Training-Conditional Coverage Bounds under Covariate Shift
Authors: Mehrdad Pournaderi
Yu Xiang
Venue: Accepted by TMLR ()

Abstract: Conformal prediction methodology has recently been extended to the covariate shift setting, where the distribution of covariates differs between training and test data. While existing results ensure that the prediction sets from these methods achieve marginal coverage above a nominal level, their coverage rate conditional on the training dataset—referred to as training-conditional coverage—remains unexplored. In this paper, we address this gap by deriving upper bounds on the tail of the training-conditional coverage distribution, offering probably approximately correct (PAC) guarantees for these methods. Our results characterize the reliability of the prediction sets in terms of the severity of distributional changes and the size of the training dataset.

Published in Transactions on Machine Learning Research (01/2026)
Training-Conditional Coverage Bounds under Covariate Shift
Mchrdad Pournaderi
m.poorraderriuuaaaa.cct
University of Utah
Yu Xiang Florida Atlantic University
griinn@@fau edu
Reviewed on OpenReview: https: // openreview. net/ forum? id= F6h
Abstract
Conformal prediction methodology has recently been extended to the covariate shift sctting, where the distribution of covariates differs between training and test data. While existing results ensure that the prediction sets from these methods achieve marginal coverage above a nominal level, their coverage rate conditional on the training dataset-referred to as training- conditional coverage-romains unexplored. In this paper, we address this gap by deriving upper bounds the tail of the "raining-ooddtiona coverage distribution, offering probably approximately correct (PAC) guarantees for these methods. Our results characterize the reliability of the prediction sets in terms of the severity of distributional changes and the size of the training dataset.
1 Introduction
Conformal prediction is framework for constructing distribution-free and model-agnostic pre- dictive confidence regions under the exchangeability assumption for the training and test sam- ples (Vovk et al. 2005) (also see Shafer and Vovk (2008); Vovk et al. (2009); Vovk (2012)) Let ((X1,1)) (X2, Y2)..... (X,, Yn), (Xeet, Ylest) )) denote a tuple of exchangeable data points, consisting of a train- ing sequence of n samples D = ((X,, Yi),.., (Xn, Yn)) E (X' (X' and one test sample (Xtest, Yrest) E Xxx.. For fixed error rate level a, the conformal prediction framework, provides a prediction set (Xtes) for Ylest that satisfies
P (Yrost E C. (Xtes))) 21-a.
(1)
where č : :'' + 2 is a data-dependent map. This type of guarantee is referred to marginal coverage, it is averaged over both the training and test data.
One natural direction to stronger results is to provide a coverage guarantee conditional on the test data point Xtest; i.c.,
(Yrres E
Xtest 2
However, when Xtest has a continuous distribution, it has been shown in Vovk 2012 Foygel Barber al, (2021): Lei and Wasserman (2014) that it is èmpossible to obtain (non-trivial) distribution-froe prediction regions C., in the linile-sample regime; relaxed versions of this type of guarantee have been extensively studied (see Jung et al. (022): Gibbs et al. (2023); Vovk et al. (2003) and references therein). As different approach to stronger guarantees, several results (e.g., Vovk (2012); Bian and Barber (2023)) have been reported on the training-come conditional guarantee by conditioning on D,,, which also more appealing than the
marginal guarantee as can be seen below. Define the following miscoverage rate as function of the training data,
(.(D.) = P (Yrra & C. (Xreet)))

---

Published in Transactions on Machine Learning Research (01/2026)
Note that the marginal coverage in I is equivalent to (D,)) S a. The training-conditionn guarantees concern the concentration of the conditional error rate below the nominal level a and they have the following form, for some small > 0,
P (D.)
or its asymptotic variants. Roughly speaking, this guarantee means that the (1 ax) -leeel coverage lower bounds hold for a generic dataset.
Recently, Barber et al. (2021) proposed modified versions of jackknife and cross validation (CV), namely jackknife+ and CV+, which can be used to compute conformal prediction sets. For the K-fold CV+ with ! samples in each fold, the
coverage bound
P (D,( 2 2a + (o/0/( 28
(2)
is established in Bian and Barber (2023) Additionally, Liang and Barber 2023 proposed training-conditional
coverage bounds for jackknife++ and full conformal prediction sets under the assumption that the training algorithm is symmetric and satisfies certain stability conditions (see Section 4 for more details). In this line of research, samples are assumed to be i.idd,, which is not only exchangeable but also ergodic and admits some nice concentration properties. However, this assumption can be violated in the application. In particular, the input data distribution during deployment can differ from the distribution observed during training. This phenomenon is called distribution shift and it is a crucial problem in trustworthy machine learning (see Section 2.3 In this regard, split and full conformal prediction methods as the central methods for distribution- free uncertainty quantification of black-box models have recently been extended to handle a popular type of distribution shift called covariate shift (Tibshirani et al. 2019). In the covariate shift setting, the distribution of covariates in the test data differs from the one observed in the training data, but, the conditional distribution of the response given the features remains the same across the training and test populations. A similar extension has been made for the jackknife++ method Prinster el al. 2022
Despite these significant progress on handling distribution shift, they primarily focus on marginal coverage. Although the weighted conformal prediction mothods (proposed in Tibshirani et al. (2019) and Prinster et al. (2022) are guaranteed to keep the marginal coverage rate above the nominal level, they often reduce the concentration of the training-conditionn coverage rate P(Yest E (Xusat (D,) (i.c., coverage rate conditional on the training data D..) around the nominal level a. In particular, the weighting scheme leads to heavier tails for the training- coniiiionooiiiinnlll coverage rate. This phenomenon is illustrated in Figure | where ordinary and weighted conformal methods with 80% target coverage rate (a = 0.2) are used for constructing prediction intervals under exchangeability and distribution shift settings, respectively. The distribution shift is introduced artificially via resampling according lo an exponential tilt as in Section 2.3 from Tibshiranii et al. 2019). It can be observed that the tails of the training conditional coverage get heavier when the weighting scheme used to handle the distribution shift. In this paper, we explore the quality of the weighted conformal prediction sets by computing upper bounds on the tails of the training-conditional coverage distribution, quantifying the relationship between the tail behavior and the extent of distribution change. In other words, we examine the efficiency of weighted conformal prediction methods for generic training dataset under distribution shift.
We present concentration bounds of the training-conditional coverage for weighted jackknife++ (JAW) Prinster et al. 2022), full, and split conformal methods. Regarding the training algorithm, no assumption is made for the split conformal method. However, full conformal and jackknife+ methods are analyzed under the assumption of uniform stability explained below. Let ÂD, denote the predictor function estimated using the training data D,,. A training algorithm is called uniformly stable if,
[AD, PiD; 1xx S
with B = (1/nn) for any two datascts (D.,, D',) differing in one (training) data sample (Bousquet and El Elisseell 2002 This is a stronger notion of algorithmic stability than the (m, n1)-tabilitty assumed in Liang and Barber (2023). Nevertheless, uniform stability is satisfied by a class of regression models known as RKHS regression [Bousquct and Elissooff. 2002), i.c., regularized empirical risk minimization over an RKHS (Paulson) and Raghupathi] 2016 Schölkopf and Smola 2002). Examples of reproducing kernel Hilbert spaces are certain Sobolev spaces of smooth functions (Wahba 1990).
2

---

Published in Transactions on Machine Learning Research (01/2026)
exchengeabiiity
exchengesbilltty
shirt
shitt
0
Jonsky
6-
Densty
9.6
0.7
0.8
0.9
1.0
0.6
0.7
0.8
0.9
10
Coverage
Caverage
exchangsabiitty
eschangesbility
0.5
Coverage
Coverage
coverage rates are presented for four datasets from the UCI Machine Learning Repository: Wine Quality (top left), Abalone (top right), Concrete Compressive Strength (bottom left), and Combined Cycle Power Plant (bottom right). See Appendix E for details of this simulation study.
Figure 1: Histograms of the
2 Background and Related Work
2.1 Full conformal and split conformal
Define the shorthand D, U (x) ==
Yn), (æ,, y)) and let ((awy) == T(D,, U (x,y) v))
denote the regression function obtained by rumning the training algorithm on D,, (x, Define the score function s(x', y/s (1) = f(((='),x)) via some arbitrary (measurable) cost function ʃ y x) + and predictor function μ: X + V. For instance, s(ø', w'ip) = ly - -=(z)) when f(y.v) = ly v/l. For the data multiset D, ((X,, Y,) X xx Wiie [m]] where [n] == (1,2. (1,2....,)) define
S(D,: = {s(x', v/))) (2' e.)
Observe that if T is symmetric, i.c., T(D) invariant under permutations of the elements of D for any data tuple D, then the elements of S(D,, U (Xrest, Yust): PAXe Yina)) are exchangeable. Therefore,
P (((eestt Yeast; (XXsa Yiua)) < f-1 S(De, Veea lifl Nenr Nean (1 - -a)) 2 1 a,
where (.) denotes the empirical quantile function of S. Thus,
P Yuest E ddull (Xtost) 2 - O2,
3

