk-averse utility equals (2, t).
B.4. Proof Theorem 3.2
start from the reparametrization RA-CPO given in (12):
maximize t:2[+[0,1]
Ex [(x(x)(()))
(Reparametrization of RA-CPO)
subject to: Ex [(x)] 2 a-a
We will further reparametrize this optimization problem and find equivalent relaxations. To do so, let us define
p(x,t) 1[ S (x:)).
(19)
Also, we will need to consider the derivative of the function (ar, t) in terms of its second argument f. Since the function 0 can be discontinuous, we will have to consider its generalized derivative (i.e. consider delta functions). More precisely, let (z..): IR + H' where R' is the space of functionals I, such that 0 (Ix, is the generalized derivative of ((x,.). In other words, for any real values and b,
L O' 0(x, 0(x, a).
We can just think of e' (z, t) as the derivative
de 0(r,t). We can then rewrite the objective of our optimization problem as
Ex [0(x+)) = umax + Ex S' 1-1
p(X,1)0' (X,
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Decision Thcorctic Foundations for Conformal Prediction
where we used the fact that ((x; 0) Umax for any s E X' by definition and ((x, O(x,0) So 0'(2, l)dl. Similarly, rewrite the constraint as,
Ex [t(X]] Ex [ p(X, t)dt.
Given the above notation and relations, (Reparametrization of RA-CPO). The optimization variable here is the function p(z, t) which is step function according to (19). We further note that any such step function defined on the unit interval can be equivalently thought of as non-increasing function on the unit interval which only takes its value in the set {0, 1.. Hence we arrive at the following integer program that an equivalent reparametrization of (Reparametrization of RA-CPO) as well as the RA-CPO.
write down the following equivalent reparametrization of
Se S plz, t)p(x)0 (x, t),t)d
maximize (0.1)
S p(z))(2, f)dedt 2
(Integer Program)
subject to:
p(x,t) = non-increasing in t
We now consider a relaxation of the above integer program to the following convex program. As we will see later, this relaxation becomes equivalent to the above integer program as every solution of the relaxed program would correspond to a solution the integer program. However, for now, let us focus on the following continuous relaxation whose variable pla; 1) can take values the interval [o, 1] (in contrast to the original integer program in which p could take its value only in the set {0, 1)):
f pla, ()(()) (x, l)dxdl
maximize
J Sr p(x))(( t)dadt 2 -
subject to:
(Relaxed Program)
p(x, t) E [0 Ve E X, [0, 1]
p(x, t) non-increasing
Here, the "optimization variable" p(z,t t) belongs to an infinite- dinnit--dimesiooall space. Hence, in order to be fully rigorous, we will need to use the duality theory developed for general linear spaces that are not necessarily finite-dimensional For reader who is less familiar with (i.e. writing the Lagrangian) for the usual linear programs in mnite-dimensional spaces.
spaces, what appears below is direct extension of the duality theory
Let F be the of all measurable function defined X' [0, 1). Note that F is linear space. Let Ω be the set of all the measurable functions on X' [0. which are non non-increasing in t and are bounded between 0 and 1; L.c.
Ω {p E F s.t. p X [0, 1] + [0, Vor e p(z, t) is non-increasing in f}
(20)
Note that Ω is a convex set. We can then rewrite the (Relaxed Program) as follows:
maximize S p(a, plx,t)p(z)) ())(t))(())))))))) (x, t)drdt
S (ee)p(,,tdddde (1 a)
subject to:
PEΩ
Moreover, let us define the functional F + as
F(g) S ple, Op(2)0 (z, Cydrdy,
(21)
and also define the functional
R
G(p) = |xmon1 p(a, (jz)))((z)dddt - (1 a).
(22)
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Decision Thcorctic Foundations for Conformal Prediction
Using the above-defined notation, our program becomes:
maximize F(A)
subject to: G(p) 2
p e f2
Note that the feasibility of the above program is non-empty, as p(x; t) = a, for all (x, E X [0, 1], a feasible
point. We can now use the duality theory of convex programs in vector spaces (See Theorem 1, Section 8.3 of (Luenberger, 1969). Specifically, let OPT be the optimal value achievable in the above linear program. Then, there exists scalar B 2 0 such that the following holds:
OPT sup (F(p) + 3G(p)) ,EN
(23)
Here, note that (3 is the usual I Lagrange multiplier.
By using (21), in order to solve the optimization in (23) we need to solve the following optimization (note that we change inf sup by applying a negative sign):
( p()))(1, (0'(), + 3) dodl + 3(1 - ar). PEL
sup
We denote the optimal solution of the above optimization problem by P3(x, 1). From the above optimization problem, is clear that the optimal solution can be determined individually for every æ E X. We will use Lemma B.1, provided below, to characterize the optimizer of the above optimization. From the lemma, and assuming that, almost surely for every X, the maximizer of (x;, + .L is unique over we obtain:
P3(2, t) = 1(t_((()))
(24)
where
[" (0(x,1) 3) di arg max (0(z,, + Bs} g(x, sE|o,1]
1" (x) arg max
aE[0,1] t=0
And the value of should then be chosen such that this optimal solution satisfies the coverage constraint.
We finally note that the optimal solution Pz(z, t) given in (24) is integer valued. As result, there is zero relaxation gap from the (Integer Program) to the (Relaxed Program).
Lemma B.1. Let 0 [0. + R. Also, let Ω be the set of all the integrable functions p [0, 1] [0. 1] which non-decreasing. Consider the following optimization problem:
[ ('())(()d
max PEl2
where 6' denotes the derivative of 8 with respect to t i.e., 0(a) - -(b) = -1º (t) Then, the the of solutions of the above optimization problem consists of functions p² such that
pt())= E ConvexHull (f11: t' arg max 8(t) ())
(E
is solution to the above optimization problem. As a corollary, if has unique maximizer t", then its corresponding p²(t) (t) 1[ < t*] is the unique solution of the above optimization problem.
Proof. For every p E Ω write using integration by parts:
| (())))))))))())))))))))))))))))) -(()0()))p(00(00) S. d'())((t)tt
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Decision Theoretic Foundations for Conformal Prediction
Let us define max [ O(1). Since p(() is non-increasing function, we have
(a) S mmax [[= p(())dt (max [ ((t))(t))tt
where the step (a) is obtained because p(t) is non-negative. As a result, we obtain
L (())) s max (p(0) ((1)) ((1))(1) ((0)((0)
= (0max ((0)) (o(0) p(1)) ((1) (0(1) -(0))
0max 0(0),
where step (b) is obtained since p(0), p(1) E [0, 1] and 0(1) & 6max-
Now, it it easy to see that if tth is such that 0(t*)= 0max then both steps (a), (b) will be equality (instead of an inequality) for the following function
= [tt
On the other hand, for step (a) to be tight we must have the following: For every point such that (1) < O, we have 0(t) = max. This shows that an optimal solution must be in the convex hull defined in the theorem, and hence, the result of the theorem follows. The uniqueness also follows similarly.
