Fast and Slow Streams for Online Time Series Forecasting Without Information Leakage
Authors: Ying-yee Ava Lau
Zhiwen Shao
Dit-Yan Yeung
Venue: ICLR 2025 Poster (2025)
Keywords: online time series forecasting
concept drift
online learning
Primary area: learning on time series and dynamical systems
TL;DR: Redefined the setting of online time series forecasting to prevent information leakage and proposed a model-agnostic framework for this setting.

Abstract: Current research in online time series forecasting (OTSF) faces two significant issues. The first is information leakage, where models make predictions and are then evaluated on historical time steps that have already been used in backpropagation for parameter updates. The second is practicality: while forecasting in real-world applications typically emphasizes looking ahead and anticipating future uncertainties, prediction sequences in this setting include only one future step with the remaining being observed time points. This necessitates a redefinition of the OTSF setting, focusing on predicting unknown future steps and evaluating unobserved data points.  
Following this new setting, challenges arise in leveraging incomplete pairs of ground truth and predictions for backpropagation, as well as in generalizing accurate information without overfitting to noise from recent data streams. To address these challenges, we propose a novel dual-stream framework for online forecasting (DSOF): a slow stream that updates with complete data using experience replay, and a fast stream that adapts to recent data through temporal difference learning. This dual-stream approach updates a teacher-student model learned through a residual learning strategy, generating predictions in a coarse-to-fine manner. Extensive experiments demonstrate its improvement in forecasting performance in changing environments. Our code is publicly available at https://github.com/yyalau/iclr2025_dsof.

Published as a conference paper at ICLR 2025
FAST AND SLOW STREAMS FOR ONLINE TIME SERIES FORECASTING WITHOUT INFORMATION LEAKAGE
Ying-yee Ava Lau, Zhiwen Shao, Dan Yeung Department of Computer Science and Engineering The Hong Kong University of Science and Technology yya 1 pallaueconnece .ust.ht hk, zhi wen hk, dyyeungecse.ust ust.
ABSTRACT
Current research in online time series forecasting (OTSF) faces two significant issues. The first is information leakage, where models make predictions and are then evaluated on historical time steps that have already been used in backprop- agation for parameter updates. The second is practicality: while forecasting in real-world applications typically emphasizes looking ahead and anticipating fu- ture uncertainties, prediction sequences this setting include only one future step with the remaining being observed time points. This necessitates redefinition the OTSF setting, focusing on predicting unknown future steps and evaluating unobserved data points. Following this setting, challenges arise in leveraging incomplete pairs of ground truth and predictions for backpropagation, as well as generalizing accurate information without overfitting to noise from recent data streams. To address these challenges, we propose novel dual-stream framework for online forecasting (DSOF): a slow stream that updates with complete data using experience replay, and fast stream that adapts to recent data through tem- poral difference learning. This dual-stream approach updates teacher-student model learned through a residual learning strategy, generating predictions in a coarse-to-fine manner. Extensive experiments demonstrate its improvement forecasting performance in changing environments. Our code is publicly avail- able at btttps://gtttuee http://wwww...ayyyaaaaaa/aaaa0/. dsof
INTRODUCTION
Accurate forecasts benefit diverse applications like electricity consumption monitoring (Zhu al. 2024). climate modeling (Mudelsee 2019), retail (Böse et al. 2017), and stock markets forecasting (Feng et al. 2019). Consequently, signilicant efforts have focused on better forecasting performance by enhancing deep neural networks to identify complex dependencies in time series data (Zhou) al. 202T; Nie et al.) 2023] Wu et al., 2023). An emerging area is online time series forecasting (OTSF) (Anava et al. 2013; Pham et al., (023), where the forecasting model generates time series sequences using moving window of and updates the model immediately after corresponding ground truth sequences arrive. This incremental update adapts the model to real-world complications like dynamic datasets and distribution shifts.
1.1 REDEFINING THE ONLINE TIME SERIES FORECASTING SETTING
Existing works that endeavor improve OTSF model accuracies (Pham et 2023 Zhang al., 2023 2024a) follow uniform task setting: at time t = the model receives input data with H +
and generates predictions from / to = i + 1; at the next timestamp = i 1, the actual data for t = + becomes available, allowing the model to evaluate its previous prediction and adjust its parameters based on this new ground truth. For instance, as illustrated in Figure [ if the input and output windows are of lengths L= 5 and H = 4 respectively, at time t = 10, the model first takes data with timestamps from t t= 3 t 7 and predicts values from t 8 to t 11. Once the actual data fort 11 is available, the model's 'sns predictions are evaluated against the ground truth. Subsequently, the ground truth from t 8 to 11 is used in backpropagation update the model's parameters. At l 12, the same
timestamps from
L.
2 to i

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procedure performed at t = 11 is repeated. advancing predictions by unit of time. This setting aligns with the conventions online supervised learning for regression and classification tasks (Hoi et al. 202T): p: it allows immediate updating of model parameters the subsequent time step. As the farthest value prediction is one timestamp ahead the currett time, all ground truth data for the predicted sequence is available the next timestamp.
We raise two concerns about the mentioned setting. First, there is an issue information leakage. Originally defined in data mining as the use of external data outside the training dataset (Kaufman et al. 201T. information leakage, formally described in Section C occurs here because the eval- uation
12 contains data points from optimize the model parameters at = 11. This overlap of time steps leads to biased evaluation overestimation of the model's effectiveness in real-world applications. Second, we question whether it aligns with the general conception of forecasting: forecasting tasks are designed predict future events (Petropoulos et al. 2022). However, in this scenario, the model only forecasts one future step any given timestamp. For example, at 10, it predicts values from t 8 to t- = 11, even though the ground truth from 8
9 to t 11, which have already been utilized to
outcomes, resulting
is already known.
Att
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At ground truth att 11 becomes available. 1=3
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1=
1
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Figure Overview of the data streaming framework in existing online forecasting studies. For illustration, in this figure, we set the lookback window to L 5 and forecast horizon to H = 4. Information leak occurs because ground truth from gation at / 11 and is re-evaluated at 12.
11 was already used in backpropa-
Consequently, redefine the OTSF problem meet two key criteria: the model is evaluated only on time steps without backpropagation for updates, and the output window should be equal to the number of unknown future time steps forecast. For example, illustrated in Figure 2 with input window L = 5 and output window H = 4, at / = 10, the model takes in data from t = 6 to t 10 for forecasting values from f 11 to 14. At 11, the model can only be updated with ground truths with timestamps up to = 11. The prediction sequence generated at t = 10 can only be evaluated on or after 14, when all actual data from 11 to 14 available.
1.2 CHALLENGES AND PROPOSED SOLUTION FOR THE NEW SETTING
cannot consequent challenge of this new setting is that f 11, the sequence predicted be fully updated in supervised manner due to the absence of ground truth from t= 12 to f = 14. This limitation would prevent the model from effectively learning the most up-to-date information in fully supervised manner, as is typically observed in conventional time series forecasting (Benidis) et al. 2022). Moreover, sparse supervisory signals offer limited feedback, hindering the model's ability to learn complex patterns. This make the model overly sensitive to limited data, which may increase the chance of overfitting if recent data contains noise or anomalies. To address the challenges of the new setting, further propose online training framework, called DSOF, that
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t-6
t-7
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=tt== ground truth at becomes available. The model usn data points with timestamps a update its paromaters.
t=6
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At ground truth sit available.
!-7
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Figure 2: Overview of the redefined OTSF setting. Att = 10, the model only forecasts the unknown H future steps from = 11 t = 14. At = 11, it can only update using ground truth observations up to 11. Predictions made t = 10 can only be evaluated on or after 14, once all actual data from = t = 14 are available.
leverages the latest and most immediate knowledge from the data stream while maintaining accurate generalization in the data distribution.
Our proposed DSOF framework employs fast and slow streams update a teacher-student model built on the residual learning approach. Given input data sequence, the teacher model generates coarse predictions based on historical patterns, while the lightweight student model refines these predictions using real-time feedback. This dual-stream mechanism benefits the online training pro- cess in adaptability and stability. The fast stream, inspired by temporal difference (TD) learning's intermediate updates in reinforcement learning, leverages the latest information from data streams to improve near-future forecasts without waiting for the complete ground truth sequence. Meanwhile, the slow stream conducts experience replay (ER) to stabilize training, penalizes overfitting the noise in the incoming data, and ultimately learn accurate generalization of the data distribution. Through extensive experiments, observe an overall improvement in prediction performance using this integrated strategy across various datasets and forecasting horizons.
