Model Uncertainty Quantification by Conformal Prediction in Continual Learning
Authors: Rui Gao
Weiwei Liu
Venue: ICML 2025 poster (2025)
Keywords: Uncertainty
continual learning
Primary area: theory->learning_theory
TL;DR: We propose a  conformal prediction based method for model uncertainty quantification in continual learning (CPCL) and theoretically prove the asymptotic coverage guarantee of CPCL.

Abstract: Continual learning has attracted increasing research attention in recent years due to its promising experimental results in real-world applications. In this paper, we study the issue of calibration in continual learning which reliably quantifies the uncertainty of model predictions. Conformal prediction (CP) provides a general framework for model calibration, which outputs prediction intervals or sets with a theoretical high coverage guarantee as long as the samples are exchangeable. However, the tasks in continual learning are learned in sequence, which violates the principle that data should be exchangeable. Meanwhile, the model learns the current task with limited or no access to data from previous tasks, which is not conducive to constructing the calibration set. To address these issues, we propose a CP-based method for model uncertainty quantification in continual learning (CPCL), which also reveals the connection between prediction interval length and forgetting. We analyze the oracle prediction interval in continual learning and theoretically prove the asymptotic coverage guarantee of CPCL. Finally, extensive experiments on simulated and real data empirically verify the validity of our proposed method.

Model Uncertainty Quantification by Conformal Prediction in Continual Learning
Rui Gao
Weiwei Liu'
Abstract
2022; Xu & Liu, 2022; Zou & Liu, 2023a). Continual learn- ing (CL) referred to as lifelong learning (Chen & Liu, 2018; Gao & Liu, 2025b), sequential learning (Aljundi et al., 2019), and incremental learning (Aljundi et al., 201 (018)-is paradigm that enables a model to learn large number of tasks sequentially, where data from previous tasks are no longer accessible during the training of new tasks. A signifi- cant challenge in continual learning catastrophic forget- ting (Kumaran et al., 2016), where standard deep learning methods tend rapidly forget previously acquired knowl- edge when learning new tasks (Kirkpatrick et al., 2017). Most works of continual learning have concentrated on mit- igating this issue. For example, the regularization-bu based methods such as EWC (Kirkpatrick et al., 2017) and MAS (Aljundi et al., 2018) measure the parameter importance and introduce a regularization term in the loss function to con- solidate previous knowledge; replay-based methods such as iCaRL (Rebuffi et al., 2017) and GEM (Lopez-Paz & Ran- zato, 2017) try to replay the knowledge of previous tasks by storing samples or generating ppeuo-samplees and then provide these replayed knowledge to the model while a new task is being learned; parameter isolation methods such as PackNet (Mallya & Lazebnik, 2018) and HAT (Serrà et al., 2018) dedicate different model parameters each task so prevent any possible forgetting. Overall, these works mitigate catastrophic forgetting of model in continual learn- ing, which is often reflected as improved accuracy of model on previous tasks.
Continual learning has attracted increasing re- search attention in recent years due to its promis- ing experimental results in real-world applica- tions. In this paper, we study the issue of calibra- tion in continual learning which reliably quantifies the uncertainty of model predictions. Conformal prediction (CP) provides a general framework for model calibration, which outputs prediction in- tervals or sets with theoretical high coverage guarantee as long the samples are exchange- able. However, the tasks in continual learning are learned sequence, which violates the principle that data should be exchangeable. Meanwhile, the model learns the current task with limited or no access to data from previous tasks, which not conducive to constructing the calibration set. To address these issues, we propose a CP-based method for model uncertainty quantification in continual learning (CPCL), which also reveals the connection between prediction interval length and forgetting. We analyze the oracle prediction inter- val in continual learning and theoretically prove the asymptotic coverage guarantee of CPCL, Fi- nally, extensive experiments on simulated and real data empirically verify the validity of our proposed method.
However, existing works to date have ignored the issue of calibration in continual learning. Calibration often deemed as important the standard criterion of accuracy in statistics and machine learning (Thelen et al., 2022; Park et al., 2024). A well-calibrated model can reliably quantify the uncertainty of its prediction (Guo et al., 2017; Hermans et al., 2021). Model uncertainty quantification crucial in many applications of continual learning. For example, in medical imaging, advancements in technology or changes in diagnostic procedures result in continuous variations in image appearance (Hofmanninger et al., 2020; Ghesu et al., 2021). In this context, the model must provide critical information to doctors, enabling them to understand the un- certainty in model predictions and make informed decisions regarding patient care.
1. Introduction
As vast amounts of data are produced, the number of new tasks is increasing overwhelmingly (Zou & Liu, 2023b; Liu et al., 2019; Gong et al., 2023b; Chen & Liu, 2023). Con- sequently, learning systems are required to rapidly adapt to the continuously emerging tasks (Gong et al., 2023a; 2021;
'Shoool of Computer Science, National Engineering Research Center for Multimedia Software, Institute of Artificial Intelligence and Hubei Key Laboratory of Multimedia and Network Communi- cation Engineering, Wuhan University, Wuhan, China. Correspon- dence to: Weiwei Liu <liuweiwei863@ gmail.com>.
Proceedings of the nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
Among various uncertainty quantification techniques, specifically concentrate conformal prediction (CP) (Vovk et al., 2005; Ndiaye, 2022; Fisch et al., 2022; Qian et al., 2024) in this paper. CP provides a general framework for model calibration, which is model-agnostic,, nonparametric and distribution--Fe (the coverage guarantee holds for any distribution). CP takes in three components: a black-box predictive model, an imput feature and potential output. It requires the design of score function to measure how non- conforming the potential output is and to calculate scores calibration set. In regression tasks, CP outputs prediction intervals while in classification tasks, it produces prediction sets. These prediction intervals or sets can theoretically guarantee high coverage probability as long as the sam- ples are exchangeable (Shafer Vovk, 2008)-a weaker requirement than the standard i.i.d. assumption. However, uncertainty quantification using CP poses significant chal- lenges in continual learning. As illustrated in Figure 1, tasks are learned by a single model in sequence. Some works have shown that changing the order tasks significantly af- feet the performance of model in continual learning (Lange al., 2022). This violates the principle of data exchange- ability. Meanwhile, samples from previous tasks limited or inaccessible, resulting in restricted calibration set for CP. Therefore, effective use of CP for model uncertainty quantification in continual learning requires further investi- gation.
Model:
fr
Test
Citecs lisited accose dits from gortous tasks
Data:
D
Dr-1
Dr
Unitchrgoole
Task:
T1
TT
TT
T un
krowledged
Figure 1. A general framework of continual learning.
how solve the catastrophic forgetting of accuracy. Some surveys (Parisi et al., 2019; Lesort et al., 2019; Pfülb & Gepperth, 2019; Farquhar & Gal, 2018) have shown that existing continual learning works can be broadly divided into three categories: (i) replay methods, (ii) regularization- based methods and (ii) parameter isolation methods.
Replay methods generally have two ways to replay in arti- ficial neural networks: partial replay (PR) and generative replay (GR) (Hayes et al., 2021; Gao & Liu, 2023). Replay methods store samples or generate pseudo-samples using generative model, and these samples are replayed while a new task is being learned (Rebuffi et al., 2017; Lopez-Pazz & Ranzato, 2017; Rolnick et al., 2019; Ayub Wagner, 2021). PR stores either all subset of previously learned inputs in a replay buffer and mixes these inputs with new samples train the classifier (Rebuffi al., 2017: Lopez- Paz & Ranzato, 2017; Rolnick et al., 2019; Ayub & Wagner, 2021) while GR generates synthetic samples for previous tasks. For example, iCaRL. (Rebuffi et al., 2017) selects and stores samples (exemplars) closest to the feature mean of each class by assuming fixed allocated memory: DER (Buzzega et al., 2020) mixes rehearsal with knowledge dis- tillation and regularization and matches the network's 's logits sampled throughout the optimization trajectory to promot consistency with its past; DGR (Shin et al., 2017) uses GAN to to generate previous samples for data replaying. In addi- tion to previous task samples, graph-based replay (Tang & Matteson, 2021) have been proven to be efficient.
In this paper, we consider the regression setting and con- struct prediction intervals for continual learning with asymptotic coverage guarantee. Our main contributions can be summarized follows:
We propose CP-based method for model uncertainty quantification in continual learning, termed CPCL. CPCL constructs a new dataset which takes the depen- dencies among original data into count and defines the prediction interval by a conditional quantile estimator. Additionally, CPCL reveals the inherent connection between prediction interval length and forgetting.
We analyze the oracle prediction interval in continual learning and provide the asymptotic coverage guaran- tee of the prediction interval of CPCL.
Extensive experimental results on simulated and real data demonstrate the validity of CPCL.
2. Related Work
Regularization-base methods add extra regularization term to the loss function and penalize changes to impor- tant parameters of model for previous tasks (Kirkpatrick et al., 2017; Zenke et al., 2017; Aljundi et al., 2018; Lee et al., 2017; Gao & Liu, 2025a). For example, EWC (Kirk- patrick et al., 2017) uses the Fisher Information Matrix to measure the importance of parameters; IMM (Lee al.,
Continual learning. Continual learning consists of se- quence of disjoint tasks that are sequentially learned one at a time. After sequentially learning all tasks, continual learning wants the model to perform well on all seen tasks. Existing research on continual learning focuses primarily on
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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
2017) estimates Gaussian posteriors for task parameters, in the same vein as EWC, but inherently differs in its use of model merging; MAS (Aljundi et al., 2018) measures the importance according to the gradients of the squared T---norm of learned network output function. Other kinds of regularization-bue methods are also efficient, such as functional regularization (Pan et al., 2020; Benjamin et al., 2019), node importance (Jung et al., 2020) and uncertainty regularization (Ahn et al., 2019).
structed in time series, while the calibration set in continual learning is limited the current task. Second, time series fo- cuses on the model's performance on data arriving at future time points, whereas the core concern of continual learn- ing is the catastrophic forgetting regarding previous tasks. The connection between the prediction intervals (sets) con- structed by CP and the catastrophic forgetting in continual learning requires further research. Overall, is necessary to study CP in continual learning.
Parameter isolation methods dedicate different model pa- rameters to each task to prevent any possible forgetting (Xu & Zhu, 2018; Mallya & Lazebnik, 2018; Serrà et al., 2018). PackNet (Mallya & Lazebnik, 2018) iteratively assigns pa- rameter subsets to consecutive tasks by constituting binary masks. HAT (Serrà al., 2018) incorporates task-specific embeddings for attention masking where the per-layer em- beddings are gated through Sigmoid to attain unit-based attention masks in the forward pass.
