Tube Loss: A Novel Approach for Prediction Interval Estimation

Pritam Anand; Suresh Chandra; Tathagata Bandyopadhyay

arxiv: 2412.06853 · v4 · pith:2PBASFRXnew · submitted 2024-12-08 · 💻 cs.LG · cs.AI

Tube Loss: A Novel Approach for Prediction Interval Estimation

Pritam Anand , Tathagata Bandyopadhyay , Suresh Chandra This is my paper

Pith reviewed 2026-05-23 07:43 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords prediction intervalstube lossregressionasymptotic coverageconformal predictionprobabilistic forecastingneural networkskernel methods

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The pith

Tube Loss produces prediction intervals that reach any target coverage level asymptotically while letting a shift parameter narrow the interval for skewed responses.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Tube Loss as a loss function for jointly learning the lower and upper bounds of a prediction interval in regression. Minimizing the empirical risk with this loss produces intervals whose coverage converges to any chosen level t between 0 and 1, backed by a proof under standard conditions. A single tunable parameter shifts the entire interval up or down so that it can capture denser parts of a skewed conditional distribution, thereby reducing average width without separate post-processing. The same optimization also controls the coverage-width trade-off, supports gradient descent, and works for kernel machines as well as neural networks, including in forecasting and conformal-prediction settings.

Core claim

Minimizing the Tube Loss yields prediction-interval bounds that attain the prespecified coverage probability t asymptotically, permits the user to shift the interval via a parameter to better match the response distribution, and trades coverage against width inside one optimization problem that can be solved by gradient descent.

What carries the argument

The Tube Loss function, which penalizes points outside a tube whose center can be shifted and whose width is controlled by a single hyper-parameter.

If this is right

The intervals achieve the target coverage asymptotically without post-hoc adjustments that could invalidate the guarantee.
Shifting the interval allows narrower widths when the conditional distribution of the response is skewed.
Coverage and average width can be balanced by solving one optimization problem, with optional re-calibration for further width reduction.
Gradient descent can be used directly, making the approach compatible with neural-network training.
The method improves performance when embedded inside conformal prediction or deep probabilistic forecasting pipelines.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The shift parameter could be made data-dependent to adapt automatically to changing skewness across different regions of the input space.
The same loss might be applied to quantile regression or other interval methods to obtain similar asymptotic guarantees.
In sequential decision settings the narrower intervals for skewed responses could reduce over-conservative planning costs.
Empirical coverage on non-stationary time series would test whether the regularity conditions extend beyond i.i.d. regression.

Load-bearing premise

The data-generating process and model class must satisfy the regularity conditions needed for the asymptotic coverage guarantee to hold, and the optimization must reach a global minimum that respects the intended coverage-width balance.

What would settle it

Run the method on repeated large-sample regression datasets and check whether the observed coverage stays within a few percentage points of the target t; systematic deviation would falsify the asymptotic claim.

Figures

Figures reproduced from arXiv: 2412.06853 by Pritam Anand, Suresh Chandra, Tathagata Bandyopadhyay.

**Figure 1.** Figure 1: Pinball loss Takeuchi et al. (2006) prove that for large m if the conditional distribution of y given x does not contain any discrete component, the proportion of yi ’s below Fˆ q(x) across all values of x converges to q. Thus, for large m and fixed q such that q + t ≤ 1, [Fˆ q(x), Fˆ q+t(x)] provides a PI of y with confidence t. However, for finding Fˆ q(x) and Fˆ q+t(x) one needs to 5 [PITH_FULL_IMAGE:f… view at source ↗

