Tube Loss: A Novel Loss Function for Prediction Interval Estimation
Authors: Pritam Anand
Tathagata Bandyopadhyay
Suresh Chandra
Venue: Accepted by TMLR ()

Abstract: This paper proposes a novel loss function called Tube Loss, developed for the simultaneous estimation of the lower and upper bounds of a Prediction Interval (PI) in regression problems, including probabilistic forecasting in autoregressive frameworks. The PIs obtained through empirical risk minimization using Tube Loss exhibit superior performance compared to those derived from existing approaches. A theoretical analysis confirms that the estimated PIs asymptotically attain a user-specified confidence level $1-\alpha$. A distinctive feature of Tube Loss is its ability to shift the PI along the support of the response distribution through a tunable parameter, allowing the intervals to better align with high-density regions of the distribution. This is especially valuable for generating tighter intervals when the response distribution is skewed. Moreover, the method allows further narrowing of PIs through recalibration. Unlike several prior techniques, the empirical risk associated with Tube Loss can be efficiently optimized via gradient descent. Extensive experiments demonstrate the robustness and accuracy of the proposed method in delivering high-quality PIs across a range of models, including kernel machines, neural networks, and probabilistic forecasting frameworks.

Published in Transactions on Machine Learning Research (04/2026)
Tube Loss:
A Novel Loss Function for Prediction Interval
Estimation
Pritam Anand
pritam anandSdduu.ac.in
Dhirubhai Ambani University (Formerly DA-IICT), Gandhinagar.
Tathagata Bandyopadhyay
tathagata büdauuac.in
Dhirabhai Ambani University (Pormerly DA-IICT), Gandhinagar.
Suresh Chandra
sureshiitdellni@@maallcom
Ex-faculty, Indian Institute of Technology, Delhi.
Reviewed on OpenReview: https: // opw.reveee......e.emmee for/f id= 1dd=BvuPPaa22hrrreeeere
Abstract
This paper proposes novel loss function called Tube Loss, developed for the simultaneous estimation of the lower and upper bounds of Prediction Interval (PI) in regression problems, including probabilistic forecasting in autoregressive framework. The PIs obtained through empirical risk minimization using Tube Loss exhibit superior performance compared to those derived from existing approaches. theoretical analysis confirms that the estimated PIs asymptotically attain a user-specified confidence level 1 a-a.
A distinctive feature of Tube Loss is its ability to shift the PI along the support of the response distribution through a tunable parameter, allowing the intervals to better align with high-density regions of the distribution. This especially valuable for generating lighter intervals when the response distribution skewed. Moreover, the method allows further narrowing of PIs through rocalibration.
Unlike several prior techniques, the empirical risk associated with Tube Loss can be efficiently optimized via gradient descent. Extensive experiments demonstrate the robustness and accuracy of the proposed method in delivering high-quality Pls across range of models, including kernel machines, neural networks, and probabilistic forecasting frameworks.
1 Introduction
In regression framework, machine learning (ML) models aim to predict the value of a real-valued random variable y, commonly referred to as the dependent variable, given the value of vector of input variables æ, known as the independent variable. However, providing only a point prediction of y without an uncertainty quantification may not be useful for applications. Uncertainty quantification (UQ) becomes particularly replacement.
critical when the stakes of prediction errors are high. Consider, for instance, the scheduling for a vital but, expensive component in a nuclear reactor, where failure could be catastrophic. Merely reporting that the expected remaining lifespan is 2.5 years offers limited utility for planning preventive maintenance. On the other hand, stating that the component's 's lifespan is expected to lie between 2 and 3 years with 99% confidence conveys significantly more actionable information. Such ranges for the predicted values of y are termed prediction intervals (PIs), and are associated with a specified confidence level (e.g., 99% in this case).
Suppose we are given n training samples ((,, Di) : 1,2,..., n,,, and we must now predict the unknown value of Bo+1 at a test point x X,+1. We assume {(x, vi) 1,2,... ... . 1} are i.idd. from an
The code available http:///gttttt..e//////ttaaaaaaa

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arbitrary joint distribution p(z.y) of the random vector (x: y), with feature vector R typically being vector-valued input, and response variable y E R. For the sake of notational convenience, we use (æ, v) to denote both the random vector as well as their specific values, whenever the distinction is evident from the context.
Given an algorithm A and a training set {aa, yi) i 1, ......... ........,........ ) a regression model fitted to the training set to generate ((x), the predicted value of y at x, obtained by minimizing the empirical risk
arg min Σ plys u(x:))
(1)
where, p(y,p((;)) p(x:)) is suitably chosen loss function representing deviation of p(x) from y-
In conventional regression set up, typically ((x) is an estimate of the conditional expectation Ey x) and (())) = (y (x))²² Often, quantile of the conditional distribution y/x, specifically the median instead of the mean, may be preferred as the predictor of especially when the conditional distribution is heavy-tailed.
In general, for quantifying the uncertainty associated with a predictor (i())) of ya+1, a widely used measure is the prediction interval (22 (2(xn++)] with specified confidence level 1 a, which defined as P P[p1(xn=) S Mh+1 S (x2(xx+1)] 2 aa
Equation (2) is formulated for any joint distribution p(æ,, y) and for any sample size n. The probability involved in this expression is marginal, taken over all samples (( y:) : 1.2,..., 1}. procedure aimed al constructing high-quality (HQ) PIs must satisfy two essential criteria.
First, should achieve the desired confidence level 1 without imposing strong distributional assumptions. Second, the resulting intervals should be as narrow possible across the input space, thereby ensuring that the predictions remain maximally informative.
The quality of a prediction interval is typically evaluated using two metrics: the Prediction Interval Coverage Probability (PICP) and the Mean Prediction Interval Width (MPIW), both computed on the test set ((x,, f/) i 1,n + ,,...,7. + m]. These measures are formally defined as
PICP 1 ntm
Σ
k
(3)
m
where,
1. if y: (2(x,)
h;
(4)
0. Otherwise,
and
MPIW
Σ [xx(x)) - P (xx)). 1
(5)
respectively. A HQ PI expected to minímize the MPIW while ensuring that the PICP remains greater than or equal to a, the desired confidence level.
To address uncertainty quantification in regression tasks, various methods for constructing PIs have been proposed in the literature. Traditional approaches such the delta method (Hwang & Ding (997]
Chryssolouris et al. (1996)), the Mean- Varionce Estimation (MVE) technique (Nix & Weigend (1994) [], and Boyesian inference- based models (MacKay (1992)) typically assume that the observational noise follows an independent Gaussian distribution. However, the performance these approaches tends to deteriorate significantly when the underlying data distribution highly skewed or exhibits heavy tails.
A more direct and model-agnostic strategy for constructing PI with confidence level of to estimate [Fg (1)), Fut1-a((t) where, Fy(2)) is the q (0, 1)thh quantile of the conditional distribution of y given
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defined by P(y S (z(2)))))) q. In quantile regression, an estimate of Fz(x)) say, Fz(z), is obtained by replacing ((()) in (1) with the pinball loss function (Koenker & Bassett Jr 1978 Koenker & Hallock (2001)), given by
a(v ((x)), (a- - 1)( - p(x)), otherwise.
if x2
Po(,,(())))
(6)
It is noteworthy that obtaining the lower and upper bounds of this interval involves solving two distinct oplimization problems to estimate (x) and F Furthermore, to ensure that the resulting interval qualifies as a HQ PI, one must select an optimal value of (within the feasible range satisfying q+1-a E (0, 1)) that minimizes the MPIW on the test set:
MPIW 1 Σ
(a))
Fg ((.
Determining the optimal value of q requires repeatedly solving two separate optimization problems over grid of candidate y values, rendering the overall procedure for estimating the PI computationally demanding.
A few questions arise naturally at this point: Given q, can the lower and upper bounds be estimated simultancously solving a single optimization problem? In other words, for a given value of q, is it possible to oblain the PI or both the quantiles as a direct output of single optimization process? Furthermore, if such a formulation feasible, the quality of the resulting interval would evidently depend on the choice of the underlying loss function. Hence, the central question becomes: what kind of loss function can yield good-quality PIs? To address these issues, this paper introduces novel loss function that produces good-quality PIs, referred to as the Tube Loss function. It may be viewed as a two-dimensional generalization of the pinball loss function, endowed with distinctive properties that enable the simultaneous estimation of both the lower and upper bounds of PI. The intervals derived using the Tube Loss are shown, both theoretically and empirically, to achieve the nominal coverage level asymptotically. A formal proof of this result is provided.
Moreover, by luning a hyperparameter of the loss function, the resulting interval can be shifted along the support of the response distribution y, thereby better capturing regions of high data density, As demonstrated in the experimental analysis, this feature proves particularly effective in reducing the MPIW, especially when the response distribution is asymmetric. Additionally, further reduction in MPIW on the test set can be achieved by incorporating a penalty term in the loss function with a user-defined parameter, particularly when the PICP on the validation set substantially exceeds the nominal coverage level 1 1-a. In the experimental results presented in Section 4, the proposed Tube Loss is employed to generate PIs across various regression frameworks, including Kernel Machines, Neural Networks,, Deep Neural Networks for probabilistic forecasting tasks. Apart from this, we also extend the Tube loss methodology for obtaining the conformal prediction sets and quantifying the uncertainty into the STS task estimation along with appropriate baseline comparisons. The PIs derived from the Tube loss are observed to outperform those obtained using existing methods.
The rest of the paper is structured as follows. Section 2 provides a review of existing approaches of UQ in regression framework. Section presents the proposed Tube loss function, describing its interesting properties and advantages. Section 4 discusses the results of extensive numerical evaluations. Finally, Section 5 outlines the possible future research directions.
2 Related Work
In this section, we briefly review existing UQ methods for regression within NN frameworks. For effective decision support, UQ methods must aggregate both data (aleatoric) and model (epistemic) uncertainty into the final estimate. Data uncertainty arises from the intrinsic stochasticity of the conditional distribution y x. In contrast, model uncertainty stems from the instability of NN training, where different parameter
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realizations can lead to varying predictions under identical settings. The PI estimation methods are typically designed to capture data uncertainly well.
2.1 Aleatoric Uncertainty through Prediction Interval Estimation
Based on their underlying assumptions, PI estimation method in NN framework can be categorized into distribution-based and distribution-free PI approaches. Within the distribution-baseed category. Mean- Variance Estimation (MVE) (Nix Weigend (1994)) and Mixture Density Networks (MDN) (Bishop) (1994)
are among the most widely used approaches. The former models the conditional distribution y as Gaussian, while the latter, mixture of Gaussian distributions, respectively. A systematic survey of PI estimation methods developed within NN frameworks, with particular emphasis distribution-based PI approaches,
is presented in (Khosravi et al. (011b)).