1.3 CONTRIBUTIONS
In summary, our contributions are as follows:
We redefine the OTSF setting by restricting the model from predicting and evaluating time steps for which it has already received ground truth feedback. This approach eliminates information leakage and aligns more closely with the forecasting concept as applied in real-world scenarios.
We develop a novel framework, DSOF, which employs dual-stream mechanism inte- grate real-time data with previously collected data streams for updating model parameters. This approach ensures the maintenance accurate, current knowledge while also provid- ing robustness and training stability in dynamic environments.
We conduct extensive experiments to validate that our framework significantly enhances forecasting performance through online updates. Additionally, we provided comprehen- sive empirical studies that explore various design choices for online training of forecasting models, offering insights into optimizing performance and adaptability in real-time scenar- ios.

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2
PRIOR WORKS
Concept Drift. Underlying dynamics in real-world time series change over time, leading to differ- ences between training and test data distributions (Tsymbal 2004). This poses a major challenge for deep neural networks (DNNs) in time series forecasting. To improve DNN robustness against such non-stationarity during test time, various model-agnostic methods and architectural innovations have been introduced. [Li et al. (2022) introduced dynamic data generator for predicting future data distributions, so that models can train on anticipated rather than current distributions. RevIN (Kim al., 2022) employs reversible instance normalization to adjust for changing statistical properties over time. Building on this, Liu et al. (2023) proposed Slice-level Adaptive Normalization (SAN), which normalizes local temporal slices to handle non-stationarity more granularly. Dish-TS (an framework, can better handle the distributional differences be- tween input and output spaces by separately modeling the distributions of each space. From an ar- chitectural perspective, Non-Stationary Transformers, proposed by Liu et al. (2022), combine series stationarization with de-stationary attention to balance predictability and complexity. Additionally, Koopman based neural operators (Wang et al.) 2023 Liu et al.) 2024bb have been incorporated to deep learning architectures to better adapt to shifting data distributions. Despite their advancements, however, these methods mainly address evolving trends and fail to learn patterns that emerge during test time, necessitating research into real-time online learning for non-stationary time series forecasting.
et al. 2023). using
Online Forecasting. Unlike traditional forecasting tasks, where training and evaluation are separate phases, online forecasting involves continuous learning over multiple rounds. Annva al. (2013) pioneered an online forecasting model by updating the parameters of the traditional ARMA model using regret minimization techniques. However, statistical models like ARMA have limited ca- pacity to capture complex temporal dependencies, prompting shift towards deep learning models for better performance. |hhm al. (2023) proposed FSNet by extending the temporal convolu- tion network (TCN) (Bai et al.) (018)) through the introduction of calibration module for weight adjustments and an associative memory for recalling events, enabling quick adaptation to recent changes. As previous models mainly employed only cross-variable approaches. Zhang et al. (2023) proposed OneNet, which integrates cross-variable and cross-time models to capture complex time series dependencies. However, none of the mentioned methods explicitly detect concept drift. The retect-and-Adaa method (Zhang et al. 22224) was proposed to first identify concept drift and then adapt the model using different learning scheme, balancing rapid adjustments with sustained per- formance. Additionally, research on reducing the computational cost of online updates has emerged, utilizing techniques like hyperdimensional computing (Mejri et al. 2024) and the Moore-Penross, inverse (Zhang et al., 2024b) for quicker online forecasting on lightweight devices. Nonetheless, all of these deep learning approaches failed address the issue of information leakage when using online learning for time series forecasting tasks, allowing the model to predict and evaluate data points that have already been observed.
3 METHODOLOGY
In this section, introduce proposed framework, DSOF, which facilitates the online training of time series deep learning models through fast and slow data streams. We describe how this dual- stream mechanism is utilized to update the teacher-student model.
3.1 PROBLEM DEFINITION
Terminology. Consider a time series dataset X = (an,.... .N...) E R-Nan Xn. compris- ing Nama observations, each with n dimensions, where m, E Ra represents the data point at time i. Given a look-back window of length L that ends at time i, denoted as X;-L+1# = (I,-L+1. B:L+2, IX., the task is predict the following I time steps of the series.
(-(1 The model's prediction at time i for the subsequent H steps is represented as 1411+ = (# L B:421" - (X,
Training Phases. We divide the dataset into two segments, using each segment in different training phases. In the first phase, referred to the batch learning phase, uses data with timestamps up
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to Noath to train the model in batch learning manner. In the second phase, referred to the online learning phase, we use data points with timestamps after Mutch to simulate a real-world scenario of sequential data streaming, updating the model in real time. Generating sequences with a moving window of 1, consequently, the number online data sequences given by Nontine = H
Ndata Nhatch
Redefined Setting in the Online Phase. As mentioned in Section 1.1 we redefine the OTSF prob- lem that the model is evaluated only on time steps without backpropagation for updates, and the output window size should align with the number of unknown future steps to forecast. Formally, the model takes in data from t = L to = and forecasts values from t = 2 1 to to = -i H. At f = i + 1, the model can only be updated with ground truths with timestamps f S + The prediction sequence generated = is evaluated only at 2 i + H, when all actual data from / = = + 1 to = + H is available.
with input window L and output window H,
Objective and Evaluation Criterion. Our objective to minimize the cumulative mean-squared loss, which also serves as the evaluation criterion, expressed as in Equation It is important to note that the model does not evaluate the predicted time steps that have already been observed. Specifically, at time i, the prediction + is made without access to the ground truth for any time step greater than i. The entire prediction sequence is evaluated only after t i + H, when all ground truth points are available.
Nau Σ
(X, - X+11++811
N Nonline V=
3.2 TWO MODELS: RESIDUAL LEARNING STRATEGY
The framework comprises a teacher-student model that is built on the residual learning strategy. The teacher model, with parameters denoted as ((T), usually a robust backbone like DLinear ZZng et al. 2023) or PatchTST (Nic et al. 2023), generates coarse predictions for given input sequences, as in Equation
(T.)) 141it 1/0 (X:-1+41100
The student model, with parameters represented as 0(5), often lightweight model such as Multi-Layer Perceptron (MLP), refines these coarse predictions. By concatenating the lookback sequence and the teacher model's predictions as inputs, as indicated Equation 3 the student model estimates the error between the teacher's predictions and the ground truth data.
X (S.0) ++1++ S (Concat( (X-nat(X:=4+1:: ((T.)) x :
The final forecasting prediction is obtained by summing the predictions from both models, as demonstrated in Equation
( # (X ( (()))))))00
(T, it1+ 8
((S.)
3.3
TWO DATA STREAMS: FAST AND SLOW
This section presents the two data streams for updating the teacher-studen model in Section 3.2 slow stream using complete ground truth via ER and a fast stream incorporating latest data through+ a TD
method. The term "TD" indicates that our approach is inspired by TD learning in reinforcement learning, particularly estimating the ground truth values at the next time step, addressing the challenge of missing ground truth in the redefined OTSF setting.
3.3. 1 SLOW DATA STREAM WITH EXPERIENCE REPLAY
ER is a common approach address catastrophic forgetting in continual learning. Unlike continual learning which aims for effective performance on both previously learned and newly acquired tasks (Rolnick et al., 2019), and reinforcement learning which seeks to break correlations from samples taken at adjacent trajectories to enhance network convergence (Mnih et al.] 2013). our focus is on adapting to the evolving temporal domain, aligning with the objective of minimizing cumulative
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mean squared enror (MSE). Thus, instead of promoting diversity in sample patterns, the replay buffer B retains only the most recent samples.
The replay buffer has fixed size Ng. and operates on first-in, first-out basis. At time i, upon receiving the ground truth data 2ú. the buffer appends the pairing of the input sequence Xi-H-L-1:-- H and output sequence X If the buffer exceeds NE, the oldest sample, with the input sequence X;-No- -- I: H and output sequence X;-N= H+1:inNe Ne is removed. This process is referred as the slow update stream, it waits for all ground truth values of the output horizon to arrive before adding the data stream to the buffer. This setup resembles a time- delayed environment in reinforcement learning (Liotet al.) 2022] Wu et al., 2024), where new predictions must be made despite delayed feedback from the environment.
3.3.2 FAST DATA STREAM WITH TEMPORAL DIFFERENCE LOSS
Performing ER in Section 3.3.1 requires complete ground truth data for all time steps in the data stream. A major drawback is that with large prediction length H, we must wait for H time steps for all ground truth values to arrive, before adding the sequence to the replay buffer However, the presence of concept drift, we aim to update the model with immediate observations even without complete ground truth for the entire forecasting horizon - to improve near-future predictions.