---

Published in Transactions on Machine Learning Research (01/2026)
where the following confidence region is referred to as full conformal in the literature
C (x) = {y: : s(x, yi ((x.))) - S(D..
- a)}
S {y s(a, yi ((x.))) - F s(D uffz) (1 - =a)),
(4)
It is well-known that this approach can be computationally intensive when y = R since to find out whether E Chull (x) one needs to train the model with the dataset including (r, One simple way to alleviate this issuc is to split the data into training and calibration datascts, namely D, = Dirain U Deal Without loss of generality (when splitting), we assume Dirain = ((X1, Yi), (X)...,,,,,... (Xn-m; Yn-m)) and D ((Xn-m+1: Yn Yn-m+1)..... (Xn,, Yn)). In the split conformal method, one first finds the regression predictor À = T(Dira))) and treats á as fixed. Note that the elements of S(Deal U (Xrest, Yest); (i) are exchangeable.
Hence, we get
P P (S((eestt Yeest' Pb) S F Mem Yem (1 21
Hence, we have beest E Čaplit (Xeest)) 2 1 a for
(x)
(1 {v s(z, v: & f o(oo)) (1
(5)
2 y/ sl vs (1) < S s(Dm)) - aa)
2.2 Jackknife+
Although the split-conformal approach resolves the computational efficiency problem of the full conformal method, it is somewhat inefficient in using the data and may not be useful in situations where the number of samples is limited. A heuristic alternative has long been known in the literalure, namely, jackknise or leave-one-out cooss-vlldatiin that can provide compromise between the full conformal and split conformal methods. In particular,
C2(2)) = (y (0)8(1 < F5 -a)),
where A T(D,), SJ
{s(X,, Y:
< i is Despite its effectiveness, no general finite-
S n) and A-
(X),,, (X--,, (Xi+1, Y++))..... (Xn, sample guarantees are known for jackknife. Recently, Barber et al. (2021) proposed jackknife+, a modified version of the jackknife for y = IR and f(y.z) = 1 1/1. and established (1 2a) finite-sample coverage lower bound for Let
(z) = (x) - - -xx)) [n]] (foo),
s St (x) (x> IY -x) (n]) u (oo).
The jackknife+ prediction interval is defined as
(x) Fst(n) a))]
In the same paper, an e-inflated version of the jackknife+
C++t(())= (F5 - P5t - a) +
(6)
is proposed which has 1 a A coverage lower bound (instead of 1 -22a, if the training procedure satisfies
max P(Iū(Xuest) - i -aai--''--ii (Xtust) E) <v.
*E(m]
Also, the jackknife+ has been generalized CV+ for K-fold vrass-aali,ttion, and (1 2a 22/n) coverage lower bound is established.
4

---

Published in Transactions on Machine Learning Research (01/2026)
2.3 Conformal prediction under distribution shift
Covariate shift concerns the setling when the covariate distribution changes between training and test data, while the conditionals remain the same. Specifically, we have
Z; = (Xi, Vi) LLd. Px x Prxx =: P
training data;
Zestt = teets Xtest) ~ Qx x Pyxx =: Q
test cata.
This notion has been extensively studied in machine learning (e.g., see Sugiyama et al. (2007
Quinonero- Candela et al. (2008): Sugiyama and Kawanabe (2012) Wen et al. (014)): Reddi et al. (2015 Chen et al. (2016) and the references therein).
To handle the covariate shift, weighted version of conformal prediction was first proposed in the seminal work by Tibshirani al. (0019). The key assumption is that the likelihood ratio dQ/dP= dQ2x /dPx, needs to be known; we follow the same assumption in this work. This reweighting scheme has been extended to various settings in conformal prediction Podkopaev and Ramdas (2021): ) Lei and Candès (2021) Fannjianga
et al. Guan 2023 The general domain adaption problem has also been an active area from a causal
perspective (e.g., Zhang et al. (2013a): Peters et al. (2016); Gong et al. (2016); Chen and Bühlmann 2021
Du and Xiang (2023)
The weighted procedures require computing weight associated with each sample,
(Zeest
0C: (rest) EiEr
dQ JP (Z.)
iEZ I and Wtest
Zrest) Evez
4Q JP (Z.)
JP
where I = [r|] for full conformal and jackknife+,, and C [n]] for the split conformal method which corresponds
to the calibration (or hold-out) dataset. In particular, generalize the full and split conformal methods distribution shift setting, according to Tibshirani et al. 2019 one needs to replace the empirical CDF used
in construction of (4) and (5)
Fs(()
11 (=
n 1
sES
by the weighted version
FZ Whest 1 (t x0) +
Σ m1 11 (s(2)) St). iEI
Similarly, extend the jackknife+ method, Prinster et al. (2022) proposes replacing fis- and Pist by their corresponding weighted versions where Wtest is associated to and oo, respectively.
3
Conditional Coverage Guarantees
3.1 Split conformal
The training-conditional coverage guarantee for split conformal method under the i.i.d. setting has been studied
Kiorok--KKKffoooooooooo (DKW) inequality. The following theorem concerns training-conditione. coverage guarantee under covariate shift, where the extension is made by developing a weighted version of the DKW inequality. Let D,, (Z,,..., Z,,) (X' 1)"" denote n-tuple of data points containing both the training and calibration data
in Vovk 22122; Bian and Barber (2023 The proof in this sctting relies on the
sets. Define the training conditional probability of error as
peplit (D.) P (Yunst & Eaplit
Cagllt (Xurs)
Theorem 1 Let m < denote the size of the calibration data set. Assume that Q is absolutely continuous with respect to P (Q < P) and dQ /PPPP < B < co. Then,
( > a + (/2B (olgo(/4/))))))) 20) 1 S.
p
5

---

Published in Transactions on Machine Learning Research (01/2026)
for all 8 > where C> is universal constont. in Dn, follows P.
The probability is taken with respect to P since each entry
Corollary 1 Relaring the assumption dQ/dP & B to IdQ P.2 = l(dQ/dP) IdQ//PPp==1(Q/pPPp
d B via Lemma
we
get
poplit Di.) > a+ 2B(2C +1)
P
<
0 0<
6/m
To assess the robustness of weighted conformal prediction methods, we compute training-conditional error bounds with a very mild assumption on dQ/dP. The proof of this theorem follows from Lemma 4 and the proof of Theorem !
Theorem Assume that Q is absolutely continuous with respect to P (Q < P) and that dQ/dP < oo Q-a.s. Pir B, 5 > O and e E (0, 1/2). It holds that,
dQ (Zear) > m
dQ (Zees)
poplit < 36 4P
+ 2 m
+ P
> B
dP
dP
with probability at least,
4e (2832)(2B2))
2e
20
2T
dQ dP
7 (eess) m
dQ
{ dQ
(Zonkis) S m))
E
Zoatibb)
1 dP
mg²
dP
where m m IP
dQ TP (aati)))
I B) m 1/2ee
3.2
Full conformal and jackknife+
Let uą X' R denote a predictor function parameterized by 8 e RP. By a slight abuse of notation, let the map T: (X x 7)" + Rº denote a training algorithm for estimating B, hence, B.. = T(D,). In this case, we have ÅD. = P3..'
A Assumption 1 (Uniform stability) For all i E [m], we have
sup 21....5p
(1 2
In the case the ridge regression Hocrl and Kennard 1970 with y= and x= fx | < b), this assumption holds with 6, = 12 f7((n), where À denotes the regularization parameter. See Bousquet and Elisseeff (2002) for the general result on the uniform stability of RKHS regression.
Assumption 2 (Bi-Lipschitz continuity) The map 3 μi is bi- bi-Liccchite with respect to the co-norms, i.e.,
K1 || - 8' S I1 loo I K2 || - 3'lloc
for some conslants 0 < K1 < Ki 00.
Remark 1 is worth noting that if the parometer space θ C IPP is compact, & U + (x) given by B r> Pg is continuously differentiable for some open U 2 θ, then fi < oc. Moreover, the inverse function theorem (for Banach spaces), gives the sufficient condition under which the inverse continuously differentiable over (U) and hence 81 0.