Recent NN based approaches typically estimate PIs in a distribution-free setting and have been shown to produce better PI quality, particularly under heterogencous and unknown noise conditions. One of the most well-established PI estimation approaches through the Quantile Regression Neural Network (QRNN), which trains two independent NNs to estimate the lower and upper quantile bounds of the PI, namely [2), (2)1- by minimizing the pinball loss with a suitable choice of the quantile level q. However, an alternative line of research has evolved independently of the traditional quantile regression framework, in which PIs are generated directly as solutions to a single optimization problem, bypassing explicit quantile estimation. This simultaneous estimation of the PI bounds often enables explicit minimization of the PI width within the training objective, leading to narrower PIs in most applications without compromising the target calibration.
2.1.1 Lower Upper Bound Estimation
In a distribution-free setting, Khosravi et al. (2011a) were the first to introduce method for jointly estimating the lower and upper bounds of the by directly minimizing empirical risk via a loss function, denoted by LLB. We call this method as Lower Upper Bound Estimation (LUBE). For given training set {(2; Vi) : 2, ... n}, the LUBE (Khosravi et al. (2011a)) trains single NN that considers the simultancous estimation of PI bound functions (1), (x) by minimizing the following problem
min (91.22)
ML = min 1 (11022)
Σ y
(7)
nR
where PICP is computed for the PI [M] (æ), [2(x)] over training samples {(x,, (4) ; n3.. and R= max(y;) min(ys) denotes the range of the response values y4- The parameter 1) user-defined constant that amplifies small deviations between PICP and a. The indicator function xPPCP) is defined as 0 if PICP 2 (1 a), and otherwise.
LUBE has been extensively used by the researchers in various NN-based application settings, such as energy load predictions (Pinson and Kariniotakis, 2013; Quan et al., 2014), wind speed forecasting (Wang et al., 2017; Ak et al.,2013), prediction of landslide displacement (Lian et al., 2016), gas flow (Sun et al., 2017), and solar energy (Galvan al., 2017).
A significant drawback of LUBE loss function is that it's function of PICP, step function, and its gradient is zero almost everywhere (with respect to Lebesgue measure). Thus, the gradient descent (GD) method could not be used to minimize empirical risk. This is inconvenient considering that GD is now the standard method for training NNs. Instead, they propose simulated annealing (SA), a non-gradient based optimization method. Later, various other non-gradient-based methods are usod, including Genetic Algorithms (AK et al., 2013), Gravitational Search Algorithms (Lian et al., 2016), PSO (Galvan al., 2017), and Artificial Bee Colony Algorithms (Shen et al, 2018).
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2.1.2 Durlity-Driven Loss
Building on the work of Khosravi al. (2011a)) Pearce al. (0118
propose the Quality-Driven (QD) loss
function, and seeks the solution of the following optimization problem:
(12(a))) - (1x)))
min 0K1
min (M102))
+ X a(1 (11-a) max(0, (1 -)) PICP)3, (8)
Σ k
where PICP is computed for the PI Jen (2), p2(2)] over training sample {(z,, 31) :i= 1,2, .... n2 and k is defined as
1. if Yi E (x,) (2(=:))
k =
0, Otherwise.
Also, X is the user-defined trade-off positive parameter As discussed in Pearce et al. (2018) (Section 3.3.1), like LUBE, QD loss also involves PICP, and thus, GD method cannot be used. To enable the use of the GD method, they propose approximating PICP as the product of two sigmoid functions, a technique previously used by Yan al. (2004). They also suggest using an ensemble of NN models instead of single NN. It is found to yield PIs with shorter average widths.
Further, Salem et al. 2020 extended the QD loss by incorporating two additional penalty terms: one penalizes the mean squared error to encourage accurate point predictions, while the other enforces the structural integrity of the prediction intervals by preventing crossings between the PI bounds. They termed this modified loss function as the Quality-Driven" (QD+) loss, which has been empirically shown to be more effective practice compared to QD loss for PI estimation tasks.
2.1.3 Simultaneous Quantile Regression (SQR)
Tagasovska and Lopez-Paz (2019) propose simultaneous estimation of a large number of quantiles by minimizing the empirical risk (ef. (1)) of standard quantile regression with a pinball loss function (cf. (6))- The minimization carried out by using a stochastic gradient descent algorithm, sampling fresh random quantile levels q ~ 24(0,) 1) for each training sample and mini-batch during training. Thus, SQR provides an estimate of the entire conditional distribution of given and thus, it enables them "model non-gaussian, skewed, asymmetric, multimodal, and heteroscedastie" noise.
Given all possible quantiles, to generate PI, any pair that satisfies the coverage constraint may be chosen. Because it estimates large number of quantiles rather than the two bounds of PI, it faces significantly more challenges in learning during training. In spite of its added flexibility, its performance in terms of attaining nominal coverage found to inferior to that of the standard interval regression methods, such as those based on QD loss function (cf. Pouplin et al. (2024)).
2.1.4 Relaxed Quantile Regression (RQR)
As an alternative to SQR, Pouplin et al. (2024) propose a novel loss function that directly outputs the bounds PI with a target coverage - a without requiring the intermediate step of estimating quantiles. This approach called Relaxed Quantile Regression (RQR). The RQR loss function (Pouplin al. (0244 is given by
C RQR (4I, U2) = a1l-a)u1 u2, if ui 2 0,
(9)
-Quu1g if U1 < 0,
where, in and th are the crrors, y - ph (x) and y p2(x), respectively, reprosenting the deviation of the y value from the PI bounds. Notice that, for U2d2 < 0, will be within PI (z())) and for U1 2 0, it is outside PI, but Li-a (4,, U2) remains positive in both cases. However, for smaller values of (such as 0.01, 0.05, 0.1), the value of CRQR (u,, u2) is much smaller for u < than for uy 2 0. The multipliers of un We chosen as to satisfy the coverage constraint. Figure] shows the plot of the RQR loss for
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225
28.6
15.0
125
25
60
Figure 1: RQR loss function Pouplin et al. (2024) for 1-a =0.9
1 0.9. Certainly, it is a nonconvex function of and ug. For the training set { y4) : 1,2, 1,2.....1.. RQR luss-baser PI estimation model requires the solution of following optimization problem
1 alv (x)) -aly (1 p=(()), if (y- (x))) =2(x)) <
-2))))))>><(())))))-)>)>>)))>>>) if (v (f- -xx()) 2
min
RQR 1-a (1,,)) (z(x.)) min
(10)
Pouplin et al. 2024 show that PI estimated through the RQR methodology enjoys several desirable theoretical properties, including asymptotic coverage, well as unbiased and consistent estimation.
However, a major limitation of the RQR estimator is its high sensitivity to the presence of outliers. This sensitivily arises because, similar to sspectile-based losses (Newey & Powell (1987)), the RQR loss is formulated as the product of two error terms representing the deviations the true value from the lower and upper bounds of the PI. Consequently, outliers are more likely to induce crossings between the PI bounds, which can adversely affect the conditional coverage of the PI, particularly in regions near such crossings. Our experimental results in Section 4 substantiate this observation.
2.2 Epistemic Uncertainty in Prediction Interval model
Even when a PI estimation model yields high-quality intervals, the estimates themselves may still be subject to substantial model uncertainty. Such uncertainty can arise from scnsitivity to training-set sampling, hyperparameter choices, and initialization conditions, leading to variability in the resulting PT estimates across different training runs. Therefore, effectively aggregating model and data uncertainties is essential characterize the overall uncertainty associated with PI estimation accurately.
Bayesian methods (MacKay (992)) quantify model uncertainty by placing prior distributions over model parameters and inferring their posterior distributions given the data, thereby propagating parameter uncer- lainty into predictive uncertainty. However, in the context of NN training, its estimates are sensitive to approximate inference and prior specification, and are often computationally expensive to implement, limiting their practicality for scalable uncertainty estimation.
Gal Ghahramani (2016) developed Monte Carlo Dropout (MC-dropout) a practical approach for estimating model uncertainty by activating dropout layers at test time. In this approach, dropout masks are randomly sampled and applied to the network weights for each forward pass, resulting in different sub-networks being activated at every evaluation. Repeating this process multiple times produces a collection of stochastic predictions, whose variability reflects the underlying model uncertainty.
Lakshminarayanan al. (2017) developed a simple yet highly effective approach for quantifying model uncertainty in NN and deep learning models through deep ensembles. The central idea to train the same NN architecture independently times under the same hyperpammeter selling, while using different random

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initialiattonn of the model parameters 0,, = i........, Each trained model produces its own prediction, and the variability across these ensemble members is used to quantily model uncertainly.
Further, Pearce et al. (2018) aggregate the data uncertainty captured by the NN-based PIs with the model uncertainty arising from parameter variability using deep ensembles to obtain the final uncertainty estimate. Specifically, the NN trained multiple times on the same training dataset with fixed hyperparameter settings but different random parameter initializations 0,, producing collection of PIs [u] (xx, n4 (x)], for i = ...,.. S. The aggregated Ensemble PI is then constructed by averaging the predicted lower and upper bounds across ensemble members and expanding them according to their empirical dispersion. This dispersion term captures model uncertainty, while the individual NN predictions account for data uncertainty, thereby yielding a unified uncertainly estimate for a given target coverage -
3 Interval Estimation Using Tube Loss Function
In this section, we present the Tube loss function, which offers several advantages over the loss functions discussed earlier. First, unlike RQR loss, the Tube loss is linear function of the errors, making the estimated bounds less sensitive to outliers. Second, in contrast to the intervals obtained through RQR, the bounds of the Tube loss-basct interval never intersect in practice. Third, unlike the loss functions employed in the QD and QD+ approaches, the Tube loss allows direct optimization of the empirical risk using the GD method. In QD and QD + loss based PT models, a smooth function must approximate the loss function well to enable gradient- based optimization, and the quality of the resulting intervals depends heavily on the quality of such approximation. The experimental results presented in Section vindicate this claim.
Fourth, distinctive feature of the Tube loss is that its output intervals be shifted along the support of y by tuning lyperparameter This flexibility allows the interval to better capture dense regions of the response distribution, thereby helping produce tight intervals, particularly when the conditional distribution of given x is asymmetric. Fifth, by introducing a user-defined regularization parameter 6 into the minimization problem, the Tube loss enables an effective tradc-off between PICP and MPIW, leading to narrower intervals in practice, especially when the PICP obtained on the validation set significantly exceeds the nominal coverage level 1 a. Finally, we demonstrate that the intervals derived from the Tube loss function asymptotically attain the nominal coverage. In other words, as the training sample size n + oo, the empirical coverage converges to the nominal confidence level. A theoretical proof of this result is provided in Lemma 1 (cf. Section 3.2).
