Multi-Condition Conformal Selection
Authors: Qingyang Hao
Wenbo Liao
Bingyi Jing
Hongxin Wei
Venue: ICLR 2026 Poster (2026)
Keywords: Uncertainty Quantification
Conformal Inference
False Discovery Control
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: Selecting high-quality candidates from large-scale datasets is critically important in resource-constrained applications such as drug discovery, precision medicine, and the alignment of large language models. While conformal selection methods offer a rigorous solution with False Discovery Rate (FDR) control, their applicability is confined to single-threshold scenarios (i.e., y > c) and overlooks practical needs for multi-condition selection, such as conjunctive or disjunctive conditions.  In this work, we propose the Multi-Condition Conformal Selection (MCCS) algorithm, which extends conformal selection to scenarios with multiple conditions. In particular, we introduce a novel nonconformity score with regional monotonicity for conjunctive conditions and a global Benjamini–Hochberg (BH) procedure for disjunctive conditions, thereby establishing finite-sample FDR control with theoretical guarantees. The integration of these components enables the proposed method to achieve rigorous FDR-controlled selection in various multi-condition environments. Extensive experiments validate the superiority of MCCS over baselines, its generalizability across diverse condition combinations, different real-world modalities, and multi-task scalability.

Published as a conference paper at ICLR 2026
MULTI-CONDITION CONFORMAL SELECTION
Qingyang Hao', Wenbo Liao!?, Bingyi Jing', Hongxin Wei
Department of Statistics and Data Science, Southern University of Science and Technology 2 Department of Mathematics, The Chinese University of Hong Kong
ABSTRACT
Selecting high-quality candidates from large-scale datasets is critically important in essurce--onssrrinee applications such as drug discovery, precision medicine, and the alignment of large language models. While conformal selection meth- ods offer a rigorous solution with False Discovery Rate (FDR) control, their applicability is confined to single-threshold scenarios (i.e., y/ > c) and overlooks practical needs for multi-condition selection, such conjunctive conditions. In this work, we propose the Multi-Cone Condition Conformal Selection (MCCS) algorithm, which extends conformal selection scenarios with multiple conditions. In particular, we introduce a novel nonconformity score with regional monotonicity for conjunctive conditions and a global Benjamini-Hochberg (BH) procedure for disjunctive conditions, thereby establishing finite-sample FDR con- trol with theoretical guarantees. The integration of these components enables the proposed method to achieve rigorous FDR-con CoR-contoolled selection in various multi- condition environments. Extensive experiments validate the superiority of MCCS over baselines, its generalizability across diverse condition combinations, different real-world modalities, and multi-task scalability.
disjunctive
INTRODUCTION
In many practical scenarios with limited resources, it is essential select a subset of candidates from a very large pool that best meet specific criteria for downstream processing. For instance, drug discovery screens compounds using properties like logP to avoid molecular docking pitfalls (Noor et al., 2024); medical monitoring prioritizes high-risk patients via predictive systems (Hatib et al., 2018); and large language model (LLM) research employs selective procedures for output validation (Cherian et al., 2024). Across these domains, controlling the finite-sample false discovery rate (FDR) is crucial for conserving resources and mitigating risks. The reliance on machine learning model predictions, due to the frequent unavailability of true test responses, necessitates quantifying uncertainty for maintaining efficiency. cfBH (Jin & Candes, 2023) thus formulates the selection problem multiple hypothesis test by employing conformal valuue (Bates al., 2023), which are constructed based on conformal prediction principles (Vovk et al., 2005). This multiple testing- based framework has been validated in areas such as LLM alignment (Gui et al., 2024) and extended in further studies such as mCS (Bai et al., 2025b).
However, existing approaches restrict the hypothesis to single condition of the form y > c, a formulation that is often insufficient in practical applicatonn. In the concept of "drug-likk properties, compounds with medium logP are chosen by most practices (Shultz, 2019), which exemplifies the need for conjunctive conditions (i.e., C1 < < c2). Conversely, disjunctive conditions (i.e., < or y > 62) are necessary in other contexts, such as early-warning systems that must activate when a target value exceeds falls below critical thresholds (Lameski et al., 7017; Bollepalli et al., 2021). These instances highlight the necessity of selection under multi-conde conditioniiion settings.
In this paper, extend the conformal selection framework to multi-condition settings, with central goal of achieving finite-sample FDR control under both conjunctive and disjunctive conditions. We first show that naive combinations via intersection union of single-condition selection results fail to preserve FDR control for multi-condition selection. To address this, we introduce a novel method MCCS equipped with rigorous theoretical guarantees. For conjunctive conditions, de-
"Coresponniing author (weihx @ @ww.stccc.ccu... .dd....)........................

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sign tailored nonconformity function that targets intervals defined by simultaneous constraints; for disjunctive conditions, we introduce a global ranking mechanism over conformal p-values from (BH) procedure (Ben- jamini & Hochberg, 1995). The approach naturally generalizes to compound multi-interval targets and multivariate response settings, while maintaining finite-sample FDR guarantees.
individual conditions, followed by application of the
Our method was rigorously evaluated on both simulated and real-world datasets." In simulations, demonstrated superior performance over baseline methods, as evidenced by tighter alignment between the achieved false discovery proportion (FDP) and the nominal FDR. For instance, in the experiment of the univariate case under conjunctive conditions with a nominal FDR of 0.3, the FDP achieved by baselines exhibited significant deviations, either excessively high (0.3766) or too low (0.2013), while our approach achieved the closest approximation (0.2874) without exceeding the nominal level. Additionally, results on textual, visual, and multimodal tasks also validate the practicality of our method in real-world applications. The selection capability was further validated in multi-class scenarios, enabling reliable selection of individual, multiple, or similar classes.
2 PRELIMINARY
Problem setup. We let x E R represent the p-dimensional features, which are accessible in the entire dataset, and y E R denote the responses 2. In this paper, we mainly focus on the regression setting, and our work can be generalized to multi-class tasks presented in Appendix D. The entire
dataset divided into a training dataset Dirain = (xi, yx) and a test dataset Diest = {xx+ [
where the corresponding test responses {yn+i m are unobserved. The additional assumption is
1n-m in the dataset are drawn i.d.d. from an unknown distribution Dxxy. that samples {x), yx)
The selection problem can be formulated as follows: Given union of intervals Target. which can be a union of multiple mee-sdded-umbon subset S S (1...., m) from Diest such that many test observations j E S satisfy yn+s Iarget as possible, while controlling the FDR (Benjamini & Hochberg, 1995) below a user-specified level q. The FDR is defined the expectation of the FDP among all selected observations:
or bounded open intervals It, our goal is to identify a
FDR = E [FDP), FDP = Σズ [ E S, & Trarget)
(I)
1
Since an excessively low FDR can be trivially achieved by selecting very few samples, it is also important to evaluate whether a method retains an adequate number selections that truly meet the target criteria. This aspect is quantified by the metric power:
=1 1 1 E R,yu+s E ITarget}
Power E
(2)
Iv IV 1 (y E [Terget)
Ideally, practical approach should simultaneously achieve an FDP that closely approximates, yet does not exceed, the nominal FDR, while maintaining a relevant high power.
Conformal selection. The cfH framework (Jin & Candes, 2023), building upon conformal predic-
tion principles (Vovk et al., 2005), is designed to guarantee finite-sample false discovery rate (FDR) control in conformal selection under a single condition. It performs multiple hypothesis testing by selecting candidates whose values exceed a pre-specified threshold c. For each test sample, hypothesis test is constructed with the following hypotheses:
Hoz : Un+s S vs. H1ş Vn+ts >
Rejection of Hoj indicates that the j-th sample is selected, it's considered satisfying the condition. With the hypothesis test, the CS framework ensures FDR control via two core steps as follows.
First, it employs a nonconformity score V(x, y) that satisfies appropriate monotonicity constraints to quantify how atypical a response is given the input. For a calibration sample with observed response, the score is computed as V. V(x., y:): for test sample with unobserved response, the score is evaluated a predefined threshold e, yielding Pn+, = ((x+t):c). Then, the conformal p-value for
'Code available at https://github.... wtatthub.com//gg www./oooooohhcccee hww.hy-/ww/cccc-crr222....
Later, will extend the univariate responses multivariate in Section 4.4.
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each test sample is derived from the rank of its nonconformity score relative to the calibration set. Lower p-values provide stronger evidence for selection, corresponding to rejection of the null hy- pothesis Hoj- Second, the framework applies the & Hochberg, 1995; Benjamini & Yekutieli, 2001), a widely adopted method in multiple testing, to these conformal p-values for FDR control. The monotonicity property of the nonconformity score, combined with the BH procedure, jointly guarantees finite-sample FDR control. However, the efBH framework and its existing derivatives are exclusively confined single-condition selection, over- looking the complexities of multi-condition conformal selection. This gap will be elaborated upon in the subsequent section.
(BH) procedure (Benjamini
MULTI-CONDITION CONFORMAL SELECTION
In this section, we introduce the task of Multi-Condition Conformal Selection with detailed formu- lations, and demonstrate why the cfBH procedures (Jin & Candes, 2023) cannot be directly extended to this setting via simple operations such intersection union.
3.1 CONDITIONS AND SELECTION TARGET
Single-condition CS refers to the process of comparing the response value y against single threshold. In contrast, multi-condition CS involves evaluation against multiple thresholds to determine target satisfaction. The fundamental types of multi-condition CS conjunctive and disjunctive conditions, with more complex structures emerging from their combinations. For conjunctive conditions, such as those requiring Yn+j > Ckz and Yn-j Cka the null hypothesis is defined as H8 Yn+J
(Ck,, Ckp). with the corresponding alternative hypothesis H Yn+z & (Ck,: Ckn). ). This approach ensures that any null hypothesis, whether constructed from a single condition be mapped to a target interval Te. Consequently, for disjunctive conditions, the selection target is logically extended to the union of the intervals associated with each individual hypothesis. That is, the composite hypothesis formed by combining all conditions may be expressed as follows:
conjunction, can
Hoj Hggmbine: Yn++j E TTarest
VS. Higmbine: ynn+j Target:
where Target U LR each 1 denotes one--idee--ue iided-unbounded bounded open interval. We present an intuitive illustration for conditions in Figure and the FDR control is applied to all (j. k) pairs, where j indexes samples and i indexes conditions, an explanation provided in Appendix A.1.
Irager
Single:
Conjunctive: Frange (2)
Disjumctive Faagge
62) (cg,
Figure I: Illustration of single- and multi-condition.
3.2 CONTORMAL P-VALUE
The conformal p-value serves as the critical link between the nonconformity score and the BH procedure. When the response Yn+j is observable, we can compute the nonconformity score as V; = y.) y.) for i = 1...., n + m, where V is a nonconformity function based on P,, then the oracle conformal p-value would be defined as:
Σ.= 1(V Vn+5) + UU (1 Ei 1
pj
(3)
n+1
where U,- On, 1) is an independent random variable for tie-breaking.
