Quadratic Direct Forecast for Training Multi-Step Time-Series Forecast Models
Authors: Eric Wang
Licheng Pan
Yuan Lu
Zi Ciu Chan
Tianqiao Liu
Shuting He
Zhixuan Chu
Qingsong Wen
Haoxuan Li
Zhouchen Lin
Venue: ICLR 2026 Poster (2026)
Keywords: Time-series
time-series forecast
Primary area: learning on time series and dynamical systems

Abstract: The design of training objective is central to training time-series forecasting models. Existing training objectives such as mean squared error mostly treat each future step as an independent, equally weighted task, which we found leading to the following two issues: (1) overlook the *label autocorrelation effect* among future steps, leading to biased training objective; (2) fail to set *heterogeneous task weights* for different forecasting tasks corresponding to varying future steps, limiting the forecasting performance. To fill this gap, we propose a novel quadratic-form weighted training objective, addressing both of the issues simultaneously. Specifically, the off-diagonal elements of the weighting matrix account for the label autocorrelation effect, whereas the non-uniform diagonals are expected to match the most preferable weights of the forecasting tasks with varying future steps. To achieve this, we propose a Quadratic Direct Forecast (QDF) learning algorithm, which trains the forecast model using the adaptively updated quadratic-form weighting matrix. Experiments show that our QDF effectively improves performance of various forecast models, achieving state-of-the-art results. Code is available at https://anonymous.4open.science/r/QDF-8937.

Published as a conference paper at ICLR 2026
QUADRATIC DIRECT FORECAST FOR TRAINING MULTI- STEP TIME-SERIES FORECAST MODELS
Hao Wang'. Licheng Pan'
Yuan Lu'
Zhichao Chen
Tiangiao Liu Shuting He
Zhixuan Chu Qingsong Wen" Haoxuan List
Zhouchen 3
Xiaohongshu Inc.
Coolleee of Control Science and Technology, Zhejiang University
Staate Key Lab of General AI, School of Intelligence Science and Technology, Peking University
*College of Engineering, Purdue University
School of Computing and Artificial Intelligence, Shanghai University Finance and Economics
"College of Computer Science and Technology, Zhejiang University
"Squirrel AI CCetter for Data Science, Peking University
9 Institute for Artificial Intelligence, Peking University
ABSTRACT
The design of learning objectives is central to training time-series forecasting models. Existing learning objectives such as mean squared error mostly treat each future step as an independent, equally weighted task, which leads to the following two challenges: (1) they overlook the label autocorrelation effect among future steps, leading to biased learning objectives; (2) they fail to set heteroge- neous lask weights for different forecasting tasks corresponding varying future steps, limiting the forecasting performance. To fill this gap, we propose a novel quadratic-form weighted learning objective, addressing both issues simultaneously. Specifically, the off-diagonal elements of the weighting matrix account for the label autocorrelation effect, whereas the non-uniform diagonals are expected match the preferred weights of the forecasting tasks with varying future steps. On this basis, we propose a Meta-learning Direct Forecast (MetaDF) learning algo- rithm, which trains the forecast model using the adaptively updated quadratic-form weighting matrix. Experiments show that our MetaDF effectively improves the performance of various forecast models, achieving state-of-thee the-art results. Code is available at ht tpts:p/// //ghttutbb////////// com/
com/Master Master- CiateerrPPC//OD
INTRODUCTION
Time-series forecasting, which involves predicting future values from past observations, is founda- tional to a wide range of applications, including meteorological prediction (Bi et al., 2023), financial stock forecasting (Li al., 2025a), and robotic trajectory forecasting (Fan et al., 2023). In the context of deep learning, the development of robust forecasting models relies on two crucial com- ponents (Wang al., 2025d): (1) the design of neural architectures for forecusting and (2) the formulation of suitable learning objectives for model training. Both present distinct challenges.
Recent research has focused intensively on the first aspect, namely, neural architecture design. The principal challenge lies in efficiently capturing the autocorrelation structures in the history sequence. variety of architectures have been proposed (Wu et al., 2023; Luo and Wang, 2024; Gu al., 2021). One exemplar would be Transformer models that employ self-attention to model autocorrelation and scale effectively (Liu al., 2024; Nie al., 2023; Piao al., 2024). Another rapidly developing direction would be linear models, which use linear projections model autocorrelation and demon- strate competitive performance (Lin et al., 2024; Zeng et al., 2023; Yue et al., 2025). These advances showcase the fast- past-paced evolution of model architectures for time-scries forecasting.
In contrast, the formulation of learning objectives remains relatively underexplored (Li et al., 2025b; Qiu et al., 2025; Kudrat et al., 2025). Most recent studies resort to mean squared error (MSE)
1 Corresponding authors.