3. Preliminaries
We mainly follow the continual learning setting of (Delange et al., 2021) and conformal prediction setting of (Xu &
Xie, 2023b). In this paper, we consider regression setup. Given an unknown model, f IRd + R, each observation Z (X, Y) is generated according the following model
Y= fx) +
(1)
where jr is distributed following a continuous cumulative distribution function (CDF) F, Y is continuous scalar variable, X E R2 denotes the feature and is the dimension of the feature vector.
Conformal prediction. As introduced in Section 1, CP is model-agnostic, nonparametric and distribution-fiee framework for the calibration of models. It designs a score function which measures how non-conforming the potential output is and evaluates these scores on a hold-out calibra- tion set. CP produces prediction intervals (or sets) where data is assumed to be exchangeable. Under the assump- tion of exchangeable data, CP has be successful in many applications. For example, Wisniewski et al. (2020) focus on the application of conformal prediction interval estima- tions to provide financial Market Makers (MMs) with some "meaningful" forecasts relating to their future short-term po- sition in given financial market; (Huang et al., 2023) pro- pose conformalized GNN, extending conformal prediction to graph-based models for guaranteed uncertainty estimates; Gui et al. (2023) adapt the framework of conformal pre- diction to propose a distribution-fres method for predictive inference in the matrix completion problem. Some works consider the situations in which the data exchangeability is not satisfied (Tibshirani et al., 2019; Gendler et al., 2022: Xu & Xie, 2023a). Tibshirani et al. (2019) show that a weighted version of conformal prediction can be used to compute distribution-fres free prediction intervals for problems in which the test and training covariate distributions differ. Gendler et al. (2022) consider that the test data may be adversarially attacked where the exchangeability assumption is grossly violated. Xu & Xie (2023a) develop the general framework for constructing prediction intervals for time series. Com- pared to time series, continual learning has the following differences when studying CP. First, in time series, samples from previous time points are available, while in continual learning, only samples from the current task are available, and samples from previous tasks are not accessible. This means that a complete calibration set can be naturally con-
Continual learning. There are Nr disjoint tasks (TI, T2; ...; TN, that are learned sequentially. The dataset of task Ti given by {2, (X;,, Y;) where N is the number of samples for each task. Each observation Z = (X,, n. Y2)) for task 7, is generated according to (1) where /", is distributed according to F*. In continual learn- ing, the model is trained on the current task while access to data from previous tasks is limited. The goal of continual learning is to control the statistical risk of the model across all previously seen tasks:
N₂ Σ Excr,,((((X'')0 Y')]
(2)
where & is the loss function and h is the predictor with parameter A.
Conformal prediction. In the context of CP, there are Neat observable samples {2 = (Xi, }:) Neat which are generated according to Eq. The goal of CP is to con- struct a sequence of prediction intervals that are as narrow as possible while ensuring a certain coverage guarantee. Given user-specified prediction algorithm, we use the observed samples to obtain trained model represented by f. Sub- sequently, we construct a prediction interval O for Ytest of a test sample, where a is the significance level. CP con- siders two types of coverage guarantees. The conditional coverage guarantee ensures that the prediction interval Öp satisfies:
P(Ytest E Chest (test) 2 a.
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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
The second type is the marginal coverage guarantee:
tion. We focus on the current task Tr- After sequentially learning the first 7 tasks, we obtain trained model fr. For given input X we mainly construct a prediction interval Ca to effectively qualify the potential prediction.
PlYiest E &Ciecl 2 1 a.
(4)
is noteworthy that Eq.(3) is significantly stronger than (4)) As established by split conformal prediction (SCP) (Papadopoulos al., 2007), the marginal coverage guaran- tee holds whenever data are exchangeable (Definition 1).
Calibration set. As discussed in Section 3, the calibration set is limited at the current task Ty (i.e., we only have ac- cess to data of current task to construct the calibration set), Inspired by the replay-based methods in continual learning (Rebuffi et al., 2017: Lopez-Paz & Ranzato, 2017: Rolnick et al., 2019; Ayub & Wagner, 2021). we try to replay the samples of previous tasks to construct the calibration set. For simplicity, maintain buffer to store Neat samples from each previous task. While other replay techniques could be considered, we do not develop them in this paper. Overall, by the replay, we collect the calibration set defined as U ( = (X!, 1
Definition 1. (.....EExeeiii (Exchangeabilitty (Shafer & Vovk, 2008)) The random variables V1, V2,..., Vn are exchangeable if for any permutation "y of integers 1,2...., n, the variables V2,V2..... where Vi = Vy(ay, have the same joint probability distribution as V1, V2,..., Vnn
In this paper, we aim to construct prediction intervals with a conditional coverage guarantee through CP in con- tinual learning setting. We consider N, disjoint tasks (TI,T2.... TN.). The dataset for task To is 12, = (X%, 1 As the goal of continual learning to con- trol the statistical risk across all seen tasks (i.e. Eq.(2)) the test sample should derive from union task that compasses all knowledge from previous tasks. Our work replaces the test sample Ztest = (Xeest, Ytest) with Zue = (Xue; Yut) for convenience, where Zue- = (Xae; Yut) repre- sents the test sample from a union task that contains knowl- edge from all seen tasks. After sequentially learning all tasks using any continual learning algorithm (which can be viewed as the user-speciffed prediction algorithm), we obtain trained model. Given Xu.. we quantify the un- certainty of the trained model by constructing prediction interval Oz for the output, ensuring that Yur E Og with probability - a. In light of this, some natural challenges arise:
Nonconformity score function. For each observation
7 (Xt, in the calibration set, we define the pre- diction error as
Y
(5)
Then we define the nonconformity score function as
(X Y2) 1+6-4
(6)
which ensures that the scores range in (0. We intro- duce this nonconformity score function for two reasons. First, the range of sigmoid-based nonconformity score func- tion is (0, 1). Our method is involved with the training of Quantile Regression Forests which requires to construct trees. This process asks us to define that the score func- tion ranges in (0, 1) for rigorous proof. Second, this non- conformity score function is invertible, which helps us to rewrite the prediction interval. Details of reasons can be found in Appendix B. We leverage (6) to calculate the non-conformity scores for all observations in calibration set U2 = (X;, ) New! This results in the score set
The samples from previous tasks are limited when learning the current task тy. This limitation implies that the calibration set reserved by typical conformal prediction methods (such as SCP) is also restricted.
Some works have shown that changing the order of tasks significantly affect the performance of model in continual learning (Lange et al., 2022). Therefore, the principle that data should be exchangeable is violated in continual learning. In this case, the courage of typi- cal conformal prediction methods (such as SCP) cannot be guaranteed (Zou& & Liu, 2024).
(S where S = s(X; Y?).. is noteworthy that the nonconformity score function defined in this paper is independent of the calibration set. Since observations in continual learning are not exchangeable, the calibration set exhibits inherent dependencies among its observations. Consequently, the scores in U._ ( Nont should be se- quentially dependent because of the independence of the nonconformity score function and the inherent dependen- cies among calibration observations. To account for these dependencies in the scores, we train a quantile estimator Neat Q" (æ;; o:) on U (S rather than using the empirical quantile directly.
To address these challenges, we demonstrate how to quantify model uncertainty using conformal prediction for continual learning in Section 4. Additionally, we establish asymptotic conditional validity of our proposed method in Section 5.
i==l"
4. Uncertainty Quantification in Continual Learning
Inspired by Xu & Xie (2023a), we show how to quantify model uncertainty by CP for continual learning in this sec-
Quantile regression forests (QRF) (Meinshausen, 2006). In this paper, we utilize QRF to achieve the quantile esti-
4

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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
mator ¹² (1; a)). Considering the dependencies over scores, aim to reconstruct new dataset based the score set to train QRF. In this new dataset, each input should encap- sulate the knowledge of dependencies over scores and each prediction should represent the conditional quantile of the scores from the union task. Specifically, we define the re- constructed dataset as Dj = {(
Then the prediction interval with significance level is
Ch. - () (X
(II)
where 5 = s(X YT) For convenience, we define (" (a = Sa. Recalling the nonconformity score
R ) where x," = [S] .....,, S and Y is an approximation of the true score She of the observation (X Yur) from the union task which contains knowledge from all seen tasks. It is noteworthy that X contains 7 sequential scores which are effective to predict the condition quantile of true score S...
function in (6), we can rewrite the prediction interval as
C -f-)( - In ();)
-
1)]- (12)
B
8 S
1 (11)) In(
β = algmin(In [0,0 $93
(In
(13)
We train QRF on the reconstructed dataset {(X,R Nr
8 +
After training QRF, we grow K trees. For each tree with separate parameter Ça. there are L leaves, where every leaf I is associated with rectangular subspace R₁ C TB. These subspaces are disjoint and cover the entire space IB, i,ee. for any x E B, there is one and only one leaf which is denoted as R((x.Gk)
Connection between intervals and forgetting. As the number of tasks increases, the phenomenon of forgetting becomes more pronounced. The model's test error on previ- ous tasks rises (Eq.(5)), while the error on the current task remains low. This leads to an increased difference between our estimated $3 and 8 According to (q,() we can conclude that the length of the prediction intervals will also increase, which is verified by Figure 3 in Section 6.2.
Based on the trained QRF, we then leverage X [S N..! Neat "........, S%.... to predict the conditional quantile of S Given the reconstructed dataset {(X,", ] Nagt-1 the trained QRF and Ne.. B, we can obtain the esti- mated conditional distribution function EI I Neat) as follows:
Conformal prediction for continual learning. Overall, we propose a CP-based method for model uncertainty quan- tification in continual learning, termed CPCL. Specifically, when learning current task Tr, we first obtain a trained model fr using specific continual learning method. Then we construct the calibration set by replaying samples of previous tasks. We collect score set by calculating non- conformity scores on calibration set. Following that, we reconstruct a new dataset based on the score set. Since the union task contains knowledge from all seen tasks, we select one entry S of X. which is multiplied by a coefficient w, and set Y.R = =5y We then train QRF on this reconstructed dataset to obtain conditional quantile estimator. Finally, for any test input X we produce the prediction interval which comprises the conditional quantile estimator. Details of CPCL. are shown in Algorithm 1. In practice, we consider Ntest test samples and obtain intervals based on these test samples.
I(X! E Buxx_,,5.
P out (k)
Hij Neat -1 1| E RI (503
K
p R
N
Pi
Neat
/
(8)
k=1
Nea!-1
FlyR...
