Boosting Methods for Interval-censored Data with Regression and Classification
Authors: Yuan Bian
Grace Yi
Wenqing He
Venue: ICLR 2025 Poster (2025)
Keywords: Boosting
Functional gradient descent
Interval-censored data
Minimax error rate
Nonparametric classification
Nonparametric regression
Smoothing spline
Primary area: other topics in machine learning (i.e., none of the above)

Abstract: Boosting has garnered significant interest across both machine learning and statistical communities. Traditional boosting algorithms, designed for fully observed random samples, often struggle with real-world problems, particularly with interval-censored data. This type of data is common in survival analysis and time-to-event studies where exact event times are unobserved but fall within known intervals. Effective handling of such data is crucial in fields like medical research, reliability engineering, and social sciences. In this work, we introduce novel nonparametric boosting methods for regression and classification tasks with interval-censored data. Our approaches leverage censoring unbiased transformations to adjust loss functions and impute transformed responses while maintaining model accuracy. Implemented via functional gradient descent, these methods ensure scalability and adaptability. We rigorously establish their theoretical properties, including optimality and mean squared error trade-offs. Our proposed methods not only offer a robust framework for enhancing predictive accuracy in domains where interval-censored data are common but also complement existing work, expanding the applicability of existing boosting techniques. Empirical studies demonstrate robust performance across various finite-sample scenarios, highlighting the practical utility of our approaches.

Published as a conference paper at ICLR 2025
BOOSTING METHODS FOR INTERVAL-CENSORED DATA WITH REGRESSION AND CLASSIFICATION
Yuan Bian', Grace Y. Yi Wenqing He!
1 Departmeet of Statistical and Actuarial Sciences, University of Western Ontario
2 Department of Computer Science, University of Western Ontario 3 Vector Institute, Canada yb2 6200 @ cumc. co umbia. edu, {gyi5, whe23)euwo.. ca
ABSTRACT
Boosting has garnered significant interest across both machine learning and sta- tistical communities. Traditional boosting algorithms, designed for fully ob- served random samples, often struggle with real-world problems, particularly with interval-censored data. This type of data common survival analysis and time- to-event studies where exact event times are unobserved but fall within known intervals. Effective handling of such data is crucial fields like medical re- search, reliability engineering, and social sciences. In this work, we introduce novel nonparametric boosting methods for regression and classification tasks with interval-censored data. Our approaches leverage censoring unbiased transforma- tions to adjust loss functions and impute transformed responses while maintain- ing model accuracy. Implemented via functional gradient descent, these meth- ods ensure scalability and adaptability. We rigorously establish their theoretical properties, including optimality and mean squared error trade-offs. Our proposed methods not only offer robust framework for enhancing predictive accuracy in domains where interval-censored data are common but also complement exist- ing work, expanding the applicability of existing boosting techniques. Empirical studies demonstrate robust performance across various finite-sample scenarios, highlighting the practical utility of our approaches.
INTRODUCTION
Boosting (Schapire| 1990: Freund 1995) is a foundational technique in machine learning, transform-
ing weak learners into strong learners through iterative refinement (Schapire Freund 2012). This
iterative nature not only increases predictive accuracy (Quinlan 1996 Bauer & Kobavi 1999 Diet-
terich| (000) but also enhances robustness against overlitting (Bühlmann & Hothorn 2007 Schapiree
& Freund 2012). making boosting popular choice for various applications. The AdaBoost algo-
rithm (Freund & Schapire 1996) was a groundbreaking development and remains a highly effective
off-the-shelf classifier (Breiman 1998). Subsequent research (Breiman 1998 1999 Mason et al.)
1999] revealed that AdaBoost can be viewed as a steepest descent algorithm in a function space de- fined by base learners. Boosting continued to grow as Friedman al. (2000) and Friedman (2001) extended its application to regression and multiclass classification within a broader statistical frame- work, and it is interpreted as method of function estimation. In this expanded context, Bühlmann & Yu introduced L2Booss a computationally efficient boosting algorithm that leverages the T loss function. More recently, Chen & Guestrin (2016) proposed XGBoost, a scalable and useful tree boosting system, and Ke et al. (2017) introduced LightGBM, an efficient tree boosting algorithm.
Despite the success of boosting methods, key limitation persists: traditional boosting algorithms assume access to fully observed random sample of data. In many real-world applications, however, data are incomplete or censored. This issue is particularly pronounced in fields like survival analysis, where interval- ceterrel data are becoming increasingly prevalent.
"Corresponding author

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1.1 LITERATURE REVIEW
Recent research in boosting has focused on handling incomplete or censored data. Most efforts
have extended boosting methods to accommodate right-censored responses (c.g., Ridgeway 1999
Hothorn et al., 2006 Wann & Wang. 2010, Mayr & Schmid| 2014 [Bellot & van der Schaar 2018
Yue et al. 2018: Bellot & van der Schaar 2019, Barnwal et al. |022]Cheen Chen & 20244 or missing
responses (e.g., Bian al. 2025a b). In theses cases, techniques like imputation and weighting
employed to construct unbiased loss functions for training.
While these approaches have addressed some issues related incomplete data, significant gap remains in handling interval-censored data, where event times are known only to lie within specific intervals. This scenario, prevalent in survival analysis (e.g., Sun 2006), is more complex than right censoring, as the response variable is completely unobserved though known to fall within an interval. Research on interval-censored data has expanded across various domains. For example. Yao et al. (222 introduced survival forest method using the conditional inference framework, while Cho al. (2022) developed the interval censored recursive forests method for non- non-parametric estimation of the survivor functions. Yang et al. (2024) leveraged the censoring unbiased transformation (Fan) Gijbels 1994 (996) to create tree algorithms designed for interval-censored data. However, these approaches do not capitalize the strengths of boosting, which could significantly enhance predictive performance and robustness.
L..g..uun
1.2
OUR CONTRIBUTIONS
We propose a framework that extends boosting methods to address interval-censored data, a critical underexplored problem in machine learning. Our contributions enhance the applicability of boosting algorithms to complex censoring structures:
We propose methods, called 1.2Boost-CUT and L2BoosttIMP, to extend boosting for handling interval-censored data. L2Boost-CUT adjusts the loss function with the censoring unbiased transformation (CUT), while L2Boost-IMP uses an imputation-based approach leveraging CUT. Both methods handle interval-censoring flexibly, avoiding restrictive as- sumptions and enabling predictions of survival time, probability, and status.
We provide rigorous theoretical analysis of our methods, evaluating their mean squared error (MSE), variance, and bias, as well as the connection between the two proposed meth- ods. Our results demonstrate that by incorporating smoothing splines as base learners, the proposed framework achieves optimal MSE rates in both regression and classification tasks. These insights extend the understanding of boosting methods, building upon and generalizing the foundational results from Bühlmann & Yu 2003) for complete data.
We validate methods through extensive experiments both synthetic and real-world datasets. Results show that LyBoost-CUT and LyBoost-IMP offer robust and scalable solutions for handling interval-censored data and enhancing the generalizability of boosting algorithms.
2 PRELIMINARIES
Let Y denote the survival time of an individual, and let X denote the associated p-dimensional feature vector, where Y E Rt and X X. with IR representing the set of all positive real values and X denoting the feature space. Our objective is to learn a predictive model that well predicts a transformed target variable g where g is a user-defined transformation and g(Y) E ), with
D C R. The choice of g depends on the task of interest. For instance, setting gYY) Y directly models the survival time; setting g(Y)= = log Y removes the positivity constraint of Y. For binary classification tasks, we can set gY) = 2T(Y s) - predict the survival status at time 8, where is prespecified threshold and is the indicator function.
{f X + V}. consisting of real valued functions. Let ynd denote y' x ... x y & f(v....., Wa) y E y for j 1, i....,d d} for positive integer d. Let L. 222 + R2 denote loss function, which quantifies the error between the predicted and true
We define the hypothesis space,
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n). we uppercase letters X, Y, X;, and Y,, with 1, I,..., n, represent both random variables Published as a conference paper at ICLR 2025
values, where R20 R U {0). For f F, define the expected loss, risk
R(f) E (Y,
(1)
where the expectation is taken with respect to the joint distribution of and Y. The goal is to find the optimal function f that minimizes the risk:
arg min R(f), fEF
assuming its existence and uniqueness.
In practice, the joint distribution of X and Y is unknown, and we only have access to finite sample of n independent observations of X and Y, say O _ {(x,, Yi} i = 1,....n). For simplicity,
and their realizations. To approximate f", we minimize the empirical risk, which serves as proxy for the expected loss:
fc arg min feF
Σ LY,,, f(x.1)
(2)
In the absence of censoring, where survival times Y2 are fully observed for all study subjects, fo can be obtained using a boosting algorithm that iteratively improves base learners. Specifically, the Lgooos algorithm, variant of boosting using the L2 loss function, minimizes the empirical risk via steepest gradient descent to iteratively refine the estimates of fo. At iteration t, given the current estimate f(t-1), the algorithm updates the model by adding an increment term, denoted h()) to form the updated estimate f();
((() f(t-1) 7(
(3)
where ((()) is function mapping from X' to V, called base learner, determined by
( = arg min
aT. (V,, (vx, - -h (x.)
(4)
with OL (Y, f(t- (x.))
(L(uo) 8v
(t- for I...., n. Here, h(0 in 44 can
be interpreted the least squares estimate of E (- ÖL (Y,, (x,))) X.). Thus, the LoBoost algorithm can be seen as repeated least squares fitting of residuals (Friedman) 200T). At stopping iteration i, determined by a suitable stopping criterion, the final estimator of f given by
f. a
f(:
f(0)
R0
where (()) represents an initial value for estimating f.
On the other hand, for classification tasks, particularly when the response gYY) is a step function, e.g., (Y) 2T > s) - 1 for given s, which maps the response Y to the set (-1,1), the Boost algorithm can be modified as "L2 Boost with constraints" (Lş WCBost) algorithm (üühlmann & Yu 2003). This modification allows us to handle binary classification problems, where the goal is to approximate (Y:)) given by (E > s) s){ - 1. In this case, f())
revised as:
((t) sign (f(() min (1, ((O)). with f(() = 1(1-1) k,())
(5)
where (u) = 1 if u < O. 0 if u 0, and if u
O. The modification of f(t) with the sign function included in (5) ensures that the final estimate f()) stays within the range [-1, 1). resulting meaningful output for binary classification.
PROBLEM AND METHODOLOGY
3.1 INTERVAL-CENSORED DATA
Interval censoring occurs when, instead of directly observing the exact survival time Y, we only observe pair of time points (T., R) such that Y lies within the interval (L,, RJ,, where & I.