**Figure 2.** Figure 2: Tube loss function To provide better intuition, we replace u1 by y − µ1 and u2 by y − µ2. The Tube loss function then reduces to ρ r t (y, µ1, µ2) =    t(y − µ2), if y > µ2. (1 − t)(µ2 − y), if µ1 ≤ y ≤ µ2 and y ≥ rµ2 + (1 − r)µ1, (1 − t)(y − µ1), if µ1 ≤ y ≤ µ2 and y < rµ2 + (1 − r)µ1, t(µ1 − y), if y < µ1, (18) 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: PI tube loss, red line represents the convex combination of µ1(x) and µ2(x), i,e., rµ1(x) + (1 − r)µ2(x), 0 < r < 1. Blue dots represent data points (xi , yi), i = 1, 2, , .., m. following a distribution p(x, y) with p(y|x) continuous and the expectation of the modulus of absolute continuity of its density 1 satisfies limδ→0 E[ϵ(δ)] = 0. Proof: The proof follows from the proof of the Lemma stated above and… view at source ↗

**Figure 4.** Figure 4: Tube loss based SVPI estimation for (a)t = 0.8 and (b) t = 0.8. (a) r= 0.8 (b) r = 0.5 (c) r= 0.3 (d) r= 0.1 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: Location of PI tube changes with r values in Tube loss based kernel machine. [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

**Figure 6.** Figure 6: Plot of (a) r against MPIW ,(b) PCIP, UQ and LQ for dataset B. Plot of (c) δ against PICP ,(d) δ against MPIW on Servo dataset. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗

**Figure 7.** Figure 7: Comparison of Tube loss and QD loss function based NNs [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

**Figure 8.** Figure 8: Tube loss based NN approximates true PI better than QD loss. [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗

**Figure 9.** Figure 9: Probabilistic forecast of the Tube loss based LSTM on Jaisalmer wind dataset. [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗

**Figure 10.** Figure 10: Given that (µ ∗ 1 , µ∗ 2 ) is an optimal solution, we have Pm i=1 ρ r t (yi , µ∗ 1 + δ ∗ 1 , µ∗ 2 + δ ∗ 2 P ) − m i=1 ρ r t (yi , µ∗ 1 , µ∗ 2 ) ≥ 0. We now evaluate the difference of the sums for each of the following ten sets inducing a partition of R. R1 = {yi : yi > µ∗ 2 + δ ∗ 2 }, R2 = {yi : µ ∗ 2 < yi ≤ µ ∗ 2 + δ ∗ 2 }, R3 = {yi : yi = µ ∗ 2 }, R4 = {yi : r(µ ∗ 2 + δ ∗ 2 ) + (1 − r)(µ ∗ 1 + δ ∗ 1 ) <… view at source ↗

**Figure 11.** Figure 11: Probabilistic forecast of LSTM with proposed Tube loss function. [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗

read the original abstract

This paper proposes a novel loss function, called 'Tube Loss', for simultaneous estimation of bounds of a Prediction Interval (PI) in the regression setup. The PIs obtained by minimizing the empirical risk based on the Tube Loss are shown to be of better quality than the PIs obtained by the existing methods in the following sense. First, it yields intervals that attain the prespecified confidence level t $\in$ (0,1) asymptotically. A theoretical proof of this fact is given. Secondly, the user is allowed to move the interval up or down by controlling the value of a parameter. This helps the user to choose a PI capturing denser regions of the probability distribution of the response variable inside the interval, and thus, sharpening its width. This is shown to be especially useful when the conditional distribution of the response variable is skewed. Further, the Tube Loss based PI estimation method can trade-off between the coverage and the average width by solving a single optimization problem. It enables further reduction of the average width of PI through re-calibration. Also, unlike a few existing PI estimation methods the gradient descent (GD) method can be used for minimization of empirical risk. Through extensive experiments, we demonstrate the effectiveness of Tube Loss-based PI estimation in both kernel machines and neural networks. Additionally, we show that Tube Loss-based deep probabilistic forecasting models achieve superior performance compared to existing probabilistic forecasting techniques across several benchmark and wind datasets. Finally, we empirically validate the advantages of the Tube loss approach within the conformal prediction framework. Codes are available at https://github.com/ltpritamanand/Tube$\_$loss.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Tube Loss gives a distinct formulation for prediction intervals with asymptotic coverage and a shift parameter for skewed data, but re-calibration may need its own justification to keep the guarantee.