3.1 Tube Loss Function
Let us denote the prediction errors (y - m1) and (y- - m22 by un and U2, respectively. For a given nominal coverage (1 a) and u2 < un (i,e., #2 2 (1), we define the Tube loss function as
(1 a)uu,
if u2 0,
aau2
if u2 I 0, u1 2 0 and ruz + (1 r)u1 2 0,
C(I-a) (1,, =
(11)
aur
if u2 S 0, un and ru2 (1 - r)ui 0, if in < 0, (1-aa)un
where, < r < 1 is a user-defined parameter. In the following, we discuss a few properties of the Tube loss function.
First, in Figure plots of the Tube loss are given for the default choice r = 0.5. Evidently, - (ui, u2) is symmetric continuous function around the line i1 u2 0. Additionally, it is differentiable almost everywhere (with respect to the Lebesgue measure) and has non-zero gradients, thus enabling the direct application of the GD method.
The default choice of r outputs centered Pls. However, by adjusting the value of r, the PI bounds can be shifted up or down, allowing the interval to capture the denser regions of the data cloud while maintaining coverage. It helps in reducing MPIW significantly, espocially when the conditional distribution of y given æ is skewed. The experimental results presented in Section 4 support this claim.
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0.6
0.4
0.5
0.2
0
-0.5
0.5
U2
U2
-0.5
0.5
0
4
U2
-0.5
(a) 1-a=0,9
(b) 1-a=0.8
Figure 2: Tube loss function
3.2 Asymptotic properties of the Tube loss function
Let y1, V/..... Yo be iid following the distribution of continuous random variable Y and (0, 1). Let define the following disjoint subsets of IR: Ri (2), = {y > μ2). R2 u2) {y μ4 < y P2, > rpe + (1 (1- - r. J. (2(21, (2) (2) {y μn < y < μ2, < rp2 + (1 - p) and R4(11 [2) = {y < p1). Notice that the sets R, (11, = 2, 3, 4 are so defined that they do not include the points on the boundaries.
Σ (1s 11, (2) with respect to (21, and ng denotes the Suppose (Mi uz) is the minimizer of a =1 number of data points in (e (k 1,2,3,4). Then have the following lemma.
Lemma 1 As oo, with probability the following results hold: (i) P1
(ti)
ny + I-a' and (iii)
14
n2 I-aa
By
I-a
The proof of the lemma is given in Appendix A.
We now consider the regression setup with training set T (x., mi), i 1,,,..., i= 1,2..... n are iid following density p(æ,1) Let rewrite the Tube loss function I1
......,, mn where (æi,
(1- (
if > pe(x),
-aly - #z(x))
ify S p2(x), 2 (z( and rly (z(x)) + (1-r)(y in (x)) 2 0,
(V pı(x), -(x()))
aly (x)),
if pa(x), y m1(x), and r(y u2(x)) r) vz < <0
- -11 a) m=)))) if y < px)))
that can be simplified as
(1- - (1-a)(u ay p2(2)), -aly (2(x)),
if > #2(#).
if (m) I p2(x) and 2 ru2(2) + (1 (2)
(1)))) =
(12)
aly (x)))
if S pa(x) and y < rp2(w) (x) -(1 a) (x)), y < (m)).
For the training set T, let (x), (x)) be the minimizer of the empirical risk Σ LT (35, Mi (æ:), p2(2;)),
[=1
where (z) and u2(x) belong to an appropriately chosen class of functions that is sufficiently regular and mcasurable so that uniform convergence of empirical risk to population risk holds. With some abuse of 2, 4) inducing a We then have the following proposition.
notation, we let ne denote the cardinality of the R4(= R& (f) (T(x)))
partition of the feature space (as shown in Figure
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R
H2 (x)
R
R
R
14 (x)
Figure 3: The Tube loss induces partition of the feature space into four regions. Red line reprosents the convex combination of (x) and (2(æ)), i,e., (2)( + (1 r)) (x)), 0 < < 1. Blue dots represent data points (I,,)), i = 1.2,....
Proposition For a E (0, as n oo the following results hold with probability 1, ti + I provided (æi, (e, 1,2, are tid following distribution p(x,y) with p(ulx) continuous and the expectation of the modulus of absolute continuity of its density satisfies lims-+0 E[((5)] 0.
- (ii) पhद -
(tti) n
Proof: The proof follows from the proof of the Lemma stated above and Lemma 3 of Takeuchi et al. 2006
3.3 Parameter r and movement of the Pl bounds
Now; we show that how the PI bounds can be moved up and down by choosing appropriately. Consider two values of say 71 and r2 such that 0 < r2 < 1. Let denote the cardinality of the set Rg (P (x), Ar (x)) by n (h: = 2, 4, i= 1,2.)
Now, Proposition (iii) entails that asymptolically 1 - a fraction of values should lie inside each of the n "/r > "r which in turn implies
intervals [2T] (a); /5' (x)] and (z). (x)) But, Fı < r2 simply entails
nt
I7 n from (i) and (ii) of Proposition for large n. Thus, changing the value of r from T1 to 12 moves nf
24
the PI tube up and vice versa.
The optimal choice of in the Tube loss depends on the characteristics of the conditional distribution When r= 0.5, the Tube loss implicitly assumes that y x is symmetrically distributed, resulting in a centered PI estimate with the narrowest achievable width. In this setting, the PI is calibrated such that, asymptotically, a fraction a/2 of the observations lies in regions R2 and R4, respectively. Morcover, the mid function r p2(x) + (1 - r) in (2r) (illustrated by the orange line in Figure 3) coincides with the median of the conditional distribution y æ.
For asymmetric noise distributions, the optimal value of is closely related to the skewness of the noise distribution. In the presence of right-skewed noise, smaller values of r shift the PI downward, enabling the construction of a narrower interval. Conversely, for left-skewed noise distributions, larger values of r are more appropriate and lead to better quality PI.
In Section 4, we demonstrate that, in several experiments, selecting an appropriate value of r results in a significant reduction of MPIW. Notice that, Li (u,, U2) is discontinuous at the scparating plane
'The modulus of absolute continuity of a function f is defined as the function e(6) = Sup E. (f(b,) f(a)), where the supremum is taken over disjoint intervals ba),b)) with a < by satisfying E, a.) < 6. Looscly speaking, the conditional density p()(z) is absolutely continuous on average. This ensures that the probability of a point lying on the boundaries of PI vanishes.
"hh asymptotic calibration guarantees established in Proposition do not necessarily lead to asymptotic individual calibration
guarantees Chung al. (2021)). ) that strictly requires ((1((x) S y M2(=)) | x) =

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[u11 u2) rug + (1 r) 0} when # 0.5. However, while running the experiments, observe that it has not caused any problem for the implementation of the GD method.
3.4 Recalibrating Prediction Intervals to Balance MPIW and PICP
After choosing appropriate value of if the attained coverage (PICP) of the output interval in the validation set exceeds the nominal coverage - a, then we go for re-calibration, which helps in further minimization of MPIW, and thus, enhances the quality the interval, keeping the attained coverage more than equal to a. For re-calibration, we employ the following loss function,
(w(1-a)(W-11())..) (22) (V
8112((r)) (1))))
(13)
The default choice of 8 is zero, unless there is room for recalibration, i.e, if PICP is more than 1 - a in the validation data set. Notice that contrast to RQR, the penalty term in our case uses a linear function of the errors, thus making it less sensitive to outliers. Section 4, we employ recalibration in some experiments to enhance the quality of the PI.
4 Experiments
In this section, we present the experimental results on generating PIs in kernel machines, neural networks, deep learning for probabilistic forecasting task and analyze the Tube loss methodology in comparison with existing losses in the literature. Apart from this, extend the Tube loss PI model for obtaining the conformal prediction set and quantifying the uncertainty in semantic textual similarity task along with the appropriate baseline comparisons.
In general, to compare two PI estimation methods, say Mi and M2, we adopt the following rule this paper: method Mi is considered superior to method M2 if either PICP(Mi) 2 1- and PICP(M)) < a, or if min min (PICP(M (M2) 2 1 and MPIW(M:) < (MP))
Furthermore, in experiments conducted with synthetic data, where samples generated from known distri- bution, the true conditional quantiles of y given are known for all æ. Consequently, the corresponding true PI, denoted by (x))), u2 (æ)).. is available. In such settings, the accuracy of an estimated PT, [fr(a)) (2))), is assessed using the Sum of Mean Squared Errors (SMSE), on the test set defined as
Σ (ai(2;) - (2)=))
1 Σ 2=1
(ag(a:) 22:)))
4.1 Tube loss in kernel machines
In kernel machines, the PI bounds are a pair of functions
(x) = Σ k(w, x)n + 7o and (2 = Σ k( x) + Bo.
(14)
where k(z,y) is positive definite kernel Mercer (909)). Using matrix notation, rewrite (1(x) and (2(x))
pi (x) = K(AT. x) + no and u2(x) = K(AT, x))3 + Bo,
(15)
7/1
31
where A is x p data matrix containing training points in RP, 9)
?/2
32
and K(AT. x))
"
Th
[k(a,,, m.), k(02, 2) -., k( m))- The Tube loss-based kernel machines consider the following optimization
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problem for outputting the PI:
A
(1 n B) + Σ - (1-aa) (Vi, (K(A", xx:)+ + +m),
(x)) (+))++8))) )))
min (7.
2
i=1
+ 8 Σ (K(A*, x;) B] + (1/0- - {0}|-
(16)
The Gradient descent solution to the above Tube loss kernel optimization problem is derived in Appendix B.
For obtaining high-quality PIs, the parameters and must be appropriately tuned using the validation dataset. The default settings r = 0.5 and = O, typically yield good performance when the conditional distribution of y given æ is symmetric, since in that case a centered prediction interval optimal. However, if the distribution is positively (negatively) skewed, selecting an r value smaller (larger) than 0.5 shifts the bounds downward (upward), thereby minimizing MPIW while maintaining PICP at the desired level. Once appropriate has been determined, if the resulting PICP exceeds the larget confidence level o, recalibrating through can further reduce MPIW without compromising the intended coverage.
4.1.1
Experimental Results
1. Choice of
We first show, in the context of kernel-based models, that the choice of the parameter is crucial for achieving narrow PI bounds, especially when the underlying data distribution exhibits asymmetry.
To illustrate this, we construct two synthetic datasets, denoted by D1 and D2. Each dataset consists of observations {( U:) i= 1,2,.... 1500), where 2; is sampled from the uniform distribution defined over the interval [0, 1], and the response variable y: generated according to
sin L
yb
Ess
(17)
where 6 represents random noise component. For dataset D1, 6, follows a symmetric normal distribution ((0 0.8), while for dataset D2, 6, follows positively skewed distribution given by x²(3) 33
The model is trained on 500 data points, with the remaining samples reserved for testing. For constructing linear PI tubes, linear kernel is employed. For nonlinear PI estimation, we adopt the radial basis function (RBF) kernel defined as
x(x1,x2) = e - -
where y is a suitably chosen hyperparameter.
