Non-Asymptotic Analysis of Efficiency in Conformalized Regression
Authors: Yunzhen Yao
Lie He
Michael Gastpar
Venue: ICLR 2026 Poster (2026)
Keywords: conformal prediction
efficiency
conformalized regression
quantile regression
uncertainty quantification
Primary area: optimization

Abstract: Conformal prediction provides prediction sets with coverage guarantees. The informativeness of conformal prediction depends on its efficiency, typically quantified by the expected size of the prediction set. Prior work on the efficiency of conformalized regression commonly treats the miscoverage level $\alpha$ as a fixed constant. In this work, we establish non-asymptotic bounds on the deviation of the prediction set length from the oracle interval length for conformalized quantile and median regression trained via SGD, under mild assumptions on the data distribution. Our bounds of order $\mathcal{O}(1/\sqrt{n} + 1/(\alpha^2 n) + 1/\sqrt{m} + \exp(-\alpha^2 m))$ capture the joint dependence of efficiency on the proper training set size $n$, the calibration set size $m$, and the miscoverage level $\alpha$. The results identify phase transitions in convergence rates across different regimes of $\alpha$, offering guidance for allocating data to control excess prediction set length. Empirical results are consistent with our theoretical findings.

Published as a conference paper at ICLR 2026
ANALYSIS OF EFFICIENCY IN CONFORMALIZED REGRESSION
Yunzhen Yao!
Lie He Michael Gastpar'
LLIX,, EPFL
Moo Key Laboratory of Interdisciplinary Research of Computation and Economics, Shanghai University of Finance and Economics
(yunzhen. yao, michael. gastpar)@@pfl..ch. (helie)@sufe.
ABSTRACT
Conformal prediction provides prediction sets with coverage guarantees. The in- formativeness of conformal prediction depends its efficiency, typically quan- tified by the expected size of the prediction set. Prior work on the efficiency of conformalized regression commonly treats the miscoverage level a as fixed con- stant. In this work, we establish non-asymptotic bounds on the deviation of the prediction set length from the oracle interval length for conformalized quantile and median regression trained via SGD, under mild assumptions on the data dis- tribution. Our bounds of order O(1/Vr + 1/(a²n) + 1//m + capture the joint dependence of efficiency on the proper training set size n, the calibration set size m, and the miscoverage level a. The results identify phase transitions in convergence rates across different regimes of offering guidance for allocating data to control excess prediction set length. Empirical results are consistent with theoretical findings.
INTRODUCTION
Deploying machine learning models in safety-critical domains, such as health care (Allgaier et al., Gui al. 2024), finance (Wisniesski al.) 2020 [astos 2024), and autonomous systems [Lindemann et al. [2023] Ren et |0,33) requiees not only accurate predictions but also reliable un- certainty quantification. Conformal prediction (CP) is a principled, distribution framework for this purpose, equipping black-box models with prediction sets achieving coverage guarantees or va- lidity (Vovk al. 2005) Balasubramanian al. 2014). Formally, given a set of data {(X,,Y) y
2023
drawn from a distribution over x V, for any user-specified miscoveruge level a (0, 1) and predictive model, conformal prediction constructs set-valued function C : X + 2 such that, for a test pair (Xm+1, Ym+1) ~ P, the prediction set C(Xm+1) covers the label Ym+1 with probability
P [X C(Xm+1)) 2-aa
(1)
Split conformal prediction is computationally efficient variant that incorporates training predictive models. It splits data into a proper training set and a calibration set; the model is first trained on the former, and its uncertainty is then quantified using the latter. During calibration, nonconformity score functions are constructed to measure the discrepancy between model predictions and true labels. The distribution of these scores is estimated over the calibration set, and quantile of them defines a threshold. The prediction set C is then obtained by collecting all candidate labels whose nonconformity scores are larger than this threshold.
central focus of conformal prediction is efficiency, commonly quantified by the expected measure of the prediction set (Shafer Vovk 2008). For classification tasks, efficiency relates the cardinal- ity of the predicted label set; for regression, it corresponds to the length (or volume) of the prediction interval (or region). Under the validity condition (I), smaller prediction sets are more informative. Early works primarily evaluated efficiency empirically, whereas recent research has shifted toward asymptotic efficiency, demonstrating that prediction sets converge to the oracle sets as the sample
Corresponding author.

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size increases (Sesia & Candes 2020 Chernozhukov al.) 2021 Izbicki al. 2022) In contrast, non-asymptotic efficiency, or finite-sample guarantees on the expected measure or excess measure of the prediction set, remains much less understood, with only partial results available (Lei & Wasser- man] 2014; Lei et al., 2018 Dhillon et al. 2024. Bars & Humbent, 2025). Existing non-asymptotic bounds typically expressed based the calibration set size m. whereas the effect of training set size n and miscoverage level remains an open question in split conformalized regression.
In this work, we analyze the efficiency of split conformal prediction in regression, focusing on conformalized median regression (CMR) and conformalized quantile regression (CQR) (Romano al. 2019). CMR uses the absolute residual as the nonconformity score, and the quantile of the calibration residuals then determines the half-width of a symmetric prediction interval centered at the estimated conditional median. contrast, CQR estimates both upper and lower conditional quantiles, defining nonconformity scores relative to these estimates. After calibration, CQR yields adaptive, asymmetric prediction intervals that naturally capture heteroscedasticity without assuming symmetric conditional quantiles.
Contributions. We present a non-asymptotic theoretical analysis of the efficiency of conformal- ized quantile regression and conformalized median regression under stochastic gradient descent (SGD) training. Our main contributions are as follows:
Fimite-sample bounds for CQR. For CQR-SGD (Algorithm D. we derive an upper bound of on the expected deviation of the prediction
order O(1/Vn O(I/v" 1/((nn))++++ + 1// + set length from the oracle interval, where is the proper training set size, m is the calibration set size, and a is the miscoverage level (Theorem 3.2 Unlike prior work that relies on assumptions on intermediate quantities, our analysis places assumptions directly on the data distribution.
Finite-sample bounds for CMR. For homoscedastic tasks, CMR-SGD produces symmetric intervals of constant length across inputs, enabling us to derive non-asymptotic upper bound of analogous order (Theorem 4.1 to CQR.
Theoretical guidance. To the best of our knowledge, our work is the first analysis establishing upper bounds on interval length deviation as a function of (n, m, a:), revealing phase transitions across different regimes (Section 3.2.1) Our results thus offer guidance allocating data between training and calibration to control excess length at a desired miscoverage level. These theoretical insights further validated through experiments.
Finally, while our theorems are presented for models trained with SGD, the analytical framework developed in this paper is not tied specific optimizer: the bounds extend directly to other opti- mization algorithms by substituting their corresponding estimation error rates.
2 PRELIMINARIES
Quantiles of random variables. Let F be the cumulative distribution function (c.d.f.) of a random variable Z. For y E (0, 1), the y-quantile of Z is defined as
(2) inf(u R: F(u) 2 7)
Conditional quantile function. 4, (Y X) : X
For (X, Y) P over X x V,, the conditional y-qunntile function
+ R is defined as
g7 (Y x := inf fu R F (w) 27), for all x e X.
(2)
Split conformal prediction. In split conformal prediction, the data are partitioned into the proper training set Dimin and the calibration set Deat. The training set is first used train a model h. With the trained model h, nonconformity score function din X' x ) + IR is then defined to quantify the discrepancy between candidate label y and the input x, where higher scores indicate worse conformity. The nonconformity scores Sm := (1)(2,, Uj) }y are computed for all calibration samples in Dent = { H;) )- The sample quantile q a). is calculated at level:
a)m [(1 am 1)1 m,
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corresponding to the [(1-a) 1)h smallest value in S., which is also known the empirical quantile. The prediction set for a new input x is then defined as
C(x)
{ 3 (2, v) - ((-m)) }.
Bachmann-Landan notation. We employ Bachmann-Landaa (or Big O) notation in the limit as n, m 20. For positive sequences or functions f.g. write f O(g) if there exists C, N such that [f(])] S C [g(k)] for all & 2 N; we write f = {2(g) if there exists C, N. > 0 such that
[s(k) (g)( for all k N. We write f = o(g) if f/g + O. and ((g) if 5/g + Xo.
3 ANALYSIS OF CONFORMALIZED QUANTILE REGRESSION (CQR)
3.1 PROBLEM SETUP FOR CQR-SGD
Data model. We consider random design setting where training. calibration, and test samples are drawn i..dd from an unknown distribution P over X' x V. Formally, for all i E [m].j] E [m]
(X/rain ytrain), (Xga) yyJa)), (Xtestt (yteet i.i.d. ~ P.
We assume the covariate space X' c Rd is bounded: there exists constant B 0 such that ||æ||2 I B, Vz E X.
(3)
Similarly, the response space y C IR is assumed be bounded interval [ymin Mmasx)
Learning objective. In CQR, the training set Durain is used to estimate the conditional ququntile function 9- (Y X) defined in (2), where 7 1 a-2222, The estimated function (: f. (7) is
obtained by solving the stochastic pinball loss minimization problem (Koenker & Bassett Jr 1978)
min (0) Exx,Y (4, Y)], 050
where the pinball loss takes the form
(y y) > t) 1y 2 t} (1- y) 1 < tt.
(5)
We consider a linear function class with convex and compact parameter space:
Lx 0) 0 x E R2, sup |0||2 K xc.
(6)
oce
Without loss of generality, we assume K S max |mmin 1/max H/B. The linear model represents
standard setting for theoretical analysis of quantile regression Koenker 2005 Pan & Zhou 2021 ensuring convexity of the objective function in
Learning algorithm. To solve (4), we consider the stochastic approximation framework (Robbins) & Monro 1951), focusing on stochastic gradient descent (SGD). The 0 is updated according
0k+1 = He (O,- m)*k),
where 'ik is the step size, Th denotes the Fuclidean projection onto +, and 9u is stochastic sub- gradient satisfying E[ee 0x] = gk. with It a subgradient of the population objective in (4) at 0g.
Let (7) denote the parameter learned by solving (4) via SGD on the training set Derain. For convenience, we introduce the shorthand notations for the learned parameters
i = 0a (a/2),
0n = e, (1 a/22, d, = (0., 0%))
Conformalized quantile regression. CQR employs two estimated conditional quantile functions, ta/2(:: (.)) and 1/-a/2 0.). Given the learned parameters On (f.. fn), the score for (X,Y) is (X, ;= max (x/2(X: Y, Y (X;a/2(X(00) 0wx) }.
(8)
Thus S 0 if Y lies outside the interval
and S 0 otherwise. tI (x))
Let Sm, cal: (n)) denote the scores the calibration data, and let 9 a) (Sm | (.)) be their empirical (1 a -quantile, i.e., the [(1 a)(m + 1)]-th smallest value of ("Dast 0z). The prediction set for a test covariate X is then
C(X)= [la/2 (X: (m) a(1-a)m m (Sm. (n), 4 41-/22 (X;0x) + Q(1-a)m. m (Sm Un)] (9)
if 11a/2 (X;2,) lx/2 (X;0.) 0.)) 2 (Sm In) 2 otherwise, C(X) 0.
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Remark 3.1. The phenomenon where the lower quantile estimate exceeds the upper quantile esti- mate is known as quantile crossing (Romano et al. 2019; Bassett Jr Koenker 1982). We show in the proof Proposition A.4 that, quantile crossing does not occur with high probability once the training set size n is sufficiently large. Moreover, because the covariate space X' is bounded, the ground-truth lower and upper quantile functions cannot cross, if they are not parallel.
The whole pipeline of CQR with SGD training is summarized in Algorithm
Algorithm
Conformalized Quantile Regression with SGD Training (CQR-SGD)
1: Input: Dataset size (n + m), miscoverage level a, new input æ
2: Split the dataset into proper training set Draain of size and calibration set Deal of size
3: Train quantile regressors (-; (,)) and & /0 fx) on 2 Ptrain by solving (4) via SGD
4: Compute nonconformity scores Deal according to (8)
5: ((1-a)m +- the (1 - amm qqantiiee of the scores on Deaal
6: C (x) t /// (a; (.) ((-a))m 11-022 (1: 0n) + ((-m))m)
7: Output: Prediction set C(x) for a new input
3.2 THEORETICAL RESULTS FOR EFFICIENCY OF CQR
To establish upper bounds on the expected length deviation of the prediction sets, we introduce the following assumptions.
Assumption 3.1 (Well-specificatiion in CQR). For E fa/2,1 - - there exists 0 (r/) E θ such that
Φ x=) (T) for all x X Rd.
Assumption 3.1 ensures that 6" (7)) is minimizer of (Takeuchi et
2006 Steinwart Christ-
mann 2011)
Similar to e. On. and Un. we introduce the shorthand notations for the ground-truth parameters
0 (a/2). 0+ = 00 x/2), 0" = (Q", 0*)
Assumption 3.2 (Bounded covariance). There exist constants < Ammnn S Amax < 0o such that
Amin T 3 Σ [[X I AmaxI
(10)
where I is the identity matrix, and A B means that (B A) is positive semi-defiite for two symmetric matrices A, B.
Note that Amax S B², since ||æ:|2 S for x X.
Assumption 3.3 (Regularity of the conditional density). For any x E Ex the conditional probability density function (p.d..)) fyix(- æ) exists and is continuous. Moreover, there exist constants
Smin Jmax <, 00 such that
Jmin < fv S Smax:
V Yyey.