R Neat
P J(Y, S (9)
v&...)
where [Noat- 1] represents the set 21. Neat 1}.
RuxB,,. (k) is the leaf of tree Sa that contains the input X If this leaf also contains the observation X,R from the reconstructed dataset, then T(X;R E Ruxg_... (w) is equal to 1; otherwise, it is equal to O. Kí E [Nout 1| E Re(x
cat H. represents the number of observa- tions included in this leaf from the reconstructed dataset. Pa P B.a. (k) measures the weight of observation XRR, while Pa Px )...) further measures the weight over K trees. In practice, we implement the above fitting and prediction process by Python according to (Meinshausen, 2006).
5. Theory Analysis
In this section, we first analyze the oracle prediction interval in continual learning. Inspired by Meinshausen (2006); Xu & Xie (2023a), we then demonstrate that the conditional distribution function estimated by our proposed CPCL (i.e., (9)) converges to true conditional distribution function. Finally, we prove that this convergence of distribution func- tion results in the convergence of quantile estimates, indi- cating asymptotic conditional validity of CPCL. For conve- nience we let T = N, represent the scenario where all N, tasks in continual learning have been learned.
Based on the estimated conditional distribution function F eat [X
Prediction interval.
Neat): ). we define the condi-
tional quantile estimator Q" (x X N ;a) as
(x= ("(x== X :o) isE (0,1): (s R_..) 22a] (10)
5

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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
Algorithm 1 CP-based method for model uncertainty quan- tification in continual learning (CPCL).
For any 3 E [0, or), we derive that
P(S E 1
N= (8), 2 Nº (8+1-a)) (B+11
(17)
Input: (Ti, T2, TN.): ): all training tasks; N,: total num- ber of tasks; Z = (x,, ) the dataset of task T&S N: the number of samples for each tasks: T:: the index of current task; Noni: the number of samples from each task in the calibration set; CLM: certain continual learning method.
According to (6) we define the inverse of the score
function as ((t) = s (u). For any a E [0, 1J, we have
Yo = f f(x + (a))
(18)
1: Initialization: RB- = 0
Then the oracle prediction interval in continual learning with the narrowest width is defined as
2: for T E [I....,NN,] do
3: Obtain fr from a certain continual learning method CLM.
Cu [v3 1133./++1-0 V3+1-al.
(19)
4: Collect 5 = (XT,
n 1T Jen I=
Neat from task Tr.
5: RB RB [Z (X,,, Y D
= (ragmin((((( ( + 11 (8+1-a)) Aε(0,a)
((("((())
(20)
6:
Construct - 1=1 [ i=1
Neat on RB by (6)
For convenience, above discussion directly assumes that the score function is the one designed by our work. The above analysis indicates that the estimated quantile å and model f determine how closely the estimated prediction interval approximates the oracle prediction interval. In continual learning, ff is the output model whose uncertainty we aim to quantify. So we focus on analyzing the estimated condi- tional quantile presented in this paper, i.e. (10). In this work, we want to seek the asymptotic coverage guarantee of the prediction interval of CPCL,, which is defined as
7: Reconstruct D {(XPR (
8: Fit QRF on Df and obtain the conditional quantile estimator by (Q0)
9: Construct the prediction interval ČarT for any test input X with significance level a by (11, (12) and (13).
10: end for Output: (Čur,1 T Ci......., T (,
5.1. Oracle Prediction Interval in Continual Learning
P(Y at E Oo X
NN¹) + 1 a
(21)
Here we discuss the oracle prediction interval in continual learning, which includes Y with exact conditional coverage of 1 a. We first define Z = (X*, Yt) as the data originating from task TG. The oracle prediction assumes perfect knowledge of f in (1). Given - and since N₂ tasks are learned in sequence during continual learning, the prediction of y-N- conditioned on X-N- and the sequence N [Z], -² ,..., ZN Therefore, the goal of CP in continual learning centers around the CDF of y,N- condi- tioning on X and the sequence Z which is defined as:
with Neat + co. ohhherefore, proving (21 is equivalent to prove that for any [0, 1]. we have
②N- (2;a) QN (a)
as Neal - 00,
(22)
where QN- (æ;a) is the estimated conditional quantile of CPCL and @N=(a) is the true conditional quantile. In Eq,( we have x = X which corresponds to the score sequence S,N, In Section 5.2, we prove (2.) for any x E IB
Fue = -P(Y,NN
ut S x N- NN
5.2. Asymptotic Coverage Guarantee
(14)
PP N. Sy- f(XN) 1
Here we prove that (2. holds for any E [0, and E B. As stated in (Xu & Xie, 2023b), is impossible to prove the conditional coverage guarantee without further assumptions. We first provide some necessary theoretical assumptions and briefly explain them.
where Y-N- f(Xxi)) + - and pnt is distributed fol- lowing N.N. According to (5). and (6), our designed score function s(-) is dependent of the data sequence Z N-
SNN- is defined as the score sequence in which St = sZz). In this case, the dependence among S arises from N
Assumption 1. The true conditional CDF in Eq.(1 is Lipschitz continuous with parameter L, i.e. for any X, x' e B.
Therefore, the conditional CDF (14) is equivalent to the following CDF based on the score sequence
sup |F((()) F((s=')) & TAI x' 111,
(23)
(s (NN) = P S s.
(15)
where x, correspond to the score sequence.
We define the true conditional quantile
Assumption 2. The true conditional CDF F(s(z) is con- timuous and strictly monotonously increasing in s. for any
ºx (a) innf (s: Fs((s
(16)
x E B.
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Model Uncertainty Quantificatin by Conformal Prediction in Continual Learning
Assumptions 1 and 2 make assumptions on the condi- tional CDF. The conditional distribution function is assumed to be Lipschitz continuous in Assumption 1 and strictly monotonously increasing in Assumption 2. These assump- tions are reasonably mild and don't require a particular parametric.
consistency of CPCL, showing that the ervor between the approximation of conditional distribution by CPCL and the true conditional distribution converges uniformly in proba- bility to zero as Neal + oc. Consequently, CPCL serves as a consistent method for estimating conditional distributions.
Based on Theorem 1, we prove that for any E [0, 1]. the estimated conditional quantile of CPCL converges to the true conditional quantile. Theorem 2 shows this theoretical result. We rewrite QN (a) as QN= (z;; a) for convenience. Theorem 2. Under the same conditions as Theorem 1, for any a E [0, and a E B, the estimated conditional quan- tile QN. (I:a) of CPCL converges to the true conditional (1;a)
Assumption 3. For any x E B, pc(x)) in (8.(8) satisfies that ps (x)) = o(1).
Assumption 4. For any x E B, the rectangular subspuce Ri(ZZCa)) S (0.,)), of leaf Uxx (k) of tree Sk is defined by the intervals [[x m, (k) S (0, 1), iee. Re(e,()) 8 Txr m, (k), where @ means direct sum. We assume that maxm (1 m, [s)] = (1(1) for Neat + oc. op(1) indicates that maxm (I m. (k)l converges in probability to zero.
quantile QN=(1; as Neal + 00, i.e.
Q (2; Neat + 0c
(25)
holds for any E [0, 1] and B.
Assumptions 3 and 4 focus the actual construction of trees. We refer to Meinshausen (2006) and provide two examples. Assumption 3 represents the case of Example 1. Assumption 4 indicates that the size of each interval is vanishing for large Neal and represents the case of Example
The proof of Theorem 2 can be found in Appendix E. Remark 2. Theorem 2 is sufficient establish the asymp- totic coverage guarantee of the prediction interval of CPCL. Specifically, Y E =C 2,N" is equivalent (,,,(XX;
to S.N. E E1ºN.(X X X.- According to Theorem 2. we know that P(S E -10N(x'' (3))), NN= :3 13+ 13+1-a00 +1) con- verges to P(S El0N~ (X':,(X',3)) 1,8),N== (+1100))))) as Neat 00. Based on Eq.( we then conclude that P(Y... E C.iI ZN + 1 a as Neat + oc. This establishes the asymptotic coverage guarantee of the prediction interval for CPCL.
1-a]] SN where
Example The minimal number of observations in a node is growing for large Neats i.e., 1/ //mini,ce wex (I) (11,, Neal + oo..
Example 2. This example consists of three situations. In situation 1, the proportion of observations in a node, rel- alive to all observations, is vanishing for large Neat i.e.,
maxi,ç Wya (1) = o(Neat), Neat + 00. In situation 2, when finding a variable for a splitpoint, the probability that variable m = 1,... N, is chosen for the splitpoint is bounded from below for every node by a positive constant. In situation 3, if a node is split, the split is chosen so that each of the resulting sub-noder contains at least a propor- tion ?f of the observations in the original node, for some
6. Experiments
In this section, we use simulated and real-world data to verify the validity of the prediction interval of CPCL. We consider typical continual learning methods, including SI (Zenke et al., 2017), EWC (Kirkpatrick et al., 2017), MAS (Aljundi et al., 2018), DGR (Shin et al., 2017)) and Fine- tuning on real-world data to explore the effect of continual learning methods. More details of these continual learning methods in Appendix A.
0 < 7 S 0.5.
Discussion of Examples 1 and 2 can be found in Appendix C. Based on Assumptions 1 144, we prove that for any x E IB, the conditional CDF F(s(z) of CPCL converges in proba- bility to the true conditional CDF F(s).r) as Neal + 00.
Theorem Under Assumptions 1, 2, and 4, for any E (0. 1) and x E B. the conditional CDF ((s)z) of CPCL (9)) converges in probability to the true conditional
6.1. Simulation
We follow the work of Zou & Liu (2024) to generate simu- lated data. In our setup, we consider two regression tasks within the continual learning process, setting X' RR. y = R. Define the oracle linear predictor G X - ) as G(z) = (w*, x) + b", where w* E Rd and 6* E R. We
CDF F(s(x) as Noat + 20, i.e.
[F(s|z) - F(s +p O Neat + oc
(24)
holds pointwise for any s E (0, 1) and x E B. The proof of Theorem can be found in Appendix D. Remark 1. Theorem I represents a crucial step toward establishing the asymptotic coverage guarantee of the pre- diction interval of CPCL. This theorem demonstrates the
define the marginal distribution of X for tasks 71 and T2 as
Fix = Nf FEx = -N oa
(26)
respectively, where U1 U2 E Rd are the mean vectors, o > is a scalar and La E Radd is the identity matrix
7

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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
0.950
0.900
Coverage S6.