To ensure finite-sample FDR control via the BH procedure. the conformal p-values must satisfy conservativeness (P(p) < a) S a) (Bates et al., 2023). While this property holds for the oracle conformal p-value pj. its computation is infeasible in practice since the test responses yn+t are unobservable. Consequently, the nonconformity score Vn+j V(Xn+jx Nn+j) cannot be computed directly. Instead, we substitute it with Putz V(xu+jj cc), where c denotes a prespecified threshold. This leads to the conformal p-value conformal p-value definition:
Σ 1[V Pats + UU (1 +1 Er 1 { = Puts 1)
=
(4)
n
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To ensure the conservativeness of practical p-values Pj, the nonconformity score V(x. y) must satisfy the following regional monotonicity:
Definition 3.1 (Regional Monotonicity). ((Bai al., 2025b)) nonconformity score V X Xxy + IR satisfies regional monotonicity if (x,') S y) for all x E X, y' RE, and E R. where R represents the target region.
As shown in mCS (Bai et al., 2025b), the practical p-values Pj. derived from nonconformity measure satisfying regional monotonicity, are conservative and thus ensure FDR control. Note that the conservativeness of Pj here is not the usual statistical conservatism, but refers specifically to the following under Ho: IP (pj S and E Ho) I Va E (0, 1). This property nonetheless suffices to guarantee FDR control (Bai et al., 2025b). The key challenge is thus to design nonconformity measures that preserve this monotonicity under multi-condition settings, especially for intersection conditions, which we will address in the next section.
3.3 ERROR ACCUMULATION IN TWO-STEP CFBH METHODS
seemingly straightforward approach for handling multiple conditions would to directly apply the cfBH/ method (Jin & Candes, 2023) at each boundary point independently and then combine the results through simple set operations as the second step. Specifically, for target interval (e1, 62). one may apply cfBH separately at each boundary, selecting units with Y C1 and Y < c22 then take their intersection (Jnter-cBBH). For the union (-00, e1) U (c2, oo), one can similarly use efBH at cı and 02 to select Y < c1 and Y C2, respectively, and combine the results via union (Union-cfBt)
However, these methods cannot guarantee FDR control, primarily due to error accumulation resulting from naive set operations. Such error propagation undermines their theoretical validity, necessitating a rigorously guaranteed approach for FDR control under multi-condition settings. We therefore present Corollary 3.1, supported by numerical results in Section 5.1 demonstrating the failure of both Inter-efBH and Union-efBH, thus providing empirical validation for the corollary.
Corollary 3.1. Applying two separate cfBH procedures (Jin & Candes, 2023) for boundary points and combining results via intersection (for the target interval (c1,02) 62)) or union (for (-oo, 01) U (02, o does not guarantee control of the false discovery rate at a prespecified level q, where C1 C2 are predefined thresholds.
Remark 3.1 (Explanation for 3.11llary3.1), For intersection selection S = nS, false discovery requires simultaneous errors in both procedures. For instance, a unit with Y S a1 may be falsely selected if erroneously included by Si yet retained by S2 (as Y S C1 x2)) Similarly, for Y c2, error necessitates incorrect exclusion by S2 but correct inclusion by Sy. This multiplicative structure inflates FDR as S decreases, violating control guarantees. For union selection S S=5 S₁ US, errors accumulate additively. Units with Y, E [e1, c2) are falsely discovered if selected by either procedure. The numerator sums overlapping false discoveries, while correlation between S and Sa prevents proportional denominator growth, leading to FDR inflation. To address this, we then propose the following DDR-ggurdnteed method for Multi- Colli--oonition CS.
PROPOSED METHOD
In this section, we introduce the method with a finite-sample FDR guarantee for Multi-Condition CS. We begin by addressing conjunetive and disjunctive conditions, then introduce the complete MCCS for diverse logical combinations and extend it to multivariate response settings.
4.1 ADDRESSING CONJUNCTIVE CONDITIONS
Central to addressing conjunctive conditions (i.e., C1 < < (2) is the design of the nonconformity score, For conjunctive conditions defined by the interval 4 (Ckz. C, the nonconformity score V* v*(x,y) in Algorithm 1 ensures regional monotonicity by leveraging the predictor ((x) and constant M. The score assigns
M min(f((x) -k., Ckn f(()))
(5)
if y E (Ckz, ). and
max(CkL -M(x), CkR)
(6)
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Algorithm Nonconformity Score for Conjunctive Conditions
1: function y)
P For the determined interval TE = (CRL CKR)
2: pred < p((x) M <- constant > 2 sup² max( jx()], - ckel)
3: if > Cby and y < Cka then return M min Ckz Cka - -redd
4: else return max(Ckz pred, pred Cks)
5: end
6:
end function
otherwise, guaranteeing lower scores for samples within Ix to promote selection. while M enforces the monotonicity, and further utility of M is formalized in Proposition 4.1.
Proposition 4.1. Let Vk be any fixed regionally monotone nonconformity score, and suppose {(x,, (x), Tm from DxxY, and define u) (V y) <) + u PVV
exchangeable from distribution Dxxy. - Let (x. y) denote random pair drawn
v) for any E R and E [0, 1]. For interval Target = Is = (CkL CkR). assuming fixed choice of c E IR, define
<= ( [0: : ((P( $9)
Suppose there exists i E (1'- t²) such that (F(V"(())))) q for any sufficiently small
>> O. Then the output S of Algorithm (where V adopts Algorithm 1) satisfies:
lim FDR (FF(V"(x.e)) cc). 1.. and
F(F(V (x.()" a.). Sns.yy
Bin Power
F(P(
Ply E 1n)
Proposition 4.1 extends Proposition 7 in CS (Jin & Candes, 2023) from a single condition to conjunctive conditions, establishing the exact asymptotic behavior of both the FDR and power. The proof, being highly analogous, is omitted. Here, F(E, u) can be interpreted as a lower and more robust measure of evidence, while corresponds to the critical threshold value selected by the BH procedure. The requirement that a small neighborhood around t" satisfies the definition in (7) ensures stability in FDR control by excluding scenarios where /" exhibits oscillatory behavior.
Under the assumption that V does not possess point masses, the meaning of t* can be interpreted s
more intuitively by introducing :*: let n² sup (w: in the definition of t" in (7) can be rewritten as:
s {) then the inequality
(x))) ST) (U
(x,)) 59 (x)
99)
C()
Hence, tightening the first inequality in (9) allows the algorithm to approach the nominal FDR more closely. The large constant M was introduced to achieve this objective, as detailed in Appendix A.2. Remark 4.1. Although this idea is similar to _cli in efBH (Jin & Candes, 2023) in spirit, and the "fixed choice" in Proposition 4.1 corresponds to a specific criterion, as in Theorem 4.1 of mCS (Bai et al., 2025b), the criterion in Proposition 4.1 is defined by two conjunctive conditions rather than a single one in Theorem 4.1 of mCS (Bai et al., 2025b). Furthermore, Proposition 4.1 can be naturally extended to multivariate response settings with conjunctive conditions, which cannot be handled by Theorem 4.1 in (Bai et al., 2025b).
4.2 ADDRESSING DISJUNCTIVE CONDITIONS
Addressing disjunctive conditions (i.c., y/ < CJ or > e2) requires a more effective utilization the BH procedure. To this end, we propose Algorithm 2, which implements via global approach that aggregates all m x valuuess P = { | I,.... m,; k = I,..., K}. sorts them ascendingly as
P()) < S P(NUM)) where NUM m K, finds the largest index /* satisfying P(-)) NUM and constructs the selection set (j. k)
9-1"
NUM
This method ensures finite-sample FDR control under exchangeability (Theorem 2.6, CS (Jin & Candes, 2023)) by integrating all hypotheses within a unified testing framework, thus particularly effective for selecting targets spanning multiple intervals as preventing the error inflation inherent in ad hoc combinations of separate efBH procedures, and providing rigorous theoretical guarantces for FDR control under disjunctive conditions.
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Algorithm 2 Global BH Procedure
1: procedure (p.
>j I..... m; ,,..., K
2: Collect all m x K p-values: P = {p 1 j =1, I,..., m: k = ......KK
3: Sort p-values in ascending order: P(1) S PP12) S P(NUM) where NUM
4: Find maximum l" satisfying: P(r)) S NUM
q-t"
5:
Construct selection set: S (j, k) pI
q-k
NUM
6: return Selection set S
> Selected pairs (j. k)) claiming Yn+j E Is with FDR S
7:
end procedure
4.3 ADDRESSING MORE COMBINATION CONDITIONS
Building the method for two-condition (conjunctive disjunctive) CS, the approach can be extended more formats and multi-condition combinations. one-sided unbounded interval, defined by single constraint, represents special case of conjunctive condition with one infinite threshold. Specifically, the nonconformity score for left-unbounded interval 47 (-00, Cka) and right- ungouned interval I (Ckz. +o0) is defined respectively as:
V (x, y) M 1(ycer)) ú(x),
(*(x)) M 1 A(x).
(10)
Therefore, we propose the complement MCCS Algorithm 2, which designs regionally monotone
nonconformity scores V*xx,, y) for each target interval IN computes conformal p-values D for all test samples and intervals based on the calibration data (as Eq. (4)), and applies the global BH
procedure in Algorithm 2 to select all pairs (j, k), ensuring finite-sample FDR control.
Algorithm Mulli-Condiine Conformal Selection (MCCS)
K Input: Deal yx) Diest (Xn+J ( Target UK 4,, FDR target g (0, j=1+ > Design nonconformity score tailored to each target interval I 2: Define / yx, satisfying the regionally monotone.
1: for k < 1 to K do
3: end for
4: for 1 K and i <- to n do
Compute nonconformity scores on calibration data
5: Compute V* yx).
6: end for
7: for k 1 K and j 1 to m do
Compute conformal p-values
8: Compute & n+j+ and then construct p/ in (4).
9: end for
9.1 Output: Selection set S (j,k) pI s q 9-1"
10: P + sort
{p/ (p))
1' + max
S 1:P
b> Global BH procedure
K
K
To enhance practicality, we state Corollary 4.1, which establishes that overlapping intervals within Target preserve the exchangeability and conservativeness required for FDR control, experimen- tally validated in Section 5.2. This property enables direct specification multiple target intervals without explicit intersection checks. Theorem 4.1 formalizes the finitessaple FDR control guarantee of Algorithm 3, with proof provided in Appendix A.4.
Corollary 4.1. Under the conditions of Theorem 4.1, intersections among target intervals I do not affect the FDR control guaranteed by Algorithm 3.