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Published as a conference paper at ICLR 2026
as the learning objective (Lin al., 2025; 2024; Liu al., 2024). However, MSE overlooks the autocorrelation effect present in label sequences, which renders it a biased objective (Wang et al., 2025e;d). Additionally, it assigns equal weights to all forecasting tasks with varying future steps, ignoring the potential of a heterogeneous weighting scheme. As a result, the learning objective design of forecast models is challenged by the label autocorrelation effect and heterogeneous task weights, which are not fully addressed by existing methods.
In this work, we first propose a novel quadratic-fome weighted learning objective that simultaneously tackles both issues. Specifically, the off-diagonal elements of the weighting matrix model the label autocorrelation effect, while the non-uniform diagonal elements enable the assignment of heterogeneous task weights to different future steps. Building on this, we introduce the Quadratic Direct Forecast (QDF) learning algorithm, which trains the forecasting model using an adaptively updated quadratic-form weighting matrix. Our main contributions are summarized as follows:
We identify two fundamental challenges designing learning objectives for time-series forecast models: the label autocorrelation effect and the heterogeneous task weights.
We propose a quadratic-form weighted learning objective that tackles both challenges. The QDF learning algorithm is proposed to apply the objective for training time-series series forecast models.
We perform comprehensive empirical evaluations to demonstrate the effectiveness of QDE, which enhances the performance of state-of-the- of-the--rt forecast models across diverse datasets.
2
PRELIMINARIES
2.1 PROBLEM DEFINITION
This work investigates the multi-step time-series forecasting task. Formally, given a time-series dataset S with D covariates, the history sequence at time step n is denoted by X [Sn-H+1, S,, RRHDD, while the label sequence is Y = [S,+I..... Sa+T] IR T'D where H and T denote the history and forecast horizons, respectively. Recent approaches predominantly adopt direct forecasting (DF) paradigm, predicting all T future steps simultancously (Liu et al., 2024; Piao et al., 2024). Therefore, the goal is to learn a parameterized model ge RR RR IR T'D that generates forecast sequence Y approximating Y, where represents the learnable parameters'.
Advances in forecasting models typically revolve around two axes: (1) the design of neural architec- tures for encoding historical inputs (Liu al., 2024; Zeng al., 2023); and (2) the design of learning objectives for effective training (Wang et al., 2025d:e; Cuturi and Blondel, 2017; Sakoe and Chiba, 2003). This study is primarily concerned with the latter--pecifiiallly the improved formulation of learning objectives. Nonetheless, we briefly introduce both aspects as follows for completeness.
2.2 NEURAL NETWORK ARCHITECTURES IN TIME-SERIES FORECASTING
The principal goal of architecture development in time-series forecasting is to learn informative representations of history sequence. The key challenge to accommodate the autocorrelation effect present in the history sequence. Traditional approaches include recurrent neural networks (Gu et al., 2021; Chen et al., 2023), convolutional neural networks (Wu et al., 2023; Luo and Wang, 2024), and graph neural networks (Cao et al., 2020; Mateos et al., 2019). In the recent literature, one predominant series are Transformer models (e.g., TQNet (Lin et al., 2025). PatchTST (Nie et al., 2023), iTransformer (Liu et al., 2024)). which show strong scalability on large datasets but a higher computational cost. Another predominant series are linear models (e.g., TimeMixer (Wang et al., 2024), DLinear (Zeng et al., 2023)), which are efficient varying history sequence length. There are also hybrid architectures that fuse Transformer and linear modules to combine their respective advantages (Lin et al., 2024; Wu et al., 2025).
may struggle to scale and cope with
'eereafter, consider the univariate case (D) 1) for clarity. In the multivariate case, each variable can be treated a separate univariate case when computing the learning objectives.
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100
400
10
21
32
0.100
0000
0.10
64
0.000
0.000
Ruturc stop
(a) Partial correlation and significance of labels.
(b) Partial correlation of extracted label components.
Figure 1: Statistics of label components conditioned X, with forecast horizon of T 96. (a) Partial correlation and conditional variance estimated from the raw label sequence Y. with colors indicating different X. (b) Partial correlation matrices label components extracted by FreDF and Time-ol (Wang et al., 2025e;d), Calculation details are provided in Appendix A.
2.3 LEARNING OBJECTIVES IN TIME-SERIES FORECASTING
The primary challenge driving the development of learning objectives in time-series forecasting is to accommodate the autocorrelation effect present the label sequence. The standard mean squared error (MSE) is widely used to train forecast models (Lin et al., 2025; 2024; Liu et al., 2024), which measures the point-wise difference between the forecast and label sequences:
Lmsc Y
go X)I
(1)
However, Cmse is known to be biased, as it neglects the presence of autocorrelation in the label sequence (Wang al., 2025e). To mitigate this issue, alternative objectives have been explored. One line of work promotes shape alignment between the forecast and label sequences (Le Guen and Thome, 2019; Cuturi and Blondel, 2017), emphasizing the autocorrelation structure, though these approaches generally lack theoretical guarantees for bias elimination. Another line of works transforms the labels into decorrelated components before alignment, thereby mitigating bias and improving forecast performance (Wang al., 2025e;d). These empirical advancements underscore the critical role of objective function design in advancing time-series forecasting.
METHODOLOGY
3.1
MOTIVATION
The design of learning objectives is central to training time-series forecasting models. Likelihood maximization provides a principled approach, minimizing the negative log-likelihood (NLL.) of label sequence. By Theorem 3.1, the NLL is quadratic form weighted by the inverse of the conditional covariance matrix E. This formulation reveals two key challenges in designing learning objectives.
Autocorrelation effect. Time-series data exhibit strong autocorrelation, where observations are highly correlated with their past values. This implies that future steps within the label sequence are correlated even when conditioned on the history X (Wang al., 2025e). This property necessitates modeling the off-diagonal elements of Σ, which are not necessarily zeros.
Heterogeneous weights. The training of forecast models is a typical multitask learning problem, where predicting each future step distinct task. These tasks often exhibit varying levels of difficulty and uncertainty, suggesting they require different weights during optimization. This property necessitates modeling the diagonal elements of Σ, which are not necessarily uniform.
Theorem 3.1 (Likelihood formulation). Given history sequence X, let Y E ET be the associated label sequence and go ()) E IR' be the forecast sequence. Assuming the forecast errors follow a multivariate Gaussian distribution, the NLL of the label sequence, omitting constant terms, is:
Lx Lx(X, 9o)
Y go (x) |I (Y gx() E(Y (X)));
where Σ RXXT is the conditional covariance of the label sequence given X. Σ is the inverse of E.
However, it is infeasible to directly minimize Ls for model training. The conditional covariance E is unknown and intractable to estimate from the single label sequence typically available per X.
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This difficulty leads to the widespread adoption of the mean squared error (MSE) objective, which in essence assumes Σ is an identity matrix (Lin et al., 2025) and therefore fails to model either autocorrelation heterogeneous uncertainty. Subsequent works advocate transforming the labels into latent components for alignment, exemplified by FreDF (Wang et al., 2025e) and Time-ol (Wang et al., 2025d). However, the transformations they employ guarantee only marginal decorrelation of the obtained components, not the required conditional decorrelation (i.e., diagonal Σ), thereby failing to accommodate the autocorrelation effect. Moreover, they assign equal weight to optimize each component, thereby failing to accommodate heterogeneous weights. Hence, existing methods fail to address the two challenges designing learning objectives for time-series forecast models.
Case study. We conducted case study on the ECL dataset to substantiate our claims (Fig. 1). The primary observations are summarized as follows:
The identified challenges are prominent. As shown in Fig. (()), the partial correlation matrix exhibits significant off-diagonal values (with over 61.4% exceeding 0.1), confirming the presence of autocorrelation effect. Additionally, the conditional variances differ considerably across future steps, highlighting the importance of using heterogeneous error weights.
Existing methods fail to fully address them. The partial correlation coefficients of the latent components extracted by FreDF and Time-ol (Wang et al., 2025e;d) are presented in Fig. 1(b). Although the non- diagonal elements are notably reduced, residual values remain, indicating that these methods do not completely eliminate autocorrelation the transformed components.
Given the critical role of the weighting matrix in elucidating the two challenges and the limitation of existing methods, it is essential to investigate strategies for incorporating the weighting matrix into the design of learning objectives for training forecast models. Specifically, three key questions arise: (1) How can the weighting matrix be estimated from data? (2) How to define a learning objective for model training with it? (3) Does it improve forecasting performance?
3.2 LEARNING WEIGHTING MATRIX TARGETING CENERALIZATION
direct approach to incorporating the weighting matrix Σ is use the NLL from (2). However, as previously established, it is impractical for training because the true conditional covariance Σ is unknown and intractable to estimate accurately from data. To overcome this challenge, we advocate to learn proxy Σ targeting model generalization. To this end, we treat Σ as learnable parameters and the associated optimization problem is formulated in Definition 3.2.
Definition 3.2.
Let Dim = inn Y in) and Dout = (Xout Y out) be non-overlapping splits of the training data, each consisting of historical and label sequences. The bilevel optimization problem is
min Lx (Xout, Y Vottigo.) where 0" = arg min Lx(X in, Yiui 90). Σ=
where Σ 2 0 mcans Σ is semi-definite positive, a fundamental property of covariance matrix.
There are two loops in the optimization problem (3). The inner problem trains the forecast model go on a data split Din using fixed Σ; the outer problem then updates Σ to improve the generalization performance of the trained model on a disjoint holdout set Dout This process ensures the learned Σ produces learning objective that drives the forecast model to generalize well.
ce-parameterization. To solve the problem (3), is crucial to enforce Σ 2z= O. We address this by matrix
reparameterizing Σ via its Cholesky factorization, Σ = LL where is a fower-triangul with positive diagonals (which can be ensured with a softplus activation). This reparameterization converts the constrained optimization over Σ into an unconstrained optimization over L, thus enabling the use of standard gradient-basee optimization methods. For clarity, in the following derivations, continue to use Σ and omit the notational complexity introduced by this reparameterizztion.
Algorithm 1 details the procedure for solving (3) via gradient descent. The algorithm begins by partitioning the dataset 'D into two disjoint subsets, Din and Dant (step 1). Within the inner loop, the objective Lx is evaluated over Din. and its gradient with respect to the model parameters 0 drives the update of 0 (steps 2-5). Subsequently, in the outer loop, the objective is evaluated on Dout. and
"This property is demonstrated in Theorem 3.3 (Wang et al., 2025c) and Lemma 3.2 (Wang et al., 2025d).
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Algorithm Atomic update procedure of QDE.
Algorithm 2 The overall workflow of QDF.
Input: go: forecast model, Σ: weighting matrix, D: dataset used to leamn Σ. Parameter: N: number of updates, vf update rate.
Input: ge: forecast model, Pusin: training set. Parameter: Ni: round of inner update, Nout: round of outer update, ry: update rate. K: number of splits. Output: L: ohtained learning objective.
Output: Σ: obtained weighting matrix.
1: Din Dont f- (D)
1: Σ f- LT. D2, ... DK t split(Persia))
2:
for n = 1. 1,2,... N do
2: while n = 1, Nout do
3:
Xin: Y x- Din
3: En+1 f- Algorithml (Eng Da, 90), l = I,....R K
4: f+ --0 VoLx Yin:
4: if [f][En1 - Enllp le break.
end for
5: end while
X ous: Y out f- Dout
6: X Ktrann Virain f- Derain
7: E + E VxLz(Xour Y Youtigo)
7: t- LI,, (Xtrain, Y Yerains gb)
the gradient with respect to E is used to update the weighting matrix (steps 6-7), It is important to emphasize that the gradient in the outer loop is backpropagated through the updated model parameters 0 to E, rather than being taken directly with respect to Σ. This approach ensures that the impact of E on the learned and thereby on the generalization performance is fully captured. Collectively, the procedure above performs a single update of Σ toward the optimum of (3), and can be repeatedly applied to iteratively refine the estimate of Σ.
3.3 THE WORKFLOW OF QDF FOR TRAINING TIME-SERIES FORECAST MODELS
While we have established a method to learn an instrumental weighting matrix Σ, is not clear how to use the obtained Σ for training forecast models. To fill this gap, we detail the workflow of QDE, which first learns Σ and then applies it to train forecast models. The principal steps are encapsulated in Algorithm 2, which consists of three primary phases follows.
The process begins by initializing Σ as an identity matrix. The training set Dirain is split chronologically into non-overlapping subsets (step 1). This partitioning is erucial for robustness: by updating Σ across different data distributions (subsets), we seek for estimation of Σ that is less likely to overfit to any single part of the training data (Nichol and Schulman, 2018).
Initialization.
Weighting matrix learning. With the data prepared, we iteratively refine Σ by applying Algorithm sequentially across the K subsets. The iteration stops when Σ converges (i.e., the change between iterations is negligible) or a predefined number of rounds completed (steps 2-5).
Model training. With the learned weighting matrix Σ in hand, the final phase is to train the forecast model go. This is achieved by minimizing the corresponding NLL objective (Lx) over the training set (steps 6-7). In practice, this minimization is performed using standard gradient descent, and the NLL objective can be estimated on mini-batches for computational efficiency.
By employing Lx for model training, QDF effectively leverages the weighting matrix E, thereby addressing the two established challenges. Specifically, the off-diagonal elements of Σ enable the model to account for the autocorrelation effect, and non-uniform diagonals enable heterogencous weights for each error term. There is no risk of data leakage, as the full training procedure (Algo- rithm 2) exclusively utilizes the training set, without using the validation or test sets. Notably, QDF is model- making it a versatile tool for improving the training of various direct forecast models (Liu et al., 2024; Zeng al., 2023; Piao et al., 2024).
The strategy of treating Σ as learnable parameters is conceptually related to the principles of meta- learning (Nichol et al., 2018; Finn et al., 2017). However, our work diverges from meta-learning in both goal and implementation. (1) The goal of meta-learning is to enable rapid adaptation to new, dynamic tasks, whereas QDF is designed to construct a static objective for time-series accommodating autocorrelation and heterogeneous weights. (2) This difference in goals leads to different validation schemes. Meta-learning validates generalization on set of new tasks, whereas QDF uses holdout dataset drawn from the same forecasting task for validation. (3) In time-series analysis, some studies accommodate meta-leane learning for model selection (Talagala et al., 2023), ensembling (Montero-Manso al., 2020), initialization (Oreshkin
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Table I: Long-term forecasting performance.
QDP
TQNer
PDF
L'retiorer
iTrassfommrr (2004)
IrelSS
TimesNer
MICN
TIDE
PaichTST
DLinear
Medel
(Ours)
(2025)
(2004)
(2014)
(2023)
(2003)
(2013)
(2003)
(2033)
(2023)
Metnis
MSE
MAB
MSB
MAB
MSU
MAE
MSE
MAB
MSU
MAE
MSE
MAB
MSU
MAB
MSE
MAU
MSE
MAB
MSU
MAE
MSE
MAB
ETTml
0,371
0.389
0.326
0.391
0.987
0.996
0.187
0.998
0411
0414
0:414
0-421
0.48
0-400
0.996
0-421
0.413
0:400
0.389
40400)
0:407
0.400
ETTn2
0,270
0.317
0,077
0.321
0,283
0.351
0,280
0.324
0.295
0.336
0,316
0.363
0,302
0.334
0.305
0.364
0,286
0.328
0,303
0.344
0,342 0.392
0.431
0,449
0,452
0/477
0.452
0,489
0,472
0.533
0.448
0,459
0,456
0.315
0.315
0337
0.395
00:24
0,409
0.600
0.378
0,300
0:299
PCT.
0.165
0.125
0.365
0.199
0.381
0.191
0.964
0.179
0.270
0.190
0.312
0.306
[0.19
0.900
0.315
0.292
0.195
0..36
0:312.
Washer 0.242
0.268
0.246
0.270
0,265
0.263
0.266
0.262
0.269
0.289
0.2499
0.203
0,271
0.295
0.264
0.321
0.372
0.291
0.267
0.288
0.365
0.317
PEMS0S
0.089
0.197
0,119
0.212
0
0.256
0.145
0.200
0.122
0.233
0.149
0.261
0,125
0.230
0.105 0.223 0,316 0.370
0,170
0.252
0.216
0.322
PUMSOS
0.120
0.321
0.139
0.ato
0.021
0,109
0.134
0,152
0,153 0,318 0,201
0.300 0:249
Note: We fix the input length as 96 following Liu et al. (2024). Bold and underiined denote best and second best results, respectively. The reported results are averaged forecast horizons: T-96, 192, 336 720. QDF employs sop-parforming TQNer forecast model.
et al., 2021) and domain adaptation (Narwariya et al., 2020), whereas QDF aims to obtain a versatile learning objective. To our knowledge, this is technically innovative strategy.
4 EXPERIMENTS
To demonstrate the efficacy of QDF, there are six aspects that deserve empirical investigation:
Performance: How does QDF perform? We compare the forecast performance of QDF against statc-of-the-art baselines (Section 4.2) and learning objectives (Section 4.3).
Gains: What makes it effective? We perform an ablation study (Section 4.4) to investigate the contribution of each technical element to its overall performance.
3. Versatility: Does il benefü different forecast models? We compare the performance of DF and QDF using different forecast models (Section 4.5), with further results provided in Appendix D.4.
4. Flexibility: Does the weighting malrix accommodate mela-learning methods? We attempt to learn the weighting matrix using established meta-lcarning methods (Section 4.5).
5. Sensitivity: Is it sensitive hyperparaneters? We conduct sensitivity analysis (Section 4.7) to show that its effectiveness across wide range of hyperparameter values.
Complexity: it computational expensive? We investigate the running time of QDF given different settings (Appendix D.7).
4.1 SETUP
Datasets. Our experiments are conducted on public datasets for time-series forecasting, consistent with prior works (Wu et al., 2023; Liu et al., 2024). The employed datasets include: ETT (consisting of 1T ETTh2, ETTm1, and ETTm2), Electricity (ECL), Weather, and PEMS. For each dataset, adopt standard chronological split into training, validation, and testing partitions. Further details on dataset statistics are available in Appendix C.I.
Baselines. We compare QDF with 10 previous methods, which we categorize into two groups (Wang et al., 2025d): (1) Transformer-base 2024), Fredformer (Piao al., 2024), PDF (Dai al., 2024) and TQNet (Lin al., 2025); (2) Non-transformer based models: DI Dinnear (Zeng et al., 2023), TiDE (Das et al., 2023), MICN (Wang al., 2023b), TimesNet (Wu et al., 2023) and FreTS (Yi al., 2023).
models: PatchTST (Nie et al., 2023). iTransformer (Liu et al.,
Implementation. To ensure fair evaluation, all baseline models are reproduced using the official codebases (Lin et al., 2025). We train all models with the Adam optimizer (Kingma and Ba, 2015) to minimize MSE the training set. Notably, disable the drop-last trick during both training and inference to prevent data leakage and ensure fair comparisons, as suggested by Qiu et al. (2024). More implementation details available in Appendix C.