(1I)
notice that Assumption 3.3 concerns only the underlying data distribution P. In particular, our assumptions agnostic to the induced nonconformity scores, unlike prior works which impose assumptions on the induced distribution nonconformity scores, which depends on the trained pre- dictive model. Assumption 3.3 satisfied by many common continuous distributions once truncated to bounded support and normalized, including the truncated normal distribution.
Assumption 3.3 implies that the conditional support of Y given any E X' is the common set D. The lower bound fr (y x) 2 fuin guarantees that ) bounded, while the upper bound X a) S fmax ensures that Y has non-empty interior. A constant H defined to characterize the flatness of conditional distribution. i.e.
H H(fuax, Smau) fmasx fmin-
(12)
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In particular, the Lebesgue measure of y satisfies 1/fmax < S f/fmin- Together with B in ( K (6. and Assumption 3.1 it yields
M S BK + 1/fmins
YyE E
(13)
The score S has bounded support, since 1/1/2 0n)] S BK and |Y] BK + +1/mmin i.e.,
SS] S R 2BK + 1/fmin-
As first step toward bounding the expected length deviation, Theorem 3.1 establishes upper bounds on both the prediction error of the quantile regressor and the parameter estimation error under SGD training, expressed in terms of the training sample size n.
Theorem 3.1 (Quantile regression error of ttained models). If Assumptions 3.1 3.3 hold, tak- ing step size The 1/ fmink) )) in SGD update 7 then
- 4 fmaxt
Ex,0n (ity (X;0n (1)) - t, (X;0* ()1))
(14)
Aain fain
4 max S
Eg, [110, (7) 0" (7) [12]
(15)
Xmin f2 Pmin
The proof of Theorem 3.1 is deferred to Appendix A.
Remark 3.2. The results of Theorem 3.7 are established under a strongly-convex assumption as they rely on Theorem [A.]] from Rakhlin al. (2012). Comparable rates can also be obtained for non- strongly-convex objectives under the assumptions in Bach & Moulines 2013), where Assumption 3.2
can be weakened to requiring only the invertibility of EEXX
Theorem 3.2 establishes non non-asymptotic efficiency guarantee for CQR-SGD (Algorithm D. bound- ing the expected length deviation of the prediction set from the oracle conditional quantile interval
C* (X) = 9a/2 (Y X), 41-a/2 (Y x)]
(16)
We measure the efficiency conformalized regression methods by the expected length deviation
E Ex Peni [ [[(()) (x)
(expected length deviation)
Theorem 3.2 (Efficiency of CQR-SGD). For CQR-SGD, suppose Assumptions 3. 3.3 hold. If
m 8H /minfa, 1 a), then for test sample (X, Y) and S 1/2,
Ex Ex,d., Dea [[[[(x)] |C (X)
S 0 n- 1/2 (a²n) + +m 112 exp( exp(-a²m)) (17)
where H is the constant defined in [2).
The explicit upper bound (41) and the full proof of Theorem 3.2 are presented in Appendix B with a proof sketch illustrated Figure
Remark 3.3. While Theorem 3.2 is presented for CQR trained using SGD, the analysis strategy applies to other optimization algorithms. In particular, one replace the SGD error bound in Theorem 3.1 with that of the chosen optimizer. This replacement modifies only the terms in the overall bound that depend the training set size Formally. suppose the upper bound in Theo- rem 3.7 replaced by Pn where y > 0 as oc, then the upper bound in Theorem 3.2 becomes m 1/2 exx((aa²m)
O
Pp +a
Remark 3.4. a random variable 7,, the density level set L(u1-a) is the optimal prediction set with coverage probability 1 (Lei et al. 2011). i.e.
[(w1-a): = {z Z f2(2) 2 v1-a7 arg min [C]. PiZEc]21-a
where u1-. = inf(u: P|Z C(w)] N 1 1-a), The oracle interval C (x) coincides with the optimal
prediction set for any y E (z) and any yf' E y C* (æ), it holds that fxix==(u) 2 fy (
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Ex Pxw ([I[.)) (x)]
Ex,
| (41-a/2 (X;0,) 1a/2 (X;0,) + 20 (Sm Un): 0}
(11-0/12 (x:0") - 1a/2 (x;/((x;0))) |
$ Ex,an [11- (x; - 41-022 (x:0') + May2 (X; 1a/2 (x
= =0 (v(r/) Quantile regression ervors of trained model (Thm. 3.1
Ban |l 0x>Il
-0 (v(I//) Population quantile of score (Prop, A.2)
Eon [] |1- (S 0,) 9(-m)m (S 0.)]]
O (1/m 1/(a2n)
Population finite-sample score-quantile gap (Prop. A.3
Eg, Deat [| (1(1-a)) (S (n)) Qm1-a)m (Sm 0.)l]
O (V(vI/nn exp(-a²m) 1/(a22))
Empirical score sore-quuntile concentration (Prop. A.5
Figure 1: Proof outline of Theorem 3.2 Full proof deferred to Section
3.2.1 PHASE TRANSITIONS OF THE UPPER BOUND
In Theorem 3.2 the upper bound on the expected absolute deviation between the prediction set length (C(X) and the oracle interval length C (X)] is expressed explicitly function of the training size calibration size m, and miscoverage level a. Unlike prior analyses that treat as fixed constant, result reveals its critical role in efficiency. Specifically, the terms (a²n)-1 and exp(-a2m) in the bound imply fundamental scaling relationship as follows.
Regimes of in general cases.
The length deviation converges to zero whenever a decays slower than n -1/2 and m m1/2 1/2 i.c., a wi w((mx(fn 1/2 m m-mmmm//mm -1/2)). Thus, Theorem 3.2 implies that if the expected prediction set length required to remain within fixed tolerance of the oracle length, not supposed to be chosen arbitrarily small.
For the two n-dependent terms in otherwise they are of order O ((a"n)
17 if a 2(n then they of order O(n 11/2);
For the two m-dependent terms, if Mi otherwise they are of order O(exp(-a²m))
llo m/ m/m), then they are of order O(m 1/2);
Thus, if = Q1maxx(1 /log ),/m))), the upper bound scales as O(n 1/2 + m 1//)),), which coincides with the rate in Bars & Humbert (2025) assuming finite function class.
Regimes of a when n, m of the same order. When n = O(m), the upper bound simplifies to On- + -1)) Figure 2 shows it in different regimes of a = 2(n consistent with the assumption 8H/ min (a, I at in Theorem 3.2
Data Allocation. If a 2(max(n 1/4 Vlo m//m)), the bound reduces to O(n 1/2 + m -1/2), so natural choice is to set n and to be of the same order. If ( /log m/m) and w(n 1/4), the trade-off is between Omm-1/2) and O(1 (an?)), and balancing them yields @(a'n())

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not vanishing
Upper-bound order
a(1)
0 Oa
O(n 1(2)
On 1/2)
@
Figure 2: Upper bound orders in Theorem 3.2
in different regimes of a when n (m)) Results
Lei et al. (2018); [ars & Humbert 2025 in the right most regime (blue).
4 ANALYSIS OF CONFORMALIZED MEDIAN REGRESSION (CMR)
4.1
PROBLEM SETUP FOR CMR-SGD
For conformalized median regression (CMR), we consider the same i..d. data model and learning algorithm (SGD) CQR in Section 3.1
Learning objective. In CMR, the training set Dirain is used estimate the conditional median function Q1/2 (Y X). which is the special case for conditional y-quantile estimation with 1/2 (see (2). The estimated conditional median function & f1/2(-:0 () is learned by solving the minimiza- tion the expected absolute егror (stochastic pinball loss with y = 1/2) via SGD:
min ( == Exx,))xxxx (X;; YI]. 6EB
(18)
We adopt the same linear model class as in CQR, namely (6).
The shorthand notations for the learned parameter 0, (1/2) and the true parameter (1/2) are:
0n = (1/2), Ø" = (1/2).
Conformalized median regression. formity score for (X,Y) is
In CMR, given the trained regressor f1/2( 6n), the noncon-
S (X,Y; == (11/2(X; -Y
(19)
which corresponds to the absolute prediction error of the estimated conditional median f1/2(-; 0.).
For the calibration set Deat. let Sm (Drat; 0n) denote the scores calibration data, and let
(()-)).. (Sm 0x) be the empirical (1 a)m -qaantle of S given O.. i.e., the [[1 am 1)7-th
smallest element in Sm (Deal; f.)) The prediction set for covariate is then
C (X) = (X/ (x;))) ((1-a)) (Sm (n), 11/2(X;; 0,)) ((1-)) (Sm 0.)]
(20)
4.2 THEORETICAL RESULTS FOR EFFICIENCY CMR
The vell-specification assumption in CMR assumes a linear 01/2; Assumption 4.1 (Well-specification in CMR). There exists 6" (1/2) E O such that
(1 X = (2/22;; 0* (1/2)), for all X.
For the CMR setting, we make an additional assumption top of Assumptions 4.1 3.2 and 3.3 Assumption 4.2 (Symmetry of quantiles). There exists Ç > such that for every X,
0Y X =)) 4Y/2(Y X = =x) 91/2(Y X = x) 9a/2(Y Y X x) c
(21)
Remark 4.1. Assumption 4.2 is standard in the analysis of conformalized regression based on single regressor, following the precedent set by Assumption Al of et al. (2018).
Theorem 4.1 (Efficiency of CMR). For CMR-SON RMR-SGD, suppose Assumption
4.13.2 hold. If
8H /min(a, 1 a.. then for test sample (X,Y) and
1/2,
Ex,0,, [ [[[[ (c* 0 1/2 + (a"n) m -122 1/2 + expl- exp(-a²m))
(22)
where H is the constant defined in (22
The explicit upper bound 422 and the full proof of Theorem 4.1 are presented in Appendix

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5 RELATED WORKS
Quantile regression. Quantile regression has attracted significant attention since the seminal work Koenker & Bassett Jr (1978) due to its robustness to outliers and ability to capture distributional heterogeneity. Early works derived the /n-coosistency and asymptotic normality of quantile re- gressors in the linear model (Bassett Jr & Koenker 1978) 1982, Portnoy & Koenker 1989 Pol-
lard (99T). Other works established statistical properties under fixed designs, where covariates are treated as deterministic (He & Shao 1996) Koenker 2005). More recent works have shifted toward non-asympiotic analysis with convergence rate O(1/v /n) under random designs, where covariates random and prediction performance
unseen data is emphasized (Steinwart & Christmann 2011 Catoni 2012 Hsu et al. 2014 Loh & Wainwright, 2015 Pan & Zhou 2021; He et al. 2023]
Liu et al. 2023 Sasai & Fujisawa (025). Median regression is special case of quantile regression,
has also been extensively studied (Chen et al. 2008). [Shen et al. (2025) analyze online quantile
regression with linear models trained via SGD, under regularity conditions elosely related to ours, including a local lower bound on the conditional density. These methods form the basis for confor-
malized median regression and conformalized quantile regression (Romano et al. 2019).
Efficiency analysis of conformal prediction. Conformal prediction was developed to equip point predictions with confidence regions that provide sampe coverage guarantees (Papadopoulos et al.] 2002) Vovk et al. 2005 2009; Vovk (2026). Research on its efficiency (Vovk et al. [006]] Gasparin & Ramdas] 2025) has evolved from early asymptotic convergence analyses, which es- tablished convergence rates toward the oracle prediction region (Chajewska et al. 2001: Li & Liu.
2008 Sadinle et al. 2019 Sesia & Candès [220 Chermozhukov et al.
2021 Izbicki et al., 2022), to generalization error- based bounds on expected set size Zecchin al.
(2024). , and recently volume-
minimization methods using data-driven norms (Sharma et al.] 2023 Correia et al. 2024 Kiyani
et al. 2024 Braun al. [2225] Bars & Humbert 2025 Gao al.] [025]] [Srinivas] 2026). Relat-
edly, Gauthier et al. (2025) propose backward conformal prediction, which directly controls the
size of prediction sets while relaxing the classical marginal coverage formulation. Complementary
to marginal coverage, Duchi (2025) investigates sample-conditional coverage guarantees in split
conformal prediction. We note that our analysis focuses on the i.i.d. setting; robustness under dis- tribution shift has been studied separately, Josh. et al. (2025).
For conditional density estimation, under 3-Hölder class and y-exponent margin conditions of the conditional density, Lei & Wasserman (2014) derived minimax-optimal rates of order O(log m/m))//////// 8/(88+1)) when 7 = 1, and showed that conditional coverage cannot generally be guaranteed in finite samples. When the quantile of Y is symmetric and independent of X (analo- gous to Assumption 4.2 Lei et al. (2018) incorporated training error into the efficiency analysis, treating a as fixed constant. In contrast, our results for CQR and CMR make no assumptions on the training error and provide explicit upper bounds (41
42 as functions of (n, m, a),, applicable
also to adaptive prediction sets.
Under the assumptions that the quantile function of the nonconformity score is locally 8-Hölder continuous, and that the worst-case empirical estimation error of the function class is bounded, Bars & Humbert (2025) derived convergence rates of the order O(m -Bx/2 + n -8/22) for some O K < I when the function class is finite. In the case of 3 = 1. this rate matches our bound when a treated as a fixed constant, namely (mm-1/ 1122 + n 1/2).. Different from analysis in Bars & Humberi (2025) that focuses on methods based on volume minimization, our work develops efficiency guarantees for CQR and CMR, without imposing assumptions on the score distribution induced by the trained model on the estimation error. Instead, we demonstrate in the proof (especially Proposition B.2p that the required regularity conditions of the score are satisfied with high probability under mild assumptions on the underlying data distribution.
