Self-Calibrating Conformal Prediction
Authors: Lars van der Laan
Ahmed Alaa
Venue: NeurIPS 2024 poster (2024)
Keywords: trustworthy machine learning
calibration
isotonic calibration
Venn-Abers predictors
conformal prediction
predictive inference
prediction intervals
Primary area: safety_in_machine_learning
TL;DR: We propose Self-Calibrating Conformal Prediction to provide calibrated point predictions and associated prediction intervals with valid coverage conditional on these model predictions in finite samples.

Abstract: In machine learning, model calibration and predictive inference are essential for producing reliable predictions and quantifying uncertainty to support decision-making. Recognizing the complementary roles of point and interval predictions, we introduce Self-Calibrating Conformal Prediction, a method that combines Venn-Abers calibration and conformal prediction to deliver calibrated point predictions alongside prediction intervals with finite-sample validity conditional on these predictions. To achieve this, we extend the original Venn-Abers procedure from binary classification to regression. Our theoretical framework supports analyzing conformal prediction methods that involve calibrating model predictions and subsequently constructing conditionally valid prediction intervals on the same data, where the conditioning set or conformity scores may depend on the calibrated predictions. Real-data experiments show that our method improves interval efficiency through model calibration and offers a practical alternative to feature-conditional validity.

Self-Calibratiin Conformal Prediction
Lars van der Laan University Washington lvdlaan edu
Ahmed M. Alaa UC Berkeley and UCSF amalaa@berkellyy edu
Abstract
In machine learning, model calibration and predictive inference are essential for producing reliable predictions and quantifying uncertainty to support decision- making. Recognizing the complementary roles of point and interval predictions, introduce Self- Delf-Calibratinng Conformal Prediction, method that combines Venn- Abers calibration and conformal prediction to deliver calibrated point predictions alongside prediction intervals with finite-sample validity conditional these pre- dictions. To achieve this, we extend the original Venn-Abers procedure from binary classification to regression. Our theoretical framework supports analyzing confor- mal prediction methods that involve calibrating model predictions and subsequently constructing conditionally valid prediction intervals on the same data, where the conditioning set conformity scores may depend on the calibrated predictions. Real-data experiments show that our method improves interval efficiency through model calibration and offers a practical alternative to feature-conditional validity.
Introduction
Particularly in safety-critical sectors, such as healthcare, it is important to ensure decisions inferred from machine learning models reliable, under minimal assumptions [38, 57, 11, 46, 56). As response, there is growing interest in predictive inference methods that quantify uncertainty in model predictions via prediction intervals [44, 28]. Conformal prediction (CP) is a popular, model-agnostic, and distribution-free framework for predictive inference, which can be applied post-hoc to any prediction pipeline [59, 49, 3, 34]. Given a prediction issued by a black-box model, CP outputs prediction interval that is guaranteed to contain the unseen outcome with user-specified probability [34]. However, limitation of CP that this prediction interval only provides valid coverage marginally, when averaged across all possible contexts with 'context' referring to the information available for decision decision-making. makingg Constructing informative prediction intervals that offer context- conditional coverage is generally unattainable without making additional distributional assumptions [58, 35, 4]. Consequently, there has been an upsurge in research developing CP methods that offer weaker, yet practically useful, notions of conditional validity; see, e.g., [43, 31, 47, 32, 23, 21].
In prediction settings, model calibration a desirable property of machine learning predictors that ensures that the predicted outcomes accurately reflect the true outcomes [39, 65, 66, 22j. Specifically, predictor calibrated for the outcome if the average outcome among individuals with identical predictions is close to their shared prediction value [25]. Such predictor is more robust against the over-or-undes estimation of the outcome extremes of predicted values. It also has the property that the best prediction of the outcome conditional on the model's prediction is the prediction itself, which facilitates transparent decision- decision-making [51]. There a rich literature studying post-hoc calibration of prediction algorithms using techniques such as Platt's 's scaling [45, 14), histogram binning [65, 25, 26], isotonic calibration [66, 40, 51], and Venn-Abers calibration [60].
Given the roles of both point and interval predictions in decision-making, we introduce a dual calibration objective that aims to construct (i) calibrated point predictions and (ii) associated prediction intervals with valid coverage conditional on these point predictions. Marrying model calibration and
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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predictive inference, we propose a solution to this objective that combines two post-hoc approaches Venn-Abers calibration [61, 60] and CP [59] - simultaneously provide point predictions and prediction intervals that achieve our dual objective in finite samples. In doing so, we extend the original Venn-Abers procedure from binary classification to the regression setting. Our theoretical and experimental results support the integration of model calibration into predictive inference methods to improve interval efficiency and interpretability.
2
Problem setup
2.1 Notation
We consider a standard regression setup in which the input X E X' C Rd corresponds to contextual information available for decision-making and the output Y E y C R is an outcome of interest. We assume that we have access to a calibration dataset C, = ((X,, Y)) comprising n i.i.d. data points drawn from an unknown distribution P PxPrIx. We assume access
black-box predictor f : X + V, obtained by training an ML model on a dataset that is independent of C.. Throughout this paper, we do not make any assumptions on the model f or the distribution P. For quantile level a E (0, 1), we denote the "pinball" quantile loss function la by to (J)), v) = 1(y 2 [()) aly [()) + 1(y [(x)) (1 (x) v)-
2.2 Conditional predictive inference and curse of dimensionality
Let (Xn+1, Yn+1) be a new data point drawn from P independently of the calibration data C., Our high-level aim is to develop predictive inference algorithm that constructs prediction interval (X,,,) around the point prediction issued by the black-box model, i.e., [(),++). For this prediction interval to be deemed valid, it should cover the true outcome Yn+1 with a probability 1 --a Conformal prediction (CP) is a method for predictive inference that can be applied in post-hoc fashion to any black-box model [59]. The vanilla CP procedure issues prediction intervals that satisly
the marginal coverage condition:
P(Y++1 E ((X+t)))2111
where the probability IP taken with respect to the randomness in C, and (Xx+1, 12+1). However, marginal coverage might lack utility in decision-makinn scenarios where decisions are context- dependent. A prediction band (z) achieving 95% coverage may exhibit arbitrarily poor coverage for specific contexts z. Ideally, we would like this coverage condition to hold for each context x X, i... the conventional notion of "conditional validity" requires
P(Y++1 E (X = x) 2 a
for all E X. However, previous work has shown that it is impossible to achieve (2) without distributional assumptions [58, 35, 4].
While context-conditiooaal validity as in (2) is generally unachievable, it is feasible to attain weaker forms of conditional validity. Given any finite set of groups and a grouping func- tion G X x y - G. Mondrian-CP offers coverage conditional on group membership, that is, P(Y.+1 E Cx, and context- conexx--onniitioal coverage, A multicalibration objective was introduced in [17] that seeks to satisfy, for all h in the property:
Yn+1) g)
a, Vg & [59, 47]. Expanding upon group-
class F of weighting functions (i.e., 'covariate shifts'),
E (((((x+))
11Y=+1 E (X.+1)3 = 0.
Gibbs et al. [21] proposed regularized CP framework for (approximately) achieving (3) that provides means to trade off the efficiency (i.e., width) of prediction intervals and the degree of conditional coverage achieved. However, Barber et al. [4] and Gibbs et al. [21] establish the existence of a "curse of dimensionality": as the dimension of the context increases, smaller classes of weighting functions must be considered to retain the same level of efficiency. For group-conditional coverage, this curse of dimensionality manifests in the size of the subgroup class G [4] via its VC dimension [55]. Thus, especially in data-rich contexts, prediction intervals with meaningful multicalibration guarantees over the context space may be too wide for decision decision-making.