Theorem 4.1. Suppose V is regionally monotone nonconformity score, and for any E (1....,m). the random variables V ......, V. Vi+, are exchangeable conditioned on {vR Vetebeej. Then, for any q E (0, the output of Algorithm 3 satisfies FDR S q-
4.4 EXTENSION MULTIVARIATE RESPONSE SETTINGS
Building on mCS (Bai et al., 2025b), Algorithm extends naturally to multivariate responses. For conjunctive conditions, the target bounded area RR; for condiiinnntiveecoiiiioonnn, Algorithm
conditions,

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is proposed, analogous to Algorithm 1. Algorithm 2 operates on conformal p-values independently of response dimension, enabling direct multivariate application. This integration generalizes MCCS for multivariate settings. See Appendix A.3 for details.
Algorithm Nonconformity Score for Conjunctive Conditions under Multivariate Response Setting
1: function y))
& For the determined region between RRinne and 2 Routter
2: pred 4- f((x), .M <- large constant
3: y & Rinner and Routter then
4:
return A min{dis(pred @Rnner)) dis dis(pred, @oguterr) 5: else return max(dis(pred, ÖRsmner), dis(pred, Risppeed,aRouteen ]
6:
end
7:
end function
EXPERIMENTS
We evaluated the method on both simulated and real-world data. Simulations demonstrated its supe- riority over baselines under conjunctive and disjunctive conditions, along with strong generalization complex conditional tasks. Real-world experiments confirmed its efficacy across various scenarios, including textual, visual, multimodal, and multi-class settings.
5.1 COMPARE WITH BASELINES
Experiments setup. We first perform numerical simulations using synthetically generated data. The covariates x are sampled from Unif(- 10, 10)", where p denotes the dimension of x. The response variable y is generated by y p(x) e, where μ represents the regression function and denotes the random noise. An SVM with an RBF kernel is used for model fitting. The dataset is split into training, calibration, and test sets with ratio of 8:1:1. Different experimental settings are constructed by varying the regression function and noise distribution. The FDP averaged over 1000 independent replications serves as the observed FDR.
Compared baselines. We established the following baseline methods for comparison. The existing dingle-conditinn method was applied separately to each individual condition, and the resulting selection sets were combined using simple set operations: intersection (denoted Int) and union (denoted as Uni). Bonferroni correction (Dunn, 1961) versions of these procedures are referred to Int-B and Uni-B, respectively. Additionally, Ind denotes an indicator method based whether the response variable lies within the target region (i.e., Vind > O when y E TTarget). Further details on settings and baselines are provided Appendix B.
Table 1: Comparison with baselines of selection performance for conjunctive and disjunctive condi- tions under univariate and multivariate response settings (Optimal FDR control bold).
(a) Conjunctive conditions
(b) Disjunctive conditions
Univariate
Multivariate
Univariate
Multivariate
Method
FDR
Power
FDR
Power
Method
FDR
Power
FDR
Power
Int
0.3766
0.9397
0.3531
0.9622
Uni
0.3766
0.9720
0.3165
0.8024
Int-B
0.1081
0.6005
0.2208
0.8549
Uni-B
0.1569
0.9224
0.1644
0.5765
Ind
0.2013
0.2126
0.1491
0.1
Ind 0.2290
0.0000
0.2172
0.0044
MCCS (Ours)
0.2874
0.9756
0.2907
0.5348
MCCS (Ours)
0.2848
0.9515
0.2628
0.7013
In Table 1, we compare the observed FDR and power with baseline methods, showing results for conjunctive conditions and disjunetive conditions a nominal FDR level of 0.3. For the multivariate response setting, we report results with response dimension of 30. As previously discussed, applying two separate single-condition methods followed by naive set operations leads to error accumulation, consistently yielding obtained FDR values above the nominal level. In contrast, the Bonferroni correction versions show an overly conservative behavior, underscoring the need for our proposed method. The Ind approach also exhibits conservative FDR and lower power. In comparison, our method robustly controls the FDR at or slightly below the nominal level while maintaining a relatively high power.

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5.2 EXPERIMENTS ON MORE COMBINATION CONDITIONS
Table 2: Configuration of argett for different Tasks
Non-Intersecting Target Intervals
Intersecting Target Intervals
Task (-20, (in) (czz. Cin)
Task 4: (Atg (czu Can) (csz. +-0)
Task (Ca L (Cag +:o))
Task 5:
(-00, (ou Oaa) U (Ca, +00)
Task 3:
(-00, U (1g, (2n) U (a.
Task 6:
(=00, (la) U ( U (63,- U (Cag (:00)
Table 3: Results across varying ST and Ns for Task5.
Intersecting la get Intervals
Intersecting larget Intervee
FDR
Power Ns=0.5
Sox Pour
ST
Ns=
Ns=0.5
Ns=0.9
N==
Ns=0.9
0.2971
0.2953
0.2954
0.9548
0,9522
0,9502
2
0,2910
0.2912
0.2912
0.8631
0.8630
0.8623
3
0.2970
0.2969
0,2969
0.8843
0.8841
0.8841
FSh
1:
0.2966
0.2963
0.2957
0.9530
0.9526
0.9483
5
0.2913
0.2912
0,2915
0.8632
0.8628
0.8630
SiweATT leve
Nar POX leve
0,2970
0.2969
0,2972
0.8853
0.8853
0.8854
Figure 2: Results across varying g for Tasks 1-6.
Basic performance. We designed six different tasks, as shown in Table 2, including tasks with and without intersecting intervals, each task corresponding to distinct form of Targtt Table presents the performance of MCCS on Task 5 across six distinct settings (S77-S76) and three noise levels (Ns=0.1. 0.5, 0.9) at q=0.3. The results indicate a mild reduction in power as noise increases; however, the method maintains robust performance across all configurations. Figure 2 presents the performance of MCCS under Setting across six tasks at Ns=0.5 across varying nominal FDR levels g, ranging from 0.05 to 0.5 in increments of 0.05. Consistent with Corollary 1, intersecting target intervals does not compromise FDR control. Our method consistently maintains accurate FDR control while maintaining high power across all tasks.
Table 4: Results on large K under ST1.
Prgprtion al boletted bample $11
Nl Inhected AT3
FDR
Power
Ns
K=10
K-20
K-40
K-10
K-20
K-40
0.1
0,2926
0.2886
0.2773
0.9942
0.9789
0.9069
0.5
0.2935
0,2882
0.2762
0.9951
0.9785
0,9052
0.9
0.2951
0.2888
0.2759
0.9946
0.9777
0,9038
Figure 3: Interval-specific contribution.
Analysis on large K. We conduct additional experiments at q=0.3 with a significantly larger value of interval number K, an order of magnitude higher than in Table 2. In these experiments, the target interval was defined as Trage - - (Skz cx ) with the Setting fixed at 1 and noise levels varied at 0.1, 0.5, and 0.9, respectively. Table 4 shows that as the number of intervals K increases, statistical power decreases slightly while remaining high, and FDR control becomes more conservative while staying near the nominal level. This occurs because the BH procedure's critical value n causes rejection regions to contract, reducing true signal detection, while the increased proportion of null hypotheses due to interval subdivision necessitates more conservative thresholds to maintain overall FDR control. These performance changes under large K represent natural consequences of multiple testing adjustments, demonstratingMCCCS statistical rigor.
Analysis on interval-specific contribution. We analyzed the selection proportions across intervals at q=0.3 in Task 6, using Settings 1 and 3 as examples. The proportions of all test samples were compared with those of the selected samples for each target interval. Figure 3 shows a consistent distribution pattern between the All and Selected samples across intervals. This pattern indicates that when selections favor certain intervals, it reflects stronger statistical evidence (c.g., greater conformity to the target regions), arising from inherent variations in the data distribution. (Note that the sum of selection proportions exceeds l-q because Task 6 involves overlapping intervals.)
5.3 REAL DATA APPLICATIONS
Our method is applicable to diverse data types, as demonstrated in the following tasks, where the FDP from 500 independent replications serves an estimate of the FDR.
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NLP. In natural language processing (NLP) tasks, we identify moderately toxic content for toxicity detection, aiming to prioritize manual review for moderate-risk cases, and automatically process high- and low-risk samples. Two datasets are used: Real-Toxicing Promppss (Gehman et al., 2020) (denoted as nlp-A) and LeaTTTxxxiit--allance (Gao et al., 2021; Korbak, 2023) (denoted as nlp-B).
CV. In computer vision (CV) tasks, we use the NYU Depth Dataset V2 (Silberman al., 2012) to select samples within a specified depth interval. This approach introduces an important spatial prior that narrows the solution space, thereby enhancing measurement accuracy and robustness in the target domain. This can be implemented by excluding training samples that fall outside the desired bounds during dataset preparation. We employ two distinct models, ResNer (He et al., 2016) and ViT (Dosovitskiy et al., 2020), for feature extraction, referred to as cv-A and cv-B. respectively.
Table 5: Results on nip and cv Tasks.
test
Toxicity
of Muull-Toxiiity-Prrmmpp
Pile-Toxicity Balanced
semples boand
30:00
sempli
Task
FDR
Power
29:30
Lunay tound
nlp-A
0.291
0.575
Uspaa tound
Mascl (oar)
13:00
nlp-B
0.289
0.512
1330
cv-A 0.261
0.892
203
530
cv-B
0.293
0.814
Toulcity
Figure 4:
Visualization of nlp.
In Table 5, we report the observed FDR and power for both the nlp and ev tasks at a nominal FDR level of 0.3. The results demonstrate that our method maintains effective FDR control tightly below the nominal level while achieving substantial power across varying data modalities. Figure visualizes the distribution of test set and selected samples in the nlp tasks, illustrating consistent performance across varying data structures. The selected samples remain concentrated in the target region, regardless of variations in the test set, highlighting the robustness of the approach.
VQA. We further evaluated our method on visual question an- swering (VQA) tasks incorpo- rating both nlp and cv modali- ties using VQA Dataset V2 (Lin et al., 2017). The goal is to se- lect test samples with predicted human agreement scores within a predefined confidence interval, focusing on moderately ambigu- ous instances that reflect partial consensus and are most suitable for model improvement and hu- man-A/ collaboration. We used BLIP (Li et al., 2022) for confidence prediction, then predicted agreement using Ridge and BayesianRidge (denoted as vqa-A and vqa-B). Table 6 reports the FDR and power achieved on the vça tasks. Figure 5 illustrates the concentration in the target area of selected samples, confirming the selection's effectiveness. These results affirm the applicability our method to multimodal contexts and indicate its potential for scalability to even larger models. Further details are provided in Appendix C.
Table 6: Results on vga Tasks.