6 EXPERIMENTS
This section presents evaluations of length deviation using synthetic data to access our theoretical results. Additional synthetic experiments and experiments are deferred to Appendix C dnd to space constraints. An overview of all experiments conducted in this paper can be found in Section 6.1
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[a] Expl: Fix na- 5-00000
(b) Exp2: Groundtruth 6
(c) Slopes in laglog plat (a). (b)
[d) Intercepts in logiog plat (a)
-0.5
you 2.24 071
Longth Doviation (1)
10
12
Stope
-0.75
Intercept
1.9
For 6377
For each
log
Jog 07
le1
1e3
lef
lel
1x2
1e3
0.00
0.05
0.10
0.15
0.20
Training (n)
# Calibration (m)
log(n)
Figure 3: The length deviation of conformalized quantile regression in synthetic data experiments.
Experiment setup. The data generation procedure is described in Appendix C.1 All experi- ments employ linear models trained with SGD for one epoch using a batch size of 64. Learning rates selected via successive halving over the range [10 1005,, 1]. We evaluate miscoverage levels a E {0.01, 0.025, 0.05, 0.075, 0.1, 0. 125, 0.15, 0.175, 0.2]. Reported results are averaged over 20 independent trials, and length deviations computed on 2000 test samples.
We denote the expected length deviation as A. We empirically assess the upper bound of A in + na? + vm exp(-a²m)) from three perspectives.
Theorem 3.2 of order O Vn
Effect of training size n. With a large calibration set (m = 5000), the calibration error is negligible, and the theoretical bound simplifies to 1/11//+1//mme The theory predicts that a linear regression of log on log n, i.e.
log ~ di log + 42
(23)
yields a slope a1 that transitions from -1 to -1/2 as increases. We confirm this trend em- pirically. For each a, we train models over n ranging from 200 to 20000 (Figure 31) and fit the regression model (23) (the inset in Figure 3h shows an example) to obtain slope aj and intercept 02). The resulting (a, a1) pairs, shown by the red curve in Figure3 validate that the slope shifts from approximately -1 to -1/2
grows, reflecting the transition of the dominant term in the bound from (11(nna2) )) to Vn). The intercept a2 depends on log a, as discussed below.
Effect of miscoverage level a. In the regime where (na²)-1 dominates, A is expected to follow pawer-law scaling of order -2 To examine this, we further regress the fitted intercepts 02 in
23 on logar. oz:
a2 log b2-
Together with 23). the estimated coefficient b = -.244 (Figure 3 implies that A ~ a -2.44 This aligns with the theoretical upper bound of order O(a 2). Appendix C.2 provides an addi- tional verification for the existence of this regime.
Effect of calibration size Using the ground-trui -ruth parameter 8", vary the calibration set size m ranging from 100 to 3000, ensuring that the resulting length deviation depends only on m and a. As illustrated in Figure the deviation decreases consistently with larger calibration sets. On a log-log scale, the slope approximately approaches -0.5, reflecting the increasing dominance of the ((1/Vm) term in the bound. Meanwhile, the exponential term exp( 2 decays quickly for modest values of m and becomes negligible thereafter.
6.1 ROADMAP OF EXPERIMENTS
We here outline the structure of experiments conducted the paper.
Synthetic experiments. Figure 3 in Section 6 and Figure 5 in Appendix C.2 assess the theoretical
results developed in this paper. Appendix C.3 further examines optimization effects: Figure 6 in- vestigates SGD with heavy-ball momentum, and Figure 2 reports the case of AdamW (Loshchilov & Hutter 2019). In Appendix C.4 Figure S presents results under nonlinear conditional quantile
functions. Finally, in Appendix C.5 Figures 9 T evaluate alternative convex loss models.

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Real-world experiments. In A.222 Figure Taaureelpppeeeee presents empirical evaluation of length deviation CMR and CQR under different optimizers on five real-world datasets, comparing SGD, SGD with momentum, Adam, and AdamW. W. In Appendix D.2.3 Figure
12 evaluates non-linear
models. Appendix D.3 empiricalyy investigates data-allocation strategies in Figure 13
LIMITATIONS, DISCUSSION, AND FUTURE WORK
Oracle intervals may not be optimal under certain distributions. Our theoretical analysis shows that prediction sets produced by CQR and CMR converge to the oracle quantile inter- val (16) as the training and calibration sample sizes and grow. However, the oracle interval optimal only when the condition in Remark (3.4 holds,
itself is not always officiency-optimall the structure of the conditional distribution. For instance, when the conditional density is multimodal or basin-shaped, the optimal prediction set is not a single interval. In such cases, the prediction sets produced by standard conformal methods such CMR and CQR do not approximate the optimal set. This limitation stems inherently from the standard non-conformity scores, which are restricted to producing single-interval prediction sets and therefore cannot capture complex distributional structures. One way to improve efficiency in these settings is to move beyond fixed score functions and consider parameterized nonconformity scores that adapt to the data. For instance, recent work such as BBrann et al. (2025) employs optimization-driven framework tar- geting volume minimization learn the parametrization. Such approaches could potentially learn transformations that adapt to complex conditional distributions, leading more efficient prediction sets. This is a promising direction for future research.
which depends
Role and limitations of the linearity assumption. Our theoretical analysis builds on the linearity assumption of the conditional quantiles. This assumption standard in the theoretical analysis of quantile regression (Koenker 2005 Pan& & Zhou 2021: [hen et al.) 2025). as ensures convexity of the objective and therefore the consistency of the SGD estimator as the training data size n grows. While relaxing this assumption is in principle possible, it typically requires additional assumptions on the complexity of the function class or on the estimation error bounds, which may be difficult to verify in practice (Bars & Humbert 2025).
CONCLUSION
This paper studies the efficiency of conformalized quantile regression (CQR) and conformalized me- dian regression (CMR) through the lens of the expected length deviation, defined as the discrepancy between the coverage-guarantee prediction set size and the oracle interval length. Our analysis explicitly accounts for randomness introduced by training, finite-sample calibration, and test evalua- tion. Under mild assumptions on the data distribution, we provide, to the best of our knowledge, the first non-asymptotic convergence
of the form: O(n-1/2 n -- + m 11/2 + exp(-a'* (m)) which highlights a fine- grained effect of the miscoverage level a. Empirical results closely align with the theoretical findings.
Ethics statement.
This work raises no ethical concerns to declare.
For our theoretical results, we provide complete proofs in Appendix A and B The empirical setups are detailed in Section 6 Appendix C and Appendix D To facil- itate full reproducibility, we include the source code and datasets as part of our submission. The provided repository contains scripts install the required Python environment, run all experiments, and generate the figures presented in the manuscript.
Reproducibility statement.
ACKNOWLEDGMENTS
The work in this manuscript was partially supported by the Swiss National Science Foundation under Grant 200364.
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A PROOFS OF RESULTS IN CQR
To proceed, we first define some notations as follows.
Ey (X, On (7)) = |L, (X; (+)) - by (X;0* (n)) 2 0;
(24)
A (X, 02) = max {Ea/2 (X,0,) &1-a/2 (X, 0m)] 2 0;
(25)
S* (X, Y) == max (la//2 (X:"") - Y, Y - 41-0/2
(26)
max {9a/2 (Y [X) Y, Y - 1111//22 (Y x)};
M (0w)) == max {ll(e. 1 -1)x=-0))12+-
(27)
Let {(m)
S)0, denote the empirical c.d.f. from m i.i.d. calibration scores given On, i.e.,
1 m m
plm) (8) S00n
1[s s8)
S lid Fsjo..
A.1 PROOF OF THEOREM 3.1
Theorem 3.1 (Quantile regression error of SGD-trained models). If Assumptions 3.1 3.3 hold, tak- ing step size U 1/ (Auinfuink) in SGD update 7 then
Ex,0. [tty (X;0, (7)) t, (X;0" (+))))]
4X2
max Saxxxmaas fmaxe
(14)
Pain Prain n
412 Smaxd
Es, [100, (7) 0r (7) [2]
(15)
A Pain
we first show that L, (0) in is strongly convex and smooth with respect 80 (7), as stated below Proposition A.1 The proof of Proposition/ A.1 further relies on Lemma A.1 and Lemma A.2 for the gradient and the Hessian of e, (0).
To prove Theorem 3.1
Proposition A.1. Under Assumption
3.3 and if E [ < co, the objective ly (0) in satisfies
Smin 2
Smax 0 0t (7) I1
0 0r (r)
1 26 E, (0" (7))
(28)
Assumption 3.2 furthermore holds, then
1|0 8" (v) 11 < e, (8) e, (0* (7)) f Fmax Xenax 10 8" (7) 12
(29)
2
where || 111s denotes the E-induced norm, i.e., |00|s: VOT5O.
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Proof. To prove this proposition, first need Lemma A.1 and Lemma A.2 and the Hessian of b (0). By Lemma A.1
caleulate the gradient
Vl, (0* (7)) Ex [(Frix (10 (1)) X xx - 7) x
Fx [(Frix (ay (Y X)) -7) x]
O.
By Lemma A.2
v2ly (0) = Ex [frix (0 Ix) xx By Assumption 3.3 Vo E Rd,
fmin || - = Ex (xT $ Ex [frIX (0 x | ux
fmmx (xTu)² fmax 1ammxe|l11 11
Hence, fmin 20 (0) 3 fmax22 for any E G. By Taylor's Formula,
(0) - C, (0" (n)) //" (1 u) (0 - 0r (7)) v2e, (0" + + 0" (+))) (0 0" (t)) du.
Since
smin||0 0" (7) 11s < (0 0r (7)) T E (0 (0 0- (7))) (0 0* (7))
Smax 10 0" (7) ||>:
and Jo (1 - u) du = 1/2, we have
J
||0 0" (7) 1/ (0) (0) L (0" (7)) s fmax 100 0" (7) [1.
2
Lemma A.1. Suppose
17) in Assumption 3. is true, if E [[[ < co, then
Vl, =x,, (11 < x] Ex [FFrjx (0 x x) x]
(30)
Proof. The key idea is to show that the interchange of differentiation and expectation is valid ac- cording to the dominated convergence theorem. For 8 θ holds that
P f 0 x = E(xx) [1 {Y = x']] Ex |Bxx [1 (Y = 0 X [
= Ex [ = 0 x]]
Since 11 in Assumption 3.3 is true, the p.d.f Sxxx (Y X) exists for each x E X. Thus,
Y = RT x=
fyix (Y Xy
[0
Thus, P[Y = L, (X: 0) Y 0T x = [E =.
For (x,y) E X x y. if y + by (x:; 0). the directional derivative of L. (0 x. y) at along vector u is
L (10+ + po) x,,) Ly (8T (0Txxy) x2
Du (OT x v) = lim
|10112
||ull dp I. (00 + pp)) x.
-11 (v 0 x} 7) x: |0|
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Moreover, since Ly (t.y) is 1-Lipschitz with respect to t,
Fy (0 pu) n., u) T, (0 xr, v)
1ul12 Ly ((0 po) 2.y) - Ly (0 x, 8)
1111120
|| (0 + po) - 0 z0122
|1l12
1 |
Since we assume ||] < oc, by the dominated convergence theorem,
Del - D,Ex, Ly (0 X,Y)
Ex,y
Ti (0 + (w)) X,Y - Ex,y [Ty (OT x, x))
lim
P-0
||0112
Ly (0 po) T X, X.Y Y Lyy X,))
= lim Ex,y p-1
110112
Ly (10 po) x, L, (O X,
= Exx
lim P-0
1|1112p
= Exxyy [Dule (0 (0'x.))
=Ex, [11 ( < <0
<0'x x3 -7) x]
||a||
Hence,
Ve, (0) = Ex,y [(1 (Y < x0
Ex [Brix [1 7)) X x]]
-xx [Byix [11Y<00Tx) -))) |x|
Ex [[Fr]x [0_x) X (x)-1 -)) x]
Lemma A.2. Under Assumption
3.3 if E [|] co, then
V2h (0) = Ex [fvxx (0 x X) X
(31)
Proof. By Assumption, E [|| S /E|||
co. Then, by Lemma A.1
Vl, (0) = Ex,y (1 (Y < -
-xx |rix
-7) X X Ex [BYix [[1 <0 -7) | x
= Ex [(Frix (0T x)-7)
To prove the lemma, the key point to show that the interchange of differentiation and expectation is valid, as in the proof of Lemma A.I
(Fyix x (0 pu x x -7)x --)==((0 (Frix (0T XX) 7)
lim 0+0
|10112p
-lim
1
(Fyix (OT + pv ouTx x x) Fxix (oTx) |x)
p ||01|2p
u xr
1
lim
(Fyix (0T x + put x) Fyix (TTx) x)).
|ill|
00
pe x
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According to the mean value theorem, there exists { (z) in (0Tx, x, 2. x po x) such that
po (Fy x (0 x + pu x xx - Fyix (0 X x)) frix (5 (2)) x)..
