Distribution-informed Online Conformal Prediction
Authors: Dongjian Hu
Junxi Wu
Shu-Tao Xia
Changliang Zou
Venue: ICLR 2026 Poster (2026)
Keywords: Conformal Prediction; Uncertainty Quantification; Distribution Shift; Time Series
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We propose Conformal Optimistic Prediction (COP), an online conformal prediction algorithm incorporating underlying data pattern into the update rule with provable coverage and tighter intervals.

Abstract: Conformal prediction provides a pivotal and flexible technique for uncertainty quantification by constructing prediction sets with a predefined coverage rate. Many online conformal prediction methods have been developed to address data distribution shifts in fully adversarial environments, resulting in overly conservative prediction sets. We propose Conformal Optimistic Prediction (COP), an online conformal prediction algorithm incorporating underlying data pattern into the update rule. Through estimated cumulative distribution function of non-conformity scores, COP produces tighter prediction sets when predictable pattern exists, while retaining valid coverage guarantees even when estimates are inaccurate. We establish a joint bound on coverage and regret, which further confirms the validity of our approach. We also prove that COP achieves distribution-free, finite-sample coverage under arbitrary learning rates and can converge when scores are i.i.d. The experimental results also show that COP can achieve valid coverage and construct shorter prediction intervals than other baselines.

Published as a conference paper at ICLR 2026
DISTRIBUTION-INFOREED ONLINE CONFORMAL PREDICTION
Dongjaan Hul; Junxi Wu! Shu-Tao Xia Changliang Zou't
1 School of Statistics and Data Science, LPMC, KLMDASR and LEBPS, Nankai University 2 Tsinghua Shenzhen International Graduate School, Tsinghua University
ABSTRACT
Conformal prediction provides a pivotal and flexible technique for uncertainty quantification by constructing prediction sets with predefined coverage rate. Many online conformal prediction methods have been developed to address data distribution shifts in fully adversarial environments, resulting overly conservative prediction sets. We propose Conformal Optimistic Prediction (COP), an online conformal prediction algorithm incorporating underlying data pattern into the up- date rule. Through estimated cumulative distribution function of non-conformity scores, COP produces tighter prediction sets when predictable pattern exists, while retaining valid coverage guarantees even when estimates are inaccurate. We estab- lish a joint bound on coverage and regret, which further confirms the validity of our approach. We also prove that COP achieves distribution-ffe free, finite-sample cov- erage under arbitrary learning rates and can converge when scores are i.i.d. The experimental results also show that COP can achieve valid coverage and construct shorter prediction intervals than other baselines.
INTRODUCTION
Uncertainty quantification is essential for making reliable forecasts, as prediction errors could lead to severe consequences, particularly in high-risk domains including epidemiology and finance. Despite the availability of various methods for uncertainty quantification, such as confidence calibration (Guo al., 2017) and Bayesian networks (Fortunato al. 2017). these approaches often fail to provide provable coverage guarantees, which significantly restricts their applicability and reliability.
To address this limitation, Conformal Prediction (CP) (Vovk et al., 2005) stands out as a powerful technique to construct prediction sets with model-agnostic and distribution-free properties. Under the assumption of data exchangeability, it theoretically guarantees that the prediction sets contain the true label with a specified probability. Consequently, CP has shown promising performance in multiple domains, such as large language models (Gui et al. 2024), robotics (Lindemann et al. 2023). image classification (Angelopoulos et al.) 2020), and decision-making (Bao et 2005).
Despite the success of CP for exchangeable data, it remains challenging to directly perform CP in scenarios involving distribution shifts and temporal dependence that violate the exchangeability assumption. To resolve this problem, significant efforts have been made to extend CP methods beyond
exchangeability (Barber al. 2023 Xu & Xic 2021 Tibshirani et al. 2019 Yang et al. 2024).
Recently, a large amount of research is emerging in Conformal Prediction for time series. Pioneered by ACI (Gibbs& & Candès 2021), adversarial online learning methods were incorporated into the CP framework to adapt to arbitrary online distribution shifts. To improve it, Conformal PID (Angelopou los et al. 2023b) incorporated PID control and ECI (Wu et al. 2025) leveraged error quantification via smoothed quantile loss. To further determine the step size for dynamic environments, other online learning techniques were used such as directly choosing decaying step size (Angelopoulos et al. 2024), expert aggregation (Zaffran et al.] [022]] Gibbs & Candès, (024). and strongly adaptive regret (Bhatnagar et al. 2023 Hajihashemi & Shen 2024). However, they focus on fully adversarial environments and adjust prediction sets simply based on the iterative steps, without considering the
"Equal contribution.
1 "orresponniig authors.

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underlying data patterns. Consequently, despite their well-established theoretical basis, they tend to construct overly conservative prediction sets in practice.
Instead of online learning methods, another line of work (Xu & Xie 2021 2023 Lee et al. 2024)
proposed methods to directly obtain prediction sets via estimated distribution function of non- conformity scores, such SPCI (Xu & Xie, 2023), which employed random forest estimator. Their theoretical results require assumptions like model consistency and distributional smoothness, and hence not distribution-free. Other methods focus on assigning weights to historical scores. HOP-CPT (Auer et al., 2023) utilized the Modern Hopfield Network and CT-SSF (Chen et al. 2024ap leveraged the inductive bias in deep representation learning to dynamically adjust weights. However, these approaches often lack theoretical results, such as finite- finite-sample coverage, making them potentially unreliable and suffer from overfitting issues.
In this work, we propose Conformal Optimistic Prediction (COP), an online CP algorithm that incorporates underlying data pattern of the non-conformity scores into the update rule. The key step is refinement update based on the linear approximation of the expected quantile loss and an estimated cumulative distribution function (CDF). COP maintains the well-grounded coverage guarantees inherent to traditional methods, while taking advantage of the distribution information to construct tighter prediction intervals. We also establish a connection to the online optimistic gradient descent (Rakhlin & Sridharan 2013 Zhao et al., 2024). with a scaled distribution -nnfuutiio----mmeedd hint. From this standpoint, we derive a joint bound on regret and coverage, which further confirms the validity of approach. Moreover, COP does not need to compute inverse of the estimated CDF as Lee et al. (2024) and WWan et al. (2025) do, thereby reducing computational cost and unnecessary numerical error. Our contributions can be summarized follows:
We propose novel method for online CP, Conformal Optimistic Prediction (COP). Com- pared to traditional methods, COP introduces refinement step leveraging the estimated distribution function, which allows for capturing potential predictable information of the non-conformity scores and providing more efficient prediction sets.
We also introduce another perspective on COP through the lens of optimistic online gradient descent, which facilitates more comprehensive theoretical analysis. We further explore a joint regret-coverage bound for COP that sheds some light on the theoretical motivation behind our approach.
We prove the distribution- free and finite-sample coverage of COP with general optimistic terms under arbitrary learning rates. Moreover, we demonstrate the asymptotic consistency of COP with i.i.d. scores and properly chosen learning rates.
Our experimental results demonstrate the superior performance of COP in time series datasets, including simulation data under distribution shift and wollworld data in the finance, energy, and climate domains. We show that COP maintains coverage at the target level and obtains tighter prediction sets than other state-of-the-an methods.
2 BACKGROUND
2.1 PROBLEM SETUP
Given sequentially collected data ((X,Y))ee C x V. at time f, our base model fi utilizes previously observed data ((x,, Y)) to produce point prediction Y, fx())) for the unob- served label Y,. The goal of CP is construct prediction set (,,)) based on Y,. Aligned with the standard CP methods, consider a conformal score function S.(:,) X' x ) - IR. which measures the discrepancy between the base model's prediction f(x.) and the true label Y,. For example, the score function can be Sz y) y (z)). Given S,, CP constructs the set in the form of:
C, ty y S,(X,,)) S ).).
where q, is the threshold to be determined.
Given nominal miscoverage level a, note that if the data sequence {(X,, hi) are i..d. or exchangeable, taking de to be the [(1 (1 + 1)] smallest elements among (S,,X, Y.))
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yields valid coverage guarantee by split conformal prediction
P(Y++1 E Cx+1(X++1)) 2 -
However, an adversarial setting where exchangeability does not hold, such as time series data with strong correlations. it is very difficult to achieve such real-time coverage guarantee. Recent works (Bhatnagar et al. 2023 Anneloopolos al., 2023b] 2024) Podkopaev et al., (024) have turned to consider the following long-term coverage guarantee:
T E
lim T-000 T uy, & C(x)))
Parallel these works, our target lies in designing online algorithm dynamically choose the threshold dr that can achieve the control in equation without any distributional assumptions on data {(X,, 12) Also, want the size of our prediction sets to be as small as possible.
2.2 CONFORMAL. PREDICTION FOR SEQUENTIAL DATA
Let s; S(X, Y) be the non-conformity score at time t. When the label Y; is observed, online CP considers updating &. with the following rule
0 = Gt + 17(erT, - a),
(2)
where err, = (s+ > 4 = (Y, & (x)))) is the miscoverage indicator and rj is the learning rate.
The idea to qr through online gradient descent with fixed step size first introduced in ACI (Gibbs & Candès] 2021). Subsequent improvements include improving update rules by introducing decaying stepsizes (Angelopoulos al. 2024), PID control (Angelopoulos al. 2023b), and error quantification (Wu et 2025).. Moreover, some methods borrow online learning techniques and consider not only coverage guarantee, but also dynamic regret analysis (Gibbs Candès 2024] Bhatnagar et al. 2023 Podkopaev et al., 2024). [aao et al. (2024) investigated online selective conformal inference problem for i.i.d. data stream. However, these methods only targeted at fully
adversarial environments and ignored the predictive part in the sequence. Another line of work aims
at learning the distribution data with estimated distribution functions (Xu & Xic 2023
Lee et al.
2024) or capturing temporal structure through deep neural networks (Chen et al. 2024a Auer et al. 2023
However, both of them cannot provide provable coverage guarantees.