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2.3 A dual calibration objective
In decision-making, both point predictions and prediction intervals play a role. For example, in scenarios with a low signal-to-noise ratio, prediction intervals may be too wide directly inform decision-makinn as their width is typically of the order of the standard deviation of the outcome. Point predictions might be used to guide decisions, while prediction intervals help quantify deviations of point predictions from unseen outcomes and assess the risk associated with these decisions.
Viewing the black-box model f as scalar dimension reduction of the context z; natural relaxation of the infeasible objective of context-conditional validity in (2) is prediciion-conditoon validity, (x (X)+1)) 2 1 - a. Prediction-conditee
validity ensures that the interval widths adapt to the outputs of the model f(-). so that the intervals can be reliably used to quantify the deviation of model predictions from unseen outcomes. Since prediction-conditions validity only requires coverage conditional on a one-dimensional random variable, it avoids the curse of dimensionality associated with context-conditione validity. In addition, as illustrated in our experiments in Section and Appendix C, when the heteroscedasticity (e.g., variance) in the outcome is function of its conditional mean, prediction-condition validity can closely approximate context-condittonal validity, so long as the predictor estimates the conditional mean the outcome sufficiently well.
i... P(Yn+1
Given the roles of both point and interval predictions in decision-making, we introduce novel dual calibration objective, self-calibration, that aims to construct (i) calibrated point predictions and (ii) associated prediction intervals with valid coverage conditional on these point predictions. Formally, given the model f and calibration data C. (X,+1), our objective is to post-hoc construct a calibrated point prediction fn+11 (X++1) and a compatible prediction interval Cn+1(Xn++ centered around 1 (Xn+1) 1) that satisfies the following desiderata:
(i) Perfectly Calibrated Point Prediction:
fat1 (Xa41) = B| [uaz(Xn41)),
(ii) Prediction-Condittonal Validity: P(Y++1 E (X)+) 2
Desideratum (i) states that the point prediction f should be perfectly calibrated or self- consistent [19] for the true outcome Y [37, 25, 51]. Is widely recognized that the calibration of model predictions is important to ensure their reliability. trustworthiness, and interpretability in decision-making [37, 65, 7, 24, 15]. Desideratum (i) also improves interval efficiency by ensuring C.+1(Xn+2) is centered around an unbiased prediction, meaning the interval's 's width is driven by outcome variation rather than by prediction bias. Desideratum (ii) is prediction interval variant of (i) that ensures the prediction interval ( calibrated with respect to the model fn+1, providing valid coverage for Y,+ within contexts with the same calibrated point prediction. We refer to predictive inference algorithm simultaneously satisfying (i) and (ii) as self-calibrating, as such procedure is automatically able to adapt miscalibration in the model f() due to, e.g., model misspecification or distribution shifts, ensuring that the interval is constructed from a calibrated predictive model.
Self-calibration can also be motivated by a decision- decision-making scenario where point predictions de- termine actions and prediction intervals used to apply these actions selectively. When point predictions are sufficient statistics for actions, self-calibration implies that the point and interval predictions accurate, average, within the subset of all contexts receiving the same prediction and, therefore, the same action.
3 Self-Calibrating Conformal Prediction
A key advantage of CP is that it can be applied post-hoc to any black-box model f without disrupt- ing its point predictions. However, desideratum (i) introduces a perfect calibration requirement for the point predictions of f. thereby interfering with the underlying model specification. In this section, we introduce Self-Calibrating CP (SC-CP), a modified version of CP that is self-calibrating in that satisfies (i) and (ii), while preserving all the favorable properties of CP, including its finite-sample validity and post-hoe applicability. Before describing our complete procedure Section 3.3, we provide background on point calibration and propose Venn-Abers calibration for regression.
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3.1 Preliminaries on point calibration
Following the framing of van der Laan et al. [51] (see also [25]). a point calibrator is a post-hoc procedure that aims learn a transformation 0, : + IR of the black-box model such that: (1) (n((((x++)) is well-calibrated for Yx+1 in the sense of Desideratum (i); and (2) 0₂ o is comparably predictive f. Condition (2) ensures that in the process of achieving (1), the quality of the model 1 Σ is not compromised, and excludes trivial calibrators such as 6, (f())) == n
2i=1 Y, [26]. To
our knowledge, this notion of calibration traces back to Mincer and Zarnowitz (1969) [39], who introduced the idea of regressing outcomes on predictions to achieve calibration in forecasting.
Commonly-employed point calibrators include Platt's scaling [45, 14), histogram (or quantile) binning [65], and isotonic calibration [66, 40). Mechanistically, these point calibrators learn 0, by regressing the outcomes {Yr);=1 on the model predictions {f(X))) Importantly, however, point calibration fundamentally differs from the regression task of learning E' [ as calibration can be achieved without smoothness assumptions, allowing for misspecification of the regression task [25]. Histogram binning is a simple and distribution- calibration procedure [25, 26] that learns 8,, via a histogram regression over finite nattoome-agnostee binning of the output space fX²). Isotonic calibration is an outcome- autcome-adaptive binning method that uses isotonic regression [5] learn 8, by minimizing the empirical mean square error over all 1D piece-wise constant, monotone nondecreasing transformations. Isotonic calibration is distribution-fres - it does not rely on monotonicity assumptions - and, contrast with histogram binning, it is tuning parameter- free and naturally preserves the mean-square error of the original predictor (as the identity transform monotonie) [51]. A key limitation of histogram binning and isotonic calibration is that their calibration guarantees are only approximate, and desideratum (i) only holds asymptotically.
3.2 Venn-Abers calibration
For binary classification, Vovk and Petej [60] proposed Venn-Abers calibration, which iterates isotonic calibration over imputations y e Y of the unseen outcome Yn+1 to provide calibrated multi- probabilistic predictions in finite samples. In this section, we generalize the Venn-Abers calibration procedure to regression, offering finite-sampi sample calibration guarantees for non-binary outcomes.
Let O/sn consist of all univariate, piecewise constant functions that are monotonically nondecreasing. Our Venn-Abers calibration procedure, outlined in Alg. 1, is derived from an oracle variant of isotonic calibration that provides a perfectly calibrated point prediction in finite samples, but requires knowledge of the true outcome Yn+1. Specifically, the Venn-Abers calibration algorithm iterates over imputed outcomes y E y for Yn+1 and applies isotonic calibration to the augmented dataset C. U ((xn1,, y)} to produce set of point predictions fr.x.41 (Xx+1) = (x((x++1.)) (X++1) y E } When the outcome space y is non-discrete, Alg. 1 may be infeasible to compute exactly and can be approximated by discretizing y. Nonetheless, the range of the Venn-Abers nulti-prediction can be feasibly computed as [f([ - (x); ff (x)) where [Vmin: Vmax]
r))) in light of the
min-max representation of isotonic regression [33].
Unlike point calibrators, Venn-Abers calibration generates a set of calibrated predictions for each context Xn+1, indexed by y E V. As we demonstrate later, this set prediction is guaranteed in finite samples to include perfectly calibrated point prediction, namely, the oracle prediction
x(xx+11Yo=11) (Xn+1) corresponding to the true outcome Y,+1. 1. Moreover, each prediction in the set, being obtained via isotonic calibration, still enjoys the same large-sample calibration guarantees as isotonic calibration [51]. By the stability of isotonic regression, as the size of the calibration set n increases, the width of this set of predictions rapidly narrows, eventually converging to a single, perfectly calibrated point prediction [60]. Venn-Abers calibration thus provides a measure of epistemic uncertainty by producing a range of values for perfectly calibrated point prediction. In cases with small sample sizes, standard isotonic calibration can overfit, leading poorly calibrated point predictions. When this overfitting occurs, the Venn-Abers set prediction widens, reflecting greater uncertainty in the perfectly calibrated point prediction within the set [30].
For the binary classification case, [60] derived (large-sample) calibrated point prediction from the Abens multi-predictio using shrinkage approach. We can similarly construct, for each x e X,

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Calibration Plot for SC-CP
Algorithm Venn-Abers Calibration
Frod sian Inteval
Input: Calibration data C, = [(X,, k.)) model f. context æ E X.
Cultrened Pretisian Diginal Pred stan
Predicted Outcame
1: for each y E y do
2: Set augmented dataset c(a.)) := Cn U [(x,
3: Apply isotonic calibration to using Ch. Cha,i))
ol argminggee,. Exechtio (Y 0 f(X:))²
fn :- gl..v) f.
4: end for Output: Multi-prediction ( (x) : E.).
03
13
28 25 15
Original Prediction (uncalibrated)
Higure Exxmppe SC-CP output
2001
Algorithm 2 Self- Calibraati Conformal Prediction
Input: Calibration data C., [(X,, Y.)) model f. context æ X, miscoverage level a E (0, 1)
1: for each y E y do
2: Obtain calibrated model (! by isotonic calibrating f on C. U {(x, y)} Alg.
3: Set self-calibrating conformity scores s(x.w) |Y, f(=-() (X.)), vi E [m] and S( = -y- f/ (x)):
4: Calculate 1 a empirical quantile pr pl (x) of conformity scores with same calibrated prediction as x: f(a.0 (X.) s/) 1
argmin 6E
Σ
lala, S(aw)) + la s
5: end for
(x) EE 7: Set Cu+i(u) == Ey: ly f (2a (x) S pm
6: Set fat1 = (1 (
(a>)) (x))).
Output: fat1(z) C conv()), (x)1(x) cy
a point prediction as follows:
(2.Smm))) (x) (2 (. (x)
Fn+1,a = In+1, (x)) Jn
mid
Un
f 1n-1, (z)}
(4)
Vmax Mmin
where [of (x) = 1/2 ( (x) + (le is the midpoint of the multi multi- prediction and V, : ET Y.. The behavior of (x) is natural; it shrinks the point prediction fmid f (x) towards the average outcome In (a well-calibrated prediction) proportional to how uncertain we (fla--max)) (2) - ((--)nia)))))) measures the sensitivity of isotonic regression to the addition of single data point to C.,. and a value closer to I corresponds higher degree of overfitting. In the extreme case where the calibration dataset is very large, we have f(= ((a-amae) (x) = p( (x), implying that (x) ze fm (z). Conversely, in the opposite extreme where the calibration dataset is very small and isotonic regression overfits, we have fl g/a.Vamx)) (x) Vmax and (((i)mim))) (a: x Ymin in which case F (x) A Vn We could replace Vn in (4) with any reference predictor, such as one calibrated using Platt's scaling or quantile-binning.
are in the calibration of fmid ( The ratio Vmaa Vyuim
3.3 Conformalizing Venn-Abers Calibration
In this section, propose Self- Self-Calibrating Conformal Prediction, which conformalizes the Venn- Abers calibration procedure to provide prediction intervals centered around the Venn-Abers multi- prediction that are self-calibrated in the sense of desiderata (i) and (ii).
simple, albeit naive, strategy for achieving (i) and (ii) without finite-sample guarantees involves using the dataset Cn to calibrate point predictions of f(-) through isotonic regression, and then constructing prediction intervals from the 1 a empirical quantiles of prediction errors within subgroups defined by unique values of the calibrated point predictions. To motivate our SC-CP algorithm, we introduce an infeasible variant of this seemingly naive procedure that is valid in finite samples, but can only be computed by an oracle that knows the unseen outcome Yn+1. In
5

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this oracle procedure, we compute a perfectly calibrated prediction fi == 0z+1 (((x+1)) by isotonic calibrating using the oracle-augmente calibration set {X,, Yi) where 0ā+1 argmingge... Ei (V, 0 (()()))))* Next, we compute the conformity scores S == 1
fā+1 (X,) as the absolute residuals of the calibrated predictions. An oracle prediction interval then given by I (X++1) := f=+ (Xx+1) + 1 where Pi++ (X++1) is the empirical 1 a quantile of conformity scores with calibrated predictions identical to X that is, scores in the set { 1 (X;) + i E [n + }]}. Importantly, isotonic regression, which is an auteoie-adaptive histogram binning method, ensures that the calibrated model fi+1 is piece-wise constant, with a sufficiently large number of observations averaged within each constant segment- typically on the order of 2/3 [16]. Consequently, the empirical quantile 1 is generally stable with relatively low variability across realizations of C,. our proofs, using the first-order conditions characterizing the optimizer 0n- and exchangeability, we show that f++ (Xn+1)
E| E[V++1lf2 (Xn41)]. so that desideratum (i) is satisfied. Furthermore, we establish that the interval Ch+i I (Xn-1) achieves desideratum (ii), i.e., P(Y.+1 E Ca+ (Xn+1) | (X4+1)) 2 1 To do so, our key insight is that PX+1(Xn,, corresponds to the evaluation of the function Pa+1 computed via
quantile regression as:
a+1
Pa+1 E
argmin 0of 0 ORR-R
to (0 (x
The first-order conditions characterizing the optimizer P7+1 combined with the exchangeability between Cn and ,,+1,Y,+1) can be used to show that C (X++)) is multi-calibrated against the R + IR} in the sense of (3). Using first-order conditions to establish the theoretical validity of conformal prediction was also applied by Gibbs et al. [21] to demonstrate the qulti-calibration of oracle prediction intervals obtained from quantile regression over a fixed class F. In our case, quantile regression is performed over a data-dependent function class Fn+,, learned from the calibration data, which introduces additional challenges in our proofs.
class of weighting functions Fn+1 := (0 1 0:
Our SC-CP method, which is outlined in Alg. 2, follows a similar procedure to the above oracle procedure. Since the new outcome Yn+1 is unobserved, we instead iterate the oracle procedure over all possible imputed values E Y for Yn+1. As in Alg. I., this yields a set of isotonic calibrated models fr.X0+1 = =Ife (x))) e E )), where (x+1(X.+) is the Venn-Abers multi-prediction of Y Then, for each E V and E [m], we define the self-calibrating conformity scores s ((X.111))
|Y; 1%-f% -__(X=+1.)) (X:) and s(Xx+1.2)) =- - -_-((x++1.) (X,+1) 1)], where the dependency of our scores on the imputed outcome y is akin to Full (or transductive) CP [59]. Our SC-CP interval is then
given by (X = (u E y/: s -(Xx1110) S Pnn ((x.t1.)) (XNIIi(Xx++))) (Xn+1)}, where pa the empirical
((x.11.)) (Xn+1) is ((xx=11.) (X,) fl. s(Xx+1.)) (Xn+1); i
quantile of the level set {S(X
[n + 1]}. By definition, C,+1 (Xa+11 covers Yu4I if, and only if, the oracle interval C7+1(Xat1) covers Y1+1, thereby inheriting the self celf-calibration of CX (Xn+1). Formally, Ca+1 (Xx+1) is a set, but it can be converted to an interval by taking its range, with little efficiency loss.
3.4 Computational considerations
The main computational cost of Alg. 1 and Alg. 2 is in the isotonic calibration step, executed for each y E V. Isotonic regression [5] can be scalably and efficiently computed using implementations of xgboost [12] for univariate regression trees with monotonicity constraints. Similar to Full (or transductive) CP [59]. Alg. 2 may be computationally infeasible for non-discrete outcomes, and can be approximated by iterating over a finite subset of y. In our implementation, we iterate over a grid of y and use linear interpolation to impute the threshold ph (z=1)) (x) and score s(r.v) for each y E y. As with Full CP and multicalibrated CP [21], Alg. and Alg. 2 must be separately applied for each context E X. The algorithms depend solely on x E X through its prediction fz), so we can approximate the outputs for all x E X' by running each algorithm for a finite number of r E X corresponding to a finite grid of the ID output space f(X' = (f(x) x E X C R. In addition, both algorithms are fully parallelizable across both the input context E X and the imputed outcome y E Y. In our implementation, we use nearest neighbor interpolation in the prediction space to impute outputs for each z e X. In our experiments with sample sizes ranging from 5000