Trus agresment dicribbtios
Task
q
FDR
Power
Upper
vga-A
0.3
0.263
0.589
vga-A
0.5
0.483
0.977
vga-B vga-B
0.3
0.285
0.726
0.5
0.477
0.991
Figure 5: Visualization of vqu.
S of
Multi-class. We extended our MCCS to multi-class tasks using the CIFAR-10 (Krizhevsky et al., 2009a) and CIFAR-100 (Krizhevsky et al., 2009b) datasets, enabling the selection of single, multiple, and similar classes. The achieved FDR closely aligns with the nominal FDR and maintains balanced class proportion during similar-class selection. Complete details are provided in Appendix D.
RELATED WORK
Conformal Prediction (CP) (Papadopoulos, 2008; Vovk, 2013; Vovk et al., 2005) constructs predic- tion sets with finite-sample coverage guarantees under the assumption of exchangeability. Despite recent progress in CP (Kiyani et al., 2024; Ge et al., 2024; Luo & Zhou, 2025; Snell & Griffiths, 2025; Kiyani et al., 2025; Zargarbashi et al., 2025; Schröder et al., 2025), they cannot be directly applied to Conformal Selection (CS) problems, which require controlling FDR in a select set rather than constructing prediction sets with coverage guarantees. This gap motivated the development of the
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CS framework cfBH (Jin & Candes, 2023) bridge CP principles with the selection tasks. Subse- quent research has expanded this foundation into specialized extensions addressing diverse practical challenges. For instance, WCS (Jin & Candès, 2023) incorporates propensity score weighting to handle covariate shifts between calibration and test data, while mCS (Bai et al., 2025b) generalizes the framework multivariate responses, enabling selection based multiple outcomes. For hier- archical data structures, conformal e-values (Lee & Ren, 2025) have been developed to ensure FDR control in settings with grouped data. Additionally, OpiCS (Bai & Jin, 2024) enables data-driven model optimization without compromising validity, and ACS (Gui et al., 2025) dynamically updates the prediction model during the selection process to enhance power. DACS (Nair et al., 2025) integrates diversity metrics to promote representative selections, and a approach (Qin et al., 2025) achieves asymptotic optimal power under covariate shift. Applications such as LLM alignment (Gui et al., 2024) and drug discovery (Bai et al., 2025a) underscore the practical value of these advancements. However, these advances primarily focus on cingle-condttion scenarios. Our work extends this line of research by addressing multi-condition settings while maintaining a theoretical guarantee, which is realistic in various real-world applications.
Pearson-inppired
7 DISCUSSION EFFICIENCY IMPROVEMENT
To optimize the identified sorting bottleneck in the global BH procedure, based on our theoretical analysis and experimental evaluation of MCCS, we identify integration of the Quickselect algorithm (Hoare, 1961) enhanced the computational efficiency. This approach, termed MCCSqqck, reduces sorting overhead by directly determining the BH threshold, avoiding full sorting, We compared MCCS and MCCSqck across varying numbers of intervals under Setting and noise level 0.5.
Results are shown in Table 7, where Time refers to the time taken for 100 replications without GPU acceleration. The results demonstrate that MCCSqck maintains comparable FDR and Power metrics to MCCS, while runtime advantages become increasingly pronounced as K grows, underscoring its reliability for large-scal applications. This optimization aligns with MCCS's rigorous framework, offering practical balance between efficiency and statistical robustness.
Table 7: Performance comparison of MCCS and MCCSqck (Improved efficiency in bold).
FDR
Power
Time
Target
MCCS
MCCSqck
MCCS
MCCSqck
MCCS
MCCSgck
K=4
0,3030
0.2971
0.9823
0.9810
54s
52s
K=10
0,2935
0,2950
0,9951
0,9951
151s
124g
K=20
0.2941
0.2945
0.9901
0.9901
294s
238s 158s
K-40
0,2762
0.2822
0.9052
0.9610
560s
469g
1914
8 CONCLUSION
In this work, we introduce a theoretical extension of conformal selection to multi-condition settings, addressing an important gap in the field: while existing methods default to single-condition targets (e.g., one-sided intervals), MCCS systematically solves this long- overooverlooked problem by provid- ing rigorous framework for multi-condition selection with finite-sample FDR guarantees. This advancement is particularly necessary because simpler alternatives, which often rely heavily on predictors, exhibit limitations such as low power or overly conservative FDR control, as empirically demonstrated in comparisons with baseline methods. The design of MCCS thus emerges a critical response to these practical shortcomings. Empirical validation across various domains confirms the practical effectiveness and adaptability of the approach, with extensions to multi-class scenarios, offering a versatile and theoretically grounded solution for selection in resource- scenarios.
Linitations. In line with standard conformal selection settings, the training and test data in our experiments were drawn from the same domain. However, domain shifts may occur in worldworld applications, suggesting direction for future research.
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ACKNOWLEDGEMENTTMENT
This research supported by the Shenzhen Fundamental Research Program (Grant No. JCYJ2023080709 809020). 10900)).))00900)000.. We gratefully acknowledge the support of the Center for Computa- tional Science and Engineering at the Southern University of Science and Technology.
ETHICS STATEMENT
This paper aimed to advance the field of Machine Learning. There are many potential societal consequences of work, none of which feel must be specifically highlighted here.
REPRODUCIBILITY STATEMENT
To ensure the reproducibility of findings, detailed implementation instructions for the proposed method are provided in the Appendix. The real-world datasets used in the paper are publicly available, ensuring consistent and reproducible evaluation results. Additionally, the source code is publicly available at the following URL: https://github.com hon/hhgynew/mmcss.o.. are intended facilitate the verification and replication of our results by other researchers.
These measures
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A ALGORITHM DETAILS
A.1 EXPLANATION FOR FDR
We demonstrate that applying FDR control to each individual interval within a multi-interval target region ensures control of the overall FDR for samples falling outside all target intervals. For a specified target Iarget, if selection is performed for each interval Ig, one can readily show:
claim: Yn+s E Is, but: : Yn+j E Target
eetthr =E
1V|5|
claim: Yntj E Tgy but Yutj T FDR.
SE
1 1
Hence, maintaining FDR control for any individual interval 15 suffices to control the overall FDR across all intervals. Therefore, we subsequently focus on controlling the FDR for each interval.
The FDR control is applied holistically to all (j. k) pairs, where j indexes test samples and k indexes conditions. This means that the algorithm governs the proportion of false discoveries among all individual claims of the form "sample satisfies condition k.", which guarantees finite-sample FDR control while allowing flexible preferences for specific intervals. This granularity unattainable with baseline methods that aggregates decisions into sample-level binary judgment. In practice, the (j, --pii control mirrors the use "person-tinee" each occurrence (e.g., ,, violation event) is counted independently, rather than aggregating across individuals. This analogy underscores why sample-level control is insufficient: just as counting "person- timetttimm" captures the frequency of actions rather than merely the number of actors, (j. -pair control accurately reflects the multiplicity of decisions in multi-condition environments, balancing theoretical rigor with practical utility.
in legal or cpidemiological statistics, where
A.2 EXPLANATION FOR THE LARGE CONSTANT M
Tightening the first inequality in (9) can be achieved by ensuring V (x,y) S v² so that y & Ig In an extreme case, V yxxx.y y) can be defined as -Ā(xx) when y & I& and to0 when y Is As
practical compromise, we define V(x, y) M [yEl+) min (ï(()) Ch2y Ckg (x(x)), where
M > 2- sup, max(|c*2 - ((x)),, (ā(x) - -cnnl). which leads to:
VK (x, y where y 22 sup max (lok, - [xxx). (f(x) -Ckn min (f(x) Ckui Ckn fxxx))
x
>2 sup max (lekz - ((x)). [ā(x) -CERl) 1) max (lā(x) ck2l. |CER fi(x))
sup max( (axx). (Ā(x) -CkR
> (x, Y, Wwhere y
This ensures that the nonconformity scores for E Ti are always larger than those for $ Ig- When q < Pyy & Ix), follows from the definition that u" lies below the g-th quantile of V (x.y/ y). Consequently, the first inequality in (9) becomes exact equality, leading to obtaining FDR closer to the nominal level and higher power.
For multivariate response settings, the absolute difference should be replaced by distance metric, ic., M 2- sup, max (exx(dds((()))) ( di R where the target region, and dis()( aR) infrER []ā(x) rlly
A.3 DETAILS OF THE EXTENSION TO MULTIVARIATE RESPONSE SETTINGS
The MCCS algorithm can be naturally extended to multivariate response settings, where the target region may comprise multiple domains bounded by several constraints. For brevity, we term this generalized approach Multi-Region Conformal Selection (MRCS). Analogous
Algorithm 3,
provide the complete procedure for MRCS below, where (ās()((), ²R) = inf []ā(x) - rllp- and project denotes a mapping function used to derive the expression for y from the boundary equation of region 2R for computation of the nonconformity scores of test samples.
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Algorithm 5 Multi-Region Conformal Selection
Input: Dealth = y:); (x k= (with boundary equations), & & (0, 1), M()
(pre-trained predictor), M > 2- sup, max (dis( Rrnner) dis(f((x), (
1: for k <- to K do
Region-specific nonconformity scores
2: if' Rs has inner boundary 0R then
3: Define (xx,,) = M 1(y R - dis(j(()), R
4: end if
5: if' Rg has outer boundary 0. then
6: Define V (x. = M 1(y E R + dis(f((x), (
7: end if
8:
if Rg has both boundaries then
Define V v*((x),,)) M min min(dis((((x)), RRieeer, dis((((x)) O d max{dis(i((( (x)))
y E R
9:
@ a1epmer) dis (((x),
() y & Ra
end 11: end for
12: for t- to K and i + 1 to n do
Compute calibration scores
13: V* t- V (x,,, y:)
14: end for
15: for k < 1 to K and j 1 to do
Compute test scores and p-values
Dy + project( (RR)) V V (x,, bx)
17: Then construct pf as in (4)
18: end for
9.1 m
19: P +- sort (1
1" max
:: [:P
Global BH procedure
q g-
Output: Selection set S (i,k) p/
m K
A.4 PROOF OF THEOREM 4.1
Exchangeability Condition. variables V/ ,.... V.. Vk n+j are exchangeable. More precisely, conditional the estimated test scores (V j 1,..., K), these scores are exchangeable. This is because the calibration scores based on i.i.d. data, and the true test score V* comes from the same distribution. Conditioning on the estimated scores other test points does not affect exchangeability, as these estimates are based independent test points and fixed model.
Given the i.i.d. data and fixed model, for any fixed (j. k). the random
Consequently, for any null hypothesis (j, k) E Ho, the oracle p-value pf* is superuniform conditional on the other p-values. That is, for any E [0, 1]. P(ut** < <1 111F S 4, where F-j,- is the r-algebra generated by all other p-values, i.c., {u/ (1, (i,k).