For dataset D1, the Tube loss function is applied with a linear kernel to construct PIs al nominal confidence levels of 1
0.8 and a 0.9, using the default parameter settings r 0.5 and d The obtained PICP and MPIW values for the test datasct corresponding to 1 0.8 (0.9) are 0.799 (0.909) and 2.10 (2.78), respectively. Figure illustaates the estimated PIs on the test data for both target confidence levels,
a 0.8 and 1 - = 0.9.
For dataset D2, we construct the PI corresponding to 1 - = 0.8 using the RBF kernel. The Tube loss- based kernel machine with r r= 0.5 and = produces a centered interval, as shown in Figure Bbb, yielding PICP and MPIW values of 0.79 and 6.47, respectively. Owing to the positively skewed nature the data distribution, such centered interval expected to produce relatively wider PI (i.e., higher MPIW) compared to HQ PI. To effectively capture the denser regions of the data, the PI bounds therefore need to be shifted downward.
Figures 5(a), 5 c), and 5 depict the PIs obtained from the Tube loss-based kernel machines for r = 0.8, 0.3, and 0.1, respectively. It. is evident. that as r decreases, the estimated PI bounds move downward, covering the denser portions of the data and thereby reducing the MPIW.
Figure 6 (a) presents the variation of MPIW with respect to r, clearly showing that smaller values of r lead to lower MPIW, as expected. Figure 0 bb) displays the plots of PICP, lower quantile (I.Q), and upper
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(K) (1, 00 0002
data
(4(0)
(2(
(1r (x) (1
d
0.0
0.0
.0.5
0.5
Figure 4: Tube loss-based PI estimation for D₁ with (a)l a 0.8 and (b) 1 0.9.
X0
3,00 -.. No. 01+11 10,00
10 19 1o H1-
1,00
pl.... 2o, 11-16,
15-
(a) r= 0.8
(b) r 0.5
(c) r= 0.3
(d) r= 0.1
Figure 5: Tube loss-based PI estimation for D2 with
0.8 and 0.1, 0.1, 0.5, 0.8.
053
0100-0
2.15
2.1
0.5
MPIV
0.51
2015
22-
03
10
PC7
0.02
0.04
0.06
0.08
0.12
1.8
011
02
53
04
05
06
07
08
(b)
(d)
Figure 6: Plot of (a) against MPIW (bb PCIP, UQ and LQ for dataset B. Plot of (c) 0 against PICP, 8 against MPIW on the Servo dataset.
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Model
(++1-a.g//ra))
PTCP
MPIW
Time (8)
D1
Q Ker M
(0.95,0.15)
0.78 1 0.017
2.088 1 0.109
219.38
Q Ker M
(0.90.0.10)
2
221.72
Ker M
(0.85,0.05)
0.79
2.151
217.38
T Ker M
(0.6,0)
0.002
2.165 0.006
56.93
T Ker M
(0.5.0)
2,156
61.50
D2
Q Ker M
(0.90,30)
0.59 1 0.032
0.163
138.05
Q Kerl M
(0.95,0.35)
0.58 + 0.023
5.5369
146.60
Q Ker M
(0.80,20)
0.59 + 0.027
3.6196
146.41
Ker
(0.3,0)
0.60 1 0.032
3.495
0.166
39.07
Ker M
(0.2.0)
0.601 0.028
+
41.55
Table 1: Quantile-based Kernel Machine (Q ker M) and Tube loss-based kernel Machine (T ker M) dataset D1 and D2. Q ker M involves parameters (q+1-a,q) and T ker M involves (r, 6).
quantile (UQ) against r, where LQ and UQ denote the proportions of ys values falling below the upper and lower bounds of the estimated PI tube, respectively. The plots indicate that changing affects both LQ and UQ but leaves PICP largely unchanged. Notably, reducing from 0.5 to 0.1 results in significant improvement in MPIW by approximately 21.80%, without compromising the desired coverage.
It is important to note that for datasel D1, where the noise lerms are drawn from a symmetric dist the default choice of r = 0.5 yields a HQ PI. In contrast, for dataset D2, characterized by a positively skewed distribution, obtaining good quality PI requires roducing the value of from its default sctting of 0.5.
Finally, we compare the performance of PIs obtained using the Tube loss with those based on the pinball loss, where the latter estimates the two bounds (x) and ((+(1-a((z)) separately. We generate 10 independent replicates each of datasets Dj and D2. For every replicate, compute the PICP and MPIW of the Pls derived from both methods at nominal confidence levels of 1 - a = 0.8 and 0.6, respectively.
It is worth noting that for quantile-based PI estimation, the choice of the parameter q, and for Tube loss-based estimation, the choice of the parameter T, are both critical for producing HQ intervals. Table summarizes the comparative results. The Tube Loss-based method not only offers a significant reduction in overall training complexity, thereby decreasing training time, but also consistently achieves the desired nominal confidence level across both datasets.
For dataset D1, the centered interval corresponding to 0.1 under the pinball loss framework is expected to perform best. Although its PICP falls marginally short of the target value of 0.8, it yields a slightly narrower interval compared to the Tube loss -bsed PI with r = 0.5. As expected for symmetric noise distributions, the Tube loss with (r 0.5, 8 = 0) provides the most reliable PI for dataset D,.
For dataset D2, however, the MPIW values of the estimated PIs can be substantially improved by reducing the value of r, which aligns with expectations since the noise component follows positively skewed distribution.
2. Choice of 8
As discussed earlier, we now demonstrate how an appropriate choice of the recalibration parameter d (cf. Equation (6b) can further reduce the MPIW on the test data. To illustrate this, we consider the well-known Servo dataset [Karl UIrich| (2016)), which contains 167 observations across five variables. A randomly selected subset comprising 10% of the data points is used the test set, while the remaining data are utilized for training with 10-fold cross-validation procedure to obtain a Tube Loss-based PI (using the linear kernel) at a nominal confidence level of 0.8, with default settings of r and 8. The mcan PICP observed on the validation set. is 0.82. Since the empirical PICP exceeds the larget level of 0.8, there exists an opport to reduce the MPIW on the test set by selecting a positive value of 8. We incrementally increase until the validation PICP reaches the target level of 0.8, as depicted in Figure (c). This adjustment results approximately a 44% reduction in MPIW on the test set, as shown in Figure (d), while preserving the desired coverage.
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4.2 Tube Loss in Neural Network (NN)
TheTube loss based PI estimation model within the NN framework aims to solve the following optimization problem using the training set T = {(x,, vi) : = 1.2,..... ,.2
min (PL)
Σ
u2(x,)) + p2(xz)) -
p1 -2
(18)
where (x) and (2(æ+) represent the two outpul neurons of the dense nelwork corresponding the lower and upper bounds of the PI, respectively. The weights of both the hidden and output layers are iteratively updated through gradient-based optimization using backpropagation of the loss function.
4.2.1 Comparison with QD Loss
Within the NN framework, Pearce et al. 2018) demonstrated that PIs obtained using the QD loss yield slightly superior performance compared to those derived from the LUBE approach proposed by Khosravi et al. (2011a). Furthermore, they observed that the QD method also pulperfomss the mean-variance estimation
(MVE) approach of Nix & Weigend 1994 several benchmark datasets.
In this study, we conduct a simple experiment to show that the Tube loss-based PI provides better performance than the QD-based PI in terms of the sum of mean squared errors (SMSE) a local measure of smoothness and tightness introduced earlier in Section 4. To this end, we generate 1000 data points (x4, y1) according to the following model:
sin(x:)
+ Es:
(19)
where n3 (m22 27) and 6 1(-1,1)
2.0
True
2.0
True Pl Estimeted
Estimated
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.5
-05
-1.0
-1.0
-1.5
-6
-4
7
4
-4
4
(a) QD loss-based NN
(b) Tube loss-based NN
Figure 7: Comparison between Tube Loss and QD Loss function- bastod neural networks.
Both theTube based and QD loss-based NN models are trained to obtain PIs at a nominal confidence level of 0.95. After hyperparameter tuning, the NN architecture fixed with 100 Re..U-activated hidden neurons and two output neurons. The Adam optimizer is employod for model training. Figure 7 presents the PI estimated using the QD loss, while Figure (b) displays the PI obtained via the Tube Loss with 0.5 8 0, alongside the true PI. It is evident that the PI generated by the Tube loss is smoother and more compact compared to that produced by the QD loss. The true PI is computed from the 97.5% and 2.5% quantiles of the conditional distribution of y|æ for various values of r.
The observed (PICP, MPIW) values for the PIs obtained using the Tube loss and QD loss are (0.95, 1.78) and (0.98, 2.17), respectively. For the QD loss proper adjustment of the softening parameter s, which governs the approximation of the step function via the product of sigmoid functions, crucial but often cumbersone. Through the tuning process, the optimal parameters for the QD loss identified 200 and 0.1.
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180
100
140
120
Mean Square
100
60
40
20
r= 0.5, 5
s= 120
s 160
==
200
240
280
Figure 8: Tube Loss-based NN approximates the true PI more closcly than the QD N.V.
20-
Tan
20
3
mi
15-
15
10-
-6
(a) RQR loss based NN
(b) Tube loss based NN
Figure 9: Comparison of RQR loss and Tube loss function based NNs
Subsequently, we computed the SMSEs of the resulting Pls. Figure g illustrates the comparison of SMSE values between the Tube loss and QD loss for various choices of the softening parameter It is evident that the performance of the QD loss-based PI is highly sensitive to the selection of 8, and consequently, to the fidelity of the step function approximation achieved by the sigmoid functions.
4.2.2 Comparison with RQR loss
As stated before, the RQR PI bounds are sensitive to outliers. The presence outliers may trigger the crossing of the two bounds, thus adversely affecting the conditional coverage of the PI near the crossing. We perform a simple experiment to show the impact of outliers RQR based PI It is worth mentioning in this context that for comparison with Tube loss, we consider the RQR-Width Width Minimizing (RQR- W) method. We generate artificial data set { W:), 1,2, .0.00 using the relation.
y; sin(z:) Ci. 2;
(20)
where is UT- 2T) and e; is from U). Next, contaminate the dataset by adding 60 (2%) data where {(2;, yi), = 300, are generated randomly. The Tube loss and RQR Loss-based NNs were trained using a single-layer NN with 100 hidden neurons, ReLU activation, a batch size of 100, 1200 epochs, and the Adam optimizer for a target confidence level of 0.8. For both loss functions, the penalty parameters presents the results of RQR- and Tube boss-based NNs with r 0.3. In I(),, the RQR-based PI deviates considerably from the true PI (black dotted line), and the PI bounds cross each other near 4.
points ((x,, 10y;), i 3002,
Tube Loss and RQR. were set zero. Figure
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In contrast, the Tube loss-based PI almost overlaps with the true PI, effectively handling the asymmetric noise introduced by the outliers. This example clearly supports our apprehension stated above.