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OrcandTiugh
GroundTruhh Predectio
Groundrruth
0.5
-0.5
2.4
24
-15
-1.5
0.0
-20
350
100
320
(a) ETTm2 snapshot.
(b) ECL snapshot.
Figure 2: The forecast sequence of DF (in blue) and QDF (in red), with historical length H = 96.
Table 2: Comparable results with other objectives for time-series forecast.
Loss
QDF
Timo-ol
FreDF
Koopman
Soft-DTW
DF
Metries
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
ETTml
0.371
0.389
0.372
0.390
0.375
0.390
0.595
0.499
0.387
0.394
0.376
0.391
TUNe
ETTh
0.431
0.431
0.437
0.432
0.432
0.432
0.451
0.442
0.453
0.438
0.449
0.439
ECL
0.165
0.257
0.167
0.257
0.168
0.257
0.166
0.258
0.623
0.524
0.175
0.265
Weather
0.242
0.268
0.245
0.269
0.244
0.268
0.282
0.306
0.255
0.276
0.246
0.270
ETTml
0.381
0.394
0
0.390
0.387
0.400
0.587
0.485
0.396
0.404
0.387
0.396
PDR
ETTh]
0.436
0.429
0.438
0.438
0.437
0.435
0.497
0.472
0.447
0.447
0.452
0.440
ECL. 0.194
0.277
0.195
0.276
0.194
0.274
0.196
0.281
0.695
0.548
0.198
0.281
Weather
0.259
0.281
0.264
0.284
0.268
0.287
0.268
0.290
1.296
0.452
0.265
0.283
Note: Bold and underlined denote best und second best results, respectively. The reported results averaged over forecast botioons: T:96, 192, 336 and 720.
4.2 OVERALL PERFORMANCE
In this section, we compare the long-term forecasting results. As shown in Table 1, integrating QDF yields consistent improvements in forecast accuracy across all evaluated datasets. For instance, on the PEMS08 dataset, QDF achieves notable reduction in both MSE and MAE by 0.019, We attribute the enhanced performance to QDF's adaptive weighting mechanism, which addresses two critical challenges in objective design: label autocorrelation effect and heterogeneous task weights.
Examples. A qualitative comparison between forecasts generated by DF versus QDF is presented in Fig. 2. The model trained with DF captures general patterns, but it often fails model subtle dynamics. For example, on ETTm2, it struggles to follow a sustained upward trend, and on ECL, it misses periodic peak around the 150th step. In contrast, QDF accurately captures these subtle patterns, which showcases its practical utility to improve real-world forecast performance.
4.3 LEARNING ORJECTIVE COMPARISON
In this section, we compare QDF against alternative learning objectives. Fach objective is integrated into two forecast models: TQNet and PDF, using their official implementations. The results summarized in Table 2. Overall, methods designed to correct for bias in likelihood estimation, namely FreDF and Time-ol, deliver consistent performance improvements. However, as we established in Section 3.1, these approaches cannot handle the two challenges and yield suboptimal performance. In contrast, QDF achieves the best performance, with its weighting matrix effectively tackling the two main challenges in objective design: the label autocorrelation effect and heterogeneous task weights
4.4 ABLATION STUDIES
In this section, we examine the technical components within QDF that address the two key challenges of learning objective design and assess their individual contributions forecast performance. The results are presented in Table 3, with key observations as follows:
QDF enhances DF by enabling heterogeneous task weights. Specifically, this variant follows the QDF procedure but sets the off-diagonal elements of the weighting matrix to while allowing
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Table 3: Ablation study results.
T=96
T=192
T=336
T=720
Avg
Model
Hettroo
Auto.
Data
MSR
MAR
MSF
MAF
MSE
MAR
MSF
MAE
MSF
MAE
ETTml
0.310
0.352
0.356
0.377
0.388
0,400
0.450
0.437
0.376
0.391
DF
X
ETThl
0.372
0.391
0.430
0.424
0.486
0.454
0.507
0.486
0.449
0.439
ECL
0.143
0.237
0.161
0.252
0.178
0.270
0.218
0.303
0.175
0.265
Weather
0.160
0,203
0.210
0.247
0.267
0.289
0,346
0.342
0.246
0,270
ETTml
0.309
0.351
0.354
0.378
0.387
0,401
0.450
0.439
0,375
0.392
QDF
V
x
ETThl
0.372
0.394
0.432
0.424
0.475
0.445
0.494
0.481
0.443
0.436
ECL
0.135
0.230
0.154
0.246
0.170
0,263
0.203
0.293
166
0.258
Weather
0.159
0,202
0.208
0.246
0.265
0.287
0,344
0.341
0.244
0.269
ETTml
0.308
0.351
0.353
0.377
0.385
0.399
0.443
0.436
0.372
0.391
QDF
X
ETTh1
0.369
0.391
0,430
0.422
0.477
0.447
0.492
0.475
0.442
0.434
ECL
0.136
0,230
0.153
0.245
0.171
0,264
0.203
0.292
0.166
0.258
Weather
0.159
0,202
0.210
0.247
0.266
0.289
0.343
0.340
0.245
0.269
ETiml 0.307
0.349
0.352
0.376
0.383
0,398
0.441
0.434
0,371
0.389
0.365
0.467
QDF
ECL
0.135
0.229
0.153
0.245
0.169
0.262
0.202
0.291
0.165
0.257
Weather
0.158
0.201
0.207
0.245
0.263
0,286
0.342
0.339
0.242
0,268
Bold and underfined denote best and accond-hert results, respectively. "Hetero" and "Auto" abbreviations helerogenouu, tack weight and label mutocorccaatonn effect, respectively. It employs the top-parfomning TQNeet as the forocast model. Arg indicates average results over forecast borizons: T-96, 192, 336 and 720.
ODF
0.32
OEF
ODF
0:33
GDF
0.34
ODF
ODF
5.0%
7
0.30
-3.3%
4.32%
2.2%
0:30
4.6%
0:30
0.37
-1.8%
0.28
0.25
0.18
6.88
0.44
(a) ECL with MSE
(b) ECL with MAE
(c) Weather with MSE
(d) Weather with MAE
Improvement of QDF applied to different forecast models, shown with colored bars for means over forecast lengths (96, 192, 336, 720) and error bars for 50% confidence intervals.
Figure 3:
the diagonal elements learned. It consistently outperforms DF, indicating that assigning heterogeneous weights to different forecast tasks can improve performance.
QDF improves DF by modeling label autocorrelation effects. Specifically, fixes the diagonal elements of the weighting matrix to one, while learning the off-diagonal elements. It also surpasses DF, achieving the second- -bett results overall. This highlights the benefit of modeling autocorrelation effects in the learning objective for forecasting performance.
QDF integrates both factors above and achieves the best performance, demonstrating the synergistic effect of addressing both heterogeneous task weights and label autocorrelation.
4.5 GENERALIZATION STUDIES
In this section, we explore the versatility of QDF model-agnostic enhancement. To this end, integrate it into different forecast models: TQNet, PDF, FredFormer and iTransformer. The results in Fig. 3 show that QDF delivers consistent performance gains across all evaluated models. For example, on the ECL dataset, augmenting FredFormer and TQNet with QDF reduced their MSE by 7.4% and 5.9%, respectively. This consistent ability to elevate the performance of various models underscores QDF's versatility for improving time-series forecast performance.
4.6 FLEXIBILITY STUDIES
In this section, we explore the flexible implementation of QDF. Since the weighting matrix in QDF is treated as set of learnable parameters, it is natural investigate whether established meta-learning algorithms can be used to optimize it. To this end, we examine several represen- tative meta-learning methods, including MAMI. (Finn et al., 2017), iMAML. (Rajeswaran al.,
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Table 4: Comparison with meta- methods on ECL dataset.
T-96
T=192
T=336
T=720
Method
MSF
MAR
MSF
MAF
MSF
MAF
MSF:
MAF
DF
0.143
0,237
0.161
0.252
0.178
0.270
0.218
0.303
iMAMI.
0. 0.155.
0.230, 2666
0. 0.154,
0.246,, 0.246,.a58% 55%G
0.263,.47%%
0.205, a0%C
0.293,.50%%
MAML
0.136, 54%6
0.230g.20585 20%6
0.230
0.154, 2.49
0,246
.47%6
7.170
0.263
5.09
0.205
03%6
0 0.293 0.2%%
MAML+ ++
0. 0.155
0.229
33364
0.154
2
0,246,
40%%
0.170
729944
0. 0,263,
6005_
0.204
0.292, 6779
Reprill
1313 00%6.
0.230
g,
7398
0.247
91194
0.264
0.206
36%94
0.2942.0088 (08944
QDF
0.135 1006)
0.229 d3564
0.153
7
0.245 8226
0.169 14%63
0.262
0.202 379C
0.290,.00%
Nom: Bold and underlined danote best and second hest results, respectively, The subsccipt denotes the relative error rediccion compared with DE:
8:18
0.13
0.18
M68
9:16
T-336
0.16
T-336
0.16
0.14
0:14
0.14
0.28
Tw100
0.25
Tw100
0.20
MAE
T::36
0.5
T-336
0.2
T::36
0.24
0.24
0.3
023
0.23
0.20
0.00
0.30
0.15
0.23
N
(a) The round of inner update (Nm.). (b) The number of splits (K).
(c) The update rate (7).
Figure Impact of hyperparameters on the performance of QDF.
2019), MAML+((Anoonio et al., 2019), and Reptile(Nichol and Schulman, 2018). Overall, all these methods outperform the canonical DF approach that sets the weighting matrix as an identity matrix, thereby demonstrating the flexibility of QDF's implementation. However, these methods do not explicitly optimize the weighting matrix for out-of-sample generalization, which a distinct advantage of our implementation that benefits forecast performance.
4.7 HYPERPARAMETER SENSITIVITY
In this section, we examine the impact of key hyperparameters QDF's performance, with results shown in Fig. 4. The main observations are as follows:
The coefficient Nin determines the number of inner-loop updates in Algorithm 2. We observe that increasing Nj, from 0 to 1 significantly improves forecasting accuracy. Further increases bring marginal gains, suggesting that the forecast model's performance after one-step update already provides valuable signals guide the weighting matrix update.
The coefficient K determines the number of data splits in Algorithm The best performance is achieved when K = 3, indicating that splitting the data enhances the generalization ability of the learned weighting matrix. Increasing it further leads to diminishing returns, as the sample size per split becomes too small to be informative given large values of K.
The coefficient 17 determines the update rate in Algorithm 2, where setting it to zero immediately reduces the method to the DF baseline. In general, using 7) > 0 to update the weighting matrix effectively improves performance, and the improvement robust to wide range of r) values.
CONCLUSION
In this study, we identify two key challenges in designing learning objectives for forecast models: the label autocorrelation effect and heterogeneous task weights. We show that existing methods fail to address both challenges, resulting in suboptimal performance. To fill this gap, we introduce a novel quadratic-form weighted learning objective that simultaneously tackles these issues. To apply this

