Conformal Robustness Control: A New Strategy for Robust Decision
Authors: Yang Hu
Jieren Tan
Changliang Zou
Yajie Bao
Haojie Ren
Venue: ICLR 2026 Oral (2026)
Keywords: Conformal prediction
Contextual robust optimization
Coverage
Decision robustness
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: This paper develops a new strategy for robust decision problems via conformal robustness control.

Abstract: Robust decision-making is crucial in numerous risk-sensitive applications where outcomes are uncertain and the cost of failure is high. Conditional Robust Optimization (CRO) offers a framework for such tasks by constructing prediction sets for the outcome that satisfy predefined coverage requirements and then making decisions based on these sets. Many existing approaches leverage conformal prediction to build prediction sets with guaranteed coverage for CRO. However, since coverage is a *sufficient but not necessary* condition for robustness, enforcing such constraints often leads to overly conservative decisions. To overcome this limitation, we propose a novel framework named Conformal Robustness Control (CRC), that directly optimizes the prediction set construction under explicit robustness constraints, thereby enabling more efficient decisions without compromising robustness. We develop efficient algorithms to solve the CRC optimization problem, and also provide theoretical guarantees on both robustness and optimality. Empirical results show that CRC consistently yields more effective decisions than existing baselines while still meeting the target robustness level.

Published as a conference paper at ICLR 2026
CONFORMAL ROBUSTNESS CONTROL: A NEW STRAT- EGY FOR ROBUST DECISION
Yang Hu'; Jieren Tan², Changliang Zou?, Yajie Bao & Haojie Ren'*
"School of Mathematical Sciences, Shanghai Jiao Tong University 2School of Statistics and Data Sciences, LPMC, KLMDASR and LEBPS, Nankai University {hu-yang, haojiirenyyeetuu.dduu tanjrêmail nankai edu. (yajiebao, zouel (auuul)enannaal edu.
ABSTRACT
Robust decision-making crucial in numerous risk-sensitive applications where outcomes are uncertain and the cost of failure is high. Conditional Robust Op- timization (CRO) offers framework for such tasks by constructing prediction sets for the outcome that satisfy predefined coverage requirements and then mak- ing decisions based on these sets. Many existing approaches leverage conformal prediction to build prediction sets with guaranteed coverage for CRO. However, since coverage sufficient but not necessary condition for robustness, enforeing such constraints often leads to overly conservative decisions. To overcome this limitation, we propose a novel framework named Conformal Robustness Con- trol (CRC), that directly optimizes the prediction set construction under explicit robustness constraints, thereby enabling more efficient decisions without compro- mising robustness. We develop efficient algorithms solve the CRC optimization problem, and also provide theoretical guarantees on both robustness and optimal- ity. Empirical results show that CRC consistently yields more effective decisions than existing baselines while still meeting the target robustness level.
INTRODUCTION
In many real-world applications, it is crucial for decision-makee makers to account for operational risks to avoid irreversible consequences. For example, portfolio management (Markowitz I952) aims to maximize returns while navigating the trade-off with risk tolerance. Similar risk-sensitive decision- making challenges are also evident in fields such medical diagnosis (Kiyani et al.) 2025 and
transportation planning (Patel et al. 2024).
Consider scenario where we observe an input X, but the corresponding outcome Y is unknown. The decision-maker needs to choose a decision = =(X) based on the input such that the incurred decision loss <(Y, 2XX) does not exceed a certain risk certificate r(X) with high probability. For- mally, the (1 llevel robustness requirement is given by
P(o(Y,=(X)) S r(X)3 21 a.
(1)
At the same time, the decision-maker seeks to minimize r(X) to improve efficiency and reduce potential worst-case losses.
Over the years, Conditional Robust Optimization (CRO), introduced by Chenreddy et al. (2022).
has become widely adopted and effective framework for robust decision-making As extension of classical robust optimization (Ben-Tai Tal et al., 2009), CRO incorporates covariate information to enhance decision quality, enabling more precise and context-aware responses in complex tasks. In the CRO framework, decisions are derived from a minmax optimization problem using a prediction (X)). formulated as (X) := arg min max Φ(y, z). By designing U(X) with regular structure, such as a box or an ellipse, the resulting minmax problem remains convex and
can be solved efficiently in polynomial time. The corresponding risk certificate value is defined
"oorresponiigg authors

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(X) max dly, xx()), To meet the robustness requirement coverage condition on the prediction set, that is
II) CRO enforces
P|Y E E((X)
Recently, Johnstone & Cox ( and Sun et al. (2023) first employed the conformal prediction Vook et al. 2005) to construct the prediction set (1(X) from historical labeled data with the target coverage level a; then substituted it into the minmax problem to make the final decision. It can be observed that the coverage property ( is sufficient condition for achieving the final robustness target I Hence, the two-step procedure above provides a statistically valid robustness guarantee
for the subsequent decisions.
However, as noted by Ben-Tail lal et al. (2009), controlling robustness via this sufficient condition often results in suboptimal and overly conservative decisions. In this paper, introduce Conformal Robustness Control (CRC), new strategy to alleviate the conservativeness of the existing CRO framework and to enable more efficient robust decisions. Our contributions summarized as follows.
(1) Unlike conventional CRO methods that enforce coverage constraint on prediction sets, CRC directly minimizes the expected risk certificate under explicit robustness constraint, significantly improving decision efficiency. The CRC procedure is amenable to efficient gradient-based optimization algorithms that minimize empirical loss using labeled data.
(2) We establish non-asymptotic theoretical guarantees both the robustness and the opti- mality gap of the resulting decisions. For a given test data point, we further develop a sample-splitiing calibration procedure to endow the optimized prediction set with finite- sample robustness guarantees.
(3) Through extensive experiments on both synthetic data and real-world applications, the pro- posed CRC consistently outperforms baseline methods across key metrics.
Figure Sngureecompee compares the conventional CRO framework with proposed method CRC, the nominal robustness level 1 a 90%. The brown circular regions the right panel represent prediction sets ((X) satisfying 90% coverage constraint and 90% robustness constraint, respectively. The CRO decision attains robustness level of 98%, which is significantly higher than the nominal requirement. This leads higher risk certificate and decision loss compared to the proposed CRC.
Prediction set
CRO
CRC
Decision:
Rubusiness: Coverager 0.50
Robustness: Coverage:
Au (+) angmin max ( re8
Target:
CRO: Coverage constreint
Risk certificate:
robustress
(x) max (x))
Placr, (O) S 22 -
CRC: Robustness
certificate: Decision loss: 0.32
cartificate: 1.25
Decision loss 0..1
Figure 1: Comparison of CRO and method CRC. Portfolio optimization problem with d(y, -y 2, = {z TR2 21 + 1, 21, 22 2 0}, and a 0.1. Blue lines show CRO solutions for the brown circular prediction sets. The shaded blue regions indicate where the loss diy. =(X) is below the risk certificate r(X). The prediction set in CRO achieves exact 90% coverage, with r(X) 1.93. In contrast, CRC meets the 90% robustness requirement, yielding more efficient decision with r(X) = 1.25.
RELATED WORKS
Robust optimization is well-established method for decision-making under uncertainty. Early work focused on approximating Value at Risk (VaR) by designing deterministic prediction sets to
induce robustness (Ghaoui al. 2003 Natarajan al. 2008 Bertsimas et al. 2018). Later studies,
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such Shang et al. (017)) Bertsimas al. (2018): Hong et al. (2021) have proposed data-driven prediction sets. With the growing size of data, Chenreddy et al. 2022) explored how covariate in- formation could be leveraged to develop more effective prediction sets, leading to the introduction of the Conformal Robust Optimization (CRO) framework. Subsequent works by JJonnstone & Cox 202T]][Patel et af. (2024); [Sun et al. (2023) incorporated conformal prediction methods construct prediction sets that satisfy coverage conditions, thereby providing finite-sample robustness guaran- lees for CRO. Kiyani al. (2025) derived the explicit form of the optimal prediction set that has the minimum risk certificate under the coverage constraint. However, the construction relies on min- imizing the VaR function, which often also leads to intractable formulations if the decision space is contimuous (Uryasev & Rockafellar 2001). In addition, Wang et al. (023) also considered opti- mizing the prediction sets in a robust optimization problem, but relaxing the robustness constraint through the conditional Value at Risk transformation (Rockafellar & Uryasev (002). Compared to existing work, we impose the exact robustness constraint rather than a coverage constraint on the prediction set, thereby enhancing the generation of more effective decisions.
Conformal prediction is a widely used method for uncertainty quantification, notable for its model- agnostic and distribution-fiee properties (Vovk et al., 2005 Lei et al.; 2018 Angelopoulos et al. 2024a). In predictive inference tasks, the efficiency measure of conformal prediction sets is the size volume. Recent research has increasingly focused improving the efficiency of these predic- tion sets. Several studies, such as [Sadinle et al.] (2019). [Bai et al. (2022), Stutz et al. (2022). and Kiyani et al. (2024b) have formulated constrained optimization problems that minimize the size of prediction sets subject to coverage constraints. In addition. Yang & Kuchibhotla (2025) introduced sample-splitting approach to select models yielding the smallest prediction sets, Tollowed by con- structing split conformal prediction sets (Vovk et al., 2005; Papadopoulos et al. (002). Differently, sattl-
Liang et al. (2024) proposed a method that avoids sample splitting while maintaining coverage during model selection. In terms of decision efficiency, since the performance of deci- sions varies significantly with different conformal prediction sets, Chenreddy & Delage (2024) and Ych et al. (2024) proposed end-to-end learning methods that train the conformal prediction sets by directly minimizing downstream expected decision risk. Moreover, [ao et al. (2025) developed new frameworks for prediction set selection in the CRO problem, which could keep finite-sample robustness control while avoiding sample splitting.
PREDICTION SET OPTIMIZATION WITH ROBUSTNESS CONTROL
3.1 PROBLEM SETUP
Let X' be the covariate space, and y be the label space. The primary goal of robust decision is to find a decision policy z(-) : X' - 2 and a risk certificate function r(.) : X -> R that minimizes
E[r(X)] subject to the robustness constraint in (I. It is consistent with the Risk Averse Decision Policy Optimization (RA-DPO) problem defined by Kiyani et al. (2025):
min B[r(X)] s.t. PP (Y, S r()) 2 - z())())
However, directly optimizing over arbitrary forms of z(-) and r(-) is generally difficult. The CRO framework provides flexible alternative by introducing prediction (()) that maps each covari- ate x E X' to a subset of the label space V, which relates to both the decision and the associated risk certificate. Specifically, for x E X,
zu (z) := arg min max o(y,2), ru(x) max Φ(y, (x()). yE((()
=EZ yEl((z)
To identify the optimal decision policy and risk certificate, it is natural to minimize the expected risk certificate under the robustness constraint:
min u()xa++2
E [X)(
s.t. (Y, S rx())) 21
(4)
The next theorem shows the equivalence between RA-DPO in 3 and the problem (4. which also means that optimizing the prediction sets will not result in a suboptimal risk certificate.