B.5. Proof of Theorem 4.1
We have:
Pr[Yn-1 E CRAc(Xn+1)) (e) Pr[Y E ((
[[ E By..il
E
1
Σ
1|M C(x BY + 1[r (x
1
(25)
where, (a) comes form the definition of the prediction set. (b) comes from the fact that
(r. .... (Xas Yn, Bynta) (Xn+1, Yn-1. Br..1)}
are exchangeable, which due the fact that (i) the exchangeabilityy of the original (n + 1) pairs {(X,, Y)) [(Xn+1, Yn+1)). and (ii) the symmetric way in which Algorithm 1 assigns By to each y E y. Finally, (c) follows from the definition of By
C. Utility function for medical experiment
Our results and findings in the medical experiment of section 5. can be reproduced with any other reasonable design of utility function. The goal of that experiment not capture a precise characterization of difficulties and consequences in medical decision making but rather pinpoint the advantages of a risk averse calibration approach in sensitive tasks like medical decision making. Of course, in real world scenarios, more comprehensive approach is needed to define principled utility function that captures the interests of all the involving parties. That being said, for the sake of proof of concept, we designed utility matrix using the ChatGPT ol model by OpenAL. The following is an AI generated text justifying the proposed utility matrix.
Clinical Justification of the Utility Matrix
21

---

Decision Thcoretic Foundations for Conformal Prediction
The utility matrix presented in Table 3 reflects the balance of benefits and harms associated with different medical actions for each true clinical condition. Each utility value is determined based standard clinical guidelines and evidence-based practices, ensuring that the chosen actions optimize patient outcomes while minimizing potential risks.
Normal (No Disease)
No Action = 10
For a patient who is truly healthy, no intervention optimal as it avoids unnecessary costs, side effects, and patient anxiety. Unwarranted use of antibiotics or quarantine measures can lead to adverse effects and resource wastage (NIHCE, 2015).
Antibiotics = 2
Prescribing antibiotics to a healthy individual can contribute to antimicrobial resistance and cause side effects without any clinical benefit (CDC, 2022).
Quarantine 2
Quarantining healthy person imposes unnecessary social and psychological burdens without providing any medical advantage (CDC, 2020).
Testing =4 4
While testing can confirm the absence of disease, routine testing in healthy individuals often not cost-effective and may lead unnecessary follow-up procedures (of Radiology, 2023).
Pneumonia
No Action
Untreated pneumonia lead to rapid deterioration and increased mortality, making inaction highly detrimental (Metlay et al., 2019).
Antibiotics 10
Timely administration of appropriate antibiotics is crucial for treating bacterial pneumonia, improving survival rates and reducing complications (Metlay et al., 2019; WHO, 2021).
Quarantine 3
While some forms of pneumonia may be contagious, standard infection control measures generally more beneficial than full quarantine, especially when bacterial pneumonia is suspected (Metlay et al., 2019).
Testing=: 7
Diagnostic tests such chest imaging and sputum cultures are essential for confirming pneumonia and guiding antibiotic therapy (Metlay et al., 2019).
COVID-19
No Action 0
Ignoring a COVID-19 infection can result in severe disease progression and widespread transmission, making inaction extremely harmful (CDC, 2020).
Antibiotics 3
Since COVID-19 is viral, antibiotics are generally only useful if there is a suspected secondary bacterial infection (WHO, 2020).
Quarantine 10
Quarantining individuals with COVID-19 is essential for controlling the spread the virus and protecting public health (CDC, 2020).
Testing= 8
Confirmatory testing is vital for diagnosing COVID-19 and guiding appropriate interventions, including quarantine and specific therapies (CDC, 2020).
22

---

Decision Theoretic Foundations for Conformal Prediction
True Label
No Action Antibioties Quarantine Testing
Normal
10
2
Pneumomia
0
10
COVID-19
10
Lung Opacity
10
Table 3. Utility matrix for the four-class chest X-ray task.
Lung Opacity
No Action =
Ignoring lung opacities can lead missed diagnoses of serious conditions such malignancies tuberculosis, posing significant risks (Rubin et al., 2020).
Antibiotics 4
Empirical antibiotic therapy may be beneficial if an infectious etiology is suspected, but it is not universally appropriate and may lead to resistance (CDC, 2022; Metlay et al., 2019).
Quarantine 4
Quarantine may necessary if the underlying cause of the opacity is contagious, but many causes do not require isolation (CDC, 2022).
Testing = 10
Comprehensive diagnostic evaluation is crucial for determining the exact cause of lung opacities, guiding targeted treatment and preventing misdiagnosis (Rubin et al., 2020).
Key Takeaways
1. Benefit vs. Harm: The utility scores balance the potential benefits of medical interventions against their associated risks and costs.
2. Specifc Standard of Care: Treatments are aligned with established clinical guidelines specific to each condition.
3. Avoidance of Unnecessary Interventions: The matrix discourages overtreatment in healthy individuals to prevent adverse effects and resource wastage.
Overall, the utility matrix aligns with standard clinical guidelines by advocating for appropriate treatment of infections, isolation of contagious diseases, thorough diagnostic evaluations for ambiguous findings, and avoiding unnecessary interventions in healthy patients.