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Published as a conference paper at ICLR 2026
objective, propose QDF learning algorithm, which trains the forecast model using the quadratic objective with an adaptively updated weighting matrix. Experimental results demonstrate that QDF consistently enhances the performance of various forecasting models.
Limitations & future works. A limitation the current QDF is that it assumes a static weighting matrix Σ suffices to enhance the learning objective. While this assumption is well motivated to address the two identified challenges- label autocorrelation and heterogeneous task weighting it constrains the expressiveness of the learned objective. promising solution for future work would be to employ hyper-network that dynamically generates input-dependent weighting matrix, thereby yielding a more adaptive and expressive formulation that could potentially lead to further performance improvements.
ACKNOWLEDGMENTS
Z. Lin was supported by the NSF China (No. 62276004), the Beijing Natural Science Foundation (No. L257007), and the Beijing Major Science and Technology Project (No. Z251 1251000008425006
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A ON THE LABEL AUTOCORRELATION ESTIMATION DETAILS
In this section, we introduce the procedure for estimating the label autocorrelation in Fig. 1. A primary challenge in this estimation is accounting for the confounding influence of the historical input sequence, X (Wang et al., 2025c; Li et al., 2024b;c). A direct correlation between labels at different time steps, such Yr and Y v' may not exist. However, failing to control for the common influence of X can introduce spurious correlations (Wang et al., 2023a; 2025a), leading to biased estimation (Wang et al., 2025b; Li al., 2024a). Consequently, standard metries like the Pearson correlation coefficient are inadequate for this task, they unable to isolate the relationship between Y, and Y, , from the spurious correlations.
To overcome this limitation, utilize the partial correlation coefficient to provide proxy of label autocorrelation. Our approach mirrors MATLAB's 'partialcorr' function', Specifically, to compute the partial correlation between two points in the label sequence, Y, and Y u', while conditioning the history sequence X (the control variables), we employ a two-stage regression process. First, we fit separate linear regression models using ordinary least squares (OLS) to predict Y, and Y from X. The resulting residuals, ez and fr, represent the variance in Y, and Y that is not explained by X. The partial correlation is then computed the standard Pearson correlation between these two sets of residuals, p(Et, Ev) This procedure effectively quantifies the linear relationship between Y, and Y y after factoring out the confounding influence the historical context.
1.000
80
37-
81
31-
81
31-
9-
0310
0010
0010
7
0010
0.001
0021
0.001
0021
8
0000
(a) Partial correlation coefficients between different steps in the raw label sequence.
1.000
-0 0000
48
-3 010
0.10
60
D421
0.01
95
56
90
96
16
32
(b) Partial correlation coefficients between different components obtained hy FreDF.
1.000
1,000
1.00
1430
30
.2
22
0.100
0.100
21
0001
2021
0001
0021
8.
8
0.000
8
2020
0 000
0.000
(c) Partial correlation coefficients between different components obtained by Time-ol.
Figure 5: The label autocorrelation effect on the original label sequence and the components extracted by FreDF and Time-ol (Wang al., 2025d;e). The datasets are ETTh1, ETTh2, BCL, and Weather from left to right. The forecast length is uniformly set to 96.
To further validate the observations from the case study in Fig. 1, we extend the analysis on four additional datasets. As illustrated Fig. 5, partial correlation matrices corresponding to the raw labels display significant off-diagonal values across multiple datasets. This pattern provides strong evidence for the widespread presence of label autocorrelation. In contrast, while the latent components extracted by methods such FreDF and Time-ol (Wang al., 2025e;d) show marked reduction in these off-diagonal correlations, they do not succeed climinating them entirely. The persistence of
'The official implementation is detailed at https:::p http::/wwwwwwwwwwe///:/ttt mac.works wwt./o/mmeceeeeeeeeeeee partialcorr html.
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these residual values suggests that these methods only partially eliminate the autocorrelation effect. Therefore, directly applying point-wise error (such as MSE or MAE) on the obtained components yields bias due the oversight of residual autocorrelation effect.
One might advocate for directly estimating the conditional covariance from data statistically. However, this approach is generally intractable due to its prohibitive computational complexity. Specifically, to estimate the partial correlation between each pair of time steps t and t', two OLS problems must be solved over the entire dataset. The scale of each OLS problem grows rapidly with the length of the history sequence and the number of covariates. Worse still, the overall complexity increases quadratically with the forecast horizon. For example, if the forecast length T 720, computing the full partial correlation matrix requires estimating 720 x 720 partial correlations. In our case study, we mitigate this complexity by subsampling only 5,000 examples from each dataset, restricting the history sequence length to 8, and limiting the forecast horizon to 96. This reduction makes the estimation tractable and affordable at the cost of accuracy, which is acceptable since the estimated results are used solely for the case study rather than for model training.
B THEORETICAL JUSTIFICATION
Theorem B.1 (Likelihood formulation, Theorem 3.1 the main text), Given history sequence X, let Y E RT be the associated label sequence and gx(X) E ITT be the forecast sequence. Assuming the label sequence given X follow a multivariate Gaussian distribution, the NLL of the label sequence, omitting constant terms, is:
Lx Y; gg) = Y (er-ge(X)) = -Y-o0(x) Σ(Y (x)))),
gx))
where E E RTxT is the conditional covariance of the label sequence given X.
Proof. The proof follows the standard derivation of negative log- likelihood given Gaussian assump- tion. Suppose the label sequence given X follows multivariate normal distribution with mean vector go ()) and covariance matrix E. The conditional likelihood of Y is:
exp(
go (xI
(5)
(2 (2.t)..T 0
On the basis, the conditional negative log- log-likelhhood of Y is:
log Pr y 2 (oog(22)) + log Σ + |Y - ge (x)]
Removing the terms unrelated go. the terms used for updating 0 expressed as follows:
(x Y; :0) YY go (x)
(6)
The proof is therefore completed.
C REPRODUCTION DETAILS
C.I DATASET DESCRIPTIONS
Our empirical evaluation conducted on a diverse collection of widely-used time-series benchmarks, with their key properties summarized in Table 5. These include:
ETT (Li al., 2021): Electricity transformer data consisting of four subsets with varied temporal
resolutions (ETTh1/ETThe at I-hour intervals, ETTm 1/ETTm2 at 15-minute intervals).
Weather (Wu al., 2021): Comprises 21 meteorological indicators recorded every 10 minutes from the Max Planck Institute.
ECL (Wu et al., 2021): Hourly electricity consumption data from 321 clients.
PEMS (Liu et al., 2022): California traffic data aggregated in 5-minute windows. We utilize the PEMS03 and PEMSOS subsets.
For all datasets, we adopt a standard chronological split into training, validation, and testing sets, following established protocols (Qiu al., 2024; Liu et al., 2024). We standardize the input sequence length to 96 for the ETT, Weather, and ECL datasets, evaluating on forecast horizons of (96, 192, 336, 720). For the PEMS datasets, use forecast horizons of (12, 24, 36,48) 48).
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Table 5:
Dataset description.
Dataset
D
Forecast length
Train validation / test
Frequency
Domain
ETTh1
96, 192, 336, 720
8545/2881/2881
Hourly
Health
ETTh2
96, 192, 336, 720
8545/2881/2881
Hourly
Health
ETTml
7
96, 192, 336, 720
34465/11521/11521
15min
Health
ETTm2
7
96, 192, 336, 720
34465/11521/11511
15min
Health
Weather
21
96, 192, 336, 720
36792/5271/105440
10min
Weather
ECL
321
96. 192. 336, 720
18317/2633/5261
Hourly
Electricity
PEMS03
358
12. 24, 36,
15617/5135/5135
5min
Transportation
PEMS08
170
12, 24, 36, 48
10690/3548/265
5min
Transportation
Note: D denotes the number of variates. Frequency denotes the sampling interval of time points. Train, Volidation, Test denotos the number samples employed each split. taxonomy aligns with (Wi et al., 2023).
C.2 IMPLEMENTATION DETAILS
All baseline models were reproduced using official training scripts from the iTTrasffommer (Liu et al., 2024) and TQNet (Lin et al., 2025) repositories after checking reproducibility. Models were trained to minimize the MSE loss using the Adam optimizer (Kingma and Ba, 2015). The learning rate was selected from the set {10-3, 5 10- 10 0,5 x 10-5). We employed early stopping patience of 3, halting training if validation loss did not improve for three consecutive epochs.
When integrating QDF into existing forecasting model, retained the original model's established hyperparameters as reported public benchmarks (Liu et al., 2024; Piao et al., 2024). Our tuning conservatively limited the QDF-specifie parameters, i.e., the round of inner update (Ni.). the number splits (K), and the update rate (1)), along with the learning rate. The final hyperparameter configuration for each model was selected based on its performance the validation set.
D MORE EXPERIMENTAL RESULTS
D.1 OVERALL PERFORMANCE
We provide additional experiment results overall performance Table 6, where the performance of each forecast horizon T is reported separately.
D.2 SHOWCASES
We provide additional experiment results of qualitative examples in Fig. and Fig. 7.
D.3 LEARNING OBJECTIVE COMPARISON
We provide additional experiment results learning objective comparison in Table 13.
D.4
GENERALIZATION STUDIES
We provide additional experiment results of generalization studies in Fig. 8.
D.5 CASE STUDY WITH PATCITST VARYING HISTORICAL LENGTIIS
We provide additional experiment results of varying historical lengths in Table 8, complementing the fixed length 96 used in the main text. The forecast models selected include TQNet (Lin et al., 2025) which is the recent state-of-the-an forecast model, and PatchTST (Nie et al., 2023) which is known to require large historical lengths. The results demonstrate that QDF consistently improves both forecast models across different history sequence lengths.
16