Properties of the BH Procedure. The BH procedure has key properties used in the proof. Let R [S], and R_j.-k [S- where S_j.-k k is the selection set obtained by applying the BH procedure all p-values except P/. Then, R 2 R_j,- because adding p-value cannot decrease the number of rejections. Moreover, if pf > R.R count, so R = R-j These properties are standard and are utilized in the proof.
then adding P does not change the rejection
Proof of Theorem 4.1. We now prove that under the exchangeability condition and regional mono- tonicity, the MCCS algorithm controls FDR < q.
Begin by decomposing the FDR:
[ n Ho
k((ik) E ES}
FDR
E
Σ E
[S]
G.
since for (j.k) k) Ho. 1 ((j,k) S and yntj & Tx) (((,k) ES.
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Let R = |S], NUM = m.K. For any (i,k) Hu, the event (j. k) S is equivalent to p by the definition of the BH procedure. Thus,
9.12
NUM
1 ( k) S}
10 S
9-R
E
NUM
R
Since Ynts & Je, regional monotonicity implies p3;" S ppj. so
1p -
q- -RR
1p
q
NUM
NUM
and hence
(p S
9-R NUM
}
1p
S
9.R
E
NUM
R
Now, introduce conditional expectation. Let be the o-algebra generated by all p-values except pf. Let R- be the number of rejections when applying the BH procedure all p-values except pf. By the properties of the BH procedure. R > R_j,-k- k. and if pf Therefore.
q.R
NUM then R j.
1n
S
9-R
1pp S
g -B-iIc NUM
NEW
R
R-j,-k
This holds because if p) S g-R2
g-R-i-h NUM then since R 2 R-j.-k, we have
gRR. NUM
g-R-j_z so NUM
1 (as R 2 R, > 0), making the left side less than or equal to the p S NUM and S j.-
right side. pf q-R- the right side is zero, and R R-j,- implies the left side is also zero. Thus,
1( I
R-R
1p I
s NUM
E
NUM
< E
R-j,--
Since 133" S pÿ, we have
9 R-ii-k S 11p < 9 R-
10g -
NUM
NUM
so
1p ( NUM
1 S
R.R
NUM
E
S
R-j,-
R-j,-
Next, apply superuniformity. Conditional on F-j.- j.-kk, by R-j fixed, and pf* is superuniform:
PV < 6 F1F--.-- S for any E [0.
Therefore,
1
1
ko <
q. R NUM
|F S q. R R.-- j. J
q NUM
E
R-j.-k
R
Taking expectation,
lp <
9-R-1.ck NUM
q NUM
E
R R-j.-k
Combining the incqualities,
1 1 <
gilt
NUM
q NUM
E
Finally, sum over all null hypotheses:
FDR Σ
E
kj E S}
R
<
Σ
NUM
q
q
NUM
Hol
< g,
(H,))Hz
since |Hol S NUM.
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A.5 ADDITIONAL EXPLANATION
In Algorithm 2 and Algorithm 5, when M is large, the subtracted term becomes negligible and can be omitted. Similarly, calibration samples outside the target selection have minimal influence and may also be excluded, thereby conserving computational resources.
B DETAILS OF NUMERICAL EXPERIMENTS
B.1 DATA GENERATING PROCESSES FOR UNIVARIATE RESPONSES SETTINGS
The synthetic datasets systematically constructed to evaluate methodological performance under controlled conditions with varying degrees of complexity, encompassing a spectrum from linear to highly nonlinear relationships paired with different noise characteristics. Settings 1-3 are designed with distinct functional structures: Setting embodies a linear data-generating mechanism, Setting introduces moderate nonlinearity through combination of exponential and trigonometric compo- nents, and Setting 3 further increases model intricacy with multi-feature nonlinear interactions.
To rigorously assess robustness under different stochastic environments, Settings 4-6 adopt the same functional relationships as Settings 1-3, respectively. but incorporate Laplacian noise instead of Gaus- sian noise, thereby introducing heavy-tailed disturbances that present additional modeling challenges. Across all configurations, independent noise, Gaussian in Settings 1-3 and Laplacian in Settings 4-6, added with zero mean and comparable variance, which corresponds to the noise level varying in Section 5.2, to ensure equitable comparison of noise resilience. Detailed mathematical formulations for each setting are explicitly provided in Table 8, enabling transparent and reproducible evaluation of model behavior under diverse data-genee generating processes and noise regimes.
Table 8: True regression functions and noise distributions
Setting
(x)
En= Byr, with By ~ 10)
N((0,02)
2
exp(x1) + sin(= x2)
N(0,0²)
3
exp(x1) + sin(2 x2)) exp(233) sin(z ( m4) exp(xs) sin(m x6)
N(0,02)
4
Same Setting
((0,A)
5
Same as Setting
((0,))
6
Same as Setting 3
L(0,A)
B.2 DETAILS OF THE BASELINE METHOD IND
The method provides unified algebraic approach transform the geometric condition of whether point lies within one or multiple target regions into an equivalent analytical condition expressed via the sign of scalar-valued function.
In the univariate case, the target set consists of intervals, such as (a, b) u (c, d). polynomial function f(t) constructed the negative product of shifted linear factors:
f(t = - - a) tt (d - t).
A point Y belongs to the union of the intervals if and only if f(Y) 0. This criterion converts the set-membership condition into an inequality constraint amenable to algorithmic evaluation.
The multivariate extension, take 2-dimension an instance, generalizes this idea to multiple curved regions-eade defined by a quadratic inequality the form (r - h,)2 + (y - k;)² r?? The overall target region is defined by Boolean combinations of such inequalities. The function f(u, v) constructed as the negative product of the left-hand sides of these quadratic expressions:
f(u,0) - II [u h + (v 2
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Then, apoint (Y1, Y2) lies within the target if and only fY1, Y2) 0. This construction effectively encodes the geometry of multiple intersecting circles into a single differentiable function whose sign indicates membership.
This method offers computationally tractable approach to transform geometric constraints into algebraic inequalities applicable to both univariate and multivariate response settings. However, its performance is often suboptimal.
B.3 DETAILS oF TASKS IN SECTION 5.2
We constructed six distinct tasks in Section 5.2, encompassing both scenarios with intersecting target intervals and those without overlaps. To detail the experimental parameters and offer schematic overview, Figure is provided below.
Task 1: Irarget = (-00, - 20) 1-,
-20
-55
10
Task 2: Frarget (-15,10) (20, s0)
Task 3: Irarget (=c0, 20) u (-15, 10) (15,40)
-20
-15
(a) Tasks without Intersecting Intervals.
Task 4: Irarget = (-15, 10) u (15, 40) (20, + m))
Overlap: (20,
Task 5: Iaarget (=20, 20) u (15, 40) U (20, + o0)
Overlap: (20,40)
Task 6: Irarget = (-00, -20) (-15, 10) u (15, 40) (20, 00)
Overlap: (20, 40)
(b) Tasks with Intersecting Intervals.
Figure 6: An Intuitive Comparison of Different Tasks.
B.4 SUPERIORITY OF MCCS OVER IND ON INTERVAL- -PPECIFI PREFERENCE
MCCS considers the value of each interval, thereby not only improving overall performance but also allowing us to control the selection ratio for each interval, which Ind-like methods, which treat all conditions uniformly, cannot do. We hereby theoretically demonstrate that this fine-grained control will maintain FDR control.
1. Problem setup
In MCCS, we consider m test samples and K target intervals, resulting in a total of N = m x K hypothesis tests. Each hypothesis H,v corresponds to test sample j and target interval Ig.
We introduce predefined and data-independent weights Wikk > to reflect the importance or prior information of different hypotheses, and define the weighted p-value as:
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weighted
Pit 1a
Pyk
Conditions Required for Weights
To ensure FDR control, the weights must satisfy the following conditions:
1. Non-negativity: Wikk > 0 for all k.
2. Normalization: $ プラ
Wikk = 1, i.c., Ej Wikk N.
3. Null Hypothesis Weight Constraint: Σ(U,)EHO Wit I N, where Ho is the set of null hypotheses (hypotheses where the true condition does not hold).
Condition 3 is equivalent to requiring that the average weight of null hypotheses does not exceed /, but this automatically satisfied because:
Note:
From Condition 2, the average weight of all hypotheses is 1.
The of weights for null hypotheses I the total weight sum N.
Eng Mik Ka
Thus,
N (though this is not strictly needed).
Ha
In practice, Condition 3 is automatically satisfied because the sum of weights for null hypotheses cannot exceed the total weight sum N.
Weighted BH Algorithm
The algorithm steps as follows:
1. Compute the weighted p-values: Pyk
weighted
Pikk /Uik-
2. Sort the weighted p-values in ascending order: P(1)
weighted
S P(2)
weighted
- ... S P(N)
weighted
3. Find the largest index | such that: P(t)
weighted
< g..
4. Reject the corresponding hypotheses.
4. Proof Process
4.1. Notation Definitions
Ho: Set null hypotheses (yn+s & 1x), Rgs 1(Hys is rejected).
= ((,k)) Rykk 1}, R S, FDR = IE
V
RVV
4.2. Proof
FDR = EGADER I
I(HHk is rejocted) RVI
Define the event Fijk
[pik
weighted
<
9-R N
Then: FDR = EG,DE E
[(E,,) Rv1
Let F-jk jkk the r-algebra generated by p-values and weights except Pys- Define R_sk as the number of rejections obtained by applying the weighted BH procedure after excluding Hjk-
From the properties of the BH procedure:
R-ik- weighted
If Pjk
> 9-R212 N then R = R R_gk.
Rmak) N
Thus:
I(Ea) RVI
RI
vogunal g. 16 pa
Take the conditional expectation: E
I(Ena) RvI
F-gh
< E
F F
v1
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Given F-jk, R- is fixed. The event Pyk
weighted
q-R is equivalent pyk I 8-
By the conservativeness of the p-values (for Hjk E Ho): (Pys I sa(F-jk) Sa Va [0,1].
Set q-R-aw- "
yielding: E
[
weighted
Fーjik] I g gaaaaa
Pik
wiighed 9:R
ik
Therefore: E
Pikk
F jk
W. N
when R-jz 2 1. When R-jh 0. both sides
R=
are zero.
Take the full expectation: E
1{Ex RVV
g N
Sum over all null hypotheses: FDR S (i.N)EH T.M.R. =q-
Egaae wyk
From the weight condition E(i,)EHo Wyyk S EEiN Wigh N.
We obtain: FDR 9 Eng 1856 59 N g.
4.3. Boundary Cases
*
When R = 0, FDR = 0, satisfying the control condition.
When all weights are equal (wz* 1), the method reduces the standard BH procedure.