23


Reviews:
[{"review_id": "YTSb0tqJNz", "reviewer": "Reviewer_c5sw", "summary": "The paper studies the decision-theoretic properties of conformal prediction sets. For risk-averse agents who want probabilistic certificates on certain actions for the utility to be greater than some value with high-probability, the paper suggests that this goal can be accomplished via conformal prediction sets that satisfy marginal coverage guarantees, in fact they show that both notions can be derived from each other, establishing that prediction sets are fundamental objects for a risk-averse agent. The paper then proposes an algorithm to devise prediction sets that result in better utility for risk-averse agents, with experiments supporting the claims.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their insightful comments, constructive feedback, and positive evaluations of our paper.\n\n**On the motivation for minimax optimality:**\nThis is an important point, and we appreciate the opportunity to clarify our choice. As you correctly pointed out, minimax optimality is indeed not the only conceivable approach. However, our rationale for selecting it is both intuitive and theory-backed, which can be summarized as follows:\n\n1) From an intuitive standpoint, minimax optimality aligns well with the perspective of a risk-averse decision maker. Specifically, the set $\\Omega$ captures all plausible scenarios faced by the decision maker, and a risk-averse agent would reasonably seek actions that are robustly optimal in the worst-case realization among these scenarios. This conservative stance naturally leads to the adoption of a minimax criterion.\n\n2) Perhaps a more complete picture emerges by looking at Theorem 2.3. There we prove the max-min rule (introduced in Section 2.1), combined with optimized prediction sets, matches the optimal solution to the risk-averse decision problem (RA-DPO). In light of that, the minimax optimality of the max-min rule (proved in Proposition 2.2) states that for any fixed marginally valid prediction set, even if the prediction sets are not optimally derived, the max-min rule is still a natural choice. This is particularly important as in practice, due to the limitations of finite sample data, we can only have \"approximately\" optimal prediction sets.\n\nNevertheless, we fully acknowledge your remark that alternative decision-theoretic approaches (such as those discussed in the paper you mentioned, which we will cite in the revised version) are also valuable and potentially yield interesting complementary results. We will incorporate a remark explicitly addressing this.\n\n**On the comment regarding improvements in presentation:**\nWe completely agree that the clarity and intuitive presentation of our results could significantly benefit from additional explanations. Indeed, this has largely been limited by the strict page constraints. To address this, we will leverage the two extra pages allowed for the revised version to provide detailed intuition and clearer explanations, especially regarding:\n\n1) The duality derivations and additional implications arising in Section 3.\n\n2) Intuitive justification and step-by-step reasoning behind Algorithm 1 in Section 4.\n\nRegarding Algorithm 1's connection to utility, the utility function enters through the definition of $\\hat{C}$ just before Algorithm 1. To summarize the intuition briefly here: Algorithm 1 uses the predictive model to approximate key quantities required for deriving optimal prediction sets, up to a scalar calibration parameter obtained from calibration data. As highlighted in the paragraph following Corollary 4.2, the safety guarantees of actions produced by our framework hold regardless of predictive model quality. Nonetheless, higher-quality models produce actions closer to the optimal choices among the safe ones.\n\n**On RAC’s coverage alignment:**\nWe believe this phenomenon arises due to finite-sample effects. For instance, in the same Figure 2(d) on the left, one can see that RAC's miscoverage is sometimes better and sometimes worse. Moreover, it is important to emphasize that RAC differs algorithmically from conventional CP methods, which rely primarily on thresholding conformity scores. Consequently, finite-sample behaviors may differ across these approaches. However, as the data size grows sufficiently large, we anticipate these differences will diminish, and coverage behaviors will become comparable across all CP methods, given their common coverage guarantees. \n\n**On tuning the miscoverage level $\\alpha$:**\nThank you for this insightful question—this is indeed a critical practical concern. In our framework, the value of $\\alpha$ is treated as a given, provided alongside the utility function. Accordingly, RAC takes $\\alpha$ as an **input**. As you pointed out, this value should be selected by the decision maker to reflect their degree of risk aversion, and choosing an appropriate $\\alpha$—much like specifying a utility function—can be non-trivial in some applications.\n\nFrom a practical standpoint, one can evaluate several candidate values of $\\alpha$ using validation data, and the decision maker can select the one that offers the best trade-off between risk aversion and utility. This process can involve inspecting plots similar to those presented in our paper. For example, in the context of medical treatment recommendations, one could tune $\\alpha$ by jointly examining Figures 2(b) and 2(c), and choosing a value that strikes the desired balance between maximizing average utility and minimizing critically harmful treatment decisions.\n\nWe will add a remark in the revised version to clarify this point.", "claims_and_evidence": "The paper provides enough support for the claims made. However, I'd appreciate more justification for the max-min policy for the agent (Equation 7). While it makes sense, and frankly an intuitive thing to do, but I assume a decision-maker might be interested in resolving the uncertainty over $\\Omega$ in a different way (e.g. from here https://arxiv.org/abs/2204.11318). So it makes sense to justify the max-min rule as employed, as why that is used. I'll frame my question differently: Instead of \"we are interested in the policy that is minimax-optimal\" (line 191), I appreciate the answer to \"why we are interested in the policy that is minimax-optimal?\"", "methods_and_evaluation_criteria": "Yes, the paper tests their methodology appropriately.", "theoretical_claims": "I have verified all the statements to the point that I agree on a high-level that they are correct, but haven't fully verified the notational writing. However, I haven't formally checked the duality arguments (or the validity of them) that leads to Equation 14.", "experimental_designs_or_analyses": "The experiments sound convincing, and make sense. However, I'd appreciate some reasoning why RAC better aligns nominal miscoverage with realised miscoverage  (Figure 2d), as compared to other scores.", "relation_to_broader_scientific_literature": "The paper is certainly relevant to the ICML community. The paper advances the decision-theoretic interpretation of conformal sets, as in how to use them, and devises an algorithm to use them better in that sense.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "The paper could be heavily improved in terms of writing, in particular Section 3 and Section 4. More intuition can be provided behind Algorithm 1. I'd appreciate how Algorithm 1 gets rid of specifying utility as everything before that $\\hat{\\theta}$ and $\\hat{a}$ are defined in terms of the underlying utility of the decision-maker agent?", "other_comments_or_suggestions": "None", "questions_for_authors": "I get the approach requires specifying the miscoverage level $\\alpha$. Is there a way for it to be connected to the level of risk aversion of a decision-maker, or how should the decision-maker choose $\\alpha$ depending on their level of risk-aversion. I see that RAC starts matching the averaged realised utility for nominal miscoverage level around 0.1 (Figure 2c), which makes sense that the prediction set sizes  might gets shrink and in the extreme case, only the true label will be in the prediction set, in which case the expected utility and the utility realised by RAC will coincide? But it does not serve the risk averse individual as they are not concerned with the average utility? I'd appreciate any clarification on this; it could be possible that my question is poorly worded. I guess I'm interested in how to design the sets that balances the risk aversion of a decision-maker with their utility maximisation?