Most relate us the LQT algorithm proposed by Areces et al. (2025) and the Conformal PID algorithm in |Aneelopoulos et al. (2023a). They consider both adversarial and predictable data. However, the application of LQT relies heavily linear autoregressive structures. For Conformal PID, the selection of its scorecaster model arbitrary and lacks principled guidance. An inaccurate excessively complex scorecaster may impair performance by introducing additional variance.
2.3 OPTIMISTIC ONLINE GRADIENT DESCENT
Optimistic online gradient descent (OOGD) has been extensively studied and applied across various domains. aims at adding prior knowledge about the sequence within the paradigm of online learning. RRahliin& & Sridharan (2013) proposed the general version of the Optimistic Mirror Descent algorithm, which was bulit upon the results in (Chiang et al.,
2012). Recently, OOGD has been
adopted in Riemannian online convex optimization (Wang et al.
2023 Roux et al.
2025), generative
models (Daskalakis et al. 2017), reinforcement learning (Ito 2 Bubeck et al.
2019). etc. Other
theoretical results of OOGD include gooblem-dependee regret bounds (Chen et al.;
2024b Zhao et al.
2024), saddle point problem (Daskalakis & Panageas 20188] Mertikopoulos et all
2018 Mokhtari
al. 2020) and other last iterate convergence properties (Gorbunov et al. 2022
METHOD AND THEORY
3.1 CONFORMAL OPTIMISTIC PREDICTION
We begin by considering eq. 2 from the view online graient descent. Let l = (q >
0} agg denote the (1 ath quantile loss, then eq. (2) be viewed online (sub)gradient
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descent (OGD) on the quantile loss 41-a (St q). For simplicity, we denote
V1 as
(s .e), and rewrite eq. (2) as:
g le + (err, a) = 0: V Qc)
(3)
One critism of eq. (3) is that the good coverage is obtained due reactively correcting past mistakes and through a cancellation of positive and negative errors. The radius d: may exhibit significant fluctuations around the true value (Gibbs & Candès 2021 Wu et al., 2025), leading to inefficient prediction intervals. An alternative way is to directly estimate the conditional quantiles or distribution function F, of the scores in an online fashion (Lee et al. 2024). Then i can directly take F - a). However, the validity of these methods fully counts the estimated quantiles CDF, which may suffer from overfitting when addressing adversarial data.
To mitigate the limitations, we consider improving eq. (3) by taking advantage of the predictable information through an estimated distribution function. Let S = (81, 8t) and we denote the cumulative distribution function (CDF) of $t+1 conditional on S, as F-+)(-(S,)
F++1(-) for
simplicity. Note that, if the conditional distribution of st SS is invariant over t, the target of conformal prediction can be approximately viewed as finding the conditional (1 a-)-th quantile of st St Hence, an intuitive way to improve the efficiency of OGD is to adopt the following refinement for g.+1. For some A++1 > O:
g++1 arg min Exyy (8-a - g)[S(] + 2A++1
11 t++ 122-
Given that we output q: at timestep t, the miscoverage indicator of interest turns into err; = 1(s; > 43,, instead of {s > q: in eq. (3). Hence, eq. (3) is substituted with:
g--1 = q + n(err; - a) = 9: 1 qr). -
The refinement serves as a re-calibration step to adjust and optimize the conservative action that update eq. (4) makes, and utilizes the predictable information non-conformity scores {sth21 to enhance the conditional performance.
Let (q) E..., (1 (8t+119){8.).. Note that V,,C+1(9)) Fi- (1 a), the optimiza- tion above does not have closed form solution. Similar issues also occur when using online mirror descent. We discuss this Appendix [A.T] As opposed to the above implicit update, we therefore turn to minimize the local approximation.
By convexity of Le+1:
L+t1(g) 2 (441((41)) + (q (4+4((++2((444) -11 a))
Denote Fit1 1 as the estimated CDF s (S. Replacing F.t1 with Fit1. . the optimization above reduces to:
Gc+1 = arg min Le+1 (q (q- (122) (fe+1(0+11) (1 (1) + 2
1 111--168 1--11111..
Equivalently, we have
2++ Gc+1 De-1 (Pet) (1-)))
(5)
We remark that our refinement also be implemented another widely employed online CP method, adaptive conformal inference (ACI) (Gibbs & Candès] (022). as detailed in Appendix A.2 There are many choices for Fit1, such as empirical cumulative distribution function (ECDF) and bernel-based estimation (Silverman 1986). As brief clarification on the benefit of our refinement, we present the proposition as follows.
Proposition Assume that ((+1)) - (1 - a) and Fe41(ge+1)
a1) have the same sign,
and Frt is L-Lipschitz continuous. With suitably small X141 > 0, we have:
E [61- (1 9 < TE (f1- (8 (+t1) St),
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Le. the expected quantile loss of qt+1 is smaller than that of 9++1 timestep t 1. This indicates that tolerate certain deviation of Fe+1 from Fr+1- It suffices that their relative ordering with respect to - a coincides at certain points and our method is preferable in this regard. We also add a small modification of eq. ( in Appendix A.3 that avoids the "same-sign" assumption. Moreover, even if the estimated distribution function F(4 not accurate, we can still obtain the basic long-term coverage guarantee, as shown in Proposition 2
Formally, we propose our method Conformal Optimistic Prediction (COP) in Algorithm I The naming reflects the "optimistic" belief that the estimated CDF Fi+1 reflects the behavior of the true CDF F.+1- We will further clarify the close connection between our method and online optimistic gradient descent (OOGD) the next section.
Algorithm 1 Conformal Optimistic Prediction with estimated CDF
Require: nominal miscoverage rate a (0, 1), base predictor f. learning rate " 0, 1, 2, T. scale factor A++1 2 0, init. IR
1: for 2 1 do
2: Observe input X, X
3: Return prediction set (x,,, (fa(X.)
yt, f((X,) gel
4: Observe true label Y, E y and compute true radius 8; = inf E R Y, ((),,)
5:
Update the primary radius
G = 4. 17 (1 > qt) a]
6: Compute estimated cumulative distribution function Fi+1 for 8141
7: Update the refined radius
4++1 Gc+l -et1 (Peti( - (1 -a))
8: end for
3.2
CONNECTION WITH OOGD
To begin, we note that eq.
44) and eq. can be rewritten in the way of OOGD by:
@+1 min min n (V ( -() + 2 a/ll
(6)
+ _ 12110- qr+ 1
4+1 arg min
n (M++1,
In the literature of OOGD, M4+1 is the optimistic term, aiming at guessing the next move and incorporating into the objective (Rakhlin & Sridharan (013). While classical OOGD often takes Mc-1 as the previous gradient, COP uses a iittibution-inffomen hint:
M4+1 (Fe+1 (1 a:) Aet1/1,
(8)
which captures potential distribution shift of the next score. Empirically, we take \411/n S 1 and denote it as the scale factor, indicating our confidence the accuracy of Fr+1- For instance, when the base model performs well, the scores 8, tend to be highly stochastic and difficult to predict, suggesting a smaller A4-1- Conversely, when Fr reliable, the scale factor can be set close to 1. Specifically, COP boils down to OGD if simply set A = 0.
Besides the coverage guarantee Section 3.3, dynamic regret often considered as an additional performance metric for online CP (Bhatnagar al. 2023) Ramalingam et al.) 2025). From the view of OOGD, we obtain joint regret-coverage bound with constant learning rate in Theorem The result for dynamic learning rates can be seen in Appendix B.2
B.2

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Theorem 1. Let (1-a(st 9). For arbitrary {ue uo 0, we have:
711 - 2a)
-1=1 eTTE
[c()))) -
- [uuu)]
aa errz Mell2
T
regret
coverage
T
(1le
del12 - | - -
21
enviruments
The term Et.1 tml 2n (1/ug - 4: || 1 1xl reflects the non-stationarity of environments (Zhao et al. 2024). Although without distributional assumption, regret bounds and coverage bounds of online CP do not imply each other (Angelopoulos
al., (2025). Theorem [ shows that properly choosing M, can simultancously reduce the bound for both regret and coverage. Specifically, note that B||a- err, - M, 1 S Elaa 1 (st > qe) M, St-1, a natural idea for sequentially
choosing Mj would be taking it close to:
((( (1 - a) arg min E a 1s; > a.)
M, 1
M,
This coincides with the choice of our method with the scale factor taken as 1.
3.3 COVERAGE GUARANTEES
In this section, we present coverage guarantees of COP. For simplicity, we consider the refinement step eq. (5) in the way of OOGD:
g++1 = ur+1 - M/+1
where M is the optimistic term, instead of the choice in Equation & All of the following results
require M4+1 to be bounded, and this holds in our case since the scale factor de+1/1h mi X and
Me [a - a. in Equation (8).
The detailed proofs can be found in Appendix B We first show the long term coverage of COP with fixed learning rate 1).
Proposition 2. Assume that for each t, St [0, B] and M, E I, then for all T 2 prediction sets generated by COP with fixed learning rate 17 satisfies
1, the
Σ emr al
B - (2+ 6M)n
(10)
T
Ta
Although we assume boundedness of scores 8 and optimistic terms M,, running COP and achieving eq. (10) does not require knowing the upper bound B and M. The knite-sample result in Proposition naturally implies the long-
long-term term coverage eq. {1 i.c. limy-+00 2 etf,
Besides, we further develop coverage guarantee with dynamic learning rate тІ: Theorem 2. Assume that for each t, E [0, B] and M, E [-M, M), then for all T 2 1, the prediction sets generated by COP with arbitrary learning rate ne satisfies
Σ err a
B (2+6 6M))
ArrlI
where A 1/1 1 A, = 1/- n. for all 2 1, and |ArTll Σt= A, maxigrsT The
Note that, || 1:7|) is closely related to the number of times we increase the step size. Specifically, ||A |I 2N,/ (min Th.,, where Ny = -Er 1 > Tt... Hence, the upper bound in Theorem 2 will converge zero as long Ny / 7t) oT)). ) In particular, if we consider de- caying 7) in (Angelopoulos et al. 2024h, then 22 = 7/, || = 1//r, Aw:/h//T & NT/TTr- Setting 7/r o(Nr/T) ensures the long-te coverage in eq.