0.975
Coverage %
Coverage
0.950
0.900
0.850
0.925
SI
EWC
MAS
Finetuning
0.875
Si
EWC
MAS
Finetuning
0.825
Si
EWC
MAS Finetuning
CPCL
methods
CPCL
methods
CPCL
(a)
0.05
(b) 0.1
(c) 0.15
0.850
0.800
*6 0.825
Coverage %
0.775
Coverage
Coverage
0.800
0.750
0.7000
0.775
SI
EWC CL methods
MAS
Finetuning
0.725
Si
EWC
MÁS
Finetuning
0.675
EWC CL methods CPCL
MÁS
Anetuning
CPCL
CL methods + CPCL
(d) a = 0.2
(c) a 0.25
(1) a 0.3
Figure 2. The violin plots illustrate the coverage for 100 experimental runs the simulated data. The red lines indicate the marginal coverage guarantees we aim to achieve. In each violin, the white median line represents the median, while the endpoints of the thick line denote the 0.25 quantile and 0.75 quantile. We present results a {0.05, 0.1, 0.15, 0.2, 0.25, 0.33.
of dimension d x d. We define the conditional distribu- tion Y given X as Fy = .V(G(z), ag)) for both tasks and 72. We draw 5000 samples from the above distribu- tion to construct the train dataset for each task. We define the trained model where á E IRd and b E R. We draw 1000 samples from the distribution of task ti and another 1000 samples from the distribution of task 72 to construct the test dataset. We Neat 1000 which is the number of samples for each task to construct the calibration set. After sequentially learning tasks Ti and 72, we quantify the uncertainty of the trained model G(z)) using CPCL. The evaluation metric is the cover- age which is the ratio between the number of test examples such that y/ O% (as) (12) and the size of the test dataset.
6.2. Real-world Data
We conduct experiments using Tiny ImageNet, subset of 200 classes from ImageNet (Deng et al., 2009), rescaled to an image size of 64 x 64. We perform 20 runs with different random seeds, randomly selecting 5 classes to form tasks each time. In this case, there are regression tasks in con- tinual learning process. For each task, have 500 samples belonging to class, subdivided into training (80%) and calibration sets (20%) along with 50 samples for testing, We utilize a pretrained AlexNet (Krizhevsky et al., 2012) (denoted as AN)(.., modified by replacing final layer with a linear layer suitable for regression) to define conditional dis- tribution Y given as Fy = (AN az) The trained model here is ResNet-18 (He al., 2016) which is also modified for regression. After sequentially learning T tasks, we quantify the uncertainty of the trained model by CPCL. We evaluate the results by calculating the coverage, utilizing testing samples from all T previous tasks, as described in Section 6.1. While calculating the coverage, we also col- lect length each prediction interval and calculate the average prediction interval length over all seen test samples.
linear predictor G(x) = (1²,x) b,
We set the significance level to values in (0.05, 0.1, 0.15, 0.2, 0.25, 0.3) and conduct 100 runs with different random seeds. Figure 2 shows the results of experiments on simulated data using CPCL. From the figure, we observe that the violins for CPCL with different continual learning methods are over the desired coverage lines. For example, the coverage results evaluated by CPCL exceed 0.95% when we use EWC the continual 0.05. These experiment
We set the significance level to values in (0.1,0.2) 0.3). Figure 3 shows the results of experiments on real-world data using CPCL. From Figures 3(a), 3(b) and 3(c), we observe that the most swarms with different continual learn- ing methods are over the desired coverage lines. These
learning method and results demonstrate the validity of our proposed CPCL on simulated data. We further study the effect of continual learning method with more tasks using real-world data.
8

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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
1.00-
0.90
0.80
Coverage %
Coverage
0.85
Coverage
0.90
0.70
EWC
Fineunin
Finetuiing
Finetuiing
MAS
0.85
0.75
0.65
2
2
(a) 0.1, coverage
(b) 0.2, coverage
(c) 0.3, coverage
12
DGR
DGR
Finetuning
EWC
finetuning
EWC MAS
Finetuning
Average Length
MAS
Average Length
MAS
Average Length
3
4
5
3
4
Task
Task
Task
(d) 0.1, length
(c) a 0.2, length
(f) a = 0.3, length
Figure The swarm and box plots illustrate the coverage and average interval length from the 20 experimental runs on real-world data. Figures 3(a), 3(b) and 3(c), the red lines indicate the marginal coverage guarantees aim to achieve. In Figures 3(d), 3(e) and 3(f), the median line represents the median, while the two edges of the box correspond to the 0.25 quantile and the 0.75 quantile. Different colors represent different continual learning methods. We present results for (0.1, 0.2, 0.3).
demonstrate the validity of our proposed CPCL on real- world data. Figures 3(d), 3(c) and 3(f) show the average interval length. As the number of learning tasks increases, find that the average interval length based any con- tinual learning method tends to increase. For example, the average length of the prediction interval can reach after learning task 5, while it remains below after learning task 2, when we us EWC as the continual learning method and a = 0.1. This phenomenon results from catastrophic for- getting. Although continual learning methods are effective in mitigating catastrophic forgetting, some forgetting still occurs the number of learning tasks increases. (q.(12) shows that the prediction interval is based on the resulted model fr. Consequently, an increasing prediction interval length indicates model forgetting. Therefore, our proposed CPCL effectively reflects the phenomenon of forgetting and measures the performance continual learning methods.
set, where scores are sequentially dependent. It accounts for these dependencies by reconstructing a new dataset based on score set. CPCL trains QRF using reconstructed dataset and defines the prediction interval the conditional quantile es- timator derived from the trained QRF Then analyze the oracle prediction interval in continual learning and provide the asymptotic coverage guarantee for the prediction inter- val of CPCL. Finally, extensive experiments on simulated and real data empirically verify the validity of CPCL.
Acknowledgements
This work is supported by the Key R&D Program of Hubei Province under Grant 2024BAB038, the National Key R&D Program of China under Grant 2023YFC3604702, the Fun- damental Research Funds for the Central Universities under Grant 2042025k10045.
7. Conclusion
Impact Statement
In this paper, we propose a CP-based method for model un- certainty quantification in continual learning, termed CPCL. CPCL collects the calibration set through replay and de- signs a nonconformity score function to construct score
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.

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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
A. Additional Experimental Setting
Here, we provide more information about continual learning methods. There are five methods in our paper. We give some descriptions of these baselines.
SI: The synaptic state tracks the past and current parameter value, and maintains an online estimate of the synapse's importance toward solving problems encountered in the past.
MAS: Redefines the parameter importance measure to an unsupervised setting and obtains gradients of the squared L2 Lg-norm of the learned network output function.
EWC: EWC introduces network parameter uncertainty in the Bayesian framework; the true posterior is estimated using a Laplace approximation with precision, determined by the Fisher Information Matrix (FIM), which shows equivalence to the positive semi-definite second order derivative of the loss near a minimum.
DGR: DGR consists of deep generative model ("generator") and task solving model ("solver"). DGR trains deep generative model in the generative adversarial networks (GANs) framework to mimic previous data. Generated data are then paired with corresponding response from the previous task solver to represent old tasks. Called the scholar model, the generator-solver pair can produce fake data and desired target pairs as much as needed, and when presented with a new task, these produced pairs are interleaved with new data to update the generator and solver networks.
Finetuning: Finetuning greedily trains each task without considering previous task performance- hence introducing catastrophic forgetting-and and represents the minimum desired performance.
B. Discussion of Nonconformity Score Function
We introduce the sigmoid-based nonconformity score function for two reasons. (1) The range of sigmoid- based noncon- formity score function is (0, 1). We train QRF on the reconstructed dataset D = {(X R y M where each entry St in X. is the calculated score. The training of QRF requires to construct trees. This process involves the intervals Tar, m, (x) C (0, 1). The interval (ar, m,
(k) is used determine whether X. R is in the node corresponding [(x; m, Ck),
i.e. if STT E [xx, m, Sk), then X is in the corresponding node. Since [xx m, Sie) S (0, 1). we need to define that the score function ranges in (0, 1) for rigorous proof. (2) Sigmoid- based nonconformity score function is invertible. In Eq. I) we present the prediction interval in terms of scores. The invertible function helps us to rewrite the prediction interval which is shown in (12). A typical score function in CP is sXX .Y)) | where μåt YÝ (X;).. It is not invertible and the corresponding range is not (0, 1). Therefore, it is not suitable in this paper.
C. Examples
We refer to Meinshausen (2006) and provide two examples. Assumption represents the case of Example 1. Assumption 4 indicates that the size of each interval is vanishing for large Neat and represents the case of Example 2.
For each tree with separate parameter ÇR. there are T. leaves, where every leaf / is associated with rectangular subspace Ri C B. These subspaces are disjoint and cover the entire space BB, i.ee for any x E IB, there is one and only one leaf which corresponds to R.(..ca) Denote the node-sizes of the leaves I of a tree by Wcg, (1) =
i e (1,...,,c, Neat - 1} X, E Ri X, E (0. 1)Nr is the observation to train QRF
We discuss Assumption by Example 1: The minimal number of observations
node is growing for large Neat, i.e.
1/ miny,c. wc. (f) = o11, Nea! + oc.. Recalling (x) = Ek= (z /x)/K, we have 0 S (z) S 1/ mini,c+ ca. wc, (1) o(1), which means that (x) o(1) for any x B. Therefore, Assumption 3 represents the case of Example
We discuss Assumption 4 by Example 2 which consists of three situations. In situation 1, the proportion of observations in a node, relative to all observations, is vanishing for large Neat i.e., maxt,c, WF,, (1) = o(Neat), Neat - co. In situation 2, when finding a variable for a splitpoint, the probability that variable m = 1, N, is chosen for the splitpoint is bounded from below for every node by a positive constant. In situation 3, if a node is split, the split is chosen so that each of the resulting sub-nodes contains at least proportion 7 of the observations the original node, for some 0 <> S 0.5.
As any IB is dropped down tree, several nodes are passed. Denote by S(x, m, (R) the number of times that these nodes contain splitpoint variable m. The total number of nodes that x passes through is denoted by S(z, (x)
13

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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
Σm= S(x, m, (R). Using situation the maximal number of observations in any leaf, max; Wcy (1) is bounded (for every tree) from below by NearrSSmin(Ge) where Smin (Cx) minngg S(x, <x). Using situation 1, max; we, (1) is on the other hand bounded from above by an o(Noaa)) Putting together, we conclude that 7 (Gx) = o(1) for Neat + cc. Hence there exists a sequence Neas with SNaat + 00 for Neal - 2o such that Smin (Sx) 2 & As the probability of splitting on variable m = I,... N₂ is bounded from below by a positive constant, by situation 2, there exists a sequence @N... with IN 7 Do for ceal + oo such that P'min, S(x, m, (x) > 9N..: } - 11 Neal - co. Using situation 3, we obtain that (i E [I....,A. Noa 1} : Xim E [xx m, (x)H)/ - 1) (1 (011-1 (1-))))11)))))m)))))) -))(e,mm.()) Putting together, we conclude that maxm Ii E {I,.., Neal 13 Xi.m [2, m. (/)ee/(Neaa - 1) = 0p(1) which indicates max [1 m, [+) (1()
for Neal 00. Therefore, Assumption 4 represents the case of Example 2.