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qDGTQajl85", "reviewer": "Reviewer_q2pE", "summary": "This paper aims to address three questions at the intersection of conformal prediction (CP) and decision making: (1) Understanding what type of uncertainty quantification is best for risk-averse decision makers, (2) How risk-averse decision makers should use prediction sets (ie, what policy they should use, given a prediction set with marginal coverage validity), and (3) How to design CP sets for such risk-averse decision makers. In this paper, “risk-averse decision making” refers to selecting an action $a$ in response to context $x$ that maximizes the “value at risk” $\\nu_{\\alpha}(a; x)$: that is, maximizing the smallest value/utility that the agent can expect to receive with high probability $1-\\alpha$ (ie, the probability/risk of receiving utility less than $\\nu_{\\alpha}(a; x)$ is at most $\\alpha$).\n\nTo address questions (1) and (2), the authors provide analysis and discussion that connects the objective of (marginal) risk-averse decision making (i.e., Risk Averse Decision Policy Optimization or “RA-DPO”) with one of risk-averse decision-making using conformal prediction sets (i.e., Risk Averse Conformal Prediction Optimization or “RA-CPO”): That is, the authors prove that from any optimal solution of RA-DPO one can construct an optimal solution to RA-CPO with the same utility, and vice versa (Theorem 2.3). To address question (3), the authors provide some analysis for the optimal prediction sets (Section 3) and a practical algorithm building on this analysis (Section 4), and they evaluate the proposed algorithm compared to an expectation-maximizing or “best-response” policy baseline on medical diagnosis and recommender system tasks (Section 5).\n\n## update after rebuttal (Note: I copy-pasted this from my response to the authors' rebuttal.)\n\nThank you for your detailed response to my questions and concerns! In particular: The concrete changes that the authors describe address the main concern I had around the framing and references; the expanded remark and discussion they describe on the distinction between marginal and conditional seems like it will be valuable for future readers (and avoiding common points of confusion on this distinction); and the explanation regarding why selection bias is avoided (ie, the quantiles are taken for each action [based on softmax outputs from the model], independently of the calibration data) was helpful to me.\n\nRe notation clarity, I also appreciate the clarification about why $\\nu$ and $\\nu_\\alpha$ are distinct (ie, the first being the optimization variable), and the different roles of $t$ and $alpha$. I still feel like $\\theta$ was perhaps unnecessary notation, and maybe something clearer could be achieved with something like $\\nu_t$ or $\\nu_{1-t}$ (ie, only changing the subscript variable), but this is a stylistic comment and ultimately up to the authors. A proof-read for a camera-ready version could help double-check that all notation and new terms (eg, $\\nu$ vs $\\nu_\\alpha$) are clearly introduced with their distinctions, and generally to address other points of confusion/questions among the reviews.\n\nI'm happy to update my score--congrats on a nice paper!", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their detailed and insightful feedback. We are grateful for the positive recognition of our core contributions—especially the novel linkage between prediction sets and risk-averse decision-making. In what follows, we detail the concrete actions we have taken to address the reviewer’s concerns.\n\n**References:**\nThe reviewer’s comment on expanding the references is extremely valuable. We agree that including prior work from the economics literature on risk aversion is crucial to understanding the challenges and collective effort in risk-sensitive decision making. In the revised version:\n\n1) We have expanded the Related Work section by adding two new paragraphs. One discusses the economics and game theory literature (including all the suggested references) to highlight classical approaches to risk aversion. The other covers additional works, including the ones mentioned by the reviewer, in CP that address decision making beyond risk control.\n\n2) We fully acknowledge that we are by no means claiming to have invented the topic of risk-averse decision-making, but rather are drawing on concepts developed in the economics literature. To better reflect this, we have renamed the section “Fundamentals of Risk-Averse Decision Making” to “Preliminaries of Risk-Averse Decision Making.” In this section, we now include an additional paragraph—along with appropriate citations to the economics literature—that briefly outlines alternative formulations of risk aversion.\n\nWe again thank the reviewer for highlighting relevant work from the literature, which has been very helpful in improving our presentation. While space constraints prevent us from discussing detailed connections to these works, we would be happy to explore them further in follow-up rounds.\n\n**Marginal and Conditional Objectives:**\nWe thank the reviewer for emphasizing this important issue. Although Remark 2.1 highlighted this gap, space constraints limited our discussion in the original submission. In finite sample settings it is impossible to learn an optimal action for every individual scenario without some type of averaging over the covariate space. A common approach in CP is to use group-conditional guarantees, where validity is ensured on average over pre-specified groups (e.g., based on patient age or sex). We have:\n\n1) Extended Remark 2.1 to clearly delineate which parts of our theory can be systematically extended to group-conditional scenarios and which present challenges.\n\n2) Expanded the Future Work section to discuss these issues in detail. Specifically, we explain that while the results in Sections 2 and 3 extend naturally to the group-conditional setting (resulting in an m-dimensional characterization), the finite sample algorithm in Section 4 becomes more challenging due to the need to calibrate an m-dimensional vector. \n\n**Conformity Scores and Selection Bias:**\nWe appreciate the reviewer’s thoughtful questions on these topics. Regarding the connection to conformity scores, it is crucial to note that our optimal prediction sets are not derived from calibrating a threshold for a conformity score. As shown in Section 3, the optimal sets do not form a nested sequence as the miscoverage threshold varies. In contrast, optimal sets for objectives like minimizing average set size are typically characterized by a threshold rule (e.g., $p(y∣x)≥q$). We will add a remark to clarify this technical difference, which is crucial for deriving optimal actions for risk-averse agents.\n\nWe do not suffer from selection bias. We construct one prediction set per test instance and apply the max-min rule directly—rather than selecting among multiple candidate sets. The calibration procedure involves a one-dimensional parameter, $\\beta$, which is determined using calibration data to ensure valid coverage (see Theorem 4.1). Although our calibration phase involves computing a quantile for each action (as in eq. (15)), these computations rely on the softmax outputs from the model and remain independent of the calibration data, preserving the statistical guarantees.\n\n**Notational Clarity:**\nNote that $\\nu$ and $\\nu_\\alpha$ are distinct: $\\nu_\\alpha$ is defined in Eq.