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R S 30. Different scenarios arise depending the values of L and R: L 0 yields a left- censored observation; R = oo leads to a right censored observation; 0 < R < oo gives a truly interval-censored observation; and when L Y and R Y, we have the exact observation, where y- _ 1o (Y - a), with - 0+ representing a approaching 0 from the positive side. Let [0, r] denote the study period, with being finite. Following standard practice for modeling interval- censored data (e.g., Zhang et al.] 2005 Cho et al.) 2022). we assume conditionally independent interval censoring, meaning that given features X, as well L = L, R T, L Y S R. the probability of the survival time Y occurring before a positive value depends only on I < Y S.
Formally,
Pr(Y -T
<Y R Pr(Y < < < S Sr
for any positive 4, and with 1 r.
Suppose for subject i ,,..., n. there are M observation times 4411 < U422 < U.M oo beyond v4,0 A O, where M random integer, with m denoting its realization. While the randomness of M does not affect calculations for given dataset, its presence reflects real-world data uncertainty with varying numbers of observations. For dataset with m 2 2 and i = 1....,,, define the censoring indicators for each subject and interval j as A.jj - I( < Y S u.
for j = I,..., m, and iim+1 A I IY; > ui.m) Clearly, Ai = -1 1- وÿ= A..j. These indicators reflect whether the true survival time Y; falls within the corresponding time interval. Let the observed data for subject i be O; - {[X,, ui.j, A. 1,..., m). and let the full observed dataset be OlC & O Alternatively, O can be equivalent expressed as O {{X,, ; L; R,
i ,,...,,,0 .1., where for each subject i, we identify the interval ( (L,, R.] containing Y; by finding the index Ji (1,.... m) such that uijj.11 < Y; S Wi.je with L, Wi.h--1 R. Wiih. The
sequence {, ] : I....,n] nn is then ordered in increasing order and the distinct values are
denoted 01 < U2
Um.
3.2 BOOSTING LEARNING WITH INTERVAL-CENSORRY DATA
To address the interval censoring effects, we modify the initial loss function L(g(Y)), f(X.)) f(X,)) to be an adjusted loss function L (O,, f(X.)), with the argument g(Yi) in L replaced by O in L*, and further, we require:
E(L" ((X.)) = ((L(g(V)), fXx)))
This means that minimizing the expected adjusted loss (O, f(X,)) is equivalent to mini- mizing the original risk function R(ff) defined in (I) treating Y; as if it were not interval censored but available. Here, we focus on the L2 loss function, expressed as:
1 L (o(Y)) - + + (f(,)))
For 1, we adjust {g(Y:)}* using the transformation:
(o,) Σ E (1o(3)))) = 1,,),,
(8)
where
E ((o(Y)))* = 1,), S(u, S(ut,s- Jug-1 (s
1
(9)
for = 1,..., and X,))X,) represents the conditional survivor function of Y, given X,
We propose modified version for 7 called the censoring unbiased transformation (CUT)-based
Le loss function, given by
(xx)) Leut (O,, f(xi)) = L -2(0.)
(
(10)
Proposition 1. For the proposed CUT-based loss function (70). we have
FATTau((O,, F(TAY,
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it leads the same This proposition ensures the validity of the CUTT-based loss function 10
risk ① as that of the original loss function. Consequently.
can be implemented with the loss function replaced by (1). where for = 1,2, (O,) in 8 is replaced by its estimate, denoted (O,),, which is derived from replacing S(y)X.) with its estimate (to be described in Section 3.3
Let I,,,, f(x,)) denote the resulting estimate of (1)) and let PCUT denote resulting estimate of 2 with L(Y, ((X,) replaced by f(
Algorithm I outlines a pseudo-code for obtaining iCUT: its implementation code is available on The algorithm modifies the usual Looosst algorithm (Bühlmann & Yuj 2003) for (2) with the initial L2 loss function L.Y;, ((x))) replaced by i(O,,. [(),,),, which directly applies to interval-censored data. Alter- natively, one may employ the usual Ly Lyoso algorithm, but replace unobserved Y with (i,)).
GitHub at https: cww sariiyyaanbian/L2Boo0.
Specifically, (12) on Line 7 of Algorithm i replaced by
Σ L
n-' ' (Y(O,), ((- Sn 1=1
together with replacing i. (O,,)) Lines and 4 of Algorithm by L (P,()))))) We refer to these two algorithms as L BoBotttC and L2Boost-IMP, respectively, with "IMP" reflecting the imputation nature of the latter algorithm. The estimator from the LyBooss-INM algorithm denoted as fIMP.
These two algorithms differ in their approach to addressing interval-censore data. The Boost- CUT method adjusts the loss function so its expectation recovers that of the original L2 loss L, as required in (6). whereas the L2Boost-IMM IMP method preserves the functional form of the original loss I, but replaces its first argument with the transformed response (O,) in Therefore, the loss functions for those two algorithms are distinct:
Lcur(O,, + (()()).
The risk from L LoBoost-CUT satisfies Proposition I (proved in Appendix D). but this property does not hold for L9Boost-INMe IMP. Nevertheless, due to the linear derivative of the L2 loss in its first argu- ment, the following connection emerges:
(11) This leads to closely related increment terms in both methods, and as such, B.2ooostCUTT and Lg Boost-IMPP mainly differ in the stopping criterion, suggesting that they often yield similar results, as observed in the experiment results in Section Fldd Appendix G Further discussions on these two methods are provided in Appendix E.3
ai (()))())))
OL (" Pi(,)) (x.))
3.3 BASE LEARNERS AND SURVIVOR FUNCTION
To outline the key steps in Algorithm ! we begin with notation related to the base learners at each iteration. For iteration t = 1, let h(O) = (XI) (X))) T
where
7,000 is the base learner at iteration 1, defined in Line 4 of Algorithm F = (Pr(O))..... (o(O))) where (0,)) represents an estimate for ( For (( in Line
For k 1, 1,2, let
of Algorithm ! we define fit-1) = (f) ((t-1) Y (t-
((,=))(x,)) and compute the residual:
(13)
Algorithm D iteratively updates the base learners that map X' to y for each iteration. In our imple- mentation, we use linear smoothers (Buja et al.) 1989) and focus particularly on smoothing splines, as in Bühhmann & Yu (2003). Linear smoothers are versatile and cover wide range of function classes, including least squares, regression splines, kernels, and many others.
At each iteration, the residual ()+)) is smoothed using a smoother matrix, represented by an n matrix V which transforms the residual into an updated base learner:
(1)) (1-1)
(14)
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Algorithm LoBosst-UUU
Take f()) arg min; n Ei=1 - h(x,))
and set 17 " for given w 1;
2: for iteration & with = 1...... do
(i) calculate a (O,, (xx:) & aL(u.e] 8v
3:
for = I,.... n
1= 0=0,,=
4: (ii) find h(* arg m0
Ei=1 { {-oL
OL (O f(t (X:)) -x.) ?
}
5: (iii) for regression tasks, update f()) (X;) as for 1,..., n;
6: for classification tasks, update (x,, as for 1,..., n2;
7:
if at iteration E
Σ f -n Σ (o, ((- In
(12)
8: then stop iteration and define the final estimator as FCUT = f(i-1).
9: end
10:
end for
Here, V determined by the chosen linear smoother, which may depend on features but not on z((-1) 1) Hastie et al. 2009 Chapter 5.4.1). We provide further details on smoothing splines in
Appendix B
The execution of Algorithm requires calculations of (O,) in (8). which hinges on consistently estimating the conditional survivor function S(y) X;); here Slyl X,) J is interpreted as Sty) x;) for any realization z; of X;; similar considerations apply for functions of X; or conditioning on X, throughout the paper, with X; evaluated its realization. While estimator of S Xi) with faster convergence rate yields a more efficient estimator iCUT consistency suffices to ensure the validity of our methods. Instead of pursuing faster convergence through parametric approaches, which are vulnerable to model misspecification, we prioritize robustness by opting for the interval censored recursive forests (ICRF) method (Cho et al. 2022), whose consistency has been established by Cho et al. (2022). ICRF a free-based, nonparametric method designed for estimating survivor data. It serves as a component within our framework for developing boosting methods for regression and classification with interval-censore data, aiming to predict transformed target variable s(Y) described in Section 2 Further details this estimation
functions for
provided in Appendix C
4 THEORETICAL RESULTS
Assuming consistent estimation of X;), we now develop theoretical guarantees for the proposed methods, both in regression and classification contexts, with the proofs deferred to Appendix D
4.1 REGRESSION
Consider the regression model
o() for 1,...,
(15)
where 6 independent and identically distributed with E(c,) = 0 and var(c)) = 0² < 00, Ø unknown smooth function that can be linear or nonlinear, and is a user-specifted transformation, discussed in Section
At iteration /. = 1.,. the BoBoost-CUTT and BoBosstIIMP methods map the interval-censored data OCC (described in Section 3.1) to f(t), following Algorithm For & = 1,2, we first utilize 88 and ICRF to construct Vi from OI then apply the conventional L2Boost method (described
in Section to {{x,, (1().) -i= 1,.....1 Specifically, for Y, defined in Section 3.3 these

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procedures be formulated as:
(16)
matrix that transforms Y1 f(() at each given t. The following proposition shows that (()) can be represented in terms of the smoother matrix V. Proposition 2. For t ., let Btt) denote the LgBoost-CUT LęBoost-IMP operator at iteration I. Let V/ represent the smoother matrix for the chosen linear smoother. Then, B - (- +)++ for 1,2,..,, where the n x n identity matrix.
where B(O) represents
Next, we examine the averaged mean squared error (MSE) for using f() (defined in Line of Algorithm T to predict Φ in 15 similar to Bühlmann & Yu (2003). The MSE is defined as
E. (110(x)) xxx))
MSE(I, V: =
(17)
where MSE(t, V: depends on v via (16), and the expectation is taken with respect to the joint distribution for the random variables in O defined in Section 3.1 Here, <(X;) is treated as constant for each realization of X; Let
var((, V) -
Σ (x,) and bias (1, v; - nn
Σ
E - -xx))
(18)
denote the averaged variance and the averaged squared bias for using f()) to predict o, respectively. Proposition 3. MSE(1, d) in 17) can be decomposed into the sum of var(1, Vf and bias (1, V; in (78):
MSE(t, v; 6) var(t, vi + bias> (t, Mi 6).
Let & denote the vector (i(X)), of (X Assume that the smoother matrix V is real, symmet- ric, and has eigenvalues {Ai, } with corresponding normalized eigenvectors {QI,.... Qn).