D. Proof of Theorem 1
Theorem shows that the conditional CDF F(s)z) of CPCL converges in probability to the true conditional CDF F(s(z) as Neat + 0c. We recall Theorem and prove it.
Theorem 1. Under Assumptions 2, 3 and 4, for any (0, 1) and x B, the conditional CDF ((((x) of CPCL converges in probability the true conditional CDF F(s(z) Neal 00, i.e.
F - F(s[z)] +p Neal 00
(D.1)
holds pointwise for any E 10, and B.
Proof. For each observation in {(X,R Mar! we define
U, F(X
(D.2)
the quantile of the i-th empirical score Y,R (corresponding to s in CDF). We replace Y.R with S.R for convenience in this proof. It is noteworthy that U; ~ Unif(o, 1] due to the assumption of continuous distribution function in Assumption 2. We then derive the point-wise wise difference between the conditional CDF F(s ()) of CPCL and the true conditional CDF F(s(r).
Neat-1
F(s(x) F(s|Ø)=
Σ
P((=) S) -F(s)a)
(D.3)
Under Assumption 2, we know that {S S $} is identical to (U S F)) Therefore, we derive that
Ne pa(x)1(U; < F()]x())) - -((s))
F(s(x) - F((/z)
Neat-1
N
P S F(s|z)) F(s[x)+
(D.4)
Taking the absolute value of both sides of (D.44. we derive that
Nent
Newt -1
[f()))) - F(s|=) &
Σ Pa(x)T(U, S F(s)=)) (s/x))++1
Σ
p:(x) < F(xxX)) - T(U, s F({+))))
(D.5)
The first term of right side of (D.5)) is variance-type part, while the second term of that reflects the change in the underlying distribution. For the first term, we derive the expectation:
Neat1
E Σ pr(T) F(s(x))) Σ
p(( & F(s(x))
(D.6)
Due to Uj Unif 1], we conclude that EE[I(U < F(s|=))] F(s(x)) According (q) of Section 4, we know that Noot-1 (x(x) Therefore, we derive that
Pr(x)I(U, S F(s|x)) = Neat-1
E
Σ
(((x) = F(s(x)
(D.7)
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Model Uncertainty Quantification by Conformal Prediction in Continual Learning
Based (q.(D. we rewrite the first term of right side of (q.(D.5)
Neat-1
Neai
Neal-1
S F(s|=)]=
Σ
s F(s(x)) R
Σ
p.(+))(( F(s[x))))
(D.8)
By Chebyshev inequality, for any O, we have
PO
F(sxx) (a(a)(( S -((l))) 26)
Neal
pr(r)[(U; (()(=E) - EI Neal-1
= P(
Σ
pi(x)[(U; S F(sxx)) 26)
(D.9)
N
< Var
(a(a))((U+= S F(st))))
Due to the way of constructing
{(XR ) we U3 for i I...., Neal independent. Therefore, we have
Neal-1
Var(
Pr(x)[((U, s F(sla))
N
Σ
Pi(c))Var([(U) F(s))))
Neat-1
Σ
(E(() < F(s[=)) BUU F(s)x)))3)
(D.10)
Neat-1
Σ
(F())) - =((()))
Newr
p=(x:)
By Assumption 3 and (8), we have <p((x) o(1) and E pi(x)) 1. Therefore, for any x E B, derive that
Nea1-1 p((x)3 + 0
Neal co.
(D.11)
Combining (D.9). (D.10) and (D.11), derive that the first term of right side of (q.5). converges to 0 Neat co.
i.c.
Neat p((T)((U, < F(s))) F(s|x) +p 0 Noat xo.
(D.12)
holds for any s E (0, and x E B. Then we turns attention to the second term of right side of Eq.(D.)). We define H(U)
as
Nea-1
H(U,)
Σ
(I(x) s F(s(x!)) I(U, s F(s(z)))
(D.13)
15

---

Model Uncertainty Quantification by Conformal Prediction in Continual Learning
Due to all U, are uniform over [0.1], it holds that
Newt =1
[[H( = E
(FII) - 5())))))s))
Nea-1
Σ p()))|IT(U, & F((X)) - I(U, & F(s(x))
(D.14)
Noa-1
[(())((((X{]) F(([2)
By Chebyshev inequality, for any c > 0, we have
(U) EIH(UII Var(H(U.))
(D.15)
Similar to the proof process of (D.10.,) derive that
Nea--1
Var(H(U:)) = Var(
- I(U, E F(s(z))))
F({({)) I(U, s F(s(=)))
Neat
( S F({()())) 1004 S F(([x)) ETI( & F(s(x;")) -I(U, s F(s(x))13)
(F)))(((({);))) - F(s|2) -F(() F(s)z))
p=(x)²
(D.16)
Combining (D.1(D. and (D), we derive that HU:)) +pp (U:) as Noal - oo. Therefore, we have
Newt
Noat
(TU) F()X)) - I(U ((s()))) +p
- as Neat - oc.
(D.17)
According to Assumption
derive that
N -1
Neat
( <
(()))(())))) - F(s|x)
Neut
pı(x) sup -
(D.18)
Na
px(x) x|11
By Assumption have max, [[x, m, (x) (1) for Neat 0o and any & E B, which suffices to show that
New-1
Σ
p(())[|IX* =1))=
(D.19)
We therefore conclude that the second term of right side of (D.5) converges to 0 as Neat + co. Overall, we prove that under Assumptions 1, 2, 3 and for any s E (0,1) and B, the conditional CDF F(s(z) of CPCL converges in
16

---

Model Uncertainty Quantification by Conformal Prediction in Continual Learning
probability the true conditional CDF F(s(æ) as Neal - oc, i.e.
[f(s)z) F(s[z)] +p
Neal
(D.20)
holds pointwise for any & E (0, 1) and x E B. Xu Xie (2023a) provide a similar proof which can also be referred to.
E. Proof of Theorem 2
Theorem 2 shows that the estimated conditional quantile Q (2;aa of CPCL converges to the true conditional quantile QN a)) as Neat - oc. We recall Theorem 2 and prove it.
Theorem 2. Under the same conditions Theorem 1, for any E [0, 1] and x E BB the estimated conditional quantile @Ne (Iza) of CPCL converges to the true conditional quantile Q (I; a) as Neal + co, i.e.
@ (x;a) + (2; Neal 00
(E.21)
holds for any E [0, 1] and x E B.
Proof. Proving (E.21. is identical to prove that
j@" (x;a) Q (2; (2; + 0 Neal
(E.22)
holds for any x E B and [0, 1). Recall that
= inf(s: F(s(x) 2a).
(E.23)
By Assumption we know that F(s(z) is continuous and strictly monotonously increasing s. We consider small
perturbation of Q (x; (x), which results in the changes of the conditional distribution function, i.e. [F(QN,
and F( (z)z) a|. We define this changes 6 min( [ [F(O. (z; -e(x)) a), FFQN (z; +e(x) al).
Therefore, we consider the estimated quantile Q (æ:a) the perturbation of Q (z; a). az;a) When the 1 (N;a);
Qx; a:) > e, the change of [(s(r) is more than 8, i.e. [ (x;a)) - al > 8. Then we derive that
P(IĀN~(2;00) Q 2 (x; a))
c) ((IF al
(E.24)
= (P(IF((N F(ON= (xza) > 8)
According to the consistency in Theorem we have
Neal + 00. P(IF((N: (Iza)) F(O (r;a)] > 8) + as
(E.25)
Combining (F.2() and (F.25), we derive that
P(IΦN= (æ;a² - (2; >
+ +0 as
Neat + 0o.
(E.26)
This means that
(Q:a - (x; + as Newt + xc.
(E.27)
Overall, we prove that the estimated conditional quantile N: (aza) of CPCL converges the true conditional quantile N (zx; a)) as Neat cc.