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to 40000, quantile binning of both f(²) and y into 200 equal-freqeency bins enables execution of Alg. and Alg. 2 across all contexts in minutes with negligible approximation error.
4 Theoretical guarantees
In this section, under exchangcability of the data, we establish that the Venn-Abers multi-prediictinn Ju,N,+1 (X++1) == [fa (x (Xn+1) E )) and SC-CP interval (Xx+())+1) output by Alg. satisfy desiderata (i) and (ii) in finite samples and without distributional assumptions. Under iid condition, we further establish that, asymptotically, the Venn-Abers calibration step within the SC-CP algorithm results in better point predictions and, consequently, more efficient prediction intervals.
The following theorem establishes that the Venn-Abers multi-prediction is perfectly calibrated the sense of [61], containing perfectly calibrated point prediction of Yn+1 in finite samples.
C1) Exchangeability: {(X,, F=) are exchangeable.
C2) Finite second moment: Ep < <00
Theorem 4.1 (Perfect calibration of Venn-Abers multi-prediction). Under Conditions CI and C2, the Venn-Abers multi-prediction fn,X..1 (Xn+1) almost surely satisfies the condition
fa f(xa-11 (Xn+1) E[{n++ I ((( (Xn+1)).
Theorem 4.1 generalizes analogous result by [60] for the special case of binary classification. Even in this special case, our proof is novel and clucidates how Venn-Abers calibration uses exchangeability with the least-squares loss in a manner analogous to how CP uses exchangeability with the quantile loss [21].
The following theorem establishes desideratum (ii) for the interval Cn+1(Xn+11 with respect the oracle prediction f of Theorem 4.1. In what follows, let polylog(n) be given sequence that grows polynomially logarithmically in
C3) The conformity scores |Y; f( ((XN+1.Nnt) (X.)). Vi [n + 1]. are almost surely distinct.
C4) The number of constant segments for fn (x441 is at most n polylog(n).
Theorem 4.2 (Self-calibration of prediction interval). Under CI, holds almost surely that
P (Yn+1 E (Xn+1) | I foXnttiNet)) (Xnn)))))))()))) 2 1 -
If also C3 and C4 hold, then E/a - (Yats & On+1(Xn+1)) f Jo (Xn+)))))))))))))))))))))))
polvlog(w)
273
Theorem 4.2 says that (x+1(X satisfies desideratum (ii) with coverage that is, on average.
nearly exact up to factor (nlylog(n). Notably, the deviation error from exact coverage tends to 2//3
zero at a fast dimensionless rate and, therefore, does not suffer from "curse of dimensionality" Condition C3 is only required to establish the upper coverage bound and standard in CP see, e.g.. [36, 21]. Although it may fail for con-continuuus outcomes, this condition can be avoided by adding small amount of noise to all outcomes [36]. The constant segment number of 1/3 polylog(n) in C4 is motivated by the theoretical properties of isotonic regression; Assuming C2 and continuous differentiability of t + Ep|Y f(X) = t). it is shown in [16] that the number of observations in given constant segment of an isotonic regression solution concentrates in probability around 2/3 In general, without C4, our proof establishes a miscoverage upper bound of + [[[N+1) where
E[Na+1] is the expected number of constant segments of ((Xx+1)).11))
The next theorem examines the interaction between calibration and CP within SC-CP in terms of efficiency of the self-calibrating conformity scores. In the following, let a E X, E be arbitrary. For each E tiso. define the d-transformed conformity scoring function Sp : (a', .yy))) + |v/ 0 f(z'). Let 0p ;= argminuge f [Sp(x', " yy) be the optimal isotonic transformation of f(-) that minimizes the population mean-square error. Define the self-calibrating conformity scoring
function ((.,,)(2'') (z'.v') == l// py- (z')), where ((x.)) is obtained as in Alg. 1.
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C5) Independent data: {(X,, KI) are iid. C6) Bounded outcomes: y is uniformly bounded set.
Theorem 4.3. Under C5 and C6, we have f(Sh ( (x', v/) Sa, //)) Op(n O/(n.
The above theorem indicates that the self-calibrating scoring function s( used Alg. asymptot- ically converges in mean--quare error to the oracle scoring function Sg, at a rate of n 2/3 Since the oracle scoring function Se, corresponds to a model On f with better mean square error than ʃ. we heuristically expect that the Venn-Abers scoring function s!*00) will translate narrower CP intervals, at least asymptotically. We provide experimental evidence for this heuristic in Section 5.
Limitations. The perfectly calibrated prediction f1 (X.41), guaranteed to lie by Theorem 4.1 in the Venn-Abers multi-p multi-predion prediction, typically cannot be determined precisely without knowledge of Yn- However, the stability of isotonic regression implies that the width of multi-prediction (x+1(X)+1) shrinks towards zero very quickly as the size of the calibration set increases [10]. Moreover, the large-sample theory for isotonic calibration in [51] demonstrates that the fi-calibration error of each model S4 ((*=11.2) with y E V is Op(n 2/3).. One caveat of SC-CP intervals is that desideratum (ii) satisfied with respect to the unknown, oracle point prediction fl x((4411)a+1)) (( (X.+1))
However, we know that this oracle prediction lies within the Venn-Abers multi- multi-prediction by Theorem 4.1, and its value can determined with high precision with relatively small calibration sets [60] - see, c.g., Figure 1. These limitations appear to be unavoidable as perfectly calibrated point predictions can generally not be constructed in finite samples without oracle knowledge [61, 60].
4.1 Related work
The work of [42] proposes a regression extension of Venn-Abers calibration that differs from ours, both algorithmically and in its objective. While our extension constructs a calibrated point prediction s(X) of Y such that ((X) = E ((x)], their approach uses the original Venn-Abers calibration procedure of [60] construct distributional prediction f((X) of (Y () that satisfies f((X) = P s (x)) for t e V.
The impossibility results of [25] imply that any universal procedure providing prediction-conditionall calibrated intervals must explicitly or implicitly discretize the output of the model f(.). The works of [31] and [29] apply Mondrian CP [59] within leaves of regression tree f to construct prediction intervals with prediction-comditional validity. However, this approach is restricted to tree- based predictors and does not guarantee calibrated point predictions and self-calibrated intervals. Mondrian conformal predictive distributions were applied within bins of model predictions in [9] to satisfy coarser, distributional form of prediction-conditioaal validity. A limitation of Mondrian-CP approaches to prediction-conditiona validity is that they require pre-specification of a binning scheme for the predictor f(-). which introduces trade-off between model performance and the width of prediction intervals, and they do not perform point calibration (desideratum (i)) and, thereby, do not guarantee self-calibration. In contrast, SC-CP data-adaptively discretizes the predictor f(-) using isotonic calibration and, in doing so, provides calibrated predictions, improved conformity scores, and celf-calibrated intervals.
Other notions of conditional validity have been proposed that, like seff-calibration, avoid the curse of dimensionality of context-conditional validity. In the multiclassi- fication setup, [50] and [18] use Mondrian CP to provide prediction intervals with valid coverage conditional on the class label (i.e., outcome). In [8], Mondrian CP is applied within bins categorized by context- -specxt- difficulty estimates, such as conditional variance estimates. Multivalid-CP [32, 6] offers coverage based threshold defining the prediction interval. For multiclassification,, [41] propose a procedure for attaining valid coverage conditional on the prediction set size [2].
validity and
Real-Data Experiments: predicting utilization of medical services
5.1 Experimental setup
In this experiment, we illustrate how prediction-conditonn validity can approximate context- conditional validity when the heteroscedasticity in outcomes is strongly associated with model
8

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predictions, thereby ensuring validity across critical subgroups without their pre- pre- sre-specffication.. We analyze the Medical Expenditure Panel Survey (MEPS) dataset [1], supplied by the Agency for Health- care Research and Quality [13], which was used in [47] for Mondrian CP with fairness applications. We use the preprocessed dataset acquired using the Python package cqr, also associated with [47]. This dataset contains n = 15, 656 observations and d = 139 features, and includes information such as age, marital status, race, and poverty status, alongside medical service utilization. Our objective is to predict each individual's 's healthcare system utilization, represented by score that reflects visits to doctors' offices, hospital visits, etc. Following [47], we designate race as the sensitive attribute A, aiming for equalized coverage, where A 0 represents non-white individuals (no- 96400 and A = 1 represents white individuals (n1 = 6016). The outcome variable Y is transformed by Y log(1 + utilization score) to address the skewness of the raw score. In Appendox B, we present additional experimental results for the Concrete, Community, STAR, Bike, and Bio datasets used in [48] and publicly available in the Python package cqr, associated with [48] and [47].
We randomly partition the dataset into three segments: a training set (50%) for model training, a calibration set (30%) for CP, and test set (20%) for evaluation. For training the initial model f()), we use the xgboost [12] implementation of gradient boosted regression trees [20], where maximum tree depth, boosting rounds, and learning rate are tuned using 5-fold cross-validation. We consider two settings for training the model. In Setting A, we train the initial model on the untransformed outcomes and then transform the predictions as ÿ + log(1 (1 + D). which makes the model predictive but poorly calibrated because overestimates the true outcomes, in light of Jensen's 's inequality. In Setting B, train the initial model on the transformed outcomes, leading to fairly well-cu well-calibrated predictions. In both settings, calibration and evaluation are applied to the transformed outcomes.
For direct comparison, we compare SC-CP with baselines that leverage the standard absolute residual scoring function S(x, y) := ly - [x)] and target either marginal validity or prediction-conditinn validity. The baselines are: Marginal CP [34], Mondrian CP with categories defined by bins of model predictions [59, 9], CQR [48] with model predictions used as features, and the Kernel- smoothed conditional CP approach of [21] with model predictions { used as features and bandwidth tuned with cross- validatii..... Due to the slow computing time of the implementation provided by [21]. we apply Kernel on subset of the calibration data of size neal 500. SC-CP is implemented as described in Alg. 2, using isotonic regression constrained to have at least 20 observations averaged within each constant segment to mitigate overfitting (via the minimum child weight argument of xgboost). The miscoverage level is taken to be a = 0.1. SC-CP provides calibrated point predictions and self-calibrated intervals, while the Mondrian and Kernel baselines offer approximate prediction-condiiie coverage. We report empirical coverage, average interval width, and calibration error of model predictions in the test set within the sensitive attribute. Calibration error is defined as the mean error of the point predictions, Ely Y [A]] within the sensitive attribute A, which measures model over- or under-confidence. For SC-CP, we use the calibrated point predictions from (4), while the original point predictions are used for Marginal, Mondrian, and Kernel. For CQR, we use an estimate of the conditional median, obtained from a separate xgboost quantile regression model, as the point prediction. We note that, since quantiles are preserved under monotone transformations of the outcome, we expect the conditional median model of CQR to be well-calibrated, at least in median sense, in both Setting A and Setting B. We include the baseline Mondrian" for direct comparison with SC-CP, in which Mondrian is applied with the same number of prediction bins as data-adaptively selected by SC-CP.
validity, and Marginal and CQR only guarantee marginal
5.2 Results and discussion
The experimental results for each setting are depicted in Figure Each panel's left-most plot showcases a calibration plot [62] for SC-CP, illustrating original and calibrated predictions alongside prediction bands. On the right, the panels display prediction bands of our baselines as function of the original model predictions. Visually, as expected by Theorem 4.2, the SC-CP bands adapt to outcome heteroscedastictty within model predictions, while Marginal lacks adaptation, Mondrian under-adapt due to insufficient bins, and Kernel adapts but offers wider intervals for large predictions where observations are sparse. The bands of CQR appear adaptive and similar to those of SC-CP, however, do not gaurnatee finite-sample prediction-condition validity. The calibration plots reveal that heteroscedasticity in outcomes is primarily driven by their mean, suggesting that prediction- conditional validity may approximate
validity. This heuristic is supported by the