Let Q denote the matrix with Qi being the /th column for 1,.... . and let μ (HI....,Py) Mn)
Q Φ be the function vector in the linear space spanned by the eigenvectors of v. Let O be collection of random variables, drawn from the same distributions as the elements of O, and let
²² var }
Proposition 4. Assume regularity condition (C1) in Appendix A Then var(t, V) and bias (1, V; Φ) in (8) can be, respectively, simplified as
varll, V) =n (1--11 and bias (1, vi n Σ p2 2022
These results align with Proposition in Bühlmann (003), and they suggest that the iteration index í can be interpreted
"smoothing parameter" that balances the bias-variance trade-offs.
Corollary 1. Assume the regularity condition in Proposition 4 If A E {0, 1} for 1...., n. then B(O) for t = 1......
This corollary implies that in special cases, such when the smoother matrix has cigenvalues of 0 or (c.g., projection smoothers (Hastie et al. 2009 Chapter 5.4). like least squares, polynomial re- gression, and regression splines (Buja al. 1989)), the L.Booost-CUT and LzBoost-IMP algorithms cease to provide additional boosting to learners.
Proposition 5. Assume the regularity condition in Proposition 4 and condition (C2) in Appendix A Then, as the number of boosting iterations increases, bias" (t, M Φ) decays exponentially and var(t, V) exhibits an exponential increase, yielding
lim MSE(t, v: = à?. 100
Similar Theorem 1(a) of Bühlmann & Yu (003), this proposition implies that running the Bo and LyBoosstIMi algorithms infinitely is generally not beneficial: the MSE will not decrease below a², and excessive boosting may lead to overfitting.

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Proposition 6. Assume the regularity conditions in Proposition 5 and condition (C3) in Appendix A Then there exists a positive integer ty. such that MSE V; @) is strictly smaller than o².
This result, complementary Theorem 1(b) of Bühlmann & Yu (200), shows that in contrast to condition (C2), when a stronger condition (C3) holds, the Ly Loosstt-UU and LyBoost-IMN -MP algo- rithms can achieve an MSE smaller than d², even with a finite number of iterations.
Theorem 1. Assume the regularity conditions in Proposition and condition (C4) in Appendix A Then for mo 2 2 in condition (C4), the first [mo] iterations of the LgBoost- A.Boost-CUT algorithm (i.e., Algorithm 7 improve the MSE over the unboosted base learner algorithm (i.e., linear smoothers), where [.] is the floor function.
Condition (C4) basically requires base learners to be weak (see Appendix A for details). This theorem suggests that the LyBoost-CUT and LyBoost-IMP algorithms consistently outperform an unboosted weak learner. This result complements Theorem ((c) in Bühlmann & Yu (2003).
Theorem 2. Let é & (i(0,) E { Assume the regularity conditions Proposition
and condition (C5) in Appendix A Then for positive constant there exists positive constant that is functionally independent of (but may be dependent on q and n) such that as - oc,
E - = =E()) + O(exp(-C1))
Y2
(19)
For q Theorem 2 directly yields Proposition 5 In the following development, may write the iteration index t as fn to stress its dependence on the sample size n.
Theorem 3. Consider the model (5) with X, E R satisfying Assumption (A) of Bühlmann & Yu 2003). If base learner (e is the smoothing spline learner of smoothing degree r and degrees of freedom df, detailed in (B.1) and (B.2) in Appendix [ & E -W(.)) (2)) with v r, then there exists an optimal number of iterations tn = O (2o4F()) such that f(ta) achieves the minimax- optimal rate, O (n-o/(22+11) for the function class W(x:22 in terms of MSE, as defined in 177
Theorem 3 shows that the LyBoost-CUn CUT and L-Boost-IMP algorithms achieve minimax optimal- ity with smoothing spline learner for one-iimensional feature X,, Even if the base learner has smoothness order r < v. the algorithms still adapt to higher-order smoothness v. attaining the opti- mal MSE rate O (2==/(22++)) asymptotically, similar to LyBoost in Bühlmann & Yu (2003). When paired with smoothing spline learner, the LgBoost-CUT and LyBoost-IMP algorithms can adapt to any eth-order smoothness of ()(,.2(x2'. For example, with a cubic smoothing spline (r = 2) and v 2, (tm) can achieve the optimal MSE rate of n --// by selecting in O 0(nn (n1/5). For higher-order smoothness, such as exceeding with 3, ((t)) can attain an optimal MSE rate of 6/7 with t, O (n²/7). While the L2Boost-CUT and LyBoost-IMP algorithms
also adapt to functions with lower-order smoothness (v < r), this adaptability may not provide additional gains in such scenarios, noted by Bühlmann & Yu 2003) for the scenario with complete data.
4.2 CLASSIFICATION
In classification tasks, the goal is to estimate the probability P(Y; > s) for given time in order to determine a predicted value for new features. In this instance, we define g(Y,) 21(Y, s) 1, and let 9, (Y;) denote it to stress the dependence on At iteration f, T=gBoost-CUT provides an estimate, denoted f! for (X))
2p,(X,) - 1. where (X,) & E(I(Y, s){X,). Here, f!() represents (() in Algorithm with the dependence on explicitly spelled out.
In line with Bühlmann & Yu (2003). estimating 2p.
can be loosely regarded as analogous
to 15
g (V.) = (X,) - + 6x for i 1, 1,...,
where the noise term e; has E(c,) 0 and var(e:) (x,, p. (X.).. }. Because the variances var(e;) for 1,..., .n are upper bounded by 1, Theorem can be modified to give the optimal MSE rates for using the BostotttUUUT and Boost- IMP methods estimate ps-
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Theorem 4. Consider the setup in Theorem 3 If Ps belongs to W(), then there exists
in O (n2r/(20t such that f(tn) achieves the minimax-optimal rate, O (n 2z/(2++)),, which minimizes MSE as defined in (7).
Next, similar to Bühlmann & Yu 2003), define the averaged Bayes risk (BR) for any s >> 0:
BR. = nnn Σ Pr (sign (X,, 1) (xi)).
Theorem 5. Assume the regularity conditions in Theorem hold. Then there exists in
O (n (222/(2+))) such that
Σ Pr (H(t)) 9x(x)) - BR, = =0 ( //-w/(20t1))) 2=1
n
Theorem 5 shows that, for LoBoost-CUT and LyBoost-IMP, the difference between the empirical misclassification rate and BR, is of order O (n (n=w/(2ot1) which approaches 0 as + cc.
5 EXPERIMENTS AND DATA ANALYSES
Experimental setup. Each experimental setup involves conducting 300 experiments with a sample size n. For = 1, ..... n, let X; (Xi........ Xp..) where the Xt, are independently drawn from the uniform distribution over [0. 1] for / = I....,P and = , n.. The responses Y, are then independently generated from accelerated failure time (AFT) model (Sun 2006), given by 15). where g = log u, and the error terms G are independently generated from either a normal distribution (0, o²) with variance o² or the logistic distribution with location and scale parameters set as 0 and 1/8, respectively. For = 1,..., n, we generate m monitoring times independently from uniform distribution over [0, r). and then order them as u1 < u422
u4.m- We set 500, and ((X)) Xx 0.5| B x + B2 sin( with 3o 1,
0.25, p 31 = 0.8, and B2 = 0.8.
6,
Learning methods and evaluation metrics. We analyze synthetic data using the proposed LoBoost-CUT (CUT) and L2Boost-IMP (IMP) methods, as opposed to three other methods: the oracle (O) method uses the oracle dataset O - {[ X,) i = 1, 1,...,n1) with true values of (X,)) the reference (R) method uses the complete dataset O (i), i 1...., mi), and the naive (N) method employs surrogate response Y, & [L, R,) if R, < and Y, e L. otherwise, together with X4. While the O and methods require full data availability, which is unrealistic in real-world applications, they provide upper performance bounds under ideal, fully informed conditions. This, in turn, benchmarks how our methods perform in realistic settings.
Synthetic data split into training and test datasets in 4 ratio. We assess the performance of each method using sample-based maximum absolute error (SMaxAE), sample-based mean squared error (SMSqE), and sasple-based Kendall's (SKDT), for regression tasks, along with sensifivity and specificity for classification tasks. Details are provided in Appendix F.I
Experiment results. Figure summarizes the SMaxAE, SMSQE, and SKDT values using boxplots for predicting survival times. The N method produces the largest SMaxAE and SMSqE values yet the smallest SKDT values, whereas the proposed CUT and IMP methods outperform the N method, yielding values fairly comparable to those of the R method. Figure 2 displays the sensitivity and specificity metrics for predicting survival status, where sensitivity plots for s and specificity plots for = are omitted because no corresponding positive and negative cases exist; and the CUT and IMP methods produce identical lines. The N method produces similar specificity values but significantly lower sensitivity values compared to the proposed CUT and IMP methods. To evaluate how the performance of the proposed methods is affected by various factors, including sample sizes, data generation models, noise levels, and different implementation ways of ICRF, conduct additional experiments in Appendix
Data analyses. We apply the proposed CUT and IMP methods well the N method to analyze two datasets, Signal Tandmobiel data and Bangkok HIV data, whose details are included in Ap- pendix F.4 In addition, implement& procedure (denoted COX) based the Cox model, though

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SMaxAE
2.0
1.0
1.5
SMSqE
0.2
0.8
1.0
0.1-
0.5
CUT
IMP
CUT
0.4
CUT
IMP
1.0
1.5
SMaxAE
SMS
0.2
3KDT
1.0
0.6
0.0
N
CUT
IMP
N
CUT
IMP
R
CUT
IMP
Figure 1: Experiment results of predicting survival times. The top and bottom rows correspond to the lognormal AFT and loglogistic AFT models, respectively.
methods
CUT
Senstiveyy
1.00
1.00
Sersstiviyy
1.00
0.75
0.75
0.75
0:75
0.50
0.50
0.50
0.50
0.25
0.00
0.00
0.00
0.00
(a) The lognormal AFT model
(b) The loglogistic AFT model
Figure 2: Experiment results of predicting survival status.
its results are not directly comparable to those three methods. Details are provided in Appendix G.2
Figure reports boxplots for the values of exp (X.)). where (X,) represents an estimate from method. Clearly, both the CUT and IMP methods yield comparable estimates, while the N method produces smaller estimates. The results from COX appear to closer to those from the two proposed methods than those from the N method.
30
velue
25-
4.5-
20-
COX
CUT
cox
CUT
IMP
(a) Signal Tandmobiel data.
(b) Bangkok HIV data.
Figure 3: Boxplots of data analysis results.
6 DISCUSSION
In this paper, we introduce the boosting algorithms tailored for interval-censored data. These meth- ods offer the flexibility in predicting various survival outcomes, including survival times, survival probabilities, and survival status specified time points. Further discussions, including computa- tional complexity and extensions, are included in Appendix E Like all methods. the validity of our approaches depends on certain conditions, as outlined Appendix A For example, using smoother matrices that fail to meet these conditions may compromise their effectiveness. Condition (C4) states the importance employing weak learners as base learners. Using overly strong base learners could violate this assumption and negatively impact the performance. Our methods inherently depend on estimation the survivor function, which typically utilizes ICRF. While this ensures robust and consistent estimation, it requires additional computational cost shown in Appendix E.2 and Table F. in Appendix F33 This computational cost reflects the price paid to achieve the methodological robustness that approaches offer.