---

Published in Transactions on Machine Learning Research (01/2026)
In the case of linear regression with X' fx: ||>||| S 6), verify that Assumption 2 holds with
and R2 /bb. Let Bn E Bn, B_t T Z1-1, Z......., (n) with Z, (X,, Y) and B_, E 3
Define
p(n-1)(t) 12,x ()ri- 1Y--4 13 (Xi) st) FQ (t) P 1²_, (X1) st)
Assumption 3 (Bounded density) F(()) < Ln and la LQn where F(()) and F(t denote the derivative of F(n) and Fln). respectively.
We introduce the following shorthand:
1
n
2p
A(n,p,e) c) == 2xz Gn-1
Ki
+
V
2kg
log
Theorem (Jackknife+ under exchangcability) Define Pel (Dn) = :-P(Yeet & O.+ (Xtest) Dm) Un- der Assumptions 1 3 for all e, > O, it holds that
log(2/0)
P P (D,))
a+ >
+ L,-1 A(n,prc)
setd.
2n
Now we present our result on jackknife++ under covariate shift when dQ/dP & B.
Theorem (Jackknife-+ under covariate shift) Assume that Q absolutely continuous with respect for all e, 0, it holds that
P (Q & and dQ/dP- < B. Under Assumplions
P P (D)( >a 2Vo 20) V + Lon - Se+8.
where C. is a universal constant and e( is the same as in Theorem
Using the same arguments in the proof of this theorem, Unlike 2 which is meaningful only if the number of samples in each fold & is large, the bound we present in the following corollary is suitable for cases where l/n + 0.
get coverage bound for the CV+ as well.
Corollary (CV+) Define PCV+ PCV+(D,)) = (Vrest & CCV+ (Xtt(())st) D..). Under Assumplions for all c, > O, it holds that
log(2/6)
2p
p cV+ (D.) > a
2n
26 Ln Cn-t
K5
V
2
log
Se+6.
The following theorem concerns the training-conditionaal guarantees for the full conformal prediction. We again introduce shorthand:
2p E(n,p,e) Cat1 + 2n log
E K
Theorem (Full conformal under exchangeability) Define pfull (D.) = P(Yoee & Chill (Xree)X)))).. Under Assumplions ] for all 6, > 0, il holds that.
log 1og(2/0) In F(m.p.e) sc+5.
pfull (full(D,) (Da) a
2n

---

Published in Transactions on Machine Learning Research (01/2026)
Theorem (Full conformal under covariate shift) Assume that Q is absolutely continuous with respect to P (Q < P) and dQ/dP= < B. Under Assumplions
1 3 for all e, 6 >> O, it holds that
(D,)) > + (V28log(110)+ 20) V + LQn F(n. Se + ō.
where C' is a universal constant and p,), is the some as in Theorem 5
We note that similar to Remark 1 one can relax assumption dQ/dP < B to |dQ/dP]] S B and in this case the bounds for the jackknife+ and full conformal methods hold with the slow rate ((1/(ó/z) instead of O(log (1/5)/vn).
4 Discussion
We have presented training-conditional coverage guarantees for weighted conformal prediction methods under covariate shift. The split conformal method has been analyzed for the black-box training algorithm while the full conformal and jackknife+ have been analyzed under three assumptions. Although Assumptions and 2 have been verified only for the ridge regression in this paper, we conjecture that they are satisfied by truncated version of the general RKHS regressions which we leave for future research. Truncated and sketched versions of the RKHS models have been extensively studied in the previous literature from the computational efficiency perspective (see Amini (022]; Williams and Seeger (2000); Zhang et al.) (2013b); Alaoui and Mahoney (2015); Corles et al. (2010) and references therein). The results in this paper quantify the training sample size (or calibration sample size for split conformal) needed for the coverage under covariate shift.
Estimation of the likehood ratio. A natural question here is how to include the error arising from estimating the likelihood ratios in the analysis. Define
V V
d0
V
d0
Σ
DiEln)
IQ
aQ Eie(n)
(Z,)
dP
and let s() denote a fixed score function. In this case, one needs to include the following term to the weighted DKW inequalities derived this paper,
Σ (w: ( 50 iein]
Σ
sup RER
sE/n]
V1
Σ
V
V
ic|m]
Σietn
(Dr.V - V.V
+ V.V - -vDD))
VV
1i - V/ + Vv vv- ( 1i)
Eie(n]
VV
Eieni
1
Vg
V
v
Eie V - Va
<2
where the concentration properties of this term around zero depends on the estimator of dQ (Q/dP(Z)) Zz).
Comparison with the (m, n)-stability. The (m, n))-tability parameters were recently introduced in Liang coverage bounds for inflated full conformal and
and Barber (2023) and used compute jackknife+t prediction intervals under exchangeability. Unlike uniform stability which is distribution-free
8

---

Published in Transactions on Machine Learning Research (01/2026)
property of training process, (m, -ittaiiiyyy depends both the training algorithm and the distributions of the data as follows
yout == 18 lAD, (Xtest) - Dm (Xtest))s
(9)
sinn m, = 1E EiDD (X1) AD (X1)l.
(10)
with Xest IL D Dn+m: Dn+m ((X1, YI), (Xn+ms \n+m)), and (D, denotes the predictor function obtained by training on Dn- Although weaker than uniform stability, these parameters are yet not well-understood in a practical sense. To elaborate on the difference between this approach and uniform stability, we evaluate their resulting training-conan conditionl bounds for the ridge regression under exchangeability.
Assume X fo ||^||| s b} and y [-I, I.. As stated in the previous section, this regression model satisfies G, 16 (/122T2/A1ng) (Xn), K1 and K2g bpb. Hence, get the following bound for both full conformal and jackknife- methods,
Pa(Dn) > a O( 1/2 (Vlog(118)+
S +
On the other hand, the following bound is proposed for the infflated +ackknife+ in Liang and Barber 2023
log(1 /5) min(m, n) +2
yout m.
wout m
P
p (D,) > a +3
+2V
I 38 +
(11)
7
7
for all 1. We get viout (men) since v/ouu < Co+1/2 2 by definition 9
and Assumption
and
yout < Sm-m-1 Vi holds according to in Liang and Barber in bound we obtain
2023
Lemma 5.2)
Substituting for goout
mm,n-
pd (DtnDD>>+0))
log(1/0) min(m,, n))
m Cn-1
MCn-1
P
>
< 35 +0
(12)
Letting m m-2 1/2
(m/n)2/3 to balance the two terms V mj
1og(1/8)
and m/mm-1/7 we get n2/5, By
plugging m 2/5 into 12 we have
IP p
Dm) > a 0 (Vlooo(())))) + x -1111) - 38 +0 11-1/5--1/3
(13)
This bound, although dimension-free (i.e., does not depend on p). is very slow with respect to the sample size. Liang and Barber (2023), the same bound as (11) established for 7-inflated full conformal method except with pm-1 instead of - Hence, the same bound as (13 can be obtained for the y-inflated full conformal method via viln O(mx)).
5 Conclusion
In this work, we have studied the training-conditional coverage bounds of full conformal, jackknife++, and CV+ prediction regions from a uniform stability perspective, which is well-understood for convexly regularized empirical risk minimization over reproducing kernel Hilbert spaces. We have derived new bounds via concentration argument for the (estimated) predictor function. In the case of ridge regression, used the uniform stability parameter to derive a bound for (m, -istabiiiyyytyy and compare the resulting bounds from Liang and Barber (2023) to the bounds established in this paper. We have observed that our rates are faster in sample size but dependent on the dimension of the problem. Even though our work is thcoretical in nature, it can potentially shod light on understanding a much broader downstream-task setups in reconstructive self-supervised learning (SSL). where the existing literature focuses on either linear regression ridge regression (e.g., Lee et al. 2021 Teng et al. (2022): Du and Xiang (2024)). ). For split.
have
9

---

Published in Transactions on Machine Learning Research (01/2026)
conformal, our result allows for flexible downstream setups in SSL well beyond simple regressions, while for jackknife++ it would be interesting to reveal the interplay between uniform stability and excess risk analysis in SSL. Another worthwhile direction is from the robustness perspective in conformal prediction and detection problems, including adversarial and label noise settings (e.g., Gendler et al. (2021): ) Einbinder et al. (2024);
Zhang et al. (2025)) For instance, one can study the robustness of training conditional guarantees under adversarial attacks during inference time.
Acknowledgment
This work was supported part by the National Science Foundation under Grant CCF-28 CCF-2611415.
References
Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic learning in a random world, volume 29. Springer, 2005.
Glenn Shafer and Vladimir Vovk. tutorial on conformal prediction. Journal of Machine Learning Research, 9(3), 2008.
Vladimir Vovk, Ilia Nouretdinov, and Alex Gammerman. On-line predictive linear regression. The Annals of Statistics, pages 1566-1590, 2009.
Vladimir Vovk. Conditional validity of inductive conformal predictors. In Asian conference on machine learning, pages
PMLR, 2012.
Rina Foygel Barber, Emmanuel J Candes, Aaditya Ramdas, and Ryan J Tibshirani. The limits of distribution- free conditional predictive inference. Information and Inference: Journal of the IMA, (0(2):455-488 2021.
Jing Lei and Larry Wasserman. Distribution-free prediction bands for non-parametric regression. Journal of the Royal Stalistical Sociely Series B: Statistical Methodology, 76(1):71 96, 2014.
Christopher Jung, Georgy Noarov, Ramya Ramalingam, and Aaron Roth. Batch multivalid conformal 15145, 2022.
prediction. arXiv preprint ar Xixi:2200. K5v:2209.15145,
Isaac Gibbs, John J Cherian, and Emmanuel Candès. Conformal prediction with conditional guarantees. arXiv preprint Xrx::2330.. 12616, 2023.
Vladimir Vovk, David Lindsay, Ilia Nouretdinov, and Alex Gammerman. Mondrian confidence machine. Technical Report, 2003.
Michael Bian and Rina Foygel Barber. coanting coverage for distribution-free predictive inference. Electronic Journal of Statistics, 17(2):2044-2066 2023.
Rina Foygel Barber, Emmanuel Candes, Aaditya Ramdas, and Ryan Tibshirani. Predictive inference with the jackknife+. 2021.
Ruiling Liang and Rina Foygel Barber. Algorithmic stability implies training-conditional coverage for distribution-fres prediction methods. arXiv preprint arXiu:2311042295, 2023.
Ryan Tibshirani, Rina Foygel Barber, Emmanuel Candes, and Aaditya Ramdas. Conformal prediction under covariate shift. Advances in neural information processing systems, 32, 2019.
Drew Prinster, Anqi Liu, and Suchi Saria. Jaws: Auditing predictive uncertainty under covariate shift. Advances in Neural Information Processing Systems, 35:35907 -59200, 2022.
Olivier Bousquet and André Elisscoff. Stability and generalization. The Journal of Machine Learning Reseorch, 2:499-520 2002.
10