17


Reviews:
[{"review_id": "txgkpZNiTw", "reviewer": "Reviewer_CUQK", "summary": "This paper explores **calibration in continual learning**, specifically focusing on **model uncertainty quantification** using **Conformal Prediction (CP)**. CP provides **theoretical coverage guarantees** under the assumption that data are **exchangeable**, but this assumption is violated in **continual learning**, where tasks are learned sequentially with limited access to past data.  \n\nTo address this, the authors propose **CPCL (Conformal Prediction for Continual Learning)**, which:  \n- **Constructs a calibration set using replay mechanisms** to address the lack of past-task data.  \n- **Designs a nonconformity score function** to quantify uncertainty in predictions.  \n- **Uses Quantile Regression Forests (QRF)** to estimate conditional quantiles for prediction intervals.  \n- **Theoretically proves asymptotic coverage guarantees** of the prediction intervals.  \n- **Empirically validates CPCL** on simulated and real-world datasets, demonstrating **robust uncertainty quantification** and a link between **prediction interval length and catastrophic forgetting**.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**C1. Lack of ...**\n\n***R to C1.*** Here we discuss the essential difference of QRF against quantile regression (QR) approaches [1] in continual learning. QR approaches estimate the conditional quantiles of the response variable over varying predictor variables. At training time, QR approaches need to minimize the pinball loss while QRF needs to construct trees. Since the pinball loss depends on the significant level $\\alpha$, QR approaches can be computationally expensive to train for multiple significance levels. In contrast, QRF is trained without $\\alpha$ and can provide the asymptotic coverage guarantee with significant level $\\alpha$ (Theorems 1 and 2). Neural quantile estimation [2] is based on conditional quantile regression. It incorporates the concept of quantile regression and considers the case of multiple dimensions. Therefore, the difference of QRF against [2] is similar to that of QRF against QR.\n\n[1] Koenker, R. and Bassett Jr, G. Regression quantiles. Econometrica: journal of the Econometric Society, 1978\n\n[2] He Jia. Simulation-Based Inference with Quantile Regression. ICML 2024\n\n**C2. Potential ...** \n\n***R to C2*** In our paper, the previous task samples stored in the replay buffer will not be changed. Therefore, even if there is a significant distribution shift between tasks, the samples stored in the replay buffer remain representative for previous tasks.\n\n**L1. Limited...**\n\n***R to L1*** In the experiments of Section 6.2, we consider a CL baseline finetuning which greedily trains each task without considering previous task performance---hence introducing high forgetting rates as the number of learning tasks increases. From Figure 3, we observe that the most swarms with finetuning are over the desired coverage lines. These demonstrate the validity of our proposed CPCL. However, the average interval length based finetuning increases as learning tasks. \n\n\n**Essential References Not Discussed**\n\n***R to ERND.*** We refer to the presented references as [3] and [4], respectively, which will be discussed and cited in the revision. The works of [3,4] leverage Bayesian approaches to consider uncertainty in continual learning. [3] studies Bayesian continual learning with uncertainty-aware priors and highlights the role of uncertainty in mitigating catastrophic forgetting. [4] successfully trains deep networks with a natural-gradient variational inference method, VOGN, on a variety of architectures and datasets. Due to the benefits from Bayesian principles, the performance of [4] for continual-learning tasks is boosted. These works and CPCL all focus on uncertainty in continual learning. Compared with these works, CPCL provides asymptotic coverage guarantee with a significant level for the prediction intervals. Moreover, CPCL shows the relationships between the length of prediction intervals and forgetting, which are not introduced by these works.", "claims_and_evidence": "###  Supported Claims  \n- **CPCL provides well-calibrated uncertainty estimates:** Empirical results confirm that CPCL maintains high prediction interval coverage across **different continual learning methods (e.g., SI, EWC, MAS, DGR, Fine-tuning)**.  \n- **Theoretical coverage guarantee:** The paper proves that **CPCL's estimated conditional quantiles converge to true quantiles** as the number of calibration samples increases.  \n- **Forgetting affects prediction interval length:** Experiments show that as **more tasks are learned, forgetting increases**, leading to **wider prediction intervals**, aligning with theoretical expectations.", "methods_and_evaluation_criteria": "### Strengths  \n- **Appropriate benchmark selection:** CPCL is evaluated on both **simulated regression tasks** and **real-world Tiny ImageNet data**.  \n- **Fair baseline comparisons:** The study compares CPCL with **state-of-the-art continual learning methods (e.g., EWC, MAS, DGR)**.  \n- **Clear evaluation metrics:** **Prediction interval coverage** and **interval length** provide useful insights into uncertainty quantification and forgetting.", "theoretical_claims": "###  Correctness of Theoretical Claims  \n- The **asymptotic coverage guarantee** of CPCL is mathematically proven, ensuring that **prediction intervals maintain the desired confidence level** as the number of calibration samples grows.  \n- The connection between **forgetting and prediction interval width** is logically derived and supported by empirical results.  \n\n###  Concerns  \n- **Lack of alternative quantile estimation methods:** While QRF is used for quantile estimation, comparisons with other methods (**e.g., neural quantile estimators**) would strengthen the theoretical claims.  \n- **Potential distribution shift issues:** The paper assumes that **previous task samples stored in the replay buffer remain representative**, but real-world continual learning often involves **domain shifts** that could affect calibration.", "experimental_designs_or_analyses": "###  Strengths  \n- **Comprehensive ablation studies:** The paper evaluates the impact of **different continual learning methods on CPCL's uncertainty estimation**.  \n- **Visualization of uncertainty evolution:** **Coverage and interval length plots** clearly illustrate how **prediction uncertainty changes over multiple tasks**.  \n- **Experiments on both simulated and real-world data:** Ensures robustness of the proposed approach.  \n\n###  Limitations  \n- **Limited discussion on failure cases:** The paper does not analyze situations where **CPCL fails (e.g., high forgetting rates, significant distribution drift)**.", "relation_to_broader_scientific_literature": "The paper makes a meaningful contribution by integrating Conformal Prediction with Continual Learning, extending existing methods while providing new insights into model forgetting and uncertainty quantification.", "essential_references_not_discussed": "The paper focuses on **uncertainty quantification (UQ) in continual learning**, yet it does not reference prior works that have explored UQ in this setting. Some missing references include:  \n\n- **Van de Ven et al. (2022)**: Explored **Bayesian continual learning** with uncertainty-aware priors. Their findings highlight the role of **uncertainty in mitigating catastrophic forgetting**, which aligns with the paper’s motivation but is not cited.  \n  - *Van de Ven, G. M., et al. \"Bayesian Continual Learning: Uncertainty-Aware Priors for Sequential Task Learning.\" NeurIPS, 2022.*  \n\n- **Osawa et al. (2019)**: Investigated **Monte Carlo dropout-based UQ** for continual learning, showing that model confidence degrades over sequential tasks.  \n  - *Osawa, K., et al. \"Practical Deep Learning with Bayesian Principles.\" NeurIPS, 2019.*  \n\nThese works **precede CPCL** in addressing **uncertainty in continual learning** but use **Bayesian approaches** instead of conformal prediction. The paper could compare CPCL to Bayesian methods and discuss their relative strengths/limitations.", "other_strengths_and_weaknesses": "No", "other_comments_or_suggestions": "No", "questions_for_authors": "No", "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": "gzR2PbAywm", "reviewer": "Reviewer_KEqT", "summary": "The paper introduces a conformal prediction-based continual learning (CPCL) method to quantify model uncertainty in continual learning models. CPCL constructs a calibration set using replay techniques and applies a nonconformity score function to measure prediction errors. Theoretical analysis and experiments on simulated and real-world data demonstrate CPCL’s effectiveness in achieving reliable uncertainty quantification.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**R to W1.** Here we conduct experiments in domain-incremental scenarios with gradual distribution shifts. We use the dataset of CORe50 [1], which contains 50 objects (classes). Each object has been collected in 8 distinct indoor sessions characterized by different backgrounds and lighting. Due to the time limit, we only consider $\\alpha=0.3$ and perform 5 runs. In each run, we randomly select one object from the 50 objects and form 8 tasks. The dataset of each task consists of samples which are collected in one session for the selected object. Therefore, 8 tasks corresponds to 8 distinct sessions, respectively. Other experimental settings are the same as those on real-world data (Section 6.2). After learning all tasks, we present the average coverages and length for 5 runs in the following table.\n\n| CL method   | Average coverage  %  | Average length  |\n| ------------------ | ---------- |---------- |\n| SI     |  71.26| 7.54|\n| EWC     |  72.76| 5.14|\n| MAS     | 73.02 | 4.93|\n| DGR     | 73.18 | 3.67|\n| Finetuning   |  71.03 | 16.38|\n\n We observe that the average coverage for 5 runs with different continual learning methods are over the desired coverage 70%. These demonstrate the validity of our proposed CPCL in domain-incremental scenarios. \n\n[1] Vincenzo Lomonaco, Davide Maltoni: CORe50: a New Dataset and Benchmark for Continuous Object Recognition. CoRL 2017.\n\n**R to O1.** We deeply appreciate your valuable comments. We will check all the images.\n\n**R to Q1.** Please refer to **R to W1.** of Reviewer KEqT.\n\n\n**R to Q2.** After learning the new task $t$, we obtain the trained model $\\hat{f} _{t}$ which is changed compared to the model $\\hat{f} _{t-1}$ at time $t-1$. According to Equation (5), for each observation $Z _{i}^{j}=(X _{i}^{j},Y _{i}^{j}), j<t$ in the calibration set, the prediction error $\\hat{\\mu _{i}}^{j} =Y _{i}^{j}-\\hat{f} _{t}(X _{i}^{j})$ is changed, which results in calculating different nonconformity scores. Therefore, QRF needs to be trained form scratch, which indicates that intervals should be recalculated from scratch too. It is interesting to explore whether they can be derived from previously computed one. We will study this issue in the future work.\n\n**R to Q3.**  Figures 3(b), 3(d) and 3(f) in the main file show the average interval length on real-world data. As the number of learning tasks increases, we find that the average interval length based on any continual learning method tends to increase. For example, the average length of the prediction interval ranges from 5 to 10 after learning task 5, while it remains below 5 after learning task 3, when we us SI as the continual learning method and $\\alpha=0.1$.\n\n**R to Q4.**  As stated in **R to Q2.**  of Reviewer KEqT, the trained model $\\hat{f} _{t}$ at time $t$ is different from $\\hat{f} _{t-1}$ at time $t-1$.  QRF needs to be trained form scratch, which indicates that intervals should be recalculated from scratch too.  Therefore, for  each test sample, the corresponding prediction interval should be updated at time $t$ and the output prediction interval at time $t-1$ is not valid.\n\n**R to Q5.