E.2
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ACKNOWLEDGMENTS
This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). Yi is a Canada Research Chair in Data Science (Tier 1). Her research also supported by the Canada Research Chairs Program.
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APPENDICES: TECHNICAL DETAILS AND ADDITIONAL EXPERIMENT RESULTS
A REGULARITY CONDITIONS
For features, we consider the same assumptions as in Bühlmann & Yu 2003). Furthermore, we assume the following conditions.
(CI) The smoother matrix v is real and symmetric having eigenvalues {A,...., An and corre- sponding normalized cigenvectors {Q1..... Qn.. with Q Q = 1 for 1,.... n.
(C2) The eigenvalues A satisfy 0 < > S for
n. (C3) There exists least one k in (1,..., n) such that A
(C4) There exists 2 2 such that for all k with λ < 1,
p > 1/(1 - Ax.) mo
(C5) For ? defined in Theorem 2 E (29) < x for 1,2,....
The smoother matrix V, which satisfies conditions (C1) and (C2), includes projection smoothers and shrinking smoothers (Hastie al. 2009 Chapter 5.4), such as smoothing splines introduced in Appendix B To clarify further, shrinking smoothers also satisfy condition (C3), whereas projection smoothers do not. Condition (C4) encompasses the condition in Theorem ((c) of Bühlmann & Yu (2003) as a special case. It can be interpreted as follows: a large value on the left-hand side suggests that Φ is relatively complex compared the estimated noise level o², while a small value on the right-hand side implies that Å is small, which indicates that the learner either applies strong shrinkage or smoothing in the kth eigenvector direction, or is inherently weak in that direction.
B SMOOTHING SPLINES
To introduce smoothing splines, we start with considering the simple case where the features X; is onc-dimensional. For X' C IR and positive v, let
(1(022 {
g differentiable up to order e and / J (0
}r d'do dx < oo )c) )
denote the Sobolev space of the oth-order smoothed functions defined over X.
Let be positive integer. At iteration /. in Algorithm I we find a smoothing spline learner of degree r, denoted 1, by solving the penalized least squares problem:
R,000 arg min A0
M at -x()) + +X f DEN
(x.)
(NOM))) ? do
n
(B.1)
where ((06)) represents the rth order derivative of 1,(), and À is a tuning parameter. Here, the dependence of i,(t) on the tuning parameter X, smoothness degree r, and v is suppressed in the notation.
Taking v - 2 often offers viable way to handle practical problems, yielding cubic smooth- ing splines (Hastie et al. 2009). Varying À from O to xo accommodates different forms of 1,0)) Setting x- 0 imposes penalty in B.... and (())) is then natural spline that interpolates for i 1,...,;; taking À = oo leads ((()) in (B. to be the rth order polynomials e 2 r [Wang] 201T). The larger is, the weaker base learner is. Though the value of À is crucial to the success of the learning process, it is difficult to decide optimal or reasonable value for À when using smoothing splines. Alternatively, noting that a more interpretable
(X,, L (O,, ((t)))
parameter, degrees of freedom, defined
df Trace (V),
(B.2)
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is monotone in X, Hastie et al. 2009 p.158) suggested to invert the relationship and specify À by fixing df.
Though we start with the infinite dimensional space (x) (() in B.1 1) is showed to be a nat- ural polynomial spline with knots at all distinct X, for i 1,....77 (Eubank 1988), which belongs to finite dimensional space (Hastie et al,, 2009) Chapter 5.4; Wang, 2011). Let [N::] 1er:==1..... 1....,7y. ;.) denote of second-order differentiable basis functions for the family of natural splines, and let N and Ω denote matrices with the (i, 1) entry equaling (X;, and Jzex (z) re-
spectively. Let 0: denote the ith element of (N N + +X2) [00)"'NTT () Further, assuming the X; are all distinct for I,...,n, 2 = r = 2 can be written as
n. Hastie al. (2009 Chapter 5.4) showed that h in B.1 with
h
Níd1.
1=1
That is, the cubic smoothing spline with a prepppeeeeeee sreespecified À is a linear smoother with in (14) equaling
N (N AQ) NT (Hastie et al. 2009
Next, we consider the general case where X' C IRP, for which we may employ (B.) elementwisely to update base learner in manner similar to Bühlmann & Yu (2003). Specifically, at iteration f, consider each component Xt, of X; - (Xi,i...,,
we employ the smoothing spline with
the selected feature X where & E [p. is determined by
1. = arg min Σ (-ot (0, (())()))))) R/ (x..) ISISP
Here, (Xis) is the smoothing spline in (B.1 obtained from replacing X; in B.I with the feature X1,5-
C ESTIMATION WITH INTERVAL-CENSOREE RECURSIVE FORESTS
Here, we describe our estimation detail with the interval-censored recursive forests algorithm. Let T and D denote the total number of iteration and the number of bootstrap samples, respectively. We describe the estimation procedure as follows.
Step
We an initial estimate for S(y)Xt), denoted (10) (x)) simple way is ((0) XX) to be the nonparametric maximum likelihood estimate (NPMLE) of uncon- ditional survivor function of Y, denoted S(y) (Turnbull 1976). Then for = 1, 1...., n, we employ the kernel smoothing technique obtain smoothed estimate of S(y) X,), denoted by (0) (x). That is,
( (,)x, 1 J. ["
1 Ku(s (e) (c),,,
where KE is kernel function with bandwidth h 0.
Step 2. At iteration I, we draw D independent bootstrap samples with size [0.95n] from Oi denoted o10) ((..,))))) where [1] is the ceiling function; and keep O Ol() the out-of-bag sample for d = 1,..., D, denoting them (OB.(() .D.. (OB.(() For each bootstrap sample 0 with
1,..., D, we build a tree using two-sample testing rules for interval-censored data based the conditional survivor function S 5((-1) (y) Xz), say the generalized Wilcoxon's 's rank sum (GWRS) test or the generalized logrank (GLR) test (Cho)
et al. 2022).
Specifically, at each node, we randomly pick [P] features, with p denoting the dimension
of features, and then we find the optimal cutoff suggested by GWRS or GLR. Let ( de- note the total number of terminal nodes of the resulting tree for the dth bootstrap sample at iteration For / = 1,...,
. I let (1) denote the /th terminal node in the dth tree. At the
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Ith terminal node of the tree, estimate the survival probabilities for each node, denoted 3:0)
using the quasi-honest or exploitative approaches. The quasi-honesty ap- proach employs the NPMLE based on raw interval-censored data, whereas the exploitative approach averages the estimates of the conditional survivor function from iteration t (Cho et al. 2022). The exploitative approach computationally efficient, while the es- timator obtained from the quasi-honesty approach exhibits uniform consistency, provided regularity conditions (Cho et al., 2022). However, the finite sample performance these two approaches varies, with neither consistently outperforming the other (Cho al. 2022).
d.i) A
(e)
d.l
To presume some degree of smoothness in the true survivor function, ((t)) is further smoothed as (() using the kernel-smoohiing technique, yielding smoothed estimate of the conditional survivor function X(yl X()) CCoo et al. 2022).
$ (vlx,) = Σ S (v14 (x, All)
and
(0)
(x(x,) = Σ 1 X (v14 (x, E A!9).
Calculate the out-of-bag error as the integrated mean squared error (IMSE)
(1)
1
n2
OOB
T (R4 (LGAT)
LiAT
(1
ds Ja. (x()))))))
R,AT
dsg
R.
where n 00B - n - [0.95n] denotes the sample size O (0B,()) and a b 2 man(a,b) b).
Step 4. Averaging the corresponding quantities over trees, we obtain that
P
D
(y)x,) D
Σ
5 (IX)))
(y(x,)
D
Σ ( a (vlx,) and (0)
Σ
(8)
D
d=1
Then the final estimate of S(y)X,) is determined as A(y)X,) 3/frr (y) (y)),,, with Rops arg min gigr (t),
Step 5, We approximate S.y (o(w)) in p as
( (7(e14lXx,) X(m])x,) I(u--1 S 411 < o mz)).
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D PROOFS OF THEORETICAL RESULTS
Proof of Proposition
!
= (2(0.) - ((00) + +1
1
F (2()0)) F. (v(100))((x))) + +
m+1
Σ A. -.
2
m+1
E S(X:) Σ (o0,)Aus = 11) [((())))
j=1
E (E [H(())) ( E
(
2 ((00)))91X3) E (I
[1(())))²
F. 2 [
-E (((()))(())) E | (1(x)))3
where the first step uses [0), the third step is due to 8 the fourth and six steps come from the law of total expectation, the fifth step is from the the property of conditional expectation, and the last step uses
To prove Proposition 2 6 Corollary D and Theorems 5
we adapt the techniques of Bühlmann
& Yu 2003) with modifications tailored specific setup.
Proof of Proposition
For 70) in Line of Algorithm
we choose V such that
f( VY
(D.1)
By 13
we obtain that for f. = 1,2,
((1-1) = -
Yi (Tt)) k(01)
= Y f(t) + v
(( (t)(t-1)
where the second step is due to Line of Algorithm I the third step comes from step due to (13) Therefore,
14). and the last
7(0 = (T - ()
(D.2)
Recursively applying (.2)). we have that for = 1,2.... 2.?...
7(t-1) 1) = - 1/--111 (7(0)
= (1 1/ (Y--
- -(0)
(I v ) 1-1 ( - VY
(I y
(D.3)
where the second step
113 and the third step is due D.
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Recursively applying Line 5 of Algorithm
[ have that for t 1,
p = f +
K
5=1
= -V Σ
VD(T - V/8
/=1
Σ Ve - wyy.
1=
(1 (I - v) w) 81,
where the second step uses (14). symmetric matrix A with (I - A) being invertible, Σ/= A = (I - A)- (I - A which may
D.I. and D.3 and the last step comes from the fact that for
be derived using the same reasoning for geometric series.
Therefore, by (16). we may that B(t) = 4
Proof of Proposition
We examine the MSE in (17
E (1) - o(xi) }"]
MSE(t, L; Ø) = nn
i=1
var (x,) - o(xi)) + {1= (1)) o(xx))1²
12
12
var (x() + ( o(X))}
(xx)
var (x()
Σ E (x)) -o(x)))) x(xx))²
where the second step comes from the property that E(U?) = var(U) + {E(U))2 for any random
variable U, and the third step dues to the fact ob (X;)) is taken a constant.