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(h) Setting B (well-calibrated f(-))
Figure 2: MEPS-21 dataset:
Calibration plot for SC-CP, prediction bands for SC-CP and baselines, and empirical coverage, width, and calibration error within sensitive subgroup.
tables in Figure 2, which display empirical coverage, average interval width, and calibration error within the sensitive attribute (A) for all methods. In Setting A, the base regression model f(.) are poorly calibrated, i.e., R[f(X) Y A] is not close to O, resulting in wider intervals, overconfidence in point predictions, and decreased interpretability for the baselines, as their intervals center around biased point predictions. In contrast, being self-calibrated, SC-CP corrects the calibration error in f. achieving the smallest interval widths and well-calibrated point predictions in both settings, as guaranteed by Theorem 4.1. In both settings, the quantile regression model of CQR appears to have worse calibration than SC-CP, which may be due to the median differing from the mean because of the skewness of the outcomes. Additionally, SC-CP predictions achieves a smaller difference in calibration eтror between the two subgroups than Marginal, Mondrian, and Kernel, suggesting they are less discriminatory and more fair [47]. SC-CP and Kernel achieve the desired coverage level of 1 a 0.9 in each subgroup and setting, whereas Marginal exhibits over- or under-coverage in each subgroup. Mondrian tends to under-cover with 5 and 10 bins and only attains good coverage when using the same binning number Lata-addppivvly selected by SC-CP, highlighting its sensitivity to the pre-specified binning scheme. CQR attains good coverage in the A = 0 group but undercovers in the = group, which may be explained by CQR only guaranteeing marginal coverage in finite samples. Even with SC-CP having higher coverage, the intervals of SC-CP are narrower than those of Kernel and Mondrian". This provides experimental evidence that calibration improves conformity scores and translates into greater interval efficiency, suggested by Theorem 4.3.
6 Extensions
Our theoretical techniques can be used to analyze conformal prediction methods that involve the calibration of model predictions followed by the construction of conditionally valid prediction intervals. Our analysis can be adapted to the general case where either the conformity score or the conditioning variable depends on the calibrated model prediction. While we use the absolute residual conformity score in our work. SC-CP can be applied to other conformity scores, such as the normalized absolute residual scoring function [43], allowing for the inclusion of context-specific difficulty estimates in the SC-CP procedure. Although we use Venn- Abers calibration in SC-CP, our analysis also applies to other binning calibration methods, such as Venn calibration [61, 60). Thus, we can replace the isotonic calibration step in Alg. I and 2, for example, with histogram binning [25]. Additionally, a group-vale -ooup form of SC-CP can be achieved by applying Alg. 2 separately within subgroups, similar to Multivalid CP [32]. Interesting areas for future work involve integrating point calibration with conformal prediction methods for predictive models beyond regression, such as the isotonic calibration of quantile predictions in conformal quantile regression [48].
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A Code
The methods implemented in this paper are not computationally intensive and were run in Jupyter notebook environment on a MacBook Pro with 16GB RAM and an M1 chip. A Python im- plementation of SC-CP is provided in the package Selrcalibratinnooofooaaa pip. Code implementing SC-CP and reproducing our experiments is available in the GitHub repository SelfCalibrating ormal, which can be accessed at the following link: https: //github.comm www.//aaswaanrrrllaaao..ooo http://awaaadderraaaaaaaaaaaaaaa Stt////effaaaaaaaa /araa//aaaaabbaaanooooooooooooe ormal.
available via
B Supplementary real data experiments
B.1 Additional results
In this section, we present the experimental results for the concrete, STAR, bike, community, and bio datasets used in [48] and publicly available in the Python package cqr, associated with [48] and [47]. For the STAR dataset, the sensitive attribute A was set to "gender." For the Bike dataset, the sensitive attribute A was set to "workingday." and for Community, it was set to "race binary." For the remaining datasets, the sensitive attribute A set dichotomization of the final column in the feature matrix, as above or below its median value.
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1-1
A-0
Dilleranx
Stepeed
0.35
0332
0.33
0.50
-0.0.3
-300001
-000311
Magunal
0337
0.00
0.315
0.223
-03543
400130
-U,253
oraggeD
0.568
0.399
0556
0:551
-0.0.3
3.00662
.0.00368
Mandina biee)
6355
0.506
0.588
0.53
-0.35-3
-091.9
.0.5990
Mondtaa ribino
01963
01800
0761
0.011
<1.0000
Mendrox: best
2.90
11394
0.911
40.593
-00193
41.2203
3.00692
Kand
0.553
0317
0.79
0.532
-001.0
-0.00002
-0.03368
Kanal
6352
0.75
0.585
0.422
.0.3543
-00130
-0,0990
0.887
0.801
0377
0.63
SC-CP
0.00
11856
01545
(b) Setting B (well-calibrated f())) Figure 6: Concrete dataset: Calibration plot for SC-CP, prediction bands for SC-CP and baselines,
(a) Setting A (poorly-calibrated f(-))
and empirical coverage, width, and calibration eтror within sensitive subgroup.
Calibration Plot for SC-CP
Mag
COR Smarkka
Calibration Plat for SC.CP
FOR DroNEbii
Predecte outcom
Pandfaan 20
Card
Original Prediction (nncaibrated)
Original Prediction (uncalirated)
Mellie
Concraat
Arcrage
1-0
mm]
2-00
Difforence
Liffienncr
Nlagina
1330
10324
1.31
1.3
3.18
D1153
WE01
Teepas
1131
1381
12h
1.3
1200117
2010)
(roeptual
Mondiaa tano
0.509
11215
1.33
1.54
3.16%
111.3
Montrua tine
13553
D302
1150322
-01191
Manahia (80kna)
0.501
11310
1.50
1.75
5.10
01155
0085
Monkina b0ot
0.853
$9005
1,40
1.59
0.00227
40001
802157
Kendl
0.92
0.981
1.53
1,53
1.6
0.133
0.655
Kanal
0.580
0.855
1.48
0.08227
-6019)
0/22137
SC-PP
0353
0.963
1.25
1.56
40091
0.03
400221
SC-CP
0.853
0906
1.58
1.52
4.0174
0.0673
.00213
(b) Setting B (well-calibrated f(-) Figure 7: Bio dataset: Calibration plot for SC-CP, prediction bands for SC-CP and baselines, and empirical coverage, width, and calibration error within sensitive subgroup.
(a) Setting A (poorly-calibrated f(-))
C Supplementary synthetic data experiments
C.1 Experimental setup
In this appendix, we perform additional synthetic experiments to evaluate the prediction-condiiional validity of our method and how translates to approximate context-conditional validity in certain cases.
16

---

Synthetic datasets. We construct synthetic training, calibration, and test datasets Dirain. Deat, Drest of sizes Tetraia Beal Ritests which are respectively used train f. apply CP. and evaluate performance. For parameters d E N, K. > 0,a 2 0, b 2 O, each dataset consists of iid observations of (X, Y) drawn as follows. The covariate vector X := (. Xa) E [0, 1] is coordinate-wis wise independently drawn from Beta x) distribution with shape parameter k. Then, conditionally on X I, the out- come is drawn normally distributed with conditional mean u(z) == d 1/2 -ー=11 {az 3 + sin(4z,))
0.02) /2)² Here, a and control the heteroscedasticcty and mecn-varianed relationship for the outcomes. For Deal and Diest- set Keal Keest 1 and, for Dirain. vary Ktrain to introduce distribution shift and, thereby, calibration error in f. The parameters d, a, and are fixed across the datasets.
and conditional variance (z) == f0.033 log + 08 + +b (1vo(z))/220
Baselines. To mitigate overfitting, we implement SC-CP so that each function in Ois has at least 20 observations averaged within each constant segment (implemented using the minimum leaf node size of argument xgboost). When appropriate, will consider the following baseline CP algorithms for comparison. Unless stated otherwise, for all baselines, we use the scoring function S(x, y) := by [=))). The first baseline, uncond-CP, is split-CP [36], which provides only unconditional coverage guarantees. The second baseline, cond-CP, is adapted from [21] and provides conditional coverage over distribution shifts within a specified reproducing kernel Hilbert space. Following Section 5.1 of [21]. we use the Gaussian kernel K (X,,x, == <sp(-4|X; -1,,), with euclidean norm || ll2 and select the regularization parameter À using 5-fold voos--valiittion. The third baseline, Mondrian CP, applies the Mondrian CP method [59, 8] to categories formed by dividing f's predictions into 20 equal-frequency bins based on Doat. As an optimal benchmark, we consider the oracle satisfying (2).
C.2 Experiment 1: Calibration and efficiency
In this experiment, we illustrate how calibration of the predictor / can improve the efficiency (i.e., width) of the resulting prediction intervals. We consider the data-generating process of the previous section, with Vetrain = Neaa Ritest = 1000, d = 5, a = 0, and b = 0.6. The predictor f is trained on Dtrain using the ranger [63] implementation of random forests with default settings. To control the calibration error in f, we vary the distribution shift parameter Ktroin for Dirain over {1, 2,,,,,,,,
Results. Figure 8a compares the average interval width across Diest for SC-CP and baselines as the e-calibration error in f increases. Here, we estimate the calibration error using the approach of [64]. As calibration error increases, the average interval width for SC-SP appears smaller than those of uncond-CP, Mondrian-CPP, and cond-CP, especially in comparison uncond-CP. The observed efficiency gains in SC-CP are consistent with Theorem 4.3 and provides empirical evidence that seff-calibrated conformity scores translate tighter prediction intervals, given sufficient data. To test this hypothesis under controlled conditions, we compare the widths of prediction intervals obtained using vanilla (unconditional) CP for two scoring functions: S (x, + ly fx)) and the Venn-
Abers (worst-case) scoring function Seal (xr, y) + maxyey ly _((20) (x))- For miscoverage levels {0.05, 0.1, 0.2), the left panel of Figure 8b illustrates the relative efficiency gain achieved by using Seal, which we define as the ratio of the average interval widths for Seal relative to S. The widths and calibration errors Figure 8b are averaged across 100 data replicates.
Role of calibration set size. With too small calibration sets, the isotonic calibration step in Alg. can lead to overfitting. In such cases, the their uncalibrated counterparts, potentially resulting in less efficient prediction intervals. In our experiments, overfitting is mitigated by constraining the minimum size of the leaf node in the isotonic regression tree to 20. For E {0.05, 0, 0.2). the right panel of Figure 8b displays the relationship between Neat and the relative efficiency gain achieved by using Scat, holding calibration error fixed (Kerain = 3). We find that calibration leads to a noticeable reduction in interval width as soon as Neal 2 50.
conformity scores could be larger than
C.3 Experiment 2: Coverage and Adaptivity
In this experiment, we illustrate how self-calibration of (X++)))))) can, some cases, translate to stronger conditional coverage guarantees. Here we take Atrain Neat 1000 and Neett 2500,
17