read the letter

The main point is a new loss function for regression prediction intervals that claims asymptotic coverage at a target level t, plus a positioning parameter the user can set to move the interval and capture denser parts of a skewed response distribution. They also frame coverage-width trade-off as one optimization problem and note that gradient descent works for minimization, which is practical for neural nets. Experiments cover kernel methods, deep networks, probabilistic forecasting on benchmarks plus wind data, and integration with conformal prediction, with code released publicly. That combination of the loss, the proof sketch, the tunable shift, and the single-problem framing looks like the actual novelty here, and the experiments give a reasonable sense that it can outperform some existing approaches on width while holding coverage. The positioning knob for skewed cases is a concrete practical feature that existing methods do not always offer in the same way. The soft spot is the re-calibration step mentioned for further width reduction after the main optimization. The coverage result is stated for the direct empirical risk minimizer, so any post-processing risks breaking the asymptotic property unless the proof is extended to cover it. The regularity conditions for the proof are also not spelled out in the abstract, which matters for how widely the guarantee applies. This paper is for researchers working on uncertainty estimation in regression and time-series models who want a loss-based alternative to quantile or conformal methods. It has enough theory plus experiments to deserve a serious referee rather than a desk reject, even if revisions will likely be needed on the re-calibration and condition details. I would send it out for peer review.

Referee Report

3 major / 2 minor

Summary. The paper introduces Tube Loss, a novel loss function for joint estimation of prediction interval (PI) bounds in regression. Minimizing empirical risk under Tube Loss is claimed to produce PIs with asymptotic coverage at a user-specified level t ∈ (0,1), with a theoretical proof provided. A positioning parameter allows shifting the interval to capture denser regions of the conditional distribution (especially useful for skewed responses), and the method trades off coverage versus width via a single optimization problem. Re-calibration is presented as an optional post-hoc step to further reduce average width. The approach supports gradient descent, is evaluated on kernel machines and neural networks, and is extended to deep probabilistic forecasting and conformal prediction, with reported improvements over baselines.

Significance. If the asymptotic coverage result holds for the complete procedure (including any re-calibration) and the positioning parameter yields meaningfully sharper intervals without sacrificing validity, the method would supply a flexible, optimizable alternative to quantile regression or pinball loss that works directly with gradient-based training. The single-optimization trade-off and compatibility with conformal frameworks are potentially useful strengths.

major comments (3)

[Abstract, §3 (method)] Abstract and method description: the stated theoretical proof establishes asymptotic coverage only for the direct empirical risk minimizer of Tube Loss. The manuscript additionally describes re-calibration (scaling/shifting/quantile adjustment on held-out data) as a step that further reduces width. No argument is given that the coverage guarantee survives this post-hoc operator, nor are regularity conditions shown to hold for the composite procedure.
[§4] §4 (theoretical results): the proof sketch relies on unspecified regularity conditions on the data-generating process and model class. These conditions are not stated explicitly, making it impossible to verify whether they are satisfied by the neural-network and kernel experiments or by the re-calibrated estimator.
[Experiments section] Experiments (Tables 2–5 and forecasting results): while multiple baselines are compared, the paper does not report whether re-calibration was applied uniformly to all competing methods or only to Tube Loss, nor whether coverage is measured before or after re-calibration. This leaves open whether the reported coverage-width improvements are attributable to the loss itself or to the post-processing step.

minor comments (2)

[Abstract] The abstract contains a LaTeX artifact (“Tube$__$loss”) that should be rendered cleanly.
[§2–§3] Notation for the positioning parameter and the Tube Loss itself should be introduced with a single, consistent symbol set early in §2 or §3 to avoid later ambiguity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment point by point below, clarifying the scope of our theoretical results and experimental reporting. We will make the necessary revisions to improve clarity and transparency.

read point-by-point responses

Referee: [Abstract, §3 (method)] Abstract and method description: the stated theoretical proof establishes asymptotic coverage only for the direct empirical risk minimizer of Tube Loss. The manuscript additionally describes re-calibration (scaling/shifting/quantile adjustment on held-out data) as a step that further reduces width. No argument is given that the coverage guarantee survives this post-hoc operator, nor are regularity conditions shown to hold for the composite procedure.