To prove Proposition 4 we use the following basic properties of matrices.
Lemma 1. Let A and B be two symmetric matrices of the same dimension. Let T be the identity matrix of the same dimension as Then the following results hold:
(a). A B and A B are symmetric matrices;
(b). A is symmetric for any integer k;
(c). If A has eigenvalues {AI..... .......,n). and corresponding normalized eigenvectors (Q,....,Q... Then
(i). for any positive integer k. the eigenvalues of A are {*i,..... A with (Qu....,Qu) being the corresponding eigenvectors;
(i)) the eigenvalues of A + are {>1 + I,...,, An + 13 with (Q..... Qn} being the corresponding eigenvectors;
(i.). the eigenvalues of Are {->1,..., ーAn3 with (Q1,.... Qn) being the correspond- ing eigenvectors.
Proof of Proposition 4 Assume that V is symmetric with eigenvalues {Ai,..., An and corre- sponding normalized eigenvectors {Q .Q....., Qn). By Proposition 2 together with Lemma [ have that (1)) is also symmetric, and its eigenvalues are {1 (1 X ) ..... (1 X,
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with corresponding eigenvectors {Qi...., Qn}- Consequently, we can decompose B(t) using thonormal diagonalization as
B = 10 Q-
(D.4)
where A A1) _ diag (1 - Xx) k = 1,..,,, and the matrix Q _ (Q1,.... Qn) is orthonor-
n)
mal, satisfying QQ" = QTQ = and Q Q
Next, we examine the variance in (18)
var(l. v) = n -1
var ( (X,))
==1
n tr
{ cov
(f
n - cov (B(O),)} }
(xi) (B(O)) }
n -n
0 cov
n ir @11000110 1021 (2110--1)
n2n 'tr (o diag [1 (1 -6) 1 k = 12 0))
6²n Σ (1-
(D.5)
where the second step follows from the definition of the trace of the covariance matrix, the third step is from (T6), the fourth step applies the property of scaling the covariance matrix when multiplied by a constant matrix, the fifth step uses (D.4) and the definition of o², the sixth step is derived from the properties of the trace and the fact that Q product with a diagonal matrix and Q===-
QJQ Q = 1, and the final step follows from the matrix
Finally, we examine the squared bias, given in 18
bias? () n- Σ E B (xx( )(x)))
=n {E (B(OY)) _0}² {o (B()))) - -0}
--- (BB - o) {BB() -1) - -ō}
n - [(( (Aw) - -1) T T (101-)) - })
= n 3 T (1 - -1)" T (A - 1) ~
5 T diag {(1 21 :1 1, n
n Σ - -1
(D.6)
where the second step is due to (16), the third step is due to E ()(()))))))))))) E EX.) = <(X)), the
fourth step uses D.4], the fifth step comes from QTQ Q and Q- QT, and the last step is due to definition of μ. given before Proposition
Proof of Corollary 7 This corollary follows directly from using the properties of diagonal matrices that have entries either 0 or 1.
Proof of Proposition 5
By condition (C2), 0 S (1 - Av) < 1, and thus, bias? o) in D.6 de-
cays exponentially with increasing and var(I, V) in (D.5 exhibits exponentially small increase
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increases. Further, by D.5)) we have that lim var(t, V)
lim
ōºn n
1 (1 11
i=1
lim (1 - 11) ? (1000
= onn -
1=1
= p -1
0²,
and by (D.6,, obtain that
lim bias (t, 2 t0000
v; ()
lim t1000
14j (1 2
lim (1 - 21) 8-+00
[=1
pf0
1=1
= 0.
Therefore, by Proposition 3
lim MSE(t, 6) lim var(t, V2 + lim bias 3 (4, Φ) 1-000
5-000
62.
Proof of Proposition 6 By propositions
and
we obtain that
MSE(L, V Φ) = 6²n
(1- + Σ (1-11 21
1=1
1=1
Considering this function of only, with other quantities treated as fixed, consider the function:
n
(1) (1- +
pe 21222
which equals
(bu) 02n
{1 2(1 >1) + (1 - 2x + -!
11 21122
1=1
1=1
= =n-1 Σ (002 + mi) (1- 11-x))=t1 - -202] (1 - Ac) + 0²
=n Σ 1(02 + ma) (1 - - 20²} (1 -xe) +
(D.7)
By condition (C3), there exists at least one such that d < Considering all those & such that A < 1, we let .i,..., Kn, denote them, where S n. For j = 1...., no.
lim {002 m²:)) (1 Ak,) uu 26 -202
x
leading to
+ 12: {0²2
(1 A1e)) uut -262| x-22
lim
#00
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Therefore, for j = 1,.... no. there exists uy such that
(62 + pk; (1 Ax,) u,t1 -26 -02.
(D.8)
Letting to = max(ui...... Uno). D.8 yields that for j = 1,...,
0² + ph; (1 Ak,)) 1t+1 -20² < -02
Therefore,
D.7 leads to
no
(b()) -n Σ 0² Na, 20t + 0² <30
and the conclusion follows.
Proof of Theorem! 1
We calculate the derivative of (u) in D.7
(u) 22n
1(02 - =1+)) (1 A6) A6)=-1 02) (1 a log(1 An).
A=
Now consider those is with A 1, condition (C4) leads to (0² + (&.) (1- A > 0². Since for o² (0, ma 1). (bu) is decreasing over (0, mo - 1). By the continuity of (b()) over [0, mg - 1). we have that (0 > (11) > ..... >
any u E (0, mo 1). (1- 2mm (1 )x) which yields that (o²+ n²²) - A4) (1-da)
for any E (0, ma 1). Therefore, (u) < for all
(mmo 1), suggesting that the first [mo - 1] iterations of the L2Boost-CUT algorithm improves
the MSE over the unboosted base learner algorithm (i.e., corresponding to (0)).
Proof of Theorem 2 For a vector u. we use (u)i. {u)i. or [u]s to denote its ith element. For i = 1...., n, let (X,) & (x)) - ((X,) denote the bias term for subject i. Let 7= (c 2n).
We examine the summands of the left-hand side of (9))
E { (x.) -x(x)) {
F. (x,) (X,) F. (()) + E F (x)x(x(((xx))
E (( - - ((x)) [ a (x)) 
E
- 125002
E
{ (200)))) ( (1a))))) 1
(
E
Iin | a (ecxoeeg
Σ ()
Σ
w)+1) 1I (11) a (oo)
1
(x))))) E
(1 -
(D.9)
where the third step is due to (16); the fourth step is due to the definition of é;; the fifth step is derived under assumption that X, treated constant; and the sixth step is due to Proposition|
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Then, examine ((9) (X,):
600) (X:) = E o(Xi)
E (B 8)
{BO) - 1) o]
((o(20) - 11) Q 0-1) の
Q diag - 1)
n2 0T
a
Σ Quelx 1)( 1k
O (exp(-Catt) t 00,
(D.10)
for some positive constant Cb. where the second step is due (16). the third step is due to E (O.)) -E(Y) = <(x)), the fourth step is from (.4). the fifth step comes from the definition of (1), and the six step comes from the definition of f Q i7 with Qa representing the (i, k k)th element of Q.
Next, we examine E ([[ - (1 - w) 10 :
E
[1 -I- 1o) )
9 - 1)
aे ((T 1)(t1))
E
k
1-1
at [Qdiaag(] >3 j 1, 1 ++1 19-1-
k:
1)
at [Qdiag ((1 (1- j = I,..., n) Q 10 1-1ー
E
k
E
c Σ (15(1-2)))
Σ ojl Ev
u=1
F 1) + O (exp(-Cat)) Cyt)) as - x
(D.11)
for some positive constant Cy.
Combining D.9 with D.10 and D.II yields
(x,) - o 3"
n
E
9 10 (exp(- (exp(- E( 0 (exp(-C,t))
n
= =E(()) + (xxp(-C))
for some positive constant C..
Let denote the smoother matrix for the smoothing spline of degree r and degrees of freedom df (equivalently expressed in terms of tuning parameter X). Given the tuning parameter X, the eigenvalues y are arranged in decreasing order and are written as:
Proof of Theorem 3
U.n for = r + 1,...,
= 1, d =
(D.12)
x qt.
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where Utn depends on 5t defined in Appendix B 2009).
Utreras 1983
Bühlmann & Yu 2003
Hastie al.
By Assumption (A) of Bühlmann & Yu 2003 Utreras
1988
showed that for large there cxists
a finite positive constant ag such that
Qn R agl
(D.13)
For f "/ (a, b]. there exists finite positive constant such that
Σ
4i
S M.
(D.14)
1=++1
Let X/ap- Then by D.13) the d for / = + 1,... Y2 in D.12 are
1-2r
A
(D.15)
1-2r
Arar + 1
Then (D6 can be bounded by
bias (1, V; =
Σ
1 (1(1 2x)2
1=++1
21+2
22
1~0
RJ 12
A/2r
+
|+++
20+2
1-
1
22
122
max
|=++,...,m
212r +1
1=++1
1
21++2
< M
max
1
(mrtl,...
i12r +1
A .M
max
exp {n(())}
(D.16)
Impel
with
(1) = (2t + 2)
APr + +1
2u log(1),
(D.17)
where the second step D.15 and the fourth step uses (D.14). Taking the derivative of DD.17 yields
2r(21 + 22
2v
(1)
1(//2r 1)
2r
2t + 2 v(Ar +1)
1(A12r +1
Now, consider any positive integer nı with r < n1 S n, and
2 {o(2n] (2)) 1.
(D.18)
Then (1) 2 0 for any < S n1, therefore, n1!) is increasing and exp(n(!)) for 0 $ n. Therefore, for any < S n1, we have that
(x()) < exp(n(n1)}
20+2
1
Ānf +
n1
21+2
1
1
n2
20
(D.19)
2n
+
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Applying [D19 to D.16 gives that for nı and t in (D18 and t 2 {u(2n +1))/((2r) have that
21+2
bias (t, vi - M
1 2
An + 1
2 Mn
2x as n2 + 00,
(D.20)
and hence, (t, () of order 0
[nj 20)
as n - 00.
Now we examine D.5 For any in
D.18 by D.12
[1 (1 (1-1)) 22
var V
r
n
62m
{1- - (1 21)
n
n
1=1+t1
= =0
11
+
11- (1)
(D.21)
n2
n
-= +1
By Bernoulli's inequality that (1 a) 2 ab for S and b 2 0, we obtain that
(1
A1)
++| - 1 - (1- (1t + 1)} = >r(1 + 1).
Therefore, for t in D.18 by D.15).
obtain that
0²
Σ
(1-
n
[=
n
n1
1=
8 (t + 1)²
n
(212 + +1) 2
1=
( + 1)₂
n
(=n++1 (x12r)
62(1 1)22
1
du
n
(u22))
2 ( 1)2
1-1r
ni
X2 -1) 1)
71
as 121 00.