In the following part, we derive the asymptotic coverage property of COP under appropriately chosen learning rates. For this, we require that the scores are i.i.d..

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Theorem 3. Assume that the scores 8t ud
P and has a continuous distribution function F. The
learning rates {me satisfy:
20. Let q" be the (1 (1-o) athth quantile of P. satisfying: for q q", F()) 1 and for 9 < q" F(g)
00,
1/1
1=1
1=1
1 - a, Then q: - q", i.e. lime-+00 P(Y, E Cx().)) = a
Choosing constant learning rates in eq. (3) will cause oscillations (Gibss & Candès 2024 Theorem demonstrates that, under the i.i.d. assmptions and choosing learning rates satistying the Robbins- Monro condition (Robbins & Monro 195T), q: exhibits convergence behavior, even if add bounded optimistic terms M, in every update.
4 EXPERIMENTS
4.1 SETUP
We evaluate five simulation datasets under changepoints, distribution drift (Barber et al. 2023), variance changepoint, heavy-tailed noise, and extreme distribution drift. Besides, we evaluate four real-world datasets: Amazon stock, Google stock (Nguyen, 2018), electricity demand (Harries et al. 1999) and temperature in Delhi (Vrao. 2017). In the subsequent sections, we will provide detailed introduction to each these datasets.
Datasets.
Base predictors. We evaluate three base predictors that have distinct levels of forecasting perfor- mance. The Prophet model, Bayesian additive model, forecasts Y, g + stt) h((t) + et. capturing the overall trend, seasonality, holidays, and noise. The AR model forecasts Y, 61 Y +.. Yr + Ez, with p = 3. The Theta model decomposes the series by adjusting the curvature with coefficients 0 0 (long-term trend) and 0 2 (short-term dynamics).
Baselines. We compare with seven state-of-the-art methods: ACI (Gibbs & Candès 2021), OGD, SF- OGD BBhatnagar et al., all]]2022 2023), decay-OGD Angelppopoooos al. (2024). Conformal PID (Angelopoulos et al. 2023b). ECI (Wu et al. 2025) and (iT(e))e) (Areces et 2025). Several follow-up works
on ACI consider adaptively setting learning rates via the expert aggregation technique (Gibbs Candès 2024 |hhanngarr et al. 2023). COP and the seven baselines are orthogonal to these works and can be naturally incorporated by serving as a single expert.
General implements. We choose the target coverage 1 - 90% and construct asymmetric prediction sets using two-side quantile scores under a// respectively. For prediction sets, baselines will output asymmetric sets ( - o/. Ye + q3) with upper score g. and lower score of under half of the coverage level a respectively.
Hyperparameters. The proposed COP has three hyperparameters, the base learning rate scale factor À 0.5 and the window length rates ? = (max(*:-w+1>>> implementations leverage the open-source Python codes from Wu al. (2025) Areces et al. (2025). The byperparameters of LQT need to be tuned via grid search. The operational ranges of 1) and more
100. Same previous works, the adaptive learning } - min($t-w+11--.. 8, }). For reproducibility, all baseline
details about 7)t for each methods can be found in Appendix [
Choices of estimated CDF. By default, we set the estimated CDF as empirical CDF, which is
Fe+1((++)) = Ei=-w41 (ss < ). ). In addition, we conduct experiments with estimated CDF based on the kernel density estimator Appendix
Evaluation metrics. The coverage rate measures the proportion of time steps where the true observation Y, falls within the prediction set C, qt). The width of the prediction set reflect the efficiency of CP, including the average width (reflecting overall performance) and the median width (robust to outliers and extreme intervals). A well-calibrated method should achieve coverage close to the predefined target level, while keeping the width as short as possible. We also evaluate recovery time in Appendix F] statistical significance Appendix and average time cost in Appendix [] For visualization, also provide the figures in Appendix K
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Overview of experimental results. We have conducted extensive experiments, including five simulation datasets in Section 4.2 and Appendix [ four real-world datasets in Section 4.3 We
have also conducted some ablation studies, including the choice of estimated CDF in Appendix C and the scale factor in Appendix H To evaluate the case that estimated CDF is inaccurate, we
have conducted the inaccurate estimated CDF in Appendix G Our code is available https
7/github. com/ ctt.ub..coccceatoorrca creator xi/Conformal--oe
tareFormal-ppp Cmpppiimisiiioo Prediction
4.2 SIMULATION DATASET
We evaluated our method on two simulation datasets under changepoints and distribution drift setting, respectively, Both datasets {X,, Y. are generated according to linear model Y, X 3, ct. X₂ ~ (0, I4), & ~ (00, 1), n = 2000.
Changepoint setting: we set two changepoints: Br - (2, for 1,... 500: 3: = g() = (0, 22 -1) for = .0., 15000; and B; = =g()) = =0,,0, 2,0,2,1) for
t == 1501, 2000.
Distribution drift setting: we set 31 (2,1, B, = (0,0,2,1) and use linear interpolation compute 3r B1 + n-1 (Bm B1).
Table 1: The experimental results in the two simulation datasets with nominal level = 10%.
Prophet
AR
Theta
Dataset
Method
Coverage
Average
Median
Coverage
Average width
Median
Coverage
Average
Median
(%)
width
width
(%)
width
(%)
width
width
ACT
89.9
DC
8.20
89.9
8.20
89.9
8.43
OGD
90.0
89.9
8.39
8.40
89.9
8.73
8.70
SF-OGD
90.0
12.48
11.56
90.0
12.58
11.69
89.9
12.70
11.88
decay-OGD
90.0
8.30
8.22
90.0
8.26
8.21
90.0
8.57
8.60
Changepoint
PID
89.7
11.02
9.64
89.9
10.83
9.35
89.7
11.23
9.78
ECI
89.9
8.16
8.25
89.9
8.17
8.26
89.8
8.55
8.68
LQT
89.8
8.49
8.31
89.6
8.54
8.29
89.8
9.29
8.75
COP
89.8
8.29
8.44
89.7
8.18
8.25
89.8
8.45
8.53
ACI
89.9
6.69
89.8
6.56
89.9
6.79
OGD
90.3
7.24
7.05
90.2
7.15
7.10
90.3
7.34
7.25
SF-OGD
90.0
11.48
10.34
89.9
11.46
10.31
89.9
11.95
10.54
Distribution
decay-OGD
90.6
7.64
6.95
90.4
7.31
6.81
90.6
7.62
6.91
Drift
PID
89.7
9.41
7.81
89.8
10.08
7.92
89.7
10.06
7.90
7.06
6.98
90.2
7.55
7.18
LQT
90.6
9.74
8.72
91.9
8.16
7.22
91.0
10.48
8.86
COP
90.6
7.07
6.89
90.0
7.09
6.97
90.9
7.30
6.99
The quantitative results are shown T ACI frequently produces infinitely wide prediction sets due to updating Otg, The overly conservative sets undermine the utility of prediction. OGD and SF-OGD partially balance coverage and width, but their performance is overly sensitive to learning rates. In contrast, decay-OGD performs better in terms of median width due the stability of decaying learning rate in the later stages. Conformal PID borrows from PID control for adjustment and needs to train scorecasters. LQT is sensitive to hyperparameters and relies on grid search. which makes its performance unstable. ECI reacts quickly to distribution shifts through aroor-uantiffictton, and hence achieves relatively tight sets. However, ECI may struggle complex data environments since it cannot capture the underlying information of data.
As for COP, it maintains the coverage rate close to the nominal 90% level and tighter widths than other methods. COP achieves this by incorporating optimistic term based on the estimated cumulative distribution function, which preserves the long-term coverage guarantee of traditional online CP while using predictable distribution information to adjust the width more precisely. The experimental results also show the generality to adapt to different base predictors across Prophet, AR, and Theta models.
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4.3 REAL-WORLD DATASETS
Moreover, we evaluated our method on four real-world time series datasets across three critical domains. All datasets retain temporal ordering to preserve real-world sequential dependencies.
Financial markets: daily opening prices of Amazon and Google from 2006 to 2014, capturing non-stationary trends, regime shifts like the 2008 crisis, and heteroskedastic volatility. The base predictors will forecast the daily opening price on a log scale.
Energy systems: New South Wales electricity demand for half an hour from 1996 to 1998 normalized to [0, 1), featuring multiscale periodicity and demand surges).
Climate science: daily Delhi temperatures from 2003 to 2017 reflecting seasonal cycles, long-term warming trends, and extreme weather anomalies.
Table 2: The experimental results in the four real-world datasets with nominal level a 10%.