Hence,
lim p-0 po x (Fy x (0 x pv (X) - Fxix (0 X [X)) = lim frxx (E (5(z) x).
p-0
Since JYIX (Y X) is continuous for Px-almost every x X, have for Px-almost every E X.
lim frxx (x()) frxx (0 x). p-0
Hence, for Px-almost every X
(Fyxx (0T x + po x) 7) x- (Fyix (0 x) - 7) Sxx
lim 0~0
1 (0 X) (Frix (0T x + po (X) - 7) x (Fyix (AT x) - 7)
||1|1
|0
Since
in Assumption 3.3 is true, for any X, Fyjx is mhaxxLippcchiz.
||1||2
||l|2e [Fyix (0T x + pv x x)) - -rix (0 X XIX)) ||0|
|111220 fmaxp]llllll
fmaxx|==|1"
Since [|] < oo, it holds that f | < co. Therefore, by the dominated convergence theorem, the directional derivative of Vl, (0) at 0 along vector v
Do (Ve, (0))
D,Ex [(Frix x0 7) x]
Ex [[Frx (0 x pu' Tx X X) -7) X -xx [(FyLx (0 x) - -7) X
lim
||1|12p
-lim Ex
(Frix (0 + po X x) - Pvix (0Tx) x1xxx)) x
p-00
[|lllap
Exx
linn p00
1
(Fyix x0 T
x xx)
FYix (0 (x)) X
||u||2
= Exx
[frix (0Tx) X x) X
||a||
Hence, vee (0) Ex |fxx (0 x X) x|
With Proposition A.I we are ready to apply Theorem A.1 for SGD and get Corollary A.1
Theorem A.1 (Section in Rakhlin et al. (2012)). Suppose the loss function E s-srrnngly conver and ymsoohth with respect to a minimizer 0" over Q, and Elllg.1 || G². Then taking me = 1/Xt, holds for any n that
Es, [f (0w,) - f < 2μG²
(32)
X2n
Corollary A.I (Upper Bound of Extra Loss). Suppose Assumption 3.1 and 3.3 hold. Let
Dirain == ((,,, H be the set of training samples and 0, be the estimator by optimizing stochastic pinball loss produced by SGD (7). Taking Tp. =1/ (Amiufminh)) holds that
Eo, [e, (0n (x)) - by (0* (7)) < 2X2 I'axd
(33)
f2
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Proof. In this proof, denote On (x) by On for simplicity. By Lemma
A.1
Vly (0)
Ex (1 (Y 0 ("x]) x]. Then,
Ex, [iee, (0.))l²)
[x [(1 { < xx x []]
Eo.,
Ex []](1 {v < x, x||]²
EE,,
S Ex [1] Amaxd.
where the last inequality is by Assumption 3.2
E [I]]]] < E [| = E trace (XX
trace (EE [XXT]) S trace (Amax I) = ddmax.
The corollary then follows from Proposition A. and Theorem A.I
Now we are ready to prove Theorem 3.1 In this proof, we denote 6, (7),0" (7) by 8,, 8", respec- tively, for simplicity. By Proposition A.I
- 2 (E(0 e(()))
1|0, - 0 S
Smnn
00, 1 S fminAmin (e(0m) - (8*)).
Since the test sample (x, sampled independently of the set of the training samples f(X,, Y.) and 6N is function of ((X;, Y;) 8n is independent of X.
(10. - -0) x){
Ea,,x (x(;00)) - (x;00))² = Ee,..
Ee,
Ex (0n 0²)T xxT (0n 0-) 10m
Ee.
(0, 0²) "Ex [Xx]] (0n, 04)
Eo, 010 011
Hence, by Corollary A.1
Eo,, [|0,,
-0
< Juin
Eo, (0m) - -((0))) <
422 Smmax di Amin (2
This completes the proof of Theorem 3.1
A.2 PROOF OF PROPOSITION A.2
Proposition A.2 (Population quantile of the score). In CQR, if Fyix (Y X = x)) is continuous for all X, then
q1 a (S 0n)l < max (1Lo, l' 11 Ō- 0-
(34)
Suppose Assumptions
3.1 3.3 hold,
2B Amax v 22fmaxe I
Es, (lai- (S1 <
(35)
S
The proof of Proposition A.2 relres relies on the following lemma.
Lemma A.3. Suppose F (Y X) is continuous for each x E X. Then,
Iq--a (S X, 0,)l < 0n), where qia ($| 0.)) denotes the (1 - a: -quantile S given X, dn.
(36)
Proof. By the definitions
24
25
26
S (X, (XXY; 0.) == max (la/2 (X;0,) - Y.Y
Y 4 4-a/2 (X;0a))
- max {Ea/2 (X.0.) + 9a/2 (Y X) - Y, E1-a/2 (X, 0.) + 91-a/2 (Y |X)]
A (X, S (X,Y),
(37)
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where the last inequality is because max(ui +11, U2 + v2) I max(u1, 22) + maxfv1. Similarly,
(X,Y; Y;d.) = max (ta/2 (X: - Y, Y h-a/2 (X;0.)}
2 max {4a/2 | -
91- 91- (V 1x) x) &1-0/2 (X,0n)}
= 5" (x, A 0n),
(38)
where the last inequality is because max(ui U1, U2 02} 2 max(ui, u2) max v2
Note that 5" (X < O is equívalent to 4a/2 (Y X) S Y I 01 01-0/2 (Y X). Since Fyix is
continuous,
IP [9a//2 (Y (X) S Y a/2 /2 (Y) x) x] 1 a.
Hence, (X, I 0| 1 a. Let 91-a (S" X) be the (1 - a) quantile of S* given X. Since X is given, and Py|x is continuous, F---px is continuous. Then, Q1-a (5 X) 0.
Conditional on X, 0n. A (X, t),) is deterministic. By (37) we have
(IS 0n) S a x, 0.1 P/A ,,,) + 5" (X, S 1 X P|S (X,Y; < (X, X, 2n) 2 PSS x0
Then, Q - (S X,000 s A By 38 we have
P ( S X P(* (X,Y) - sx)
7 P =
X, [|
Then, 8-a |
X, 0n) 2 -,,
For 7 E =12,1 1 )
E, (X, 0n (r)) = (0, (7) 00 (n)) x < (10,, (n) - 0" (x))1la S B ll0., (x) - 00 (r))lla
where the last inequality is from the fact that the norm of x E is bounded by B. Then,
A (X, 0x)) S B max {Ile, 1') (0,,
|| = B M (0x).
By Lemma
A.3
[1 (S X, 0.)) s (X, 0.) - B M (0.). Then, P (X Y; 8,)) < B (D.) 0,) 2 1 1 a;
(X, Y:0,) 0,)) 2 B (M,M X, S 1 a.
Then, removing the conditioning on X,
YY; < B M (9.) 0.]
= Ex.ria. [S (X, d,) s B-M (0x) [O,]
=Exi, Ey|x, (X - B M (D.)) | X, 0x 0m] B- M (f,) X, 0.]] d.]] = -xxe, (;,)) B.
2 Exie. [1 a1-al [x]] =-
Hence, 91-a (8 (n) s B M (0.). And by similar arguments as below, q1-a (8)))) 2 -B. M (0x).
FIS(X,Y; 0m)) -B- [0,]
Ex,Y [1 (S ( 2 -B-M (0n)) | [x,]
=Exja, [Exxx, (X; -B. M (9W) | X, 8n] 0m]
=Exiom (, 2BB ((a) Xx00] 00.]
- Lxxo 1 1-a.
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Therefore, 191-a (8 vn)l I B M (U,). Then,
Es, 1 (S 0,,)l - B Es, [M (dn)
[((O, - -0")11 + [ 8-) 1
I B Es.
Es. [I[en 22)2 + I 0-) ||2]]
S B V
012 Es. [Iien 00)1 + Eo [1 0-)
2 V
8X2 max Imaxt
2B Amax V ma 1
S BV
V
IminJJiia
T2
where the second inequality is from max(a, bj < va² + +b, the third inequality is by Jensen's inequality, and the last inequality is from Theorem 3.1
This completes the proof of Proposition A.2
A.3 PROOF OF PROPOSITION A.3
Proposition A.3 (Population finite- sample score-quantile gap). In CQR, Suppose Assumptions 3.7 3.3 hold, if m > 8H/ min a, 1 -a} for H in 12), then
1056R 3 22 B2d
Es, [1(1-))... (s)
Q1 a (S 0,) S fmin min(a?, a.) []] min min 0,))
max max
To prove Proposition A.3 we first need the following critical proposition: Proposition A.4. Suppose (0, is constant. Define
a
1-a
4x Fraaxfmaxc Fmaxd
2A
B : min
A =
== B
2ma
2f
X min
nō
Under the same setting of Theorem 3.1
if en < 3/44 (equivalently n. > 32AB² ), then for ò E (0, 1), R28
with probability at least 1 - d over Un, the following (denoted by event V) hold simultaneously:
1. For with |s|
fsje, en
(s 1n) 25
2. 11 (S 8, S fn < 3/4.
Proof. By the definition of S in la/2 (I; 2n) -SSV S b//0 (x: 0n) + X, 0n
PIS < s|X,
and > (x/2(2:0m) t1 -1-0/2(2;8 (2;0,
Hence,
if < Sova(mren -2
O. Fsxx,o, (s) Frix an (4--//2 (1: +) Frix (1a/2 (z: - s). otherwise.
(39)
We now show that with high probability, it holds for in the neighbourhood of 0 that 8 ta/2 (a; (m)) 11-a/2 (n))
11-0/2 (c; 0w) + 8 &EE la/2 (x; - y.
Let Vnax == suply E }} and Ymin = =inf E V}. Then, under Assumption 3.3 Imax > Umin-
2a/2 (Y X =) 41-1/22 (Y | = x) E [ymins Vunax]:
2a/2 (Y X x) - Umin 2 2f
2 B,
Umnx 91-0/2 (Y X = x) 2
2 3
2 2
41-a/2 (Y) X = x) Ua/2 2 (Y X = x)
1-0
>
2 B.
2
2 ffma
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By Theorem 3.1 Es,, [110,, (v) - 0" (7) [8] I A for {2
1
2). By Markov's inequality,
2A
6 2'
P
10 (1) (7) - 00 (7) 112 2
V
2 1
no
Applying the union bound, we have
2A
P
max NE(2,1-2) 2)
(00 - 0² (7) 12 S
2 11 8.
nd
Since for each x E X,
Eg (n (1)) ,t, Oz On (7)) - by (z; 0r (1)) = [ (7) 0" (7)) T
S 110n (7) 0* (1)) |اا|| S B [10(0, (7) 0" (7)))l2-
we have that with probability at least - 5,
/2A
sup
(z, an)) S
max 1 (7) - -0" (7) ll2 S B
=: En.
- }
nd
and by Proposition A.2 it also holds that
1 (x)] S cn
(40)
Then, w.p. 2 1 - 8, for any x E EX,
ta/2 (2; (x) 2 Ga/2 (Y) X = =x) x) - (x) 2 Ymin + Exi
a///2 (x; 0n) S 91-0/2 (Y) X = =x) + (x. vn)) S Vmax 3 + Cui
0n) ti aa (1;6n)
ta/2 (x: (n) 91-a/2 (Y X =x) 4a/2 (Y X σ)
(rr, 0n,) 2 En-
2
2
The last inequality above shows that with high probability, quantile crossing will not occur given n. is large enough.
In this case, for any s with [s] < In = B Cns we have Vo E X,
la/2 (x; (n) > Umin
fn Fn yminn
ta/2 (z: (n)
9a/2 (Y
X x) cn I'n S 29-0a/2 (Y X x) B S Vmaxi
11-//2 (z: 0z) < ymax
en I' S ymaxi
11-/22 (x: 0m) + 8 > 01-0/2 (Y X = - 6n I' 2 4a/2 (Y X = x) - -3> Ymin;
ta/2 (x; 0 - -1aa122 (x; 0m)
> -1 |s| - -n En 3
2
Since Y' is an interval,
1/ (x: - s y,
4 4--/2 (I;0n) + + &EE..
Therefore, by (39.. conditioning on nn. for with |s| <r = 3 En.
ssja. (8 (n) = Exje. I frix.o. (la/2 (n) - s X, Un) +frix,0, (t (z; 8n) + X, 0n)]
2 2fmin
Suppose n > 324B² 626 which is equivalent to en <3/4 Then, On = 3 - c, 2 38/42 En. By 40
Ia--a (S) 0..l < B
The proof of Proposition A.3 also relies on the following useful lemma, which is a classical result (Bobkov & Ledoux 2019). We include the proof here for completeness.
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Lemma A.4. Let F be c.d.f. with p.d.f. f. Suppose there exists an interval I E R and a constant
Co > 0 such that f(s) 2 for all 8 E I. For E (0, 1), 9p == inf{u F F()) 2 p} E I, define
To = min [gp inf I, sup dpl 2 0. Then, for any pl such that 'p' pl < coTo. it holds that
qy E I, and lap qpl = In'-nl
co
Proof. By assumption,
F (qp ro) - F (qp) coro = p CorO:
F (gp ra) 2 F (qp) + coru = p + COTO.
Since Ip' pl coro. either p S p' < p + Coro pl p p' coro- If S p' < coro. then p S pl < F (4p + ro). Since is non-derreasing, 4p S 9p < 4p tro- Similarly, if pf coro < p'
S
p. then F (qp ro) p' S p. and 9p To 4p S 9p. Hence, 9p' I, and 1gp' -9 9p S v'-rl
8
With Proposition
A.4 we apply Lemma A 4.4 and get Lemma A.5
Lemma A.5. Under the same setting of Proposition
A.4 if event Proposition A.4 occurs, and
4 if m. JminB: then it holds that (9(1-a),,. (S 0,,)
S
8/2, fsyon (9(1-)),,. (5 0 2 2
and (11-a),,. (S (n)) 91- (S 0n))
Juin TM
Proof. For simplicity, in the proof we denote 9p (8 Un)) by 9p-
a1 (m + 1) & [(1 a) (m + 1)] < (1 a) (m + 1)+1 +1
(1- (m + 1) (1 - a) [(1 a) (m 1)1 - a) 100000000000
+ +0< 1-a
2 2
I a) - (1 a1 a)
m
m
m
Since En < 3/4, from Proposition A.4 with probability at least 1 - 8, for with |s| < 33/4,
fsys (8 0n) 2 and 4/1-a| < 3/4.. 4. In this case, ro min (inn/g111 -a 33/4, 91-a3 >
3/2. If m. > m then [1 a)m - (1 a) < 2 < 2f 7 < 2fmin $ < 252min -r- Then
by Lemma A.4
(1-a),.. | S 3///4, fsjo, (q11 (S 0n)) 2 2 and (9(1a) 91-aa <
((1m))
(1-a]
Fmin S 3/4. Hence, (( S | 1 19(1a) -1-al 8/4 4 + B/4
fmii
3/2.