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Published as a conference paper at ICLR 2026
1.
1.
10
10
-0.5
-05
-1.0
-1.0
-10
-1.0
103
230
300
403
100
200
500
402
100
200
300
400
100
200
800
400
(a) Example with TQNet
(b) Example with PDF
-0.5
-0.5
-0.5
-0.5
-1.5
-15
-15
-1.
-20
-20
-20
(c) Example 2 with TQNet
(d) Example 2 with PDF
GroundTruth Prediction
GroundTruth Fredittioo
GroundTiuh Prediction
GroundTiuh
Prediction
-05
-0.5
-0.6
0.55
-10
-1.0
-15
-15
-16
-20
-20
-25
-2.
103
250
403
200
400
100
200
400
100
200
400
(c) Example with TQNet
(f) Example 3 with PDF
Figure 6: The forecast sequences generated with DF and QDE. The forecast length set to 336 and the experiment is conducted on ETTm2.
GroandTiuth Predkction
Gragestnuth
Groundtouhh Prendieae MMmm
Grouped
Predition
Prendcenie Ma
1.2
12
12
-12
-12
100
200
200
100
200
2000
100
200
200
100
200
300
(a) Example with TQNet
(b) Example with PDF
MMe
1.8
1.8
1.8
1.0
0.0
-0.
(c) Example 2 with TQNet
(d) Example 2 with PDF
2.4
MWW MM
GroundTTrah Prodiction
MW MM
Predittion
GrounsTruhh
GroundtTuuhh Prediction W00
2.4
Prediction Moaaooblee
GroundtTium
-1.2
100
303
100
200
802
100
200
300
100
230
350
(c) Example with TQNet
Example 3 with PDF
Figure 7: The forecast sequences generated with DF and QDE. The forecast length set to 192 and the experiment is conducted on ECL.
17