Prophet
AR
Theta
Dataset
Method
Coverage
Average width
Median
Coverage
Average width
Median width
Coverage
Average width
Median
(6)
width
( (6)
(6)
width
AC1
90.2
00
46.97
00
13.77
89.7
00
12.31
OGD
89.6
55.15
30.00
89.9
19.10
15.00
89.8
18.07
14.50
SF-OGD
89.5
61,47
31.75
89.9
24.44
21.05
90.0
23.88
21.14
Amazon
decay-OGD
89.9
97.22
36.20
89.7
20.23
14.01
89.2
17.49
13.46
Stock
PID
89.8
52.56
39.09
89.6
59.22
37.93
89.5
61.19
40.20
ECI
90.1
47.00
34.84
89.5
17.12
12.73
89.7
17.46
12.49
LQT
89.3
31.42
15.26
21.61
18.92
27.17
15.13
COP
89.6
39.86
27.91
89.5
17.09
12.90
89.6
17.21
12.23
ACI
90.0
66.83
89.8
18.64
90.5
32.78
OGD
89.7
57.60
46.00
90.7
33.76
23.00
89.9
31.49
29.50
SF-OGD
89.6
58.92
47.78
89.9
28.31
24.42
90
34.04
31.48
Google
decay-OGD
89.9
77.23
50.18
90.2
46.53
26.77
90.2
55.32
33.71
Stock
PID
$7.47
63.58
$4.05
ECI
89.9
56.06
46.96
89.7
19.95
17.19
89.6
30.92
29.53
LQT
89.9
57.31
47.00
90.5
41.80
25.00
89.6
41.70
37.58
COP
89.7
49.72
42.09
89.6
19.87
17.04
89.3
30.25
28.24
ACI
90.1
DC
0.443
90.1
0.105
90.2
0C
0.055
OGD
89.8
0.433
0.435
90.0
0.133
0.115
90.1
0.081
0.075
SF-OGD
89.9
0.419
0,426
90.0
0.129
90.3
0.106
Electricity
decay-OGD
90.1
0.531
0.521
0.122
0.099
0.100
0.059
Demand
PTD
90.1
0.207
0.177
90.0
0.434
0.432
89.9
0.413
0.411
ECI
90.0
0.384
0.382
90.0
0.117
0.098
89.9
0.071
0.055
LQT
90.1
0.221
0.218
90.1
0.144
0.138
90.0
0.113
0.111
COP
90.1
0.385
0.376
90.0
0.117
0.098
89.8
0.069
0.052
ACI
91.0
(C
8.49
90.0
00
6.06
90.2
00
6.48
OGD
90.4
7.54
7.60
6.10
6.36
6.30
SF-OGD
90.0
7.17
7.08
90.1
6.37
5.91
90.1
6.75
6.43
decay-OGD
90.1
8.84
8.35
90.0
6.36
5.67
89.9
6.56
6.18
Temperature
PID
90.1
7.65
7.65
89.7
8.92
8.86
89.7
8.77
8.79
ECI
90.0
7.20
7.22
90.1
6.39
6.10
90.0
6.41
6.27
LQT
90.2
8.57
7.30
6.78
90.1
7.51
7.08
COP
90.1
7.05
7.07
89.9
5.85
5.58
6.27
6.18
The experimental results under the real-worke dataset demonstrate the performance of each method in real-world scenarios. As can be seen from the data in Table COP shows superior performance. ACI generally maintains the coverage rate at nominal level, but its prediction intervals often infinitely

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Published as a conference paper at ICLR 2026
wide, which makes the results lack practical application value. The OGD series of methods achieve more balanced trade-off between coverage and interval width, but they are highly sensitive to learning rate selection. Conformal PID trains scorecasters compensate for the base predictor's accuracy and often improve when the base predictor is less accurate. ECI produces tight prediction sets through error quantification while maintain the coverage rate. Overall, these results highlight the advantages and limitations of each method in real-world applications.
5 CONCLUSIONS
In this work, we introduce Conformal Optimistic Prediction (COP), a novel online CP algorithm that leverages estimated cumulative distribution functions of non-conformity scores. Viewing COP through optimistic online gradient descent enables a comprehensive theory, including joint re- gret-coverage bound that clarifies its motivation. Theoretically, we also prove distribution-free. finite-sample coverage for general optimistic updates under arbitrary learning rates, and show asymp- totic consistency with i.i.d. scores under suitable rates. Experiments on synthetic distribution shift and real time-series data from finance, energy, and climate demonstrate that COP attains target coverage while producing tighter prediction sets than state-of-the-art alternatives.
ACKNOWLEDGEMEN
We would like to thank all anonymous reviewers and the area chair for their helpful comments. Shu-Tao Xia was supported in part by the National Natural Science Foundation of China under Grant 6257 1298. Changliang Zou was supported by the National Key R&D Program of China (Grant Nos. 3022YFA1003705 2022FA11038000, and the National Natural Science Foundation of China (Grant No. 12231011).
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A SOME DISCUSSIONS
A.1 DISCUSSIONS ON ONLINE MIRROR DESCENT
We remark that if can obtain the estimated distribution function F; in online fashion, direct way to combine it with eq. is to leverage online mirror descent (Orabona 2019 Wang et al. 2025
V 7 Vibe(ge) + me(er; a), where err, (s J (g) Es, [41-a (8: - S)( + on²² /2, and o > 0 is a tuning
(11)
parameter. Equation (11) equivalent to:
Fe41(0(+1) + 04++1 Fi(g() + 0q: + ne(erT a)).
(12)
Further, if 8&+1 has a probability density function (PDF) that bounded away from zero almost surely, directly use:
(441(1-1)+ = Fe((:) + ne(err, o.)
(13)
suffices to use F to substitute F above. However, both eq. 12 and eq. do not have closed
form solutions for Q4+1, which makes the update difficult to compute.
A.2 DISCUSSIONS ACI
Adaptive Conformal Inference (ACI) (Gibbs & Candès 2021) is another widely used online CP update besides eq. Let (-) be the fitted quantiles of the non-conformity scores in calibration
set D Deal:
@(p) = inf s:
Σ
((..(X..)))5)
2P
(Deal
(X
For prediction set (a()) = (v (xt,,) S Q1 - a)), define:
= = 8: : Yi (,(3)).
Denote err: = (6 > Be), then ACI follows the iteration:
u++1 = 1 - d; + n)rr, - a)
= 1 - âi V - (1 âz)).
Note that 1 - Se can be viewed as in eq.
3 the similar refinement is as follows:
a4+1 = aet1 det1 (Feti(1 - -0e11) - -a))
i..
a(+1) 0++1 = but1 + det1 ((-1)())a+))(111)))) (11-a))
where Fr+1 is the estimated CDF (1 - 1. However, ar < <0 a > can happen frequently for some 0 and may output infinite null prediction sets. Hence, we do not adopt this kind of update.
A.3 DISCUSSIONS ON THE "SAME-SIGN" ASSUMPTION
In order to clarify the superiority of the refinement step, introduce a "same-sign" assumption in Proposition I which is unverifiable in real-world data. However, note that
(F+1(((+1)) (1-a)] (Fex (1 a))
follows from
(841((011) - (1 - a)) 2 sup (Ft1 P.t1(g)) A 4111
13

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Hence, an intuitive way to avoid the unverifiable assumption is to replace eq. with:
g++1 0++1 A 1
(1- (1-a) ec-1) (Fi+ ( - (1 (1-a)).
(14)
For deployability. c++1 can be viewed as a hyperparameter that depends on the temporal properties of data and the accuracy of Fe+1. Following the same proof of Proposition [
we have
( gt+1) S E [e1-a ($t41 l++1) S.], as long as Fi+1 is L-Lipschitz and A > 0 small.
B PROOFS
B.1 PROOF OF PROPOSITION 1
Proposition 1. Assume that ( - (1 - a) and Fo+1(4e+1) - -(1 ax) have the same sign,
and F(+1 is L-Lipschitz. With a suitably small > have:
(51-a - ((+1 S E [61-a Stl.
Proof. Let L(g) 1- 4) Note that the T-LLipchhizness of Fi+1 implies that VL((2)
(8
L-Lipschitz continuous. Hence:
[(q++1) L((e+1)= s VL(l++1((44) - ((t1) + |1 90+1112
- ( (1 a1 V (g( + T.22-11 (1+2(((+1) - (1 -a)]
2
-d++1 ((+1((+11)- (1 (1-) F(1+41) - -1 a)
L71+1
Fe-1(0u+1) - -1 2
To satisfy E [[(1-a(8e41 - - < (E(e-o (0+1) [ it suffices that Ae+1 satisfies:
(1-a))
A+11 L L(Fi+1(014) - - a))
which can be achieved since (e+1(0 (1 aa) and Fe+1(1e+1) - -(1 - a.) have the same sign.
Denote Ex+11 sup, |F-1 - F(+1 = |f+1100+11 - (1 a) & 1 a, then
L
C(g(t1) - [(q(+1) I V ac+1) + 119041 -i++1
L
< (6t + +E(+1) + 1e+108+11
_
2
< <)41(1 (1+ 2
Hence, even if the same-sign sign assumption in Proposition I does not hold, the "instantancous" perfor- mance of the refinement step bounded by A1+1 and e++1
B.2 REGRET GUARANTEE WITH CONSTANT LEARNING RATES
Theorem 1. Let (g) li-a (s. 4). For arbitrary (u. uo 0, we have:
(((1)) - [e(we)]+ 111 - 2a)
- ert,
C
<
7
T
|a err: - -M
4
T
t=1
regret
coverage
T
(1lux drl13 - | - (.l1
20
environments
14

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Proof. We begin by presenting the Bregman Proximal Inequality (Chen & Teboulle 1993): 1
Lemma 1. (Bregman Proximal Inequality): Let be a convex set in a Banach space, and f X -R be closed proper convex functtin. function.. Given convex regularizer vp X ++ I. denote its Bregman divergence by D. Then, qt under the update:
arg min f(g) Do((. gEX
satisfies for any E R.
- - f(u) S (u gc-1) - Dw qt) - Dvlai,
In our case, we take & be (x)) /2, then Des (z, y) - yy1//2 By convexity of l, we upper bound the instantaneous dynamic regret to sum of three terms:
(() - ((() & (Vlr(g)) q: uz) ((e(()) Mr q: q(11) (M,, 91 9++1
Vee(ee) 9+1 we)
(a)
(b)
(c)
For term (a), note that
qt 90+1 = q: -mM, qt+1 17(a- erTi - -M),
hence
(a) = (V((4))) - Mr. 90 - 9e+1) = n/1aa - err Mel1.
For term (b):
(b) (M,, q: 90+1)
2/2 (llerr; al - ||a - err, - M,l1 - M2)
2 ller: a12 a err: - Mr + Mrl
"
Ilerr: all
4
For term (c), using Lemmal 1 the update gc+1 arg min, {1 (g)) 9) + +11l4 [el1?] implies:
1
(c) (Vle(g)) un) < (1luz
Ge ||u 4 112
119: qu+1 12)
21
1
(1/4== - àell2 -Ilu - $ 112)
? lerrz - 0:11
2m
2
Combine the three upper bounds:
17 lerri - al12+ 1
(g(g)) - ((4) S n1aa- a - err; M
(1]u - 9ul12 - -1/4 - -c++1l2)
2n
< 1/|aa err: Melle 7 (1 20) - a)
1 1(a a²)
4
1 2m
(1lue -cll? - 1lue (+1112)
Summing over from 1 to T, we have:
T
(1 2--a()errre--ae - a)+ nT T
[(((
E((ue)] + " I
Σ
aa - a²)
4 1=1
4 t=1
T
T
<
||0 eTT, - -M +
(1/ug - ne - ||u, c 11)
2
[=1
t=1
T
||a erT: -
+
(llu
Ge 11 ||u Grll2) (uo 0)
21
[=1
t=1
Dividing both sides by T and ignoring the constant
2 (a a²) completes the proof.