Notice that ((-a)) ($1 --- (S) 0,,) is bounded by 2R. Let V denote the event in Propo- sition A.4, and VC its complement. Then, by Lemma A.5
Eg, [1q0-a) (S ) - 91-a (S o,,) |]
P[V] Es, [19n-a),,. (S (m) q-a (S 0.) || V]
+ PP [V9. Rod, [l((1-)).. (5 0m)) - --aa (S 0n) Vg
+RR..
mmin
Picking 5 = 3"
33
completes the proof of Proposition A.3
A.4 PROOF OF PROPOSITION A.5
Proposition A.5 (Empirical score-quantile concentration). In CQR, Suppose Assumptions 3.1.35 hold, if m > 8H/ min(a, 1 - -a} for H in 12) then
Eon.D.w [1q1-a),. (Sm (x,) - 9(1 9) (S (n))
VT
min (1 ff m
1056R max M B²d min(a², (1 a)² }) A
4Rexp
2fum V2m
8f2
min min
To prove Proposition A.5 first prove the following lemma:
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Lemma A.6. Under the same setting of Lemma A.3 A.4 occurs. for any u E [0, 3/4), if
if the high probability event V in Proposition
sup Fsjo.. (s) - Sldn (s)
flm)
S 2fmin
then ((1 (Sm (n.) - a)) (S [n)] S u.
Proof. For simplicity, in the proof we denote qp (S 0n,) by gp- By Lemma A.5 for [0, (04)/4). 19 a)) - u/ 5 3 and (4(1-a),. + ul I 33/4. /4. Hence, in this case,
Fs 0. (4(1-)),, - -u) s Fs (911-)).. - 2f u = (1 - a)m 2f u; Fs 0. (9(1-a). + u) 2 Fs (9 + 2fiin (1 a)m + 2f
By assumption,
0. (9(1 - uu) - f(mm
S)v. (9(1-a),, -u) S 2fmin
Fs (g(1-a)). + +u)
((m)
9)0n (1(1-a) + +u] - 2fmin
Then
p(m) S. (9(1a)).. -u) S a1
p(m) Sid,
(q-1a),, + 2 (1- a)m-
Since f(mm is non-decreasing, we have ,
(1-a) (Sm (n.) == inf Sm. F(m) (u') (1 a) E [q--a)... u, ((1-a) + +
where Sm is the set of scores of the calibration data.
Then, 1 (5m (n)) q(1-a) (S 0..)
Lemma A.7 (Dvoretzky-Kieee -ienekky-KKeeer-We Kiefer-Wolfowwie Inequality (Dvoretkyy al.) 1956 Massart 1990 Given natural number m, let XI...., Xmm be real-valued i..d. random variables with c.d.f. F.).. Let p(m) denote the associated empirical distribution function defined by
m
((m) (x)
1(, S a)),
x EER. I..
m
Then,
sup (mm (x) - (x) > s 26 2ms=
Ve 2 0.
.ER
By the DyorerAty Inequality (Lemma A.7
sup Fsjo. (+)
(2)-f((m)) f((m) (s) 2 2/min u
< 2exp (-8mfan u²).
Thus, by Lemma A.6 given that the event V occurs,
P [ā(--)),. (Sm 0n) (1-a)),.. (S (m))m) 2 vl < 2exp
u E 30,
Specifically.
[((1-)) (Sm | (2)) - 9(1-0) (5) 2 3/4 v| < 2exp (an
Then, for any > B/4.
P [((1-0) (Sm 0n)) 9m (5|
vl 2 (B84)
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Since |S| - RR 1o() (Sm | 0,) - -(-a),, (8 (s)) 2R. By the layer cake representation of
the expectation of a non-negative random variable Z, which [ Jo PIz 2 u/ du,
Deal
Deal
(/(1-))
(Sm vn.) - ((1-)) (S 0,)) v
2R
P [(@1-o), (Sm (n)) am) (S vn)) 2 u v du
8/1
22
2exp (-8m Fmim u² du) +
2 2exp (-8mf (3/4)² du
8//1
<
exp
(-8m )) 4Rexp (m/4))
V
+ 4Rexp ( (a
2i 22m
Therefore, have
Dea [1(11-a),, (5m (n)) - ((1-a)... (S 0n)) []
< [V] Eg,, 1u (5m [n) - q(1-a),.. (S Un)l v) +P pIv 2R
V
(-1
1 p mmin m + 2R8.
+ 4R
2 V 2m
Picking 6 8=
33 8
completes the proof of Proposition A.5
A.5 PROOF OF THEOREM 3.2
Theorem 3.2 (Efficiency of CQR-SGD). For CQR-SGD, suppose Assumptions 3.7 3.3
3.3 hold. If
> 8H /a, 1 a). then for test sample (X and 0 $ 1/2,
Ex,d,,Daal Dew [|[(()) - CC (X)
| O
1/2 + (a?n)-1 + m 112 + exp( (ap(-a²ml) (17)
where H. is the constant defined in 72
Proof. By the definition of the prediction set (9)
[C(x)) = max {0, 1 f1-022 (I;0n) ta/2 (z; + 20 am)m
S | 1//0 (x; 0n) to/2 (m)) f..) 1-
We further bound the right hand side by 1/10/22 (x; 0m) - 1a/2 (z; (l.) + 20(1 20(11-a)ml
= =|t1-a/2 (x; 8m) - t t1-a/2 (I; (I:00) + t1/a (z;0*) ta/2 (z: 0,,) + tta/2 (x: ta/2 (z:0*)
+22(1-a)m
< | 1110/2 (x; 0n) -/12 (1; 0+1 + 1/o2 (x;; fm)) t2/ (=)]")] + |
+ 11 (x; 8¹) ta/2 (x: {
= 11-a/2 (x; 0m) 11-a/2 (x; 0*) + 1aa2 (z; Qn) - ta/2 (x;4²) + ((
+ (t1-a/22 (x; 0+) ta (x; 0*))
where the last equality follows because
t t1-0/2 0*) q1 4/2 / (Y X) 2 9a/2 (Y X) ta/2 (z:
Hence,
[C(X) - (11-0//2 (x:6*) ta/2 (1; f')
11-0/2 (z;0n) 21 11-a/2 (x:0*) + Ita/2 (.)) ta/2 -a/2(2;;")) + 2 (0)1-))m.l
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We also have
(Ic(X) - (41 (1;8) - ta/2 (x;
= (ti-a/2 (2;6") - f//2 (x;;") max {0, ti a1 (x; 0n) - t//2 (z; mnn) + 20 mm)m
< 11-0/2 (2; 0-)) la/2 (1:(") a1-0/2 (1; 0m) + la/2 (1; 2n) 2Q 1))m
< 11-0/2 / (1; 0n) 11-022 (2; 0*) Ma/2 (2;; (n) 4//2 (2: + ((1
Therefore,
||c(x)| (ti-a/2 (z:0*) 2 (x; ))
$ 1120/22 (1; 0m) 11/2 0*) + Ma12 (13 la/2 (2; ((1-a)m
Hence, for test sample (X,Y),
Px, Pat [||((x) ( 1- (X:0*) - la/2 (x:
< Ex [14-0/2 (X; 0.) - 11-/22 (x: + Ex,0n [| (X:0,) - la/2 (x:
+ 2E,,, Pul [19 (Sm dn)l
< Ex, dn [14-0/2 (X; 11-/22 (X; 0*) + Ex,0, ||/a (x:2n) laf2 (X: 2
2Eg [1-1 a (8 0.)]l 2E Pai [| 911 (5 9..) Q(2 ((2-a)m (Sm }1
< ex,dn ||41-//2 (X;0.) 11-0/2 (x: 0²) |] + 2x,,, |la/2 (X; la/2 (X:
+ 2m,,, [l91 a (S 0..)ll + 2E.,, [/ (S 0m) 91-a)m (5 0m,)
+ 2E,,, Dal [1 (S 0..) a1-a) (Sm 0m.)l]
By Theorem 3.1
Ex,0, II,, (X; (7)) ty (X;0* (t)) S v Ex,0, (tr (X;0n - (X:0* (7)))²
2)max V Fmaaed
<
Amiua Sin v Amiun
By Proposition A.2 A. 3
A.5
Ex Dei [|((xx] 11-0/2 (X:0* 1//2 (x;
4dmax v Fmaxd d
2B Amax v /2fmaxd
1
Amiu V /mmin
Pania Smin
S
66 66AB2R R
+ 4.eepp
22 82m
m 22m
B2n
4dmax v fmaxd
2B Amax 22fmaad d
1
1
m +
I V Admin
Pmiaiiin
2fain v
J m
min (1 a) J
1056X2 2 L
expexp
8/2
m
min(a², (1 -=) min
(41)
This completes the proof of Theorem 3.2
B PROOFS OF RESULTS IN CMR
To prove Theorem the goal is to upper bound
Ex, Deal [|2 (() (S) 8m) (g1-a/2 (Y X) 4a/2 (Y [x))
= 2 Ex x 0,, Deal [1 (S 0n) - (q1/2 (Y XX) 2a/2 (Y |X)]
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Further decompose it, and we have
Â (S 0n) (11/2 (Y X) - 2a/2 (Y) x))
Q a) (S 0m) 9(1-a)m (5 0m) Q(1-a)m (810)) -1-a (5 (51)))
+Q1-a (S 0m) (41/2 (Y | Qa/2 (Y xx)]
1 ()) mm (S 8m) Q(1-)).. (5 0m)] + 9 mm) (S Bn.) - -q-- (S 0,))
+ - (S 8.) (91/2 (Y XX) 9 (Y) x))|
Thus, the expectation is decomposed into three parts follows, and we will upper bound each of and B.1
them in Proposition B.4 B.3
E X.0..Deai ||2 ((1-))m (5| (m)) (q1-0/2 (Y) X)- - 90/2 (V) x))]
2 BB, Deal []
(S (m) 9(1-)).. (S) B..) ]
+ 2 Es, [1q(1-a),. m (S 0m) - -q-- (S Bm)
+ 2 E3 x,0 [191-a (S 0n) (q1/2 (Y (X) qa/2 (Y [x)]
VT
fim min (1 - a) m min Pmin min min (1 )
2056R max fo B²d
<
+ 8Rexp
fmin 22m
8
2
4B Amax F d
/
(42)
m
X2 Muinnmin
To proceed, we define some random variables for simplicity.
A (x, |t1/2 (X: Bm) 112 (x:6*) | 0;
(43)
S" (X,Y) |911;2(8| XX) YY;
(44)
M (B.) : =1( 812
(45)
B.1 PROOF OF PROPOSITION B.I
Proposition B.1. In CMR, suppose Assumption 4.2 holds, we have Q-aa (S X, x. 5) s B M (0.)
(46)
If Assumptions 4.1 3.2 3.3 further hold, then
2B Amax v Fmax
Ex,6, [-- (S (s) - -911/2 (Y X) 2a/2 (Y x))[] -
(47)
Suin
n
Proof. Notice that
S (X,Y;Om,)
= 1112 (X;0m) -Yl
$ //91/((Y XX) -Yl + |t1/2 (X; -5122)) |x)
= = (X,)) A (x,
Similarly, (X.YY: St (X,Y) (X, B.). Hence,
(x,Y S*
s A < | (0,, - -0))11 S B. 1100n 112
Now show that - (8* X)= (Y X) 4/./2 (Y X). Note that given X,
5 (X,Y) < / (Y X) - (V X))
- (1 x)= (Y xx) <Y 41 X)) S 91/2(Y (X) - 42 (X)
- (Y x) < Y < 91 ( X),
where the last step is from Assumption
4.2 Since Fyix continuous.
TP [ga/2 (Y X) Y S 11-0/2 (Y X) | x] 1 a.
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Hence,
P(S (X,Y) S 91/2(Y [X) - 4x/2(( X) = 1-c a. Let 91-o (S* | X) be the (1 - qu--nnne of 5* given X. Since X is given, and FYix is continu- ous, F is continuous. Then, Qı-a (5* X) = (1 X) - 4a/2(Y X).
Conditioned on X, f,, A (x,6,) is deterministic. Thus,
P S (x, Y; 0.)) <u u X, P (x, Y) + A (x, 0n) S X, 0, P ( (x,6, + Q1/2(( XX ///22 (Y) X) X, (m] a.
2 (X,Y)
X) (a |X) X]
Then, a1-a (S x,) s (x.6)))) XX) - (Y | X). Similarly, we have (X, 5u X, P (X - A (x, 0m)) S X, 0.] (Y |X) 4a/2(( (Y] xX) x, 6.] - [ s 2//2(( - 9a/2(( |x) X] =1 -
+ P 5
Then, 91-a (S X, 0n) 2 - (x, (1 XX- (V/2(( (V X). Thus, by Assumption 4.2
1 (S X, 0.) (4)/2(Y XX) -9//2(Y xx) S Sx(x6
+ |1-a ($ X, (m) - 5l S B M (6m).