Dataset
TUBE
RQR-W
QR
SQR-C
SQR-N 85.97 (0.48)
QD loss
miami
89.34 (0.45)
88.89 (0.32)
90.50 (0.57)
87.36 (0.46)
90.26 (0.49)
0.49 (0.02)
0.51 (0.01)
0.51 (0.01)
1.90 (0.05)
1.52 (0.03)
1.74 (0.12)
wine
91.21 (0.37)
90.16
89.97 (0.36)
91.94 (0.67)
88.19 (0.97)
88.03 (0.92)
0.26 (0.017)
0.33 (0.01)
0.28 (0.00)
0.49 (0.03)
0.48 (0.03)
1.18 (0.12)
power
90.31 (0.54)
89.34 (0.55)
89.92 (0.54)
91.81 (1.34)
89.11 (1.40)
62.47 (13.63)
0.04 (0.009)
0.07 (0.01)
0.06 (0.01)
0.15 (0.03)
0.15 (0.03)
1.14 (0.23)
yacht.
89.32 (0.41)
90.32 (0.42)
91.45 (1.23)
85.97 (1.18)
84.68 (1.69)
90.32 (0.96)
0.16 (0.019)
0.33 (0.02)
0.32 (0.02)
3.56 (0.38)
2.91 (0.26)
1.66 (0.29)
kin8nm
89.00 (0.32)
89.27 (0.40)
91.89 (0.27)
87.16 (0.49)
85.53 (0.53)
89.52 (0.46)
0.42 (0.01)
0.42 (0.01)
0.50 (0.01)
1.14 (0.01)
1.10 (0.01)
1.61 (0.12)
protein
92.16 (0.29)
88.47 (0.30)
90.40 (0.23)
89.01 (0.20)
86.01 (0.32)
90.38 (0.30)
1.68 (0.02)
1.50 (0.01)
1.61 (0.01)
2.19 (0.02)
2.05 (0.01)
1.96 (0.09)
boston
91.58 (0.48)
88.14
86.67 (1.06)
82.55 (1.48)
81.18 (1.58)
90.29 (0.77)
0.44 (0.025)
0.40 (0.02)
0.38 (0.01)
1.09 (0.03)
1.01 (0.02)
1.41 (0.15)
energy
90.08 (0.42)
89.68 (0.70)
93.05 (0.88)
89.42 (0.99)
86.30 (0.99)
89.42 (0.76)
0.23 (0.016)
0.23 (0.01)
0.29 (0.01)
1.31 (0.01)
1.23 (0.01)
1.39 (0.17)
sulfur
92.00 (0.36)
88.87 (0.20)
88.30 (0.34)
87.83 (0.47)
87.40 (0.50)
89.41 (0.53)
1.06 (0.027)
1.02 (0.01)
1.05 (0.01)
1.09 (0.03)
1.02 (0.02)
1.36 (0.07)
cpu act
91.03 (0.35)
89.14 (0.37)
88.94 (0.48)
90.85 (0.75)
86.73 (1.10)
90.32 (0.62)
0.44 (0.014)
0.44 (0.00)
0.41 (0.01)
0.78 (0.01)
0.76 (0.02)
N/A* (N/A*)
concrete
91.12 (0.52)
(8.74
87.77 (1.09)
86.02 (1.29)
85.29 (1.39)
89.37 (0.68)
0.46 (0.026)
0.47 (0.03)
0.44 (0.01)
1.39 (0.03)
1.33 (0.02)
1.28 (0.16)
naval
91.01 (0.34)
88.81 (0.22)
88.34 (0.25)
90.70 (1.68)
88.86 (1.77)
91.16 (0.35)
0.02 (0.008)
0.03 (0.00)
0.02 (0.00)
0.26 (0.04)
0.26 (0.04)
2.93 (0.18)
Table 2: Comparisons of Tube loss based intervals against that of based on RQR-W and other existing baseline NN methods on 12 datasets for the PI estimation task with target 1 - a =0.90. The first row reports the PICP, while the second row reports MPIW. All results represent the test set mean over 10 runs (+ standard error). Best results are in bold; the best PI method allains average confidence level at least 0.90 with the lowest MPIW. Tuned parameters are listed Appendix C.
4.2.3 Comparison on benchmark datasets
Next, we assess the performance of the Tube loss-based PI in comparison with the distribution-free PI bascline approaches discussed in Section 2, using twelve benchmark datasets. In particular, we examine the PIs generated by RQR-W (Pouplin et al. (2024)), QRNN (Quantile Regression Neural Network), SQR-C (centered intervals) (Tagasovska & Lopez- Paz (2019)). SQR-N (potentially non-centered and typically the narrowest) (Tagasovska & Lopez-Pax (2019) and QD (Paacee et al. (2018)). The numerical results summarized in Table are produced under the experimental sctup described in (Pouplin et al. (2024)
Details regarding the selection of the tuning parameters and are presented in Appendix C.
Among the twelve datasets considered, the Tube loss-based intervals exhibit superior performance in nine cases, whereas the QR-based intervals perform best in the remaining three. These results clearly demonstrate the advantage of Tube loss-based intervals over existing approaches in generating high-quality prediction intervals (PIs) within the NN framework.
4.3 Tube Loss based deep Probabilistic Forecasting
Further, we use the Tube loss function in Long Short. Term Memory (LSTM) Network (Hochreiter & Schmidhuber (1997)) for obtaining the probabilistic forecast upon popular benchmark time-series datascts namely Electric (BP Ember. (2016)). Sunspots (SIDC Quandl.), SWH (NDBC), Temperature Female Birth (datamarket.com) and Beer Production (Australian (1996)). ) Also, we compare the Tube loss based LSTM (T-LSTM) with Quantile based LSTM (Q-LSTM) on
mmachnelannnggnastery.com
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these datasets. The 70% initial data points considered for training and rest of them were test set. Out of training sets, last 10% of observations were used as validation set. For each dataset, tuned the LSTM
Dataset
PICP
MPIW
Training Time (SEC)
Improvement in Time(s)
Q-LSTM
T- LSTM
Q-LSTM
T-LSTM
Q-LSTM
T-LSTM
Better
Electric
0.95
0.95
18.23
17.02
145
58
60 %
Sunspots
0.93
0.95
121.09
113.98
839
442
47 %
SWH
0.96
0.96
0.52
0.35
5119
2733
Temperature
0.95
0.94
24.82
15.56
1135
447
60 %
Female Birth
0.95
0.96
28.20
28.09
118
43
63 %
Beer Production
0.94
0.95
134.8
42.91
132.8
89.6
33 %
Table 3: Comparison of Quantile based LSTM (Q-LSTM) and Tube loss based LSTM (T-LSTM) on
real-world time-series datasets with target coverage - Q = 0.95. Tuned parameters and plots obtained by the T-LSTM are listed in Appendix C.
architecture, dropout, and window size based on prior literature, then applied Tube loss and the quantile approach scparately to target a 0.95 PI. Table 3 shows that the T-LSTM model obtains better performance on five cases out of datasets. It also lends to oblain lower MPIW than Q-LSTM model, as the parameter facilitates the re-calibration for capturing the narrower PI in T-LSTM model. But, main advantages of T-LSTM over the Q-LSTM is the significant improvement training time is because that the Q-LSTM needs to be trained twice where as T-LSTM requires only single cycle of training.
In distribution-fres setting, quantile- based deep learning models are the most popular and established choice for probabilistic forecasting. However, we also need to compare the performance of the Tube loss based probabilistic forecasting with that of the deep learning models trained with the QD loss function Pearce et al. (0188) )- A significant, practical challenge in training deep learning models with the QD loss function (Pearce et al. 2018 is the frequent occurrence of NaN losses during computation. However, the extended Quality Driven (QD* +) loss function (Salem al. (2020)) solves this problem up to some extent. Consequently, we have implemented the QD+ loss function based deep forecasting models for obtaining the probabilistic forocast. We tune the parameter values of QD+ loss function based LSTM well and compare its performance with the Tube loss based LSTM model on time-series benchmark dataset at Table 4 Unlike, the Quantile and Tube loss based LSTM models, the QD+ loss function based LSTM model fails to obtain the consistent performance. Out of six datasets, it fails to obtaín the target coverage on three datascts. In case of Sunspots and Temperature datasets, the QD+ + loss function based LSTM obtains very poor performance. Additionally, across all datasets, the Tube loss LSTM models outperform the QD+ loss
LSTM models.
Dataset
PICP
MPIW
Improvement in MPIW
QD+
-SSM
T- LSTM
QD*-LSTM
T-LSTM
Better
Electric
0.96
0.95
54.63
17.02
68.84 %
Sunspots
0.43
0.95
24.06
113.98
NA
SWH
0.96
0.96
0.36
0.35
2.78%
Female Birth
0.91
0.96
38.98
28.09
27.94%
Temperature 0.799
0.95
5.94
15.56
73.13
Beer Production
01.66
0.95
159.71
42.91
NA
Table 4: Comparison of Extended Quality Driven loss based LSTM (QD* LSTM) and Tube loss based LSTM (T-LSTM) on real-world lime-series datasets with targel coverage 1 = 0.95
For numerical results presented the Table 3, the tuned parameters are detail in the Table g Also, the plot obtained by the Tube loss based LSTM for benchmark datasets reported in Table 3, are shown in the I1
4.4 Application to wind probabilistic forecasting
We have also employed the Tube loss function in LSTM, GRU and TCN for probabilistic forecasting of wind speed. Table lists the performance of the Tube loss based deep learning models along with
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the recent benchmark models used for probabilistic forecasting in recent wind energy literature on San
Francisco wind dataset containing 26,304 hourly observations. last. 30% of observations were used as the test set, while the final 10% of the training data was reserved for validation.
Rank
Model
PICP
MPIW
MPIW /PIPP
I
TCN + Tube
0.9543
4.560
4.78
2
LSTM + Tube
0.9561
4.627
4.84
3
GRU + Tube
0.9507
4.857
5.11
4
GRU Quantile
0.951
4.955
5.21
5
LSTM + Quantile
0.9594
5.407
5.64
6
TCN + QD+
0.9505
5.490
5.78
7
LSTM + QD
0.9734
6.043
6.21
8
Deep AR
0.9855
6.2023
6.29
9
GRU QD+
0.9724
6.309
6.49
10
MDN
0.949
4.620
4.87
11
TCN + Quantile
0.9428
4.908
5.21
12
TimeGPT
0.9357
11.235
12.00
Table 5: Ranking of different deep probabilistic forecasting methods including Deep AR (Salinas et al. 2020 ) Mixed
Density Network (MDN) (Bishop (1994). ), QD based deep learning models (Salem et al. (2020)) and Time GPT
(Garza et al. (2023 ) on San Francisco Dataset with target coverage 1 - a 0.95. DeepAR (Salinas et al. 2020
has recently gained traction in probabilistic forecasting, including wind power prediction (Arora et al. 2002 while
studies (Yang et al. (2021). Zhang et al. (2020), Men et al. (2016)) highlight the effectiveness of Mixture Density Networks (MDN) for wind forecasting.