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B.3 REGRET GUARANTEE WITH ARBITRARY LEARNING RATES
To ensure that the regret guarantees with arbitrary learning rates match the form of Theorem 1 and presented more transparently, we consider the original OOGD:
ar+1 q: + (err; aa)
a++1 h++1 M++1M++1.
Note that in our algorithm, the learning rate used for the second update is instead of 11++1- For arbitray learning rate (11,, we have:
E erri
((( [e(u)] + Cr
(1 2a)
+
a²
T
T
1
Mi ||aaa err:
Mell2
(1
âell3 1
gc+ 1
2n
where CT is the harmonic mean of the learning rates, i.c.
T
OT
El= mJ
T
Proof. Similar to the proof of Theorem
!
we obtain:
l((g)) (u)) - [li(g)) 9:
ut)
= [le(()) - Me, at - @ct1) + (Me, dt - @+11 + (Vee(t)), 4++1 - we).
(8)
(b)
(0)
and:
(a) = (Vee())) Mr, 9: (u+1) = 17/ err: - Mell2-
(b) v ? errt - a:12-
4
(c) S 2m
1 (11ue - dell2 - 1 ac+1 12)
1)t 2
|erri - all3
Combine the three upper bounds:
((g)) ((( S The ||a err, Mell3 11 leerr aa13
1 (lue dell2 - 1| (0+1112)
2111
Summing over f. from 1 to T, we have:
T Σ
T Σ
Σ [((()) - (x(4))) =
Σ
aa - -err, - -M 1
ne(err; -a)
(11u4 - 2. 1 - -
217
t=1
t=1
ΣE
lerr: T
a/2
$
ne |||aa erri -M TCr
T
(11uz - gell3 - 1|14 - ge+1 l12)
(Cauchy-Schwarl incquality)
2m
ΣT erri
2
Mr T/ err; MMll3 TCT (1 (1-20 2a)
a
+ a
a²
(1lue dell3 || - 2t+1 12)
277
which completes the proof.
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B.4 PROOFS OF COVERAGE GUARANTEES
only prove the general result of Theorem 2 We first prove the lemma below that shows the boundedness of qe. Lemma is essential in the proof of Theorem 2
Proposition is simply special case of Theorem 2
Lemma 2. Fix an initial threshold q1 E [0, BJ. Then COP in equation 1 with arbitrary nonnegative
learning rate Th satisfies that
-2, 1) 41 B + 9,2M 11) 1121,
where G2g 0, and Q4 maxigrst fr for 1.
Proof. We first prove the upper bound. Combine eq. and eq. ( we get:
f --11 + 1 a.) + Tp- M₁ T/ M++1,
(15)
where M2 is the optimistic term defined in eq. (8) For any t, if q: < we have at < B < B + (2M + 1). If qt > St, denote as the largest integer below satisfying qi S 8t, then dr > 8y for I < r S t. Hence,
gr 9r-1 11- + Th- 1M, M,H1, lersh.
Through iteration obtain:
q: = q
[err + (Mr-1Mr 17, Mp+1)
[-1
qi + 1
n + (m-1Ms - M++1)
S 81 + TM + (M Mi - 1/:-+1 Mc+1)
B + (2M + 1).
For the lower bound, if g: > sr, we have q: 0 (2, gf S st, denote I as the largest integer below satisfying qi > st, then 4r S 8r, for < S Hence,
q: Ω +
(ern a)n + 1 - Mr+1))
q m Σ
7r a) + (m-1M) M++1)
r=l
81 1 + (M Me+1)
2 - U2 (2M + 1).
Theorem 2. Assume that for each t, s; [0. B] and M, e [-M, then for all T > 1, the prediction sets generated by Algorithm I satisfies
B (2+ |
emr
<
T
T
where ||1:T||
1
Σ
11p 1 Ωr maxigrsT Tr-
17

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Proof.
T
emr; - a)
Σ
Te (err; - a)
T
T
T
T
N1 (eTT: -a)
T
Dr (qT+1 - -r mr ny Mo+1)
T
1
Σ A - 9r) -9t)
+ Σ (n1 Mr+1 Mr+1)
T
B + (2+ 4M 2
2M | Arrlll
| 11
T
B (2 6M)
|
T
Specifically, if 1/2 = 11, we have
B + (2 + 6M)
err, a)
5
T
Tq
iid Theorem 3. Assume the scores ~ P and has continuous distribution function F. The learning rates {nd} satisfy:
20 T4 = 00; Σ 1/2
Σ
x.
Let q" be the (1 - a-) quantile of P. satisfying: for q-, F(g)) > 1 and for < F(()
1 - a, Then d: - q", i.e. lim4-100 P(Y, E Cx(X.)) -1-
Proof. Denote random variable e, err, Eerr: erri 1 + F(g:), E 1/ic, Ar
E TaC, Fi oSs, ... st). We first prove A, 8.a.
Note that E( Fi) and for j B(G,e) E (&GC E (EE 0.
For each 2 1, B( = St + F1
S.. Hence, {S,h21 is a martingale with
respect the filtration F,. Further,
S < E|S
Eep
111 oc.
i=1
Applying Doob's first martingale convergence theorem, obtain that {Sih21 converges almost surely. Therefore, A, a.s.
Next, let g(x)= 1 - a - FF(x), + M, + A.. We have:
Pt+1 Pt 91+1 qi + M 7- + Ac-1 Ar
(err,, - a) ne(er,, - F(q:))
(by 15
=T((1 - F(q:))
=mg(pp- - A, Th
By Lemma 2 convergent subsequence (Pu.). ). We prove lim Pu. by contradiction.
9: is bounded, hence {p.} is bounded, a.s.. By Bolzano-Weierstruse Weierstrass theorem, it has
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Suppose lim Pu; a.a. 0, there exists to s.t. Vt to. |A, - 6, the ò, Pu. q" > Since A, + 1/-11 26. For t > to.
9¹ 35, 6 > O. The case of lime-+o Pu. 0 follows by the same argument.
Puc-1 Pu. TJ, T,, -19Pa Au U/, - 22 2 Pm. 1/4-11 1 > 9- +
hence
Pus Pa--1 Thu 711--190 (igp,, - Thu n
S Puc-1 + 1lu--1((2) PP 6) S Pui-1 + Tu 19((9))) = Puu
Le. Pu--1 2 Pur By induction we can prove that for any t > to Per-i 2 Put-1+1 2 ... Pu. q² + 26.
Next, let (ip) = --- then for any Mtg we have:
((P+1)) - (p.)) (p: + me(p: A. 7i- T/- M,) 9-¹² (p. q²)²
(Pr - a (p) - A: - -1t-1 + th lp(p - AA - Mo)]
I 2m 25g(1' + +6) + n
= -4 +6) - FF)"")) + T/.
(1)
By assumption, F(q" + 8) F(4²) > 0. As Ei=u, 1)t oo and Σi=ur "Ie co, summing equation | from ut, to co gives that V - -c, contradicting the convergence of {Pu, h2i
Finally, we conclude that any convergent subsequence of {pr}121 converges q² Therefore. lom Pt q² Using A, - 0 and 7/ Mr + 0,t + completes the proof.
C IMPLEMENT OF COP WITH BERNEL-BASED CDF
In this section, we compare the performance of using ECDF and Gaussian kernel density estimator (KDE) for estimating the CDF in the COP framework Fi+1. The KDE is implemented using a sliding window approach to incorporate recent data points and adapt to distribution shifts. We set the scale factor = 0.5, the length window w = 100, and the bandwidth h of Silverman's rule:
h = 0.9 x a 0.2
where a is the minimum of the standard deviation of the window data and the interquartile range (IQR) divided by 1.34. Then the CDF value at is estimated using the Gaussian kernel:
Be+1(1e4))
9:+1
Si
h
i=f-w+1
where the cumulative distribution function of the standard normal distribution, s; is the non- conformity score at timestep t.
We evaluate the performance of the choices of estimated CDF, including empirical CDF (denoted ECDF) and dernel-based CDF (denoted Kernel), across six datasets: Changepoint, Distribution Drift, Amazon Stock, Google Stock, Electricity Demand, and Temperature. The results are presented in Table We can see that even with simple ECDF, COP can achieve tight prediction sets and maintain coverage rates close to the nominal level. The kernel-based CDF shows comparable performance in some cases, but may require further tuning of parameters such as bandwidth for optimal results.
D LEARNING RATE SELECTION IN THE EXPERIMENTS
In our experiments, the default setting is to select four learning rates on all datasets and then pick the one that yields the best performance. Considering the high sensitivity of LQT and <GD-based methods to the learning rate, we carefully select more than four candidate learning rates for these methods across various datasets. The following is the list of all learning rates used, which
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Table 3: The experimental results on the choices of estimated CDF with nominal level a 10%.