---

Published in Transactions on Machine Learning Research (01/2026)
Vern I Paulsen and Mrinal Raghupathi. An introduction to the theory of reproducing kernel Hilbert spaces, volume 152. Cambridge university press, 2016.
Bernhard Schölkopf and Alexander J Smola. Learning with kernels: support vector machines, regularization, optimizalion, and beyond. MIT press, 2002.
Grace Wahba. Spline models for observational data. SIAM, 1990.
Masashi Sugiyama, Matthias Krauledat, and Klaus-Robert Müller. Covariate shift adaptation by importance weighted cross validation. Journal of Machine Learning Research, 8(5), 2007.
Joaquin Quinonero-Candell,, Masashi Sugiyama, Anton Schwaighofer, and Neil D Lawrence. Dalasel shift in machine learning. MIT Press, 2008.
Masashi Sugiyama and Motoaki Kawanabe. Machine learning in non-stalionary environments: Introduction to covariate shift adaptation. 2012.
Junfeng Wen, Chun-Nam Yu, and Russell Greiner. Robust learning under uncertain test distributions: Relating covariate shift to model misspecification. International Conference on Machine Learning, pages 631 931-639. PMLR, 2014.
Sashank Reddi, Barnabas Poczos, and Alex Smola. Doubly robust covariate shift correction. In Proceedings of the AAAI conference on artificial intelligence, volume 29, 2015.
Xiangli Chen, Mathew Monfort, Anqi Liu, and Brian D Ziebart. Robust covariate shift regression. In Artificial Intelligence and Statistics, pages 1270-1279. PMLR, 2016.
Aleksandr Podkopaev and Aaditya Ramdas. Distribution-free uncertainty quantification for classification under label shift. In Uncertainty in artificial intelligence, pages 844- 853. PMLR, 2021.
Lihua Lei and Fanmanuel J Candès. Conformal inference of counterfactuals and individual treatment. effects.
Journal of the Royal Statistical Society Series B: Statistical Methodology, 83())9111 938, 2021.
Clara Fannjianga, Stephen Batesa, Anastasios Angelopoulosa, Jennifer Listgartena, and Michael T Jordana. Conformal prediction for the design problem.
Leying Guan. Localized conformal prediction: A generalized inference framework for conformal prediction. Biometrika, 110(1):33 50, 2023.
Kun Zhang, Bernhard Schölkopf, Krikamol Muandet, and Zhikun Wang. Domain adaptation under target and conditional shift. In International Conference Machine Learning, pages 819 827. PMLR, 2013a.
Jonas Peters, Peter Bühlmann, and Nicolai Meinshauscn. Causal inference by using invariant prediction: identification and confidence intervals. Journal of the Royal Statistical Sociely. Series B (Statistical Methodology), pages 947-1012, 2016.
Mingming Gong, Kun Zhang, Tongliang Liu, Dacheng Tao, Clark Glymour, and Bernhard Schölkopf. Domain adaptation with conditional transferable components. In International Conference on Machine Learning, pages 2839 2848. PMLR, 2016.
Yuansi Chen and Peter Bühlmann. Domain adaptation under structural cansal models. Journal of Machine Learning Research, 22:1 80, 2021.
Kang Du and Yu Xiang. Learning invariant representations under general interventions on the response. IEEE Journal on Selected Areas in Information Theory, 2023.
Arthur E Hocrl and Robert W Kennard. Ridge regression: Biased estimation for nonorthogonal problems. Techmomelrics, 12(1):55 67, 1970.
Arash A Amini. phoctrally-truncatee kernel ridge regression and its free lunch. Electronic Journal of Statistics, 15(2):3743 3761, 2021.
11

---

Published in Transactions on Machine Learning Research (01/2026)
Christopher Williams and Matthias Seeger. Using the nyström method to speed up kernel machines. Advances in neural information processing systems, 13, 2000.
Yuchen Zhang, John Duchi, and Martin Wainwright, Divide and conquer kernel ridge regression. In Conference learning theory, pages 592-612 617. PMT.R, 2013b.
Ahmed Alaoui and Michael W Mahoney. Fast randomized kernel ridge regression with statistical guarantees. Advances neural information processing systems, 28, 2015.
Corinna Cortes, Mehryar Mohri, and Ameet Talwalkar. On the impact of kernel approximation on learning accuracy. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pages 113 120. JMLR Workshop and Conference Proceedings. 2010.
Jason D Lee, Qi Lei, Nikunj Saunshi, and Jiacheng Zhno Predicting what you alrcady know helps: Provable self-supervised learning. Advances in Newral Information Processing Systems, 34:309-322 2021.
Jiaye Teng, Weiran Huang, and Haowei He. Can pretext-based self-supervised learning be boosted by
downstream data? theoretical analysis. In International conference artificial intelligence and statistics, pages 4198-4216. PMLR, 2022.
Kang Du and Yu Xiang. Low-rank approximation of structural redundancy for self-supervised learning. In Causal Learning and Reasoning, pages 1008-1032. PMLR, 2024.
Asaf Gendler, Tsui- Wei Weng, Luca Daniel, and Yaniv Romano. Adversarially robust conformal prediction. In International Conference on Learning Representations, 2021.
Bat-Sheva Einbinder, Shai Feldman, Stephen Bates, Anastasios N Angelopoulos, Asaf Gendler, and Yaniv Romano. Label noise robustness of conformal prediction. Journal of Machine Learning Research, 25(328): 1-66, 2024.
Daofu Zhang, Mehrdad Pournaderi, Hanne Clifford, Yu Xiang, and Pramod Varshney. On the adversarial robustness of learning-based conformal novelty detection. arXiv preprint orXim:2510.00400 2025.
Aad W Van Der Vaart and Jon A Wellner. Weak convergence and empirical processes: with applications to statistics. Springer, 1996.
John Lafferty, Han Liu, and Larry Wasserman. Concentration of measure. On-line. Available: https://www.sstttccm cow.eu//uaaed/u//........ hrrg/=sml/Cnneentrtiinn. 2008.
Rick Durrett. Probability: theory and examples, volume 49. Cambridge university press, 2019.
Colin McDiarmid et al. On the method of bounded differences. Surveys in Combinatories, 141(():148-185 1989.
Aryeh Dvoretzky, Jack Kiefer, and Jacob Wolfowitz. Asymptotic minimax character of the sample distribution function and of the classical multinomial cstimator. The Annals of Mathematical Statistics, pages 642-669, 1956.
A Technical Lemmas: Weighted DKW Inequalities
A.1 Background
In this section, we first brielly introduce the notion of brackeling namberr and four foundamental results, which we will be applying in our proofs.
Brackcting Numbers. mcasure of complexity for class of functions is the bracketing number, For a pair of functions f and g, the bracket [S.]] is defined as follows,
[f.g] (h: 2 5g)
12