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Published as a conference paper at ICLR 2026
Table 6: Full results on the multi-step forecasting task. The length of history window is set 96 for all baselines. Avg indicates the results averaged over forecasting lengths: T=96, 192, 336 and 720.
Modele
QDF (Durs)
TQNNet (2025)
PDJ (2024)
Predformer (2024)
ilranddomme (2024)
F (2023)
TimesNet (2023)
MICN (2013)
TiDE
PS (2023)
DLinear (2013)
Merrics
MSF:
MAR
MSE
MAR
MSE
MAR
MSR
MAR
MSR
MAR
MSE
MAR
MSB
MAR
MSR
MAR
MSB:
MAR
MSE
MAR
MSB
MAR
96
0.307
0.349
0.510
0,352
0.326
0:361
10.366
0,961
0.358
0.37
0.342
0.318
0.06
0,994
0.319
0. 366
0.353
0.324
0.335
0.964
0,346
0.373
192
0.382
0.376
0.383
0.380
0.308
0.357
0/402
0.404
0,427
0.166
0.411
0,395
0/423
0.414
0.413
0.414
00:41
0.434
01450
0.437
0.458
0.457
(0,899
0,644
0.496
0.461
0.513
0.409
(0.22
0,424
0,50
0.099
0.448
0.448
0.063
0.442
0.472
0.450
Avg
90.371
0.389
0.376
0.391
00.387
0.296
00.387
0.358
0.411
0.414
0.421
[0:435
0.430
0.396
0.421
[0.413
0.407
0.359
|0,403
0.407
96
0.170
0.253
0.125
0.256
0.176
0.266
[0.177
0.260
0.162
0.365
0.188
0.274
0..184
0.362
0.178
0.277
0.182
0.365
0.150
0.266
[0.888
0.281
192
0.234
0.294
0.245
0.310
0,300
0.264
0.329
0.257
0.308
0.256
0.345
0.247
0.304
0.339
0.280
0.356
336
0:305
0:322
0.343
0.387
0.389
0.404
0.403
0.397
0.423
0.411
0.489
0.482
0.452
0.421
0.489
0:405
0.395
0.422
0.526
0.508
Avg
00.270
0.317
0.277
00,283
0.351
002300
0.524
0.295
0.316
0.365
00330
0.334
0.308
0.364
[0.286
0.325
0.303
0.344
00,342
0,292
96
0,365
0.389
0.372
0.391
0,338
0,400
0.377
0.396
0
0,405
11330
0.405
0.399
0.415
0.351
0.416
0,387
0,345
0.381
0.400
0,,399
0.404
192
0.427
0.421
01430
0.460
0.428
0.637
0.425
0.410
0.437
0.451
0.442
0.452
0.451
0.480
0.439
0.425
0.450
0.645
0.442
0.440
336
0.466
0.449
0,483
0.436
0.501
0.435
0.482
0.488
0.467
720
0.466
0.467
0.507
0.445
0.482
0.467
0.504
0.600
0.571
0.515
0.005
0.484
0.471
0.892
0.505
0.502
Avg
00.33
0.431
0.440
0.439
0.452
0.440
0.447
0.434
0.452
0.445
0.489
0.474
0.472
0.463
0.533
0.519
00.448
0.435
0:459
0.451
[0,456
0.453
96
0.286
0.338
0.209
0.545
0.200
0.340
0.205
0.546
0.301
0.349
0.315
0.314
00321
0.356
0:351
0.396
000001
0.540
0.299
0.54
0,330
0.081
192
0
0.385
0.364
0.390
0.374
0.291
0,372
0.391
0.383
0.397
0.466
0.467
0.418
0.417
0.492
0.459
0.376
0.392
0.383
0.404
0.439
0.450
336
0.405
0.422
0.414
0.426
0.420
0.435
0.425
0.522
0,502
0.464
0.454
0.676
0.417
0.427
0.439
0.644
0.389
0,538
720
0.419
0.439
0.456
0.792
0.643
0.450
0.429
0.446
0.757
0.626
Avg
0.366
0.397
0.375
0.400
0.3755
0.299
0.337
0.402
0.386
0,407
0.524
0.49%
00,409
0.420
0.620
0.546
003376
0.401
0.390
0.413
[0,529
0.499
96
0.135
0.229
0.143
0.237
0.175
0.259
00161
0.258
0.150
0.242
0.180
0.266
0.170
0.272
0.170
0.251
00.99
0.274
0.170
0.264
0.199
0.282
192
0.155
0.245
0.161
0.242
0.132
11266
0.174
0.269
0.168
0.259
0.154
0.272
0.185
0.292
0.185
0.297
0.197
0.277
0.179
0.273
0.197
0.296
336
0.169
0.262
0.178
0.270
0.087
1.282
0.194
0.200
0.182
0.274
0.199
0.290
0.303
0.302
0.190
0.298
0,312
0.292
0.195
0.258
0.209
0.301
720
0.202
0.290
0.257
0.319
0.234
0.365
0.254
0.325
0.245
0.334
Avg [0.165
0.257
0.125
0,200
00.198
0,281
0.191
0.284
0.179
0,270
0.199
0.268
|02212
0,306
0.192
0,302
00.215
0.292
0.195
0.256
00222
0,201
Weaher
192
10.158
0,207
0.201
0.245
0.100
0.210
0.205
0.242
0.188
0.133
0:221
(0.000
0.222
0.220
0.258
0.171
0.246
0.22%
0,210
0.174
0:213
01224
0.185
0.342
0.276
0.229
0.179
02422
0.310
0.244
10.192
0:240
0.270
0.252
0.189
0.218
0.262
0.230
00.19
0.238
0.253
0.296
336
0,263
0.286
0.267
0.259
0.283
11200
0.283
0
0.296
0.313
0.270
0.316
0.293
0,312
0.233
0.330
0.292
0.307
0.258
0.305
0.282
0.332
720
0.360
0.348
0.358
0.337
0,353
0.347
0,385
AAg
00.242
0.265
0.246
0.270
00:265
0.283
002211
0.282
0.260
0.280
01240
0.203
002271
0.205
0.264
0.321
10,272
0.291
0.267
0.258
00.265
0.317
PEMS09
0,064
0.167
0.07)
0.180
10.00
0.204
[00081
0.191
1.022
0.179
00000
0.198
0.004
0.201
0.096
0217
10:117
0.226
0.02
0.210
10.005
0:220
0,880
0.189
0.099
0.206
0.149
0.261
0.121
0.240
0.104
0217
0.129
01244
0.116
0.221
1005
0210
0.233
0.322
0.144
0.263
0.183
0.297
0.095
0.205
0.125
0.230
0.210
0.314
0.180
0.292
0.157
0.251
0.173
11266
0.134
0.237
1107
0.23
0.379
0.415
0.200
0.309
0.258
0,361
0.112
0.223
0.157
0.256
0.275
0.364
0.201
0.316
0.174
0.283
0.207
0.315
0.161
0.362
0125
0.34
0.535
0.516
0.245
0.344
0.319
0.410
Avg
0.009
0.197
0.139
0.217
0.188
0.286
0.146
0.260
0.122
0.233
0.149
0.261
0.126
0.230
0.106
0.223
103316
0,370
0.170
0.262
00.316
0,322
PEMSUS
0.874
0.176
0.079
0.153
0.100
0.209
0.091
0.199
0.084
0.147
0,096
1.200
0.111
0.205
0.161
0.744
[0:21
0.233
0.106
0.223
0,013
0.225
0.104
0.205
0.137
0.222
0.168
01273
0.138
0.345
0.123
0.27
01254
0.139
0.232
0.17
0257
0.202
0.325
0.162
0.275
0.199
0,202
0.134
0.237
0.158
0.260
0.244
0.333
0.199
0.563
0.170
0.268
0.203
0.203
0.168
0.260
0.148
0.252
0.376
0.427
0.234
0.331
0.295
0.371
0.168
0.263
0.203
0.295
0.327
0.289
0.255
0.338
0.218
0,305
0.247
0.334
0.189
0.272
0.175
0.270
0.543
0.527
0.301
0.382
0.389
0.429
Avg
(0.120
0.221
(0119
0.240
|0310
0.301
|0.171
0.211
0.149
0:347
0.174
0.275
0.152
0.343
01553
0.258
10:318
0.378
0.201
0.908
|0:349
0.393
39
0
D.6 RANDOM SEED SENSITIVITY
We provide additional experiment results of random seed sensitivity in Table 9. The results include the mean and standard deviation from experiments using five different random seeds (2021, 2022, 2023, 2024, 2025) in Table 9, which showcase minimal sensitivity to random seeds.
D.7 COMPLEXITY
We provide additional experiment results of the running time of QDF in Fig. 9. Specifically, we investigate (1) the complexity of each inner-loop update, i.e., calculating Zx with fixed Σ for updating 0, and (2) the complexity of each outer-loop update, i.c., calculating Es with fixed 0 for updating Σ. The forward phase calculates LE while the backward phase performs updates.
As expected, the running time for both forward and backward phases increases with the forecast horizon T, since T determines the size of the weighting matrix Σ involved in the learning objective. Nevertheless, the running time remains below 2 ms even when T increases to 720. Moreover, QDF's additional computations are confined exclusively to the training phase and are entirely isolated from inference.
18

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Published as a conference paper at ICLR 2026
Table 7: Comparable results with different learning objectives.
FreDF
Koopıan
Solt MTT-DDTW
Metries
MAE
MSE
MSE
MAE
MSE
Forecast model:TQNet
96
0.307
0.349
0.309
0.351
0.114
0.355
0.806
0.578
0.315
0.353
0.310
0.352
192
0.352
0.376
0.353
0.375
0.619
0,515
0,360
0.377
336
0,383
0.398
0.383
0.398
0.382
0.396
0.507
0.468
0.398
0.402
0.388
0,400
720
0.441
0.434
0.444
0.436
0.444
0.432
0.450
0.437
0.476
0.446
0.450
0.437
Avg
0,371
0.389
0.372
0.390
0,375
0,390
0.595
0.499
0.387
0.394
0,376
0.391
96
0.365
0.389
0.381
0.395
0.369
0.391
0.415
0.425
0.379
0.390
0.372
0.391
E1Thl
192
0.427
0.421
0.427
0.424
0.425
0.422
0.430
0,422
0.437
0.424
0.430
0.424
336
0.466
0.449
0.471
0.444
0.467
0.445
0.474
0.445
0.488
0.453
0.486
0.454
720
0.466
0.467
0.469
0.466
0.468
0.469
0.483
0.474
0.510
0.487
0.507
0.486
Avg
0.431
0.431
0.437
0.432
0.432
0.432
0.451
0,442
0.453
0.438
0.449
0.439
96
0.135
0.229
0.136
0.228
0,136
0.228
0.137
0.231
0,162
0.258
0.143
0.237
192
0.153
0.245
0.154
0.245
0.155
0.245
0.154
0.247
0.446
0.449
0.161
0.252
BCL.
336
0.169
0.262
0.171
0.262
0.172
0.263
0.171
0.264
0.912
0.675
0.178
0.270
720
0.202
0.290
0.208
0.293
0.209
0.293
0.204
0.292
0.971
0.715
0.218
0.303
Avg
0.165
0.257
0.167
0.257
0.168
0.257
0.166
0,258
0.623
0.524
0.175
0.265
96
0.158
0.201
0.159
0.201
0.158
0.199
0.223
0.268
0.161
0.202
0.160
0.203
Weather
192
0.207
0.245
0.209
0.246
0.209
0.246
0.269
0.304
0.212
0.247
0.210
0.247
336
0,263
0.286
0.268
0.290
0.266
0.288
0.291
0.309
0,270
0.289
0.267
0.289
720
0.342
0.339
0.344
0.341
0.344
0.341
0.346
0.343
0.378
0.365
0.346
0,342
Avg
0.242
0.268
0.245
0.269
0.244
0.268
0.282
0.306
0.255
0.276
0.246
0,270
Forecast model:PDI
96
0.320
0.358
0.326
0.361
1,051
0.663
0.323
0.362
0.326
ET0ml
192
0.361
0.380
0.371
0.386
0.372
0.388
0.420
0.414
0.371
0.388
0.365
0.381
0.401
0.409
0.415
0.408
720
0.451
0.437
0.448
0.439
0.453
0.443
0.456
0.448
0.480
0.454
0.458
0.437
Avg
0.381
0.394
0.386
0.399
0.387
0.400
0.587
0.485
0.396
0.404
0.387
0.396
96
0.380
0.403
0.632
kT1h]
192
0.423
0.419
0.422
0.425
0.423
0.426
0.424
0.429
0.430
0.432
0.440
0.428
0.463
D.441
0,450
0.453
720
0.484
0.468
0.485
0.483
0.475
0.476
0.476
0.478
0.511
0.496
0.495
0,482
Avg
0.436
0.429
0.438
0.438
0.437
0.435
0.497
0.472
0.447
0.447
0.452
0.440
96
0.173
0.253
0.278
0.164
0.250
192
0.177
0.261
0.1
0.262
0.179
0.261
0.173
0,260
0.387
0.410
0.182
0.266
DCT.
336
0.192
0.277
0.196
0.282
0.196
0.278
0.189
0.276
0.966
0.698
0.197
0.282
720
0,234
0.312
0.229
0.307
0.237
0.312
0.228
0.310
1.263
0.834
0.237
0.315
Avg
0.194
0.277
0.195
0.276
0.194
0.274
0.196
0.281
0.695
0.548
0.198
0.281
96
0.178
0.202
0.178
Weather
192
0.225
0.260
0.236
0.267
0.235
0.268
0.225
0.258
0.232
0.262
336
0,280
0.299
0.284
0.304
0.274
0.295
0.280
0.302
0.281
0.296
0.285
0.300
720
0.357
0.347
0.357
0.348
0.356
0.350
0.353
0.347
4.502
1.036
0.360
0.348
Avg
0.264
0.284
0.268
0.287
0.268
0.290
1.296
0.452
0.265
0.283
19