(2) while $\\nu$ is the optimization variable in RA-DPO. In RA-DPO, the decision maker jointly optimizes for a utility certificate ($\\nu$) and an action policy ($a$). This distinction clarifies why Theorem 2.3 is non-trivial, as it is not obvious that the optimal utility certificate can be obtained as the max-min value of a prediction set. \n\nRegarding the notations $\\theta$ and $\\nu_\\alpha$, while they are equivalent when $1-t = \\alpha$, it is important to note that $t$ and $\\alpha$ play different roles. Here, $\\alpha$ is the fixed miscoverage threshold (as in standard CP), whereas $t$ is a tunable variable during calibration to ensure that the final prediction sets achieve the desired coverage. We will clarify this point in the revised version to prevent confusion.", "claims_and_evidence": "Mostly yes: All of the main formal theoretical claims appear to be solid, the experimental evaluations appear to be reasonable and sound, and the high-level takeaway from the paper should be very valuable to the broader CP and ML communities (i.e., roughly that “prediction sets are a good form of uncertainty quantification for risk-averse decision making”). \n\nHowever, I have a couple concerns or questions about main claims:\n- **(1) Ensuring proper contextualization relative to related work:** For example, although there are many conformal prediction papers cited in the Related Work (Sec 1.1) and Further Related Work (Appendix A), I do not see any references to or discussion of classic literature on risk-averse statistical decision making, which there is much of, e.g., in economics (see “Relation To Broader Scientific Literature” and “Essential References Not Discussed”). Additionally, the Further Related Work may miss some CP references more closely related to decision making while instead citing many CP papers that are influential but not as relevant to the current paper.\n- **(2) “Minding the gap” between marginal and conditional objectives/decision making:** The paper does acknowledge the gap between the *conditional* objectives/decision making/coverage guarantees that one would ideally like and the “Marginal Version” (see Sec 2) that is realized in practice with CP, but there are some parts of the methods section where this distinction seems less clear, and the paper seems to lack a thorough discussion of the limitations of marginal decision-making, which would be valuable for readers, practitioners, and future researchers building on this work.\n- **(3) Question about whether using same calibration set to select action affects CP guarantee:** See “questions for authors”", "methods_and_evaluation_criteria": "Overall, the experimental settings and evaluations seem to sufficiently support the main message of the paper, that is regarding how prediction sets and the proposed Risk-Averse Calibration (RAC) algorithm can be used for risk-averse decision making. \n\nThat said, it appears that the only baseline is an expectation-maximizing “best response” policy, and it would likely improve the paper if the proposed RAC algorithm were compared with other methods that use CP for decision-making--such as methods proposed in Vovk and Bendtsen (2018), who have a method based on conformal predictive distributions for asymmetric classification, similar to the experimental setting in this paper, or perhaps methods in Lekeufack et al. (2024), if relevant--however, since I view the main message of the paper to be one that’s more broad and conceptual regarding prediction sets, I don’t view this as absolutely necessary.", "theoretical_claims": "Yes, I made an effort to check the proofs and they appear to be sound, though it is possible there are some details I did not understand given time constraints.", "experimental_designs_or_analyses": "I overall reviewed the experimental settings and they seem to be reasonable. E.g., it was important that the experiments not only be regression or classification, but also have some sort of actions and corresponding utilities (e.g., Table 1 for medical diagnosis experiment).", "relation_to_broader_scientific_literature": "There is growing work on leveraging conformal prediction (CP) to improve decision making, and understanding the theory of doing so; separately, there is also a large literature on risk-averse statistical decision making, though admittedly much of it is in the economics, finance, and game theory literature (eg: https://en.wikipedia.org/wiki/Risk_aversion). In my view, this paper draws insightful and useful connections between CP/prediction sets and risk-averse statistical decision making, although it unfortunately seems to not discuss this latter literature at all (aside from citing Duffie and Pan 1997 to introduce the definition of “Value at Risk”); and regarding the CP literature, there are arguably highly relevant references missing, and superfluous references provided. Please see the following “Essential References Not Discussed” Section for specific details.", "essential_references_not_discussed": "**Literature on risk-averse decision making:** Apart from citing Duffie and Pan 1997 to introduce the definition of “Value at Risk” in Section 2, there appears to be no mention or discussion of prior literature on risk-averse decision making (eg, https://en.wikipedia.org/wiki/Risk_aversion); I think it is essential to at least mention/acknowledge this literature, if not discuss some connections more thoroughly in the appendix. Eg, whereas currently Sec 2 “Fundamentals of Risk Averse Decision Making” could be read by many reasonable readers as if this paper is inventing this topic, I think there should be some mention of prior literature such as the following, and potentially further discussion in appendix. Following are some examples:\n- *Arrow-Pratt measure of absolute risk aversion / coefficient of absolute risk aversion:*\n- Arrow, K.J. (1965) Aspects of the Theory of Risk Bearing\n- Pratt, J. W. (1978). Risk aversion in the small and in the large. In Uncertainty in economics (pp. 59-79). Academic Press.\n\n- *Fundamental connections between second-order stochastic dominance and risk aversion (e.g., see https://en.wikipedia.org/wiki/Stochastic_dominance#Second-order):* \n- Hadar, J., & Russell, W. R. (1969). Rules for ordering uncertain prospects. The American economic review, 59(1), 25-34.\n- Meyer, J. (1977). Second degree stochastic dominance with respect to a function. International Economic Review, 477-487.\n\n**Arguably highly relevant CP+decision making papers not cited/discussed:**\n- *Papers on CP for counterfactual treatment-effect estimation for personalized decision making [paper-specific context in brakcets]:*\n- [CP for counterfactual treatment assignment decisions] Lei, L., & Candès, E. J. (2021). Conformal inference of counterfactuals and individual treatment effects. Journal of the Royal Statistical Society Series B: Statistical Methodology, 83(5), 911-938.\n- [Extension for robustness to potential confounding] Yin, M., Shi, C., Wang, Y., & Blei, D. M. (2024). Conformal sensitivity analysis for individual treatment effects. Journal of the American Statistical Association, 119(545), 122-135.\n- [Extension for robustness to potential confounding] Jin, Y., Ren, Z., & Candès, E. J. (2023). Sensitivity analysis of individual treatment effects: A robust conformal inference approach. Proceedings of the National Academy of Sciences, 120(6), e2214889120.\n\n- *Papers on CP under distribution shifts induced by decision making/actions of an AI/ML agent (e.g., black-box optimization or contextual bandits):*\n- [CP for contextual bandits decision making, off-policy setting] Taufiq, M. F., Ton, J. F., Cornish, R., Teh, Y. W., & Doucet, A. (2022). Conformal off-policy prediction in contextual bandits. Advances in Neural Information Processing Systems, 35, 31512-31524.\n- [Theory & expts on CP under feedloop shifts induced by active learning or black-box optimization ML decision process] Prinster, D., Stanton, S., Liu, A., & Saria, S. (2024, July). Conformal validity guarantees exist for any data distribution (and how to find them). In Proceedings of the 41st International Conference on Machine Learning (pp. 41086-41118).\n- [Empirically studies CP under feedback-loop shifts induced by BayesOpt decision making] Stanton, S., Maddox, W., & Wilson, A. G. (2023, April). Bayesian optimization with conformal prediction sets. In International Conference on Artificial Intelligence and Statistics (pp. 959-986). PMLR.\n- [CP under one-step “feedback covariate shift” decision making] Fannjiang, C., Bates, S., Angelopoulos, A. N., Listgarten, J., & Jordan, M. I. (2022). Conformal prediction under feedback covariate shift for biomolecular design. Proceedings of the National Academy of Sciences, 119(43), e2204569119.