---

Published in Transactions on Machine Learning Research (01/2026)
A set of brackets 9il...... [fr 9n] is c-L*(P) bracketing of function class F if F S UIsienlie 9il and (J [ff - gildP))/e I c for all E [m]. The brackoting number Nú (G.F))(())) is the size of the smallest. (P)(P) brackeling of F.
Theorem (Van Der Vaart and Wellner (1996) (Theorem 2.14.2)) Let F be a class of measurable functions with a measurable envelope F. Then, for some universal constant > 0,
< <CIFilee2 ('' V1+ log No (c., (P())) de. vn J d( (aPPA-PP1)
E
(14)
FEF
Theorem (Lafferty et al. (2008 TTherrmm 7.86)) Let F a class of measurable functions with supsez IIfllr, and = supser so Then, for < 2A/3,
/ d(P,,
[d(Pa- P) p) 20) S S (e/8, exp (- (76AAB)
96nc
sup fEF
(15)
Theorem 9 (Durrett (2019) (Theorem 2.2.11)) For each n. let Xn.k, 1 k S be independent. Let bn > with bn + 00, and let YR.k = X I bn). Suppose that, as n + 00, (i) Ek IP[[ > by) + O, and (ti) bn2 Ex_ E EX + 0. If we let Sn = X1,0 +... + Xnn and put an Σx= E EX then
(Sn, ann/bn in probability.
Proof: Fix and let Sn =x1,
X1,n
X n,n- We note
Sn an
Sn
IP
>E)
SPSS # 33) +P
> >
bn
We bound the two terms separately. For the first term P(S, * Sn), observe that the event (Sn, * BB, occurs if and only if at least one sumnaan is truncated, that is, {S, & Sn} = Uh_ [1xm.l > bn Hence, by the union bound and assumption (i), IP(Sn # 3,) s Σ K== ((
bn) - 0 n co.
For the second term, P(((S, ab)l > =), note that an E S, By Chebyshev's inequality,
Sn On
Sn On
)
P
E
Var(3,.).
bn
bn
222
Because the variables ,....... X n. are independent and Var(Y) s EY² we have Var(S.) $ En=1 EX² n.k- Combining these inequalities with by assumption (ii), we have
P
S
On bn
Σ
EX D. +0, as n 00
>
E
222
We can now apply this theorem to obtain the following result, also known as the Feller's weak law of large numbers without first moment assumption.
Theorem 10 (Durrett (009
(Theorem 2.2.12)) Let X1,X2,.. be i.i.d. with x P([[ > x)) - 0 as
x 00. Let S, X1 +... X, and let pn Ex S n). Then Sn/n - pn - in probability.
Proof: We apply Durrett's Theorem 2.2.11 (i.e., Theorem 9 to the triangular array with entries X,, X for < and the normalizing constants 6n n. It suffices to check the two conditions of that theorem.
(1) Σ P(Xo, > m) nP([x)) n) 0 by the assumed tail condition xPP([X1 > x) 0.
13

---

Published in Transactions on Machine Learning Research (01/2026)
(ii) Let x, X n). We need to show n= -== B(X mx)
n-' (-'B(Xn,)) - 0. By
Lemma 2.2.13 of Durrett (2019) with Y = [ and p p= 2,
R n,
2y [ n, > y) dy.
Since nn,1 I n, the integrand vanishes for y 2 n, and for S y S n we have
P([m.11 > 8) P( > y).
Set ( = 2y ([ > y). Hence
E n,1) S S g(v) dy.
By assumption, we have g(v) + 0, as y + oo, so does its average.
A.2 Weighted DKW Inequalities
With these results in place, are ready to proceed with our prools. Let Fo(x) = (
S x)) where
s(-) denotes some fixed score function. For Zi iid P, we define
dQ aP
(7.)
(x)( = Σ (s(2))
S }}
8
(16)
dQ Eietai
aP (7,)
(E(M]
Lemma 1 (Bounded likelihood ratio) Assume that Q is absolutely continuous with respect to P (Q < P) and dQ/dP= < B. Then, for all 8 > 0
P sup (2n(x))F(((1)) > 5+ 201 #ER
B
< 4e n8 1288
where C > 0 is some wniversal constant.
Proof: By substituting the formula for wy in the definition of Fe we get
dQ Eieni
dp (Z.) s x
FO..
1 Eien di (Z.)
Hence we have
dQ dP
sup ER
sup
Σ
Z (2)]((((2)
2z) - Fe(x) +
zER
11
(c(R)
dQ
sup
Σ
(2,) [2)) xz)
1 Σ
iE|n dr (2.)
dP
iEin
sup
1
Σ
dQ
(s(2) S < Fo(x))
+
1
Σ
dQ
(%))
(17)
dp
dP
=ER
iE|n]
<E(M)
Regarding the first term, we have,
sup
dQ dP
(Z))(s(())
=sup
/
( sx)
dP
- -P)
ZER
(E(n]
zER
14

---

Published in Transactions on Machine Learning Research (01/2026)
=sup / la d(Pa - -P)/ zER
where Ja (* x 3, P) is defined as f=(=) 03 (=)1{<(>) S <.), and (A(A) = = 1(2 E A} the
empirical measure.
We note,
dQ/dPll = (dQ//P)3 dP S |dQ/dPlls
/ dQ/dP dp S
according to the Holder's inequality. Hence, |dQ/dP S VB and by Theorem have
E
sup ZER
/ fo d(P, - d(Pp.-P) P1)) - CV
B J.
V 1+ log Ni log
de,
(18)
with some universal constant C > O, and Ni pel (PP)) denotes the bracketting number for function class F = [fx E RJ. It remains to compute upper bound for Nn (e.F, LP Let to -00 and define,
4+1 == sup (ER: / ((3)(1((3)) (t,, dP 2
(19)
where f = dQ/dP. We note that there exists l < 2o such that 4 = 00. This is true since 1j+1 < co implies
f (())(18(2)) e (ta,4441)) dP dP2c. e, 0 Osisj.
according to 19 and dominated convergence theorem. Hence,
(j 1+11)²2 S Σ P(x)1{s(2) e (t.teil) dP d / fz dp= |IdQ/PP13..
(20)
and therefore j 1+15
Let k = minfe e Z = -o}. Now define e-brackets as follows b; (g(=)) f())11=(2) S 4 s g()) f())1{s(>) < 44133
where 0sisk k- 1. Clearly, F c b with by = {f). Brackets by...., b have size
( (())11))))) (tut++)) dP) Se. 0sisk-1,
1/2
according (19 and bracket by has size Hence, we have is brackets in total with size smaller than e. By
220), we have h (IIdQ/dl / which implies that N1 (c. I.2(P)) k + < 2 (Id2/dPllna2/)] Plugging this into 20
get
1
Nn (1dd0/dPlp2...1 <
Therefore we have,
PJ) Ja d(Pr, d(Pr--P)) s 20 /
/ n
B
B
sup [EE
C'
(21)
n
which follows from the observation that
I' V1++lg < (1+ (1+/ log(2 (012++1/2) S 1+2 (e
1/2
11+22((2110)))
loge) (2-
1/2
22
Now let p(()) denote the empirical measure obtained after replacing Z, by some arbitrary Z/. In this case, we observe that
/ fa d(Pr. P) - sup sup
/ fa d(p()) - p)
d(P, / ss de - p(()) -p(i))
sup #ER
sup ZER
15

---

Published in Transactions on Machine Learning Research (01/2026)
1 sup [a( - f2(21) S
B
n zE
n
Therefore, by McDiarmid's inequality McDiarmid et al. (1989) we get
P
sup zER
/
fo d(P, - P)
E sup zER
/
fa d(P. d(Pp--1))
s 2e 2nd2 =6]
Combining with 21 we obtain,
/ fo d(P, - p) > 3 08+
B
S 2 2n02/
P
sup =ER
Back to
17 for the second term, Hoeffding's inequality implies
1 | Zi)
P
S 2e- 22n82/8BB
(22)
xE(m]
Combining the bounds for the two terms, we obtain,
sup
(2) -F(())(((()))) > C²
B
S 4e /(2B2)
xER
The following version relaxes the assumption dQ/dP - B to |IQQ/PPI ||| r2 S K at the cost of slower rates.
Lemma 2 (Alternative version: bounded second moment) Assume Q <<< P and |dQ//PPIpp2 S K. Then, for all > 0
4CK
4(K² 1)
sup - > S
8/8
ns²
where C some universal constant.
Proof: By the same argument that led to
21 in the proof of Lemma
[
we get
La d(Pa - -p))
2CK
C"K
E sup ZER
Vn
vn
Combining this with Markov's inequality, we obtain
/ Jo d(P, -P) P) >
C'K Ovn
IP
sup >ER
(23)
This bounds the first term of
17 Regarding the second term, Chebyshev's inequality gives
Var 1 12
dQ (Z,)
dQ
K22
P
(2)
<
(24)
n xEin
82
nd2
The result follows from combining In the following, we present yet another version by dropping the dependency on the constant C.
23 and
24
Lemma (Alternative version: bounded likelihood ratio) Assume Q & P and dQ/dP S B. Then,
P sup (z)(()) - Fo(x)) > =0) < (72/8)e ((1B) (18BBB + 2e -18 -0_-n2/(28)) /(2B²)
for all > 0.
16