At time i - 1. the teacher-studen model generates X (4-1) -11 Right after the prediction, at time i, the ground truth x; becomes available. To compensate for the absence of ground truth for the future
timestamps = ++ 1 to = i H 1, the teacher model's prediction at time t = i, i.e. X (T,55)
is utilized to generate the pseudo labels x(() ii+ -1 as outlined in Equation
X 6++
(T.)) 441++
(5)
In practice, the pseudo labels can be generated with different proportions of ground truth data and teacher model predictions. Consequently, we extend Equation [S) by introducing a parameter k which signifies the use of the latest k ground truth data points along with the initial H - k steps of predictions from the teacher model to form the pseudo label. The impact of using different & values, as well utilizing multiple & values, are discussed in Section A.6.2
is assumed that the teacher model's predictions for the distant future are considerably more un- certain and less significant to the current prediction than those for the near future. Considering the temporal nature of time series data, we introduce a geometric decay factor y, to reduce the influence of prediction errors and pseudo labels for each step further from the current observation, as outlined in Equation 6 We further elaborate on establishing the relationship between the TD approach used in our method and that found in reinforcement learning Section B
H
(X x TD (x
10
- 10
(6)
H
3.4 OVERVIEW OF THE FRAMEWORK IN ONLINE TRAINING
This section outlines our framework in the online training stage, with detailed algorithm presented in Algorithm| During the online training phase, upon receiving each new data point, first update the replay buffer B and then conduct an ER, as detailed in Section 3.3.T This process involves randomly selecting a small batch of size N₂ from the buffer to train and update the teacher- student model. By default, ER performed whenever a new data point arrives, but this frequency can be adjusted two data points. After completing ER, the student model is updated using the TD loss described in Section 3.3.2 To maintain training stability, the teacher model's parameters ((T) remain fixed, while the student model is updated with pseudo labels.
for instance, can be triggered twice per new data point once for every

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Algorithm Overview of DSOF during the online training phase.
1: Initialize:
Replay Buffer B with capacity Ns Batch Size Nb
Learning rates OT, 4s, n.
2: for i == Nosch + to Nam H do
3:
Received ground truth æ;
4: B 4- ( -------+----- X H+1+))
Append the sequence. The buffer exceeds its capacity. vo Remove the oldest sequence.
if |8 2 No then
6:
B B-BBY H- i X Na
7:
end if
8:
if |8|
b The huffer has enough sequences sample. O , 0 +- REAERIENNEE,ELLLYYRRRE OLT), ((T))(()) .(())
No then
9:
10:
end if
TEMPORAL DIFFERENCE UPDATE(X, L.s 86-1) 1
12:
(
0 ((5))
b Teacher-student model forecasts the next step.
4144
13:
end for
14: procedure EXPERIENCE, REPLAY UPDAAEEBB (()))
15:
NH) ~ B
b Sample mini mini-batch from the buffer. Teacher-stuuee student model makes a prediction. Update teacher model's parameters. Update student model's parameters.
Ry = ===ffi1]] f(i-1) (x2; (T) (()) (Equation
( + ((T)
ar Vorr E(K
18:
018) 0(8) Vala ((H,
19:
return )
20:
end procedure
21: procedure TEMPORAL TEMPORAL.DIFEEREEEE WWWFFEREENEEUUUAAEE...CCCC
X 1144-11 (T)) (()))
22:
Freeze the teacher parameters 0 (T)
23:
(T::) Ha++ = p (X; (T)) (Equation
> Teacher model makes a prediction utilizing æ.. Generate the pseudo-label.
24:
1i+H-1 [r., (T.))
++1++ (Fquation
25:
Compute the temporal difference loss Ef (X
2(6)
1-+H-1) using Equation 6
26:
(9))
+ (()) ao V Voss 6it return '
(x(-1)
b Update student model's parameters.
27:
28:
end procedure
4 EXPERIMENTS
In this section, we demonstrate the effectiveness and adaptability of our proposed method. Our experiments involve comparing our framework against batch learning approaches, utilizing a diverse range of model architectures baselines. We also evaluate the effectiveness our method against other online learning frameworks, which we have modified to avoid information leakage. Further analyses are conducted in Section
4.1 GENERAL FORECASTING RESULTS
In this section, DSOF compared to batch learning, demonstrating the advantages of online training and the framework's s ability to adapt batch methods to online scenarios for improved performance.
Datasets. We utilize benchmarks from Autoformer (Wu et al. 2021 which are primarily designed for long-term forecasting tasks. For our experiments, we adhere to the data distribution outlined in Pham et al. (2023). splitting the dataset chronologically into 20% for training, 5% for validation, and 75% for testing, simulate the temporal distribution shift in real world scenarios. The datasets used our experiments include the following: the Electricity (ECL) dataset tracks the electricity consumption of 321 clients from 2012 to 2014; the ETT benchmark tracks oil temperature and six power load features over two years, with ETTh2 recorded hourly and ETTml at 15-minute inter- vals; the Exchange (Ex.) dataset includes daily exchange rates for eight countries from 1990 to

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2016; the Weather dataset, collected every 10 minutes throughout 2020, features 21 meteorological indicators, such as air temperature and humidity: the Traffic dataset provides hourly road occupancy rates from the California Department of Transportation, reflecting measurements from various sen- sors on freeways in the San Francisco Bay Area. Detailed information about the datasets, including the number of features and timesteps, is presented in Table T
Table 1: Statistics of benchmark datasets used.
Electricity
ETTh2
ETTml
Exchange
Traffic
Weather
Features
321 26304
7
7
7 7396
862
21
Timesteps
14400
57600
17544
52696
Baselines. We set prediction lengths H to 1, 24, and 48, with a lookback length of 96, following the experimental setting used in the previous works (Pham al. 2023 /Zhng et al. 2023). In results, batch learning depends exclusively on the teacher model, whereas our method integrates a three-layered MLP with hidden dimension of 16 as the student model, built on top of the teacher model. A variety of backbone architectures for the teacher model are utilized for comparison with batch learning. These included linear backbones such DLínear (Zeng al. 2023 FITS (Xu et al. 2024). and Mixer (Wang et al.] 2024). Additionally, we employed convolution back- bones including FSNet (Pham et al., 2023) and OneNet (Zhang et al. 2023). Lastly, for transformer backbones, we used i'Transformer (iTrans.) (Liu et al.] 2024a)) PatchTST (Nie et al. 2023). and
Non-Stationary Transformer (NSTrans) (Liu al. (022).
Results. To account for variability from random weight initialization, we ran experiments with five random seeds. Table reports the average MSE, while standard deviations are in Section E.] Our framework generally outperforms batch learning across datasets and architectures, effectively ad- dressing distribution shifts online forecasting. Notably, on the ECL dataset, FSNet and NSTrans- former teacher models achieve learning outperforms online approaches. One possible factor affecting performance could be the normalization choices and hyperparameters of the backbone, which are kept the same in batch learning and not further optimized for online learning.
50% MSE reduction. However, in some cases, batch
Visualizations. Figure presents visualizations of forecasting results, illustrating 48-step predic- tion sequence on the Traffic dataset. It is observed that at step 300, where the distribution closely resembles the training distribution, the performance of batch learning similar to that of online learning. However, at later time steps, such as 5000 and beyond, the performance of online learning surpasses that of batch learning.
4.2 EVALUATING OTSF FRAMEWORKS WITHOUT INFORMATION LEAKAGE
Baselines. Previous works, such as FSNet (Pham al. 2023) and OneNet (Zhang et al. 2023 adopted a different OTSF setting that suffers from information leakage, leading to an overestimation of model effectiveness during evaluation. To ensure fair comparison, eliminate information leakage by delaying ground truth feedback. At time t = i, the model receives data from f t= i- H. Unlike before, where predictions were evaluated and immediately backpropagated using the ground truth at t + 1. this procedure performed after H steps, specifically at i H. Here, we name this framework as "Delayed Gradient" (DGrad).
modified this setting to
L + 1 to & i, and generates prediction with time steps from
+ 1 to
We also incorporated continual learning methods, TFCL (Aljundi al., 2019) and DER+ (Buzzega et al. 2020), for comparison, However, these approaches are not tailored for time series forecasting. They were originally utilized in vision tasks rather than forecasting and do not account for labels with temporal contexts. Consequently, the techniques they propose may also be susceptible to infor- mation leakage. We mitigate this risk by delaying the ground truth feedback, employing the same strategy in DGrad.