(D.22)
Applying (.200) D.21) and (D22 to Proposition we obtain that
MSE(t, vr O + (n 20)
as n4 oc.
Treating the order as a function of n1. it is minimized as O (n 20/(2x+1) by taking nı O (1)/(2x+1)). Therefore, for this n L in D.18) can be taken as 4, A O (12r\
Proof of Theorem have that
By Theorem 2.3 and the discussion on Page 102 of Devroye et al. (199)), we
((a)) +Y.) BR < 2 MSE(t, fz)
n
i=1
o (1=o/(e0+t))
as n + 00,
where the last step is due to Theorem 4
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E DISCUSSIONS AND EXTENSIONS
E.1 LEARNING RATE FOR BOOST AND OUR PROPOSED ALGORITHMS
In traditional boosting methods, learning rate, denoted as ( is introduced to control the contri- bution of (()) each iteration for t 2,.... scaling how much it corrects the prediction error of f(t);
f f ta
where
Σ L (Y, (Ye, + +
a0 = arg min attER
n
However, in our algorithms, which are based on the L2 loss function, the learning rate a is inher- ently incorporated within the optimization process for h()) Specifically, when using the L2 loss, the minimization problem for (+) simplifies to
a arg min G
Yi (x))
which is integrated naturally into the computation of (t) because of 11
7 = arg min 4O)
Σ Yi ((1 (x.-
n
As result, Algorithm does not require explicit learning rate parameter, as (() is effectively determined as part of the optimization of (t)
E.2
COMPUTATIONAL COMPLEXITY
Our proposed BrBoosttCUU method in Algorithm 1 basically comprises two components: ICRF and boosting. The computational complexity of ICRF O(n)), with 1 < y For smoothing splines, when implemented efficiently, the complexity can be O(n) (Hastic et al. 2009. Chapter 5). With & boosting iterations and smoothing splines as base learners, the total computational complexity Offn). Therefore, the overall computational complexity of the Lg BoosttCU method On fn).
(Cho et al.
2022).
E.3
POSSIBLE EXTENSIONS
The L2Boost-CUT framework can be extended to Ly loss functions for handling interval-censored data, where y an integer, and the L, loss function given by
L (o(Y)), x {g(Y)) f(X.)3
Σ (O(X))
k=
with { replaced by its transformed form (8) together with 9 Here, is extended to take any value in (1,....4).
As shown in (T). the linear derivative the T.2 loss with respect to first argument suggests closely related increment terms in both BoBosst-CUT and Boost--IMMP thus often leading sim- ilar results. However, this connection does not hold for the loss function Lg when q 2 Conse- quently, the L, Boost-CUUU and L.y Boost-ssMMM IMP methods likely yield more different results, where the L.g Boost method is obtained by replacing the L2 loss the LyBoost-IMMe method with the Lg loss.
While extending the LoBoost-CUT method to accommodate the L, loss for q 2 3 is straightforward, adapting it to any general loss function for constructing adjusted loss function like LCUT in
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10 to ensure Proposition [ presents significant challenges. In contrast, generalizing the LyBoost-
IMP method to any loss function is straightforward by using imputed values determined by the transformed response in (8)
For example, considering widely used loss functions, such as exponential loss function L(u, v)
exp(-wx) ww) (Schapire & Singer 1998) and the binomial deviance loss L(u, = log (1 +
exp(-2uv) (Friedman et al. 2000). one may apply the censoring-unbiased transformation ( to
these loss functions and adapt the proposed LgBoost-IMP method enable boosting algorithms
like AdaBoost (Freund & Schapire| 1996) and LogitBoost (Friedman et al., 2000) to handle interval-
censored data. For XGBoost (Chen & Guestrin 2016). one may replace 7 in (2) of Chen & Guestrin
(2016) with the transformed unbiased loss function (10). This extension would allow XGBoost to handle interval-censored data.
While L-Boost-CUT can be extended to boosting frameworks with Ly losses (q 2 3) and Boost IMP can be extended to accommodate any loss function procedurally, establishing theoretical prop- erties for these extensions is nontrival. Unlike the L2Boost-CUT method, optimal learning rates would need to be estimated iteratively, complicating updates and disrupting the elegant form of the boosting operator in Proposition 2 Developing theoretical guarantees for these extensions presents substantial challenges and remains an open problem.
The principles behind our methods could potentially be adapted to other machine learning frame- works, such as deep learning or ensemble methods. Exploring this adaptation could an interest- ing avenue for future research. Furthermore, while Theorem demonstrates that the LzBoost-CUUT and Ly LyBoost-IMP algorithms consistently outperform unboosted weak learners terms of MSE, this result is established under the assumption of weak base learners (as stated in Condition (C4)). Quantifying the extent of improvement provided by boosting over unboosted learners and investi- gating how this improvement depends on the form of weak learners, particularly in the context of interval-censore data, would be valuable directions for future research.
F DETAILS OF EXPERIMENTS AND DATA IN SECTION 5
F.1 DATA SPLITTING AND EVALUATION METRICS
The dataset is divided into OIR A ((Y, X,, ((Xi), wij)
1, ..... M1: 1,-.., m) and ......,) m) in 4 ratio, where n1 = 400 and n2 = 100. Take OF & {{Y, X,, u.y :1 1,.... = I....,m) as training data and O A {(x,, (XX)) : i- n1 + I,...,,, + +22 as test data. The training data O are used to obtain estimated fo in (2). denoted ft, using the proposed methods introduced in Section while the test data OJT are employed to evaluate the performance of fa, For classification tasks,
TE & {(Y,, X,, (X,),, [ny i n1 + 1,... 01 n2: -l....
fa, 1-,, derived from the E.BCooost-basee algorithm.
For regression tasks, the first metric represents the sample-based maximum absolute error (SMaxAB), defined as the infinity norm of the difference between exponential of the estimate and exponential of the true function with respect to the sample:
ji of
exp [in (X.)) exp (o(X.))
max +i.
X,
and the second metric reports the sample-based mean squared error (SMSqE), defined as:
P of n2
Σ
fi (x.) exp (o(x,))}
exp
The smaller these metrics, the better the performance of the estimator ft,. addition, we consider the cample-based Kendall's (SKDT), defined as
C nD
n2 - 1)/2'
where n.. and n denote the numbers of concordant and discordant pairs, respectively. For i, ny + 1, ...... Bi + n2, pair is called concordant if fr, (x,)) > fr, (Xv) and d(X)) ((x,), and
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discordant if fi (X,) fú, (Xp) and <(x)) o(X;). This metric evaluates the concordance between fü, and its target function @ from a different perspective. The bigger this metric, the better the performance of the estimator fn.
For classification tasks, we write fü, as f (X,) explicitly to show the dependence of the estimates on time s. We predict the true survival status at time with 2, 3, 4, based on using whether fa (Xi) is greater than O for 71 + 1, .....,.7............. R2 + n2. Specifically, for i ni + ....;........................... 18 + n22 if fü, (X,, > O, we predict the survival status at time s as 1; otherwise, we predict it as -1. We evaluate classification performance by using the test data Oje to calculate the sensitivity, defined as the proportion of correctly identified positive cases among the true positive cases, indicated by {i M1 + 1,...,77, 741 + n2;eexpp exp(x(X,))
8), and the specificity, defined the proportion of correctly identified negative cases among the true negative cases, indicated by {i : nh + I,..., 11 + n2g (xx))(X.)) I s}. Sensitivity and specificity assess classification results from different perspectives. The larger these metrics, the better the performance the estimator fx
F.2 LEARNING METHODS IN EXPERIMENTS
Regardless of the value of n, we set w = 5 for Algorithm T as a stopping criterion, and take cubic smoothing splines base learners with weak base learners. Bühlmann & Yu (2003) showed that the shrinkage strategy (Friedman, (001) make base learners weaker by multiplying small constant
2 in Section B Suggested by Theorem [] take
the smoother matrix V. In other words, for a small constant u, the linear smoother learner with smoother matrix 4, u x y weaker than the linear smoother learner with smoother matrix v. Thus, as in Bühlmann Yu (2003), for (B.1). we set df 20, and replace & in (14) with V and = 0.01. The shrinkage strategy equivalent replacing Line 5 of Algorithm with
(((+1) = (()) + (1+1)
For ICRF, specify the splitting rule GWRS, described in Appendix C adopt an exploitative survival prediction approach, use a Gaussian kernel with bandwidth en 1/5 and take and D = 300. Here, e is the inter-quartile range of the NPMLE, and Munin is the minimum size of terminal nodes, set 6.
F.3 COMPUTING TIME COMPARISON
To access computational complexity, we record the computing time for one experiment by applying the five methods to synthetic data generated from the lognormal AFT model with 0.25 in Section[5] Computing times (in second) are reported in Table F. for three sample sizes, where we separately display computing time for implementing ICRF from that for implementing both unbiased transformation and boosting (UT + B). The implementation of the proposed methods requires a lot longer time than that for the O, R, and N methods, expected.
Method
R
N
CUT
IMP
Size
ICRF
UT B
ICRF
UT
500
1.037
0.969
1.053
493.462
3.262
494.319
2,611
1000
2.200
2.190
2.148
2633.012
11.058
2668.756
7.935
1500
4.671
4.756
4.564
8709.231
18.440
8725.549
14.509
Table F.1: Computing times (in second) using a cluster with I node, where UT and B represent the procedure corresponding to unbiased transformation and boosting, respectively.
F.4 SIGNAL TANDMOBIEL" R DATA AND BANGKOK HIV DATA
The first dataset, called the Signal Tandmobiel R data, arose from a longitudinal prospective oral health study, conducted in the Flanders region of Belgium from 1996 to 2001. This study initially
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sampled 4,430 first-year primary school children who underwent annual dental examinations per- formed by trained dentists. Further details can be found in Vanobbergen et al. (2000). Our analysis focuses subset of the data with 3737 subjects, whose features were fully observed, specifically examining the emergence times of the permanent upper left first premolars (tooth 24 in European dental notation). Following Komárek & Lesaffre (2009), manent teeth do not emerge before this age. The response variable Y; represents the emergence time of tooth 24 since age for child Because the dental examinations take place annually, the observed Y; is inherently interval censored by design. Among the participants, 1611 children are right-censored and others are truly iherval-censored The features are defined follows: X₁1 if the child is a girl and I otherwise; X12 O if the primary predecessor was sound, and 1 if it was decayed, missing due to caries, or filled; and X represents the scaled age at which the child started brushing teeth.
the origin time at age 5, as per-
The second dataset, called Bankok HIV data, includes the information on the incidence of Human Immunodeficiency Virus (HIV) infection. To identify associated risk factors to guide prevention efforts, the Bangkok Metropolitan Administration conducted cohort study (Vanichseni et al.) 2001) in Bangkok from 1995 to 1998. The study enrolled 1124 participants who were HIV negative the time of enrollment. These participants were repeatedly tested for HIV at approximately four-month intervals over the study period. The response variable Y, represents the time when participant i was first tested positive for HIV. Among those participants, 991 were right-censored, meaning they were never tested positive during the study period, while the remaining were interval-censored, meaning the exact time of seroconversion is only known to occur between two testing intervals. The features are defined follows: X₁1 if the participant is a female and otherwise; X22 if the participant has a history of injecting drug use and 1 otherwise; and X represents the scaled age at enrollment.