---

Published in Transactions on Machine Learning Research (01/2026)
Proof: To prove this lemma, compute the upper bound for the first term of 17 differently. According to
Theorem g we get
/ Ja d(P. (P) (NN(()/8FF1111)))) ( e -
IP
sup WER
(25)
for 2/3, where (e/e/88 F, (P)) denotes the bracketting number for function class F= (fa x RR.. [ it can be shown that Nn L' (P)) S 2 1/e. Therefore,
Similar to the proof of Lemma
sup P
/
p) > e) s 8(1+ +/c) e -ne v (36/e)e n
La d(P,,
FER
Combining with 22 for the second term of 17 we obtain
P sup (.) - Fo(x)) 6) - (72/6)e-nd /(1B)
+ 2r -no" ((2B*)
ZER
nd²
< (74/6) exp
2(B 1)3
Lemma 4 (Alternative version: unbounded likelihood ratio) Assume Q < P and that dQ/dP < oc Q-a.s. Fix B, > 0 and e e 1/2) holds that,
( [Q dp
B n
dQ
(rear) > B B
sup FER
Fo(x) (() 38 + 4P
(ress) > >n) + +C¹
P dp
with probability at least,
dQ (Zrest) =1)
d6 46'8 22n -F
dP
8
((2))
dQ
(Poosssa) )
"1(00 dQ
1
(Zoath) S 5m))
ng²
dP
dP
where n' nP (d0 caali)) S B
n2222e
Proof: To drop the assumption dQ/dP, we start by writing X XB U th. where X's fx (Qx < B). Define
s(Zuss)) S =P((((uux))
dQ
FQIxs =
(nest) S SB
dP
ΣiEr dP Zz (s)()) S x1
(2.) S
JP
Fo
Σ 28 (2))
40 7.) S B
where the summations are taken over calibration data points generated from P. We note,
Fo(z) Fl)(z)) s sup FQ(x) FQix2 (x) + sup =ER
Foix, (z) - FQ.
sup
=ER
xEB
f (x) f
sup
=ER
upper-bound the first term, we observe that
(s(2ron)) -ss(nen0)))
dQ (Zrse) S B
Fo(x) Forx (x)) = P (5))))
<
dP
P ((Zrost)) <
cQQ <
(1rott)
P
dQ
(Zrest) < SB -1)
cPP
dP
17

---

Published in Transactions on Machine Learning Research (01/2026)
dQ
dQ
+ 8 (Zess)
S x.
(?uest)
> B P
(Zesst) ) > B
dP
dP
dQ
dQ
S P
(Zeott
> B)
P
s (s
S dP
(eest)
>>
dp
dQ S x.
P (s(zust)
(Zrest
B
dP
dQ < P
(rest) > B)
Regarding the second term, using the same argument as the proof of Lemma 1 we get
P
sup JER
FQIx (x) - (x) $ $d+
C² BB/nB
nB 2 1 4e 1 /22B))
where ng = EsEz 1
dQ dP (2,) s B]. Now let n' =P ( dP
(Q
(Zallib) I B) -n 1/2t and observe that
PP
sup RER
Fqixn (x) f(.m (r)) s 58+C'
TP
sup sER
FaiNa (a) P (om
s 50+ + C' B/nB, us 2 2n)
E
IP
sup #ER
Eqixa (x) (x)
58+C' BB/nB nB 1(n 2 n'3
2 E
(1-4e-
4e
1(mB 2 n'))
4e 82 1(2B9)
P Pnn== 2 n')
) 1 4e n 1222))
(1 2e 2n
2 1 4e n''
n'8 8² 112B²) 2c 2n
where (x) holds according to Hoeffding's s inequality.
For the third term, we note
WS(x) (x+ Wz(x)
P (2)
- PZ))(x))==
=
W
W
(<()) - W<W>(z)
W<WW< + W>)
WSW>
W
2 W<(W++ + +W>
(26)
W + WS
where
(x(x)= = Σ dQ
dQ
(Z.) S B B).
Z1 (z)) S <x)1
dp
dP
IET
w>(x) = Σ dQ
(2) sa)1
dQ
dp (Z,) > B
dP
IEI
W Σ dQ
(7,)
dQ
(7,) B)
dP
dP
IE
dQ
W>
Σ dp
(2,))
(4)) B
TP
18

---

Published in Transactions on Machine Learning Research (01/2026)
and the inoqualty 26 follows from the fact that (x(x) - W and (x) W> for all R. Hence,
sup
F ( - (x) 22 =ER
1Q EiEz
aP
[2,]
dP
1Q
(Z,) >
+ 2
dQQ
(test) B)
Σ1EI dp (Z,)
dP
We note,
7 Σ
IEI JP [i))
1Q
(Z.) >
(Q
TP
dQ
- P
(Zeest
> B
5
1
LieT
dQ
(Zi)
dP
n
dP
dQ
dQ
dQ
1
(7:) >
B
TP
(Zeest) ) >> +
11
dp
(P
dP
IEI
1
-
Σ
dQ
{
dQ (Zi) > B) s
(2:)1
{iP
DiEz dP (Z.)
dp
VEI
1 Σ
dQ (2))
dQ (Z.) B
IP
dQ
(rest)
B)
+
1
Σ
dQ
(Zi)
- 1 S
71
dP
n
dP
1 Σ
dQ
dQ
dQ
(rest) S 5nl
(2,)1
(24) > B B] PP
B
12 1EI
dP
dP
dP
dQ
(Zrest) ) >> B, dQ
1
dQ
dQ
dQ
P
(reest) ) > m)
(i))
- P
(Zrst) S 5m)
+
(?rest) >
dP
dP
dP
dP
dP
IEI
dQQ (rest)) >
< 6 +
2P 82P
dP
with probability at least
2 " (100 Zeatt)) ) dP
dQ (troat > mn)
dQ
(Zram) 5
1 - 1
no2 E
(27)
dP
where in the last step we have used (the proof of) Theorem with e 5/2 and b, = twice; once for the first term with Xn.k (z,)1 { de (Zx) > B}, k E I, and the other time for the third term with Xn.* (Zs), ke I. The requirements of the theorem are satisfied by
dQ dP
(ealth) > x S dQ dP
{ [Q dP
(Zaath) > x x) P /dQ (Zest)))
(p(d) (rest) > >x) + 0
x
Zram)1
(28)
x - and then according to Theorem corresponds to the calibration datasct. Putting everything together, we have
10 also recall that I C [n]] for the split conformal method
dQ (Zest) > m) + C'y
B
dQ (Zrest) > B).
sup Fo(x)) (a) =ER
38 + 4P dP
+
P
n
dP
with probability at least,
dQ
1 - 4e 4e- (282)
-2n2 - 22
(Zrest)
> n2
dP
8 E
/QQ
(Zoalib)
1
dQ
(calth) S sm))
nd
dp
dP
19

---

Published in Transactions on Machine Learning Research (01/2026)
B
Proof of Theorem
Recall from Section 2.3 wC have
dQ (Zi)
(a) = Weest 111 = oo) + Σ
[wil(s(2)) $
Wi
dQ dP
Z
test
AP
EiEz
dp
(Z:)
(29)
IEI
where Zi
P. Treat Q, and I denotes the set of indices corresponding to the calibration dataset. We
note
(D(D) =P =P(Yiest & Caplit (Xtest)
IP (( Ytest (i) > F (1 a) D.,
() Q.m (1 a) Dm)
< P s ((eeest Ylesti ft) > f-1
Fo Ff (1)))))
where (x(x)) Pz~o(s(Z) x) and f2 is defined according to the weighted DKW inequality from Lemma ! we get
16 with index set I instead of [n]. Using
B
B
Fo -a)) Fm (F (1- -a)) -5 -8
5 Cy
whp.
m
Hence, we have
B < 4e mmB8
P
P. P.(D)) P.D.))a+&+C'
m
for all 8 > 0, or equivalently,
2B
4 C²
B
Pc(D.)) > a (V2B log 4/6 C²)
B
P P.(Dn) > >a
log + 0
P
1 \m
s 6.
This completes the proof of Theorem
I
B.1 Proof of Remark !
To note that by using Lemma 2 instead of Lemma
we get
6CK 8mm
4(K2 - 1)
P(P.(D)) a a
m8²
for all 8 > 0. Hence,
2CK + VO(K2-1 1)
))
P P.(D.) > a+2
58
5/m
for all 8 > 0, which implies
2K(2C' + +1)
P P.(D,) > a
0<851.
8/m
20

---

Published in Transactions on Machine Learning Research (01/2026)
C Proof for +ackknife+
Lemma 5 If Assumption
1 and
2 hold, then
IP (113 - E 13 26) S 2pexp
2x3E²
ne
Proof: Assumption 1 and 2
imply that
sup P
(4.............
............................ S
On
Ri
By McDiarmid's inequality McDiarmid et al. (9899) we get.
_2niee
PP (1 E. 2 2c) -P (II ..... Zn) - T(Z1. ... Za)l 2() 2p cppxp
nc²
for independent Z₁ and all c 0.
Lemma Under Assumptions and are have
2n
IP max Ma.. Ma_ 2 S 2pexp
h2Cn-1
K5
Proof: From Assumption ! and 2 it follows that
[| - 301100 S max 4
Cn-1 161
(30)
Also, according we have 113-1-1 - 1011-1111 301 e with probability at least note that,
2xxxpp(22xe/(mmm) We
(+)
P max MA.
M3
2c) S P K2 max
B - B- 00 2c)
(..)
Cn-1
+
| - 3
26
2x
6
< epexp
n
620n-1
where (*) and (**) hold according Assumption and 30 respectively.
C.1 Proof of Theorem 3
We note,
8J+ (Xest) 2 E :: Σ
Yi - 11 (x,) 2 y- -14,((Xoss) (Xima) > a
i=1
2 y
VER:
Y 133 (xx)
| (X:) - 1/z_ (x:) 2
y1 M3_,, (Xrest) + ua (Xrost) -1B_, (Xuos) }
21