Then we can remove the conditioning on X,
P [x, Bm) S (+ + B M (B.) 6m]
Ex,
[1 [s < B M (0.)) } | 6.]
Exia. [yixx, (1 S(+ B M (8..) } | | 0x] |82]]
= =xxi,, [P
xx SC+ B. M (6.) X, 0,0n| 0%)
2 ExjØ, [1-a| a 0m] a
Hence, Q1-o (S) 0.) SC+B B. M (0.). And by similar arguments as below, 21-a (S 0.) >
<- B- M (0n). Specifically,
P (X, Y: 0n) 2 ( - B. (0.) 0.]
Ex,ro [1 (X, 2 B M (0.)) | 6.]
=Ex [ (1 (x((x) - B. M (0.)) xx 0.] [..]
Exjo, [P[S (X, (X.Y: 25-2.. B. (6.) [x,x] [%.]
S E x 6., [1 - a 0m] = =1
Therefore, 101- (s) cl s B M (0m).
-
Then, by Theorem 3.1
E3g [1 (S 0m) - <1] S B. E., [M (0.) S B V. [ice, - 02)1
4X2 Smaxe
2B Amax v Smaxed
S B
Min P2
n.
i.c.,
2B Amax v Smaxd
Ex,â [101- (510m) - -(9/22 (Y XX) 2a/2 (Y Xx)) <
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B.2 PROOF OF PROPOSITION B.2
B.2
Proposition B.2. In CMR, suppose Assumptionn..1...2 3.3 hold. Define
{
1-a
A
8 3:=min
en :n := BV
(48)
2mmxx 2fmax
n
If c, < 3/4, then with probability at least for any such that for E EI fs
Is
5
cn) fs1é.. (s) 2 m and qq--a (S B.)) 5) S cn 3 En.
Proof. By the definition of
P p < s|X, 0m] [t/2 (X:0n) - - Y st// (x:6m) + |
X,
Hence,
F... (s) Fyxx,s. (1;; +) Fy, (41/2 (z: 0n) s). We now show that with high probability, holds for 8 in the neighbourhood ( that f1 (2: B,))
(49)
(1; (x) - sEY.
By Theorem 3.1 E [110,, 01] _ for A
Smard fmand A P2
By Markov's inequality,
A
P |00, 0-1
2 1 1 8.
no
Hence, with probability at least
-6, 6,
sup A (x, 0n) S <Bli0n 00112 S BB
=: Cn-
nō
In this case, by (46).
91 (5) -| S cn-
(50)
Then, for every s such that -1-51553 - En ie., 5 I, it holds that
t1/2 (z; 6m + S 91/2 ( = 91/2 ( 91 /2
(Y|X)+ S Ymaax
11/2 (1; 0n) + 2 91/ ( = 212 (Y|X)+ 41 n1-a/2 (YIX)-3 2 Yminn
11/2 (1; 0m) s S 41/2 ( -nn tn = 41/2 (Y|X) - - 22/2 (IX)+ + B S ymaxx
6-3= = qa/2 (Y(X) 3 2 Ymiin 11/2 (1; 0m) 8 2 41/2 (V|X) -1, - = 01/2 (Y|X) 3
Thus, 21/2 (I; 8m) + s V,t// (x;6,) - y.
By 44 if en 3/4, then with probability at least 1- 8, we have for any such that |s -s En
f a. (s) = frix,0 (t1/2 (z; 8m) + +) + frix,A, (ti/2 6,.) - s) 2 2fmin- (51)
Since 1910 (S 0m) -cl S En S 3 en < 3. after taking expectation over X, we have
fsjā, (41-a (S 0m) - 2 f
B.3 PROOF OF PROPOSITION B.3
Proposition B.3, In CMR, suppose Assumption
3.2 3.3
3.3 4.2
hold. If
8
m Jmin min min(a, (1 (1-
(52)
then
514RX2 f XA f2 min { a 2m)
max Fax B²d
Esg ( (n)) - 92-a (S 0m) fmiam
(53)
256 a Pu Bdd
and farthermore
Pain min (a² (1-)(33) then with probability at least 1 - 6,
3 19 () (00) a1-a (S (10000 1 smim 4
1
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Proof. Notice that
1 1-a a
2-a a
1 - a) - -11 - -a]l
let m 35mm
for 3 defined
(48 then
2
8
(11 a) (1 (1-a)
2f min
m
According to Lemma A.4 since 91-0 (S 8n) -c S En & by Proposition B.2 the distance
from I is ro > 2. Thus, by Lemma A.4 a(1-a),,. (S 0m) QQ-a (S 0m S
8 and
Fmime
4
hence, m (S 8m) - 5l < 2
Therefore, if cn < 8/4 and m > Iuin then
4
1 Jmin 2RR.
Eg [112-)),, (S - aa-a (S 0,) S
Taking 8 maa L B²d
257A
X min mii (1-a)= and we get
1 514R max f Smxx B2d
Eg, [19(1-a),, (S ())) 91-a (5 0m) || S
+
fmin
A Mii minn min(a (1 a)
B.4 PROOF OF PROPOSITION B.4
Proposition B.4. In CMR, suppose Assumption
4.1 3.2 3.3 4.2
hold. If
8H
m min (1 a))
(54)
for in (2). then
Egg, Deat [0)(1-)),, (5m 0m)
((1-)),. (.)
VT
P minfa?, (1,,(11111 a)2)
514R / > > B2d +
+ 4R 4Rexp
maxe
2. 22m
8aax
min (1 (1)=)) aa)
The proof of Proposition B.4 is essentially the same the proof of Proposition A.5 here for completeness.
We include
Proof.
Lemma B.1. In CMR, under the same setting of Proposition B.2 if the high probability event V in Proposition B.2 occurs, for any u. E [0, 1/4), if
sup F Sq (8)
f((m) (s) S 2fmin slō,
then ((11))m (Sm (m) - 4(1-a), (S ($10m) S u.
Proof. For simplicity, in the proof we denote dp (S 0m) by q.. By Proposition [0. 8/4], (q(1a)) - - ul < 33/4 /4 and - and ((1-)) + I for defined in Proposition B.2 Hence, in this case,
B.3 for
ul < 33/4, i.e., q1 (1-m)).. u E I
Fs (1(1-a),, - -u) < Fsjē (4(1-a)..) 25min (1 a), 25min U5 F 900 (11-a)), + +u) 2 Fsjo. (1(1-)).) + 2fmin 24 = (1 a)m + 2f U.
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By assumption,
Fsie. (9(1 -u) -f(m) (9(1 -u) _ 2f u;
F S0. (1(1-a)), + +u) f((m) (g(1-0),. + +1)
s 2fmin u.
Then
f(m) (q(1a)m - u) S am 516,
p(mn) (9(1-a), +2) 2 (1 a)m-
Since f(m) is non-decreasing, have 5/d,
(()-)),. (Sm 0m.) == inl E Sm plm) sid,,
(v)) (1 am [u(1-a),. U, a(1-a),.. + tu]
where Sm is the set of scores of the calibration data. ((1-a).,. (5) 0n) u. By the Dvoretzky
Then, (/(1-)),. (Sm 0m)
Inequality (Lemma A.7
TP sup Fsje. (s)
((() S/a.
(s) 2 2min S 2exp (-8mfmin u²)
Thus, by Lemma A.6 given that the event V occurs.
IP| @uu-a),. (Sm 0m) ((1-a),, (5)0m) 2 v| < e (-8mf u E [0,8/4]
Specifically,
P [iá (Sm (n) a) (51 2 3/4 v| S 2exp (-8mfmin (8/4)²) Then, for any B/4.
P P((ā1-0).. (Sm (m)) 9(1-a)m (S (m) V < 2exp (-8m (3/4)3).
Since |S| S R, a)) (Sm Bm)) - 4(1 a) (S B.)) - 2R. By the layer cake representation of
the expectation of a non-negative random variable Z, which is [=|2| So PIZ 2 u| du.
Ei_
[a(1
(Sm (m) q(1 a) (S 0m)) vl
2R
P [Q@1-a) (Sm (m) 911 9))
(S 0m) 2 vldu du
8/1
22
2exp (-8m u² u2)du+
2 exp (-8m (8/4)2) 2) du
<
exp
(-8mf " u²) + 4Rexp (-8 (3/4)²m
+ 4Rcxp (-1 1m
2mmin 2m
Therefore, we have
Eg [lá(1-)),,
(Sm (n)) ((1-a)m (S
$ P [V]
Bjg,
(q(--)) (Sm 0m) 9(1-a3m (S 0m)
v.] P [VV. 2R
V
Lanl min(a2, (1 (1 2)
+ 4Rexp
m + 2R8.
2 2m
8 822
Picking 5 = 257 2 Bdd
Amin 2im min (1-a)) completes the proof of Proposition B.4
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4g
d0
1 dg
Ro
4
4
4
c-
x 0g
xx 0
x7 0
x70p
a-x
0o
Ro
01-
1
T00
1
Figure The probability density function of Y|X for synthetic dataset.
C ADDITIONAL EXPERIMENTS ON SYNTHETIC DATA
C.I DATA GENERATION IN SECTION
The sampler of the data distribution P is constructed as follows. A vector 0o is first drawn from 8 ~ Uniform [1, 2)²). The covariate X sampled uniformly from X = [1. 20'2, i.e. X ~ Uniform([1, 20 Then, the probability density function of the conditional distribution Y X constructed over support [ymin; Mmax], where y/max [20, 20 0g and Y/mi -max The conditional p.d.f., illustrated in Figure 4 is piecewise affine with five segments, symmetric about zero. The central segment carries probability mass (1 (o) and each the other four segments carries a0/4 4, where ag 0.005 is chosen to be smaller than the smallest miscoverage level considered in the experiments. The model is well-specified (Assumption 3.1 for E {a/2, 1 a/2} and all (av, 1/2) by taking (7) 1-2(1-7) 0o. and hence the true quantile functions (x; 0" (?)) Dia
1-2(1-22 Then can draw y Y|X æ. from reject sampling obtain (s, v)- I-a0
C.2 VALIDATING REGIME OF ((1((na²)
In the regime where a 0/441 and 1/2), theory predicts that the length deviation should scale as ((1((na²)) corresponding to the middle regime (green) in Figure To validate this dependence,
pick several small values size n. plotting the length deviation against 1/ 1/(na²) on a log-log scale. The fitted regression line (red) in Figure yields slope of approximately 0.92, which is close to the theoretical value of 1. The empirical results support the predicted theoretical scaling, indicating the upper bound accurately captures the observed dependence.
(0.01, 0.02, 0.025, 0.03] and vary the training
10
10
titted slope 00.92
<lope
10a1
l/nx²)
Figure 5: Log-log regression of length deviation A versus 1/(na²) for relatively small a.
C.3 TRAINING VIA ALTERNATIVE OPTIMIZERS
To demonstrate that our analytical framework extends directly to alternative optimization algorithms by substituting the corresponding estimation error rate, Figure 6 reports the empirical results ob-
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tained using SGD with heavy-ball momentum (Polyak] 1964). Theoretically, SGD with momentum achieves the same convergence rate as vanilla SGD, up to improved constants. According to Remark 3.3 the efficiency with SGD with momentum scales in the same order SGD. Consistent with this prediction, the empirical results show that the phase transition phenomenon identified in our analy- sis persists under SGD with momentum as well. Specifically, in Figure (c), the slope of curves in Figure 6 (a) changes from - to -0.5 as a increases.
Moreover, to demonstrate that theoretical insights not tied to optimizers with established convergence guarantee, Figure 7 reports the empirical results obtained using AdamW (Loshchilov & Hutter, 2019). From Figure (c), we observe the phase transition phenomenon, where the slope of curves in Figure 7 (a) changes from - to -0.5 as a increases.
(a) Expl: Fix m==0000
(b) Exp2: Groundtruch 6*
(c) Slopes loglog piot (a). (b)
.0.5
Length Dewiation (A)
1n2
0.11 0.
82
0,225
0.75
lsl
Slope
For cach Co
For each Oo
ln0
log à logn
log
[b]
1n.1
1e3
1e4
lel
1e2
1e3
0.0
0.3
0.4
Training (n)
# Calibration (m)
Figure 6: The length deviation of conformalized quantile regression with training via SGD with momentum.
(a) Expl: Fix m00000
(b) Exp2: Groundtruth
(c) Slopes in loglog plot (a), (b)
162
Length Deviation (2)
3.75
-0.
0.12
0.
Stope
-0.75
(b)
100
log
log
Agg
16-1
le3
164
lel
1e2
1c3
0.00
0.05
0.10
0.15
0.20
Training (n)
# Calibration (m)
Figure 7: The length deviation of conformalized quantile regression with training via AdamW.
C.4 NON-LINEAR GROUND-TRUTE QUANTILE FUNCTIONS
To empirically show that our theoretical insights extend beyond linear models, we conducted exper- iments in a setting where the ground-trute -trntthtthhhh quantile functions are longer linear. This is achieved by applying Gaussian convolution kernels to the original linear conditional probability density func- tions, thereby introducing controlled non-linear distortion. As shown Figure g even with this non-linear distortion, the phase transition phenomenon persists, indicating that our theoretical in- sights remain valid in a broader setting.