Table 5 shows that the Tube loss based deep learning models consistently achieve the best overall performance among all compared baselines.
4.5 Conformal Prediction with Tube Loss
Here we present an empirical study to compare the performance of Tube Loss based conformal PI with that of based on quantile regression.
As introduced carlier, let T 11)] denote set of training samples drawn from a joint distribution p(æ,1) v). The objective is construct a PI for future response Ur+1 of the form (#x41), (xx44] with nominal coverage a. Assuming exchangeabilityy of the augmented dataset Romano et al. (2019) proposed Conformalized Quantile Regression (1), v2). (I conformal framework. By introducing quantile-based nonconformity (CQR), a quantile-regresion--aess
scores, their method yields a fully adaptive conformal prediction set C(#.+1) for yo+1 that provably achieves the target coverage level 1 - a for any finite sample size.
Under the split conformal regression (CR) framework (cf. Papadopoulos et al. (2001 0: Papadopoulos (2008)), the original training set T is first divided into two mutually exclusive subsets: J, which is used to fit.
the predictive model, and 12, reserved for calibration. Subsequently, nonconformity scores are evaluated
each instance in the calibration set Iz. These scores quantify the deviation between the model's prediction for input and the corresponding ground-trut truth response V4-
Within the NN framework, the CQR approach begins by training two separate models on the training set Ii each minimizing the pinball loss to estimate the g-th and (1+g- at)thh conditional quantile functions, respectively, where 0 0 q, (1 q a) < Let the resulting estimators be denoted by Fz(x) and (z) Using these estimates, the nonconformity scores E, are computed on the calibration set Tz
E. = max maxxffflta Fg(xz) - yss yi - Pita-a(z.)).
(21)
Subsequently, following the conformal prediction framework, the empirical (1- (1-a)(1+ a)(1+ |/a quanttie
QI 12) of the set (Fa I2] is evaluated. The adaptive PI for new test input Fn +1 is then
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constructed C(En+1)= Po(xn+1) QI-a Fite-o(En41) QF,
(22)
Similar to the CQR method discussed above, in Tube loss based conformal PI estimation, we replace the quantile bounds (x) and (1+)-a(r) by the corresponding Tube loss based bounds fi (x) and (
respectively in (35) and (36).
Dataset
PICP
Error
MPIW
Time [sec)
CQR
TCQR
CQR
TOQR
CQR
TCQU
CQR
TCQR
Concrete
0.90 1 0.04
0.91 1 0.03
17.11 1.76
19.89 I 1,46
34.04
2.66
17.32 1 .99
Bike
0.90 0.01
0.90
+ 0.01
3
192.51 43.23
171.77 + 9.12
74.36
7.05
40.21 4.31
Star
0.90 + 0.02
0.90
t 0.02
3
1022.19 40.99
982.83 + 43.13
32.06
2.36
16.28 1.16
Community
0.90 1 0.02
0.91 1 0.02
5
1
0.43 1 0.02
0.47 -: 0.02
17.30
3.61
9.01 1 0.18
Facebook
0.91 0.01
0.90
18.51 4.46
17.78 1.63
182.81
5.48
99.35 3.53
Boston Housing
0189 + 0.03
0.92 + 0.03
6
2
9.38 + 0.73
13.12 3.70
4.73 - 0.06
2.66 - 0.07
Naval
0.90
0.01
0.02 0.00
0.02 0.00
83.10
6.92
4.28
Yatch Hydro
0.86 1 0.04
0.90
1 0.05
2.37 1 0.71
2.47 1 0.56
41.72 1 8,54
20.30 1 3.39
Auto MPG
0.92 0.04
0.93
+ 0.04
10.03 + 0.94
9.90 1.24
53.21
4.64
28.11 4.18
Table 6: Comparison of the Tube loss based Conformal Regression (TCR) method with Conformalized
Quantile Regression (CQR) (Romano et al. (2019)) across multiple benchmark datasets for target coverage level of 1 0.90.
To benchmark the proposed Tube-loss based Conformal Regression (TCR) against Conformalized Quantile Regression (CQR) (Romano et al. (2019)); conduct experiments nine widely used benchmark datascts (ef. Table 10). Each dataset is randomly partitioned into training, calibration, and test subsets following 3:1:1 split. PIs are constructed at the target confidence level 1 0.9, and the entire procedure is repeated 10 times.
Table reports the mcan PICP and MPIW computed on the test sets. In addition, we present under the column labeled "Error", the nunber of trials out of the 10 repetitions in which a method fails to attain the desired coverage level of other in terms of PICP and MPIW, indicating broadly comparable performance, with the TCR exhibiting CQR in several cases. Finally, with respect to computational efficiency, TCR clearly demonstrates superior runtime performance.
0.9. As shown in Table 10, neither approach consistently outperforms the
marginal improvements
4.6 Application to Semantic Textual Similarity task
Given a sentence pair (S1,S2) the Semantic Textual Similarity (STS) task predicts their semantic relatedness on a continuous scale [0, k], where regression problem,
positive integer. As STS is inherently a subjective affected by annotation noise and semantic ambiguity, making point predictions alone insufficient for reliable inference. To address this limitation, recent work advocates PI estimation principled form of uncertainty quantification, explicitly accounting for both data (aleatoric) and model (epistemic) uncertainty (Wang et al. ). Within this framework, PI-based approaches aim to learn lower and upper bounds that enclose the true similarity score denoted with w. Such uncertainty-aware STS predictions enhance trustworthiness improve robustness, and better support downstream decision-making in practical applications.
To facilitate comparative assessment of the proposed Tube loss NN against existing PI estimation approaches on the STS task, we employ the widely used STS-B benchmark dataset Cer et al. (017)). For each PI model, model (epistemic) uncertainty is quantified using ensemble based strategy. The resulting model uncertainty is subsequently combined with the corresponding data (aleatoric) uncertainty estimated by each Pl method, following the aggregation procedure described in Pearce et al. 2018) and also briefed at Section 2.2.
For PI estimation, we adopt two widely used text representation approaches: TF- IDF and Sentence- BERT (SBERT) (Reimers Gurevych (2019)). TF-IDF features are extracted using 1-2 word n- grams and subsequently compressed to 128 dimensions via Truncated SVD. For SBERT, employ the
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pattaaroettaassferrrrraa--pppppp model to encode each sentence pair into embeddings and v, from which pairwise representations [4, Iu - 1/. u @ v| are formed. The resulting feature vectors are then fy-normalized, projected to 256 dimensions using PCA, and cached to improve computational efficiency.
Table reports the performance of the different PI models after careful hyperparameter tuning aimed at obtaining high-quality PIs with target coverage a = 0.90. To account for model uncertainty, all PI models are trained over 10 independent runs under identical hyperparameter settings, and data uncertainty is aggregated with the ensemble outputs to obtain the final uncertainty estimates. We have used the ensemble approach along with different PI estimation models namely Quantile Regression through the Gradient Boosting (QRGB), Quantile Regression Neural Network (QRNN), Heterosccdastic Two-Sided (HTS) (Kendall & Gal (2017)) PI, Direct Uncertainty Prediction (DUP) Zerva et al. 2022); Zerva & Martins| 2024)), Mean Variance Estimation (Nix & Weigend (1994)), Mixed Density Network (MDN) (Bishop (1994)) and proposed Tube loss NN. These baselines are often used for the UQ in STS task (Wang et al. (2022)). Among the evaluated approaches, the Tube loss based NN yields the best quality PIs for the STS lask, oulperforming the other PI models across both text embedding strategies.
TF-IDF Features
SBERT Features
Method
PICP
MPIW
Method
PICP
MPIW
Ens QRGB
0.9369
4.4402
Ens QRGB
0.9007
3.9463
Ens QRNN
0,8600
3.3970
Ens HTS
0.1864
0.4052
Ens HTS
0.8738
3.5094
Ens DUP
0.9101
3.4634
Ens MVE NN
0.9594
5.3387
Ens Bns-MVE NN
0.8811
2.4637
MDS NN
0.8380
3.3230
Ens- MDN NN
0.2350
0.4960
Bus-TTube NN
0.9289
4.0905
Tube NN
0.9152
2.9324
Table 7: Comparative analysis of the Tube NN against existing PI cstimation models for the STS task using TF ID- and SBERT embeddings with target coverage o 0.90. Ens QRGB: Ensemble Quantile Regression via Gradient Boosting; Ens QRNN: Ensemble Quantile Regression Neural Network; Ens HTS: Ensemble Heteroscedastic Two-Sided PI Kondall & Gal (2017;; Ens DUP: Ensemble Direct Uncertainty Prediction Zerva et al. (2022]: Zerva & Martins (2024); Ens- MVN: Ensemble Mean Variance Estimation
NNxx & Weigend 1994]), ) Ens MDN: Ensemble Mixture Density Network Bishop (1994).
5
Future Works
The Tube loss function evidently offers superior method for PI estimation in kernel machines and NN. Although we present natural extension of the Tube loss for constructing conformal prediction sets with finite-sample coverage guarantees, deeper theoretical investigation of its advantages within standard and adaptive conformal frameworks remains an important direction for future rescarch in view of its recalibration ability. Furthermore, it would also be worthwhile to implement the proposed methodology for probabilistic forecasting across a range of neural architectures employed in diverse application domains, including solar irradiance, cryptocurrency valuation, exchange rate prediction, stock market trends, ocean wave modeling, pollution estimation, and weather forecasting. many text processing applications, the target variable is inherently multidimensional. A compelling direction for future work would be to extend the proposed prodiction interval estimation methodology to accommodate such multidimensional targets.
Acknowledgments
The authors thank Shyam Saktawat, Manush Sanchela and Harsh Sawaliya for extending their help in conducting the numerical experiments.
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Funding
This work supported by the Smart Energy Learning Center at Dhirubhai Ambani University (Formerly DA-IICT), Gandhinagar through grant no.- CSSR25/BBSESAAAPPA//ELL
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A Proof of the Lemma 1
Let us assume that # - 12) = ki, # Hwilys - rug + (1 - r) = k2, and # - mî) = ka, where # represents the cardinality of set. In other words, k1, k2 and K3 represent the number of points on the boundary sets.
Thus, we oblain nı + ki + + li2 na K3 + 114
Let us choose, 8f E 00, E1); 82 E [0, E2), where E1 and are positive real numbers, such that e1 < min (1/4 Bil),
C2 < min [y/ - ,15ي and rc2 + (1 T)E < min
y - (rpp + (1 r)Hi)), which entails
# Hv: : 2n < y: S 12 00 # 12 < y: S 12 62) = 0, # 14 < y 5 + +,] = =0,
# 14 < 3 si -1)= =0, #(rØž (1- (1-r)p <;; S r(12 + 65) (1 () + 0,
#trp++ (1- r) < 5:(6 83) +1- 61)} O.