---

Intovval Matea/Nsshi te =0.11
SC-CP (ouns)
12
tond-CP
nelabvv (caltmoat))
Lovel
1.0-
Mondrian-CP
0.5
Dracle
0.225
5.50
0.05 Colbration omor in
D075
0.193
12
120
Calibration
Calibratin setsiee
(a) Avg. interval width with varying calibration error.
(b) Relative efficiency change from calibration.
Figure Figure 8a shows the average interval widths for varying callibratio errors in f. Figure 8b shows the relative change in average interval width using Venn-Abers calibrated versus uncalibrated predictions, with varying e (a-calibration error in f (left), and calibration size (right). Below the horizontal lines signifies efficiency gains for SC-CP.
and no distribution shift (Kiraii = 1) in Dtrain- We consider three setups: Setup A (d =d= 5, 0, 0.6) and Setup B (d 5. 0, 0.6) which have strong mean-variance relationship in the outcome process; and Setup C (d = 5, a = 0.6, = 0) which has no such relationship. We obtain the predictor from Derain using generalized additive model [27], that it is well calibrated and accurately estimates To assess the adaptivity of SC-CP to heteroscedasticity. we report the coverage and the average interval width within subgroups of Dreat defined by quintiles of the conditional variances {o²(X,) (X,, Y,) E Dresth.
Setup A Id-51
3.50-
1.
235-
2.8
115
2.74
130
07>
6''
0.1
0'3
a'suwival
Setup B (d-20)
Covrrge- (2 -0.3)
Intenvll Witm
2
805
0':
0.1
07
0/3
o'sMoieol
Setup C (adverserial d=5)
3.50
3.0-
180
6.
02
c'3
dA
04 d' oa c'suairul
Quittles ox))
CE-
SC CP luss
Cord
Urcand C+
Mondion CF
Clock
Coloritt
SC-CP (ous) vo..
Cord-CP
Uncond-CP
Orace
(a) Coverage and interval width for o² (X) quintiles
(b) Adaptivity of SC-CP with Deal
Figure 9: Figure 9a displays the coverage and average interval width of SC-CP, marginally and within quintiles the conditional outcome variance for setups A, B, and C. For a example, Figure 9b shows the adaptivity of SC-CP prediction bands (a 0.1) across various calibration set sizes.
18

---

Results. The top two panels in Figure 9a displays the coverage and average interval width results for Setup A and Setup B. In both setups, SC-CP, cond- and Mondrian- exhibit satisfactory coverage both marginally and within the quintile subgroups. In contrast, while uncond-CP attains the nominal level of marginal coverage, exhibits noticeable overcoverage within the first three quintiles and significant undercoverage within the fifth quintile. The satisfactory performance of SC-CP with respect to heteroscedasticity Setups A and B can be attributed to the strong mean-vale variance relationship in the outcome process. Regarding efficiency, the average interval widths of SC-CP are competitive with those of prediction-bel binnneed Mondrian-CP and the oracle intervals. The interval wider than those for SC-CP and Mondrian-CL. especially for Setup B. This
widths for cond-CP difference can be explained by cond-CP aiming for conditional validity in a 5D and 20D space, whereas SC-CP and Mondrian-CL target the ID output space.
Limitation. there is no mean- variance relationship in the outcomes, SC-CP generally not expected to adapt to heteroscedasticity. The third (hottom) panel of Figure 9a displays SC-CP's performance in Setup C, where there is no such relationship. this scenario, evident that the conditional coverage of both uncond-CP and SC-CP are poor, while cond-CP maintains adequate coverage.
Adaptivity and calibration set size. SC-CP can be derived by applying CP within subgroups defined by data-dependent binning of the output space f(X), learned Venn-Abers calibration, where the number of bins can grow with Real- In particular, if + E[y f(X) t] is asymptotically monotone, which is plausible when f consistently estimates the outcome regression, then the number of bins selected by isotonic calibration will generally increase with Neal, and the width of these bins will tend zero. As consequence, in such cases, the self-calibration result in Theorem 4.2 translates to conditional guarantees over finer partitions of f(X) Teal increases. For d = 1 and a strong mean-variance relationship (a = 0, b = 0.6), Figure 9b demonstrates how the adaptivity of SC-CP bands improves neal increases. In this case, for Real sufficiently large, we find that the SC-CP bands closely match those of cond-CP and the oracle.
0.92
SC-CP (aurs)
Coverage (a =0.1)
cond-CP
uncond-CP
0.90
Mondrian-CP
Oracle
0.88
0.050
0.075
0.100 Calibration emrrr in f(])
0.125
(a) Avg. interval width with varying calibration error.
Figure 10: Figure 10a shows the marginal coverage corresponding to the interval widths of Figure 8a for varying e calibratioo emrors in f-
D Proofs
Proof of Theorem 4.1. I. It can be verified that an isotonic calibrator class ORoo satisfies the following properties: (a) It consists of univariate regression trees (i.e., piecewise constant functions) that are monotonically nondecreasing: (b) For all elements 0 E O and transformations y : + RR it holds that g o E Ora Property (a) holds by definition. Property (b) satisfied since constraining the maximum number of constant segments k(n) does not constrain the possible values that the
19

---

function may take in a given constant region. We note that the result of this theorem holds in general for any function class is that satisfies (a) and (b).
Recall from Alg. that fn ((xu+1)N+11) = 00 ((xtt)))) o f. where ofxu+1)0n+1) E +iso is an isotonic
calibrator satisfying fn s(Xn+t)/441) 0% (( f.. By definition, recall that the isotonic calibrator class Seo satisfies the invariance property that, for all g R - E, the inclusion 0 E ② Oiso implies g 0 E iso Hence, for all g : I - lK and & > 0, it also holds that (1 + Eg) (...111 1) lies in
issn- Now, we use that (1 eg) ( of = (1+ o [(Xx+tiNn++)) and that f ((4441. is an empirical risk minimizer over the class {g ʃ g E Oioo). Using these two observations, the first order derivative equations characterizing the empirical risk minimizer f/ (x44)))a41) imply, for all g: IR + R, that
Σ gg N (Y:- -/ 1 (xi)]
1+1
d
Y, (1 Eg) f(x No+1) 2
de
n
0.
(5)
Taking expectations of both sides of the above display, we conclude
a+1
Σ
B (g fl ((t)))
(xi (xi- f (x.))
(6)
We now use the fact that {(x,, Yi, ff (n (X.) : i E [n + 1B} are exchangeable, since f(X;, D are exchangeable by C1 and the function fn (((0+11Nat1) is invariant under permutations of ((x, ) Consequently, Equation (6) remains true if we replace each (X,, Yu J1 (( (X,)) (X,)) with E [n] by (Xn+1, Y f M That is.
(g f(Xn+t (X (X { (Yn+1- f s(unssiaat (X
n+1
(g o fl (a1)) Yru1 f( (o }
n +
(g (X ( {y; f (1
0.
By the law of iterated conditional expectations, the preceding display further implies
E (gg o ((xnttnY+))))Xxx.1 [[Yata ff( (((n+uya+t)) (xaet+(((+))))) (Xat1)] f(x ((Xatt. Ya+1)
(Xnti))] (ott) 0.
Taking g R- + R to be defined by (go fi ((Xnt (Xn+1) = E[Yn+1 1. (oNnnttaats) (Xn+t)]- fl ((Xa=1) (Xn+1), we find
(Xx+1)))))))))
BlYa4i | ((tnntnatt))XXam.. (Xn+1)) - 1(x (X 2²1 =0.
E
The above equality implies E[Yn41 ((Kass (Xn+1)) fu ( as desired.
(Xn-1) almost surely,
Proof of Theorem 4.2. Recall, for a quantile level a E (0, 1), the "pinball" quantile loss function lo given by
(1- ()())), fa(s ( - s) if < f(z).
if
As established in [21], each of of given by:
in the direction g, for some B E (a 1,
Gapa (f(()) + tg(()), 8)) I-z0 == 1(f(x))))))))))))(()))))))))11)_ sa 1(f < s)) + +(f(x) = 8))g(z).
20

---

For (z,z)= E X' x y/ define the empirical risk minimizer:
(( E
argmin ( -.B-RR
la (0 f(2.)) (X,), s(awi)) + (0 (x), s
Then, since the isotonic calibrated predictor fn <(2.9) is piece-wise constant and the above optimization problem unconstrained in the map R R. holds that the evaluation pn (2..0 (ax) lies in the solution set:
argmin /ER
K(((D 2.) la (q, S(a.1))) la ((11))
(((m))) where K2 K,(h 2) = 1
fn ((. (X,) (/ (z)) Consequently, we see that the empirical quantile
Ph (r (x) defined Alg. 2 coincides with the evaluation of the empirical risk minimizer ph (( (-) at a, suggested by our notation.
We will now theoretically analyze Alg. by studying the empirical risk minimizer x' Pn (Xn=1=Ytt1)2z)) To do so, we modify the arguments used to establish Theorem 2 of [21]. As
[21], we begin by examining the first-order equations the convex optimization problem defining ( (x').
x' + Pm
Studying first-order equations of convex problem: Given any transformation 0: IR R, each subgradient of the map
E e la (phk (X,) + EB fl Nat1) (Xi). S ((Enta)utt)
is, for some B E ja 1,a of the following form:
1.2,11) depal {ea (pl (phNntiiitt) (Xi) =0 f(xx-11.x11)) (X,), sf s(xmaniatti) le=0
0o fl (l (x 1 (ph ( (x,) < of s(anan Bi ph
if s(XaniNaas)) * Pn (nn11) (Xi),
if of = Pn Xx 1: (1111)1
(X,, (Xi).
Now, since On (Nn+z.Na41) is empirical risk minimizer of the quantile loss over the class {0 fa ((Xx-11 that
+ R}. there exists some vector 3 = (Si...., Bh+1) E [a - L,a]"t1 such
n+
u1s (f Nn41 # ph (e 10 f(x N xx - (n
(Xi) < S s((ntsntatill)
n1
i (x.11111 = = (o
13 ( ((xan)
The above display can rewritten as:
o p P( (X:)(a 10ph (x))).) < s s((nnii)ots))))
n+1 n
1 Is ( -nt [1 (1-871) (1-
(0) f( knt)))
Now, observe that the collection of random variables
(f(( (X,), S( ((..111.411 1) , ( : : (m [m+ + 11}
are exchangeable, since {(X,,, Yi) : i E [n + 1}} are exchangeable by and the func- tions <(X (x+1)) and p (xxo1x(t1)()) are unchanged under permutations of the training data
21