Prophet
AR
Theta
Dataset
Method
Coverage
Average
Median
Coverage
Average
Median
Coverage
Average
Median
(6)
width
width
( (6)
width
width
(6)
width
width
ECDF
89.8
8.29
8.44
89.7
8.18
8.25
89.8
8.45
8.53
Changepoint
Kernel
89.8
8.29
8.45
89.7
8.11
8.21
89.8
8,45
8,53
Distribution
ECDF
90.6
7.07
6.89
90.0
7.09
6.97
90.9
7.30
6.99
Drift
Kerel
90.6
7.07
6.89
90.0
7.10
6.98
90.9
7.30
6.99
Amazon
ECDF
89.6
39.86
27.91
89.5
17.09
12.90
89.6
17.21
12.23
Stock
Keree
89.6
40.32
27.98
89.4
17.23
12.95
89.6
17.35
12.56
Google
ECDF
89.7
49.72
42.09
89.6
19.87
17.04
89.3
30.25
28.24
Stock
Kernel
89.7
50.85
42.33
89.6
19.92
16.94
89.3
30.36
28.40
Electricity
ECDF
90.1
0.385
0.376
90.0
0.117
0.098
89.8
0.069
0.052
Demand
Kernel
90.1
0.384
0.377
90.0
0.118
0.099
89.8
0.069
0.052
ECDF
90.1
7.05
7.07
5.85
5.58
90.0
6.27
6.18
Temperature
Kernel
90.1
7.06
7.08
89.9
5.85
5.59
90.0
6.25
6.17
determined through this systematic selection process to ensure fair and effective comparison.
ACI 1) {0.1,0, 05, 0.01, 0.005):
OGD : (10,5, 1,0.5, 0.1, 0.05, 0.01, 0.005),
SF-OGD : = {1000, 500, 100, 50, 10, 1,0 0.1,0. 100,
decay-OGID : (2000, 1000, 100, 20,
1,00,,000
Conformal PID : = (1,0.5,0.1,0.05,
ECI :7 (1,0,,,0.1,0005,
LQT 1 = (10.55.1..55.0110000,,,,0 0.1,0,00.,...00001,,
COP: : (1.0.5,0.1,0.05,
For ACI and OGD, they do not use adaptive learning rates. For SF-OGD:
(q))
(e)0
where (q)) quantile loss and q is the predicted radius at time t. For decay-OGD:
n
where the hyperparameter 0.1 follows Angelopoulsss al. (2024). For conformal PID, ECI and COP:
T) = 7) (max ( (1 ". 8t) min -t- ................................ 8{))) where s; is the non-conformity score at time and the window length w = 100 follows Angelopoulos al. (2023b).
E MORE EXPERIMENTAL RESULTS IN SIMULATION DATASETS
We also evaluated our method in three other simulation datasets under variance changepoint, heavy- tailed noise, and extreme distribution drift setting, respectively. Both datasets (X,, Y;); are generated according linear model Y, X Be + X, N'O0, T4), NN(0, n 2000.
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Variance changepoint setting: set fixed 3 (2,1, and two variance change- points: o, 1 for = I,..., 500; de = for 551....., 1500; and or = 0.5 for .. 2000.
Heteroskedastic and heavy-tailed noise setting: we set fixed 3 = -2,1, -0.5) and the noise ε, is (2),, with standard deviation + 2 3| [E[ [ 3)
1, TT_ 3,, = 1,, 10) and use linear interpolation compute Br - 31 A (3n
Extreme distribution drift setting: we set 31 = (20. 10
Table 4: The experimental results in the three other simulation datasets with nominal level 10%. Note that, since PID, ECI and LQT methods completely fail on the Extreme Drift dataset, we did not include them in the table.
Prophet
AR
Theta
Dataset
Method
Coverage
Average
Median width
Coverage
Average width
Median
Coverage
Average
Median
( (8)
width
(%)
width
(%8
width
width
ACI
91.0
11.06
91.0
10.74
91.0
10.90
OGD
90.1
10.57
10.65
90.0
10.43
10.25
90.0
10.44
10.07
SF-OGD
90.0
14.75
14.06
90.0
14.80
14.18
90.0
14.47
13.82
Variance
decay-OGD
90.2
10.71
11.21
89.9
10.41
11.00
90.2
10.76
11.12
Changepoint
PID
89.7
13.17
11.81
89.7
13.07
11.77
89.7
13.26
11.99
89.9
10.66
10.39
89.8
10.35
9.75
89.9
10.60
10.04
LQT
11.77
88.7
10.84
10.97
89.8
11.64
11.38
COP
89.9
10.78
10.79
89.8
10.37
9.87
89.9
10.66
10.10
ACI
90.3
10.03
90.2
OC
10.01
90.3
0C
9.82
OGD
90.0
9.94
9.95
90.0
9.91
9.95
90.2
10.18
10.25
ST-OGD
90.0
15.22
14.05
90.0
15.10
13.93
90.0
15.43
13.91
dccay-OGD
90.3
9.97
10.01
89.9
9.69
9.77
90.8
10.10
9.94
Hcavy-tailed
89.7
13.60
11.47
89.6
13.14
11.36
89.6
13.12
11.19
ECI
89.9
10.54
10.49
90.0
10.34
10.27
90.0
10.55
10.36
LQT
92.0
13.21
12.32
89.4
10.13
10.09
92.8
15.79
13.74
COP
89.8
9.56
9.79
89.9
10.38
10.27
90.4
9.87
10.03
ACI
90.4
oC
79.01
89.7
OC
61.85
90.3
0C
70.06
OGD
91.1
275.87
280.00
89.9
64.03
62.00
91.3
213.10
212.00
Extreme
SF-OGD
423.68
445.98
68.23
63.62
92.4
376.03
388.03
Drfit
decay-OGD
89.7
265.81
274.93
90.0
64.50
60.32
91.6
246.51
248.24
COP
91.1
275.82
279.84
89.9
64.10
62.18
91.3
213.54
211.91
The results shown in Table We observe that COP consistently maintains the coverage rate close to the nominal level of 90% in all scenarios while producing competitive interval widths. Notably, in the Variance Changepoint and Heavy-tailed settings, COP achieves average widths comparable to or tighter than OGD and ECI, without the instability seen in ACI (which yields infinite widths). In the Extreme Drift setting, traditional methods like PID, ECI, and LQT failed to construct valid prediction sets due to the severity of the shift, and are thus excluded from the table. In contrast, COP successfully adapts to the extreme drift, maintaining valid coverage similar to OGD but with slightly tighter intervals in the Theta base predictor case. These results further validate the robustness of COP in handling complex distributional shifts.
F POST-SHIFT COVERAGE RECOVERY TIME
To quantify the responsiveness of COP and competing baselines after an abrupt distribution shift, we define shit-shift recovery time based on the stability of short-horizon empirical coverage. Let to denote the changepoint index. For each method, we compute sliding-window
window coverage rate
1
cog, (t)
1ss < 9:
Wr +1
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where wP 20 is the coverage- window (independent of any internal calibration window). Recovery declared at the earliest time t, > te such that
1 w, I cug, (t) S 0.9 and l 10. This criterion requires the local empirical coverage
1/w,, for & consecutive indices E....... ty+--1
with nominal coverage 1 to remain around the target level for multiple consecutive checks, preventing spurious early detections due to stochastic noise.
Table 5: Post-shift coverage recovery time (in steps) in the Changepoint datasets.
Prophet
AR
Theta
Method
Recovery
Recovery
Recovery
Recovery
Recovery
Recovery
Time I
Time 2
Time
Time 2
Time
Time 2
ACI
35
73
50
0
35
66
OGD
12
40
0
0
SF-OGD
12
12
dccay-OGD
6
153
73
73
PID
40
12
0
40
0
ECT
12
8
40
0
40
0
LQT
156
73
60
73
60
73
COP
12
40
0
G ROBUSTNESS INACCURATE ESTIMATED CDF
To evaluate the robustness of COP in adversarial environments where the estimated CDF may be inaccurate, we performed an additional experiment in which the ECDF is corrupted by randomized noise. Specifically, at each time step, we construct noisy ECDF by
Fnoiey = + ()-
where is the ECDF, (O, denotes a uniform random noise, and {1, 0.9, 0.5, controls the level of the corruption. The case 7 1 corresponds to the ECDF, while 7 represents purely adversarial environment in which the estimated CDF contains no information about the data distribution. We apply this noise independently at each time step and use Pnoisy in place of F in the COP update rule.
Table 6: The experimental results under different noise level y in the two real-world datasets with nominal level 10%.
Prophet
AR
Theta Average width
Dataset
Coverage
Average width
Modian width
Coverage
Average
Modian
Coverage
Median
(6)
(6)
width
width
(6)
width
1.0
89.6
39.86
27.91
89.5
17.09
12.90
89.6
17.21
12.23
38.95
89.3
17.14
12.71
89.6
17.45
13.12
Amazon
0.5
89.7
42.24
29.56
89.4
17.24
13.08
89.6
17.20
12.76
Stock
0.1
90.1
46.93
30.89
89.2
17.45
12.44
89.6
17.60
12.73
0.0
90.0
62.65
53.09
89.7
22.07
18.26
89.5
30.76
28.56
1.0
89.7
49.72
42.09
89.6
19.87
17.04
89.3
30.25
28.24
0.9
89.8
49.99
42.06
89.6
19.81
16.92
89.4
30.43
27.79
Google
90.0
54.57
19.84
17.33
29.82
27.27
Stock
0.1
61.25
51.91
89.6
19.98
17.23
89.5
30.68
28.18
0.0
90.8
86.26
55.43
90.0
30.04
20.32
89.9
30.24
20.76
Table 6 shows the results for Amazon and Google stock datasets across different noisy levels y. The coverage rate remains stable across all values of 7. Even when the CDF is fully replaced by uniform
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noise (y = 0), the coverage rate varies by most 0.5%. In contrast, the average and median interval widths exhibit a relatively monotonic trend: as decreases and the CDF becomes less informative, COP widens its intervals adaptively for validity. This behavior demonstrates that COP can flexibly utilize the CDF information to righten intervals when the CDF accurate but to revert to conservative intervals when the CDF is adversarial.
Overall, these results indicate that COP retains near-nominal coverage even when the CDF is corrupted or completely uninformative, thereby confirming its robustness under adversarial noisy CDE.
H SENSITIVITY ANALYSIS OF THE SCALE FACTOR
In this section, we will analyze the sensitivity of the scale factor A. As defined in the update rule
q++1 - (F++1(0++1) (1 (1-a)).
this parameter governs the magnitude of the refinement step derived from the estimated CDE. We evaluated the performance in the Amazon Stock and Google Stock datasets for three distinct values: {0.1, 0.5, 1.0).