While DSOF includes a student model, the methods, DGrad, TFCL, and DER++ do not. To ensure fair comparison, additionally present an alternative version of DSOF that excludes the student model. In this version, instead of combining the outputs from both the teacher and student models, rely solely the teacher model's output - this modifies the prediction results from Equation 4
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Table 2: Comparison of MSE results between our framework and batch learning across various datasets and backbones. Our method integrates MLP as the student model built on top of the teacher model, while batch learning relies exclusively on the teacher model. The better results from the two settings are highlighted in bold.
DLinear
FITS
FSNet
OneNet
iTrans.
PatchTST
NSTrans.
H Batch DSOF Batch Learn.
DSOF
Batch
DSOF Batch
DSOF
Batch
DSOF
Batch
DSOF
Batch DSOF Learn.
Learn.
Learn.
Learn.
Learn.
Learn.
1
2.842 2.065
2.870
2.235
3.8c+2 2.330
3.2c+1
4.733
1.976
2,430
4.770
2.244
3.9c+1
2.703
24
1.5c+1
4.737
4,509
4,597
1.7c+255.47
8,5c+1
4,510
4.119
5.155
1.7c+1
5.169
4.0c+1
8.668
2.7e+1 48
6,181
5,257
5.433
1.8e+27.000
1.5e+2
5,943
4.936
6,015
1.7e+1
6.665
3.8e+1
9.981
1 0,470
0.365
0.522
0.375
1.1c+1 0,431
2,531
0.548
0.872
0.384
0.915
0.382
3.088
0.415
24
2.269
1.701
2.189
1.757
1.9c+1 3.114
7.017
2.363
2.688
1.869
5.213
1.925
3,973
2.127
48
3.389
3.082
3.275
2.988
2.3e+
5.318
9.790
4.037
3.769
3.465
6.566
4.473
4.840
4.729
1 0.112
0.105
0.123
0.111
0.190
0.121
0.156
0.096
0.179
0.152
0.155
0.108
1.101
0.141
24
0.628
0.525
0.732
0.542
1.500
0.609
1.094
0.418
1.049
0.618
1.170
0.582
1.503
0.904
0.818
0.695
0.900
0.716
2.279
0.843
1,588
0.554
1.320
0.856
2.103
0.805
1.547
1.119
1 0.009
0.009
0.012
0.011
0.034
0,010
0.024
0.009
0.011
0,010
0.024
0.011
0.234
0.015
Ex.
0.098
0.095
0.095
0.093
0.754
0.120
0.487
0.152
0.117
0.110
0.443
0.103
0.335
0.127
48 0.194
0.192
0.178
0.176
1,366
0.270
0.815
0.289
0.209
0.218
1.107
0.213
0.449
0.280
Weather Traffic
1
0.302
0,302
0.342
0.313
0.599
0.228
0.265
0.264
0.243
0.242
0.280
0.232
0.762
0.257
24 0.649
0.608
0.617
0.607
0.761
0.366
0.576
0.327
0,459
0.422
0.573
0.432
1.102
0.565
48 0.769
0.681
0.707
0.680
0.824
0.403
0.683
0.369
0.517
0.457
0.621
0.454
1.001
0.604
1 0.357
0.337
0.359
0.341
0.728
0.388
0.478
0.296
0,474
0.336
0.423
0.361
2,489
0.336
24 1.182
1.043
1.236
1.086
1.760
1.020
1.377
0.671
1.492
1.019
1.494
0.877
2.881
1.037
1.702
1.447
1.634
1.434
2.620
1.415
2.699
0.909
1.740
1.435
2.033
1.308
3.039
1.419
to Equation
while keeping other parts of the algorithm unchanged. We denote this version as
"DSOF (w/o 0 (l)yy
1 4414 = X (() (XI O(T))
(T. (T.0
Results. Similar to the comparison with batch learning in Section 4.1 experiments were repeatedly trained using five different random seeds. Table [ shows the results, where DSOF, with or without the student model, consistently ranks among the top two methods. Notably, for ECL. at a prediction length of 48, our approach reduces MSE by at least 15%, regardless of the teacher model used, outperforming both the baseline and typical continual learning methods.
In some cases, omitting the student model yields better results. While its inclusion does not always guarantee optimal performance, our study in Section A.7t suggests it is less sensitive to learning rate variations, reducing hyperparameter tuning efforts.
5 CONCLUSION
In summary, we first redefined the online forecasting setting eliminate information leakage, ensur- ing that the model does not predict or evaluate time steps for which it has already received ground truth feedback. We then introduced model-agnostic framework, called DSOF, enabling various types of models to adapt to temporal shifts in real-time data. Our extensive experiments demonstrate that DSOF significantly improves forecasting performance through ongoing updates. Particularly, when compared to batch learning, the ECL dataset demonstrates the most significant improvement, with teacher model backbones such as FSNet and NSTransformer, achieving over a 50% reduction in MSE. Additionally, we have provided detailed ablation studies on design choices for training forecasting models online, offering valuable insights for better forecasting performance and adapt- ability. However, there still areas to explore, such as identifying better model architectures suited for online forecasting, using more advanced semi semi-supervised techniques to generate pseudo labels, improving ER methods, and extending forecasting to longer time periods.
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Ours
Batch Leamning
Ground Truth
Forecast Wilue
Fanocast Volloe
0.2
Forecant Value
200
200
340
300
203
402
420
440
5020
5282
3005
5050 5220
5:.0
3340
5300
14000
12320
32540 22000 22000
22303 saser 12140 Timestampe
Timesstamps
Timestamps
(a) DLinear at step 300.
(b) DLinear at step 5000.
DLinear at step 12000.
Forecaat Valee
Forecaast Valu0
Foreccast Value
100
300
300
300
300
405
470
110
5020
500
5062
5000 Timestamps
2000
5:50
5140
$160
12000 17600 32540 1766 12660 13103 12120 12
Timestamps
Timestampe
(d) PS at step 300.
(e) FSNet at step 5000.
(f) FSNet at step 12000.
Fowroest Value
Forruct Vilne
0.30
11.00
3.0
340
200
402
420
440
5020
5202
5002 5055 5200 5:20
3240 Timestamps
5380
12000 12520 12540 12000 12009 :2302 22120 12140
Timestaimps
Timestamps
(g) PatchTST at step 300.
(h) PatchTST at step 5000.
(i) PatchTST at step 12000.
Figure 3: Visualization of the online learning process for Traffic at steps 300, 5000 and 12000.
Table 3: Comparison of the MSE results between DGrad, i.e. the modified online training frame- works from earlier studies (Pham al. 2023 Zhang et al.) 2023), as well as TFCL. (Aljundi al.) 2019) and DER++ (Buzzega et al. 2020). All configurations have been modified to prevent infor- mation leakage.
ECL
ETTh2
Traffic
Teacher Model
Student Model
Framework
24
48
24
48
24
48
DGrad
2.187
9,954
1c+01
0.386
3.038
6.251
1c+15
1.189
1.461
DLinear
TFCL
3.033
6.897
9.746
0.718
2.240
3.457
0.324
0.697
0.745
DER++
2.172
5.369
7.339
0.385
1.986
3.168
0,300
0.623
0.709
DSOF (w/o ((5))
2.066
4.759
6.250
0.365
1.694
3.015
0.299
0.603
0.685
MLP
DSOF
2.065
4.737
6.181
0.365
1.701
3.082
0.302
0.608
0.681
DGrad
3.197
Se+01
Se+01
0.470
2.956
5.359
le+18
0.470
0.574
FSNet
TFCL
2,968
7.482
8.512
0.726
3.469
6.256
0,315
0.418
0.467
X
DER+
2.902
le401
le+01
0.432
2.726
5.239
0.286
0.411
0.449
DSOF (w/ ((5)
2.567
5.611
7.130
0.427
2.944
4.846
0.265
0.364
0.394
MLP
DSOF
2.330
5.475
7,000
0.431
3.114
5.318
0.228
0.366
0.403
DGrad
3.863
9.453
8.607
0.392
2,759
5.721
0.251
0.446
0.480
PatchTST
TFCL.
3.805
7.653
1e+01
0.714
3.275
7.763
0.272
0.558
0.607
DER++
3.403
6.846
8.025
0.386
2.163
5.849
0.242
0.478
0.502
X
DSOF (wo ((
3e+03
5.253
6.427
0.373
1.838
3.568
0.231
0.434
0,459
MLP
DSOF
2.244
5.169
6.665
0.382
1.925
4.473
0.232
0.432
0.454
ACKNOWLEDGEMENTT
This work has been made possible by a Research Impact Fund project (R6003-21) and an Innovation and Technology Fund project (ITS/004/21FP) funded by the Hong Kong Government.
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Table 4: Wall time comparison across various frameworks and datasets, measured in milliseconds iteration (ms/itr). Lower values indicate higher efficiency.