5. Experimental results
If no weights are set, the proportion of samples in each interval of the selection is consistent with the trend of the overall sample distribution. If needed, weights can be integrated to control the proportion of each interval in the selection set. We take the results of Task under Setting 3 as an example in Figure! 7.
Proportion Samples
Proportion
Samples
Proportion
Samples
Al Sampiia
Sempla
All Sempla
CEIN
24.5%
13:25
54%
54%
KAN
(a) No weights.
To focus specific interval.
(c) To balance across intervals.
Figure 7: Results of different interval preference settings for Task 6 under Setting 3.
B.5 EXPERIMENTS ON MORE BASELINES
To strengthen the claims, we evaluated another two baseline methods: the Kenjamini-Yekuttee (BY) procedure and the Prediction Interval (PI) method, which provide valuable contrasts to MCCS.
conservative adaptation of the Honjjamini-Hochbee (BH) method that accounts for potential dependencies among p-values by incorporating a harmonic number correction. Specifically, it adjusts the rejection threshold to 9.
-ekkmiii-YYekuteli (BY) Procedure. The BY procedure
m- m-K-o(m-KJ where c(m) = Em 1/1 accounts for dependence structures.
Prediction Interval (PI) Method. The PI method relies on parametric regression- basedd prediction intervals, such those derived under normal error assumptions, select samples whose intervals overlap with the target region. For instance, for a test sample zj, the method computes a prediction interval [L.j, Ug] the 1 q level and selects samples satisfying [Z.j, Uz] Taaget F 0.
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We compared the performance of these two baselines and MCCS on Tasks 1-6 with Setting of and noise level of 0.5. The results are shown in Table 9, with the best results (closest to the nominal FDR and highest power) highlighted in bold.
Table 9: Comparison with More Baselines on Task 1-6 under Setting (Best results in bold).
FDR
Power
Task
BY
Pl
MCCS
BY
Pl
MCCS
0.0353
0,5039
0.2987
0.6242
0.8906
0.9954
0.0369
0,4057
0.2995
0,6522
0.9848
0.9952
3
0.0335
0,4436
0.3037
0,2181
0.9652
0.9697
4
0.0375
0,4106
0.3009
0.2452
0.9942
0.9732
5
0.0243
0.3562
0.2965
0.1428
0.9641
0.9487
0.0239
0.4021
0.3030
0.2240
0.9692
0,9823
While the BY procedure theoretically ensures FDR control under broader conditions, our experimen- tal results show that it often leads to excessively conservative FDR estimates (i.e., FDR significantly below the nominal level) and substantially reduced power. This conservatism stems from its strin- gent threshold adjustment, which diminishes the ability to detect true positives, particularly in multi-condition settings where the number of hypotheses (m x K)) is large. contrast, the PI method relies traditional regression-basee error distributions, to select samples whose intervals overlap with the target region. This approach is computationally straightforward and can achieve high power by leveraging model-based uncertainty directly. However, our experiments reveal that PI methods frequently exceed the FDR control target, especially when model assumptions are violated or in finite-sample scenarios. This lack of robust- ness highlights the limitations of parametric methods in complex, real-world applications where distributional assumptions may not hold.
prediction intervals, such those assuming normal
Only our MCCS demonstrates balanced performance, maintaining FDR control at or near the nominal level while preserving high power across diverse conditions. This achieved through its innovative use of p-values with regionally monotone nonconformity scores and the global BH procedure, which adaptively handles dependencies and multi-condition structures without requiring stringent assumptions. The experiments confirm that MCCS outperforms both BY and PI methods: avoids the excessive conservatism of BY and the FDR inflation of PI, thereby offering a robust and practical solution for selective inference. Incorporating these baseline comparisons not only enriches the experimental section but also underscores the unique value of MCCS. By demonstrating superiority methods that represent different approaches, we provide comprehensive evidence for MCCS's effectiveness. This expanded analysis would solidify the claims that MCCS achieves an optimal trade-off between FDR control and power in multi-condition environments.
B.6 DETAILS MULTIVARIATE RESPONSE SETTINGS
Data Generation. We use dataset generation function to produce multivariate regression data with systematically controlled feature-response relationships and correlated noise structures. It enables the evaluation of multidimensional regression methods under varying conditions of dimensionality and noise correlation. For each observation, the predictor variables x are uniformly sampled from the interval [-1, 1]. forming
x d design matrix where d denotes the feature dimension (default 5). The response matrix y E Bank is constructed through a structured linear combination: each dimension Y, is generated y, zz 0.5 Ij+1 Zj+2 + 1.5, with j mod d. This results in a rotating feature selection pattern across response dimensions, establishing a consistent yet varied linear relationship between predictors and responses.
To emulate realistic correlated noise, multivariate Gaussian noise introduced via a covariance matrix Σ with specific structure: all diagonal elements equal, and all off-diagonal elements are also equal, reflecting a homogeneous variance and uniform correlation across all response dimensions. Here, controls the overall noise level and sets the magnitude of correlation between
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any two dimensions. This setup allows fine- fine-grained control over both the signal-to-non ratio and the inter-dimensional correlation pattern. The data generation strategy supports rigorous evaluation of regression methods across several key aspects: the number of response variables k, the noise intensity a, the correlation strength p. and the predictor dimension d, providing a transparent and reproducible framework for assessing model performance under controlled settings.
Task Settings. We define the conjunctive conditions for multivariate responses as a ring-shaped region centered (2, 2, ..,.,, 2) across all dimensions, forming symmetric reference frame. This annular region is bounded by two concentric hyperspheres whose radii are scaled proportionally from a predefined dimension -ppiioii base value. A point considered within the region if its Euclidean distance to the center lies between 0.6 and 1.0 times the reference radius, thus creating a well-defined annular subspace for conjunctive conditions under multivariate settings.
We define the disjunctive conditions for multivariate responses multi-sphere region, constructed from several hyperspheres, each with distinet centers and radii. The primary sphere is centered at (2, 2, ............ 2). while subsequent spheres are displaced via controlled random offsets uniformly sampled from [-0.5, 0.5)4, with offsets growing proportionally to the sphere index. The radii increase gradually from 80% of the base value, thus forming a cluster of intersecting hyperspheres with systematically varying positions and sizes that produce complex overlapping regions suitable for evaluating multivariate methods for disjunctive conditions.
Based on the aforementioned framework, we performed series of simulation studies with response dimensions set d 10, 30, and 50. The corresponding radii used in each configuration summarized in Table 10. In these configurations, the task involving the union of conditions is referred to Spherical Shell; we provide the outer sphere radius, while the inner sphere radius is set to 0.6 times the outer radius, as described earlier. For the union of multiple spherical conditions, experiments were conducted with varying numbers num = 2, 4, and 8 of spheres, and the reference sphere radius is reported for each case.
Table 10: Radius Settings for Multivariate Tasks with Varying Dimensionality of y.
Task
d = 10
d = 30
d = 50
Spherical Shell
5.1
8.1
10.5
2
Spheres
4.4
7.1
8.9
Spheres
4.4
7.2
9.5
8 Spheres
4.9
8.2
10.5
More Results. Under the experimental framework outlined above, which aligns with the univariate configuration, we employed an SVR model with an RBF kernel as the regression method. The observed FDR was estimated by averaging the FDP over 100 repeated experimental trials. In this section, we present more comprehensive set of results than those discussed in Section 5.
For the conjunctive conditions, compared our approach with a baseline method, denoted Int, which directly applies the intersection of two single-condition selection procedures. The experi- mental outcomes, summarized in Table 11, reveal that our method consistently achieves effective FDR control across varying response dimensions. In contrast, the Int baseline consistently produced FDR values exceeding the nominal level, underscoring the deficiency in its theoretical guarantees. Although the Int method occasionally attained higher statistical power than our approach, this ap- parent advantage came at the expense of loss in FDR control. It is important to note that pursuing higher power alone, for instance, by selecting all samples, is trivial and statistically uninformative. Hence, the ability to maintain high power under strict FDR constraints represents a meaningful and advantageous property of our method.
For the disjunctive conditions, we compared our approach with a baseline method denoted as Uni, which simply applies the union of two single-condition selection procedures. Experiments were conducted across varying response dimensions, each involving target sets composed of different numbers of spheres. The results, summarized in Table 12, demonstrate that our method consistently maintains effective FDR control with relatively high power under all dimensional and number of spheres settings. In contrast, the Uni baseline consistently produced FDR values exceeding the
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Table 11: Comparison with Baseline of Selection Performance for Conjunctive Conditions under Multivariate Response Settings (Optimal FDR control in bold).
FDR
Power
Dimension of
Inf
MCCS
Inf
MCCS
d=10
0.3534
0.2998
0.9696
0.8878
d=30
0,4195
0.2940
0.9761
0,4737
d=50
0.3469
0.2931
0.9787
0.6338
Table 12: Comparison with Baseline of Selection Performance for Disjunctive Conditions under Multivariate Response Settings (Optimal FDR control in bold).
(a) SuurTep =
FDR
Power
Dimension of y
Uni
MCCS
Uni
MCCS
d=10
0.3079
0.2679
0.7641
0.6977
d=30
0.3164
0.2788
0.7611
0.7718
d=50
0.3133
0.2833
0.7484
0.7794
(b) numap 4
FDR
Power
Dimension of
Uni
MCCS
Uni
MCCS
d=10
0.3125
0.2683
0.7584
0.6759
d=30
0.3178
0.2822
0.7877
0.7236
d=50
0.3094
0.2751
0.8512
0.7905
(c) numppp
FDR
Power
Dimension of
Uni
MCCS
Uni
MCCS
d=10
0.3159
0,2791
0.7614
0.6663
d=30
0.3172
0.2706
0.7964
0.6438
d=50
0.3166
0.2636
0.8199
0.6305
nominal threshold, underscoring the inherent limitations of such heuristic strategies due error propagation and affirming the theoretical soundness of our proposed approach.
C DETAILS OF REAL DATA APPLICATION
C.1 DETAILS OF TASKS OF cv
The cv tasks employ pipeline for monocular depth estimation from RGB images, utilizing pre- trained ResNet and ViT models, respectively. This is combined with the MCCS approach based on conformal inference identify samples falling within specified target depth interval.
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Experiments conducted the NYU Depth V2 dataset, which comprises 1,449 RGB-D images of indoor scenes along with depth maps captured using a Microsoft Kinect sensor. Each image has resolution of 640x480 pixels, with depth values representing physical distances in meters. The dataset is partitioned into training (74%), calibration (13%), and test (13%) sets. Depth values are obtained by averaging the center crop of each depth map, resulting in single sealar depth value per image. We then use Ridge regressor to predict the depth for the following steps.