---

f(,, YDe Thus, the expectation of the left-hand side of Equation (7) can be expressed as:
0 f( (X;) (o.N)) < S s((un1
E
(X.)fa (xth)) < S s/Nnin)anr)l]]
0
n
= E 0 ( (Xn+1 (a - ( (Xn+1) < Sttl ((
where the final inequality follows from exchangeability. Combining this with Equation (7), we find
E 0 fl ((
(Xtetttt(Xnnnnnn (xn+1)fa 1 (pp (x (Xx+1) < S ( }}]
(8)
E HIs ( Pi
(Xn+1 ( (X:)
[n1 (1- (1-3))(0- Sa (xx
(9)
Lower bound on coverage: We first obtain the lower coverage bound in the theorem statement. Note, for any nonnegative 0: R - R., that (9) implies:
E
f/ P.X-- (xa+1) (pl (X Nntil < S (
2
x+1
This inequality holds since (1 - 3 2 and (0 o Nn (Xi) 2 almost surely, for each [n + 1).
i E
By the law of iterated expectations, we then have
E 0o 11. 11) (Xn+1) {a - IP (p (1 < Sat1 s( | f( 11)) H 2 0.
(Xn+1)
Taking 0: RR + R as a nonnegative map that almost surely satisfies
0oʃ fof (aarn (X = =1 1f S P (hh (o < s s(fnainKat)) p (( (Xnti)) (X
we find
E {a-PP (pfpontts. (N (Xn+1) 141
((Xm11,Xu12)
S f(XnttXa+t1) (X
{ ] 20,
where the map L {1}- = 14111= < 0) extracts the negative part of its input. Multiplying both sides of the previous inequality by -1, 1, we obtain
0 s B (a-aa-P(p
PP (plKntii (Xn+1) < s41l
aa-
((4111, ((Xntt (Xan)) S..
We conclude that the negative part of { a - PP (pr (((x111))11)) (X < S ((X1112)a11)) | sf (Xntt)) is almost surely zero. Thus, it must be almost surely true that
(Xtt))
a 2 PP (ott
(Xn+1)= < n+1
((x-11)8m-1) 1) | ((XottNatt) (Xut1))
Note that the event Yn-1 E ((n+1(Xn+1) = fy E y:: : s(Xx+1w) < pm (Xx+1.)) (Xn+1)} occurs if,
and only if, pr (Xa+1,Yn41) bound:
(Xnt2)= 2 S s((att2in+1))
a we obtain the desired lower coverage
1-a S P (Yzt1 E (xt1(X 1( (Xxti)).
Deviation bound for the coverage: We now bound the deviation of the coverage of SC-CP from the lower bound. Note, for any f IR + R. that (9) implies:
E 0 o f(xx (o- (Xnz)aa - 1pp! 11 (X < s((ntniontil))
E ((nK(n)))))))))) pn
(Xo+i )n+ss (xi)) (1- (0 Kaxn)) ||
(10)
22

---

Using that (1 [3) E [0, and exchangeability, we can bound the right-hand side of the above as
n+1
115 | (ns((o-1 = ph p.x Nott (Xi)) (1- 8,)(0 o 11)
E
max (E(++1]
(0 l 1
11 = =1 11
Next, since there are no ties by C3, the event 1 ((S(An-nXatt) Par (Kuttitaa1) (X₁)) for some index pn ((n4tsluts) since, otherwise, ((4n+1 Pm (x (X,) Pn (xXx+ = s s(Xn=1, for some + j. However,
i [n + 1] can only occur once per piecewise constant segment of
On (Nn-1) is transformation of fr (a-1)) and, therefore, has the same number of constant segments fü ((x1)) Thus, holds that
1(s ( (ainkaa
(Xi)) N(Xn+11
where N(4m) is the (random) number of constant segments of Sn f((nn1s)eas)) This implies that
a+1
E max 1(0 o (X,) [[+++]]
1 1 ( = ((
n2 n+ E N(Xnntaaat))
10 o f/ Nute))
max VE[+1]
Combining this bound with (10), we find
E 0o (( (X 1) 110 (Xn+1) S ((t2l1)
E
max iE(n+1]
(00 f P (Xxi)
By the law of iterated expectations, we then have
(Xn+1) E 0 o fl ((tnt ( (P (Xn+1) < S((xn-1Xn41)) I
Snti
1 N(Xn411
1(0 o Ynt) (E[++1]
max
n +
Next, let V C denote the support of the random variable fl (( (Xx+1). Then, taking 0 be
t - 1(t E V)sign {a TP (p (x (Xn+1) < s411 ((Xa+1 I ((xnnt)aa+1) ( = =
which falls almost surely in (-1, 1), obtain the mean absolute error bound:
(DCKntniaaiiiXx.. (Xn+1) < S s(Xn41Naat)) pl
E [ACCANttioon))] S n +
1
Since the event Yat1 & Cntz(Xa41)= = fy y: (( I ph (x (Xn+1)) occurs if, and only if, pn (n11) < Spt1 ((8111)8111) we conclude that
E3 (( E Ea- - P (Yut1 & (n+z((xn1)) (xtt)) <
n +
Under C4, E 1.8111)) E. & n 1/3 polylogn, n, such that
E a - P (Yntt & (Xnm)) I (( (Xnti))
1// polylog
polylog n n/3
n 11
as desired.
23

---

Proof of Theorem 4.3. Let Pa+1 denote the empirical distribution of {(X,, Y)) [= and let Pa de- note the empirical distribution of (.) For any function g X x x R: we use the following empirical process notation: Pg = ((x.,)dd(r,,)) v). Pat1g = J g(x, y)dPu+t
and Png == ʃ g(z, y)dd (z.
Define the risk functions Rh (0) E (S, YI)2 + ITI (2 (z. y)} 10) := Ex Y.))? and Ro(0) == [I y)) y). Morcover, define the risk min- imizers as 0/--10) argmingg@.... R and 6o argminggo... Ro(0). Observe that R (( (0(-.0))- p! (0o) s 0 since fn minimizes R., over iis. Using this inequality, it follows that
(o(
Ro(Oo) = (o( () (0(ew)))
+ R( (0(2.0)) R (0o) + R/ - Ro(
s Ro(0 Ro((/22211 R( (0) (0(2.11)) { ( v) (0o) Ro((u) }
S (o( - Ru+ (0(2.)) - [ (0o) Ro(0o)}
+ R (0(2.0)) - Ru+1 (4=+10//:211 (0/=))) - (la - Ro01(
The first term on the right-hand side of the above display can written as
Ro(0((:0)) Rn+1(0((22)) [Rn+1(00)) Ro(Oo)} (Pa+1 - P) [S {Se0 J21 (0!4.00)) - (1 (0o) Bo+1 For
We now bound the second term, R/l (ofwww)) Ru+ 1
any 0 E iioo observe that
(0) n [1 - 0o - ((xx)))3-(((++111- - 00 -00 [(Xn+1)33)
We know that w/-))) and lo. being defined via isotonic regression, are uniformly bounded by B = supyey (u), which is finite by C6. Therefore,
(e - Ra++ Rn+1((( (0(2:2)) - { (R( (0o) Ra+ (00)
s
n n = O(n
8B2
Combining the previous displays, obtain the excess risk bound
Ro(O(2:))) - (oo) < (Pa+1 - P) 15 ( )² - (5o, 32| + O).
(11)
Next, we claim that Ro Ro(0 (On) 2 ((((2)) o f (0o o I11 To show this, expanding the squares, note, pointwise for each x E X' and y E y, that
{S w) (z, (x,,)}² - (Sa, (x, (c) = 10(=00) f(x))² 10o f(x))² - 2y { ((0(=0) f)() (0o o [(x }
= ((f(:10) fz) + (0o fz))) - 22 {(0(=20) f)(z) (Gu f)())
Consequently,
Ro (o(e-w))-Ro((k = / (100020) o f())) + (0. o f))) - 2y} (0( o fz)))
(Oo o f)(z) } dP(x).
(12)
The class O..o consists of all isotonic functions and is, therefore, convex space. Thus, the first-order derivative equations defining the population minimizer 6n imply that
/ g(aw)) f(x) Pu o 5(x)) tv Pu o f(x)) f(x)) dP(x, S 0.
(13)
24

---

Combining (12) and (13), we find
Ro (0( Ro(Oo) = / { )
o f)()) - (0o o (x)) dP(x)
2 / ((0 (C) - n) { (0 ())x (0o o Max }
dP(x)
z / { (10 o (()) - (0o o Mar) dP(x),
as desired. Combining this lower bound with (11), we obtain the inequality
/ (Ola-w) f)))) (0o a )) dP(x)= s Ro( - Ro(Oo) S (Pr - -P) (( - }²|
+ O(1mn)
(14)
Define on V 1()) (0o o [a)) dP(x)), the bound B suppey |v). and the
()x)
function class,
11,1 == ((.,1)) - {(1 + 02) f 22y - o
Using this notation, (14) implies
/ ((0) + 02) f(x) 2y 02) s(o)dd(P, - P) ()(,)))
6%
I sup
61.0,50sro:1000 021 56,
sup
(Pr P)h + 0(1
Using the above inequality and C5, we will use an argument similar to the proof of Theorem 3 in [52] to establish that 6n = Op(m-1/3), This rate then implies the result of the theorem. To see this, note, by the reverse triangle inequality,
(y x') So(a', m/)
|0/ "la:0) 10/ -0(x')))
$ 00(2z') - (20)
Squaring and integrating the left- and right-hand sides, find
/ s(( (y/, x²) So(z', v) aP(x' S |00((') -
2 dP(x') = 6% ((6(==))((xxv)))) "lx:)
as desired.
We now establish that on Op/(n/)/)). For function class F, let N(e, F, L2(P)) denote the covoriing number [53] of F and define the uniform entropy integral of F by
I(5 = S. sup Vlog Ne,, (2())) de, Q
where the supremum is taken over all discrete probability distributions Q. We note that 01,)) = 1 sup sup
SupD/NGe /N(, I1. |l. (1)) de == / sup / /Nee, Oisos ll Ilquf 1) de = (ō, Qisa)
where f- is the fush-forward probability measure for the random variable f(W). Additionally, the covering number bound for bounded monotone functions given in Theorem 2.7.5 of [53] implies that
J6. 11... = Jô, @ise) S V8.
Recall that f is obtained from external dataset, say En, independent of the calibration data. Noting that f is deterministic conditional on a training dataset e,,, Applying Theorem 2.1 of [54] conditional on En, we obtain, for any ò > 0, that
(1
01.,01.n)
E
sup
(P, Phh En S n 1/1
Vnd²
AEO 4BB
n 1/- Oioo) (1+
[(ō. 2
Vnd²
25