(x) + (1 (x)
(2(
11(x)
Figure 10
Given that (Hi,P²) is optimal solution, we have Em Li (Vt, 11 +5 - CT (V6, (²)) 2 0.
We now evaluate the difference of the sums for each of the following ten sets inducing a partition of R.
R1 {y:: :W::0 > > + 87), R2 = (,:11 < +1 55), R3 = = =1
R4 (1/ (1- <) <2), Rs (u: r)}+(11r)){ < yi S (((US++)))(--(((++)))
Rs {yi W/ rpg + - (1-r) R7 Ry (m: (1:: + < < (1 r) Rs : < 1 +
Rg {I 3: pi). R10 (31 : ys Hn
Denote the difference Ewer, Li (Vi, pi + 0 + +5) E L (M, by A. The simple
calculation leads to the following values of A,, i,2,. 10: A -1--a)m188 Ag = 0, = (a)kk A =
10:
(a (a)nz282, A, = 0, AS = (5) + (2rr)pī - (1 r)-r), 4, = (a))n3 A. = 0, Ag = (1 a)k38 A
(1 a)n
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Thus, obtain
a
Ci Ca ui 05 02)
Σ
-i- (al We: pi) (1 aan + + ( (a)k2(6²
i=1
+(2
r)u (1 r)p) (a))30) + (1- a)k3 a)n 2 0. (23)
If we assume 8 = 0 and 82 > O, the above inequality entails
n1
a n2z22 + h1 - (2 - r)4 - (1-r) (1-))
Given that yı ' are the realizations of a continuous random variable with no discrete probability mass, ki/n, k2/n + with probability 1 as n + co. Thus, as n + 00 with probability 1,
(24)
f12
Arguing on a similar line, if we consider 62 = 0 and 07 > in 23
then we have the following inoquality
(k3 n0
2 a
nz0 - k2 + (2 - r) - (1 - r)15)
As n + oo, we can state that the following inequality holds with probability 1,
(25)
n3 1-a
Again, starting with (ui, as an optimal solution, we have Σ.T (u μi - 1 - o2
Ems -i-a (Br,
(3-a (5) 2 0. For computing this, let us evaluate the difference of the sums for each of the following ten sets inducing a partition of R.
R² (y:: 3i:3 pb), R2 = (y 1/ = R2 {y: : 16- Su < pz) = (m: < <
82). Rt {y: W = =((()+++++)))))))(()))) r)p]) Rg {yr r(p) 65) (1- ([]-))(pi-- 1 S и <rb + r)
RG {uT <i < <r(n - 83) + ( - 88), Rg ( = ni). Rg = {ui : < v1 s 19 87).
11 = {y : y1 < < 85
Denoting ΣnER R. C (yi, |1 - 83) - - EyER, C (7. by A we obtain the following values
of / 1, .... 10: A = (11-a)mm6p A/ -a))k18 A/3 =0 .A. (a)n25 A/S -a ,/b-
0 4 (f) A/S (o)k30 ,
= 0, A -(1 a))4d].
Thus, we obtain
Σ CT-a + +5 +8)) - Σ ==1
CT (v. (3. = (1 (4))) (a)m20 (a))20
i=1
+(a)nz0t + (a)kso - (1 a)n 2 0.
(26)
Now, if we assume oj = 0 and 83 > 0 in 266), then we obtain,
hn + n2 + 12 a
which entails
(27)
f1 0
with probability 1.
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Published in Transactions on Machine Learning Research (04/2026)
Similarly, considering 62 0 and 57 0 in 26
obtain as oo,
a
(28)
n1
a
holds with probability 1.
Thus, 24 and 27
imply that as 71-000
n1
a
(29)
72
1
holds with probability
Similarly, combining (25) and 228
we obtain n oc,
n44
(30)
n2
a
holds with probability 1.
Thus, combining 29 and 300. we obtain, as 72 - 00;
711 + 714
(31)
72 + 723 a
holds with probability 1.
B Gradient Descent method for Tube loss-based kernel machines
The Tube loss-based kernel machine estimates pair of functions
M (x) = k( I) + no and u2(x): = Σ k(0,, + So.
(32)
where k(x,v) is positive definite kernel. For the sake of simplicity, we rewrite /4 and (2(x)) in vector form
A1 (z) K(AT x)n + no and pz(z)= == K(AF, ,,), Sg.
(33)
where A is the nx p data matrix containing n training points in R", K(AT, .x) 2(, x), (((n))))
7
B1
17
722
and B
B2
The Tube loss based kernel machines considers the problem
[n_
> - (nTTT BTB) + Σ CT-a (v:, (K(AT, ,,)) no), (K(AT, ,2:)8 + 90) )
min (9.R.no.96)
Jo B, 10, (0)
+6 Σ IK [B m/ (Bo no)l,
(34)
where CT-a the Tube loss function given in
12 with the parameter
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For given point (2s, ve), let us compute the gradient of Li (K(A", 22)) (2))))))))) + bi), (K("", ,2)3 + bu) )
first.
02; (v.. (K(AT (K(a"
08
(A,
if ((((A",=))++))) if (K(A", ,2)n m)) y (K(A",2;)) Bb) and yi < 2r) (1-))))) r))))
2.,++ m) < di < <K((A, -2))++8) and y4 > (K(A" ..) (1-)))))+++++))+)00 (r96 (1))))))
0,
if K(A",x;))) +x)y + >y
-11
- (11-o) (A,,)))
((TA ,x)8 + 90 y.
(vz.
830
if (K(A") (,dy 1 n1) < < (K(A7 =x)8 (B)) and > (K(A", a.)(18 1 (1 ()r))) (n50 1 (1 -r)o).
0,
if (K(AF, 2,)y + no) < y < (K(A" ,x:) + Bu) and y < (K(A2 (z)(r8 + +(1 (1-r)y) + (r80 + (1 (1)r)
if K(A", ([A", + 100 > V.
(1 a):
if K(A", 22:33 90 < y.
0CT a
(ve.( adntt)))) (K(Aa #0848))
84
if (K(TT, 25)n + no) yi (x)) + (6) and vs > (K(AT, 2:) + (1-))m) + (+0 (1-r)m))
ak(AA (1 ().())))))))
if (K(A", æz)9 + nu) < y (K(1F, xx;8 + (u) and ys < (K(A7 .x))(88 + (1 r)n) + (r80 + (1 r)
if K(AT,a,) + 70 >9
if (AA" .2,33 + 90 < y.
(v.. (K((T (onnTanntt)).... (s<Ca" (x<Ca 208+8))
O2T
0,
if (K(AT, (2(A" 70) < < <KKA"" +2:)8+ 96) and yi (K(A" (K(AT,2))(1)) (2:)(++ (1-)))) (1 (1 m)))
=2)n+100 (0)))
< y1 < (x() .=)8 and v. > (K(AT,2;)(B8) (2) (1
ar
(1-a),
if ((AT,2;))+ + 11 >.
0
((T,, ,a.88 + <.
For data point (zx, that lies exactly on the boundary of the PI or upon the surface r(K(AT, )x + Bo) (1 (K( zx2) + no), the Tube loss is not differentiable. In this case, the gradient is not unique, and any valid sub-gradient may be used for computation.
Now, for given data point (I,, we consider the width of PI tube (()((",,,)(( - -n) + (3o -0) and denote it as J(B. n, Bo 7o, x;) and compute
aJ Go No, vo) sign((K(AT (B:)((-n))++++ -Bo (00--n)))) (A,,)) (K xx:) 17) +Bo no) + 0
11)
03
(J n,, 3o, No, Fzi sign((K(A" (3: - -)) + +Bu - -n)) aBu
if (K ,z:)(8 - -m) + (g0 - no) * 0 (K(A", ,x,((( - -1) + (80 - 7o) / ir (K(A", xx,)(B 11) +B0 7o) 0
(J n7, fo. No, R.) sigg(((((A" ()))((((",,;)(( - -1) + (Bo - (A,,)
On
(3 n, Bo, no. F2)
-iinn((((A" (x, - 11) + (B0 -n))
Ono
For data point (I;, satisfying ((AT, x3: (AT,,:)) - 1) + (Bo - nn) = 0 the function J(3, So. no; )2; is not differentiable. In this case, the gradient is not unique, and any valid sub-gradient may be used for computation..
Now, we state gradient descent algorithm for the Tube loss based kernel machine problem.
Algorithm 2:-
Input:- Training Set T= {zz U1) 24 E RP, E R 1, 1,2,..m target coverage 1 - a E (0,1). shift parameter r,, re-calibration parameter 5, learning rate 7 and tol. Initialize:- .77 E R² and g e R.
Repeat
27

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Published in Transactions on Machine Learning Research (04/2026)
>i= 86 (5--.(u.(H (x(4 (KCA")
B(++1) B(kk) 7a (A
+
Σ
8.(8.n.8o.10..
8500
80 g(++1) 8/4) 76 EE=1
OC (v. (2:)8+88))
(K(A (a))tt))))
+8
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BJ(B So
0.%
K(A" m.)0+B00) (K(A" (2,))++))))))))
17 (k+1) (k) 76 (An
Ei=1
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(2)(t)
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86 2Cinn(mn (KIA", (.)8+20)) (K(A"
(R+1) 110 7he X
(R)
+
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(6)
8mg
(5)
g(--1) 1) 8(0)
1/a tol.
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7 1k-1) ()
1)-1) 76 (A)
In our implementation the gradient descent algorithm for the Tube loss kernel machine initially utilizes only the gradient of the Tube loss function. After a certain number of iterations, once we confirm that the upper bound of the PI, ((ATx:)) + So, has moved above the lower bound, ((A"x,;) x,)n + 1/, of PI, we incorporate the gradient of the PI tube width into the training of the Tube loss kernel machine.
C Additional Experimental Details
Tuning r' parameter:- Idcally, the tuning of the r parameter the tube loss function should align with the skewness of the distribution g/a. However, real-world datasets often exist in high dimensions. Consequently, for a specific value of z, there may be only a few of y; values in the training dataset, leading to potential inaccuracies in estimating the skewness of the distribution yla. In practical benchmark experiments, we have tuned the values for the tube loss machine from the set 0.2, 0.9].