Theorem 3.1. Let RA.DPO (.), (.. be the optimal solution of RA-DPO in 3,, and le be the optimal solution of (4). It holds that RA (X) = Elru- (X)}. which means problems (3) and are equivalent in minimizing the expected risk certificate while maintaining robustness control.
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We defer the proof of Theorem 3.1 to Appendix B.2 The prior work of Kiyani et al. (2025)
also derived a formulation equivalent to RA-DPO in Eq. (3. termed Risk Averse Conformal Prediction Optimization (RA-CPO). This formulation optimizes the expected risk certificate over all possible prediction sets subject to a coverage constraint, that is, replacing the constraint in 4 with (2)- To construct the "optimal" prediction set that solves RA-CPO, [Kiyani al. (2025) proposed method based on the minimizer and minimum value of a contextual VaR problem: 2" (x) arg mingez Va =)|X x) and r (x) mingez -a (o(, X = r). Here, VaR(-|X = r) denotes the conditional 1 - a population quantile given the covariate X = x. In the case of classification with a finite decision space, Kiyani al. (2024a) approximated the opti- mal prediction set by first estimating the conditional distribution of Y Y X, then the associated VaR problem can be solved by traversal. However, this approach does not extend to continuous decision spaces 2, where the VaR problem generally becomes intractable (Uryasev & Rockafellar 200T).
To address this limit, we consider solving problem 4 over the parametrized prediction set Ulg where 0 E θ refers to the model parameters. In regression settings with X' RR, , 129 two
commonly used types of prediction sets are box and ellipsoidal sets (Johansson et al. 2017 Sun
et al. 2023 Their parametrized forms can be defined as follows.
Box prediction set. A box-shaped prediction set is constructed by componentwise lower and upper bounds for the response vector. Let (.) Rp + R2 and hy (-) IR" - R2 be models with parameters A E 6, then
Uo( = - E E (x) Sy S h2 (z))
Ellipsoidal prediction set. Unlike box sets, ellipsoidal prediction sets account for correla- tions among components of the response vector. Let (a) TR" - R and E-(-) IR"
IRex denote the mean and covariance model with parameters 0 O, then
(x(x) = {u R (v - (==())))) En (x) (v (v- =
In Appendix [.6] we also provide the example of a parametrized polyhedral set based on the defini- tion Bärmann al. (2016).
For parametrized prediction set (.), we denote the corresponding decision policy and risk cer- tificate functions as z-(-) 2u, (-) rol)) rue (-) for short. We then consider the parameterized
version of problem
min [xr] s.t. P S ro())) 2 a. 8E0
Even though the parametrized optimization can also be applied to the RA-CPO framework, we show that our proposed problem (5) yields lower risk certificate in Appendix B.3 which further confirms the benefit robustness constraint over the coverage constraint. In the following subsections, we investigate the optimization problem (5) based on the collected labeled data and provide theoretical results for the robustness and optimality guarantees. Moreover, we also developed a differential algorithm to solve the optimization problem for the continuous decision space.
3.2 EMPIRICAL OPTIMIZATION WITH CONFORMAL ROBUSTNESS CONTROL
Suppose we have collected an i.i.d. labeled dataset D., = ((X,, } drawn from some distribu- tion P. We first optimize the prediction set by addressing an empirical version of the problem (5). and then apply this prediction set for decision-making, By approximating both the objective and the constraint in (5) via sample averaging, we obtain the following empirical counterpart:
arg min BE
(x()))
s.t.
Σ 1 (66, S rx())) 2 1 a 1=1
(6)
To distinguish from coverage control methods and to emphasize the explicit robustness constraint, refer to this procedure Conformal Robustness Control (CRC).
A natural approach to solving problem (6) is to consider its dual formulation. Define the La-
grangian function as L():0) = [(0) + (0), where X > 0 is the Lagrange multiplier, f(0)
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Ei= (X) and ( is typically differentiable if the CRO problem for (z) can be reformulated into a convex pro- gramming. In such cases, its gradient can be computed using existing implicit differential tools, see Amos & Kolter (2017) and Agrawal et al. (2019). In contrast, the term g(0) is non-smooth approximate the indicator with smooth surrogate {a < b} 1 + erf( (=2 )), where erf(z) -े Jo e- dt the Gaussian error function and a > 0 controls the smoothness. Replacing the indicator in g( with this surro- gate yields a smoothed constraint function O(0). Similar smoothing techniques have been employed in the optimization problem of conformal prediction (Bai et al. 2022 Kiyani et al. 2024b Wu
2x())) rx))).) The function f()) - a Ei= 1 (o
due the indicator. To enable gradient-based optimization,
al. 2025). The resulting smoothed dual problem is given by: mingge maxxzo (xxx; 0), where L(x;0) = f(0) - AO) This smooth approximation enables numerical solution via an alternating gradient descent algorithm. refer to Davis et al. (2020) and Bolte et al. (202I) for the convergence analysis of similar optimization problems. Implementation details are summarized in Algorithm
Algorithm Prediction Set Optimization with CRC
Input: Loss function é, robustness level a, labeled dataset D, parametrized set set (do(.) with 0 E O, smooth surrogate function i, learning rate rj > 0.
{(X, 1=
2: Initialize 0 <- 0g and X = 0.
3: Compute (X,) and zæ (XX)) for i [m].
4: Define empirical objective f0) and set smooth constraint ī(0).
5: Form the smoothed Lagrange multiplier L(): f(0) + Ağ(0).
6: while no converged do
7: Perform few steps of gradient descent on 0 to minimize Σ(>; 0).
8: Compute g(0) and perform projected gradient asent À max (0, (g))
9: end while
10: 0 05-0.
11: Output: Prediction set Ug(-).
In learning problems, Angelopoulos et al. (2024b) proposed the framework named conformal risk control by extending the miscoverage risk to general monotone risk functions. The robustness constraint can be regarded as a special risk. whereas it is not monotone in the model pa- rameter. In addition, the objective function in Angeloppolos et al. (2024b) is the threshold parameter of prediction sets, but we consider the risk certificate function r(.), which is more complex.
Remark 3.1.
3.3 THEORETICAL RESULTS
This section presents the theoretical guarantees for the solution problem (6) The analysis for the smoothed variant (Algorithm) II. being conceptually analogous. are deferred to Appendix D.2 We equip the parameter space ④ with the supremum norm and state the underlying assumptions.
Condition 3.1. Loss function & is La- the decision zo(x)) is -.p-hipschitz in O. The risk certificate re ra(x) is Lipscchti in E O, and uniformly bounded by positive conslant B, > 0 0.ror any x. E X and 0 E θ.
in decision 2 for any y V. For any E X,
These regularity conditions are mild and typically satisfied in practice. For example, in portfolio optimization with the loss function <(y, =>= -y 2, the Lipschitz condition holds if y is bounded. For decision function 2g and risk certificate function roo if the CRO problem for prediction set Uo can be transformed into smooth convex optimization problem, then the Lipschitz property can be derived from the KKT conditions and the implicit function theorem, see Boltt et al. (2021) and Amos & Kolter 2017). The next assumption introduces a mild distributional assumption.
Condition 3.2. Let 0 E θ the density of Vo ) is uniformly bounded by constant po >> 0.
Y) = ó (Y, z)(X)) (X(
) for data (X, Y) ~ P. Suppose that for all
The bounded density condition is often needed for concentration guarantees in the conformal pre- diction literature (Kiyani al. 2024b Jung et al. 2023 Lei & Wasserman 2014).
Definition 3.1 (Covering number). Let θ be a parameter space with the supremum norm || lbo. Given any 0, the subset O θ is called an e-covering of θ if for every 0 E Q there exists
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e such that ||0 0. loo c. The covering number M(@, || llac. is the smallest cardinality of any e-covering of O.
some 02
Covering numbers quantify the complexity function class and are a fundamental tool sta- tistical learning theory and convergence analysis (Van Der Vaart & Wellner 1996). The next two theorems provide a non-asymptotic characterization of the robustness and expected risk certificate value for CRC.
Theorem 3.2 (Robustness gap). Let O be the solution to optimization problem (6) Under Condi- tions 3.7 and 3.2 for any independent data (X,Y) ) ~ conditioning on the labeled data Das the following inequality holds: with probability at least -n
P { (V,==(X))) (x)( D. 21 An:
where the robustness gap A,, =V 1oR(2N'(0,11 1111-) 4(IgIa+Le) Lr
Theorem 3.3 (Risk certificate optimality). Let 8r, denote the optimal solution of problem ( at the robustness level
An. Under the same conditions as Theorem 3.2 conditioning D
with probability at least 1 - 2n
log(2 (2,
|| |1,,,
log n 4L,
(X) - T0 (x) Dr SAB,
2n
For finie-dimmnnsones parameter space θ of dimension d, the covering number scales approxi- mately as II,, Iloe,n x n², so both the robustness gap and the expected risk certificate converge to zero at rate /lg n/n). ). In Appendix D we provide more comprehensive theoreti- cal results, such as in the setting where the function class has finite VC dimension.
Remark 3.2. It is worth noting that 0',, denotes the optimal model under slightly relaxed robust- ness level 1- a + An rather than the exact level 1 1- a. This relaxation introduced to ensure that 82 is feasible the problem (6) with high probability, thereby guaranteeing that the empirical risk certificate of ê is less than that of 0',, with high probability. Finally, leveraging relevant theories of empirical process, problem (4.. If additional assumptions are imposed regarding the 10 may no longer be needed.
establish the bounds in Theorems 3.2 and 3.3
Let 0* be the solution the
10 such relaxation
TEST-TIME DECISION WITH FINITE-SAMPLE ROBUSTNESS CONTROL
In this section, we turn to the practical task of making decisions a test point Xn+1 with unknown label Y A straightforward approach is to output the decision 24, (Xn+1), where 0 is solution to the problem (6). Theorem 3.2 shows that the robustness of the decision z24, (Xx+1) converges to the target level asymptotically. To achieve finite-sample robustness control for the decision of the specific test point, we further calibrate the prediction set obtained from Algorithm I using both the test data Xn+1 and the labeled data {(X,, Y.)