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Published as a conference paper at ICLR 2026
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0.42
0.40
0.38
0.90
0.38
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(d)
ETTh2
Figure 8: Performance of different forecast models with and without QDF. The forecast errors are averaged over forecast lengths and the error bars represent 50% confidence intervals.
Table 8: Varying input sequence length results on the Weather dataset.
Models
QDF
TQNet
QDF
PatehTST
Metries
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.158
0.201
0.160
0.203
0.180
0.224
0.189
0.230
192
0.207
0.245
0.210
0.247
0.226
0.262
0.228
0.262
336
0.263
0.286
0.267
0.289
0.279
0.300
0.288
0,305
0.347
0.242
0.268
0.246
0.270
0.260
0.283
0.267
0.288
Input sequence length
96
0.152
0.199
0.151
0.197
0.161
0.208
0.163
0.209
192
0.198
0.241
0.198
0,241
0.207
0.248
0.207
0,249
0.252
0.282
0.283
0.287
0.268
0.293
0.324
0.332
0.327
0.334
0.337
0,511
0.451
0.231
0.263
0.232
0.264
0.240
0.270
0.287
0.301
96
0. 148
0. 198
0.149
0.198
0.160
0.214
0.158
0.208
192
0.195
0.240
0.196
0,243
0.204
0.253
0.235
0.291
0.279
0.287
0.252
720
0.313
0.327
0.318
0.331
0.324
0.338
0.326
0.336
Avg
0.225
0.261
0.263
0.273
0.243
0.280
96
0.148
0.199
0.155
0.206
0.161
0.217
0.153
0.205
0.192
0.203
0.251
0.205
0.255
0.205
0,254
0.295
720
0.310
0.329
0.319
0,339
0.315
0.337
0.317
0.339
Avg
0.264
0.273
Table 9: Experimental results (mean with varying seeds (2021-2025).
Dataset
BCL
Weather
Models
QDF
QDF
Metries
MSE
MAE
MSB
MAE
MSE
MAE
MSE
MAE
0.135
+0.000
0.229
0.000
0.237 o. ooe
16160
0.001
0.203+a.001
0. 0.160=a..oo
0.203 o 001
192
0.153
0.245
0.161
0.252 ooe
0.208
0.246
0,21 oo1
0.248 001
336
0.169
1. 060
0.262 0.26220.000 n00
0.17820.000
200000
0.270 one
0.264
0.287 001
0.266 001
0.289 001
0.202-40.00
0.291
0.218 200000
0.303 000
0.343g0.0m1
0.340
0.345
0.342.4
Avg
0.165 10.001
0.257_0..o0
0.175.00.00
1 ooe
0.244_p.oor
0.269 2o.00
0.24620.com
0.271 001
20

---

Published as a conference paper at ICLR 2026
(118)
1.6
Time (ms)
1.75
1.4
Timo (mc)
1.50
1.00
1:25
1.0
(a) Running time in the forward phase.
(b) Running time in the backward phase.
The running time of the QDF algorithm given varying forecast horizons (T). In each sub- figure, the left panel considers the complexity of each inner-loop update (i.e., step 4 in Algorithm 1), the right panel considers the complexity of each outer-loop update (i.e., step 7 in Algorithm I),
Figure 9:
Table 10: The comprehensive results of different learning objectives on the MAPE metric.
Datasel
QDF
Time al
FreDF
Koopman
Dilate
Soft DTW
DF
Forecast MoNel:TQNet
ETTml
2.305
2.315
2.313
2.783
2.338
2.319
2.338
ETTh1
9.619
9.697
9.875
10.488
10.036
10.283
10.290
ECL
2.578
3.121
Forecast model:PDF
ETTml
2.232
2.420
2.412
2,876
2,421
2.391
2.283
ETTh1 9.565
10.563
10.468
11.255
10.648
10.980
9.846
ECL
2.750
2.795
2.787
2.757
2.749
5.199
2.869
Weather
3,156
3.226
3.189
3,293
3,244
5.064
3.228
Therefore, QDF introduces no additional complexity to model inference, and the extra computational cost during training is minimal.
D.8 PERFORMANCE ON THE MAPE METRIC
We provide additional experimental results of the MAPE metric in Table 10 on four datasets. The underlying forecasting model is selected as TQNet (Lin al., 2025) and PDF (Dai al., 2024) for their competitive performance. Overall, QDF achieves the best performance on 7 out of 8 cases, thereby further substantiating its effectiveness on the MAPE metric.
D.9 SHORT-TERM FORECASTING RESULT
We provide additional experimental results of the short-term forecasting task the M4 dataset in Table 11. The forecasting architectures are selected as TONet (Lin et al., 2025) and PDF (Dai et al., 2024) for their recency and competitive performance. Overall, QDF consistently performs best in 21/30 cases, yielding the best overall performance.
D.10 CONVERGENCE OF ALGORITHMS
We visualize the loss curves of Algorithm and 2 in Fig. 10 to demonstrate the convergence of the two algorithms.
D.11 VISUALIZATION CORRELATION MATRIX
We provide additional experimental results to visualize the correlation matrix learned by QDF in Fig.
E STATEMENT ON THE USE OF LARGE LANGUAGE MODELS (LLMS)
In accordance with the conference guidelines, we disclose our use of Large Language Models (LLMs) in the preparation of this paper follows:
21

---

Published as a conference paper at ICLR 2026
Table 11: The comprehensive results on the short-term forecasting task.
Loss
Time
PreDE
Koupman
Menric
SMAPE
MASE
OWA
SMAPE
MASE
OWA
SMAPE
MASP:
(WWA
SMAPE
MASE
(WA
SMAPE:
MASE:
(WW
Forecast
ToNel:TQNet
Yearl
13.355
3.015
0.788
13.377
3.004
0.787
13.404
3,022
0.790
22.588
5.512
1.385
13.502
3.074
0.800
Quarterty
10.018
1.174
0.883
10.174
1,200
0.899
10.116
1,196
0.895
17.713
2.415
1.685
10.132
1.192
0.895
Monthly
12.756
0.884
12.776
0.949
0.889
12.786
0.952
0.891
18.655
1.506
1.355
12.777
0.945
0.887
3.285
Average
0.857
Forecast
model:PDF
Yearly
13.426
3.044
0.794
13.524
3.014
0.793
13.479
3.052
0.796
23.515
5.695
1.436
13.532
3.036
0.796
Quanerly
10.361
1.224
0.917
10.690
1,282
0.953
10.367
1,241
0.923
19.090
2.572
1.804
10.646
1.279
0.950
Monthly
12,930
0.961
0.900
13.181
1.003
0.928
13.023
0.987
0.916
20.595
1.756
1.540
13.208
0.999
0,928
Others
4.891
3.262
1,029
5.012
3.256
1.04]
5,381
3.579
9.890
8.213
2.336
5.698
3.735
1,188
Average
12.026
1.618
0.866
12:254
1.645
0.882
12.108
1.653
0.879
20.370
3.181
1.583
12.292
1.672
0.890
0.25
0.72
LO88
0.48
0.20
0.38
0.24
0.18
0.63
500
300
Step
Step
Step
0.96
0.96
Loss
0.70
0.72
0.84
0.72
www.mmm
0.35
Step
Step
Step
(a) The dynamics of loss functions in Algorithm across different epochs.
0.21
0.42
Loss
Loss
0.20
Loss
0.15
0.40
0.18
0.38
2
7
2
3
8
7
Epoch
Epoch
Epoch
0.17
0.38
Loss
0.16
0.16
0.34
0.15
0.32
10
16
22
10
Epoch
Epach
Epach
(b) The dynamics of loss functions in Algorithm 2 across different epochs.
Figure 10: The dynamics of loss functions ETTml, ETTm2, ETThl, ETTh2, ECL, and Weather from left to right. The forecast horizon 336.
Table 12: Running complexity of QDF algorithm under task splits K=3.
Direction
Loop
T=64
T=96
T=128
T=192
T=256
T=336
T=512
T=720
Porward
Inner
199 1.175.0.01
1.1762a oos
1.409 _0002
1.41 20.00
1.523 1.590go.01
012
1.763go.011
Outer
10.1100010
1.168 014
1.16240 ooe
1.350_0.om
1.36620.001
1.357
1.361
1.67220.012
Backward
000
1.14440.000 005
1.13740.000 000
_8.7-
1.492+0.008
1.647 164740.000 0o7
ao.00 1.770=n.010
Outer
0.95440.006
0.967 00008
0.950
1.190=p.0m
1.94400.009
1.343
1,296
1.477=0.000
22