G ADDITIONAL EXPERIMENTS
To comprehensively evaluate the performance of the proposed methods, here we conduct additional experiments examine how their effectiveness may be influenced by various factors, including sample sizes, data generation models, noise levels, and different implementation ways of ICRF. The details are presented as follows.
G.1 ALTERNATIVE SAMPLE SIZE AND DATA GENERATION MODEL
To assess how the sample size may affect the performance of methods, conduct additional experiments in the same way as in Section but replace n = 500 by = 1000. Results of predicting survival times are reported in Figure G.I demonstrating the same patterns observed for Figure
1.0
SMaxAE
1.5
SMS
0.2
SKOT
1.0
0.1
0.8
0.5
0.7
0.0
N
CUT
IMP
CUT
IMP
R
CUT
IMP
Figure G.1: Experiment results of SMaxAE (left), SMSqE (middle), and SKDT (right) for predicting survival times with 1000, for the lognormal AFT model with 0.25. O, N, CUT, and IMP represent the oracle, reference, naive, CUT, and IMP methods, respectively, as described in Section
5
In contrast to the experiment setup in Section 5 we take p 5, r 12, m ((x,)
|o| 0.5| + X,X3, + 82 (xs.).s.)) with Ag 1, 31 = 0.8, and 83 0.8, where X2,; and X4,; are inactive input variables for model (5). but they are still involved the boosting procedure. Figure G.2 summarizes the values for the metrics, SMaxAE, SMSQE, and SKDT, across 300 experiments for predicting survival times. The N method results in the largest SMSqE vet the smallest SKDT, though the SMaxAE for the N and proposed methods are similar. Figure G.3 reports the values for two classification metrics, sensitivity and specificity, across 300 experiments for predicting survival status. The N method produces the worst results at = 2 and = 3, with
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the lowest specificity at 4. In contrast, the proposed methods only show reduced sensitivity at
s
0.8
SMaxAE
0.9
SKDT
0.4
0.7
0.2
0.6-
0.0
0.5
0
N
CUT
IMP
N
CUT
IMP
R
CUT
IMP
41
SMaxAE
SMSqE
0.6
SKDT
0.8
0.2
0.4
CUT
CUT
CUT
IMP
Figure G.2: Experiment results of SMaxAE (left), SMSqE (middle), and SKDT (right) for predicting survival times with different survival models. The top and bottom rows correspond to the lognormal AFT and loglogistic AFT models, respectively. O, R, N, CUT, and IMP represent the oracle, refer- ence, naive, CUT, and IMP methods, respectively, described in Section 5
methods
CUT
IMP
Senebaity
1.00
0.75-
Specifiity
1.00
0.75
Seneebitt
1.00
0.75
Speetfitt
1.00
0.75
0.30-
0.30
0.30
0.30
0.25-
0.25
0.25
0.25
0.00
0.00
0.00
0.00
(a) The lognormal AFT model
(b) The loglogistic AFT model
Figure G.3: Experiment results of predicting survival status with different survival models. O, R. N, CUT, and IMP represent the oracle, reference, naive, CUT, and IMP methods, respectively, as described in Section 5 Specificity plots for = 1 are omitted because negative cases exist.
G.2 NOISE LEVEL COMPARISON AND Cox MODEL
To access the sensitivity of our methods to the noise level of data, addition to a = 0.25 considered in Section 5 for model 15 with N (0, )²), we further consider = 0.5, 1, or 1.5. Increasing makes survival times more variable, thus spanning over a wider interval. Consequently, T is set as 15, 80, 100 to generate interval-censored survival times. The results reported in Figure G.4 in the same manner as for Figure T in Section 5 The O, R, N, CUT and IMP methods reveal the same patterns those observed in Figure The N method performs the worst, the O method performs the best, and our proposed CUT and IMP methods outperform the method. When the noise level a is more substantial, the differences between our methods and the N method are considerably enlarged, and the performance of our methods becomes very close to, or nearly the same as, that of the O and methods.
We further consider two additional methods here. The first method, denoted as YAO, employs an existing ensemble approach for interval-censored data: the conditional inference survival forest method proposed by (Yao et al. 2021), where predicted survival times are provided by the R package ICeforest. The results from the YAO method are in good agreement with those produced from our proposed CUT and IMP methods. However, the SKDT values from the YAO method appear slightly more variable than those from our methods.
In the second method, denoted COX, we manipulate synthetic data to create right-censored data ((Y,, A,,X) : 1...., n²), with useudo-survival time Y; defined as in Section 5 and an arti- ficially introduced right-censoring indicator A, Here, we consider the best-case scenario where no subject is censored, with A; set to for all ...... We then fit the data with the Cox model, where predicted survival times are taken as the medians of the estimated survivor functions by extracting the "median" column of the survfit.coxph object in the R package survival. While the results from the COX method are not dircctly comparable to the other six methods, which are primarily nonparametric-based it is interesting that the COX method can sometimes outperform the
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R method, especially when a is small with value 0.25, as shown by the SMaxAE and SKDT values. However, when a is large with value 0.5, 1 or 1.5, the COX method does not outperform the O and R methods our proposed CUT and IMP methods, as shown by the SMaxAE and SMSqE values. Nevertheless, its SKDT values remain better than other methods, except for the O method; this may be attributed to the absence of censoring in the COX method. Suggested by the SMSqE values, the COX method can even perform worse than the N method when a is not small.
2.0
0.3
1.0
SMaxAE
1.5
SMSqE
0.2
SKD
0.8
1.0
01
0.6
0.5
0.0
0.4
COX YλO
R
CUT IMP
COX YAO
0
CUT IMP
COX YAO
N CUT IMP
1.00
SMaxAE
SMS
1.0-
SKOT
0.75
0.50
0.5
0.25
0.00
COX YAO
R
N
CUT IMP
COX YAO
0
R.
N CUT IMP
COX YAO
R
CUT IMP
40
100
1.0
SMaxAE
30-
SMSOE
75
SKOT
20
50-
-0.5
-1.0
COX YAO
0
R
CUT
IMP
COX YAO
R
CUT IMP
COX YAO
CUT IMP
1.0
SMaxAE
40-
SMSqE
150
SKDT
+
100
0.0
20
-0.5
-1.0
COX
YAO
0
R
CUT IMP
COX YAO
R.
CUT IMP
COX YAO
R CUT IMP
Figure G.4: Experiment results of SMaxAE (left), SMSqE (middle), and SKDT (right). for predicting survival times with varying noise levels. From the top to bottom, the four rows correspond to the lognormal AFT model with a = 0.25, 0.5, I, and 1.5, respectively. COX is the procedure of fiting the Cox model to pseudo-survival times; YAO represent the method Yao et al. (2027); O, R, N, CUT, and IMP represent the oracle, reference, naive, CUT, and IMP methods, respectively, as described in Section
G.3 SURVIVOR FUNCTION ESTIMATOR COMPARISON
As detailed in Appendix C the implementation of our methods employs ICRF to provide consistent estimation of the survivor function, and we take K = 5 and D 300 to run experiments in Section 5 (as well as those additional experiments in Appendix G). To see how different choices of K and D may affect the performance of the proposed methods, here synthetic data generated in Section S using ICRF with different values of K and D, where we set == and D 1; K and D 100; K and D 300; and K 3 and D 300; and we denote the resulting CUT methods as CUTI, CUT2, CUT3, and CUT4, respectively. In addition, we implement ICRF using quasi-honest survival prediction method, as discussed in Appendix C and the comprehensive greedy algorithm (Breiman| 2001). and we denote them as CUTS and CUT6, respectively. We report the results in Figure G.5 in the same manner as for Figure [ The results demonstrate that the CUT method with K 5 and D = 300 (the one with heading CUT in Figure G.5 tends to perform the best, although all other methods produce fairly close results.
implement the CUT method to
H CONVERGENCE ANALYSIS OF EXPERIMENTS
This appendix assesses the convergence of the proposed methods. For f E F, let R(f) denote the approximation of the empirical risk function. In Figures H.1 H.3 we plot the values of ft (f(1)
and ft (f!)) against the number of iterations for the experiments in Section 5 The results clearly
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SKDT
0.7
0.5
0.6
Figure G.5: Experiment results of SMaxAE (left), SMSqE (middle), and SKDT (right) predicting survival times with varying ICRF estimators.
show that R (f(t) and R (11²) approach zero as increases, confirming the convergence of the proposed algorithms.
0.
0.4
0.4
0.4-
0.3
0.4-
0.3
R(/0)
0.2
0.2
0.1
0.3
100
100
300
200
Iteration
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The top to bottom rows correspond to the lognormal AFT and loglogistic AFT models in Section respectively. From left to right, the columns represent the O, R. N. CUT, and IMP methods, respectively. Here, O, R, N. CUT, and IMP represent the oracle, reference, naive, CUT, and IMP methods, respectively, as described in Section
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Published as a conference paper at ICLR 2025
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Figure H.2: Predicting survival status the lognormal AFT model in Section 5 Plots of R (1!