Specifically, we split the labeled dataset D, into training set Dursin = ((x,, } and a calibra- tion set Deal {(X,, Y.) where no < n. We first obtain the optimized prediction set Ug using only Derain in Algorithm Next, we apply full conformal prediction (Vovk et 2005 Lei et al. 2018 to calibrate the prediction set Ug. (-) based on Deat and X
computationally expensive and often unnecessary. Instead, adjust the prediction set Ugg (-) by tuning single radius parameter E R++ which controls the size of the set and provides an efficient way of model calibration. Following the framework of nested prediction set in Gupta et al. 20022), we call the family (1),(()))EE>> nested sets if t1 I tz implies that Ue, e, (z) Uo.,, (2) for any E X. For the two examples of prediction sets in Section 3.1 the nested versions are given as follows.
Calibrating the entire parameters
Nested parametrized box set:
Uo E RD (1) 15 (3 +1};

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Nested parametrized ellipsoidal set:
Uo (o,e(a)) {y E IR (y m)())) T (x)(x) (y-nu(x)) S 1).
Let E y be a hypothesized value for the test label Y,+1, and denote the augmented calibration set as {(x,, (,") where Y," = Yú for no + - S n and Y = y. Given the prediction set ()) the hypothesized radius threshold is computed by
A min tERt n no
Σ s=n
(0(1,"
(b,, I roo. (Xi) 2 21-a (7)
where 29,((2) arg min maxceulo.(et) o(c, 2) and (0,) maxcello Øic, 20, (x)). Then the calibrated prediction set is given by
Ucal (ca((Xn+1) = {u E E (ly, Fo. (Xn+1)) Sro (Xn+1)
Finally, the decision for test point made by Adea (Xn+1). Calibrated CRC (Cal-CRC), and summarize it in Algorithm
We name the procedure above as
Algorithm 2 Cal-CRC
1: Input: Same as Algorithm I size of training set no. and test point X
2: Sample splitting:
Dirain M and Dext = {(X,, Y,)) -=
3: Training: Obtain the prediction set Ug (.) by running Algorithm on Derain.
4: Calibration: (ca((Xn+1) Ø.
5: for y do
6: Define {(X,, y 1= where Y = Y, for no +1 S i and Y y.
7: Calculate the hypothesized threshold f via (7)
8: if (V, 20 (Xn+1)) I 6o (X++1) then
(Ca((Xn41) Uca (XX+1) {y}. 10: end if
9:
11: end for
12:
Make the decision: Zuca (Xa+1) &- arg mingez maX,etees(XXx-) (ly, z).
13:
Output: the decision zuca (Xn+1)-
Theorem 4.1. the labeled data {(X,, Y.) and test data (Xx+1: Yut1) are i...d, then we have the fimite-sample robustness guarantee
PP { 11n+1, (Xn+1)) S Tica (X n- 1)} 2 a
The finite-sample robustness relies solely on the exchangeability of data, which identical to that in classical conformal prediction theory (Lei al. 2018). For implementation, note that the calibrated prediction set Uca((Xn+1) is obtained by traversing all possible values of E y In practice we can apply the discretization technique (Chen et al. 2018) avoid exhaustive search. The complete implementation is provided in the Appendix BB4 The decision optimality of zuc, is analyzed in the Appendix B.5 and corresponding simulation results will be provided in Section E.3
EXPERIMENTS
In this section, we compare our proposed CRC with two baseline methods for robust decision- making: (i) CRO with conformal prediction sets (Sun et al. (023): (ii) End-to-end (E2E) method (Chenreddy & Delage 2024. Yeh et al. 2024) to minimize the expected risk certificate. For clarity, we refer to the application of CRC to ellipsoidal prediction sets as CRC-E, and box prediction sets
CRC-B, The same naming convention is applied the CRO and E2E methods for consistency. The implementation details of each baseline method are given in Appendix E.2
We utilize the following metrics to evaluate the performance of three methods. (i) Risk Certificate:
The average of (X) across all test samples; (ii) Decision Loss: The average of o(Y, 2te (X)) across all test samples; (iii) Robustness: The proportion of test samples where the o(Y, 24 (X)) is less or equal to (X); (iv) Coverage: The proportion of test samples where the true label Y is covered by the prediction set (X).
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5.1 SYNTHETIC DATA ON PORTFOLIO OPTIMIZATION
In this simulation, we define the loss function as Φly: =) = -y" where y TR2 and 2 (=
[0, 112 ||=|| 1). The labeled data and test data are generated by:
Y = - - 2X c1,
Y2 = 33x X2 62
where (2,, X2), and e = (e1, The covariate X ((11,) 2.25 12). where 12 is 2-dimensional identity matrix. The noise N(0, I2) is independent of X. We only consider ellipsoidal prediction sets since the oracle prediction set of X is ellipsoidal under the normal noise setting. Further experimental details will be presented in Appendix E.1 All methods are evaluated over 100 trials, and the average results are reported.
CRC-E
EZE-E
CRO-E
Robustness
Coverage
7.35
1.00
7.30
0.95
7.25
Loss
10
7.20
0.05
0.1
0.15
0.2
0.25
0.05 6.1
0.15
0.2
0.25
0.05 0.1 0.15 0.2 0.25
0.05 0.1
0.15 0.2
0.25
Nominal level
Nominal level
Nominal level
Nominal level
Figure 2: The results of risk certificate, decision loss, robustness, and coverage synthetic data when varying nominal level a with identical sample size n = 1500. The horizontal gray dashed lines refer to robustness levels. The prediction sets are ellipsoids.
Results. We evaluate the decision performance of CRC and the baseline methods by varying the nominal level As shown in Figure CRC consistently outperforms the baselines in both risk certificate and decision loss. addition, CRC also maintains the robustness level to the nominal target, while the baseline methods tend to be more conservative. With respect to coverage, CRC attains a much lower coverage rate than the robustness level, which verifies the motivation of our method. Additionally, the results for varying sample sizes are shown in Figure 3 CRC-E continues to show strong performance across all metrics, demonstrating its stable advantage. In Figure 6 of Appendix E.3 present the density plots for risk certificate and decision loss when 0.15. The overall density of CRC is shifted towards the lower loss region, further validating its superiority.
CRC-E
Risk Certificate
Decision Loss
Robustness
Coverage
7.30
Locs
Rate
0.8
7:15
7:05
0.4
1000
1500
2000
2500
3000
1000
1500
2500 Samples
2000
3000
1000
3500
2000
2500
3000
1000
1500
2000
2500
3000
Samples
Samples
Samples
Figure 3: The results of risk certificate, decision loss, robustness, and coverage on synthetic data when varying sample size with identical nominal level a 0.1.
Since the RAC method proposed Kiyani et al. 2025)) is applicable to discrete decision space in classification problem, conduct the simulation RAC method by discretizing the label space Y and decision space Z. The results are provided in Appendix EE.4 addition, we also report the simulation results under the polyhedral prediction set in Appendix E.6
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5.2 US STOCK PROBLEM
We conduct an additional experiment on the portfolio optimization problem using a real-world dataset, following the experimental design outlined in Chenreddy al. (2022). The dataset com- prises historical US stock market data from January 1, 2012, to December 31, 2020, covering 64 stocks across eight different sectors. Daily percentage gains or losses are computed from the ad- justed closing prices of consecutive trading days and used as labels. To enhance the input informa- tion for the model, we also incorporate the trading volume of individual stocks and several market benchmark indices as covariates. To evaluate the robustness of the methodology, lect 15 stocks from the pool of 64 as the investable asset set in each experiment and repeat the process multiple times to mitigate the influence of random chance. We define the loss function as 6 2) = -y 2, where y R and Z {2 E [0. 1|9 | = 1, 20. 0}.
randomly
Table 1: The results of risk certificate, decision loss, and robustness under nominal levels = 0.1 and a 0.2 on the US stock problem.
Nominal level a 0.1
Nominal level 0.2
Method
Risk Certificate
Decision Loss
Robustness (%)
Risk Certificate
Decision Loss
Rohustness (%)
CRC-B
1.160
-0.055
90.9
0.731
-0.059
80.6
CRO-B
3,794
-0.051
99.9
3.017
-0.054
99.5
EZE-B
2.129
-0.046
96.7
1.512
-0,041
92.7
CRC-E
1.028
-0.077
90.8
0.701
-0.075
80.6
CRO-E
6.345
6.195
-0.046
E2E-E
4.995
-0.071
98.6
4.503
-0,070
96.4
Results. As shown in Table 1 CRC outperforms the baseline methods in both risk certification and decision loss. In terms of robustness, CRC maintains a level close to the target 1 a, demonstrating strong stability and adaptability. In contrast, E2E and CRO frequently exceed the nominal robustness target, which indicates the adoption of overly conservative strategies that lead to higher losses and risks. Overall, CRC achieves superior balance between risk control and decision performance.
5.3 BATTERY STORAGE PROBLEM
In this subsection, consider battery storage control problem based on the frameworks of Donti ctt al. (2017) and Yeh et al. (2024). Given hourly electricity price forecasts y E RT and contextual covariates over hour horizon, the controller determines the charging power 2 IR. discharg- ing power gut E IR.,, and the resulting state of charge state E R..," subject to the constraints for t ,,..., T.
20
state
B state state
21
21-1
2t + 724 2t < B.
0 5 in S in 05 2t < out
Here B denotes battery capacity, denotes charging efficiency, and etx out denotes per-hour power limits, The objective balances three key factors: (1) Profit from arbitrage, which involves buying and selling energy based on the prices E RT, (2) Flexibility, which is encouraged by maintaining the battery's state of charge close half of its total capacity; (3) Battery health, which is preserved by penalizing large charging and discharging magnitudes. The resulting loss function is:
state 1 | + E(11="111
o(y 2) Σ
(2
2t =) + 3
+|2 uu 112)
Following Donti et al. (017) and Ych al. 2024 we also set T 24 hours, B = 1, 0.9.
cin 0.5, out 0.2, 0.1, and 0.05.
Results. Figure 4 presents a comparative analysis of the CRC method against the baselines using ellipsoidal prediction sets. For clearer visualization, negative indicator values are mapped onto the positive half-axis via sigmoid transformation. The results demonstrate CRC's consistent superi- ority over both E2E and CRO across all three key metrics. As the nominal level increases, CRC effectively mitigates risk in all measures, sustaining the lowest risk and loss values at higher levels.