ECL
ETTh2
Traffic
Framework
Mean
SD
Mean
SD
Mean
SD
DGrad
5.58
0.18
4.63
0.14
11.03
1.06
DER++
13.79
1.08
12.71
3.50
23.82
3.37
TFCL
23.28
1.69
13.02
0.33
30.89
2.76
DSOF (w/o (())
23.16
1.40
8.13
0.12
51.66
29.09
DSOF
30.76
1.06
12.04
0.16
63.96
12.42
Table 5: Processor time comparison across various frameworks and datasets, measured in ms/itr. Lower values indicate higher efficiency.
ECL
ETTh2
Traffic
Framework
Mean
SD
Mean
SD
Mean
SD
DGrad
4.90
0.36
3.67
0.18
166.87
19.09
DER+
193.29
17.46
155.89
26.68
373.74
30.84
TFCL
337.87
17.38
137.98
3.51
551.64
79.99
DSOF (w/o ((S))
358.31
18.71
7.32
0.32
596.36
58.51
DSOF
478.45
31.96
10.91
1.20
800.70
10.74
A MODEL DESIGN CHOICES AND ANALYSIS
This section examines the effects of different learning strategies on the teacher-student model. It also analyzes experience replay (ER) parameters, including buffer size and update frequency, and it ex- amines the impact of temporal difference (TD) loss on mean squared error (MSE) results. Addition- ally, it also compares how the framework performs with and without the student model, specifically by examining the influence the learning rate.
A.1 COMPUTATIONAL. COST COMPARISON ONLINE TRAINING FRAMEWORKS
A.1.1 RUNTIME COMPARISON
Online learning consists of two phases: training with batches of data and testing with streaming data. We focus the testing phase, where the model updates parameters each time new sample arrives. We measure the time each framework takes to perform an online update per sample using an Intel Xeon Silver 4214R processor (12 cores, 24 threads. 2.40 GHz) and an NVIDIA GeForce RTX 3090 GPU (24GB memory).
In this experiment, the teacher model used DSOF DLinear. The mean and standard deviation of the runtime are calculated over 250 runs. Wall time and processor time are reported in Table Table respectively, in milliseconds per iterations (ms/itr). For instance, 100 ms/itr indicates that the framework can perform parameter updates for one sample in 100 ms. Therefore, a lower number indicates better performance. Both tables show that our method, DSOF, is less efficient compared to others, highlighting a weakness of our approach. This inefficiency is especially evident in the Traffic dataset, which, as stated in Table I has significantly larger feature dimensionality than ECL and ETTh2.
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Table Comparison GPU memory usage during the online phase across different online learning frameworks (measured in MiB).
Framework
ECL
ETTh2
Traffic
DGrad
386
346
436
DER+
424
346
542
TFCL
392
346
440
DSOF (w/o 0(5)
386
346
436
DSOF
400
348
508
A.1.2 GPU MEMORY USAGE COMPARISON
One downside of our approach compared to other online learning frameworks is that it involves using a student model for residual learning, which adds extra model parameters. As the other experiments conducted in this study, the student model is 3-layered MLP with a hidden size of 16, and the replay batch size in the ER procedure set to 32. Using DLinear as the teacher model for DSOF, the GPU memory is evaluated during the online phase, as shown in Table with units measured in mebibytes (MiB). The results suggest that our method does not require significantly more GPU memory.
A.2 COMPONENTS OF FRAMEWORK: EXPERIENCE REPLAY AND TEMPORAL DIFFERENCE LOsS
A.2.1 QUANTITATIVE RESULTS
Experiments are conducted to investigate the necessity of ER and TD loss. Table summarizes the comparison between the MSE performances with and without ER and TD loss. The first row of the table, which represents the setting without either ER or TD loss, corresponds to the method, DGrad, mentioned in Section 4.2 These results verify that both components are critical for the effectiveness of our approach.
Table Impact of ER and TD loss components on model performance.
ECL
ETTh2
Traffic
Teacher Model
Student Model
ER
TD
24
48
24
48
24
48
X
X
2.842
14.705
26.795
0.470
2.269
3,389
0.302
0.649
0.769
2.118
13.530
30.725
0.523
2.199
3.535
0.300
0.650
0.762
DI Linear
MLP
2.068
5.038
6.405
0.372
1.748
3.255
0.302
0.605
0.682
2.065
4.737
6.181
0.365
1.701
3.082
0.302
0.608
0.681
381.629
473.965
481.403
10.555
19.326
23.412
0.599
0.761
0.824
33.746
282.664
314.527
2.296
13.387
11.011
0,400
0.667
0.764
FSNet
MLP
V
X
2.228
5.514
7.132
0.465
3.220
5.267
0.231
0.364
0.402
V
2.330
5.475
7.000
0.431
3.114
5.318
0.228
0.366
0.403
X
X 4.770
16,594
16.943
0.915
5.213
6,566
0.280
0.573
0.621
3,952
24.128
13.939
0.900
4.262
4.626
0.270
0.584
0.619
PatchTST
MLP
X
2,409
5.109
6.978
0.378
2.022
4.514
0.232
0.432
0.456
2.244
5.169
6.665
0.382
1.925
4.473
0.232
0.432
0.454
A.2.2 MSE ANALYSIS
In the ETTh2 dataset, shown in Figure 4a DGrad struggles quickly adapt to new distributions from timestamps 3300 to 3450, resulting in higher emrors compared to other methods. The advan- tages of ER and TD loss suit different situations: between timestamps from 3300 and 3450, using
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DGrad
ER
TD Loss
ER Loss (Ours)
Cumulative
145E
20003
1.7
10003
9010
1030
3302
130
15.0
1700
Timesta mas
Timestampe
(a) MSE individual timestamps.
(h) Cumulative squared emoor over time.
Figure MSE analysis of ER and TD loss components using the ETTh2 dataset.
the TD loss outperforms, whereas ER excels in timestamps from 3450 to 3900. In the long run, ER helps the model generalize with complete labels and batches of data. Again, the cumulative squared error plot Figure 4b) shows that using both components results in slower MSE growth.
A.2.3 INTERACTION THE COMPONENTS
To explore the interaction between ER and TD loss components, we conduct two case studies using the ETTh2 dataset. Experiments are conducted across five runs with different random seeds. In Figure the shaded area represents the deviation of the five runs.
Ground
-10
Foree and Value
1.5-
-2.0
-2.5
-30
2000
2200
2045
2280
2200
2000
2325
2140
Timestamps
(a) Sequences with predictable eycles: ER offers more stable and confident predictions while TD loss exhibit large variance.
Ground Truth
loss
loss (Ours)
1.5
1.0
Foreraest Value
0.5
0.0
-25
-10
-1.5
1203
1300
1300
Timestamps
(b) Sequences with sudden changes: may form outdated assumptions about the current distrihution while TD adapts to the shift.
Figure 5: Visualization the interaction between ER and TD loss components in ETTh2.
By training the model with batches of historical sequences, the ER component allows for more assured predictions, especially when similar patterns are encountered. contrast, the TD loss component relies pseudo labels defined in Equation 5 for updates, where only the initial data
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point is based ground truth, and the remaining points less certain predictions from the teacher model. This approach results in high sensitivity to minor fluctuations in sequences, causing greater variance in predictions. As illustrated in Figure Sa relying solely TD loss can result in significant variance in predictions, even for predictable sequences with regular cycles. In these cases, ER helps reduce variance by offering more stable and confident predictions.
For immediate changes in data distribution, TD loss can make rapid adaptations. As shown in Figure [bb it effectively identifies sudden changes around timestamp 4300, leading lower error in predictions. In comparison, the ER component, reliant on historical data, may make outdated assumptions about the current distribution. may make confident predictions that are inaccurate in rapidly changing environments.
The interaction between ER and TD loss resembles bias-variance trade-off. By utilizing both com- ponents, the model can harness their strengths to balance this trade-off, resulting in more accurate and reliable predictions across different data distributions.
A.2.4 RUNTIME EFFICIENCY ANALYSIS OF THE COMPONENTS
We measure the time taken for online training per sample the main components of the net- work, ER and TD loss. Consistent with Section the wall time statistics TableB processor time statistics in Table are based 250 repeated runs, using ms/itr the unit of measurement, where a lower number indicates shorter runtime. Note that a replay batch size of 32 was utilized in this experiment. The results clearly show that ER contributes more to runtime than TD loss. A potential future direction is to develop method with lower runtime complexity to replace the cur- rent ER strategy, or smaller replay batch size as in Section A.5.2 This particularly relevant for datasets with very high feature dimension such as Traffic.