---

Published in Transactions on Machine Learning Research (01/2026)
where the first relation holds according to Bian and Barber (2023) This first step can also be recovered by letting wi 1/(nn + 1) and vi; = 1/n in the first four steps of C.2. Assuming max; Il 13 |l 2o < 6, we obtain
+ +20 > a ) Yi - MB (x.) 2 y 113 1B (Xtest)
+J+ (Xtest) 2
y E
R
Σ
n
+ 26 > za )
yf E R: p(n-1) 1) (lu- 11-113_ 133 (Xtost)
2
Assuming
[f(-1] - ((n-1)
<8, 8, obtain
E (Xest)) 2
y E : F(n-1) (1V- (10-15_, 13__ (Xust) + 22) a +b]
F(n 2 y VER:1 1- (10- (1-M3. 1- (Xust)) >a+ 2cIn-1
Therefore,
Po(Dn) = P(Yes A+ (D)
Dn) S IP 1)(-1) (Vrest Uz (Xeest) s a + 22eln-1)
=a + 8 2cTen-1
for Dn E AAB where A = {D. max, 11,__ U3 1llo < <e) and = (D: p(m-1) (D:|(((u1)) p(a-1)
<6].
From Lemma we know
1 --)))) )
(-2mi 2n (-m K22n-1 N6
)
P(D, A 2pexp
Also, according to Ovoretzky-Kiene KKifer-Wolloower inequality 2e 2265 Thus,
Dvoretzky al.
1956),
have P(D,, & B) s
a s 2e-2-22 2nd
+ +ppexp
2n²
1
n
2Ln-1K220-
Kil
equivalently,
1og(2/8)
2p )
IP P,(D))>at a
+ 2 L,, K2 Cn-1
V 2
log
Set 6.
2n
Kg
C.2
Proof of Theorem 4
We note,
OJ+ (Xectt) = y E R: Σ w.1 [2 (Xrest) + Ys - (1,, <w} < 1 - a
--aan
1=1
WE R. Σ
wil [1 (1 (X
Yz 1x, > >8/ s a
i=1
2 y y R Σ
wi
Yi V u4 (xi)
< 3y --1a (Xums)) <
22

---

Published in Transactions on Machine Learning Research (01/2026)
y R Σ
a
10 { [ (X -0/ -//> Yi 13 (x:) < a
i=1
y E :
wil Yi M4_ (Xi)
< y - -M (X } < <1-a -a
1=1
2
y R:
0, Y, MB_, (xx)
< y/ - MB_ (X
} <
<11aa - a
w { Y, Ms_, ,,,, (x1)
2 8 - MB_. (X } =a
y R: :
1=1
2
YE R
0111
Yi - 13_ 1V-13_ (xi) - | (Xi)
(Xi) - 4z (Xi) 2
1=1
{ } /
y 13_ (Xust) + | (Xeest) - -13_, (Xass))
where the first and last relations hold by the definition of Cd (Xtest) and triangle inequality, respectively. "B alo00 < e, we obtain
Assuming max; 1B,
VER: Σ 0 { Vi- 1²_, (xi) 2 y V-13_ (Xeest)
n
E (Xuet) 2
+ 2c +20 a
n 1/ E R
+ 26 )
2
10
13 Vi= - 1,3_ ,x,,(x,) 2 |y vー1 (Xuest) )
Y;
+20
> a
1=1
2
y E R
m(a-1) (IV ( (1-1
MB_ (Xusst) + 26 > a
where P
(1=
= Em= 0;1 0;1 f|rs- 13_, (Xi) { and, wi and wy are defined in
16 and
29 respectively.
Define,
Fg ((n-t)(() (t) P (1r 1y5-15 1-_,,,,, (x1) $t).
Assuming
f(a-1) -F(a-1)
< 8, we obtain
AT+ (Xtest) 2
1 E R :R:11 1-
FQ Fda-!) (Iy 1z (Xtes))
(X ) + +20) > a aat
2 VER:I :- 1 (1- 11 (X > >a+
2eLgm S
Therefore,
Pe(D.) = P(Yest & El+ (Xrest))
S P (1-F =a =a 8 2eL
F(l-1) (Krest - Uz (Xiest) s a a 6 2eLQ
for D, E AnB(() where A { D: max, 11,__ U3_ 1loo <c) and B(8) = {D: f(n-1) 1) -F
(n-1)) 1) p(n
so].
x
From Lemma G. we know
{_2x
2x
4 )
e
P(D. ( < 2pexp
K2Cn-1
Ki
23

---

Published in Transactions on Machine Learning Research (01/2026)
Also, according to weighted DKW inequality from Lemma] 1 have
PP
Dn $ B
8 8+20
B
s de n52 ((289)
Thus,
B +0
s de /(2B²)
P
2x²
+ exp
2LQ,n-1826n-1
equivalently,
4 P PD(D)) a+ (V28 log + 20'
B
1
2p
V
+ 2 LQ
K2 cn-1
+
log
2x
Se+ 8.
D Proof for full conformal
Lemma 7 Under Assumptions and
have
2k7e²
P [u... 13 2e) < 2p epexp
n
Proof: According to Lemma 5 we have 1|3,, - -Balo < 6 with probability at least 1 2pexp'
2x7
Tt.
follows from Assumption that,
2xqe² 2
(113 17. |1 20
x P
62
Bn 3n 2() S epo
We need the following notation for the proof of Theorem
8xien. == T(((X,, Yi), (Xn, Vn), (Xrest: 1)
D.1 Proof of Theorem 5
We note,
Sfulll (Xrest)
3 E R
Σ
Y 1xx (xi) 2 2|v 14 18x (Xrest) a =
t=1
Y; 1B, (xi) - | (Xi) - 11 Mxxamen (Xi)
2
3/ WER
R
yy MB, (Xrest) + [u3, (Xrest) 1e B (Xecst))
> }
Σ 1 Y; uB, (x,) > | v -13 11 (Xrest) cutz } a
2
E R
24

---

Published in Transactions on Machine Learning Research (01/2026)
where the first and last relations hold according to the definition of Ch (Xtest) and Assumption 1,.. F3 10 < we obtain
Assuming
(Xeest) 2
1 VER RR::
Yi 1B, (xi) 2 y y-1% (Xeost) Cut1 + 26 22 } 17,
1/ R: fd(m) (10- HBg (Xenst) + Cx+1 26 > za..
Assuming
[ F(a)
8, we obtain
Chulll (Xeest) 2
Y VER: Fr(n) (Iu- Mz,
1,, (Xueat + cnt1 + +20) > r
2 VER: 1 - F(m) 1 (1v- M3, (Xcest)
> a - 6+ (2e + Cn
Therefore,
P. (D.). = IP $ Calll (Xtest)
< TP ((n) (1x |rost 1B, (X S 10+5+ 026 (2+1)
8 + (2e + Cat1)lnn
for D,, E EAB where A: = {D: 11s. - MB, 1l < <e) and B ::= D: | | Fr(n) <o}. From Lemma 6
2n nniee
know P(D,, $ A) S 2p exp
Also, according to Dvoretzky Kiefer Wolfowitz inequality, we have
P(D,. $ B) < 2c 2n82
Thus,
(-
(Kl(e/Ln (Mile//m -m+t1)) )r)
P(P.(D.)) (T(
2e 2n8 2pexp
V2nK220
or equivalently,
P P.(Dn>>ar a
log(2/5)
I'n Cat1 + v 2n log
2p K2 Cn c K2
)
S se+&
2n
D.2 Proof Theorem
We note,
Afull (test) =
y E R
w;1 { {xr Yi - M (xi) < y - MBx (Xemst) } < a
i=1
2
8 E R Σ
0;11 { Yr, - Max (xi) < y - MBx (Xerst)) } < - a
iml
w;1 { Y; - Maxiaa (xi) 2 1Ax (Xeest) | }20
8 E R
i=1
2
DER R
W;
Y; Mgg (xx) - -1m3. (Xx) Pxxiea (Xi)
1=1
14.. (Xrest) + No, (Xrost) - 11 18xemm.. (X
> a
25