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In addition, CRC maintains robustness values close the nominal target, highlighting its stability and adaptability. In contrast, E2E and CRO consistently exceed the robustness target, leading to decisions characterized by excessive conservatism. The results of the box prediction set is presented in Figure
CRO-E
Risk Certificate
Decision Loss
Robustness
1.0
0.4
1.00
0.95
Sigmold(Loss
Sigmold(Loss)
0.3
0.90
0.4
0.80
0.1
0.75
0.70
0.05
02
0.15
0.25
0.05
0.2
Nominal level
Nominal level
Nominal level a
Figure 4: The results of risk certificate, decision loss, and robustness when varying nominal level on battery storage problem. The prediction sets are ellipsoids.
CRC-B
E2E-B
CRO-B B
Risk Certificate
Decision Loss
Robustness
0.45
1.00
0.40
0.24
0.95
Sigmoid(Loss)
Sigmaid(Loss Loss)
0.22
0.90
0.20
0.85
0.30
0.18
0.80
0.25
0.16
0.75
0.20
0.14
0.70
0.15
0.12
0.65
0.05
0.1
0.15
0.2
0.25
0.05
0.1
0.15
0.2
0.25
0.05
0.1
0.15
0.2
0.25
Nominal level
Nominal level
Nominal level
Figure 5: The risk certificate, decision loss and robustness when varying nominal level a on battery storage problem. The prediction sets are box.
CONCLUSION AND DISCUSSION
This paper introduces Conformal Robustness Control (CRC), a novel framework that optimizes the construction of prediction sets for robust decision- making by directly minimizing the expected risk certificate under robustness constraints. Unlike existing conditional robust optimization methods with conformal prediction sets, CRC adopts robustness constraints instead of coverage constraints, expanding the range of feasible prediction sets and enabling more efficient decisions. Theoretical guarantees for both robustness and optimality are provided, and empirical results on real-world data demonstrate significant improvements over baseline methods. Our work also identifies future research directions in data-driven robust optimization, such as developing more efficient strategies to solve the optimization problem and designing domain-specinf parameterizations of prediction sets to achieve higher-quality decisions in practical applications.
ACKNOWLEDGEMENTT
We would like to thank the anonymous reviewers and the area chair for their helpful comments and suggestions. Changliang Zou was supported by the National Key R&D Program of China
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(Grant Nos. 2022YFA1003777 2022YFA1003800) and the National Natural Science Foundation of China (Grant No. 12231011). Yajie Bao was supported by the National Natural Science Foun- dation of China (Grant No. 12501408), the Postdoctoral Fellowship Program of CPSF (Grant No. GZC202519999 1999), and the fellowship from CPSF (Grant No. 2025M773046). Haojie Ren was sup- ported by the National Key R&D Program of China (Grant No. 2024YFA1012200 the National Natural Science Foundation of China (Grant Nos. 12471262, 12522115), and Shanghai Jiao Tong University 2030 Initiative.
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A USAGE STATEMENT OF LARGE LANGUAGE MODEL
We used a large language model solely for improving the fluency and readability of the manuscript. The model was not involved in research ideation, experimental design, data analysis, or result inter- pretation. All scientific contributions and substantive content were solely produced by the authors.
B MORE DISCUSSION ON RELATED WORK
B.1 RELATIONSHIP BETWEEN THE CRC PROBLEM AND THE VAR PROBLEM
The following proposition illustrates the relationship between the VaR problem and the CRC prob- lem
Proposition B.1. Let (x) = arg min VaRı-a (0(Y,2) X) be the unique minimizer of VaR
problem. There exists prediction set u such that 24 29. Moreover, the robustness constraint is satisfied:
o (Y, zn S TuA (x)] 21
Furthermore, there exist cases where z = zus with Ut being the solution to optimization problem and cases where 29 # zy..
The above conclusion indicates that, at least in some cases, decision zy- and decision 20 are consis- tent. When decision zy- and decision z are inconsistent, the expected risk certificate generated by decision zy- will also be lower than that of 2 indicating that zu- still holds practical significance.
Proof of Proposition B. 1 To find a prediction set Ut such that 20 equals zuo, is sufficient to define prediction set 77 in the following form:
ue {v y: (y, (x)) s VaR (o ( X = =)} e X.
Next, will proceed with the verification. Since the coverage constraint is sufficient condition for the robustmess constraint, we have
P ( (Y, (u S (X)] 2 P(Y EUR(x)}
= (Y (x)) VaRi-a (x,, xx}
a-
Thus, we verify that robustness holds. Secondly, based on the definition of quantiles, have P(Y E (x))))
a, Vz E X. Therefore, we can control the upper bound of VaRi-a (Y, (X)) | z) in the following way:
VaRi-a (0 (Y, =xe X x) < max & (1, VE(R(()
o (g,,0(z)) (=))(())
max
< VaRi-a (6 (Y,2 (x)) X = x)
That is, Zng is also the optimal solution in the sense of minimizing the a quantile. Therefore, tye equals 20 if Q is the unique optimal solution.
For the scenario where decision z is equal to decision zy., consider the following example. Let X' {0, 1} and Z {0, 1}. Suppose that the density of @YY, 2) given X = is as follows. For E (0, 1), = 0, the density is
1
f()) 25
00100 10]].
For E (0,1), 1, the density is
f(0) = 25
[[0
31 << 10)
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It can be verified that when 0.1, the optimal solutions of VaR and problem 4 the same:
z.- = -000
[0, if is
0. is
For the scenario where decision 20 is not equal decision 2y,, consider the following example.
Let X {0, }} and Z {0, 1j. Suppose that the density of @YY, x) given X = is as follows.
For (0, 1), 0, the density is
f()) 15
1 1
1(4h 350cck
2 +
1(4h 15o< 4h +1 + < <0)]
For (0,1), 1, the density is
f(() 15
2I <
- 3(06-350<4
2 + 2
- 15 4k + 10).
k=1
can be verified that when a 00.4 4e/30
(e -04--4e/30
sufficiently small), we have
0, is
0,
if
On the contrary, the solution of the problem
is different
1, {
0, if 0
1 4- (y E e(v, 6.5 6.5 {y e 1) s 5},
is is
Note that the example above, X' and Z are discrete spaces. We can naturally extend them to the continuous spaces [0. 1] while keeping the conclusions unchanged. The specific details are omitted here.
B.2 RELATIONSHIP BETWEEN CRC AND RA-DPO, RA-CPO IN KIYANI ET AL. 2025
For classification problems, Kiyani al. optimal decision:
2025 proposed the following RA-DPO framework for the
min E[r(X)) =))(()
s.t. PP (Y,,()) S r(X)) 2
This optimization problem can be viewed as marginal version of the VaR problem. In addition, Kiyani al. (2025) also defined an optimal decision framework based on prediction sets, called RA-CPO, as follows:
min
(X) s.t. PPY U(X)) 2 a-a.
The difference between RA-CPO and CRC lies in the fact that the former employs a coverage con- straint rather than a robustness constraint. We can leverage the idea from Theorem 3.2 in Kiyani al. 2025 to prove the equivalence between the CRC problem and the RA-DPO, RA-CPO problem.
Proof of Theorem 3./ Since RA-DPO and RA-CPO have been proved be equivalent in Kiyani et al. (2025). it suffices to establish the equivalence between RA-DPO and CRC.
Let (2 RAA-DPO (zz), RA RA- (x)) be optimal solution to RA-DPO. Define the uncertainty set
4" (w: {u= (v. RA-DPO (x)) RA RA--DO (A)P))()))
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Then,
PP (Y, 2y (X)) S (x)
2 P(Y (X) P ( (Y, RAA (X)) I RA (ARA-DPO(X)) (X))
Thus, Ut satisfies the constraint of CRC. Moreover, by definition,
ru- (x) arg min max (U, ZEZ (El
max & RA- S RA-DO- yEll(()) (a)
Hence,
R[ru- (X) I Bl.RA-DPO(X" (X)).
This shows that any optimal solution of RA-DPO induces feasible solution to CRC with risk certificate least as good. Conversely, let 22 be an optimal solution to CRC. Define
RA-DPO (x) zu- (z), y .AAA- (AAD)P((z)) (I) -Tu (z).
This pair is feasible for RA-DPO and satisfies B[ (AAAAADD)O)) = B[rz-- (X)). Therefore, RA-DPO and CRC are equivalent, and the thcorem follows.
B.3 SUPERIORITY OF PARAMETRIZED CRC OVER PARAMETRIZED RA-CPO
The parametric formulation of RA-CPO in 9 is given by:
min E [X)( IE0
s.t. P(Y E (, 211))11-- 21- a.
The difference between parametrized RA-CPO and parametrized CRC (5) lies the fact that the former employs coverage constraint rather than robustness constraint. The relationship between the two frameworks is formalized in the following proposition.
Proposition B.2. For any parameterized prediction set Uo()), it holds that
ElygeRc (X)] S BlygRA cpo (X)],
where OCRC and OBA-CPO denote the theoretical optimal solutions of the parametrized CRC and parametrized RA-CPO problems, respectively. Moreover, there exist cases which the inequality is strict.
In fact, no constraints are imposed on the prediction set, then as proven in Section B.2 the RA- CPO and CRC frameworks equivalent. However, in regression settings, prediction sets gen- erally required to satisfy certain structural properties-sut as convexity and boundedness- ad- dition to being parameterized to render the problem tractable. As consequence, once the prediction set is parameterized, the solution derived from the CRC problem typically outperforms that obtained via RA-CPO.
Proof. We first show that E|rgenc (X)] < E[paRA-cpo (X)). Let UGRA-CPO be the optimal solution to the RA-CPO problem. Since also satisfies the constraints of the CRC problem, the incquality follows directly from the definition of the CRC problem.
We now proceed to construct case where the inequality strict. Consider a parameterized predic- tion set of the form:
Ua(x) = (y R ()-m((x)) (xx:)) (z))) < 0}. R
where Y X ~ (H)()), (()). Let the loss function be @(v,2) Tz. Then the coverage probability given by:
P(Y 4. = P(X
where X denotes chi-squared random variable with degrees of freedom. To analyze the robust- ness constraint, we derive the dual of the inner maximization in the CRO problem:
max -y xX) = 1 - x(x) 2x()). (x)
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By the definition of the robustness level, we have:
P O(Y (x)) = max Q(v o(y, (x)))
vE(b(X)
P Y (x) - S
x =o
P =x( (Y ((X)) S V0 (x)=0)(
=o(X)T(2 (Y ((X))
(X) S <VD - P[ 2 V0..