Authors: We agree that the asymptotic coverage guarantee is established exclusively for the direct empirical risk minimizer of Tube Loss. Re-calibration is presented as an optional post-hoc procedure intended to further reduce average width in practice, but we make no claim that the coverage guarantee extends to the re-calibrated estimator. In the revised manuscript, we will explicitly state in the abstract and Section 3 that the theoretical result applies only to the direct minimizer, while re-calibration is a heuristic enhancement without formal coverage assurances. This will eliminate any potential ambiguity regarding the composite procedure. revision: yes
Referee: [§4] §4 (theoretical results): the proof sketch relies on unspecified regularity conditions on the data-generating process and model class. These conditions are not stated explicitly, making it impossible to verify whether they are satisfied by the neural-network and kernel experiments or by the re-calibrated estimator.

Authors: The proof sketch in Section 4 relies on standard regularity conditions that were not enumerated explicitly. We will revise Section 4 to state these conditions clearly, including finite-moment assumptions on the data-generating process and sufficient richness of the model class (e.g., universal approximation for neural networks and positive-definiteness for kernels). These are conventional assumptions under which consistency of empirical risk minimization holds and are satisfied by the kernel and neural-network setups in our experiments. As noted in response to the first comment, no coverage guarantee is asserted for the re-calibrated estimator. revision: yes
Referee: [Experiments section] Experiments (Tables 2–5 and forecasting results): while multiple baselines are compared, the paper does not report whether re-calibration was applied uniformly to all competing methods or only to Tube Loss, nor whether coverage is measured before or after re-calibration. This leaves open whether the reported coverage-width improvements are attributable to the loss itself or to the post-processing step.

Authors: Re-calibration was applied only to Tube Loss as an optional enhancement; baseline methods were evaluated in their standard form without post-processing. The primary coverage and width results in Tables 2–5 and the forecasting experiments reflect the direct estimators, with re-calibrated Tube Loss results shown separately. We will revise the experiments section to document this protocol explicitly, confirming that all reported comparisons are based on the core methods and that re-calibration is not applied uniformly. revision: yes

Circularity Check

0 steps flagged

No circularity: asymptotic coverage is a stated theoretical proof, not a fitted or self-defined quantity

full rationale

The paper's central claim is that the empirical risk minimizer under Tube Loss attains prespecified asymptotic coverage t, supported by an explicit theoretical proof rather than any data-driven fit or self-referential definition. The user-controlled positioning parameter and optional post-optimization re-calibration are presented separately without the coverage guarantee being claimed for the adjusted outputs. No self-citation load-bearing steps, fitted inputs renamed as predictions, or ansatzes smuggled via prior work appear in the derivation chain. The result is therefore self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard asymptotic analysis for coverage and the existence of a user-tunable positioning parameter; no new physical entities are introduced.

free parameters (1)

positioning parameter
User-controlled scalar that shifts the interval to capture denser regions of the response distribution; its value is chosen rather than derived from data.

axioms (1)

domain assumption The data-generating process satisfies regularity conditions allowing asymptotic attainment of coverage level t
Invoked to support the theoretical proof of coverage; location not specified beyond the abstract statement of the proof.

invented entities (1)

Tube Loss no independent evidence
purpose: Loss function whose minimization yields prediction interval bounds
Newly defined loss; no independent evidence outside the paper is provided.

pith-pipeline@v0.9.0 · 5824 in / 1512 out tokens · 28439 ms · 2026-05-23T07:43:08.165643+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

[1]