---

Published in Transactions on Machine Learning Research (01/2026)
2 - Mg, (Xtest) cnt1 } 2 a
)
2 y e Σ 00 vi M3, (xi)
where the first and last relations hold according to the definition of Ch (Xrest) (under covariate shift) and Assumption Assuming llM3, - M3 oll0 < e, we obtain
+2 ] sa
y R: Σ 01 Vi 13,, (Xi) 2 y- - -13, (Xrest) Cu+1 + 26 Chull (test)
)
y R: [(n] (1W -Me,
Mo (Xrost) + Cn+1 + 26 > >aj.
Assuming
F(( p(a)
8, we obtain
20
Chulll (Xtnst)
y yER:1 1- F(!N) (1V 13, (Xrost) + cntl +20) a aat
BER: 1 Fg (Iu U7z (Xrest) > a (2x (attt
Therefore,
Pr(D,,) P(Yrest $ Cfulll (Xenst)
P p(N) (Krest - 1², (Xrest) sa+b+ (2c+ cxt1)
a + 8 (2r + Cn+1) T.2.
for D, E B() where A {D: |11g, - -17,, 1100 < <e) and B(5) = {D: | p(a) <o). From
2x3e²
Lemma
we know P(D,, & A) < 2pexp
Also, according to weighted DKW inequality from
Lemma
we have
B )) 8 8+ 202 [/
4e 1c-15/2(28)
P
Dn $ B
Thus,
S de n88 (188)( + exx (-( (kile/bban -Cn12) ))
P a 20
+ +6
equivalently,
Pc(Dn) >a a (V2Blog 4/8 + 4 +2)) V
B
+ LQn cn+1 + v 2n log
2p R2 Cn
))
s Se+ 8.
e Ki
E
Experiment Details for Figure 1
The sizes of the training, calibration, and test datasets are 525, 225, and 247, respectively, for the Wine Quality, Abalone, and Combined Cycle Power Plant datasets. For the Concrete Compressive Strength dataset, the corresponding sizes are 360, 155, and 169. Distribution shift introduced following the approach of Tibshirani al. (2019), by resampling data points with probabilities proportional to exp(x B), where
Bwine quality (0.5, 0,0,0,0,,0,0000000
26

---

Published in Transactions on Machine Learning Research (01/2026)
Oracle weights
Estimated weights
4-
0.9
Coverage
Figure 2: The likelihood ratio is estimated using kernel density estimator with Gaussian kernel.
Babslone [0, 10, 10, 10,000,0,0000,
Bccs = 0,00,0001,,,,,0,00000 BCCPP = [0.2,0.0.0].
As a simple illustration of the role of likelihood ratio estimation, as discussed in Section 4 Figure 2 shows the impact of likelihood ratio estimation on the distribution of the training-conditional error for the Concrete Compressive Strength dataset.
27


Reviews:
[{"review_id": "hVNdI0XgSA", "reviewer": "Reviewer_n8tU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper establishes data set conditional guarantees on the confidence intervals obtained by several techniques, including jackknife/cross-validation.", "claims_and_evidence_yn": "No", "claims_explanation": "The paper describes proofs of the claims, but there are a few points where I could not verify the proofs. The most significant is in the proof of Theorem 1, at the top of p.18. There it is claimed that conditioned on the dataset, the probability of a value falling outside a quantile of the true distribution function is at most the probability that the value falls outside a quantile of the empirical distribution function. I don't see why this should be true. Later theorems include an analysis of the difference between the empirical and true distribution functions, so I imagine that this can be addressed, but it might involve some additional loss in the bounds.\n\nIn the other places, the presentation is simply incomplete. The most significant example of this is in the proof of Lemma 4 (p.17) that references the proof of a theorem in another work. Since this is the appendix of a journal submission, length is not at issue, and I would have expected to see the relevant portion of the argument here. (Naturally, the authors should still be acknowledged.) Without the full argument, I can't verify it. In other cases, the proof relies on a theorem proved elsewhere and for the sake of being self-contained, I would have expected to at least see the statement of the claim (not the proof in this case) from those works reproduced here.", "audience_yn": "Yes", "audience_explanation": "The paper considers some well-known methods for obtaining estimates of parameters and derives some new confidence bounds for these. I believe this could be of reasonably broad interest.", "requested_changes": "It is critical that the gap in Theorem 1 is addressed, and that the relevant argument in Lemma 4 is included. The statements of the several other theorems that are used, in Lemma 1, Lemma 3, and Theorem 3.\n\nThere are also several points in the main paper where the notation is not clear. Section 4 discusses \"(m,n)-stability\" without explaining what it is: I don't know what the psi functions in equations 9 and 10 mean. In assumption 3, I don't entirely understand the notation -- why is a prime attached to F (and not to F^{(n)}, which is what I would expect if it was a derivative bound). If this is simply a matter of ugly notation, the assumption seems to come down to the function being Lipschitz, and perhaps it should just be explained in those terms. It was not clear if the bounds here are simply constants. In Theorem 2, there is an expression (dQ/dP)^2(Z_test), and it is not clear if this means that the output is squared or if the function is iterated.\n\nThe rest are less critical, but still important:\n\nThe error function for Jackknife+ is used in Theorems 3 & 4 without being defined (and similarly for CV+ in Corollary 2, etc.). The definition should be included. Relatedly,  I assume that data-conditional probability of error in these functions is w.r.t. Q, but please clarify this.\n\nIn remark 1, it is stated that the function should be continuously differentiable for some open set that contains the entire parameter space. Is this what was intended? (If so, can you include an example?)\n\nI can guess what you mean by ||.||_{P,2} in corollary 1, but I would not advise using this notation. At minimum, it should be recalled. Incidentally, a significant portion of the audience won't be familiar with absolute continuity and won't be able to make sense of the notation \"<<\" in Theorem 1. (I don't think you could even google this.) It isn't clear that you even need to state it since you're already assuming the derivative exists.\n\nIt would also be helpful to recall the definition of the bracketing number in the appendix before you start reasoning about it, at minimum for the sake of fixing the meaning of the notation.", "broader_impact_concerns": "No concerns. This is a theoretical work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "fMnA0KuikW", "reviewer": "Reviewer_ZWFG", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper mainly focuses on conformal prediction. The authors introduces the first training-conditional coverage bounds for conformal prediction under covariate shift, extending beyond marginal coverage guarantees. They derives PAC-style bounds for split and full conformal prediction and jackknife methods. A novel weighted DKW inequalities is proposed to handle non-exchangeable data under covariate shift. They have also conducted experiments to demonstrate the problem of degraded training-conditional coverage under covariate shift using UCI datasets", "claims_and_evidence_yn": "Yes", "claims_explanation": "Strengths:\n1. This paper is solving practical and essential problems in conformal prediction.\n2. The proposed method enjoys better sample dependence than (m,n)-stability baselines.\n\nWeaknesses:\n1. The method requires known likelihood ratio dQ/dP. Uniform stability only holds for simple models.\n2. The given bounds depend on unknown constants (κ1, κ2, Ln) and stability parameters (cn). The presentation can be improved.\n3. Impact of likelihood ratio estimation (Sec. 4) or stability parameters not tested.\n4. It would be better to conduct experiments on some real-work data benchmarks.", "audience_yn": "Yes", "audience_explanation": "This paper tackles ​distribution shift—a fundamental challenge for real-world deployment. It provides theoretical guarantees for uncertainty quantification (conformal prediction), directly supporting ​robustness and safety.", "requested_changes": "Please refer to the weaknesses.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "l0rp8yGymE", "reviewer": "Reviewer_UBba", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Conformal prediction is a framework for constructing distribution-free and model-agnostic predictive confidence regions under the exchangeability assumption for the training and test samples; such guarantees are more powerful when they are training-conditional training-conditional guarantees (Bian & Barber, 2023; Liang & Barber, 2023). Another, related line of research (Tibshirani et al., 2019; Prinster et al., 2022) developed \"weighted conformal prediction\" methods which adapt standard conformal prediction to situations with distribution or covariate shift and guarantee correct marginal coverage. The re-weighting scheme is clever but the fix isn't free. By giving more importance to certain data points in the calibration set, it makes the performance for any specific training set less reliable or \"heavier tailed\". Instead of just noting that this problem exists, as prior work has done, the authors compute upper bounds on the tails of this training-conditional coverage distribution and use these bounds to quantify the relationship between how bad the tails get and how severe the distribution shift is.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The authors appear to be extremely competent and well-versed in their subject area. Their related works are helpful and thorough, and are cited throughout the paper. The theorems appear to be supported by robust and extensive proofs -- on this point, however, I am prepared to defer to other reviewers who are more active in this subfield than I am, should their opinion differ from my own. In short, I am convinced both that the bounds presented here are not anticipated by prior work, and that they are correct.", "audience_yn": "Yes", "audience_explanation": "While there exist foundational papers on training-conditional guarantees and weighted conformal prediction, the specific combination of deriving **training-conditional concentration bounds for weighted conformal methods under algorithmic stability assumptions* is, to the best of my knowledge, unique to this work. It is also significant, not only from a theoretical standpoint (although that would be sufficient) but from a practical standpoint. The bounds presented by the authors in 3.1 are extremely sobering; even with relatively conservative values of B, the bound would quickly become vacuous in practice.", "requested_changes": "I implore the authors to consider ways to make this work more approachable. The paper lacks a conclusion section, a layman's overview for the discussion section, and any significant experimental results in the main body. While I do not consider any of these failings disqualifying, the absence of these sections impacts the clarity of the evidence presentation, and makes it harder to convince readers who are not already fully immersed in the specifics of this subfield that this result is important.\n\n* Nitpick: they often reduces the concentration -> they often reduce the concentration", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)