Table The experimental results under different scale factor the two real-world datasets with nominal level 10%.
Prophet
AR
Theta
Dataset
À
Coverage
Average
Median
Coverage
Average width
Median
Coverage
Average
Median
(6)
width
width
(6)
width
(%)
width
width
1.0
89.4
40.31
28.96
89.0
16.97
12.88
89.5
17.01
12.2
Amazon
0.5
89.6
39.86
27.91
89.5
17.09
12.9
89.6
17.21
12.23
Stock
0.1
89.7
40.98
29.06
89.3
16.90
12.62
89.6
17.26
12.66
1.0
89.7
52.20
42.83
89.3
19.93
16.98
89.3
30.49
28.56
Google
0.5
89.7
49.72
42.09
89.6
19.87
17.04
89.3
30.25
28.24
Stock
0.1
89.8
52.47
43.72
89.6
19.85
17.34
89.6
30.66
28.26
The results are shown in Table 7 Overall, COP demonstrates high robustness to variations in the scale factor. Across all base predictors (Prophet, AR, and Theta), the coverage rates remain stable and consistently close to the target, regardless the specific chosen. Regarding efficiency, the setting of X = 0.5 generally yields the most favorable trade-off, achieving tighter average and median widths compared to the more conservative À = 0.1. While X = 1.0 also produces competitive widths, it occasionally results in slightly lower coverage rates (e.g., AR Amazon Stock). These empirical findings justify our default hyperparameter selection of À = 0.5 used in the main experiments.
I STATISTICAL SIGNIFICANCE ANALYSIS
We regenerated the Changepoint and Distribution Drift datasets using ten different random seeds. For each method and each base model (Prophet, AR, Theta), we evaluated the coverage rate, the average interval width, and the median width on every dataset and then reported the mean and standard deviation across the ten generated datasets.
As shown in Table g and Table 9 the variability across multiple data generation seeds is small: coverage rate varies by less than 0.5%, and interval width variations vary slightly. This indicates that performance differences between methods not attributable to any single random dataset. In both experimental settings, COP consistently achieves coverage rates close to the target level while yielding tight interval widths.
Moreover, we conducted paired t-tests to reflect the statistical significance of shorter width of COP. All of the baselines above show p-values less than 0.05 (c.g., OGD has p-value=0.014, SF-OGD has p-value==.0000 00000, and ECT's p-v0lue=0.03). These results confirm that the observed differences in width between our methods and the COP baseline statistically significant.
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Table The experimental results the Changepoint datasets with nominal level a = 10%. Values represent the mean + standard deviation of coverage Rate, average width, and median width across ten independent runs using different random seeds.
Pruphet
AR
Method
Coverage
Average
Median
Coverage
Average width
Median width
Coverage
Average width
Median
width
width
(8)
width
ACT
90.0=0.01
00
8.27=0.04
90.0=0.01
8.15=0.04
90.0=0.01
00
8.33=0.05
OGD
10.120002
8.6340.09
8.62&0.10
90.020.02
8.4620.100
8.48
90.120.03
8.6820.12
8.64.0.13
SF-OGD
90.0+0.01
12.65=0.12 12
11.59=0.12
90.0=0.01
12.66=0.15
11.6300.15
90.0=0.01
12.68+0.25
11.71+0.28
decay-OGiD
90.610.10
8.52x0.20
8.34g0.15
90.120.09
8.1720.22
8.11
90.520.30
8.4320.30
8.29.0.20
PTD
89,7+0.01
11 11.06=0.15
9.36+0.18 18
89.7+0.09
10,86 10
9.18=0.13
89.7+0.02
10,99+0.20
9,32=0.25
ECT
89.9+0.01
8.3240.
8.34+0.19 19
89.9s0.09
8.22+0.23
8.26+0.22
89.9+0.02
8.38+0.17
8.43=0.19
LQT
90.420.30
10.72a1.20
9.70=1.00
90.020.30
10.23a1.00 100
9.40s0.90
90.2a0.50
10.4940.900
9.70.0.95
COP
89.8+0. 15
8.29+0. 15
8.35+0.15 15
89.4+0.50
8. 1.12+0.22 20
8.23=0.20
89. 39.8+0.30
8. 8.35+0.15
8,38+0.18
Table 9: The experimental results in the Distribution Drift datasets with nominal level 10%.
Values represent the mean + standard deviation of coverage Rate, average width, and median width across ten independent runs using different random seeds.
Prophel
AR
Theta
Methed
Coverage
Average
Median
Coverage (8)
Average
Modian
Coverage
Average
Median
(%)
width
width
widh
width
(8)
width
width
ACI
8:0.02
6.6120.03
89. 79..0
6.53=0.02
820.02
6.64:0.02
OGD
90.2+0.01
7.0420.03
6.91:0.03
90.120.02
6.8920.04
6.8420.03
90.2a0.01
7.08
6.98:0.05
SF-OGD
90.0=0.00
11.50=0.01
10.35=0.02
90.0=0.00
11 11.4420.06
10,28+0.08
90.0=0.00
11.50+0.08
10.28+0.05
decay
90.4s0.12
7.38
6.76x0.03
6.9420.06
6.60s0.03
90.2a0.08
7.24a0.05
6.7420.02
PID
89.700.00
9.5920.06
7.79:0.02
89.620.01
9.67+0.10
7.75a0.02
89.720.00
9.70=0.05
7.86:0.01
BCI
90.0=0.00
7.01=0.05
6.8420.03
89.9=0.03
6.83e0.04
6.81=0.04
90.100.03
7.10=0.08
6.93+0.04
LQT
90.7+0.05
9.47+00.7
8.53g0.03
8.00=0.20
90.520.59
9.50a0.33
COP
90.5+0.11 11
6.77+0.08
6.70e0.04
89.920.03
6.67+0.18 18
6.67=0.10
90.5+0.14 14
7.12=0.16
6.87=0.05
Overall, these repeated-data experiments confirm that our findings are not sensitive to the choice of a single random seed. The overall patterns reported in the main text remain the same across independently generated datasets.
J COMPUTATIONAL COMPLEXITY ANALYSIS
To compare the practical efficiency of the baselines and COP, we measured the average computation time for each method. All timings were obtained using single CPU core (Intel(R) Xeon(R) Platinum 8269CY CPU @ 2.50GHz) on the same machine, with each method running for 3020 steps. As shown in Table 10 the reported value represents the mean per-step time cost.
Table 10: Average time cost per update step.
Method
ACI
OGD
SF-OGD
decay-OGD
PID
ECI
LQT
COP
Time Cost (ms)
0.043
0.001
0.012
0.001
0.013
0.007
0.010
0.011
The simplest gradient-based methods (OGD and decay-OGD) are the fastest, requiring only 0.001 ms per step. Methods that consider all past data, such as SF-OGD and PID, incur higher overhead (around 0.012 0.013 ms). Methods that consider the past data in certain window, such as ECI, LQT, and COP fall in a similar range (0.007 0.011 ms). Overall, the differences between methods remain small in absolute terms, and all methods run easily in real time for typical streaming applications.
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Published as a conference paper at ICLR 2026
K VISUALIZATION OF COVERAGE AND INTERVAL
Fsooss the corresponding rolling-windows window coverage for each method. The coverage trajectories show how frequently each approach stays close the target level and how often it experiences deviations. The worse methods tend to display larger swings in coverage, including occasional periods of undercoverage. contrast, COP and several ECI variants maintain a more stable coverage across the entire time period, even during intervals with sharp changes in the underlying series.
In Figure 2 we plot the interval widths for all methods, together with zoomed-in panels that highlight periods of rapid market movement. These plots show clear differences in how each method responds changes in volatility. Some approaches, such as PID or ECI-based variants, exhibit sharp jumps or sudden drops in interval width when the underlying series becomes more volatile. Others, including ECI-based methods and COP, adjust their intervals more gradually and maintain a smoother trajectory.
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Figure 1: Comparison results of coverage rate on Amazon stock dataset with Prophet model. The coverage is averaged over a rolling window of 50 points.
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Published as a conference paper at ICLR 2026
ACI
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Figure Comparison results of prediction sets on Amazon stock dataset with Prophet model.