\n\nIt feels important to mention many if not all of these, especially given that in Appendix A (second to last paragraph) there currently appears to be superfluous (ie, 5 whole lines of in-text citation) references to many CP papers that are (in my view) far less related to decision making than the ones I have provided here....", "other_strengths_and_weaknesses": "**Other Strengths:** Overall, the paper provides an insightful perspective on the connections between prediction sets and risk-averse decision making. Additionally, the writing style is very appealing for a wide audience, with broader implications that can be understood beyond the scope of conformal prediction, so it seems like it has potential for high impact. \n\n**Other Weaknesses / opportunities for improvement:**\n- *Occasionally oversimplified notation: Eg, $\\nu(X)$ should ideally be $\\nu_{\\alpha}(X)$ in many instances including RA-DPO def:* The paper seemingly makes an effort to simplify notation for readability, but at times this may be oversimplified, which can make close-reading of technical details difficult, confusing, or perhaps misleading. For example, the value-at-risk quantity $\\nu_{\\alpha}(a; x)$ is defined to depend on $\\alpha$, $a$, and $x$, but sometimes it is stated only as $\\nu$, sometimes it is stated as $\\nu_{\\alpha}(x)$, and sometimes it is stated as $\\nu(x)$. Sometimes this is done explicitly (eg, Eq (2)), but other times, eg, the $\\alpha$ is dropped casually without explanation. In my view, this is not without consequence, because I think that $\\nu(x)$ actually depends on $\\alpha$ as $\\nu_{\\alpha}(x)$ makes Theorem 2.3 far less “surprising.” That is, just prior to Theorem 2.3, the paper states “One might expect a-priori that passing from the actual distribution to a lossy prediction set representation would discard information that is critical to finding the optimal policy. However, the following theorem shows, perhaps surprisingly, that this is not the case…”; however, once one realizes that in the RA-DPO def that $\\nu(X)=\\nu_{\\alpha}(X)$, it is easier to see how RA-DPO is effectively only leveraging the level-$\\alpha$ quantile of the full utility distribution, i.e., the same information provided by the prediction set. \n- Limitations of focusing on marginal vs conditional objective could be discussed further.\n- *Potentially redundant notation (?):* It is unclear to me why $\\theta(x, t) := max_{a\\in A}\\ quantile_{1-t}[u(a, Y) | X=x]$ (Eq. (9)) needed to be introduced as a new quantity that is distinct from $\\nu_{\\alpha}(x) := max_{a\\in A}\\ quantile_{\\alpha}[u(a, Y) | X=x]$ (Eq. (2)), as clearly they are equivalent when $1-t=\\alpha$. An explanation is welcome, but so far my impression is that this difference in notation has only caused confusion for me, and perhaps served to make the reformulation in Eq. (12) seem more “surprising”...\n- *Potentially missed opportunity to draw connections with choice of conformity score(?):* Where the prediction set $C^*(x)$ is defined in Eq. (13) and similarly for $\\hat{C}(x;\\beta)$ at the end of page 6, to a CP-minded reader, it seems like the optimal/proposed prediction sets are defined in a way that would be equivalent to having conformity scores corresponding to the utility function, for an optimal/chosen action (or from a pessimistic viewpoint, could be nonconformity score corresponding to a cost/loss function). In other words, it seems like a takeaway from the paper could be rephrased for a CP-minded person as ‘design your (non)conformity score to correspond to utility (or loss) that you care about for decision making, and then pick action/policy s.t. quantile on the utility (or loss) is best.’", "other_comments_or_suggestions": "- Subheadings could sometimes be added or revised for clarity. Eg., prior to reading the sections closely, it initially thought sections 2.1 (“A Prediction Set Perspective”) and 2.2 (“An Equivalent Formulation via Prediction Sets”) seemed very similar.\n- In Section 3, the paragraph before Prop 3.1, should $Pr(Y \\in C(x))\\geq t$ actually be $Pr(Y \\in C(x) | X = x)\\geq t$? As, the latter is what is written in Prop 3.1, and in the paragraph it says the probability is over $p(y|x)$.\n- In Section 3, I found it confusing whether $t$ needed to be referring to conditional coverage, or whether it could be marginal coverage. That is, $t$ is introduced as “conditional coverage probability,” but then in (12) it is used as a marginal coverage constraint.", "questions_for_authors": "**Question about whether there is selection bias that may affect CP guarantee (?)** At a high level, eg based on the max-min decision rule, Figure 1, and the prediction sets defined in Eq (11) and (13), it seems like the big picture story from a CP view is: (i) Chose conformity scores that correspond to utilities, (ii) for each action, compute utility distribution over calibration data and take an appropriate quantile to get the value-at-risk for that action, and (iii) select the action with maximum estimated value-at-risk for the worst-case label. Is this roughly accurate, or is there a more accurate way to rephrase the proposed procedure in the terminology connected to conformity scores? In either case, I am wondering if there is an issue of (or if so, how it is avoided) of creating selection bias, that is when an action is taken based on observing score distribution over the same holdout calibration set, I think this could invalidate CP guarantees, as in the potentially analogous case of selecting a model based on efficiency of CP sets, eg, see: Liang, R., Zhu, W., & Barber, R. F. (2024). Conformal prediction after efficiency-oriented model selection. arXiv preprint arXiv:2408.07066. I appreciate and look forward to your response!\n\n**Rationale for current recommendation:** Broadly, I view the paper as having an insightful contribution connecting prediction sets to risk-averse decision making. I have attempted to do a detailed review of the paper for correctness, clarity, and relation to broader literature, among other factors, in part due to its potential. I want to recommend accepting this paper due to the valuable perspective it offers and broad potential appeal, but at the moment I think it my concerns make me think it is most prudent not to at least until I have heard more from the authors at the discussion period. To reiterate my concerns, the main one is about further stating relation to literature (especially prior work on risk-averse decision making, and secondarily about CP+decision making papers that may be more relevant than many CP refs provided in App A: Related Work); other concerns are those stated around some notational clarity and the above question on possible selection bias. If these concerns are addressed, I’d be happy to consider improving my score.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4TiuPaGDGY", "reviewer": "Reviewer_DRsr", "summary": "This paper studies the decision-theoretic foundations of conformal prediction sets. It shows that prediction sets characterize the optimal strategy of risk-averse decision-making agents, and then connects it to a specific form of conformal prediction sets. Besides population-level characterization, based on the optimal prediction sets for this specific risk-averse strategy, methods for building finite-sample valid conformal prediction sets are established. The efficacy of the proposed methods is demonstrated by diverse numerical experiments.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their careful and detailed evaluation, their supportive stance toward our work, and their insightful and constructive questions, which allow us to clarify and deepen our results significantly.\n\n**Questions 1, 3, and 4:**\nThank you for these great questions! As the reviewer rightly points out, the most practically meaningful objective is the per-instance optimization (Eq. 2), which ensures optimal value-at-risk guarantees conditional on each individual instance $x$. However, there are two main considerations justifying our chosen marginal approach (RA-DPO):\n\nDistribution-free, finite-sample methods generally cannot produce simultaneous per-$x$ guarantees without additional assumptions. Hence, some form of marginalization is inevitable. Our specific choice, $E[\\nu(X)]$, is indeed a deliberate choice we made, as there is no canonical choice and we had to fix an objective. We also agree with your observation that marginal guarantees might indeed result in uneven coverage across individual instances. To address precisely this issue, we included Remark 2.1, pointing toward extensions involving group-conditional guarantees, where validity can be enforced within predefined groups of covariates (e.g., patient demographics). Such group-based guarantees enable practitioners to ensure that important covariate subgroups retain meaningful coverage\\risk. In the revised version, we will:\n\n1) Extend Remark 2.1 to clearly show which parts of our theory can be systematically extended to group-conditional scenarios and which present challenges.