Table Wall time comparison for the two main components of the framework, measured in msfitr. Lower values indicate higher efficiency.
ECL
ETTh2
Traffic
Teacher Model
Student Model
ER
TD
Mean
SD
Mean
SD
Mean
SD
X
X
5.58
0.18
4.63
0.14
11.03
1.06
V
X
27.98
2.18
8.12
0.10
43.74
3,45
DLinear
MLP
X
8.32
0.51
6.53
0.44
17.85
1.60
V
V
30.76
1.06
12.04
0.16
63.96
12.42
Table 9: Processor time comparison for the main components of the framework, measured in ms/itr. Lower values indicate higher efficiency.
ECL
ETTh2
Traffic
Teacher Model
Student Model
ER
TD
Mean
SD
Mean
SD
Mean
SD
X
X
4.90
0.36
3.67
0.18
166.87
19.09
X
434.26
16.37
7.58
0.68
637.02
19.47
DLinear
MLP
X
V
7.00
0.22
5.30
0.05
289.61
27.26
478.45
31.96
10.91
1.20
800.70
10.74
A.3 FREEZING STRATEGIES TEACHER AND STUDENT MODELS
The effects of three freezing settings are examined in Table 10 freezing both the teacher and student models (i.e., batch learning), freezing only the teacher model, and not freezing either model. Re- sults indicated that updating both models yields the best performance, followed by updating only the student model. Existing forecasting methods learned through batch learning may struggle to gener- alize future distribution shifts due to the continuous emergence of new patterns. Integrating these methods into an online learning framework can improve adaptability and performance in dynamic environments.
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Table 10: MSE comparison across different freezing strategies for teacher and student models. Three settings are analyzed: freezing both the teacher and student models (batch learning), freezing only the teacher model, and not freezing either model. All ((S) denotes the parameters of MLP.
ECL.
ETTh2
Traffic
Teacher Model
(
(S)
I
24
48
24
48
24
48
417.3
478.1
485.2
11.84
21.83
19.3
0.592
0.749
0.789
FSNet
153.1
396
411.3
5.944
8,558
9.014
0.267
0.518
0.580
2.233
5,446
6.475
0.669
3.392
5.188
0.334
0.420
0.398
A.4 LEARNING STRATEGIES OF STUDENT MODEL
The of student model learning strategy is investigated in method. By employing residual learning strategy, the student model utilizes the predictions of the teacher model along with the hori- zon window as input, subsequently fine-tuning the teacher's predictions, illustrated in Equation 4 In contrast, the separate learning strategy provides the student model only the horizon window, and its predictions are directly used for evaluation, independent of the teacher model's prediction. A comparison of the performance of our method (i.e., residual learning) and separate learning is con- ducted, with the results presented Table
Table 11: Comparison of student model learning strategies: Residual learning vs. separate learning. In the "Residual" column, a cross ("x") indicates separate learning. where the student model relies solely on the horizon window, using its predictions as the sole output for evaluation. Conversely, tick ("V") indicates residual learning, enabling the student model to refine the teacher's predictions by utilizing both the teacher's output and the horizon window.
ECL.
ETTh2
Traffic
Student Model
Residual
24
48
24
48
24
48
X
181.7
193.5
8.743
9.828
0.872
0.894
MLP
5.446
6.475
3.392
5.188
0.420
0.398
FSNet
X
39.31
40.66
3,496
6,858
0.388
0.417
FSNet
V
8.242
12.67
3.435
5.419
0.407
0.444
X
6.321
7.065
11.74
12.15
0.451
0,490
PatchTST
<
6.93
7.283
3.461
5.761
0.367
0.401
5,476
6.765
2.39
3.719
0.578
0.666
MLP
4.962
6.113
1.979
3.905
0.419
0.447
PatchTST
79.59
47.52
2.703
4.953
0.388
0.420
FSNet
8.633
10.52
2.248
4.215
0.411
0.443
5.874
6.82
11.62
12.95
0.445
0.483
PatchTST
24.11
53.55
1.996
3.52
0.480
0.473
Two observations can be made:
1. Residual training enables simpler models, reducing computational costs and memory age. Using an MLP the student model in residual learning outperforms more complex models trained with separate strategies. This is likely because correcting distribution shift errors is easier than mastering the main task.
2. When choosing student model for residual learning, simpler model is preferred. Replac- ing the MLP with the more complex FSNet shows similar performance, but using PatchTST leads to deterioration. This may be due to PStchTST's complexity, making ill-suited for the simpler task of learning distribution shift errors. Thus, when using PatchTST, separate learning approach is more effective.
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A.5
KEY ELEMENTS OF EXPERIENCE REPLAY: BUFFER SIZE AND UPDATE FREQUENCY
A.5.1 BUFFER SIZE
We investigate the impact replay buffer size on our method's performance across three datasets (ECL,, ETTh2, and Traffic) using various teacher model backbones (DLinear, FSNet, and PatchTST) and an MLP as the student model. As shown Figure 6 all plots exhibit an elbow-shaped pattern, indicating rapid performance increases with initial buffer size increments. However, beyond size of 300, the improvement performance becomes marginal. It is observed that, in general, a larger replay buffer size leads to better performance. However, when the buffer size exceeds 300, the improvements become marginal. Generally, larger replay buffer sizes enhance performance, but excessively large buffers may lead to declines due to training on outdated out-of-distribution data.
DLincar
FSNet
FatchTST
20
0.0
1.
0.6
200
300
1003
120
300
5.0
2002
102
300
1200
Buffer Size
Buffer Size
Buffer Size
(a) ECL
(b) ETTh2
(c) Traffic
Figure 6: Performance analysis of varying replay buffer sizes, using DLinear, FSNet, PatchTST teacher models, with MLP the student model.
Table 12: MSE comparison across different batch sizes in ER.
ECL
ETTh2
Traffic
Teacher Model
Student Model
Batch Size
24
48
24
48
48
2.104
4.803
6.433
0.373
1.744
3.091
0.299
0.600
0.685
2.075
4.771
6.364
0,368
1.715
3.072
0,300
0.602
0.683
DLinear
ML.P
32
2.065
4.737
6.181
0.365
1.701
3,082
0.302
0.608
0.681
64
2.047
4.758
6.085
0.367
1.682
3.116
0.303
0.609
0.681
8
2.200
5,409
6.763
0.476
3.297
5.555
0.223
0.367
0.403
16
2.144
5.401
6.916
0.562
3.467
5.795
0.224
0.364
0,400
FSNet
MI.P
2.330
5.475
7.000
0.431
3.114
5.318
0,228
0.366
0.403
2.482
5.513
7.225
0.603
3.063
5.784
0.235
0.370
0,409
2.680
5.275
6.318
0.383
1.884
3,819
0.231
0.436
0,460
2.583
5.109
6.173
0.373
1.872
3,900
0.231
0.435
0,455
PatchTST MLP
32
2.244
5.169
6.665
0.382
1.925
4,473
0.232
0.432
0.454
64
2.382
5.150
6.978
0.381
1.976
4.832
0.235
0.433
0.455
A.5.2 REPLAY BATCH SIZE
The impact of the replay buffer batch size on model performance is explored by testing batch sizes of 8, 16, 32, and 64. As noted in Section A.2.4 ER tends to occupy long runtime. This led us to consider the trade-offs of modifying the batch size, aiming pinpoint the scenarios where increasing the batch size would result meaningful performance gains that justify the extra computational expense. By identifying these scenarios, make informed choices about when it is beneficial
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to choose larger batch sizes, weighing the improved performance against the downside of extended runtime.
The MSE results are presented in Table 12
Overall, batch size of 16 is more effective for the ECL dataset, while the ETTh2 dataset benefits from a smaller batch size of 8, especially for a prediction length of 48. For the Traffic dataset, no specifie pattern emerges, and the results remain relatively consistent across different batch sizes. Given the lack of significant performance gains with larger batch size like 64, is recommended maintaining a smaller batch size of 16 in ER to minimize the trade-off between efficiency and accuracy.
A.5.3 MODEL UPDATE FREQUENCY WITH ER
The impact of update frequency in ER on method performance is investigated. Performance com- parisons are made using various update frequencies, shown in Table T3 Whole numbers indicate more frequent updates, while fractions denote less frequent updates. For instance, "3" signifies three updates using the ER method, whereas "1/3" indicates one update for every three data streams.
Table 13: Impact of ER update frequency on method performance. In the "Freq" column, whole numbers indicate more frequent updates, while fractions represent less frequent updates. For exam- ple, a "3" in that column signifies updating the model three times using the ER method, while "1/3" implies updating the model once for every three data streams that arrive.