Tuning 6 parameter:- Initially, we initialize 5 to zcro in the Tube loss-based machine for benchmark dataset experiments and aim to achieve the narrowest PI by adjusting the parameter, either moving the tube upwards or downwards. Once the r parameter is set and the Tube bass- machine achieves a coverage higher than the target on the validation set, we gradually fine-tune the parameter from the set (0.001, 0.005, 0.1, 0.15, to minimize the tube width while maintaining the desired target coverage
For the numerical results in Table 2, the tuned parameters for the Tube Loss summarized in Table
dataset boston concrete
(tube r)
delta (penaliy)
Ir
batch
dropout
epochs
0.5
0.03
0.05
10000
0.2
400
0.25
0.05
0.015
64
0.25
150
spt_actt
0.5
0.03
0.05
10000
0.2
400
energy
0.9
0.01
0.05
10000
0.2
kin8nm
0.1
0.01
0.05
10000
0.2
400
miami
0.3
0.05
0.05
10000
0.2
naval
0.3
0.05
0.05
10000
0.2
400
power
0.3
0.05
10000
0.2
400
protein
0.1
0.01
0.05
10000
0.2
400
sulfur
0.1
0.01
0.05
10000
0.2
400
wine
0.5
0.05
0.05
10000
0.2
yacht
0.5
0.03
0.01
10000
0.1
400
Table Best hyper-parameters for Tube Loss based NN for results generated Table 2.
IA MATLAB implementation of gradient descent with Tube loss is provided in supplementary material code.
28

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Published in Transactions on Machine Learning Research (04/2026)
Actuaa Pred
1.30
1.00
0.5
(a) SWH
300
200
(b) Sunspot
Auhunl
100
Electric Production
Actrual
220
Autuaal
860
100
160
120
(d) Beer Production
Figure 11: Probabilistic forecast of LSTM with proposed Tube loss function.
29

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Published in Transactions on Machine Learning Research (04/2026)
Dataset (Size)
LSTM Structure
Batch Size
Up Low
r,6
Window Size
Learning Rate
Q= LSTM
T-LSTM
Q- LSTM
T T-LSTM
Q-LSTM
T-LSTM
Sunspots (3268)
(.3() (28(0.2)
128
Electric Production (397)
61
32
16
0.98,0.03
0.5,0.01
12
0.01
0.001
Daily Female Birth (365)
100
64
128
0.98,0.03
0.5.0.01
0.01
0.005
SWH (2170)
300
300
0.98. 0.03
0.5,
100
0.001
0.001
Tempeerturre (3651)
16,8
300
300
0.98, 0.03
0.5, 0.01
100
0.001
0.0001
Beer Production (464)
61,32
0.0001
Table 9: Tuned parameter values for Q-LSTM and T-LSTM model for considered benchmark datasets in
Table The LSTM architecture (40.,) means that three hidden layers with neurons 128, 64 and 32 respectively and each hidden layer followed by drop out layer which are 0.4,0.3 and 0.2 respectively.
30


Reviews:
[{"review_id": "nHuDsN4LYm", "reviewer": "Reviewer_RuFc", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "### Summary:\nThis paper proposes a new loss function that simultaneously estimates prediction interval bounds. It introduces a tuning parameter to shift intervals for skewed distributions and uses linear penalties for robustness against outliers. Experiments cover kernel machines, NNs, and LSTMs.\n\n### Weakness:\n- The novelty appears increamental. The motivation is to get both bounds from one optimization, but in practice training two separate quantile models is already standard and computationally efficient.\n- In a related spirit, the proposed method needs a grid search for the hyperparameter $r$ through multiple runs, so it is also computationally expensive and not a single optimization as claimed.\n- Conformal prediction is a state-of-the-art baseline in uncertainty quantification, while the authors excludes them due to reasons like different framework or computation cost.", "claims_and_evidence_yn": "No", "claims_explanation": "- The claim of \"single optimization\" is misleading. The proposed method requires a grid search for the hyperparameter $r$ through runs, which makes the total training cost higher than standard quantile regression.\n- The authors claimed that conformal prediction requires retraining from scratch. However, standard methods are efficient post-hoc steps that do not require retraining.\n- Computation-time reporting is not comparable as presented. For a fair comparison, the paper should report end-to-end computation time including hyperparameter tuning, especially since the proposed method appears to require extensive tuning across multiple runs.", "audience_yn": "Yes", "audience_explanation": "Uncertainty quantification of machine-learning predictions is an interesting topic, especially for safety-critical applications.", "requested_changes": "- Compare with the state-of-the-art conformal baseline.\n- Report computation time including tuning for fair comparison. The proposed method requires tuning across multiple runs.\n- Provide stronger theoretical or empirical justification for the proposed method.", "broader_impact_concerns": "None", "additional_comments": "None", "extra_scores": {}, "extra_text": {}}, {"review_id": "20zQO8pF4Z", "reviewer": "Reviewer_QADY", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors proposed a new loss function, called the tube loss, for directly learning prediction intervals. The tube loss can be optimized using gradient-based methods, and is linear in the difference between the observation and the upper/lower bound. The resulting interval from the tube loss does not encounter crossing issues and can be adjusted to the density of the observed responses in the training set. They also allow users to tune the interval according to their preference for coverage or the length of the interval. The authors also theoretically proved that the resulting interval will asymptotically attain the nominal coverage. Empirical experiments are conducted to support their findings.", "claims_and_evidence_yn": "Yes", "claims_explanation": "I chose \"yes\", but I want to emphasize that the paper has too many typos and notation inconsistencies (see details later) to judge it effectively. I firmly believe that the paper should be rewritten to be better assessed. The empirical results in this paper are promising.", "audience_yn": "Yes", "audience_explanation": "Distribution-free prediction intervals are of interest to practitioners in both statistics and machine learning. Methods based on conformal inference or based on quantiles are quite popular nowadays. The tube loss proposed in this paper seems to have several interesting features to serve as a promising direction. In summary, I believe there should be individuals in TMLR's audience interested in this paper.", "requested_changes": "The paper has some merit. However, in my own opinion, it requires significant changes.\n\n(1) There are many grammatical errors. For example, on page 6, there is one sentence, \"... some good theoretical properties, such as, it attains coverage, and it is unbiased and consistent\", containing multiple grammatical issues. It is the authors' responsibility to make a significant effort to improve the writing.\n\n(2) There are also many notation inconsistencies. For example, again on page 6, equation (12), there is no input for $\\mathcal{L}_{1 - \\alpha}^{RQR}$. \n\nI also highly recommend that the authors make some effort to make the mathematical notation used in the paper aesthetically appealing. For instance, $\\mathcal{L}_{LUBE}$ looks quite annoying.\n\n(3) In Figure 1, what is $t$? A visual comparison between the tube loss and the RQR loss is more recommended than displaying the RQR loss in almost the end of the paper.\n\n(4) The theoretical results should be stated with more care. For instance, in both Lemma 1 and Proposition 1, some of the notations are not defined in the paper, such as $n_{1}, \\cdots, n_{4}$.\n\n(5) In section 3.3, what is $\\mathcal{M}$? Should it be $\\mathcal{L}$ instead?\n\n(6) Numerical comparisons should be more comprehensive. In the current version, the tube loss-based interval is only compared against one or two alternative methods.\n\n(7) The theoretical results are somewhat preliminary. This is not a very serious issue for TMLR. But I would like to see that the authors could discuss more caveats about the theoretical results. For instance, when gradient-based methods are used, even though the conditional density satisfies certain smoothness conditions, it is not necessarily the case that the eventual output converges to the true quantile regression functions, and thus the cited results of Takeuchi et al. may not be directly applicable. Proposition 1 should therefore be stated with more caution.", "broader_impact_concerns": "NA.", "additional_comments": "NA.", "extra_scores": {}, "extra_text": {}}, {"review_id": "UfmfvBEr13", "reviewer": "Reviewer_MNi4", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces Tube Loss, a new loss function for directly learning prediction intervals in regression models. The method estimates the lower and upper bounds simultaneously in a single optimization problem and can be trained efficiently using gradient-based methods. A tunable parameter allows the interval to shift along the response distribution, which is particularly useful for asymmetric or skewed data. The authors provide theoretical results showing that the learned intervals asymptotically attain the target coverage level. Empirical results across kernel methods, neural networks, and probabilistic forecasting demonstrate improved interval quality compared to existing approaches.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The submission provides both theoretical and empirical evidence to support its main claims. The proposed loss function is clearly defined, its asymptotic coverage properties are analyzed under explicit assumptions, and the empirical results across multiple model classes and datasets are generally consistent with the stated objectives. While the guarantees are asymptotic rather than finite-sample, the claims are framed accordingly and do not overreach the presented analysis.", "audience_yn": "Yes", "audience_explanation": "The paper addresses prediction interval estimation, which is a core topic in uncertainty quantification and regression modeling. The proposed loss-based approach offers a practical alternative to quantile-based and ensemble-based methods and may be of interest to researchers working on uncertainty-aware learning, probabilistic forecasting, and applied machine learning.", "requested_changes": "Major points\n\n1. Broader and stronger baselines for uncertainty quantification\n\nThe experimental comparison in the neural network setting is currently limited to a relatively narrow set of PI-specific methods. There exist several widely used and competitive uncertainty quantification approaches that are not included, such as Monte Carlo Dropout [1], deep ensembles, and likelihood-based regression models that learn Gaussian (or mixture) predictive distributions by estimating means and variances [2]. These methods are standard baselines in the uncertainty estimation literature and should be discussed in the Related Work section and, where feasible, included in the experimental comparisons to strengthen the empirical claims.\n\n2. Evaluation on non–time-series regression tasks beyond standard benchmarks\n\nWhile the paper evaluates kernel regression, tabular benchmarks, and time-series forecasting, it would be valuable to demonstrate the effectiveness of the proposed method on regression problems with qualitatively different structures. For example, regression-based tasks in robotics (e.g., behavior cloning or inverse dynamics learning) or computer vision (e.g., monocular depth estimation) are common applications where uncertainty estimation is important and data characteristics differ substantially from time-series settings.\n\n3. Interpretation of the parameter $r$ in relation to statistical functionals\n\nThe shift parameter $r$ plays a central role in the proposed method, yet its statistical interpretation remains somewhat heuristic. It would significantly improve clarity to discuss how different values of $r$ relate to classical point estimators or distributional characteristics, such as the mean, median, or mode of the conditional distribution. Even an intuitive or approximate characterization (e.g., which regimes of $r$ correspond to more expectation-like or median-like behavior) would help readers better understand and interpret the method.\n\nMinor\n\n1. Notation consistency and clarity\nThe paper uses notation inconsistently in several places (e.g., bold versus italic symbols for inputs, reuse of symbols with different meanings, switching between $\\alpha$ and other symbols for coverage). A careful pass to standardize notation and clearly define conventions would improve readability and reduce potential confusion for readers.\n\n[1] Gal, Yarin, and Zoubin Ghahramani. \"Dropout as a bayesian approximation: Representing model uncertainty in deep learning.\" international conference on machine learning. PMLR, 2016.\n[2] Kendall, Alex, and Yarin Gal. \"What uncertainties do we need in bayesian deep learning for computer vision?.\" Advances in neural information processing systems 30 (2017).", "broader_impact_concerns": "The paper focuses on methodological advances in prediction interval estimation and does not raise immediate ethical concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)