Therefore, when q 2 obtain:
gCRC Φ 2p xg.1- = BRA
where X9.1-a and 2 denote the (1 qquantils of the X₃ and NO, 1) distributions, respec- tively. In this case, it follows that:
E[rgeRe (X)) < B(rgRa cro (X)),
which completes the proof.
B.4 IMPLEMENTATION OF CAL-CRC
Algorithm Discretization construction of Cal-CRC
1: Input: Same Algorithm 2 Discretized space D with finite cardinality. Discretization map-
ping A(- : y + y. Step size To > O.
2: Discretization: Obtain the discretized calibration set Deal = {(X,, D by discretiza- tion mapping A.
3: Calibration initialization: Uca (Xm+1) 0.
4: for E do
5: Define the augmented calibration set {(Xi, P.1) n+1
:=no+1'
6:
t
st n- En for (iv (x., s To (X))
7:
8:
while 11-aa do
9:
1 + 1+
5 + n-no+I Zimmot {f {o (8," (X,", S ren,((X))
nit1
10:
11:
end while
12:
iu t.t.
13:
if o U, 20 (( S STo To (Xn+1) then
(ca((En11) Uca (Ca((X+11) {g}. end
15:
end for Anti-discretization: (Cca((nn1)) t A- (Mccil
17:
18:
Output: (X++).
B.5 OPTIMALITY ANALYSIS OF CAL-CRC
Under certain conditions, the discrepancy between Fuca (Xn+1) and ru, (Xn+1) is expected to be negligible. For instance, if ev 0 for any y E y. then by the definition of Ucal, we have Ug (X++1) (Xn+1). Consequently.
max Vet 1)
Mly AMoa (Xn+1))
max yells (x,11)
H(y, 2 (en max Hlv. Ri, (Xn+1)); yell, (X.11)
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where (a) follows from the inclusion relationship between the two prediction sets, (b) holds due to the optimality of 2u40 (Xn+1) over L4, (Xn+1). On the other hand, from a different perspective,
max yEMCAa Nn
Oly, HMw (X++1))
max yEMeaa(Nn11)
(yy, 2Mso (Xn-1))
max NEM (xa11)
Q(y. Mlp (Xn+1));
where (c) is due to the optimality of (uaa(( (++1) over Mcat, and (d) follows from the definition of the Mcal- Combining these results yields Tug, (Xn+1) Tuea (Xn+1). We now consider more general setting. First, state the generalized conditions, and then present the corresponding theoretical results.
Condition B.1. Assume that for all y E y. we have [+a) S to. where to is a positive constant.
Condition B.2. Loss function & is Lø-Lipschitz in decision 2 for any E y. The decision =Bo.1 (Xn+1) I.--iipschitz S lo- The risk certificate r r (Xn+1) T.p-Iipsciitz S lo-
Theorem B.1. Suppose that 0u is obtained by running Algorithm on the training dataset D Dirsin- Under conditions B.1 and B.2 in the calibration process. the following result holds:
TUcal (Na+1) S Tuao (Xn+1) lo(Lo L.).
Proof. For any E Ucat (Xn+1), we have
Φly, 20 (Xu+1)) I dly. 20 (Xa+1)) + to Lo
S fv (Xn+1) + to nHa (Xn+1) + lobbbbz Lo).
Therefore,
(X oly, zca(x+1))))
max
max vEMca(Xn41) 28, (Xn+1) S S (Xn+1) + Lo L.).
Note that T is monotonically increasing in for any x E X. Hence, if the initial model Ug already approximately satisfies the a robustness requirement, the calibrated threshold i will generally remain small for y E y. As a result, Ucal can maintain risk certificates and decision losses comparable to those of the initial model Ug.- Conversely, if the initial model's 's robustness is significantly below 1- then although Ucat still guarantee 1 - a robustness, it may produce relatively conservative results.
C PROOF OF MAIN RESULTS IN SECTION 3.3
C.1 PROOF OF THEOREM 3.2
By leveraging the finite covering property of the function class and large-sample probability in- equalities, we aim prove that the empirical estimates converge uniformly their expected values, thereby establishing the conclusion of the theorem. First, given an cov-ringg O with smallest cardinality of the function class, and applying the Dvoretzky-Kiefe Kiefer- Sofffr-Woofooitz (DKW) inequality [Massart 1990), we have
Σ 1Vo.
(Vo, (x, M)) (Vo, (X,Y) < 2 6in 20 |l oo, <2n)e 2nořn.
sup 1ER.0EE@ram
where éin is the tolerance error, whose specific value will depend on the covering number N/ booo (nx) and will be specified later. According to the definition of -overrin,,,,, for any given
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8 E O there exists 0u Degn such that 1|0 Bull I C2n- Therefore, the upper bound the
deviation between the empirical estimate and the expected value can be derived as follows:
1
(V, X) 0 (V s 0}
T L(Vo, (X,, K) s 0}
n (XV Y,) 5
(10)
n
1o. (x Y2) < 03 (Vo, (X, Y) 50)
72
+
(Vo (X,
[ IP (Vo Y) - 0H
(11)
Leveraging the Lipschitz condition, TT) and 11 can be bounded as follows:
Vo Yi) [0
1 ( (X,, Yi) - 0}
n
(X,, Yi) & (L + L,) - 0o.1
Σ
(oo, (X,, Y,) 50)
n1 1=1
t=1
+
(Vo, (X,, s - +.)10 - 0o1l
Σ
(Vo (X,, 8) o0}
Σ 14Vo Yi) - (X,, 5
S 4
sup
10
1=1
+ sup
P ((-(Tob + 12 S Vo, - (L + L,p (2m)
and
(VVo, - S OHl
$ (
+ -0) (V,,, < (LgL, + 1, 011}
(x,, v -LgL, + L,) [r] 00 01}
< sup PI-(LOL. + L,)cn, Vo, (X (LoL, L, L,)c2n) 0e
Finally, we consolidate the above results and obtain
(X, Y)) <0) (V](XX < 0}
n2
5 sup
Σ 1lVo. (X,, Yi) S {t) (VV,, st}
EER can
n
2 sup 0
TP (- + T.p 122n (X,,) (Tagten T.p)e2n)
Let 61m
(e.0 Hoog 2n
We have
S5/ log (2N(O,
|loo, 62.)) + log(1/0)
sup dee
1(Vo(X,,))) s 03 (Vo(XX) 50)
n
2n
+ 4(LLL. + Lr)Pucens
(12)
with probability at least - 5. Furthermore, we have, with probability at least - 5,
log (2N || b1 <2n)) + log(1/8)
P (V, S (x)(X) D.) a 5V
2n
4( + Lr)puezn,
where 0 is the solution the CRC problem dataset D,,.
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C.2 PROOF OF THEOREM 3.3
Let 6" be the convenient notation of 2.. The following formula gives the risk difference between estimated model 0 and the optimal model 0%:
E [r((X) P.] - E [ro- (X)] - (() [n)]
-
ro(Xi)
(13)
1=1
ra(X;) Σ ro (Xi)
(14)
11
+
To- (X,) E[ro (x)
(15)
For formulas 13 and (5). we adopt proof strategy similar to that of Theorem 3.2 to demonstrate that the empirical estimates converge uniformly their expected value. Given O let On E θ be the approximation of 0 in 62n covering θ e2n
We have
(X.) E[ro(X)] 5
(X.)
Σ 11
1
rea (Xi)
n
re (Xi) - 13 (x)
n
[B(ra (X) [x(o(X)]]
1
n
sup e
rao (X.) Elro, (X)
n2
22,
where the last term is derived by applying the Lipschitz condition. equality, we have:
According to Hoeffding's in-
n
log(2N((O, || boy <2n) log(1/0)
sup
ro (X+) - E[ro, (X)]
S 2B,
2n
with probability at least 1-8. Furthermore, we can derive upper bounds for formulas 13 and 15
For formula [14), we assume that event (12) the proof of Theorem 3.2 holds. At this point, since i is the solution to the finite-sample CRC problem, we deduce the following result:
n T (X.) S O.
ro(X,)
Integrating the above conclusions, we can obtain the following result:
log (2N( || |loc, 62n) + 1og(1/0) + 44,
EXra(X) Pn] - -Brr- (X)] < 4B,
2n
holds with probability at least 25.
C.3 PROOF THEOREM
Suppose that the calibration set is Deal Y.) the test data is (Xn+1, Yn41) and model O has been trained from the training set Dtrain = {(x,, Yi) First, we demonstrate that the prediction set (ca((.) achieves a coverage. Note that
P(Yn+1 E Ucaa = pP (Ynti, ( E (Xn+1)) S Pr (Xnti))
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Let W n(xn++1; Yno-1),... (Xn+1. Yn+1)) } be unordered set. Note that f is measurable with respect to statistic W. We will complete the proof by leveraging the symmetry of the data.
p Yn+1, i011 (Xn+1)) S r ii (Xn+i)}
{ (Yn+1, 28 Piis (Xut1)) S To (Xuti)) [w]]
E
E
1
a+1
E
1
Σ i=no+1
1 (oo (i (Y. S Tooie (ii))
no
The first equality stems from the law of total expectation. The second equality arises from the symmetry the data, a technique frequently employed in proofs within conformal prediction meth-
ods Vovk et al. 2005 Liang et al. 2024). The final inequality is derived from the definition of threshold (*++1, as referenced in Algorithm 2 Finally, since the coverage constraint is a sufficient condition for the robustness constraint, we can obtain the robustness guarantee, i.e.,
I o (Yn+1, Zuca (Xn+1)) S ruea (uea (Xn+1)) 2 PP E (e 2 1-a.
D ADDITIONAL THEORETICAL RESULTS
D.I THEORETICAL RESULTS FOR VC/RADEMACHEER CLASS
In this section, present theoretical results on robustness and optimality when the function class has a finite VC dimension. Additionally, we discuss a decision-making method based on parti- tioning the covariate domain (Chenreddy al. 2022). Under this approach, the corresponding function class possesses a finite VC dimension, thereby exhibiting relevant convergence properties. Let VC(C) (C(((z, y) 1 {p(y, 20 (z)) < (x(x)) : Q}) denote the VC dimension of the robustness- induced classifier class.