** Here we conduct experiments on simulated data by split conformal prediction (SCP) [1]. Due to the time limit, we only consider Finetuning as the CL baseline. Other experimental settings are the same as those on simulated data (Section 6.1). We present the average coverages for 100 runs in the following table.\n\n| $\\alpha$   | Average coverage  %   ||\n| ------------------ | ---------- |---------- |\n|   | SCP   | CPCL |\n| 0.05     |  92.58| 96.51|\n| 0.1     |  86.47| 91.74|\n| 0.15     | 81.69 | 86.23|\n| 0.2     | 76.08 | 81.30|\n| 0.25   |  70.82 | 77.05|\n| 0.3   |  65.53 | 71.29|\n\nWe find that CPCL succeeds at all significant levels $\\alpha$, but the average coverage for each $\\alpha$ doesn't reach the desired value when using SCP. SCP requires the principle of data exchangeability which is violated in continual learning. Therefore, the coverage of SCP can not be guaranteed with significant level $\\alpha$.\n\n[1] Vovk, V. Conditional validity of inductive conformal predictors. Mach. Learn., 2013", "claims_and_evidence": "The paper introduces a Conformal Prediction-based Continual Learning method for quantifying model uncertainty in continual learning. The main claim is that CPCL can provide asymptotic coverage guarantees for prediction intervals. In addition, the confidence intervals proposed are agnostic to the method used.", "methods_and_evaluation_criteria": "The authors assess the performance using both simulated data and real-world datasets for different coverage probabilities.", "theoretical_claims": "The proposed method provides theoretical guarantees. For instance, the authors show theoretical results for the relationship between the conditional distribution of CPCL and the true conditional distribution.", "experimental_designs_or_analyses": "Experiments evaluate CPCL across five continual learning methods such as  SI and EWC.", "relation_to_broader_scientific_literature": "The paper builds on conformal prediction literature, particularly adapting it for continual learning where data from previous tasks is inaccessible.", "essential_references_not_discussed": "The literature review is adequate.", "other_strengths_and_weaknesses": "Strengths:\n\nProvides theoretical guarantees. \n\nExperimental evaluation across multiple methods. \n\nWeaknesses:\n\nDoes not explore scenarios with task overlap tasks or distribution shift.", "other_comments_or_suggestions": "Just to let the authors know that in Adobe, I can’t see either the legends or the axis numbers. I can only see them if I open the PDF with Preview. I have the same issue on two different lap tops, so it might be a problem with the images.", "questions_for_authors": "Can the method be extended to handle domain-incremental scenarios with gradual shifts in data distribution?\n\nAre the intervals recalculated from scratch each time a new task is obtained, or could they be derived from previously computed ones?\n\nHow does the number of tasks affect the width of the intervals?\n\nAt time t-1, we obtain confidence intervals with a coverage probability of 1-\\alpha, for example. At time t, we obtain confidence intervals again with the same coverage probability. Are the intervals from t-1 still valid? In other words, are they simultaneous confidence intervals?\n\nCould the given intervals be compared with those obtained using other conformal prediction methods or alternative approaches?", "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": "ML6suNX7b7", "reviewer": "Reviewer_vzZS", "summary": "The authors propose a Conformal prediction-based methodology to address the calibration problem, which is reliably quantifying model prediction uncertainty in continual learning settings. The authors first enumerate reasons why a standard conformal prediction method cannot be extended to continual learning settings, including performance changes due to the order of tasks, violation of data exchangeability, where this issue is caused by the continual learning setting, and limitations in constructing calibration sets due to inaccessibility of samples from previous tasks. To address these constraints, the authors constructed a calibration set, which is made by the replay samples in the continual learning, and proposed a sequentially dependent score function, which is used for continual setting. Through this, the authors demonstrate the connection between prediction interval length and forgetting and experimentally prove the significance of the proposed method in experiments with real/synthetic data.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**R to EDOA1.** We conduct experiments on simulated data by split conformal prediction (SCP) . Please refer to **R to Q5.** of Reviewer KEqT.\n\n**R to EDOA2.** Here we conduct experiments on real-world data by creating the scenarios where tasks are ordered from easy to hard and hard to easy. In Section 6.2 of the main file, we conduct experiments using Tiny ImageNet and perform 20 runs with different random seeds. For each run, we form 5 tasks. To order the task, we utilize a pretrained AlexNet and calculate the accuracy of this pretrained AlexNet on each task. We can obtain the order of tasks from easy to hard by sorting the accuracies on different tasks from high to low. By reversing this order, we get the order of tasks from hard to easy. Due to the time limit, we only consider $\\alpha=0.3$. Other experimental settings are the same as those on real-world data (Section 6.2). After learning all tasks, we present the average coverages and length for 20 runs in the following table.\n\n|    | Easy to hard  | | Hard to easy  | |\n| ------------------ | ---------- |---------- |---------- |---------- |\n| CL method   | Average coverage  %  | Average length  | Average coverage  %  | Average length  |\n| SI     |  72.63| 6.44| 73.43| 8.97|\n| EWC     |  74.81| 4.65| 73.19| 6.52|\n| MAS     | 72.20 | 3.05| 72.33| 5.65|\n| DGR     | 73.59 | 2.26| 74.09| 4.88|\n| Finetuning   |  72.12 | 12.10|   71.25 | 17.41|\n\n We observe that the average coverage for 20 runs with different continual learning methods are over the desired coverage 70%. These demonstrate the validity of our proposed CPCL in the scenarios where tasks are ordered from easy to hard and hard to easy. It can be observed that the average interval length in the scenario with tasks ordered from easy to hard is greater than that in the scenario with tasks ordered from hard to easy. For example, when we use SI as the CL baseline, the average interval length on easy to hard task order is 6.44 while that on hard to easy task order is 8.97. These results indicate that task order significantly affects model uncertainty in CL.\n\n **R to EDOA3.** We find that our proposed CPCL cannot be directly applied on time series data. There is a sequence of tasks in CL. The CL setting requires that data within a task is exchangeable, while data between different tasks is not exchangeable. According to this setting, to train QRF, we obtain the reconstructed dataset $D^{R} _{T}=\\{(X^{R} _{i},Y^{R} _{i})\\}^{N _{cal}-1} _{i=1}$, where $X^{R} _{i}=[S^{1} _{i}, S^{2} _{i}, \\dots, S^{T} _{i}]$ and each entry $S^{t} _{i}$ corresponds to a sample of task $t$ in the replay buffer. In contrast, time series setting requires that all data is not exchangeable, which means that it is hard to obtain the reconstructed dataset. Meanwhile, it is interesting and we will study this issue for time series setting in the future work.\n\n**R to OCOS1.** We will include conformal prediction in the title.\n\n**R to Q1.** We introduce the sigmoid-based nonconformity score function for two reasons. (1) The range of sigmoid-based nonconformity score function is $(0,1)$. We train QRF on the reconstructed dataset $D^{R} _{T}=\\{(X^{R} _{i},Y^{R} _{i})\\}^{N _{cal}-1} _{i=1}$, where each entry $S^{t} _{i}$ in $X^{R} _{i}$ is the calculated score. The training of QRF requires to construct trees. This process involves the intervals $I(x,m,\\zeta _{k})\\subseteq(0,1)$. The interval $I(x,m,\\zeta _{k})$ is used to determine whether $X^{R} _{i}$ is in the node corresponding to $I(x,m,\\zeta _{k})$, i.e., if $S^{m} _{i} \\in I(x,m,\\zeta _{k})$, then $X^{R} _{i}$ is in the corresponding node. Since $I(x,m,\\zeta _{k})\\subseteq(0,1)$, we need to define that the score function ranges in $(0,1)$ for rigorous proof. (2) Sigmoid-based nonconformity score function is invertible. In Equation (11) we present the prediction interval in terms of scores. The invertible function helps us to rewrite the prediction interval which is shown in Equation (12). A typical score function in CP is $s(X _{i}^{t},Y _{i}^{t}) =|\\hat{\\mu _{i}^{t}}|$ where $\\hat{\\mu _{i}^{t}} =Y _{i}^{t}-\\hat{f} _{T}(X _{i}^{t})$. It is not invertible and the corresponding range is not $(0,1)$. Therefore, it is not suitable in this paper.\n\n**R to Q2.** In Section 5, we provide the asymptotic coverage guarantee. Specifically, Theorem 1 demonstrates that the conditional CDF $\\hat{F}(s|x)$ estimated by CPCL converges in probability to the true conditional CDF $F(s|x)$ as $N _{cal}\\to \\infty$, i.e. $\t|\\hat{F}(s|x)-F(s|x)|\\to _p 0 \\quad N _{cal}\\to\\infty$. Therefore, our proposed asymptotic coverage guarantee ensures that QRF can accurately estimate quantiles.", "claims_and_evidence": "* The authors demonstrated convergence to the oracle prediction interval as the calibration set size increases, from the perspective that accurate calibration becomes possible for each task when storing many replay samples.\n\n* The authors' proposed uncertainty measurement technique observes the forgetting phenomena of continual learning models, commonly known as deep learning models, performing poorly in continual learning settings. It serves as experimental evidence supporting their theoretical foundation.", "methods_and_evaluation_criteria": "The authors verified model uncertainty across various continual learning approaches using simulated data. They demonstrated validity by showing the algorithm's coverage for given significance levels. They also conducted similar experiments on real datasets to prove their effectiveness.", "theoretical_claims": "The authors provide theoretical evidence for the significance of their method through two theorems. Typical continual learning scenarios are often compared to an oracle setting where all data is accessible. Similarly, the authors proved these two theorems from the perspective that if a large number of samples are stored for all tasks, it approaches the oracle setting. Additionally, they specify the essential assumptions necessary for these theorems.\n\n(Theorem 1) The authors prove that as the number of samples in the Calibration Set increases, the estimated conditional CDF for a given input x converges in probability to the Oracle conditional CDF.\n\n(Theorem 2) The authors prove that as the number of samples in the Calibration Set increases, the estimated conditional quantile also converges to the true conditional quantile.", "experimental_designs_or_analyses": "* Comparative experiments are needed to evaluate the performance when using simple quantiles directly, even if it violates basic assumptions of the standard conformal prediction. \n\n* To examine how uncertainty changes according to task order, I believe experiments are needed to evaluate whether the proposed method can effectively interpret situations using simulation data that creates scenarios where difficulty changes from easy to hard and hard to easy across various continual learning algorithms. \n\n* Additionally, I would like to see validation on datasets like time series data.", "relation_to_broader_scientific_literature": "This method can be used across various fields as it can measure uncertainty in regression-type methodologies in continual learning. As data increases, a growing number of models are being trained in continual learning scenarios. Since this method can measure uncertainty regardless of model type, it can be used to analyze prediction models used across various fields.", "essential_references_not_discussed": "To my knowledge, this paper adequately covers relevant papers in the field.", "other_strengths_and_weaknesses": "* This paper applies the conformal prediction framework to regression tasks in continual learning. The authors propose a method to construct prediction intervals with conditional coverage guarantees to overcome the constraint of being unable to access samples from previous datasets. The authors explain their proposed method clearly and comprehensibly. Additionally, this approach is technically well-justified as an extension of existing methods.", "other_comments_or_suggestions": "* I suggest revising the title of the authors' paper. The current title is too broad. I believe \"conformal prediction\" should be included in the title.", "questions_for_authors": "* The authors should provide a more detailed explanation for introducing the sigmoid-based nonconformity score function.