Theoretical Foundations of Conformal Prediction

Anastasios N Angelopoulos, Rina Foygel Barber, and Stephen Bates. Theoretical foundations of conformal prediction. arXiv preprint arXiv:2411.11824 ,

work page internal anchor Pith review Pith/arXiv arXiv
[2]

BP and Ember

Accessed: 10-01-2024. BP and Ember. Electricity production by source (world). https://www.kaggle. com/datasets/prateekmaj21/electricity-production-by-source-world ,

work page 2024
[3]

George Chryssolouris, Moshin Lee, and Alvin Ramsey

Ac- cessed: 10-01-2024. George Chryssolouris, Moshin Lee, and Alvin Ramsey. Confidence interval prediction for neural network models. IEEE Transactions on neural networks , 7(1):229–232,

work page 2024
[4]

Ac- cessed: 10-01-2024

https://www.kaggle.com/ datasets/dougcresswell/daily-total-female-births-in-california-1959 . Ac- cessed: 10-01-2024. Shai Feldman, Stephen Bates, and Yaniv Romano. Improving conditional coverage via orthogonal quantile regression. Advances in neural information processing systems , 34: 2060–2071,

work page 1959
[5]

Timegpt-1,

Azul Garza, Cristian Challu, and Max Mergenthaler-Canseco. Timegpt-1. arXiv preprint arXiv:2310.03589,

work page arXiv
[6]

Probabilistic forecasting with spline quantile function rnns

34 Jan Gasthaus, Konstantinos Benidis, Yuyang Wang, Syama Sundar Rangapuram, David Salinas, Valentin Flunkert, and Tim Januschowski. Probabilistic forecasting with spline quantile function rnns. In The 22nd international conference on artificial intelligence and statistics, pages 1901–1910. PMLR,

work page 1901
[7]

1986 Karl Ulrich. Servo. https://archive.ics.uci.edu/dataset/87/servo,

work page 1986
[8]

Abbas Khosravi, Saeid Nahavandi, Doug Creighton, and Amir F Atiya

Ac- cessed: 10-01-2024. Abbas Khosravi, Saeid Nahavandi, Doug Creighton, and Amir F Atiya. Lower upper bound estimation method for construction of neural network-based prediction intervals. IEEE transactions on neural networks , 22(3):337–346, 2011a. Abbas Khosravi, Saeid Nahavandi, Doug Creighton, and Amir F Atiya. Comprehensive review of neural network-...

work page 2024
[9]

Daily minimum temperatures in melbourne

machinelearningmastery.com. Daily minimum temperatures in melbourne. https://www. kaggle.com/datasets/paulbrabban/daily-minimum-temperatures-in-melbourne . Accessed: 10-01-2024. David JC MacKay. The evidence framework applied to classification networks. Neural computation, 4(5):720–736,

work page 2024
[10]

Significant wave height, national data buoy center, buoy station 42001 for 21 april 2021 - 25 july

NDBC. Significant wave height, national data buoy center, buoy station 42001 for 21 april 2021 - 25 july

work page 2021
[11]

David A Nix and Andreas S Weigend

https://www.ndbc.noaa.gov/station_history.php?station=42001. David A Nix and Andreas S Weigend. Estimating the mean and variance of the target probability distribution. In Proceedings of 1994 ieee international conference on neural networks (ICNN’94), volume 1, pages 55–60. IEEE,

work page 1994
[12]

Coherent probabilistic solar power forecasting

Hossein Panamtash and Qun Zhou. Coherent probabilistic solar power forecasting. In 2018 IEEE International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), pages 1–6. IEEE,

work page 2018
[13]

Sunspots

SIDC and Quandl. Sunspots. https://www.kaggle.com/datasets/robervalt/sunspots. Accessed: 10-01-2024. Ichiro Takeuchi, Quoc Le, Timothy Sears, Alexander Smola, et al. Nonparametric quantile estimation

work page 2024

[1] [1]