Theorem D.1 (VC class robustness). Suppose VC(C) = H < oc. Then there exists an absolute constant C O such that, with probability at least 1 5,
H
log (2/8)
( S
rrg(X) ( Dn) 2 a
a- CV
2n
Proof. By McDiarmid's Inequality, with probability at least 1 - 8,
1 1s (0 S (Y xx() S ro(X)}
sup 8ce
2=1
< E
sup
(Y, (xo)) ( (x))
(16)
log(2/6)
+
2n
The expectation in (6) be bounded using the standard VC-elass Rademacher bounds (Ver- shynin, 2018, Theorem 8.3.23); there exists a constant C' such that
E sup EEB
Σ
(0, & re( P (o 20()) s re(()}}
sCV -ー
1=1
Combining these results yields that, with probability at least 5:
H
log (2/8)
P f (x)))(X)) S rr((X) Daj 2 1- - a - C
2n
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Theorem D.2 (Rademacher risk). Assume additionally that [ro(=)] M for all E θ and x E X.
Let 0r be the optimal solution of problem robustness level a An where An C 11 +
n
log (2/6)
Then, with probability at least
26,
2n
log log(4/0)
E [xx re- (X)
2M =
2n
where R. (-)(( :0 } denotes the Rademacher complexity for function class (ro(-) : }).
Proof. Let 0* - We bound the risk difference between the estimated model & and the optimal
model H* follows:
E (x((X) Dr] -E (X)] <
(x)a(X)
[D,]
re (X:)
11
1
rx(X)) Σ ro- (Xi)
n
ro- (X.) B[ra- (X)
n
1
sup
ro(X;) 13 [ra(X)]
(17)
8E0
71
1
O (X.)
Σ (Xi).
(18)
For the term 17 by McDiarmid's inequality, with probability 1 1 8:
log(2/8)
sup IE
To (X()
E (X)]
$ E
sup
re (X,) 18 [[[o(X)]]
+ 2M
7
2n
(19)
The expectation in 19 is bounded via Rademacher complexity:
E
sup 0:6
(x) E [ro(x))
< 2IE
sup
(rro(x)) 22 O)).
OE
1=1
For the term
18)
whenever
H
log (2/()
LY 20- (x.,))) S re- (X.)) (o z0- z- (X)) rx- SCV
n
2n
the definition of problem 6 implies
Σ rx(x))) 1 n
Σ ro- (Xi)
By Theorem D.I this event holds with probability at least 5. A union bound gives that, with probability at least 1 25:
1oo(2//)
E [r)(X) Da. E (ro-(x)) < ((ro()))) 0]) + 2M
2n
Remark D.1. In Chenreddy al. (2022). the authors introduce decision-maiiee makinng framework that leverages data driven learning of underlying structures to categorize individuals into classes based their covariates. For each class, prediction is constructed, which in turn induces
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specific decisions and risk certificates. Denote the trained classifier by A: X - [K] and the model parameters by 8. The decisions and risk certificates take the following forms:
K
(2) = 2/ ({Ax)) =k
To(x) = Σ ()){((z)= k}},
1=1
k=1
where Zk 2,rk I for each k [K] Considering portfolio optimization problem with
loss function o(y,z) = -y and = 12, the set {(I, + {0(y, r becomes a subset of the following family:
(1,1) - 1
Σ rK) 1(A(x) (()
= : an E R for all E [K]]
This family corresponds to a finite-dimensioal linear space of functions and therefore has VC di- mension at most (q+ 1)K. Applying Theorem D.I obtain the following convergence guarantee: with probability al least -
(g 1)K
12g(2/8)
P (o I (x(X) 21-a-
2n
Similarly, the function class (re(z) 0 E θ) is uniformly bounded and has VC dimension at most
K 1. Hence, its Rademacher complexity satisfies Ru((ro E θ}) S C' CVVK |K+ for some constant C'. This leads the following bound on the excess risk: with probability least 28,
n
K K+
log (2/6)
E [rra(X) - ro- (X) Dn] S 4C's
2M
n
2n
It is worth noting that, under the finite VC dimension condition, the resulting convergence rate achieves the order O( OI/I/)).
D.2 THEORETICAL RESULTS UNDER SMOOTH CONSTRAINT
In this section, we we analyze the theoretical properties of the optimal solution the following smoothed optimization problem:
1 n
s.t. Σ a
arg min 8ce n
Σ re()))
iio0 20 S ro(X))) 2 1 a.
(20)
2 So e is the Gaussian error function (af(bba)) )), where erf(.x) VR Here, i(a < b) + Vio
and a controls the smoothness of the surrogate. This formulation provides a smoothed approxi- mation of (6) and serves as the direct optimization target Algorithm T The next two theorems provide a non-asymptotic guarantees of the robustness and expected risk certificate value of the resulting solution.
Theorem D.3 (Robustness). Let O₂ denote e-covering of the θ with coverage number (omberr((G) N ||- 1loo e), and let be the solution of problem 20) Under Conditions 3.1 3.2 for any independent data (X,Y) ~ P and conditioning on the labeled data Dn,. we have
log(2N((:) || (b,0,)) c) log(1/8)
P (Y & rx(x) Dm) Dm121-0 21-a-
2n
Ls Lr)e
V opo.
V2TTO
with probability at least
8.
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Proof. For any A E O, let 0g E Oe such that ||0 - Bull < We decompose the deviation as follows:
i 1 (Y, (X.:)) S (x()) E (Y, (Y, zx)) S rx(1)
x
i (0 (o 20 5 rx()) Σ t0 20, 26. (X:)) s rob (X.))
(21)
7
+
1 (Y, 280 (x,) S re 1 (x,) [1 (6) (Y, 20. (X)) S roo (X)]
(22)
n i=1
+
E
(6b (Y, =0, (x) - 00 [(] E { (Y, 20 xx)) & sx)())]]
(23)
We can apply the Lipschitz condition to bound term
21:
i to =0 ( 1
Σ
i
S ra (X1))
i=1
[ (Y,, zo(:)) rro().)) - 1 10 (X, =8, (Xi)) S re, (X.))
1 z0 - o z0,
20, (x:)) + (xa(X.) - re (x:)
no 2m
(2) (I, Lg L,
V2na
where (a) is due to the fact that function }1:--} is both components, and (b) is derived from condition method to derive its upper bound:
Lipschitz continuous with respect to its
gv/25
3.1
For the term 23 we can apply the same
E [ ( (Y, 20, (X)) & ro, ro, (X)) - E (o (Y, < {x){]
I E [] 10 (Y, z0. (X)) S Ta, (X)] - 1 (6 (x)) s [X))]
11 (Y, 26 (X)) o 2o, (X)) + (x((X) re, (x)
/2ma
LL Lr)e
V2nT
For term
22p. by Hoeffding's inequality and a union bound over 0g E Oe, with probability at least
1 1-6 5,
1 i (Y,,, 20, (x,)) S re, (X+) - [i 20, (X)) T8, (X)} 1og(2N((,))) 1l.c c)) + log(1/8)
sup 0,
n i=1
2n
Combining these bounds yields:
(X, S rr((X)) Dm] 2 a log(2N((,,
Il,o, E)) + log(1/8)
2L Lr)e
2n
22TO
with probability at least 1 8. Finally, we quantify the diserepancy between the robustness E ( (Y, 20 (X)) rx)] and its smoothed version E. [1 (6 (Y, xx()) $ ro(X))] Let fo(-)
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denote the density of Vo(X, Y). Then:
E [] 20 (X) - rx() - -(0((X,20(X)) & [X)H]]
1
1 + erl( -t /20/ )) so(t)dt+ f-
t.0
1
(1
)
1
2
-
2
(1+
erf(
/20
fo(t)dt
1AB
P0
erf( -t
dt + Pu
-
1teff(
1 erf(
dt
220
2
V20
000
WV
2
P0:
where (a) follows from the bounded density assumption 3.2 and (b) is derived via standard Gaus- sian integral identities. Incorporating this bound into the previous result, we conclude that with probability least 1-
P f zx()) s <r((X) Pa} 2 log(2N((O,11 -1::0
e)) + log(1/0)
2n
2(L2Lo + +L,)c
π
V2TO -V 2000
opo
The key difference from the non-smoothed case the presence of the term a which quantifies the bias introduced by the smoothing. Below, we directly present the relevant optimality theorem, as its proof and conclusions are almost identical to the non-smoothed case.
Theorem D.4 (Optimality). Let 0%,, be the optimal solution of problem
(5) at the robustness level
1 + An where An conditions of Theorem
v oopo. Under the same
conditioning on Dn, we have
E rx(X) rog, (X) Pa <4B, log(2N((,,) 1 e)) + log(1/8)
4Lr
2n
n
with probability at least 1 28.
E EXPERIMENTAL DETAILS AND ADDITIONAL RESULTS
E. EXPERIMENTAL DETAILS OF CRC
To accelerate the alternating optimization of CRC and promote stable convergence, we partition the labeled data into two mutually exclusive parts: the first part is used for pretraining CRC, with the resulting parameters serving initialization for subsequent alternating optimization; the second part is exclusively dedicated to the alternating optimization phase.
The baseline method employs the same partitioning strategy: the first portion trains the prediction model, while the second portion is used for calibration or solving downstream optimization tasks. To ensure comparability, all methods uniformly employ the same scoring function and optimization objective in experiments.
Pre-training Pre-training of the CRC can be approached in two ways depending on the shape of the prediction set: For ellipsoidal prediction sets, the neural network outputs the parameters of multivariate Gaussian, namely the mean vector (a(-) and the covariance matrix Σ(-). We pa- rameterize Σ(-) via Cholesky factorization, Σ(- L()) where L(-) lower triangular. To guarantee positive definiteness, we add small diagonal jitter to the predicted covariance, i.c., E' E+el which raises the eigenvalue floor and ensures numerical stability of the Cholesky factor- ization. Additionally, our training objective is to maximize the Gaussian ligeliiid equivalently minimize the negative log-likelihood:
La
exp -
T T (y (w))
(2T)
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For box prediction sets, quantile regression to directly estimate quantiles. Concretely, train a neural network fa) to output the a-level quantile for input z. The 1 a confidence interval constructed (x), fg /1-a/2 (x)) Benefiting from quantile regression, training objective to minimize pinball loss. Given quantile level (0, 1) and prediction ğ [a(r), the loss for target y/ is
a (y 0), if > 9. (1 aa
Lo (v.