---

Published as a conference paper at ICLR 2026
Table 13: The performance of varying ranks of the inverse covariance matrix Σ.
Rank
Ranke1
Rank=D08
Rank=0.6
Rank=0.4
Rank=0.2
DF
Metrics
MSE
MAE MAPE
MSE
MAE MAPE
MSE
MAE MAPE
MSE
MAE MAPE
MSE
MAE MAPE
MSE
MAE MAPE
Forecast
TQNet
96
0.307 0,349 2.156
0.308
0.350
2,191
0,310
0.351
2.174
0.309
0.351
2,200
0.310
0.352
2.201
0.310
0.352
2.212
192
0.352 0.376 2.281
0.354
0.377
2.288
0.355
0.379
2.308
0.354
0.378
2.298
0.352
0.377
2.295
0.356
0.577
2.288
336
0.383
0.398 2.529
0.387
0.399
2.349
0.387
0.399
2.344
0.387
0.400
2.356
0.387
0.401
2.357
0.388
0.400
2.338
720
0..44
0.434
2.522
0.445
0.437
2,534
00,446
0.437
2.546
(0.443
0.437
2,530
0.445
0.437
2.538
00.450
0.437
2.516
Avg
[0.371
0.389
2.322
2.341
[0.375
0.392
[0.373
0.391
2.346
0.392
|0.376
0.391
2.338
96
[0.158
0.201
2.619
0.158
0.201
2.621
0,,15
0,202
2.635
00.158
0.201
2.653
0.159
0.202
2,602
00.160
0.203
2.642
Weather
192
0.2007
0.245
3.047
0.208
0.246
3.035
0.208
0.246
3.051
0.207
0.246
3.049
0.207
0.246
3.065
0.210
0.247
3.087
336
0.263
0,286
3.902
0.264
0.287
3.343
0.264
0.287
3,130
0.263
0.287
3.323
0.264
0.287
3.337
0.267
0.289
3,352
720
0.344 0.339
3.372
0.342
0.339
3.354
0.342
0.339
3.378
0.342
0.339
3.355
0.342
0.340
3,380
0.346
0.342
3.403
Avg.
[0.242 0,268 3.085
0.268
3.088
00.243
0,269
3.098
00.243
3.095
0.243
0.269
3.096
00.246
0.270
3.121
Forecust
mode::PD
96
[0.320 0.358 2,176
0.314
0.355
2.142
00,315
0.353
2,066
0.317
0.358
2.105
0.314
0.356
2.126
0.326
0.363
2.179
[0.361 0.380 2.210
2.249
0.358
0.376
2.191
0.379
2.194
0.380
0.365
0.381
2.209
336
0.390 0,401 2:277
0.389
0:401
2.253
0.389
0.402
2.276
0.389
0.399
2.245
0.389
0.398
2,280
0.397
0.402
2.279
720
00:451
0.437
2.487
0.449
0.434
2.469
[0:447
0.434
2.467
0.448
0.435
2.468
0.447
0.435
2.475
00.458
0.437
2.467
Avg
[0.381
0.394
2.287
0.377
0.392
2.278
|0.377
0.391
2,250
[0.378
0.393
2.253
0.378
0.392
2,283
|0.387
0.396
2.283
96
10.176
0.218
2.708
0.178
0.219
2.762
00.181
0.221
2.805
0.179
0.220
2.776
0.179
0.220
2,826
001181
0.221
2.812
0.225
0.260
3.164
0.260
3.142
0.227
0.261
0.260
3.165
0.260
0,232
0.262
3.213
336
0.280
0.299
3.391
0,280
0.298
3,408
0,281
0.298
3.387
0.279
0.299
3.407
0.280
0.299
3.408
0.285
0,300
3.407
720
0.357
0.347
3.443
0.357
0.347
3.445
0.355
0.347
3.457
0.357
0.348
3.472
0.356
0.347
3,440
0.360
0.348
3,482
Avg.
(0.259
0.281
3.176
0.260
0.281
3,189
|0.261
0.282
3.205
|.2660
0.282
3,205
0,260
0.282
3.208
|0.265
0.283
3.228
1.000
0
1.000
0
1.000
96
0.100
0.100
0,010
0.010
0,010
192
192
0.001
0.001
0.001
288
268
288
0.000
0.000
98
192
192
281
144
208
(a) The learned correlation matrix given forecast horizon
336
1.000
1.000
1.000
144
0.100
192
0.100
0.100
268
0.010
384
0.010
0.010
432
676
0.001
576
0.001
0.001
0.0000
0.0000
720
-0.000
96
192
285
384
480
576
672
192
288
384
480
578
672-
144
452
578-
720
(b) The learned correlation matrix given forecast horizon
720
Figure 11: The learned correlation matrix different datasets: ETTm1, ETTm2 and Weather from left to right.
We used LLMs (specifically, OpenAl GPT-4.1, GPT-5 and Google Gemini 2.5) solely for checking grammar errors and improving the readability of the manuscript. The LLMS were not involved in research ideation, the development of research contributions. experiment design, data analysis, or interpretation of results. All substantive content and scientific claims were created entirely by the authors. The authors have reviewed all LLM--ssistee text to ensure accuracy and originality, and take full responsibility for the contents of the paper. The LLMs are not listed as an author.
23


Reviews:
[{"review_id": "gXspFxabJK", "reviewer": "Reviewer_LpcK", "summary": "The authors propose a method named Quadratic Direct Forecast (QDF) for time series forecasting. The key idea is to optimize the training objective via a quadratic-form weighting matrix to capture both label autocorrelation and weighing different forecasting tasks. The authors apply their proposed method to baseline models and show that it improves the performance in prediction accuracy.", "questions": "1. How does the model performance change with the dimensionality of the time series? \n\n2. Related to above, is there a curse of dimensionality in the training objective of QDF? If so, how can this issue be effectively addressed?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Overall this paper is clearly written and easy to understand. The technical details are sound and mostly sufficient. \n\n2. The proposed method to improve the training objectives for time series forecasting is novel and applicable to related problems in this domain. \n\n3. The authors provide the source code of their implementation. After reviewing the source code, I did not find major issues. \n\n4. The authors perform rigorous evaluations and helpful ablation studies to understand the impact of different components in the proposed QDF framework.", "weaknesses": "1. In addition to MAE and MSE, the authors should evaluate their proposed method with MAPE (mean absolute percentage error) which is robust under different scales of the time series values.\n\n2. The authors should also evaluate their proposed method on standard benchmark datasets for time series forecasting, such as the M4 competition dataset.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sAZt4mJTka", "reviewer": "Reviewer_VKq3", "summary": "This paper presents QDF, a method for training forecasting models using a trained covariance matrix, thus solving the problem of assuming uncorrelated residuals. The authors formulate learning $\\Sigma$ as a bilevel problem, which they solve sequentially over $K$ splits of the data, then train the forecasting model using the learned (static) $\\Sigma$. They empirically compare the QDF objective with other time series losses, explore design choices for learning the covariance and comment on the sensitivity of the introduced hyperparameters.", "questions": "- Are some architectures more prone to converge to correlated residuals than others? \n- In the Ablation  study (4.4) how are the two QDFs integrated, by taking their average?\n- Visualizing the correlation matrices of the residuals after training with QDF would be interesting to see.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- The paper introduces a technically novel strategy to solve an empirically and theoretically motivated problem that is much relevant in neural, (direct) non-autocorrelational models for time series forecasting. \n- Promising empirical evidence of QDF on a comprehensive set of datasets, against transformer and non-transformer based methods, and against a variety of time series optimization objectives.\n- Ablation studies and the effect on various architectures is provided.", "weaknesses": "- A theoretical result or comments on the convergence of Algorithms 1 and 2 would make the presentation stronger.\n- Emphasizing the above point, the proposed method may be too computationally expensive for some models, given the bilevel optimization, over $K$ splits required to perform the optimization to find $\\Sigma$. \n- Some parts of the text are not very clear: For instance, on the results in Section 4.2, which model is used to compare the different forecasting objectives? Similarly in Table 3 (Ablation study).\n- The clarity  of the last paragraph of Secton 3.2 could be improved.\n- The connection of this method with meta-learning is not very clear as it is not formally stated.\n- There's the unmentioned assumption that the time series is non-heteroscedastic and thus is characterized by a single $\\Sigma$", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GuSo7vrHYp", "reviewer": "Reviewer_wfuZ", "summary": "This paper investigates the learning objective design of time-series forecasting models. Specifically, it proposes a novel quadratic-form weighted training objective (QDF) to address two issues of existing methods: (1) oversight of the label autocorrelation effect among future steps, and (2) failure to set heterogeneous task weights for different forecasting tasks.  Experiments showcase the validity of QDF on public datasets.", "questions": "1. In Definition 3.2, the optimization problem optimizes $\\theta$ in $D_{in}$ and then $\\Sigma$ in $D_{out}$. Compared to using the full dataset, i.e., the concatenation of $D_{in}$ and $D_{out}$ to update $\\Sigma$ and $\\theta$, what is the advantage of the splitting strategy here?\n\n2. Whether the proposed methodology is directly applicable to general multitask learning tasks? Is there any limitation or necessary adaptations when generalizing beyond the TSF context?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper improves the learning objective for TSF models, which is an insufficiently explored yet important research problem in time-series forecasting.\n\n2. Experiment results are effectively presented and structured to support the paper's claims.\n\n3. Code is provided which facilitates reproduction.", "weaknesses": "1. The paper could benefit from a more analytical exploration why the proposed weighting matrix could possibly improve existing formulations, especially FreDF and Time-o1. \n\n2. For time-series datasets, careful design of data splitting strategies is essential to avoid information leakage. It is not very clear whether the risk of leakage is fully bypassed in the hybrid splitting strategy of this paper (e.g., validation, meta-update, etc.). A detailed analysis is needed to analyze the leakage issue as well as the strategies or designs to avoid it.\n\n3. Although the code is released, the  introduction to key components is lacking which impedes verification: how to reproduce the results with the used hyper parameters? where is the key code components? and what is the required software environment for reproducing? whether the code is built upon established repos?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)