\n\n2) Expand our \"Future Work\" section to highlight that while Sections 2 and 3 extend naturally to $m$-groups setting (yielding an m-dimensional characterization), the finite-sample algorithm in Section 4 faces additional challenges due to the complexity of calibrating an m-dimensional vector (along with potential approaches to address these challenges).\n\nIt is also worth mentioning that moving to group-conditional guarantees, also makes our framework less sensitive to the choice of $E[\\nu(X)]$, as eventually in the fully conditional setup, the problem reduces to per-x problems. \n\nWe will explicitly clarify these points, naming the deliberate choice of $E[\\nu(X)]$ compared to alternatives and the important role of group conditional guarantees for a more meaningful decision making pipeline, in Section 2 of our revised version.\n\n**Question 2:**\nTo clarify, we focus on deterministic prediction sets in this paper, meaning a deterministic map from X to sets. Also in CP, prediction sets are deterministic when calibration data is fixed (which is the case in practice). Although, due to their mathematical convenience, the CP guarantees are usually formulated as an expectation over calibration data (similar to our Theorem 4.1), but one can also derive PAC guarantees for the same algorithm, which provides coverage guarantees conditional on the calibration data.  We will discuss this important matter in the revised version.\n\n**Question 5:**\nThank you for highlighting this important practical question. Indeed, in Section 4, our current algorithm assumes a predictive model providing softmax probabilities, implicitly targeting classification tasks. Several promising approaches can be adopted in practice:\n\nAs you suggest, one natural approach would involve training separate quantile-prediction models for each action and then choosing among them. Alternatively, a single regression model predicting the maximum quantile over all actions (as defined in Eq. 9) could also be learned directly. Both of these are similar ideas to Conformalized Quantile Regression (CQR). Other methods, such as regression-diffusion models that generate multiple samples per instance, could also approximate the quantiles in Eq. 9. We will leverage the additional two pages allowed to discuss explicitly these practical regression approaches and provide guidelines for practitioners.\n\nThanks again for providing an interesting approach for handling regression.\n\n**Question 6:**\nWe are grateful for bringing up this valuable literature. We see selection conditional conformal prediction as a complementary line of works to ours.\n\nIn the revised version, we will include a paragraph discussing this direction including the works you pointed out ([1-4]). Briefly, selective inference addresses scenarios where multiple prediction sets are constructed, and one selects a subset of these sets (e.g. choosing proteins with high predicted affinity), which then creates selection bias that one has to account for. In contrast, our current formulation constructs a single prediction set per instance $x$, with the action directly derived via a max-min strategy from that set, thereby avoiding selection bias.\n\nAn interesting future work could be to look at the intersection of these works where one might want to select a subset of actions, while both remaining statistically valid and optimizing for a form of value at risk.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. The theoretical results appear correct to me.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "How to use conformal prediction for actionable, transparent and reliable decision making is an important yet less noticed problem in the literature. This paper lays out theoretical results that may help advance research in this direction.", "essential_references_not_discussed": "No to my current knowledge. See \"questions\" part for some conceptually related works.", "other_strengths_and_weaknesses": "This paper is sound in general, and the results are interesting. However, I am not fully convinced by the marginal objective which is fundamental to the results throughout. Please see the \"questions\" section.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. Is the RA-DPO formulation equivalent to the per-x optimization? If so, it would be nice to point that out. (My understanding is no?)\n2. Does the fact that C(x) is a random (set-valued) function change the interpretation of the minimax optimal policy result? How should one interpret these results in the context of conformal prediction, given that conformal prediction set is a random mapping? \n3. Why is the objective chosen as the expectation of the lower bound $\\nu(X)$? It seems risk-averse agents may also consider a tail-probability related quantity in $\\nu(X)$, or the infimum of $\\nu(X)$ over $X$?\n4. Related to Question 1, changing from per-x problem to the marginal formulation makes the objective and constraints less interpretable. For example, asking for a marginally valid lower bound $\\nu(X)$ gives opportunities to sacrifice some \"hard\" instances $X$, which (considering the average of $\\nu(X)$) leads to something weird (equation 12), where $t(X)$ can be unequally distributed across $X$. I'm not sure whether this is still fully meaningful from a decision making perspective. Would it be possible to derive a marginal problem which is equivalent to the per-x ones? \n5. In Section 4, instead of using a density estimator (which can be difficult especially for continuous outcomes), can one just directly estimate the conditional quantile of the utility for any action $a$, and let the prediction sets prioritize high-utility regions?\n6. I like the perspective of decision making with conformal prediction, which is a prevalent issue but has thus far mostly been implicit and heuristic. I wonder how this perspective is related to several recent works with related concepts, where people use selective inference ideas to \"pick out\" units to act upon with false discovery/coverage rate control (e.g., [1,2]), or build prediction sets after decision making processes that have picked out some interesting units (e.g., [3,4]). \n\n[1] Jin, Ying, and Emmanuel J. Candès. \"Selection by prediction with conformal p-values.\" Journal of Machine Learning Research 24.244 (2023): 1-41.\n[2] Gazin, Ulysse, et al. \"Selecting informative conformal prediction sets with false coverage rate control.\" arXiv preprint arXiv:2403.12295 (2024).\n[3] Bao, Yajie, et al. \"Selective conformal inference with false coverage-statement rate control.\" Biometrika 111.3 (2024): 727-742.\n[4] Jin, Ying, and Zhimei Ren. \"Confidence on the focal: Conformal prediction with selection-conditional coverage.\" arXiv preprint arXiv:2403.03868 (2024).", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "I1MZ1Pg1XM", "reviewer": "Reviewer_jVCC", "summary": "This paper considers the marginal version of the value at risk problem. This problem is called risk-averse decision policy optimization. \n\n- The authors derive optimal policy for risk-averse decision makers given prediction sets, which takes a max-min form. \n\n- They establish that prediction sets are a sufficient statistic for safe decision-making.\n\n- They characterize the optimal prediction sets for risk-averse conformal prediction by formalizing a risk-averse conformal prediction optimization problem. \n\n- They propose a risk-averse calibration algorithm with a distribution-free safety guarantee.\n\n## update after rebuttal:\n\nI have read the author response as well as the other reviewers and author response to them. The authors clarified my major concerns related to the positioning of the paper with respect to the related work, technical novelty, group-conditional setting, and dependence on sample size. I would like to thank the authors for their good work. The changes that are mentioned in the author rebuttal should be incorporated into the final version of the paper.