---

Noting that the right-hand side of the above bound is deterministic, we conclude that
(Pr - P)h nn (0, (1+
J(ó, Oiso)
E
sup allll
Vnd=
We use the so-called "peeling" argument [53] to obtain our bound for on. Note
P 8 2n
2
Σ
P
(2
2
2 N 2
m=M
8
2m+1 2 2/233/ 2m 80 -
(Pr - P)h PPh +((Ln))
sup
heen i : |
20
Σ
2m+1 2 2 2m 22 n 2/3
(Pr P)h +0(/))))
sup
20
22m n 22/
sup
(Pa - Pch-)h++01111
m=M
hee :b0])::11||1541222 -n-18
(nn - P)h + ((1/n)
22m 2/3
M=M
J(2m+1 n 1/3 -1/3 -iso)
1 m222m 2, 2/3 0,0o)
/m2m+1 113
_0000000))
20
O(L/n)
Vn22mm 2/3
22m 2233
m=M
m=M
2(m+1)/2 (/-4t)/22-11/0 1/6
00
o(1)
22m 1/6
22m
2m/+1//2
22m
M
Since 2mm1 21/2
20m+1i/2 23m
00, we have that 2m
as M + oc. Therefore, for all
é > 0, we can choose M > 0 large enough so that
p 2 2 2 SE. P
We conclude that 8, On 1/3) as desired.
26