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Reviews:
[{"review_id": "MB6I0CtpAI", "reviewer": "Reviewer_JBdV", "summary": "This paper introduces COP, an online conformal method that injects predictable structure via an estimated CDF of nonconformity scores to perform an “optimistic” radius refinement—tightening sets when patterns exist while remaining robust under misspecification. The theory provides a joint coverage–regret bound together with distribution-free, finite-sample coverage under arbitrary learning rates and convergence when scores are i.i.d. Empirically, COP attains target coverage and yields consistently shorter intervals than strong baselines across synthetic shift scenarios and real-world datasets.", "questions": "1. Table 2 shows COP attains competitive coverage with narrow intervals, but Proposition 1 hinges on the unverifiable ``same--sign'' assumption.\n\n  (1) Please report how often $\\mathrm{sign}(\\hat F_{t+1}(\\hat q_{t+1}) - (1 - \\alpha)) = \\mathrm{sign}(F_{t+1}(\\hat q_{t+1}) - (1 - \\alpha))$ holds on the real-world datasets (e.g., by sliding windows), and correlate this rate with coverage/width. This would clarify whether COP's gains are explained by the assumption or by broader robustness.\n\n  (2) Provide a refinement-step analysis based solely on the observable CDF error \n$\\varepsilon_t := \\sup_q \\lvert \\hat F_t(q) - F_t(q) \\rvert$ that yields\n$$\n\\text{coverage} \\;\\le\\; \\Phi(\\varepsilon_{1:T}, \\{\\eta_t\\}, \\{\\lambda_t\\})\n$$\nregardless of the sign, that is, a non-asymptotic worst-case bound that does not rely on the same-sign assumption. For i.i.d. windows, a DKW-type inequality (and for time series, its mixing/martingale analogues) controls $\\varepsilon_t$, aligning the strong empirical results with equally robust theory.\n\n\n\n2. The simulation studies assume homoscedastic Gaussian noise. To better assess the robustness of COP's CDF-based refinement, please include experiments with heteroscedasticity, variance changepoints, and heavy-tailed noise (e.g., Student-$t$). Please report coverage and width as well as post-shift recovery time (steps to return to target coverage) across these settings, and summarize observed failure modes.\n\n\n3. For deployability, could you briefly report the time and space complexity of COP’s per-step update (with windowed ECDF/KDE) and a small scaling plot of per-step wall-clock versus window size $w$? If the implementation is $O(w)$ per step (vs. $O(1)$ for OGD/PID), please state this explicitly; if a data structure is used, please indicate the amortized complexity (e.g., $O(\\log w)$) and the memory footprint $O(w)$.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper is original in recasting online conformal prediction as optimistic online gradient descent, blending a standard feedback step with a CDF guided optimistic refinement to reduce conservativeness. The theory is solid, delivering a joint coverage regret bound, distribution free finite sample coverage for arbitrary learning rates, and convergence under i.i.d. scores, while the presentation is clear with precise setup and actionable pseudocode. Experiments across simulated shifts and multiple real world domains show target coverage with consistently tighter intervals, underscoring practical significance and easy deployability, using simple ECDF or KDE plugins.", "weaknesses": "The central theory relies on an unverifiable same sign assumption and boundedness that may fail under nonstationary or heavy tailed regimes; an assumption free, non asymptotic refinement bound based on observable CDF error would strengthen the claims. The simulations omit heteroscedasticity, variance changepoints, and heavy tails, lack ablations isolating the optimistic step and ECDF versus KDE, and do not report recovery time or conditional coverage. Practical clarity and scalability are under specified: window and bandwidth choices are sensitive, per step cost appears $(O(w))$ time and $(O(w))$ space without specialized data structures, and the fairness of hyperparameter tuning is unclear; adding adaptive selection rules and explicit complexity with wall clock scaling would improve deployability. Minor exposition gaps remain in mapping the trust region to the step size and in indexing consistency.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xFpvCcKMZ9", "reviewer": "Reviewer_CEXt", "summary": "This paper proposes Conformal Optimistic Prediction (COP), an online conformal prediction (CP) algorithm that integrates distributional information of nonconformity scores into the update rule. COP leverages estimated cumulative distribution functions to refine the prediction radius dynamically. A joint coverage–regret bound and finite-sample distribution-free coverage have been established.", "questions": "- How to choose the scale factor in real practice? How sensitive is COP’s performance to the choice of the scale factor?\n\n- In adversarial settings where estimated CDFs are inaccurate, how robust is COP compared with purely adversarial CP methods like SF-OGD?\n\n- How does COP behave under extreme concept drift or abrupt regime changes, where the empirical CDF is no longer representative of recent data?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper introduces an elegant refinement step using estimated CDFs. Theoretical contributions include finite-sample and asymptotic coverage guarantees, as well as a joint regret–coverage bound under general learning rates.", "weaknesses": "- Although both empirical and kernel-based CDFs are considered, the impact of different estimators or misspecification on performance and validity is not deeply analyzed.\n\n\n- The presentation could be improved for greater readability. For instance, abbreviations should be used consistently once their full forms have been introduced. Moreover, Sections 3.1–3.3 are mathematically dense, which may make it challenging for readers unfamiliar with OOGD or online conformal prediction to grasp the underlying intuition behind each step.\n\n- Computation time of the methods in experiments.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "iVY3gXrmMW", "reviewer": "Reviewer_LFYw", "summary": "This paper introduces an online conformal prediction algorithm that incorporates distributional information of non-conformity scores into the update rule. The proposed method leverages an estimated cdf of scores to anticipate predictable patterns and tighten prediction sets while preserving coverage guarantees. By establishing a connection between the proposed method and OOGD, authorss demonstrate a joint bound on regret and coverage.", "questions": "1- How were the hyperparameters chosen for COP?\n\n2- Could the authors include a computational complexity analysis of the proposed method and a comparison with previous baselines, such as ACI?\n\n3- Would authors show empirically whether performance degrades significantly with poor CDF estimation or not?\n\n4- Could the authors clarify whether $q_t$ in line 121  is a type or not? It's not consistent with eq(2)", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1- The idea of incorporating distribution-informed optimistic updates into online conformal prediction is a meaningful advancement in this area.\n\n2- The paper provides solid theoretical foundations for both coverage and regret.\n\n3-The paper is well written and easy to follow, with clear motivation and smooth storytelling that connects prior work to the proposed method.", "weaknesses": "Please check the questions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0uBCDG41iW", "reviewer": "Reviewer_nxNU", "summary": "This paper introduces Conformal Optimistic Prediction (COP), an online conformal prediction algorithm designed to address the over-conservatism of existing methods in non-stationary environments like time series. \n\nThe core innovation of COP is a two-step update mechanism that refines the standard gradient-based adjustment for the prediction set size. After a primary update based on the current miscoverage error, COP applies a refinement step using an estimated cumulative distribution function (CDF) of the non-conformity scores. This estimated CDF acts as an \"optimistic hint\" about the data's underlying patterns, enabling the algorithm to produce tighter, more efficient prediction sets when such patterns exist. \n\nCrucially, the method is robust; it maintains provable, distribution-free, finite-sample coverage guarantees even if the CDF estimate is inaccurate. The authors frame COP within the Optimistic Online Gradient Descent (OOGD) paradigm, deriving a novel joint bound on regret and coverage that clarifies its theoretical motivation. \n\nExperiments on synthetic datasets with distribution shifts and real-world time series from finance, energy, and climate show that COP consistently achieves the target coverage while generating significantly narrower prediction intervals than state-of-the-art baselines, demonstrating its superior efficiency and practical utility.", "questions": "**On the Limits of the CDF Estimator's Robustness:**\n\nA major strength of the paper is that COP maintains coverage guarantees even with an inaccurate CDF estimate. However, a poor estimate could still harm efficiency, potentially making the intervals wider than those from a simpler baseline like OGD.\n\nCould you discuss the practical boundaries of this robustness? Is there a point where a consistently misleading CDF estimate (e.g., during a prolonged, adversarial shift) makes COP perform worse than the baseline OGD in terms of interval width?\nAn experiment on a synthetic dataset with a deliberately miscalibrated or noisy CDF estimator could help characterize this failure mode and provide practical guidance on when the optimistic step is most beneficial.\n\n**On Statistical Significance of Experimental Results:**\n\nThe experimental results in the tables are promising, but they report point estimates from what appears to be a single experimental run. To strengthen the claims of superiority, especially when the performance differences with top baselines are small, could you provide measures of uncertainty for the reported metrics (e.g., standard errors or confidence intervals calculated over multiple runs with different random seeds)? This would be crucial for confirming that the observed improvements are statistically significant.\n\n**On the Impact of Noise and Low Predictability in Scores:**\n\nCOP's advantage stems from exploiting predictable patterns in the non-conformity scores. This raises questions about its performance in settings where such patterns are weak or heavily obscured.\n\nHow does COP perform in high-noise or low signal-to-noise ratio settings, where the underlying predictable structure in the scores is minimal? In such a scenario, does the optimistic update, which is based on a noisy and potentially uninformative CDF estimate, risk making spurious adjustments that could degrade efficiency (i.e., widen intervals) compared to a more conservative baseline like decay-OGD?\nIt would be very illuminating to include an experiment on a synthetic dataset where the level of i.i.d. noise added to the non-conformity scores is systematically varied. This would help to characterize the performance trade-offs and define the boundaries of where COP's optimism is a clear advantage.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "*   **Originality**\n    *   Presents a new synthesis of online Conformal Prediction (CP) with Optimistic Online Gradient Descent (OOGD).\n    *   Uniquely frames distributional information as a proactive \"optimistic hint\" rather than simply replacing the robust reactive update rule.\n\n*   **Quality**\n    *   **Comprehensive Theory:** Provides a full suite of rigorous guarantees, including a novel joint regret-coverage bound, essential distribution-free finite-sample coverage, and asymptotic consistency, establishing strong reliability.\n    *   **Thorough Experiments:** Validated on a wide range of synthetic and diverse real-world datasets, using multiple base models against a strong set of seven state-of-the-art baselines, convincingly demonstrating its superior performance and general applicability.\n\n*   **Clarity**\n    *   The paper is exceptionally well-written, with a clear logical flow that makes complex ideas accessible.\n    *   **Excellent Background Explanation:** The background section effectively explains all necessary concepts, providing the context needed to fully appreciate the paper's contribution.\n\n*   **Significance**\n    *   Addresses the critical and practical problem of over-conservatism (excessively wide intervals) in online CP, a major barrier to real-world adoption.\n    *   The proposed method (COP) is effective, theoretically sound, and simple to implement, giving it high potential to become a standard practitioner's tool.\n    *   The theoretical framework introduces new analytical tools that can inspire future research.", "weaknesses": "1.  **Lack of Statistical Significance in Experimental Claims:** While the experiments are extensive, the tables present only point estimates (mean and median) for the evaluation metrics from a single run. This makes it difficult to ascertain whether the observed improvements of the proposed method over baselines are statistically significant or merely due to experimental randomness. For example, a small difference in average interval width between two methods might not be meaningful without a measure of variance. The empirical claims would be substantially strengthened by including standard errors or confidence intervals for the reported metrics, which could be derived from multiple runs with different random seeds.\n\n2.  **Over-reliance on Tables for Presenting Results:** The paper relies heavily on large, dense tables (Table 1 and 2) to display the primary experimental outcomes. Such a presentation format makes it challenging for readers to quickly parse results and identify performance trends across different datasets and models. The paper's readability and impact would be greatly enhanced by using visualizations. For instance, it is suggested that the authors consider replacing or supplementing the tables in the main text with figures like **boxplots**. Boxplots would more effectively illustrate the distribution, median, and spread of the prediction interval widths, while also providing a natural way to incorporate the standard error or uncertainty estimates mentioned in the first point, thereby increasing the perceived reliability and clarity of the experimental validation. The detailed tables could then be moved to the Appendix for reference.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)