To intuitively demonstrate the applicability of our proposed method on ev tasks, Figure illustrates the implementation workflow of the algorithm across different aspects of the dataset. The visu- alization component provides critical diagnostic and interpretive tools for evaluating the method's performance: I) The p-value distribution histogram assesses conformity to uniform distribution assumptions under the null hypothesis; 2) The Benjamini-Hochherg procedure plot illustrates the relationship between sorted p-values and the critical threshold line, demonstrating the selection mechanism that controls false discoveries; 3) Depth distribution histograms contrast selected versus unselected samples, revealing how effectively the method identifies targets within the specified in- terval; 4) The prediction verssns--rrsuuuutth scatterplot visualizes regression accuracy while highlighting correct/incorrect selections relative to the interval boundaries. These visualizations collectively offer an assessment both statistical properties (FDR control validity) and operational characteristics (depth estimation accuracy), enabling comprehensive evaluation of the MCCS's effectiveness in identifying samples within the target depth range. Figure shows samples selected in the cv tasks.
Distribution of p-values on Test Set
Benjamini Hochberg Procedure
25
a=0.3
0.3
BH Threshold
Sorted
20
Selection Threshold
freueecc
15
0.2
10-
0.0
0.20
50
150
Sort Indicas
(a) p- value Distribution
(b) Benjamini Kenjamini-Hochberg Procedure
Distribution of real depth on test set
Relationship of Real and Prediction
All samples
Selected samples
Lower bound
15- frequents
20-
Upper bound
Predicted
All samples
Selected samples
10-
Wrong
Lower bound
Upper
6
Depth(m)
Real Depth (m)
(c) Depth Distribution
(d) Prediction-verrsu--Tuut..
Figure 8: Comprehensive illustration the implementation workflow on cv tasks.
Selected samples within depth rarge
Fasl dipirs 1i0i
tee 200m
Basl depov doy
Nen cauth 2.99
2.04 Prideced 3.500
Prediceed 1i1
Figure 9; Selected Samples in the cv Tasks.
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C.2 DETAILS TASKS
The nlp tasks employ a conformal prediction pipeline with FDR control to identify text prompts within target toxicity range. Using text embeddings and regression models, toxicity scores are predicted: the MCCS method then selects samples within the desired interval (e.g., (0.12, 0.7)) while controlling errors. Applied to the Real-Toxicity-Pronpp and Hile-Toxicity-Bllanee datasets, the framework encompasses data loading, feature extraction via sentence embeddings, regression, noneonformity scoring, and the Henhaeini-Hochherg procedure.
The Texicitt--roompeeeee dataset contains 100K online texts, each annotated with continuous toxicity scores (0-1) from Perspective API, capturing diverse real-world language use. The Pile- Toxicity-Balanced subset includes 40K text segments from The Pile, with balanced binary toxicity labels (50% toxic / 50% non-toxic) achieved through stratified sampling to mitigate class imbalance. Together, these datasets support robust toxicity detection: the first enables fine-grained analysis of real-world toxicity, while the second ensures equitable evaluation via controlled class distribution.
The datasets split via stratified sampling into training (60%), calibration (20%), and test (20%) sets to preserve toxicity distributions. For efficiency. up to 30,000 prompts are subsampled while maintaining score distribution. Text embeddinns are generated using MiniLM-L6 (384-dime)sional), a lightweight transformer, and used as features in a Ridge regression model trained only on the training set, with calibration and test sets reserved for subsequent steps. Figures 10 and 11 intuitively demonstrate our method's s applicability by illustrating its workflow as Figure 8a and 8b.
Distribution of p-values on test set
BH procedure
250
0.5
200
0.4
Frequency
Value
0.3
150
0.2
100
0.1
poalues
50
BM critcal volue
Selection threshold
0.0
0.1
0.2
0.3
0.4
0.5
0
1000
2000
3000
4000
5000
6000
p valuu
Sort indices
(a) p-value Distribution
(b) Benjamini-Hochherr Procedure
Figure 10: Illustration of the implementation workflow on nlp task of Real-Toxiccit-Prompts
Distribution of p-values on test set
BH procedure
0.6
300
Prequency
0.4
200
Value
100
scrced pvalues
Selection threshold
0.1
0.2
0.5
0.6
1000
2000
3000
4000
5000
6000
p-value
Sort Indices
(a) paluue Distribution
(b) Senjamini-Hochbor Procedure
Figure 11: Illustration of the implementation workflow on nip task of Pile-Toxieity-Balancee
C,3 DETAILS OF TASKS vQA
The VQA tasks are implemented through a pipeline that leverages conformal prediction to identify samples exhibiting high human annotation agreement within predefined confidence interval. The
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The VQA v2 dataset consists of 214.354 tion set. Each question is associated with 10 -aniitatd answers, facilitating the calculation of an agreement score, defined as the proportion of annotators who provided the most frequent response. This metric serves as a robust measure of consensus and label reliability. The implemented pipeline automatically downloads the COCO validation images, totaling 40,504, along with the corresponding questions and annotations. These components are processed through a customized VQA v2 Dataset triplets, ensuring consistent Published as a conference paper at ICLR 2026
method specifically targets agreement score range from 0.28 to 0.82, designed to capture instances with moderate to strong annotator consensus. At the core the system, the BLIP language model is employed to generate answer predictions along with confidence scores. These scores subsequently mapped to estimates of human agreement using two distinct regression techniques, Ridge and BayesianRidge regression, each providing complementary approach to modeling the relationship between model confidence and human consensus. The final selection of samples is performed using the MCCS (Multi-Class Calibrated Selection) method, which ensures rigorous control over the false discovery rate (FDR) throughout the inference process.
pairs derived from the COCO 2014 valida-
class that systematically extracts and structures maaag-question-amee and efficient data handling for subsequent modeling and analysis.
To ensure computational efficiency while preserving the essential statistical characteristics of the data, the implementation randomly samples maximum of 1,000 image-question pairs. These sam- ples are partitioned into training (80%), calibration (10%), and test (10%) subsets using random sampling, maintaining representative distributions across splits. The BLIP-VQA base model gen- erates answer predictions along with confidence scores, obtained by averaging token probabilities during the text generation process. Meanwhile, human agreement scores are quantitatively defined as agreement count of the most frequent answer reflecting the consensus of annotators for each question. The 10, implementation workflow for the vqa tasks is visually summarized in Figure 12, which follows the same structured format as Figure to facilitate comparison and interpretation, thereby illustrating the method's practical applicability and scalability. Additionally, Figure 13 presents concrete examples of selected samples from the VQA tasks.
p-value Distribution in Test Set
Benjamini Henjamini-Hoohhber Procedure
175-
DC-
Sorted pookes
Blt crtical vauues
Selection thrashldd
Frqquency
75
0.-
5.4
25
00
00
20
60
300
p-value
Someed Index
(a) p-value Distribution
(b) Benjamini-Hochberg Procedure
True agreement distribution on test set
True vs Predicted Agreement
camples
All samples
Selected semples
Selected samples
Lower bound
Incoriecty ected
bourd
bound
Interval upper bound
Froquency
3.e
Predictedt
E
C.A.
16
18
Agreement score
True agreement
(c) Depth Distribution
(d) Prediction-veree versus- versus-Truth
Figure 12: Comprehensive illustration of the implementation workflow on vga tasks.
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Selected samples with moderately ambiguity
KALANE: nough
Arawor ytu
Prodiced Confeerce: 2003
Ageemment 0,003 Ag
Occaioon: What drodtion por
stlly??
The Knoven kitt
Confieace 0.003
Figure 13: Selected Samples in the vça Tasks.
D DETAILS OF MULTI-CLASS TASKS
D.I. SELECTION FOR SINGLE-CLASS
The implementation of the class tasks on the CIFAR-10 dataset utilizes either ResNet ViT architectures with FDR control in the MCCS approach to select specific images from multiple classes. The method accurately identifies target category samples (e.g.. airplanes) while guaranteeing that the FDR remains below a predefined threshold, making it suitable for scenarios where only partial feature information of test samples available. The core idea involves treating the multidimensional response not as set of individually meaningful dimensions but rather as a holistic feature vector, thereby overcoming limitations inherent in existing approaches.
The CIFAR-10 dataset comprises 60,000 32 x 32 RGB images across 10 classes: airplanes, automo- biles, birds, cats, deer, dogs, frogs, horses, ships, and trucks, with 6,000 images per class, divided into 50,000 training and 10,000 test images. Data are partitioned via stratified sampling preserve class balance: the test set contains 500 samples with equal representation of target/non-targer classes, while the calibration set (also 500 samples) maintains 1 target-tonnon-taree ratio of CIFAR- 10. The remaining samples comprise the training set, ensuring an adequate number of positive examples for prototype computation while maintaining realistic class distributions.
The class prototyping approach establishes discriminative decision boundaries within the fine-tuned feature space, drawing inspiration from ProtoNet by representing each class through an embedded prototype. The target class prototype P is computed as the centroid of all target-class samples: / - f. A class-specific radius predefined according the feature dimension, defines the acceptance region, with sample membership determined the Fuclidean distance metric
The regression component bridges the representational gap between feature domains via Multi- OutputRegressor with Ridge regression. This model learns mapping: Redermane -> da from pre-trained to fine-tuned feature spaces. Crucially, this mapping simulates the transformation from accessible features (e.g., pre-trained embeddings) to ideal but often unattainable features (e.g., fully fine-tuned representations), which can be substituted as needed in practical applications. Trained exclusively on the training partition, the regression model enables the inference- time prediction of task-adapted features from generic representations, enhancing applicability in real-world scenarios. By ensuring that the proportion of incorrectly selected samples remains below a predefined threshold (e.g., q = 0. this pipeline enables the reliable deployment of high-confidence predictions.
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D.2 SELECTION MULTI-CLASS
The selection of multiple target classes, representing disjunctive conditions, naturally extends the application of MCCS in multi-class tasks. This approach facilitates the extension from single-class multi-class selection via parallel hypothesis testing using class-specift prototypes. The proportion of target class samples the calibration set is set to 0.65 to ensure adequate positive samples for prototype computation. Figure 14 illustrates sample selection outcomes using the target class of "airplanes" in the Single-Class Task and the target classes "airplanes, automobiles, and birds" in the Multi- Clati- Task as examples.
(a) Class "airplanes".
(b) Classes "airplanes, automobiles, birds".
Figure 14: Selected Samples in the Single-Clan Class and Multi-Class Task.
D.3 SELECTION FOR SIMILAR-CLASS
The selection of similar target classes, representing conjunctive conditions, naturally extends the application of MCCS in multi-class tasks as identifying samples within a specific distance threshold from given class prototype. This experiment utilizes the CIFAR-100 dataset due its inherent superclass structure, which provides ground-truth semantic groupings for evaluation. The proportion of target class samples the calibration set was set 0.5 to ensure balanced representation.