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Reviews:
[{"review_id": "PI4mRqCtke", "reviewer": "Reviewer_bFVX", "summary": "This paper introduces Self-Calibrating Conformal Prediction a novel uncertainty estimation framework combining Venn-abers calibration with conformal prediction. The output provides calibrated point predictions with associated prediction intervals with validity conditional on these model predictions.", "questions": "1. Can the authors provide more insight into the trade-offs between computational complexity and accuracy when using the approximations suggested for non-discrete outcomes?\n\n2. How sensitive is the method to the choice of the initial predictor f? Are there certain types of predictors that work particularly well or poorly with SC-CP? E.g. an xgboost was used how would the performance differ for other predictors f", "limitations": "The authors have done a good job of addressing the limitations of their work. They discuss potential computational challenges for non-discrete outcomes and suggest approximations. \n\nOther limitations are briefly sprinkled throughout the paper. To improve this aspect, a suggestion is to include a dedicated \"Limitations\" section that consolidates and expands upon the current discussion of limitations scattered throughout the paper to make readers more aware of these factors.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "1. Originality: The paper presents a novel combination of Venn-Abers calibration and conformal prediction, extending Venn-Abers to regression settings. This approach to achieving both calibrated point predictions and conditionally valid prediction intervals is novel.\n\n2. Quality: The theoretical foundations are well-developed, with clear assumptions and complete proofs provided in the appendix. The authors prove important properties of their method, including perfect calibration of the Venn-Abers multi-prediction (Theorem 4.1) and self-calibration of the prediction interval (Theorem 4.2).\n\n3. Clarity: The paper is well-structured and clearly written. The motivation and formulation are very well done.\n\n4. Significance: The proposed method addresses an important ML uncertainty estimation problem - providing reliable uncertainty estimates that are both calibrated and conditionally valid. This problem is important in the context of trustworthy ML.", "weaknesses": "1. Limited experimental evaluation: While the MEPS dataset experiment is a nice case-study, the paper would benefit from evaluations on a wider range of datasets as is common in conformal prediction works. E.g. the widely used conformal regression datasets from CQR (https://arxiv.org/abs/1905.03222). The synthetic setups in the appendix help, but more real world datasets would be useful\n\n2. Comparison to other conformal regression approaches: besides the baseline marginal CP, it would be helpful to see where other robust conformal approaches like CQR fit in compared to SC-CP— as they all should have good coverage and width, but it would be useful to see their calibration relative to the proposed method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Response to Weaknesses\n\n1. Thank you for this suggestions. In the revised paper, we will include additional real data experiments, specifically the \"bike,\" \"bio,\", \"star\", \"concrete\", and \"community\" datasets used in CQR reference. \n\n3. In the revised real experiments, we will include CQR as a feature-conditional baseline to investigate for which datasets prediction-conditional validity provides an adequate (or poor) approximation of feature-conditional validity.\n\n## Response to Questions\n\n1. Thank you for this question. In the revised version of the paper, rather than approximating the algorithm via discretization, we instead propose running the algorithm exactly using a discretized outcome and a discretized model. The outcome can be discretized by binning the outcome space into, say, 200 bins, and the model can be discretized similarly. In this case, the algorithm can be computed exactly, and the coverage and calibration guarantees are with respect to the discretized outcome. Discretizing the model output into 100-200 bins is generally sufficient to preserve predictive performance, especially isotonic regression/calibration already bins the model output. Discretizing the outcome allows the user to directly control how much the discretized outcome can deviate from the true outcome. We also discuss how to increase the width of the prediction interval by the outcome approximation error to guarantee coverage for the true outcome.▍\n\n2. We have added a discussion on model choice to the problem setup and experiment section. Our method, like most conformal prediction methods, provides tighter prediction intervals as the model's predictiveness (e.g., MSE) improves. In our algorithm, isotonic regression learns an optimal monotone transformation of the original predictor (Theorem 4.3) and, therefore, can asymptotically only improve the MSE of the original predictor. However, if the original predictor is poorly predictive, the calibrated predictor, albeit calibrated, will typically also not be predictive. In addition, the usefulness of prediction-conditional validity may depend on the predictiveness of the model. An example where prediction-conditional validity is not useful is when the predictor is constant, in which case prediction-conditional validity reduces to marginal validity.▍\n\n## Response to Limitations.\n\n1. We have moved our discussion of the limitations to its own subsection in Section 4. This paragraph includes all the limitations discussed in the checklist, as well as other limitations mentioned throughout the submitted version of the paper.", "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": "Ewlpa2ZKpD", "reviewer": "Reviewer_QroB", "summary": "The paper proposes a method that jointly calibrates point predictions and provides prediction intervals with valid coverage given the point predictions. This is performed by combining two existing post-hoc processing procedures, Venn-Abers calibration and conformal prediction. The analysis of the method provides guarantees as in CP methods and a convergence rate for the conformity scoring function. The method provides a calibrated model that is piecewise constant and adapts to outcome heteroscedasticity.", "questions": "What is the meaning of the subscript of $f_{n+1}$ in the desiderata (line 90)?\n\nThe colour (blue) of the results in Fig. 2 is not helpful. Would it be better to highlight which methods perform best?\n\nIs there a reason you are not comparing to the method in [42]? It seems to be the closes related method.\n\nThe example presented in the experiment section is interesting and helpful in explaining the methodology. However, could you provide other examples of datasets or problems where your methodology would offer significant advantage over existing methods? And when would it fail in comparison to existing methods?\n\nDid your theoretical analysis offer any general insights about the class of problems that you are addressing in this work? Do you consider the theoretical analysis to be the main contribution of this work? Can the analysis be reused or easily adapted to other methods?\n\nMinor:\nThe citations do not seem to be linked.", "limitations": "The paper contains a discussion of the limitations (in Sec. 3.4) as well as appendix B.3. However, I believe that the limitations discussed in the Paper Checklist should be moved to the main part of the paper as it would contribute to the exposition of the methodology.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is clearly written and provides a good overview of the background material on both CP and Venn-Abers calibration. The motivation for the methodology is sound. The main contribution of the paper is the theoretical analysis that combines existing results in a novel way. The algorithm is well-presented and appears relatively easy to implement.", "weaknesses": "One of the main disadvantages of the proposed approach is the computational complexity, as discussed in Sec. 3.4. Due to the high computational complexity, it may not offer significant advantage over other existing methods, e.g. the Mondrian CP. It is not immediately clear from the paper what is the advantage offered over existing methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Weaknesses\n\n1. **Weakness:** One of the main disadvantages of the proposed approach is the computational complexity... it may not offer significant advantage over other existing methods.\n\n   **Response:** While our method has greater computational complexity than the split-CP approach for marginal validity, it is faster compared to methods like full CP and the conditional split CP approach by Gibbs et al. (2023). It is scalable to large datasets, as isotonic regression can be computed with XGBoost. In our experiments with calibration datasets (n=10000-50000), the computational time ranged from 1 to 5 minutes. Given that training the initial/uncalibrated model can take much longer, we believe the computational complexity of our method is not a significant weakness.\n\n    Advantages of our method over existing methods are discussed in Section 4.1 and shown empirically in Section 5. Mondrian-CP approaches can only provide prediction-conditional validity and do not achieve our objective of self-calibration, which offers both calibrated point predictions and prediction-conditional validity. Another limitation of Mondrian-CP is the need for pre-specification of a binning scheme for the predictor f(·), introducing a trade-off between model performance and prediction interval width. In contrast, SC-CP data-adaptively discretizes the predictor f(·) using isotonic calibration, providing calibrated predictions, improved conformity scores, and self-calibrated intervals.\n\n## Questions\n\n1. What is the meaning of the subscript of $f_{n+1}$ in the desiderata (line 90)?\n\n\n   **Response:** The subscript indicates that $f_{n+1}$ is obtained by calibrating the original model $f$ using the calibration data $(X_i,Y_i): i \\in [n]$ and the new context $X_{n+1}$. We have clarified in our objective how $f_{n+1}$ is obtained and explained this notation in the revised paper.\n\n\n\n\n2. ... Would it be better to highlight which methods perform best?\n\n    **Response:** We have changed the coloring in the figure to highlight which method performs best.\n\n \n\n3. \"Is there a reason you are not comparing to the method in [42]?\"\n\n    **Response:** We did not include the method of [42] because their aim is to construct cumulative distribution function (CDF) estimates for a continuous outcome with marginal calibration guarantees. Our objective is to construct prediction intervals centered around a calibrated point prediction with prediction-conditional coverage guarantees. Inverting the CDF estimates from [42] yields quantile estimates and prediction intervals. However, these intervals are not centered around a point prediction, don't use conformity scores, and do not ensure prediction-conditional validity. While both approaches use Venn-Abers calibration, we use it to calibrate the regression model and employ conformal prediction techniques to construct prediction intervals. In contrast, they use Venn-Abers calibration with the indicator outcome 1(Y ≤ t) to construct calibrated CDF estimates.\n\n4. \"The example presented in the experiment section is interesting ... \"\n\n    **Response:** In the revised paper, we plan to include additional real data experiments, specifically the \"bike,\" \"bio,\", \"star\", \"concrete\", and \"community\" datasets used in [1]. Our method generally will not fail to achieve self-calibration, barring failure of standard CP assumptions like exchangeability. However, there are scenarios where the prediction-conditional validity of our approach may lead to poorly adaptive prediction intervals relative to CP methods targeting feature-conditional validity.\n \n    In Appendix B.3, we use synthetic datasets to illustrate how prediction-conditional validity can approximate feature-conditional validity when the heteroscedasticity/variance of the outcomes is related to its conditional mean. In such cases, our approach offers narrow interval widths regardless of feature dimensions, leveraging model predictions as a scalar dimension reduction. We also show that in scenarios without a mean-variance relationship, our approach may provide poor feature-conditional coverage.\n\n    In the revised paper, we discuss these scenarios and point to the synthetic data experiments. To strengthen the real-data experiments, we will include the CQR method of [1] as a baseline for feature-conditional validity to investigate if prediction-conditional validity approximates feature-conditional validity in real data.\n\n\n   [1] Romano, Yaniv, Evan Patterson, and Emmanuel Candes. \"Conformalized quantile regression.\" Advances in neural information processing systems 32 (2019).\n\n   [2] Gibbs, Isaac, John J. Cherian, and Emmanuel J. Candès. \"Conformal prediction with conditional guarantees.\" arXiv preprint arXiv:2305.12616 (2023).\n\n5. \"Did your theoretical analysis offer any general insights...\"\n\n    **Response:** Our main contribution is integrating two areas of trustworthy machine learning: (1) calibration of model outputs and (2) uncertainty quantification via prediction intervals, proposing self-calibration and self-calibrating conformal prediction. Our theoretical analysis offers further insights and potential extensions. Our techniques can analyze conformal prediction methods that involve calibrating model predictions followed by constructing conditionally valid prediction intervals. One could apply a feature-conditional CP method with conformity scores and/or the conditioning variables depending on calibrated model predictions and derive feature-conditional validity guarantees using a straightforward modification of our arguments. We plan to explore generalizations of our procedure in future work and have added a concluding paragraph discussing these insights and extensions.\n\n## Limitations\n\n**Response:** We have moved our discussion of the limitations to its own subsection in Section 4. This paragraph includes all the limitations discussed in the checklist, as well as other limitations mentioned throughout the submitted version of the paper.", "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": "NKT3ZXvLve", "reviewer": "Reviewer_qyB7", "summary": "The authors propose a new scheme to obtain prediction intervals that are valid conditioning on the model output. The approach leverages the notion of Perfectly Calibrated Point Predictors and an extension of Venn-Abers calibration to the regression setup.", "questions": "- Intuitively, how does conditioning on $f(X)$ help? How does this avoid the *curse of dimensionality*? What is the difference compared with conditioning on $Y$?\n- Is $1({Y \\in C})$ the indicator of ${Y \\in C}$? Why is $f$ called a *covariate shift* in (3)? How is this the same as saying that $f(X)$ is the model output?\n- Has prediction-conditional validity been used before? \n- How is $f^{X, y}$ defined? Should $f_{n+1} =  ( f_n^{X_{n+1}, y}(X_{n+1}), y \\in {\\cal Y} ) $ be interpreted as a recursive definition? How does $f_{n+1}$ depend on $x$?\n- Would it be possible to compute the marginal prediction band centered on the Venn-Abers calibrated predictor (the black lines in Figure 2, if I understand the plots correctly) instead of the original one?", "limitations": "The authors should have said why they do not compare with other approximations of context-conditional validity, e.g. Conformal Quantile Regression, Ref 22 in the paper, or the Error Reweighting method [3]. \n\n\n[3]\nPapadopoulos, Harris, Alex Gammerman, and Volodya Vovk. *Normalized nonconformity measures for regression conformal prediction.* Proceedings of the IASTED International Conference on Artificial Intelligence and Applications (AIA 2008). 2008.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- I like the interpretation of $f(X)$ as a \"scalar dimension reduction\".\n- Generalizing self-consistency to prediction sets may inspire future investigation in the broader CP community.\n- The idea of leveraging Venn-Abers calibration to the regression setup is a valuable contribution.", "weaknesses": "- The authors do not explain, in an intuitive way, why conditioning on $f(X)$ is a good approximation of conditioning on $X$. They could describe their 1D dimensional-reduction argument better, possibly with a concrete example. Imagine $(X, Y)\\in {\\mathbb R}$ and $Y \\sim ({\\bf 1}(X<0) + 10 {\\bf 1}(X<0)) {\\cal N}(0, 1)$. A perfect point-predictor is $f(X) = {\\rm E}(Y|X) =0$ for all $X$. In this case, conditioning on $X$ or $f(X)$ does not look equivalent.  \n- The description of Venn-Abers calibration can be improved. I have not fully understood the meaning of these two sentences [1][2].\n- The authors may explain more intuitively why they need a Perfectly Calibrated Point Prediction to compute a Self Calibrated Prediction Interval.\n- The proposed model is only compared with Mondrian CP.\n\n[1] *Venn-Abers calibration accounts for overfitting by widening the range of the multi-prediction in such scenarios, thus indicating greater uncertainty in the value of the perfectly calibrated point prediction.*\n\n[2] *Each point prediction in the set enjoys the same large-sample calibration guarantees for isotonic calibration.*", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Response to weaknesses\n\n1.  We have added clarification on when our proposed self-calibration objective can approximate feature-conditional validity. Our self-calibration objective aims to provide a relaxation of feature-conditional validity that is feasible in finite samples (Section 2.3). A key benefit is that it involves conditioning on a one-dimensional variable, thus avoiding the curse of dimensionality (Section 2.2). Self-calibration is not a replacement for feature-conditional validity. Prediction-conditional validity can approximate it when the outcome's heteroscedasticity/variance is a function of its conditional mean. Appendix B.3 experimentally confirms this using synthetic data, and Section 5 provides additional evidence for this using real data that appears to have a mean-variance relationship.▍\n\n2. We have redrafted the description for clarity and provided the algorithm before describing its qualitative properties. The revised sentences include:\n   \n    1. \"Isotonic calibration can overfit, leading to poorly calibrated predictions. When this occurs, the Venn-Abers set prediction widens, reflecting greater (epistemic) uncertainty in the perfectly calibrated point prediction within the set.\"\n    2. \"The Venn-Abers calibration set is guaranteed to contain at least one perfectly calibrated point prediction in finite samples, and each prediction, being obtained via isotonic calibration, still enjoys the same large-sample calibration guarantees as isotonic calibration.\"\n\n3. We clarified the role of calibration in the self-calibration objective, which aims to obtain prediction intervals that are centered around calibrated point predictions and provide valid coverage conditional on the calibrated point prediction. Point calibration ensures that the prediction interval is centered around a conditionally unbiased point prediction. Prediction-conditional validity is only attainable in finite samples for predictors with discrete outputs. Venn-Abers calibration discretizes the predictor, enabling prediction-conditional validity while mitigating the loss in predictive power due to discretization.\n\n4.   Along with Mondrian-CP, we also included the kernel smoothing approach of Gibbs et al. (2023). Our baselines are sufficient to illustrate the benefit of our approach combining point calibration and prediction-conditional validity. The baselines for prediction-conditional involve estimating a one-dimensional quantile function, where Mondrian-CP uses histogram regression and Gibbs et al. use kernel smoothing. Kernel smoothing and histogram regression are minimax optimal for 1D functions under weak assumptions. Also see our response to Limitation #1.\n\n## Response to Questions\n\n1. See also our response to Weakness 1. On the difference between conditioning on $ Y $ and $ f(X) $:  $ Y $ is typically a noisy signal $ f_{true}(X) + \\varepsilon $. Thus, two contexts $ X_1 $ and $ X_2 $ with $Y_1 = Y_2$ can have very different signals $ f_{true}(X_1) $ and $ f_{true}(X_2) $. Outcome conditional validity is primarily useful in classification settings where the outcome $ Y $ is a ground-truth label and thus not subject to noise. We have added this discussion to related work.\n\n2.  Yes, it is the set indicator. The covariate shift (CS) terminology for the multicalibration objective in (3) is from Gibbs et al. (2023). In equation (3), $ f $ is not the predictor but an arbitrary element of $\\mathcal{F}$. Thank you for pointing out this confusion in notation. We now denote the CS $ f $ in (3) by $ h $. The CS terminology arises in (3) because if $ h $ is a density ratio between a source and target distribution then multicalibration with respect to $ h $ in the source population implies marginal coverage with respect to the target distribution as well.\n\n3. Prediction-conditional validity as a formal objective has not been proposed before. Some works have used bins based on predictions as an application of Mondrian CP to provide a coarse form of prediction-conditional validity. Our contributions include extending Venn-Abers calibration to regression and proposing the self-calibration objective and our solution, combining prediction-conditional validity with data-adaptive output discretization with the calibration of model outputs.\n\n4.   $f_n^{(x,y)} $ is defined in Algorithm 1. The subscript $ n $ indicates that the model depends on the calibration data and the superscript $(x,y)$ indicates that the model also depends on the context $ x $ and the imputed outcome $ y $. In our notation for $ f_{n+1} $, we suppressed the dependence on the context $ x $. To avoid confusion and make the dependence on $ x $ explicit, we have now changed the notation from $ f_{n+1} $ to $ f_{n,x} $. We fixed a typo on line 153, where $ f_{n+1}(x) $ is now $ f_{n+1}(X_{n+1}) $.\n\n5.  It is not possible to compute the marginal bands obtained using split CP around the calibrated predictor without sacrificing finite-sample validity, as Venn-Abers calibration is performed using the same data used to construct CP bands.  \n\n## Response to limitations\n\n1. We have clarified our choice of baselines, focusing on those targeting prediction-conditional validity as that is part of our self-calibration objective. We will add Conformal Quantile Regression as a feature-conditional baseline in the revised experiments. For direct comparison with our method, our baselines employed the commonly-used absolute residual error conformity score. While different conformity scores can improve feature-conditional validity, our primary goal is to show how calibrated point predictions and prediction-conditional validity enhance interval efficiency and adaptivity. Our method can be adapted to any conformity score, including the error reweighting score, to provide variance-adaptive prediction bands. We now discuss this in the conclusion and provide an explicit algorithm for the Error Reweighting modification in the Appendix.", "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:

Meta-review: This paper combines conformal prediction with a regression extension of Venn-Abers calibration to provide calibrated point predictions and prediction intervals with finite-sample guarantees. Overall, the paper is written clearly, self-contained, and well-motivated. The extension of Venn-Abers calibration to regression settings is a novel and significant contribution. The empirical evaluation in the submission is somewhat lacking, focusing only on a single case study, and there are some clarity issues around the motivation for their approximation of feature-level conditioning.

While the authors promised additional experiments and clarifications, they did not include new experimental results during the rebuttal period. I am recommending that this paper is accepted despite these omissions, as I believe it will be of interest to the NeurIPS community. However, I urge the authors to put careful effort into the camera ready to deliver the promised additional results and address clarifications that came up during the reviewing process.
Decision: Accept (poster)