- (5-1)) y),
if S 9.
the cvxpylayers (Agrawal et al. 2019) Python package to implement the implicit function differentiation. The optimization is performed using the Adam optimizer, and select the optimal combination of learning rates (le-2, le-3, 1e-4) and L2 weight decay values (0, 1e-2, 1e-3) to minimize the optimization loss. Moreover, to mitigate overfitting and ineffective training. 20% of the data used for optimization held out as a validation set. Early stopping is triggered when the loss on the validation set fails to decrease for 10 consecutive iterations when the predefined maximum number of iterations reached.
Optimization For CRC optimization,
Smoothing parameters sensitivity For CRC method, approximate the indicator with smooth surrogate fa < b} = 1 (1 + erf( Vie )). We compared the sensitivity of different smoothing parameters a on CRC. The experimental results are summarized in Table 2
Table 2: The results of different smoothing parameters sensitivity of CRC at the nominal level a = 0.1, where the sample size is n 1500. The prediction sets are ellipsoids.
Method Smoothing parameter
Risk Certificate
Decision Loss
Robustness (%)
Coverage (%)
0.01
8.678 + 0.299
7.072 1 0.220
89.8 1 0.7
60.8 5.6
0.05
8.633 = 0.295
7.070 0.219
89.9 = 0.6
59.6 = 5.5
CRC-E
0.10
8,641 0.306
7.071 = 0.221
90.3
59.4 5.7
0.15
8.643 = 0.315
7.070 0.221
90.5
59.6 5.7
0.20
8.649 = 0.308
7.070 + 0,220
90.2 0.5
59.6 5.7
Lagrange multiplier update schedule sensitivity For dual variable we investigated the results of CRC Lagrange multiplier update schedule sensitivity which refers to the number of model parameter optimization steps performed before each update of x. The experimental results will be shown in Table B
Table 3: The results of lagrange multiplier update schedule of CRC at the nominal level a = 0.1, where the sample size 1500. The prediction sets are ellipsoids.
Method À update schedule
Risk Certificate
Decision Loss
Robustness (%)
Coverage (%)
1
8.641 +: 0.334
7.109 + 0.251
89.9 + 0.5
58.7 + 5,8
2
8.528 0.302
7.106 0.251
90.4 + 0.6
56.4 + 5.5
CRC-E
8.478 0.278
7.105 0.251
90.1 0.5
55.4 + 4.8
8
8.452 + 0.282
7.105 + 0.251
89.7 + 0.6
55.2 + 4.9
F.2
BASELINE METHODS
The CRO method is our implementation of the Sealict-then-Calibree framework proposed by Sun et all (2023). Specifically, we first train a predictive model to parameterize the uncertainty set (e.g., outputting the mean and covariance of ellipsoidal prediction sets). Subsequently, we construct a prediction set on the calibration set that satisfies the target coverage requirement. Finally, the prediction set is directly embedded into downstream robust optimization problem to solve for decisions and minimize task loss. Thus, this method reduces task loss while enhancing solution stability, all while ensuring coverage.
CRO
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E2E E2E is end-to-end robust optimization method proposed by Chenreddy & Delage (2024)
and Yeh et al. 2024). Unlike CRO, E2E aims to bridge uncertainty calibration with downstream task objectives by minimizing target loss through global optimization. Specifically, E2E first trains pre- diction model capable of outputting parameters of uncertainty sets. then computes non-conformity scores the calibration set, determines the threshold y that satisfies the nominal coverage a. and constructs the uncertainty set accordingly. Finally, under this uncertainty set, the robust optimization problem is solved to obtain the current task loss. The gradients of the task loss with respect model parameters are backpropagated through the differentiable optimization layer to the prediction model, enabling collaborative updates of model parameters and task objectives. Consequently, the model achieves better alignment with real-world decisions while ensuring coverage and reducing task loss. For fair comparison, we set the loss function in E2E method as the expected risk certificate.
E.3 DENSITY PLOT OF SIMULATION SECTION
The density plots of the risk certificate of three methods are given in Figure Compared with other baseline methods, CRC has achieved the best performance.
CRC-E
E2E-E
CRO-E
Risk Certificate
1.4
1.2
1.0
0.8
0.6
0.4
0.2
Loss
Figure 6: The densities of risk certificate on synthetic data when a 0.15 and 1500.
E.4 COMPARISON RESULTS OF RAC AND CRC
Based on the RA-CPO/RA-DPO framework, Kiyani et al. (025) proposed the Risk-Averse Cali- bration (RAC) method to solve decision-making problems classification settings. However, this method strictly relies on the finiteness of the label space and the decision space, i.c., |>/] < 00, Z < D0. Consequently, the RAC method is more suitable for classification problems and
applicable to regression problems since constructing the prediction set in Kiyani et al. (2025) requires solving the Value-at- Risk optimization problem, which is generally not tractable when the space is con- tinuous. In contrast, our method is grounded in the CRO framework and derives final decisions by directly optimizing over the space of prediction sets, thereby maintaining applicability to continuous decision spaces.
To evaluate the performance of the RAC and CRC methods in regression tasks, we have to make certain adjustments the RAC method. Specifically, simple regression problem can be converted into a classification problem via discretization is, by partitioning the response and decision space into discrete bins, thus allowing RAC to be applied. However, it is important to note that in general regression settings involving figg--imenssonal response (such as the 15-dimensional U.S. stock problem in Section 5.2) discretization often leads to substantial computational overhead and considerable information loss, making the application of RAC infeasible. To ensure the validity of the RAC method, consider the following simple regression problem with loss function Φ(y. -y² and decision space Z = {z E [0, 1 || = 1}. The data is generated by
Yi -1.33
-1+
where Y (Y1, E RR, and (e, R². The covariate X (00, I2) is the spurious feature and noise é1, E2 are independent standard Gaussian random variables. When implementing the RAC method, need discretize both the decision space and the label space as follows.
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For the decision space Z, divide the first dimension [0. 1] into J equally-spaced
points {21.1...., _1 Due to the constraint 21 + 22 = 1, the dicision space discretized into the finite set Zdis {21111 21,1)...... (21,,. (2.,))
-...)).
For the label space y/ we first restrict each dimension of Y to the interval between its 1% and 99% quantiles. This creates a bounded wwo-dimensional box, which benefits the RAC method by ensuring a bounded loss. This box is then divided uniformly into I. I. regions, and the discretized label space dis is composed of the top-right endpoints of these regions.
The experimental results are reported in Table
Table 4: The simulation results of CRC and RAC the nominal level 0.1, where the sample size is n = 2000. For the abbreviation RAC(J, L), the numbers J, L refer to the discretization refinement of decision space and label space, respectively.
Method
Risk Certificate
Decision Loss
Robustness (%)
Coverage (%)
CRC-E
1.384 0.049
0.541 + 0.065
90.0 + 0.8
36.1 + 1.7
RAC(6, 2)
1,730 + 0.039
0.803 + 0.021
89.0 + 1.0
89.8 + 1.1
RAC(6, 4)
1.687 0.032
0.612 + 0.072
90.7 = 0.9
89.9 + 1.0
RAC(6, 8)
1.592 = 0.038
0.725 + 0.042
91.6 0.9
89.8 + 1.0
RAC(11,2) 2)
1.732 0.038
0.803 + 0.021
89.1
89.8 + 1.0
RAC(11,4
1.684 + 0.033
0.576 + 0.066
91.0
89.9 + 0.9
RAC(1188)
1.583 0.036
0.697 + 0.042
9
89.9 + 1.0
E.5 SIMULATION RESULTS ON CAL-CRC
The experiment results of Cal-CRC under ellipsoid prediction set shown in Figure 7 where the simulation setting is the same as that in Appendix E.4
Cal-CRC-E
Cortificate
Decision Loss
Rabustness
2.1
6.90
1.0
1.0
1.5
0.60
1005
1.2
0.45
0.8
0.7
0.4
0.6
0.3
0.00
0.5
0.0
02
Nominal
Nominal
Figure 7: The results of risk certificate, decision loss, robustness, and coverage on synthetic data when varying nominal level a with identical sample size = 2000. The horizontal gray dashed lines refer robustness levels. The prediction sets are ellipsoids.
E.6 SIMULATION RESULTS ON POLYHEDRAL PREDICTION SET
In this section, adopt the methodology from Bärmann et al. (016) to construct parametric formulation for polyhedral prediction sets and integrate it into our proposed CRC framework. Simu- lation experiments demonstrate that under polyhedral prediction sets, method still exhibits better performance compared to baseline approaches.
Following Bärmann et al. 2016), the derivation of parametric form for polyhedral prediction sets is inspired by the parametric representation of ellipsoidal prediction sets. Let B = {y E R ||vi|a < 1} denote the unit sphere in IR?, and let pa(-)) : R2 R2 and Ex(-) RP [ax4 represent the parameterized mean and covariance functions with parameters 8, respectively. The parametric ellipsoidal prediction can be equivalently defined as:
ux(x) {u E R Σg 1/2 (x)
(y 1/2((x)
na (x)) E B
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Now, let 8" be polyhedral outer c-approximation of BY, defined by
B = {v Ky - k),
(24)
where K R k E R are fixed matrix and vector, respectively, and denotes the number of polyhedral facets. The corresponding parametric polyhedral prediction set is then given by:
up (x) = {u y E IR: 2. 1/11((2)) (x) (y plz)) = }
= {u E RR K - + KΣ 1/2 (z))o(a))
(25)
The construction of 8 depends on the dimension q and the approximation tolerance e. For instance, when q 2 and = 0.01, an = 23-facet polyhedron ensures that the approximation error remains are specified as:
below e. this case, the components
a1
2
2T
k = 123,
K
where 0g
cos
sin
for i = 1,... 23.
23
23
a22
The polyhedral prediction 2et can be directly incorporated into the CRC framework. For example, a portfolio optimization problem with loss function ((y,2) 2) -y and decision space 2 { : [0, 1]2 || = =, 2 0}. We can establish that both the decision 2²(x) and the risk certificate r=( are differentiable with respect to 0. This enables the search of optimal prediction sets via gradient-based optimization. Furthermore, the theoretical conditions outlined in Section 3.3 continue to hold, ensuring the validity the corresponding theorems in this extended setting.
In this simulation, compared the performance of CRC with other methods. The experimental setup remains consistent with that described in Appendix E.4 The experimental results are summa- rized in Table
Table 5: The simulation results under polyhedral prediction set with nominal level a 0.1, where the sample size is n 2000.