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Figure H.3: Predicting survival status the loglogistic AFT model in Section versus the number of iterations. From top to bottom, each row corresponds to = 1,2,3 3, and 4, respectively, From left to right, the columns correspond to the O, R. N, CUT, and IMP methods, respectively. Here, O, R. N. CUT, and IMP represent the oracle, reference, naive, CUT, and IMP methods, respectively, as described in Section
32


Reviews:
[{"review_id": "aKremC8TfV", "reviewer": "Reviewer_4h4i", "summary": "The paper presents boosting algorithms for regressors and classifiers  for the censored data. The censoring naturally introduces a bias into the estimator as it collapses  the mass of the tails of the underlying distribution on the censoring boundary.  In particular, the main contribution of the paper is in presenting the novel boosting framework with censoring unbiased transformation which focuses on modifying the loss function. Authors also present a model for imputing the missing data in censoring context. Authors investigate the theoretical properties of the algorithms and conclude that incorporating the spline estimators into the base learner results in optimal MSE rates for the predictor.", "questions": "1) What are the scaling capabilities of the models to the large datasets? (provided datasets are of order of magnitude 10^3? It would be good to see experiments for varying sizes of the dataset.\n2) The imputation algorithm and the CUT boosting in the presented experiments (especially Figure 3, E.3) seems to produce similar results. Can you provide more insights into why that is?\n3) It would be great to see performance for other real-life datasets with already known benchmarks to demonstrate that the presented method performs well against other well explored survival dataset problems. \n4) For the synthetic dataset, e.g. scenario 1, how does the performance changes based on the amount of the noise in the data, so when you change the variance in the error term $epsilon_i$, e.g. instead of only 0.25 variance for normal, when you increase the noise for to 0.5, 1., 1.5?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The paper is mathematically well written. The notation is consistent and precise. Definitions and proofs are provided in the Appendix.\n- The theoretical results are novel, clearly structured and formulated. The implications of each theoretical result are well discussed and framed within the literature context if relevant. \n- The proposed algorithms are experimentally tested on the synthetic dataset under various scenarios. The method is further applied to real-life dataset.", "weaknesses": "1) The underlying part of the model is estimating survival function. It would be interesting to include the experiment with different survival function estimators to assess how sensitive the method is to the potential biases of the estimator of  the survival times as this part of the model is not properly covered by presented theory. \n2) It is not clear how sensitive the proposed method is to the noise in the underlying data.\n3) Authors did not provide much insights into how well the algorithm scales to the larger datasets.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JJUyPdu4Zg", "reviewer": "Reviewer_mmSc", "summary": "In this work the authors propose a new algorithm based on boosting for interval-censored data. They provide an extensive theoretical analysis and an empirical comparison of both simulated and real data, for which their proposed methods have a performance similar to that of the state of the art.", "questions": "see limitations\n- are there any additional performance metrics that could be used? like the c-index (or a variation of it)?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The article is clearly written and provides a thorough theoretical analysis of the proposed algorithm, which is really important to ground further applicative work using this method in practice.", "weaknesses": "There are several limitations of this work.\n- my main concern is that the paper does not clarify the empirical improvement compared to the use of ICRF. It is slightly different indeed to use boosted trees compared to random forests (bagging), but the article fails to show a clear gain in performance, both on simulated and real data\n- there should be more baselines, like Cox models, even if they are designed for right-censored data rather than interval data. There are very few baselines included in the benchmark presented here. It's unclear whether the I method is similar to ICRF or a novel method. If not, ICRF should be included as a baseline to demonstrate the contributions of this work clearly.\n- I don't think there is any mention that the code to use L2Boost-CUT or L2Boost-Impute will be made available, is it?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bPAXrDEGPh", "reviewer": "Reviewer_uzby", "summary": "The manuscript introduces a framework that extends boosting techniques to effectively handle interval-censored data. The authors offer a thorough theoretical analysis, examining mean squared error (MSE), variance, and bias, building on foundational results from Bühlmann & Yu (2003). The proposed methods are further substantiated through experiments conducted on both synthetic and real-world datasets, demonstrating the framework’s applicability and effectiveness.", "questions": "- In page 3 line 108, should the survival probability be P(Y>s)?\n- In page 3 Equation (5), what is the update compared to Equation (3)?\n- In traditional boosting methods, a learning rate is typically included. Is there a similar parameter in Algorithm 1?\n- In Condition (C1), what does $Q_i^2$ mean? Does it denote $Q_i'Q_i$ as $Q_i$ is an eigenvector?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The manuscript is clearly written and accessible, making the methodology easy to follow.\n- The authors conduct a rigorous theoretical analysis of their proposed methods, assessing mean squared error (MSE), variance, and bias to establish a solid foundation for their approach.", "weaknesses": "- In Proposition 2, using a distinct notation for the smoother matrix would help prevent confusion with the survival function $S$.\n- Adding experiments for the Cox proportional hazards model would strengthen the manuscript's applicability and support its conclusions.\n- Additional ensemble methods exist for interval-censored data, such as:\n    - Yao, W., Frydman, H., and Simonoff, J.S., 2021. An ensemble method for interval-censored time-to-event data. Biostatistics, 22(1), pp.198-213.\n\n  Including comparisons with such methods would enhance the evaluation of the proposed approach.\n- In both synthetic and real data experiments, the proposed CUT method and the imputation approach (Bian et al. 2024a) yield comparable results, with the imputation method occasionally performing better (e.g., in real data). Could the authors further elaborate on the unique advantages of their proposed method?\n- In Appendix A, the authors outline several conditions primarily related to the smoother matrix $S$. Providing relevant examples of $S$ that meet these conditions would improve clarity.\n- All theoretical results assume consistent estimation of the survival function. How does the convergence rate of the survival function estimator impact the final estimator?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zZZOnsn6fB", "reviewer": "Reviewer_VgwR", "summary": "The boosting method is widely used in machine learning field, but it is not obvious to extend it to interval-censored data, that is, the outcome value is not specifically given but its interval is given. To apply the boosting to such datasets, the paper applies the method called the \"censoring unbiased transformation\" (CUT) to the loss function. As described in Proposition 1, the application of CUT does not change the expected loss between the interval-censored data and the true (uncensored) data, if we know the conditional survivor function $S$ (in reality we need to estimate it from the dataset). With this conversion of the problem, they showed that we can apply L2boost, an existing method of boosting for regression datasets (it can be extended to binary classifications).", "questions": "Key questions\n\n-   It uses ICRF to estimate the conditional survivor function, but perhaps can we just impute interval-censored outcomes by ICRF (instead of using (Bian et al., 2024a))?\n-   Section 1.1: The paper presented several tree-based interval-censored regression methods, but is it difficult to extend these methods to boosting? If so, why?\n-   Section 1.2: It states that L2Boost-Impute is a proposed method, but the procedure appears only in Section 5, and the imputation method is just a reference. What is the novelty?\n-   Section 5, Figure 1: Why the methods \"O\" (oracle) and \"R\" (reference) are compared in these plots? It is true that these results uses unknown information in reality and they can produce better results, but it looks that these results are not discussed in the paper.\n\nMinor questions and suggestions\n\n-   Some overlaps of letters found, so please consider replacing either of them with another letter.\n    -   $L$: overlapped between the \"loss function\" and the \"left of the interval\"\n    -   $S$: overlapped between the \"conditional survivor function\" and the \"smoother matrix\"\n-   Section 3.1, line 182: Should $L = l, R = r, l < Y \\leq r$ be $L = l, R = r, L < Y \\leq R$, as far as reading the succeeding equation? (Does it mean that the distribution of $Y$ depends on the observation of the random variables $L$ and $R$ but independent of the distributions of $L$ and $R$?)\n-   Section 3.1, line 187: It looks that the randomness of $M$ is not used anywhere else. How does the randomness work?\n-   Section 3.2, line 238: It states that we need to estimate $S(y|X_i)$, but the procedure has not been described at this point. Please consider referring Section 3.3 as the section that describes how to estimate it.\n-   Section 5: The \"sample-based maximum absolute error\" is abbreviated as \"S-MAE\", but please consider another abbreviation, since \"MAE\" also stands for **mean** absolute error. (It is confusing since \"S-MSE\" represents the \"sample-based **mean** squared error\".)\n-   Section 5, line 471: adpots --> adopts\n-   Section 5, line 510: boxlplots --> boxplots\n-   Section 5, Figure 1(b): lines for \"CUT\" are not visible since they look overlapped with other lines, so please change line style or transparency so that the lines appear.", "limitations": "", "rating": 5, "confidence": 2, "soundness": 4, "presentation": 2, "contribution": 2, "strengths": "-   The paper found that, by CUT, existing L2boost algorithm can be applied to interval-censored datasets.\n    -   There is a constraint that we need to estimate the joint distribution of $X$ and $Y$ without knowing $Y$ itself but knowing only intervals. They showed that estimating the distribution by ICRF experimentally worked well.\n-   Theoretical convergence rates and lower bounds of MSE (for regressions) and misclassifications (for classifications) are presented.", "weaknesses": "-   The procedure itself looks somewhat simple; first we apply CUT and then L2boost. If there is a difficulty or interesting results of these combinations, please emphasize.\n    -   Perhaps, the combination of boosting and interval-censored data is the novelty? If so, please emphasize the discussion on the novelty (e.g., limitations in existing methods).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "abLhmG3GQS", "reviewer": "Reviewer_Npps", "summary": "The manuscript extends the boosting framework to regression and classification tasks with interval-censored data. The framework is particularly applicable in survival analysis, due to the prevalence of interval-censored data. The approach combines the L2Boost method with the censoring unbiased transformation (CUT) approach to form L2Boost-CUT, which is implemented via functional gradient descent. The manuscript also introduces an impute method based on L2Boost. L2Boost-CUT is applied to both synthetic and real data.", "questions": "1. How easily could the framework be extended to other boosting approaches (such as XG)?\n2. Relating to the weakness point above: what's the benefit of L2Boost-CUT if it provides the same results as imputation + standard boosting?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is generally very well written and easy to follow. The authors excellently introduce the problem and previous methods. The manuscript also clearly motivates the problem. Specific strengths are:\n\n1. The manuscript contains detailed and extensive theoretical results, providing strong theoretical guarantees. \n2. The theoretical results are described very well. Each theorem and proposition is described in terms of its broader importance and the purpose for it being used. This is very important in theoretical papers, to allow the reader to follow without getting bogged down by extensive theory.\n3. The manuscript tackles an interesting problem and the contribution is important. The method is applicable to many important settings.", "weaknesses": "My main concerns are regarding the empirical results. I'm not convinced by both the scope and outcome of the results. Specifically:\n\n1. Both the synthetic and real data studies are very limited in scope. For an approach that is so widely useful, it is a shame that it has not been applied to more settings to showcase its usefulness. It would be insightful to find and present scenarios showing the extreme ends of performance (very good and very bad performance)  of the approach and discuss why this is.\n2. The results from the studies are not overly convincing. The method performs the same as imputation and standard boosting (which already exists) and does not provide a large benefit over the naive approach in the synthetic study. I have asked a clarifying question around this in the Questions, so this weakness can be addressed. \n3. I think there should be more discussion on the assumptions and limitations of the method. This would help to understand under which scenarios we might expect the method to not do well.\n4. There is no discussion on computational cost or complexity. This could either be a theoretical complexity analysis or even just simple timing tests. Either would provide useful insights for practitioners who wish to decide on whether to apply the method to their problem.\n5. There is no reproducibility. It would be helpful to have reproducible code, even for just a basic example. This would also partly help alleviate weakness 4, as I could run the code myself and observe the execution time. \n\nI would like to see more evidence that the approach is beneficial in practice. I think the paper could do with some rearrangement: some of the theoretical results moved to the appendix to make space for further empirical results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)