C.5 ALTERNATIVE Loss FUNCTIONS
To provide empirical evidence that similar efficiency scaling behavior persists for other convex mod- els satisfying our assumptions, we report results using l fj-reuularization during training in Figure and Huber penalty (Huber 1964) during training in Figure 10 In both cases, the phase transition phenomenon remains clearly visible (the slope of (a) changes from -1 to -0.5 as a increases), further validating the generality of our theoretical insights.
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[e] Exp:: Fix 5000
(6) Exp: Grouddrudd 6
Stopes liging slke (o). (b)
0.22
Longht Dovadi (5)
Stou
.075
Lc3
0.03
0.10
Figure 8: The length deviation of conformalized quantile regression on the data distribution with non- non-linear conditional quantile functions, where Gaussian convolution kernels are applied to the linear conditional probability density functions. We set a to be 0.1x conditional quantile.
(a) Expl: Fix 50=0000
(b) Exp2: Groundtruth 0"
(c) Slopes in loglog plat (a). (b)
162
00.5
Length Deviation (A)
DUGS
0,
035
175
lel
0.755
Slape
(b)
140
kug kg
log A
log 07
1e-1
1o4
1e2
le3
0.10
0.15
0.20
Training (n)
# Calibration (m)
Figure 9: The length deviation of conformalized quantile regression, training with e regularization. We set the coefficient of the regularization term be 0.001.
(a) Expl: Fix m=05000
(b) Exp2: Groundtruth 6*
(c) Slopes loglog plet (a), (b)
-0.5
Length Deviation (A)
0.60
025
EJ0TE
0:55
0.255
lel
0.
(+)
.0. 75
(b)
le0
For each
For each
log
A
1611
163
1e4
le1
1e2
1e3
0.00
0.05
0.10
0.15
0.20
Training (n)
# Calibration (m)
Figure 10: The length deviation of conformalized quantile regression, training with Huber penalty. We set Huber to be 0.5, and Huber À to be 0.1.
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D EXPERIMENTS ON REAL-WORLD DATA
D.1 STATISTICS OF DATASETS
We list the statistics of multiple popular real world regression datasets used in this paper in Table !
The Medical Expendiuure Panel Survey (MEPS) Panels 1 D and 2 2 are standard datasets used for benchmarking and comparative analysis in the quantile regression literature. Each sample consists of 139 features, including 2 categorical features, 4 continuous features, and 133 boolean features.
Table I: Statistics of datasets.
Dataset
# Features
# Number Samples
MEPS 19
139
15,785
MRPS 20
139
17,541
cpusmall Chang & Lin 2011
8,192
abalone (Chang & Lin 2011
8
4,177
California Housing mg (Pace & Barry 1997)
8
20,640
D.2 EMPIRICAL EVALUATION OF LENGTH DEVIATION
D.2.1 EXPERIMENTAL. SETTINGS
We examine the effect of the training set size n and the calibration set size m on the prediction set length, comparing the empirical results with the theoretical bound in Theorem [32]] Since the oracle quantile interval length (C)( = (ilaa/(((XX) - 2a/2 (Y X) depends on a, we evaluate the for a E [0.01, 0.05, 0.1, where the interval length |C* (X)] is approximated by its estimate with same and largest training and calibration sample sizes. We reserve 20% of the dataset for testing length deviation. The remaining 80% data was partitioned for 80% training data and 20% calibration data: the training size n varied from 10% to 80% in increments of 10%, while the calibration m was chosen from 5%, 10%, 15%, 20% of the remaining data. Throughout experiments, models trained with step size tuned by successive hal halving for epoch.
expected absolute deviation
D.2.2 EMPIRICAL RESULTS WITH VARIOUS OPTIMIZERS
Figure presents an empirical evaluation of length deviation of CMR and CQR under different optimizers real-world datasets, comparing SGD, SGD with momentum (SGDM) (Polyak 1964), Adam (Kingma & Ba, 2013), and AdamW (Loshchilov & Hutter] 2019). Although our theory based on analyzing SGD and directly extends to SGD with momentum, include Adam and Adam W due to their widespread practical use.
The results confirm two key insights from our theoretical analysis. First, increasing the calibration set size m reduces the expected length deviation. Second, for a fixed sample size, a larger miscov- erage level a leads to smaller deviation with lower variance, which aligns with the a-dependence in the theoretical rate. Consistent with Theorem 4.1 we observe that smaller values of a yield significantly larger length deviations.
Among the optimizers, we observe that Adam and Adam W generally achieve better efficiency (lower length deviation) but exhibit higher volatility, likely due to their sealed gradient norms. SGD de- cays more smoothly, providing a more consistent reduction in length deviation as the number of training samples increases. For Adam and AdamW, the benefit of additional training data can be less pronounced or even reversed certain datasets (c.g., MEPS), where fast convergence makes efficiency more sensitive to stochasticity. On other datasets such as abalone and cpusmall, however, Adam also exhibits clear decreasing trend, indicating dataset-dependent behavior.
japp2boPaffuummer=Hc-111
jsp? Sg07cbooPFffuumbe-H--111
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dblle SHE0
delen SHEOS
Melle aod
supec
supech
sugesi
sepedd
Lupii C2
17-
03 650 Vio
650 Tralle
03e Tieere
olo
05 Thur
# Trinne
Touree
(a) CMR
(b) CQR
Figure 11: Efficiency of conformalized regression under different training optimizers real-world datasets: MEPS 19, MEPS 20, California Housing (Pace & Barry 1997). cpusmall 1 1, and abal one (Chang & Lim 2011]. For each optimizer, the learning rate is selected via successive halving, while all other hyperparameters (e.g., momentum=0.9 for SGD with mo- mentum) follow the PyTorch defaults.
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D.2.3 EMPIRICAL PROBING ON NON-LINEAR MODELS
We conduct empirical probing of non-linear models on real-world datasets, and report the results in Figure 12 We observe that the length deviation remains consistent across non-linear and linear model architectures, suggesting potential practical relevance of our findings beyond linear models.
D.3 EMPIRICAL DATA ALLOCATION GUIDANCE
We empirically investigate how to allocate data cpusmall 11 dataset from LIBSVM (Chang &
Lin 2011). The training ratio In takes values from [0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99], the calibration ratio I'm is set rus the miscoverage level a takes values from [0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06,
0.07, 0.08, 0.09, 0.1, 0.15, 0.2].
The left plot in Figure 13 shows the length of the prediction interval versus a of CMR, grouping curves by training ratio (0%-20%, 40%, 80%-100%). We observe that two "elbows" occur at approximately a = 0.045 and a 0.003, at which points, reducing leads to a substantially sharper increase in interval length than before. Notably, before the first elbow, e.g., when reducing from 0.2 to 0.05, the prediction interval length increases only mildly.
The right plot in Figure 13 shows the length of the prediction interval versus the training ratio, with each curve corresponding to a different miscoverage level (lighter color representing smaller a). We observe that:
The curves largely concentrate around interval lengths of approximately 2.5 and 15, spectively, which correspond to the two elbow locations in the left plot.
For most cases where a is not extremely small, the interval length stays below 15, and the curves exhibit a wide U-shape. This indicates that allocating excessively large portion data to either training or calibration tends to degrade efficiency, whereas a more bal- anced split yields better efficiency. For reasonably large a, say > 0.04, the number of calibration samples has less influence on the interval than the number of training samples, suggesting that allocating more data for training is generally beneficial.
For very small miscoverage levels (a < 0.003), which correspond to the three curves above the dashed line at length = 15, the interval length behaves erratically and no longer follows the U-shaped trend observed for larger It is likely due insufficient sample size such small a, and the prediction interval length is a trivial upper bound of the oracle interval length rather than its approximation. This phase of extremely small may correspond to the regime w(n" 1/2), where our upper bound is non-vanishing (Figure 2)
Takeaway. The empirical results suggest that practitioners may leverage the elbow points in the left plot of Figure[33] to select a values that yield good efficiency while maintaining reasonable coverage guarantees. In particular, for extremely small a. the prediction interval becomes trivially large due to insufficient sample size, whereas in the regime of large a, decreasing the miscoverage level results in only a mild increase in interval length. In terms of data allocation, the results are consistent with the practical rule of thumb that the amount of training and calibration data should be roughly of the same order, while allocating slightly more data for training is generally beneficial.
E THE USE OF LARGE LANGUAGE MODELS (LLMs)
This paper uses a large language model to polish writing.
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Alskoac
slakows: alolonas
alaloue:
10a
Length deviation (A)
10-3
callxrsing:
u-0.06
aillowing cillooiing a-0.
Longh deviation (A)
10P
Model
lawer
thres laper
10"1
epasenall:
Longht deviakko (a)
10-1
0.25
0.50
0.70
0.25
0.50
0.76
0.25
0.50
0.76
0.25
0.50
0.75
Training
Training
Training
Training
(a) CMR
abakone: a-1l.115
abakone: a-lll
abalone:
Longth deviation (A)
x/
calloosingg a-00t
calboraing ouloosing: a-013
calborsing: a-1122
Lenghh deviation (A)
Madel
liner
Jayee
layer
spasmaill a-11.0
spasmall:
epaamull::
epusmull:
Length deviation (A)
0.25
0.50
0.75
0.25
1.50
1.75
0.225
11.00
0.75
0.25
0.50
0.75
Triining
Training
Training
Training
(b) CQR
Figure 12: Efficiency of conformalized regression with linear and non-linear models trained via SGD on real-world datasets: MEPS 19, MEPS 20, Californ's Housing (Pace & Barry 1997). cpusmall, and abalone (Chang & Lin 201T). The two-layer neural network has one hidden layer with 10 ReLU neurons, and the three-layer network has hidden layers, each with 10 ReLU neurons.
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Training Ratio
35
0% 20%
0.04
20% 80%
0.08
30
2nd elbow
80% 100%
0.12
0.16
Length of prediction interval
a 0.003
25
length
20
0.0025 0.0050-0.0075
length 15
length 15
15
elbow 0.045
length - 2.5
length = 2.5
lngth 2,5
0.05
0.10
0.15
0.20
0.0
0.2
0.4
0.6
0.8
1.0
Training ratio
Figure 13: The effect of training ratio Tn- calibration ratio Tmm and miscoverage level on epusmall dataset. The training ratio takes values from 0.01 to 0.99, the calibration ratio I'm is 1-rn, and takes values from 0.001 0.2. See Appendix fo3 detailed discussion on empirical data allocation. The left plot shows the length of the prediction interval versus grouping curves by training ratio (0%-20%% 10%-80% 80%-100%). We observe that there are two "elbows" around 0.045 and 0.003, which points, reducing leads to sharper rise of the interval length than before. The right plot shows the length of the prediction interval versus the training ratio, with each curve corresponding different miscoverage level (lighter color representing smaller a).