ECL
ETTh2
Traffic
Teacher Model
Student Model
Freq
24
48
24
48
24
48
1/3
2.066
4.769
6.892
0.373
1.849
3.302
0.246
0.514
0.563
1/2
2.072
4.75
6.562
0.370
1.78
3.187
0.244
0.516
0.562
DLinear
MLP
2.07
4.767
0.378
1.732
3.058
0.243
0.546
0.609
2.074
4.799
6.035
0.362
1.704
3.088
0.250
0.521 0.559
3
2.09
4.814
6.085
0.362
1.699
3.116
0.248
0.532
0.567
1/3
2.425
5.829
7.025
0.555
3.227
5.942
0.334
0.406
0.445
1/2
2.384
5.623
7.075
0.629
3.415
6.1
0.320
0.411
0.443
FSNet
MLP
2,233
5,446
6.475
0.669
3.392
5,188
0.334
0.420 0.398
2.389
5.469
6.833
0.614
2,955
5,311
0.325
0.417 0.464
2.334
5.468
6.783
0.711
3.081
5,889
0.316
0.423
0.471
1/3
2.562
4.782
6.4
0.470
3.349
9,962
0.236
0.425 0.456
1/2
2.409
4.852
6.291
0.415
2.622
8.649
0.235
0.422 0.453
PatchTST
MLP
2.357
4.962
6.113
0.400
1.979
3,905
0.231 0.419 0.447
2
2.211
5.193
6.761
0.383
1.859
4.214
0.229
0.418 0.446
2.175
5.167
7.152
0.378
1.803
4,438
0.230
0.418 0.447
Contrary to the common belief, a higher frequency of ER updates may not necessarily lead to im- proved performance. As an example, Figure shows four distinct temporal distributions from the Electricity dataset: the first spans from time step 1600 to 1660, the second from 1660 to 1715, the third from 1715 to 1830, and the fourth from 1830 1875. Following the distributions transitions at steps 1660, 1715, and 1830, a lower frequency of ER updates is associated with better perfor- mance, as shown Figure [6] where the MSE values are smaller these time steps. This may be attributable to the model's increased focus on TD loss when ER is less frequent, allowing for greater emphasis on immediate ground truth over more distant data streams.
On the other hand, lower frequency of ER updates limits the model's adaptability to the new on- going distribution after a transition. Between steps 1775 and 1830, as depicted in Figure 7a lower ER frequency leads to greater deviations from the ground truth. Meanwhile, shown in Figure 7b high MSE values persist for a longer duration. This decline can attributed to insufficient training on in-distribution samples, which hampers generalization able to strike a balance between learning from ER and TD loss, suggesting a moderate ER update frequency, such updating once for each data stream.
near-future data. Therefore, it is advis-
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sktee 32
Farecast Value
IR Feq. 12
BA frce -1
BKTeg =
ER Freq =
Ground Truth
2002
2250
1700
L750
2000
3050
Timestarnps
(a) Prediction results based different ER update frequencies.
035
Teq.
Trq.
232
Freq
freq
108
Timestamps
(b) MSE loss at individual timestamps.
Figure 7: Impact of ER update frequency model performance across four temporal distributions from the Electricity dataset: (600-1660, 1660-1715, 1715-1830, and 1800-1875.
A.6 USE OF TEMPORAL DIFFERENCE Loss AND ITS REGULARIZATION EFFECTS
A.6.1 REGULARIZATION EFFECTS
The impact the TD loss, as described in Equation and its regularization factor on forecasting performance is examined using three datasets (ECL,, ETTh2, ETTml) with FSNet. Regularization is controlled by adjusting the parameter 7 of the TD loss. In Figure 8 7 = indicates that the TD loss is not used, and the model updates solely with ER during the online phase. In contrast, for 1, the model updates with the fast data stream using TD loss without any regularization.
5.4
MSE
0.810
3.455
0.800
04
0.7
38
0.6
0.6
0.7
0.6
3.8
3.73
0.5
0.7
0.0
0.6
Value of
Value of
Value of
(a) ECL
(b) ETTh2
(c) ETTml
Figure 8: MSE results for various TD loss parameter choices. The parameter y = 0 signifies that the TD loss is not utilized, meaning the model updates solely based on ER during testing. In contrast, when 7 = 1, the model updates with the fast data stream without any regularization applied.
Figure 8 iillustrates that the best results are not achieved with y 0 or y= 1. Optimal performance attained with specific degree of regularization. However, a drawback of our method is its sensi- tivity to the choice of the regularization factor. For instance, in Figure Sal failing to select = 0.9 may result in worse performance than either not using the TD loss or using any regularization at all.
A.6.2 ENHANCING PSEUDO LABEL. CONSTRUCTION
Extending the method of constructing a pseudo label as described in Equation new parameter introduced in Equation 8 This parameter allows for adjusting the balance between ground truth
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data and the teacher model's predictions. The value & indicates the of the most recent ground truth data points and the first H - steps of teacher model's prediction to construct the pseudo label.
H [X X (T.0)
14)
Using l = makes the pseudo label unreliable due to limited ground truth data. We aim to explore whether the student model can enhance its learning effectiveness the value of k increases, or by utilizing multiple values of k simultaneously.
Altering the values of k. In this experiment, our aim find the best ratio of ground truth values to the teacher model's predictions for constructing the pseudo label. Setting H = 48 and using DLinear conduct the experiment by selecting a single value of instead
the teacher model of DSOF, of multiple values for the fast stream, and testing with evenly distributed & values. Specifically, we chose multiples of 4, i.c. & E {1, 4, 8, .2...., 40, 44}. As shown in Figure 9 smaller values of are preferred. The results align with our expectations, indicating that the model should be updated more promptly with the most recent data points.
3.
60
35-
3.4
62
20 Vallue of
Value of
(a) ECL
(b) ETTh2
Figure 9: MSE analysis for different values of A in pscudo label construction. The value i indicates the use of the most recent i ground truth data points and the first k steps of teacher model's prediction for pseudo label construction.
Using Multiple Values of Simultaneously. In this experiment, instead of relying solely the most recent data, we utilize a range of values, specifically powers that are less than the prediction sequence length. Specifically,
the fast data stream uses the batch of pseudo labels (a) i-jjk+tl 1- 1:4 - lk E K.), where = (2 < H, p E N), to update the student model. For example, if H- 24 and = 20, the fast data stream uses ls E {1,2, 4, 8, 16), resulting the batch (20) X (20) (20) x (20) X 1
[X 20: 19:12: 17:10: 13:36
Table 14: MSE comparison of between using single & value (k and utilizing multiple k values concurrently (k 2²).
DLinear
FSNet
PatchTST
Datasets
H
k =
1
k
2F
24
4.737
4.803
5.475
5.323
5.169
5.191
Electricity
48
6.181
6.022
7,000
6.821
6.665
6.566
96
10.661
10.393
10.023
9.839
11.248
10.249
24
1.701
1.713
3.114
2.742
1.925
1.874
ETTh2
48
3.082
3.015
5.318
4.959
4.473
3.970
96
5.609
5.323
9.526
9.446
17.363
13.351
In Table 14 the approach of using powers of is denoted k 2". The results indicate that this method is particularly beneficial for larger values H. For instance, applying to the ETTh2 dataset using PatchTST with H 96 provides significant improvements.
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A.7 LEARNING RATE SENSITIVITY ANALYSIS
This section explores the sensitivity of the learning rate in our method. The online learning rate do in Algorithm [ varied while keeping the batch learning rates constant. Performance comparisons are made with learning rates ranging from 0.0001 to 0.003 at five intervals using four teacher model backbones, DLinear, FSNet, PStchTST, and iTransformer, on ECL, shown in Figure TO
with student
without student model
20
2014
4.7
30%
24:107
1458
M45E
MM5E
105
10"
6-10
164
30.5
66-4
16-3
365
164
30.5
90-4
10.5
164
30:6
90-4
12:5
505
164
30-6
90-8
16-5
Loaming
Leaminng
Learning
Learning
(a) DLinear
(b) FSNet
(c) PatchTST
(d) iTransformer
Figure 10: Sensitivity analysis of the learning rate in our method, highlighting performance across various online learning rates the ECL dataset. By default, the student model is MLP. In cases where the student model is not used, the teacher model is continuously updated with ER and TD loss, without separate forecasting heads.