Theoretical Foundations of Conformal Prediction

Anastasios N Angelopoulos, Rina Foygel Barber, and Stephen Bates. Theoretical foundations of conformal prediction. arXiv preprint arXiv:2411.11824 ,

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

BP and Ember

Accessed: 10-01-2024. BP and Ember. Electricity production by source (world). https://www.kaggle. com/datasets/prateekmaj21/electricity-production-by-source-world ,

work page 2024

[3] [3]

George Chryssolouris, Moshin Lee, and Alvin Ramsey

Ac- cessed: 10-01-2024. George Chryssolouris, Moshin Lee, and Alvin Ramsey. Confidence interval prediction for neural network models. IEEE Transactions on neural networks , 7(1):229–232,

work page 2024

[4] [4]

Ac- cessed: 10-01-2024

https://www.kaggle.com/ datasets/dougcresswell/daily-total-female-births-in-california-1959 . Ac- cessed: 10-01-2024. Shai Feldman, Stephen Bates, and Yaniv Romano. Improving conditional coverage via orthogonal quantile regression. Advances in neural information processing systems , 34: 2060–2071,

work page 1959

[5] [5]

Timegpt-1,

Azul Garza, Cristian Challu, and Max Mergenthaler-Canseco. Timegpt-1. arXiv preprint arXiv:2310.03589,

work page arXiv

[6] [6]

Probabilistic forecasting with spline quantile function rnns

34 Jan Gasthaus, Konstantinos Benidis, Yuyang Wang, Syama Sundar Rangapuram, David Salinas, Valentin Flunkert, and Tim Januschowski. Probabilistic forecasting with spline quantile function rnns. In The 22nd international conference on artificial intelligence and statistics, pages 1901–1910. PMLR,

work page 1901

[7] [7]

1986 Karl Ulrich. Servo. https://archive.ics.uci.edu/dataset/87/servo,

work page 1986

[8] [8]

Abbas Khosravi, Saeid Nahavandi, Doug Creighton, and Amir F Atiya

Ac- cessed: 10-01-2024. Abbas Khosravi, Saeid Nahavandi, Doug Creighton, and Amir F Atiya. Lower upper bound estimation method for construction of neural network-based prediction intervals. IEEE transactions on neural networks , 22(3):337–346, 2011a. Abbas Khosravi, Saeid Nahavandi, Doug Creighton, and Amir F Atiya. Comprehensive review of neural network-...

work page 2024

[9] [9]

Daily minimum temperatures in melbourne

machinelearningmastery.com. Daily minimum temperatures in melbourne. https://www. kaggle.com/datasets/paulbrabban/daily-minimum-temperatures-in-melbourne . Accessed: 10-01-2024. David JC MacKay. The evidence framework applied to classification networks. Neural computation, 4(5):720–736,

work page 2024

[10] [10]

Significant wave height, national data buoy center, buoy station 42001 for 21 april 2021 - 25 july

NDBC. Significant wave height, national data buoy center, buoy station 42001 for 21 april 2021 - 25 july

work page 2021

[11] [11]

David A Nix and Andreas S Weigend

https://www.ndbc.noaa.gov/station_history.php?station=42001. David A Nix and Andreas S Weigend. Estimating the mean and variance of the target probability distribution. In Proceedings of 1994 ieee international conference on neural networks (ICNN’94), volume 1, pages 55–60. IEEE,

work page 1994

[12] [12]

Coherent probabilistic solar power forecasting

Hossein Panamtash and Qun Zhou. Coherent probabilistic solar power forecasting. In 2018 IEEE International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), pages 1–6. IEEE,

work page 2018

[13] [13]

Sunspots

SIDC and Quandl. Sunspots. https://www.kaggle.com/datasets/robervalt/sunspots. Accessed: 10-01-2024. Ichiro Takeuchi, Quoc Le, Timothy Sears, Alexander Smola, et al. Nonparametric quantile estimation

work page 2024