38


Reviews:
[{"review_id": "E67QglBGgJ", "reviewer": "Reviewer_N2xt", "summary": "This paper studies the expected length of prediction intervals produced by split conformal methods for regression, focusing on conformalized quantile regression (CQR) and conformalized median regression (CMR). Authors examine the case of linear models  trained with SGD and derive a bound on expected deviation between the learned prediction interval length and the oracle length given by the true conditional quantile. Specifically, authors provide non‑asymptotic upper bounds depending on training and calibration set sizes as well as miscoverage level $\\alpha$. The analysis assumes bounded covariates and outcome, as well as conditional density bounded above and below.", "questions": "- Figure 6 in the appendix shows a very strange behavior: performance oscillates with increasing training set size. Why is that happening?\n\n- Figure 7 in the appendix is much more stable, but also has problems: length deviation increases with increasing training set size for many $\\alpha$ and calibration set sizes. Can you explain this behavior?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Originality: to my knowledge, the first explicit non-asymptotic bound of this type (depending on train and calibration test sizes and miscoverage).\nQuality: explicit versions of the bounds (equations 41 and 42) make the dependence on problem constants transparent. Experiments showcase key theoretical findings.\nClarity: Assumptions are explicit and clear, and key constants are defined.\nSignificance: finite‑sample results relating dataset splitting and miscoverage level are immediately actionable for conformal prediction practitioners, complementing earlier work on CQR and asymptotic efficiency.", "weaknesses": "- Abstract mentioned “offering guidance for allocating data to control excess prediction set length\". I assume the Data Allocation and preceding sections cover that. In my opinion, the practitioners would benefit from a more clear and explicit recommendations on choosing sizes/miscoverage level, e.g., starting from a fixed training (or calibration) set size $n$ ($m$) or a required level of $\\alpha$. In fact, authors themselves follow similar setups for their own experiments in the Appendix.\n- Large constants in equations 41 and 42. Computing them for the presented synthetic data model may improve the understanding and presentation of the bounds.\n- Experiments are mainly linear with one real dataset family (MEPS). Including at least one nonlinear (e.g., NN) backbone (even if theory doesn’t directly cover it) would illustrate the practical applicability of the bounds beyond linear models.", "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": "BQNWTtPU9s", "reviewer": "Reviewer_fXxb", "summary": "This paper gives finite-sample efficiency bounds for split conformal regression, both CMR and CQR under stochastic gradient descent training. They derive an upper bound on expected deviation of prediction set length from an oracle interval for CQR-SGD. Additionally they derive a non-asymptotic upper bound for homoscedastic tasks within CMR-SGD by leveraging the fact that intervals are symmetric and of constant lengths across inputs. The upper bounds derived are functions of training and calibration set size, and miscoverage level, which allows them to identify phase transition with respect to miscoverage levels. The authors show how to use this to “guide” train and calibration data splitting both theoretically and empirically for a limited regime of model and data types.", "questions": "Your experiments primarily use a small subclass of linear models (i.e. ridge regression) on MEPS data, could you provide empirical evidence that the same alpha-scaling behavior persists for other convex models that satisfy the outlined assumptions?\n\nMEPS are primarily tabular with many bounded/indicator features (likely roughly aligning with your assumptions after some light preprocessing). Could you discuss sensitivity to datasets with genuinely heavy-tailed or unbounded covariates, where boundedness barely holds approximately (at best)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "Main ideas and contributions are presented well and the proofs/the story the authors are telling with the results are clear. The paper combines strong theoretical backing, but also provides some practical implications of this analysis relating to data splits that practitioners can use. To the best of my knowledge, theoretical results differ enough from previous works mostly in the subtleties involving relaxation of assumptions on the score distribution and the fully non-asymptotic nature of their derivations. While other works have presented in depth analysis on expected size of conformal prediction sets and “optimal” data splitting, I think the theoretical backing makes this work novel enough.", "weaknesses": "Empirical results are lacking. The paper is absent of thorough and diverse experiments across broad model and data types. While I understand the upper bound derivations rely on standard error rates of *optimizer of choice (i.e. SGD) I think more can be done empirically to show a) robustness of method and b) the extent to which one can relax some assumptions on the SGD error rates or characteristics of the data distribution and still have upper bounds empirically hold. It’s unclear to me the scope of real world data in addition to the model type that their theoretical findings would truly apply to. Further empirical probing of a broader variety of both linear models on more complex data types would make this analysis more convincing. \n\nThe authors don’t provide any empirical evidence on the optimizer-agnostic claim. While it makes sense in theory, some small experiments showing this would be a nice addition.", "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": "adAerFgzrF", "reviewer": "Reviewer_oFDM", "summary": "This paper investigates the length of prediction intervals obtained from quantile regression models trained via stochastic gradient descent (SGD). The main contribution is a bound on the deviation of the empirical interval length from its true (population) counterpart. The analysis assumes a linear conditional model for $Y|X$, together with regularity and compact support of the underlying densities. While these assumptions may appear restrictive, they are reasonable in the context of deriving precise theoretical guarantees.", "questions": "1. On the oracle interval and multimodality. You control the expected length deviation, but the optimality of the oracle interval is only briefly discussed in Remark 3.5. Under multimodality, the interval defined by quantiles does not coincide with the highest-density regions. A discussion of this limitation would be valuable.\n\n2. On notation and readability. The paper uses numerous notations, some of which could potentially be simplified. In $t_{\\gamma}(x,\\theta) = x^{\\top}\\theta$, is the subscript $\\gamma$ really necessary? The notation $\\overline{\\theta}$ and $\\underline{\\theta}$ already indicates whether the upper or lower quantile is considered. In addition, it might improve clarity to make explicit in Assumption 3.1 that the model is linear by writing, for instance, $X^{\\top}\\theta(\\gamma)$.\n\n3. On definitions and practical details.  Line 90: Is a set-valued definition of quantiles necessary? The standard generalized inverse given in Eq.(2) is well known and likely sufficient for your analysis.\nLine 146: Please specify the learning rate used in the proofs of the main theorems. The expression seems to depend on parameters that are probably unknown in practice. What step size do you use empirically?\n\n4. On related literature.The discussion around Theorem 3.1 could be expanded to better situate the work within the existing literature. Several studies investigate quantile regression trained by SGD under linear assumptions (see Shen et al, 2025 and references therein). It would be useful to compare your setting and assumptions (e.g., Assumptions~3.2--3.3) with these works, especially regarding the role of strong convexity, which Shen et al (2025) claims is typically absent.\n\n5. On missing or unclear notation. In Theorem 3.1, the constant $d$ appears for the first time without prior definition. As later sections suggest, $d$ denotes the dimension of the covariates $X$; this should be stated explicitly before the theorem. Similarly, the notation for the $\\Sigma$-indexed norm should be defined when it first appears.\n\n6. On quantile crossing and model assumptions.  Remark 3.1 and Proposition B.4 are somewhat confusing. The SGD procedure learns two parameters, $\\overline{\\theta}$ and $\\underline{\\theta}$, corresponding to the upper and lower quantiles. Proposition~B.4 shows that, if $n$ is large enough, quantile crossing cannot occur, i.e., the lines $x \\mapsto \\overline{\\theta}^{\\top}x$ and $x \\mapsto \\underline{\\theta}^{\\top}x$ do not intersect.\nDoes this imply that the two ground-truth lines are parallel? Does this follow directly from Assumption 3.1, which (as I understand it) corresponds to the linear model $Y = \\theta^{\\top} X+ $noise?\n\n7. On classical results.  Lemma B.4 is a standard result; see, for instance, Proposition A.25 in [2]. Please include appropriate references and mention when results are classical.\n\n8. On writing and presentation. Throughout the paper (particularly in the appendices), punctuation should be added at the end of equations (commas or periods) so that each displayed equation is integrated into a grammatically correct sentence.\n\nReferences\n\n\n [1] Online Quantile Regression, Shen et al. (2025), arXiv:2402.04602.\n\n [2] One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances, Bobkov & Ledoux (2016).", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The main strength of this paper lies in the completeness of its analysis. The authors derive bounds on the expected deviation of the prediction interval length that explicitly account for all key parameters — namely, the size of the training data, the size of the calibration data, and the confidence level.\n\nThe paper is clearly written and well structured, leaving a positive overall impression both in the main text and in the detailed appendices and proofs.", "weaknesses": "The paper would benefit from a dedicated discussion of its limitations at the end. In particular, while the analysis focuses on the expected length deviation, it would be useful to comment on situations where the oracle interval itself may not be optimal — for instance, in the presence of multimodal conditional distributions. Furthermore, it would be interesting to discuss whether and how the linearity assumption on the conditional distribution Y∣X could be relaxed or extended.", "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": "vWqEiZA6lf", "reviewer": "Reviewer_3qyg", "summary": "This paper presents a rigorous non-asymptotic analysis of the efficiency of conformalized regression, specifically for Conformalized Quantile Regression (CQR) and Conformalized Median Regression (CMR) trained via Stochastic Gradient Descent (SGD). The core objective is to quantify the efficiency, measured by the expected deviation between the length of the constructed prediction interval and the ideal oracle interval length.\n\n**Contributions**\nThe main contributions focus on a novel theoretical upper bound for this length deviation. Unlike prior work that often treats the miscoverage level $\\alpha$ as a fixed constant, this paper derives the first bound that explicitly captures the joint dependence on the training set size $n$, the calibration set size $m$, and $\\alpha$. The derived bound is of the order $O(n^{-1/2} + (\\alpha^2n)^{-1} + m^{-1/2} + \\exp(-\\alpha^2m))$. \nThis result yields several key insights. \n\n1. It introduces the term $(\\alpha^2n)^{-1}$, revealing that efficiency degrades significantly when requiring very high confidence levels (i.e., very small $\\alpha$). \n\n2. It uncovers phase transitions in the convergence rates depending on the regime of $\\alpha$; for instance, the rate can shift from $O(n^{-1/2})$ to a much slower $O((\\alpha^2n)^{-1})$ as $\\alpha$ becomes smaller than a threshold related to $n$. \n3. These theoretical findings provide concrete guidance for optimally allocating data between training and calibration to achieve a desired level of efficiency. The analysis is supported by empirical results that closely align with the theoretical predictions.", "questions": "1.  **On the Impact of Model Misspecification:** Your theoretical analysis relies on the assumption of a well-specified linear model. This provides a very clean and insightful result. However, in practice, all models are misspecified. Could you elaborate on how your efficiency bounds might change in the presence of model misspecification? Specifically, if the true conditional quantile function is non-linear, an unavoidable approximation error term arises. How would this error propagate through your analysis? Would it be a simple additive term, or would it interact with the statistical rates you derived in a more complex manner?\n\n2.  **Generalizability to Residual-Based Conformal Methods:** The paper provides a deep analysis of Conformalized Quantile Regression (CQR). A very common and often simpler alternative in practice is to train a single point predictor (e.g., for the conditional mean) and then calibrate using the absolute residuals, i.e., scores are $S_i = |Y_i - \\hat{\\mu}(X_i)|$. The efficiency of this method is known to be sensitive to heteroscedasticity. Do you hypothesize that your core finding—the critical dependence on $(\\alpha^2n)^{-1}$—would also emerge in an efficiency analysis of this simpler residual-based method? \n\n3.  **Implications for Classification:** While your work is focused on regression, the fundamental insight about the trade-off between the required confidence level ($\\alpha$) and the training data size ($n$) seems universally important. Could you speculate on how this principle might manifest in the context of conformalized classification, where efficiency is measured by set size? For example, would the analysis involve the steepness of the model's softmax output distribution, and would a smaller $\\alpha$ similarly demand much more training data to reliably distinguish between classes with very close scores?\n\n4.  **Tightness of the Bound and Practical Data Allocation:** Your guidance on data allocation is derived by balancing the dominant terms in your theoretical upper bound. This is a powerful and practical contribution. However, since it is based on an upper bound, the tightness of the various terms matters. Could there be situations where the (often unknown) constant factors hidden in the Big-O notation are substantially different for the training and calibration error terms? If so, the truly optimal split might deviate from the one suggested by balancing the rates alone. Could you comment on the potential gap between the \"optimal split for the bound\" and the \"true optimal split\" in practice?\n\n5. **Practical Relevance of the $\\alpha = O(n^{-1/4})$ Regime:** Your analysis reveals a critical \"phase transition\" around $\\alpha = \\Omega(n^{-1/4})$. This is theoretically fascinating. However, in many practical applications, $\\alpha$ is a small, fixed constant (e.g., 0.1, 0.05) and does not scale with the sample size $n$. Could you comment on the practical settings where such a scaling of $\\alpha$ with $n$ might be relevant? For example, are there high-stakes domains (like autonomous systems or medical diagnostics) where practitioners might demand increasingly higher confidence levels as more data becomes available, thus creating a scenario where these extreme $\\alpha$ regimes and their efficiency implications are of practical concern? As I know, there is some literature about extreme error rate cases in conformal prediction. Maybe you should discuss them.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "*   **Originality:** The paper shows the first non-asymptotic efficiency analysis for conformalized regression that jointly models the impact of training size ($n$), calibration size ($m$), and the miscoverage level ($\\alpha$). The identification of the crucial $(\\alpha^2n)^{-1}$ term and the resulting \"phase transitions\" in convergence rates is an original theoretical contribution, moving beyond prior results that typically treated $\\alpha$ as a fixed constant.\n\n*   **Quality:** The technical quality of the work is very high. The theoretical analysis is rigorous, providing a complete, end-to-end derivation under clearly stated and relatively mild assumptions on the data distribution. The quality of the empirical validation is exceptional; experiments are designed to isolate and verify specific components and scaling behaviors predicted by the theory, which provides strong evidence for the claims.\n\n*   **Clarity:** The paper is written with outstanding clarity. The problem setup, main theoretical results, and their implications are presented in a logical and accessible manner. The section dedicated to \"phase transitions\" is particularly effective, translating a complex mathematical bound into intuitive regimes and actionable insights. The figures are well-designed and clearly illustrate the alignment between theory and empirical results.\n\n*   **Significance:** The work carries significant practical and theoretical implications. For practitioners, it offers the first concrete, theory-backed guidance on how to optimally allocate data between training and calibration based on the desired confidence level. For the research community, it fundamentally deepens the understanding of efficiency in conformal prediction, revealing a critical trade-off that was previously not formally understood. The framework is also potentially extensible to other optimizers and models, enhancing its long-term impact.", "weaknesses": "*   **Linear Model and Well-Specification Assumption:** The theoretical analysis is developed under the strong assumptions of a linear model class and well-specification. This limits the direct applicability of the derived rates to modern, often overparameterized, and non-linear models, such as neural networks. The paper would be strengthened by a discussion of the challenges in extending the analysis to these more complex settings, perhaps by outlining how model approximation errors would enter the final efficiency bounds.\n\n*   **Focus on Quantile-Based Methods Over Simpler Alternatives:** The analysis centers on CQR/CMR, which requires training quantile regressors via the pinball loss. A widely used and often simpler alternative is to train a single point predictor (e.g., for the conditional mean) and calibrate using the absolute residuals $|Y - \\hat{\\mu}(X)|$. The paper does not discuss how its theoretical findings, particularly the critical $(\\alpha^2n)^{-1}$ term, would apply to this common residual-based method. A comparison would be highly valuable, as the efficiency of the residual method often relies on different assumptions (e.g., homoscedasticity).\n\n*   **Exclusion of Classification Tasks:** The paper's scope is strictly limited to regression, with efficiency measured by interval length. An analogous and equally important problem exists in classification, where efficiency is measured by the size of the prediction set. The work misses an opportunity to discuss how its core insights—especially the critical role of $\\alpha$ in determining the required training data, might be applied to the classification setting, where the relationship between model confidence scores and the final set size is a key factor.\n\n*   **Empirical Validation Primarily on Synthetic Data:** The main experiments that provide the most direct validation of the theoretical scaling laws are conducted exclusively on synthetic data. While Appendix includes real-world experiments, they play a minimal role in the main paper and cannot be used to verify the precise rates since the oracle is unknown. Adding an experiment on synthetic data that mildly violates the assumptions (e.g., with slight non-linearity) would provide valuable insights into the robustness of the theory.", "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)