Method
Risk Certificate
Decision Loss
Robustness (%)
Coverage (%
CRC-P
1.493 + 0.067
0.852 0.037
90.3 0.5
23.5 1.8
CRO-P
1.844 + 0.028
0.978 + 0.012
95.9 + 0.6
90.1 + 0.9
E2E-P
1.689 + 0.024
0.954 0.026
93.3 0.9
89.9 0.9
E.7 ABLATION EXPERIMENTAL RESULTS ON CRC
In this section, we conducted ablation experiments on CRC to compare the performance of CRC and calibrated method. For CRC, we used the parameters of the pre-trained model as the initial values for iteration. For Cal-CRC, we calibrated the model after CRC optimization. For Cal method, we calibrated the pre-trained model. The experimental setup is the same as Section 5.1 and the experimental results are summarized in T
Table 6: The results of CRC ablation experiments with nominal level = 0.1, where the sample size n 1500. The prediction sets are ellipsoids.
Method
Risk Certificate
Decision Loss
Robustness (%)
Coverage (%)
CRC-E
8.660 + 0,561
7.053 + 0.465
89.5 + 0.8
59.8 5,9
Cal-CRC-E
9.413 - 0.516
7.108 = 0.469
90.9 + 1.1
90.5 - 1.2
Cal-E
9.581 + 0.589
7.081 + 0.462
93.5 + 1.3
93.2 + 1.4
29


Reviews:
[{"review_id": "2NylzhywzH", "reviewer": "Reviewer_Cx82", "summary": "This paper proposes Conformal Robustness Control (CRC), a framework for robust decision making that directly enforces a robustness constraint on the loss, rather than the traditional coverage constraint used in Conditional Robust Optimization (CRO). CRC optimizes parameterized prediction sets so that the probability the actual loss exceeds a risk certificate is below a target threshold. The authors provide a differentiable algorithm, non asymptotic theoretical guarantees, and a test time calibration procedure.", "questions": "1. Can you provide a more rigorous theoretical comparison between CRC and RA-DPO/RA-CPO or CVaR-based robust optimization, especially in non-convex or discrete decision spaces? Are there cases where CRC is strictly more efficient or yields better optimality gaps?\n2. How does the choice of smoothing parameter in the surrogate constraint affect the theoretical robustness guarantee and optimality gap? Is there a principled way to select this parameter, and can you bound the bias introduced?\n3. Is it possible to enforce both robustness and coverage constraints simultaneously in the CRC framework? What are the theoretical tradeoffs or limitations in doing so?\n4. How does CRC extend to more general uncertainty sets (e.g., polyhedral, Wasserstein, or discrete sets)? Do the non-asymptotic guarantees still hold, or are there new challenges?\n5. Can you provide statistics on the frequency and magnitude of robustness constraint violations during training and after convergence, compared to CRO and end-to-end baselines?\n6. How sensitive is CRC to the smoothing parameter, Lagrange multiplier update schedule, and calibration split size? Does constraint satisfaction degrade for certain settings?\n7. What is the computational cost and convergence behavior of CRC compared to RA-DPO, CVaR, and end-to-end CRO baselines?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. CRC is formulated as a constrained optimization problem with a direct robustness constraint, and solved using a smoothed dual approach suitable for gradient-based learning.\n2. The paper provides non asymptotic bounds on robustness and optimality, and extends these to finite sample guarantees via test time calibration.\n3. The algorithms and theoretical results are clearly described, and the motivation is well articulated.", "weaknesses": "1. The paper refers to CRO as \"Contextual Robust Optimization\" whereas the foundational literature (e.g., Chenreddy et al., 2022) defines CRO as \"Conditional Robust Optimization\". This is not just a semantic issue, the \"conditional\" aspect is central to the theoretical guarantees and the structure of the uncertainty sets. The paper does not clarify this distinction, which may confuse readers and obscure the technical lineage of the framework.\n2. The paper claims equivalence between CRC and RA-DPO/RA-CPO (Kiyani et al., 2025) in regression settings, but does not provide a rigorous theoretical or empirical comparison of solution quality, tractability, or optimality gaps in more general settings (e.g., non-convex or discrete decision spaces).\n3.  The robustness constraint is enforced via a smoothed indicator (error function), which introduces bias. Theoretical analysis of the impact of the smoothing parameter on constraint satisfaction and optimality is limited, and there is no discussion of feasibility restoration or dual variable stability.\n4.  While CRC can achieve robustness with lower coverage, the theoretical implications of this tradeoff are not fully explored. In some applications, coverage may be required for interpretability or regulatory reasons, and the paper does not analyze the joint feasibility or Pareto frontier of robustness and coverage constraints.\n5. Theoretical results and experiments are limited to box and ellipsoidal prediction sets. It is unclear how the CRC framework extends to more complex uncertainty sets (e.g., polyhedral, Wasserstein balls, or discrete spaces), and whether the theoretical guarantees hold in these cases.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "7BtHfgxaO6", "reviewer": "Reviewer_KBcd", "summary": "This paper proposes a new methodology for the contextual robust optimization problem, where the goal is to optimize the expectation of some worst-case performance for a high probability robust set. To solve this problem, a typical method is to first construct this robust set and then optimize. Instead, in this paper, the authors proposed a new method of combining robust set construction and objective optimization together, which enjoys a better performance. The authors also use theoretical and numerical results to illustrate the strength of the algorithm.", "questions": "In addition to the question in the weaknesses section,\n\nIn Algorithm 1, what would be a good number of gradient descent steps?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The setup, methodology, and results are very clear.\n2. The methodology is novel, and the results look solid.\n3. Both finite-sample theoretical results and numerical illustrations demonstrate the strength of the new methodology.", "weaknesses": "1. The paper mainly focuses on box/ellipsoid robust sets. Is this algorithm also compatible with other forms of robust sets?\n\n2. Theorem is still an overall coverage statement. I wonder whether the authors could show a conditional statement, like with high probability, P(\\phi(y,z(X))\\leq r(X)|X)\\geq 1-alpha-Delta", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "vwpCaXSKua", "reviewer": "Reviewer_jy9S", "summary": "The paper proposes a prediction set construction method that directly provides robustness in downstream decision-making, thereby bypassing the need for coverage guarantees. The authors present a differentiable loss function to train a predictive model to output parameters for a prediction set with asymptotic robustness guarantees. To provide finite-sample guarantees, they calibrate the prediction sets using conformal prediction. This two-step process is empirically validated on US stock and Battery Storage data, along with synthetic data generated for portfolio optimization.", "questions": "* It’s clear that the calibration procedure is necessary to obtain finite sample guarantees, but can we just perform this calibration procedure on any model that outputs point estimates? What is the benefit of using this specific training procedure before calibration? Were there any ablation studies to support this? Similarly, were there any ablations that highlighted the usefulness of the calibration procedure?\n* It’s not immediately clear why coverage is included as a performance measure (I understand it’s used to show that coverage isn’t a necessary condition). It would help if there were some foreshadowing on how this will be used.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper is well-written and organized. The theoretical contributions are meaningful in validating the proposed method. The experiments use a suite of relevant evaluation metrics that further strengthen the paper.", "weaknesses": "This paper shares the same motivations as Kiyani et al (2025) which is why I think CRC and Kiyani et al (2025)’s method Risk Averse Calibration (RAC) belong to the same line of research. But I found the explanation of RAC deficient in this paper. Specifically, highlighting and explaining the weaknesses of their method is quite important in helping the reader juxtapose CRC with RAC. This comparison is especially difficult to do since RAC was not included as a baseline in experiments. \n\nIf the authors can explain the weaknesses of RAC in detail and compare the empirical performance of RAC with CRC (treat RAC like another baseline), I would be willing to increase my score.\n\n_References_\n* Kiyani et al. (2025), Decision theoretic foundations for conformal prediction: Optimal uncertainty quantification for risk-averse agents. https://arxiv.org/pdf/2502.02561.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "uvnikwG1Lp", "reviewer": "Reviewer_HPyM", "summary": "There have been a number of works proposing to leverage conformal prediction to do robust decision-making in the context of two-stage decision-making, where an upstream model will specify the parameters of a downstream problem. In the nominal case, for a true problem $\\min_w f(w, c)$, a predictor will specify a surrogate parameter estimate $\\widehat{c} := g(x)$ against which the optimization is performed as $\\min_w f(w, \\widehat{c})$. Instead of this, a conformalized approach will produce prediction regions $\\mathcal{C}(x)$ such that $\\mathcal{P}(C\\in\\mathcal{C}(x)) \\ge 1-\\alpha$, from which the robust problem $\\min_{w} \\max_{\\widehat{c}\\in\\mathcal{C}(x)} f(w, \\widehat{c})$ has guarantees of being a valid upper bound on the true optimal value. However, many of these initial works proposed this approach from the perspective that the predictor had been separately conformalized only to be later leveraged in this downstream decision-making task, likely as one of many uses of the conformalized predictor. If, however, the goal is to specifically formulate a robust optimization problem that lends itself to creating an informative, probabilistically valid upper bound, this two-step procedure can be unnecessarily conservative. This paper, therefore, proposes a formulation that directly seeks to define a maximally informative valid upper bound and demonstrates the empirical benefits of doing so compared to two-stage approaches.", "questions": "The paper is a strong contribution with a core insight that is clearly communicated. There are some minor improvements that could likely be made to the theory, but those are minor compared to the overall paper.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "This paper does a great job identifying a gap in the existing conformal decision-making literature: the idea to directly aim to optimize for the upper bound is very worthwhile. The paper is also very clearly written and the notational cleanly presented, making the insights straightforward to glean from reading. The experimental results were also very clearly presented, demonstrating how the approach produces the desired valid upper bounds while sacrificing coverage, as was the proposed thesis of how the approach was formulated.", "weaknesses": "The paper has a couple minor weaknesses that could be improved. First, the theoretical results seem like they would likely be vacuous in many scenarios: Theorem 3.1 has a professed lower bound of $1-\\alpha-\\Delta_n$, but $\\Delta_n$ seems like it would likely be much greater than 1 in any (non-asymptotic) regime. The covering number or Lipschitz constants would likely push it above that point, making this statement not particularly informative. Nonetheless, this statement does have the benefit of providing an asymptotic intuition, which is nice. Overall, this is not a big deal, since the main contributions to me are the insights of directly targeting the prediction region for producing an informative upper bound and the empirical validation more so than the theoretical guarantees. Also, not so much a weakness, but an interesting extension would be to consider prediction regions that are not necessarily just box or ellipsoids but instead unions of such regions (which can still be formulated as a convex program). It would be interesting to know how robustly these findings translate to such prediction regions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Oral)