To summarize, the MCCS multi-class tasks is validated using the CIFAR-10 (Krizhevsky et al., 2009a) and CIFAR-100 (Krizhevsky et al., 2009b) datasets, enabling the selection of individual (class-A), multiple (class-B), and similar (class-C) classes. Results nominal FDR level of 0.1 are presented Table 13. Our method maintains rigorous FDR control across all the class tasks.
Figure 15 displays selection of retrieved samples for the query class "dolphin" within the target superclass "aquatic mammals," and the class proportion within the selected samples. Notably, some incorrectly selected samples, such "shark", which does not belong to the superclass yet exhibits visual similarity to dolphins, were also identified, illustrating the method's ability to capture perceptually coherent features despite occasional semantic mismatches, thereby underscoring the rationale behind the approach. Furthermore, the balanced selection across categories rather than focusing on a single class demonstrates the method's effectiveness in handling similarity, a property ensured by the circular region defined around class prototypes.
Table 13: Observed FDR and power for class Tasks at the nominal level 0.
Task
FDR
Power
class-A
0.084
0.994
class-B
0.094
0.733
class-C
0.096
0.643
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Selected samples for Aquatic Mamnnal
Class proportion of selected samples
30
Marine Marmals
25
Other Classes
Percentage
15
10
whale
dolphin
seal
otter
beaver
shark
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(a) Selected Samples.
(b) Class Proportion.
Figure Selected Samples and Class Distribution in the Similar-Class Task.
E ACKNOWLEDGMENT OF THE USE OF LARGE LANGUAGE MODELS (LLMs)
Large Language Models (LLMs) were used to aid in the writing and polishing of the manuscript. Specifically, we used an LLM to assist in refining the language. improving readability, and ensuring clarity in various sections of the paper. The model helped with tasks such sentence rephrasing, grammar checking, and enhancing the overall flow of the text.
It is important to note that the LLM was not involved in the ideation, research methodology, or experimental design. All research concepts, ideas, and analyses were developed and conducted by the authors. The contributions of the LLM were solely focused on improving the linguistic quality of the paper, with no involvement the scientific content or data analysis.
The authors take full responsibility for the content of the manuscript, including any text generated or polished by the LLM. We have ensured that the LLM-generated text adheres to ethical guidelines and does not contribute to plagiarism scientific misconduct.
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Reviews:
[{"review_id": "TnO4fd8Ci5", "reviewer": "Reviewer_GiWV", "summary": "The paper considers the problem of selecting samples with FDR control, where multiple conjunctive or disjunctive intervals characterize the selection criterion. The proposed method generalizes *conformal selection* by Jin and Candès, and is evaluated in numerical experiments.", "questions": "Refer to the \"Weaknesses\" section.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "The writing of this paper is clear and easy to follow; the technical derivation is rigorous.", "weaknesses": "The problem raised in this paper seems to have already been solved by existing works in full generality. \nFor example, [1] considers the selection criterion characterized by a general set and allows it to \ndepend on the covariate, where the idea is to design the score reflecting the likelihood of $Y$\nfalling into the property set. The proposed (adaptive) method therein achieves \nfinite-sample FDR control, with cfBH as a special example. It would be helpful if the author \ncould clarify the contributions of the work.\n\n**References** \n\n1. Gui, Yu, et al. \"ACS: An interactive framework for conformal selection.\" arXiv preprint arXiv:2507.15825 (2025).", "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": "mLek0nSS9A", "reviewer": "Reviewer_yTUb", "summary": "This paper studies the conformal selection (CS) problem, that is, to select --- among many test points --- those samples whose unknown responses satisfy a pre-specified property. While previous conformal selection method addresses a one-sided property, this paper considers multiple conditions such as an interval or the combination of two one-sided intervals. The authors first propose that taking the intersection/union of two conformal selection sets invalidates the FDR control. Then, for conjunctive conditions, they propose a method that designs a conformity score specialized for addressing such properties in the CS framework. For disjunctive conditions, they propose to select from all p-values for each test point and each property with BH, and show the FDR control. The proposed methods are demonstrated in extensive simulations and real-world applications.", "questions": "1. What does the FDR control for selecting among all the $(j,k)$ pairs in Algorithm 3 mean? The definition is not explicitly given, and based on my guess, the practical interpretation is a bit weird to me. Ideally we want an FDR over *samples* so that at least $1-\\alpha$ fraction of them satisfy any of the conditions. However, in the current setup, consider 2 conditions, and suppose a sample j is selected for both $(j,1)$ and $(j,2)$, its meaning would be a bit strange, and I'm not sure the FDR over all the pairs can reflect the practical need very well. \n2. I was wondering if the problem can be solved via a simpler strategy. Suppose we want to select samples with $Y\\in (-\\infty, c_1] \\cup [c_2,+\\infty)$. Can we define a property as $Y^* = \\mathbf{1}_{Y\\in (-\\infty, c_1] \\cup [c_2,+\\infty)}$ and directly apply the conformal selection method?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is tightly structured and clearly written. \n2. The problem addressed is of practical relevance. \n3. The discussion on technical challenges is precise and convincing.\n4. The numerical experiments are extensive and solid.", "weaknesses": "1. The FDR guarantee for disjunctive conditions is hard to interpret (see my Q1 below).\n2. The solution seems over-complicated (see my Q2 below). Seems a simple strategy can greatly simplify the proposal.", "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": "ZzPvjpjvb3", "reviewer": "Reviewer_eRb5", "summary": "This paper introduces the Multi-Condition Conformal Selection (MCCS) algorithm, an extension of the conformal selection framework designed to address scenarios involving multiple selection criteria. The method specifically tackles conjunctive conditions and disjunctive conditions, which frequently arise in practical applications where decisions depend on simultaneous or alternative constraints. The central innovation lies in the development of a tailored nonconformity score that ensures regional monotonicity for interval-based selections, alongside a global adaptation of the Benjamini-Hochberg (BH) procedure to manage disjunctive constraints. These components are underpinned by rigorous theoretical analysis, establishing finite-sample control of the False Discovery Rate (FDR) under exchangeability assumptions.", "questions": "1. **On the Core Technical Contribution and Methodological Necessity:** My primary concern pertains to the fundamental architecture of MCCS. Please refer to weakness 1.\n2. **On Experimental Design and Interpretation:** \n    - The experiments primarily feature a small number of conditions (e.g., Tasks 1-6 in Table 2). A key claim of the method is its generalizability to diverse condition combinations. How does the power of MCCS scale as the number of intervals $K$ becomes large?\n    - The paper reports aggregate FDR and power. It would be insightful to analyze which specific conditions (which intervals ) contribute most to the discoveries. Does the global BH procedure lead to a balanced selection across all target intervals, or does it favor intervals with certain characteristics?\n3. **On the Equivalent Interval Representations:** Consider a target interval $I = (a, b)$. This can be equivalently expressed as a single interval or decomposed into two sub-intervals $(a, c] \\cup [c, b)$ (for $K=2$). While these two representations are semantically equivalent in defining the target set, will they lead to different algorithmic constructions? Can you explain on this?\n\nI will raise my score if these concerns are properly handled.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1.  **Theoretical Soundness:** The paper provides rigorous theoretical foundations, including proofs for finite-sample FDR control under exchangeability conditions (Theorem 4.1). The extension of conformal selection theory to multi-condition settings is non-trivial and well-articulated.\n2.  **Comprehensive Evaluation:** The experimental section is thorough, demonstrating the method's applicability across diverse modalities (text, images, multi-modal) and tasks (single-class, multi-class, similar-class selection). The comparison with relevant baselines (Inter-cfBH, Union-cfBH) is appropriate and highlights the limitations of naive approaches.\n3.  **Clear Presentation:** The paper is generally well-written, with a logical flow from problem formulation to methodology, theory, and experiments. The figures, such as the illustrative diagrams, aid in understanding the core concepts.", "weaknesses": "1.  **Perceived Marginal Technical Contribution (Major Concern):** The fundamental idea can be viewed as generalizing the single-threshold selection $Y > c$ to a more general form $Y \\in I$. A significant portion of the theoretical and methodological machinery seems to be a direct adaptation of the established cfBH (Jin & Candès, 2023) framework.\n     - The problem could potentially be reframed by defining a suitable function $\\phi(Y, I)$ that encapsulates the multi-condition logic, reducing it to a single-threshold problem on $\\phi(Y, I)$. For instance, for an interval $I = (a, b)$, one could define $\\phi(Y, I) = \\min(b-Y, Y-a)$, where $\\phi(Y, I) > 0$ is equivalent to $Y \\in (a, b)$. A similar construction could be devised for disjunctive conditions. The core conformal p-value machinery and BH procedure from cfBH could then be applied to $\\phi(Y, I)$, potentially obviating the need for the newly proposed nonconformity scores and the specific \"global BH\" narrative. While the authors' approach of designing a regionally monotone score is one valid path, the paper would be significantly strengthened by directly addressing this alternative, more unified perspective. A discussion justifying why their specific decomposition (separate scores per interval + global BH) is preferable to a unified $\\phi$-function approach is crucial. The current presentation makes the contribution feel more like a specialized extension rather than a fundamental generalization.\n\n2.  **Clarity on Theoretical Novelty:** The proof techniques, particularly for Theorem 4.1, heavily rely on and extend the principles established in cfBH. The paper could more clearly delineate the specific novel technical challenges overcome in the multi-condition setting compared to the single-condition base.\n\n[1] Jin Y, Candès E J. Selection by prediction with conformal p-values[J]. Journal of Machine Learning Research, 2023, 24(244): 1-41.", "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": "9nheYUwwkJ", "reviewer": "Reviewer_2sng", "summary": "The paper introduces multi-conditional score for conformal selection (MCCS). Current approaches are restricted to single-threshold scenarios and MCCS proposes a novel nonconformity score for conjunctive and disjunctive conditions that is capable of dealing with multiple conditions. The experiments show the effectiveness of the technique in multiple applications and also for multi-variate scenarios.", "questions": "- Can the authors add some discussion regarding computational requirements of MCCS, complexity and memory estimates will clarify and possibly amplify the applicability of the approach. \n- Briefly discuss extensions of the current method to domain shift will strength the paper claims. \n- Regarding algorithm 2 (Global BH procedure) $m \\times K$ sorting seems a bottleneck, is there a way to make that part more efficient?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper presents a novel extension of conformal selection to multiple conditions, it addresses an important gap in the area.\n- The paper present theoretical guarantees via finite sample FDR control. \n- MCCS general framework is capable of handling both conjunctive and disjunctive conditions, multiple intervals and multivariate responses. That broad scope is a strength of the work.", "weaknesses": "- The experimentation section in terms of baselines look not fully explored. Experiments against other FDR control methods will strength the claims.", "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 (Poster)