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their valuable feedback and thoughtful questions, which will help us significantly improve the clarity and contribution of our manuscript.\n\n**Question 1:**\nOur contributions are twofold, as outlined below: (1) we introduce a novel question within the conformal prediction literature, and (2) we develop mathematical techniques to provide meaningful answers.\n\n**Regarding the novel question:** Despite extensive literature on conformal prediction (CP) as uncertainty quantification beneficial for downstream decisions, little was previously known about whether prediction sets are the ideal means to communicate uncertainty within decision-making pipelines and how decisions should optimally incorporate them. We hope our findings bring new insights to the community in this regard.\n\n**Regarding mathematical and technical contributions:** A key contribution is deriving an explicit solution to RA-CPO (Section 2.2), accomplished in Proposition 3.1 and Theorem 3.2. To highlight the challenges, note that RA-CPO is a non-convex optimization over the set function $C(.)$, lying outside conventional duality-based analysis. We first reparametrize RA-CPO equivalently (Eq. 12), maintaining non-convexity. Next, in the proof of Theorem 3.2, through another reparametrization to a mixed-integer program, followed by a convex relaxation over function spaces, we use a key technical lemma (Lemma B.1, Appendix) to show this relaxation exactly solves RA-CPO. This approach is novel, and we will emphasize these contributions further in the revision.\n\nAdditionally, our manuscript includes: Theorem 2.3 (prediction sets as sufficient statistics), Proposition 2.2 (optimality of max-min decision rule given marginally valid sets), and Theorem 4.1 (finite-sample statistical validity of our algorithm).\n\n**Question 2:**\nThis is an excellent point. Two distinct estimation steps exist in the finite-sample algorithm: calibration-data-based and model-output-based (softmax probabilities). The only parameter influenced by calibration data (thus sensitive to sample size $n$) is the scalar parameter $\\beta$. Similar to standard CP, the sensitivity to $n$ is minimal, stabilizing once $n$ reaches a few thousand samples. Under mild smoothness conditions on the conditional distribution of $Y|X$, finite-sample upper bounds scale as $1 - \\alpha + O(1/n)$ (Theorem 4.1), a standard assumption in CP literature.\n\nRemaining computations, notably quantiles (Eq. 15), rely solely on softmax probabilities and are independent of calibration data and $n$, preserving statistical guarantees. We will further clarify this point by expanding the paragraph after Corollary 4.2.\n\n**Regarding group-conditional validity claims:**\nAlthough briefly noted in Remark 2.1, our original discussion was limited by space constraints. To clarify: the theoretical results in Sections 2 and 3—such as the characterization of optimal decision rules and prediction sets—extend naturally to the group-conditional setting, resulting in an $m$-dimensional formulation (where $m$ is the number of groups). However, the finite-sample algorithm in Section 4 poses greater challenges, as it requires calibrating an $m$-dimensional vector, which introduces both computational and statistical complexity. In the revised manuscript:\n\n- We have expanded Remark 2.1 to explicitly distinguish between the components that extend systematically and those that present additional challenges.\n\n- We have enhanced the “Future Work” section to outline potential approaches for addressing the difficulties in the finite-sample setting and to motivate further investigation of this important extension.\n\n**Regrading bayesian methods:**\nWe thank the reviewer for highlighting relevant Bayesian literature. We emphasize our approach is complementary—not competitive—with Bayesian methods:\n\nMain theoretical contributions up to Section 3 (two-thirds of the paper) remain neutral regarding Bayesian or frequentist approaches, focusing instead on general roles of prediction sets in risk-averse decision-making. The equivalence formulation (Theorem 2.3) and optimal prediction set characterization (Theorem 3.2) assume complete knowledge of underlying distributions, applicable equally to Bayesian posterior distributions (e.g., from Gaussian Processes).\n\nIn scenarios where Bayesian models (e.g., Gaussian Processes) approximate the distributions well enough, one can directly use our theoretical results without employing the finite-sample calibration of Section 4. Alternatively, even when Bayesian assumptions' precision is uncertain, one can still start from Bayesian posteriors and further calibrate prediction sets using our approach, ensuring robust safety guarantees.\n\nWe will explicitly clarify these points in the revised manuscript, along with a discussion about the existing bayesian approaches, including the ones suggested by the reviewer.", "claims_and_evidence": "It is said that the results are extendable to group conditional validity constraints. Perhaps this is the case with more realistic applications. It is said that the marginal formulation in the paper “naturally extends” to this case, but no roadmap is provided. It sounds as if this extension is trivial. If not, please specify what challenges lie ahead.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "The theoretical claims make intuitive sense. I did not check the proofs.", "experimental_designs_or_analyses": "Experiments look valid.", "relation_to_broader_scientific_literature": "Risk aversion and safety have been frequently studied by utilizing Gaussian processes as surrogate models. See, e.g., \n\n- Sui, Yanan, et al. \"Safe exploration for optimization with Gaussian processes.\" International conference on machine learning. PMLR, 2015.\n\n- Nguyen, Quoc Phong, et al. \"Value-at-risk optimization with Gaussian processes.\" International Conference on Machine Learning. PMLR, 2021.\n\n- Demirel, Ilker, et al. \"Escada: Efficient safety and context aware dose allocation for precision medicine.\" Advances in Neural Information Processing Systems35, 2022.\n\nPlease comment on the pros and cons of your approach in risk-averse decision-making compared to this line of literature.", "essential_references_not_discussed": "N/A.", "other_strengths_and_weaknesses": "I am positive about the contribution of this paper. However, a more detailed discussion is required on why conformal prediction sets are the right way to approach the problem (perhaps by comparing them with other approaches, like GP-based approaches in an experimental setting). In addition, novel steps of the technical proofs should be highlighted.", "other_comments_or_suggestions": "N/A.", "questions_for_authors": "1) This paper proposes a new model for risk-averse decision-making using conformal prediction sets. To avoid problems commonly encountered in conformal prediction, it considers the marginal version of the problem. In order to assess the technical contribution of the current work, please discuss if any new or out-of-the-box technique is used to arrive at the conclusions of the paper. Is it the reparameterization used to derive an equivalent reformulation of RA-CPO? Is it the structure discovered using duality?\n\n2) Can any further insights be gained for the finite-sample setting? It seems that everything works fine when replacing true functions with their estimated counterparts. What is the effect of the sample size n on the quality of the estimates and prediction sets? Is it possible to get some convergence rates based on n?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper presents a decision-theoretic framework for risk-averse prediction using prediction sets. It defines optimal uncertainty quantification, proposes a max-min policy for action selection, and introduces Risk-Averse Calibration (RAC), a practical algorithm that balances safety and utility. Experiments show RAC’s effectiveness in medical diagnosis and recommendation systems. All reviewers recommended acceptance, recognizing the paper’s strong theoretical contributions and practical relevance. The minor concerns raised were satisfactorily addressed in the rebuttal. Overall, this is a well-executed and timely contribution, and I strongly support its acceptance.
Decision: Accept (spotlight poster)