\n    - Could they elaborate on potential concerns when using different functions?\n* I have a fundamental question about whether Quantile Regression Forests (QRF) can accurately estimate quantiles in data with sequential dependencies through learning. I would like to know the authors' thoughts on this.", "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": "I73BmKPEBV", "reviewer": "Reviewer_kvQq", "summary": "The paper addresses the problem of continual learning with calibration guarantees. More precisely, the purpose is to train a model to address a series of tasks, in a sequential way (i.e., one task after the other). The datasets used to train the model on the successive tasks are not exchangeable, and may even be forgotten. Then, building a calibration set becomes challenging. In a nutshell, to overcome this issue, the authors propose to use a replay-based strategy to construct the calibration dataset; this dataset is used to train a quantile regression model, based on which nonconformity scores can be predicted: the dataset is first \"reconstructed\" (features being obtained from the past nonconformity scores and outputs from the average current nonconformity scores). Thus, for any test instance based on the forest, nonconformity scores can be obtained, and therefore prediction intervals. \n\nThe introduction recalls the setting, motivates the paper and provides the main ideas of the contribution. Section 2 presents the related works, starting with continual learning and proceeding with a short refresher on conformal prediction. Section 3 provides a formalization of the problem addressed. Section 4 presents the main contribution of the paper, explaining how the calibration set is obtained, presenting the considered nonconformity score, and explaining how the quantile regression forest can be trained to predict NC scores and how decisions can subsequently be made. The whole pipeline is summarized in an algorithm. Section 5 provides a theoretical analysis of the proposal, providing a consistency property based on four assumptions. Section 6 reports the experiments realized on a synthetic dataset and on a real dataset (Tiny ImageNet). Section 7 concludes the paper. \n\n### update after rebuttal \n\nI would like to thank the authors for their aswers to my comments and questions. I updated my score accordingly.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**R to OCOS1.** The end of the \"Nonconformity score function\" paragraph in Section 4 mainly discusses the score set calculated by the nonconformity score function. We are sorry that our notations is hard to follow. We will revised the confusing notations.\n\n**R to OCOS2.** We will improve our writing of all equations in our paper to makes them easy to read. No norm is missing in Equations (5) and (6). Equation (13) aims to calculate the $\\hat{\\beta}$ in Equation (12).\n\n**R to OCOS3 and OCOS4 .** We will fix these typos and incomplete references. \n\n**R to Q1.** We introduce the sigmoid-based nonconformity score function for two reasons. Please refer to  **R to Q1.** of Reviewer  vzZS.\n\n**R to Q2.**  Assumptions 1 and 2 make assumptions on the conditional CDF. The conditional distribution function is assumed to be Lipschitz continuous in Assumption 1 and strictly monotonously increasing in Assumption 2. Assumptions 3 and 4 focus on the actual construction of trees. For each tree with separate parameter $\\zeta _{k}$, there are $L$ leaves, where every leaf $l$ is associated with a rectangular subspace $R _{l}\\subset\\mathbb{B}$. These subspaces are disjoint and cover the entire space $\\mathbb{B}$, i,.e. for any $x\\in\\mathbb{B}$, there is one and only one leaf which corresponds to $R _{l(x,\\zeta _{k})}$. Denote the node-sizes of the leaves $l$  of a tree by $w _{\\zeta _{k}}(l)=\\|\\\\{i\\in\\{1,\\ldots,N _{cal}-1\\}:X _{i}\\in R _{l(x,\\zeta _{k})}\\\\}\\|$. $X _{i}\\in(0,1)^{N _{\\tau}}$ is the observation to train QRF.\n\nThe key of Assumptions 3 is \"For any $x\\in \\mathbb{B}$, $p _{i}(x)$ satisfies that $p _{i}(x)=o(1)$\". We discuss Assumption 3 by  **Example 1**: The minimal number of observations in a node is growing for large $N _{cal}$, i.e., $1/\\operatorname*{min} _{l,\\zeta _{k}} w _{\\zeta _{k}}(l) =o(1), N _{cal}\\to\\infty.$. Recalling $p _{i}(x)=\\sum _{k=1}^{K} p _{i}(x,\\zeta _{k})/K$, we have $0\\leq p _{i}(x)\\leq 1/\\operatorname*{min} _{l,\\zeta _{k}}w _{\\zeta _{k}}(l) =o(1)$, which means that  $p _{i}(x)=o(1)$ for any $x\\in \\mathbb{B}$. Therefore, Assumption 3 represents the case of **Example 1**.\n\nAssumptions 4 is \"For any $x\\in \\mathbb{B}$, the rectangular subspace $R_{l(x,\\zeta_{k})}\\subseteq(0,1)^{N_{\\tau}}$ of leaf $l(x,\\zeta_{k})$ of tree $\\zeta_{k}$ is defined by the intervals $I(x,m,\\zeta_{k})\\subseteq(0,1)$, i.e. $R_{\\ell(x,\\zeta_{k})}=\\otimes_{m=1}^{N_{\\tau}}I(x,m,\\zeta_{k})$, where $\\otimes$ means direct sum. We assume that $\\max_{m}|I(x,m,\\zeta_{k})|=o_{p}(1)$ for $N_{cal}\\to \\infty$\". We discuss Assumption 4 by  **Example 2** which consists of three situations. In **situation 1**, the proportion of observations in a node, relative to all observations, is vanishing for large $N _{cal}$ i.e., $\\operatorname*{max} _{l,\\zeta _{k}} w _{\\zeta _{k}}(l) =o(N _{cal}), N _{cal}\\to\\infty.$.  In **situation 2**, when finding a variable for a splitpoint, the probability that variable $m=1,...,N _{\\tau}$ is chosen for the splitpoint is bounded from below for every node by a positive constant. In **situation 3**, if a node is split, the split is chosen so that each of the resulting sub-nodes contains at least a proportion $\\gamma$ of the observations in the original node, for some $0<\\gamma\\leq0.5 $. \n\nAs any $x\\in \\mathbb{B}$ is dropped down a tree, several nodes are passed. Denote by $S(x,m,\\zeta _{k})$ the number of times that these nodes contain a splitpoint on variable $m$. The total number of nodes that $x$ passes through is denoted by $S(x,\\zeta _{k})=\\sum _{m=1}^{N _{\\tau}} S(x,m,\\zeta _{k}).$ Using **situation 3**, the maximal number of observations in any leaf, $\\operatorname*{max} _{l} w _{\\zeta _{k}}(l)$ is bounded (for every tree) from below by $N _{cal} \\gamma^{S _{\\min}(\\zeta _{k})}$ where $S _{\\min}(\\zeta _{k})=\\operatorname*{min} _{x\\in  \\mathbb{B}} S(x,\\zeta _{k})$. Using **situation 1**, $\\operatorname*{max} _{l} w _{\\zeta _{k}}(l)$ is on the other hand bounded from above by an $o(N _{cal})$-term. Putting together, we conclude that $\\gamma^{S _{\\min}(\\zeta _{k})}=o(1)$ for  $N _{cal}\\to \\infty$. Hence there exists a sequence $s _{N _{cal}}$ with $s _{N _{cal}} \\to \\infty$ for $N _{cal}\\to \\infty$ such that $S _{\\min}(\\zeta _{k}) \\geq s _{N _{cal}}$. As the probability of splitting on variable $m=1,...,N _{\\tau}$ is bounded from below by a positive constant, by **situation 2**, there exists a sequence $g _{N _{cal}}$ with $g _{N _{cal}} \\to \\infty$ for $N _{cal}\\to \\infty$ such that $P\\\\{\\min _m S(x,m,\\zeta _{k})>g _ {N _{cal}} \\\\}\\to1\\quad N _{cal}\\to\\infty$. Using  **situation 3**, we obtain that $|\\\\{i\\in\\{1,\\ldots, N _{cal}-1\\}:X _{i,m}\\in I(x,m,\\zeta _{k})\\\\}|/( N _{cal} -1) \\leq(1-\\gamma)^{S(x,m,\\zeta _{k})}$. Putting together, we conclude that $\\max _{m} | \\\\{ i\\in\\{1,..., N _{cal}-1\\}:X _{i,m}\\in I(x,m,\\zeta _{k})\\\\}|/(N _{cal}-1)=o _{p}(1)$ which indicates $\\max _{m}|I(x,m,\\zeta _{k})|=o _{p}(1)$ for $N _{cal}\\to\\infty$. Therefore, Assumption 4 represents the case of **Example 2**.", "claims_and_evidence": "The paper presents a strategy for continual learning with calibration guarantees. This claim is backed by a theoretical analysis, and more precisely formalized into two theorems. The experiments realized are rather simple (a synthetic dataset, and a real-world dataset), but nevertheless back up the proposal. \n\nOverall, the evidence provided rather convincingly supports the claim, even if the strength of the assumptions in Section 5 is not discussed. Some components in the proposal are also rapidly presented, and could have been better justified or at least clarified.", "methods_and_evaluation_criteria": "The experimental setting and the datasets considered make sense for the problem at hand, even if a more thorough experimental analysis would have been appreciated.", "theoretical_claims": "I only briefly checked the soundness of the proofs (provided in the appendices); they seem correct.", "experimental_designs_or_analyses": "The experimental study conducted seems valid; the experimental setup is convincing and the results are as the reader could expect.", "relation_to_broader_scientific_literature": "The discussion on the related works seems to include the main references pertaining to the problem addressed.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The paper is overall well written, but sometimes difficult to follow (see below for some comments).", "other_comments_or_suggestions": "I'd suggest to improve the writing. \n\nSome parts are difficult to understand, e.g. the end of the \"Nonconformity score function\" paragraph in Section 4 (page 4). As well, the paragraph dedicated to quantile regression forests (mainly the part page 5) is difficult to follow due to the notations, to the lack of explanations, and to an extensive use of math in the text. \n\nIn Equation (1), $\\mu$ would have been formally defined, and the assumptions pertaining to it could have been made explicit. It seems that a norm is missing either in Equation (5) or in Equation (6). Equation (13) seems to be disconnected from what precedes. \n\nThere are some (minor) typos. A couple of words are missing here and there, e.g. \"In continual learning setting\" (page 2), \"the scores in score set\" (page 4), \"score set\", \"on calibration set\", \"that conditional distribution function\" (page 5). Page 2, in \"(IMM) (Lee et al., 2017)\", there should not be parentheses around \"IMM\". In Section 3 (page 4), $Z_{ut}=(X_{ut},Y_{ut})$ is not properly defined. \n\nSome references seem incomplete, e.g. \"Krizhevsky, A., Sutskever, I., and Hinton, G. E. Imagenet\nclassification with deep convolutional neural networks. In NeurIPS, 2012.\"", "questions_for_authors": "Could you elaborate on the nonconformity score function such as defined by Equation (6) ? \n\nCould you discuss the assumptions made to establish the consistency results in Section 5.2 ?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This submission studies the issue of calibration in continual learning which reliably quantifies the uncertainty of model predictions. Starting from conformal prediction, the authors improved the inherent principle violation for continual learning context, and then proposed a CP-based method for model uncertainty quantification. Through the theoretical analysis and extensive experiments, it shows the promise of CPCL.

The submission received four reviewers' review, converging to the ratings 3, 3, 4, 4. The initial concerns of the reviewers focus on the lack of discussion about the failure cases, extension to the domain-incremental scenario and some technical details in the theoretical analysis. After the substantial efforts in rebuttal, the authors well addressed the reviewers' concerns and all reviewers maintain positive to this submission. Based on the reviewers' suggestion and interaction between the reviewers and authors, AC tend to recommend "Acceptance", and hope the authors carefully incorporate the advice into the manuscript.
Decision: Accept (poster)