We conduct comparisons under two different settings, In the first setting, similar previous experi- ments, by default, we use an MLP as the student model when one is employed. In the second setting, when a student model is not used, the teacher model continuously updated with ER and TD loss without separate forecasting heads. During fast data stream updates, the teacher model adjusts its parameters based on its predictions compared pseudo labels, rather than freezing its training and backpropagating it to the student model.
is observed that training transformer backbones like i'Transformer and PatchTST an online setting without a student model is more challenging and makes it harder to reach convergence. This particularly true for datasets like ECL, which exhibit significant fluctuations and large values, even after standardization in data preprocessing. In such cases, student model enhances robustness to the learning rate. However, for non-transformer models like DLinear and FSNet, the teacher- student model may be unnecessary. While the across various learning rates, a sufficiently small learning rate (e.g., less than 0.0001) can still yield performance comparable using only the teacher model.
model approach shows robustness
A.8 IMPACT OF LOOK-BACK WINDOW LENGTH
In this section, we explore how the length of the look-back window impacts the forecasting per- formance of various models. We consider the look-back window length as hyperparameter and perform grid search over values L {24, 48, 96, 192} using selected datasets, models, and fore- casting horizons.
Contrary to the common belief that longer look-back windows provide more information and thus enhance forecasting performance, our findings suggest otherwise. The results shown in Table 15] reveal that, in most cases, model performance declines as the look-back window length increases. Interestingly. look-back window length of 24 often results in better performance, regardless of the horizon window. This may be due to the ER strategy used. Recall that the slow stream must wait for L+ steps ensure it only includes data pairs for which all ground truth values have been received. With larger L. the data sequences have to wait longer before being added to the buffer, which might lead to less informative slow stream and reduce the effectiveness of the ER procedure.
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Table 15:
Impact look-back window length L E {24, 48, 96, 192}
MSE results across datasets
and forecasting horizons H.
DL DLinear
iTransformer
PatchTST
Datasets
H/L.
24
48
96
192
24
48
96
192
24
48
96
192
2.802
2.258
2.065
2.010
2.341
2.282
2,430
2.349
2.603
2.265
2.244
2,476
ECL
24
5.396
4.845
4.737
4.980
5.651
5.101
5.155
5.571
5.247
5.128
5.169
5.516
48
6.563
6.270
6.181
6.778
6.736
6.324
6.015
7.734
6.333
6.366
6.665
8.461
0.364
0.369
0.365
0.371
0.371
0.398
0.384
0.374
0.385
0.388
0.382
0.394
ETTh2
24
1.664
1.666
1.701
1,891
1.658
1.704
1.869
2.103
1.859
1,850
1,925
2.135
48
2.839
2.951
3.082
3.552
2.801
2.972
3.465
4.035
3.506
3.244
4.473
5.259
0.009
0.009
0.009
0.010
0.009
0.011
0.010
0.014
0.010
0.009
0.011
0.010
Ex.
24
0.092
0.093
0.095
0.101
0.095
0.100
0.110
0.115
0.093
0.101
0.103
0.107
48
0.184
0.186
0. 091
0.205
0.206
0.228
0.218
0.305
0.188
0.202
0.213
0.242
B CONNECTING TEMPORAL DIFFERENCE LEARNING IN REINFORCEMENT LEARNING TO DSOF
the fast stream leverages ideas from TD learning to quickly process latest data and improve near- future forecasts. While the learning goals of RL and OTSF differ, both fields share the challenge of updating models without complete access of ground truth informa- tion. In this section, drawing inspiration from TD learning in RI., we demonstrate how the idea of estimating the value of the next state be applied to facilitate intermediate updates in OTSF.
As outlined in Section 3.3. 3.
Recall the definition of temporal difference learning in the context of reinforcement learning (Sutton & Barto 1999) given in Equation 9 where V(S,) is the value of state S at time i, is the learning rate, Rb+1 is the reward given at + and (V(+++1 is the discounted value of the next state.
V(S)) + V(S;) + az R241 + (V(+++1) - V(S))]
We draw connection between our scenario and TD learning in reinforcement learning in the fol- lowing way:
The agent corresponds to the student model.
The state S;-1 corresponds to the teacher model's prediction output (T,i-1 4 at time i 1.
The action corresponds to the student model's prediction output (S.i-1) based on the "state" {((T,-1)
1:44-11 at time i - 1,
1-+HH-1-
The value ((S-1)) corresponds to the prediction output of the teacher-student model ((-1)) 1-+H-1 at time 1.
The reward R; corresponds to the negative MSE between the first prediction point and the ground truth æ;, once becomes available time i.
The discounted value of the next state, <V(S,), corresponds to the importance of the teacher model's prediction output at time (T.)) 1 relative to the teacher-student
model's prediction at time 1, (i-1) --1 In reinforcement learning, the discounted factor is essential because the agent faces a probability of failure uncertainty during ex- ploration at each time step. This similar to our OTSF scenario, where predictions for data points further from the current observations become increasingly unreliable.
While insights from TD learning in RL can potentially enhance OTSF performance, OTSF should not be framed as RL problem because their primary objectives are fundamentally different. RL focuses on optimizing sequence of actions for maximizing long-term rewards, whereas OTSF seeks accurate near-future predictions for data streams which may have changing distributions. This dis- tinction leads to unique algorithmic settings; for instance, in RL, the current action influences the next state, whereas in OTSF, previous forecasts do not impact subsequent ones.
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Ours
Batch Leamning
Ground Truth
Forecatt
Farecant
5902 3000 1040
5083 2007
5200
3220
5240
0302
8922
3040
5020 Timestamps
9083
2300
0:20
3340
10030 17000
19340 19385 :5000 12100 12120 19043 Timestamps
Timestamps
(a) DLinear at step 5000.
(b) DLinear at step 8000.
(c) DLinear at step 19000.
Forecast Vilue
Forcentt Wite
Forocast Value
5200 5000
5000
5363
5380
5100
$120
5340
8206
9520
8000
1800
0300
8760
8120
4740
5500
1902
1609:
1enss 19680 19ine 19100 19353 Timestaraae
Timesta mps
Timestarp ps
(d) FSNet at step 5000.
(e) FSNet at step 8000.
(f) FSNet at step 19000.
Farecca Value
Value
3.5
100
3.2
WW WWW W
5002 5020 1040
5003
5007
5200
5220
5240
8202
5040
2000
2102
8220
25000
12t00 13540 1939: 10000 13100 12120 19503 Timestampss
Timesta mps
Timestamps
(g) PatchTST at step 5000.
(h) PatchTST at step 8000.
ii) PatchTST at step 19000.
Figure Visualization of the online learning process for ECL at steps 5000, 8000 and 19000.
C FORMULATION OF INFORMATION LEAKAGE IN EXISING ONLINE LEARNING FRAMEWORKS
Information leakage occurs when data points previously used for model parameter updates also used for current evaluation.
In existing online learning frameworks, at time / = i, the model first takes in input sequence from t t= L + 2 to t = + 1, then predicts values from t = + 2 to = i
Att 1, the ground truth at i 1 becomes available. and the most recent prediction made f = is utilized for evaluation. The model then updates its parameters using the ground truth from f == H + f + and predicts values from i H 3 to i 2.
At t = i 2, ground truth at = i + 2 is available. The last prediction made at + 1, with timestamps from t = H to t = i 2, is being used for evaluation. Information leak occurs here because ground truth with timestamps from t = H + 3 to = - 1 was already used for parameter updates at i 1.
D ADDITIONAL. VISUALIZATIONS
In this section, we offer additional insights into the prediction performance of DSOF in comparison to batch learning, illustrating how it addresses various time series patterns across different datasets. In addition to examples from the Traffic dataset Figure 3 we also present examples from the Electricity and ETTh2 datasets in Figure and Figure 12 respectively.
25

---

Published as a conference paper at ICLR 2025
Table 16: Detailed comparison of MSE results between our framework and batch learning across various datasets and backbones. The better results from the two settings are highlighted in bold.
SD
0.066
0.612
1.076
0.017
0.075
0J104
0.051
0.87
0.001
0.007
0.021
0.002
0.010
0.017
0.101
0.044
0.025
DSOF
2,703
9
0.415
2.127
0.141
0,904
1.119
0,015
0.127
0.280
0.257
0.565
0.604
0.336
LO37
1,199
NSTras..
3,258
3,286
0.030
0.102
0.290
0.006
0.003
0.023
0.013
0.025
0.103
0.190
0.126
0.083
0.085
0.029
SD
Buchh
Leam
3.9c+1
4.0c+1
3.80+1
3,0%8
3.933
1.101
1.503
1.547
0,234
0.335
0.449
1.102
1,001
2.48
2.881
3.099
0
0.053
0.398
0.008
BLOSS
0.006
0.111
0,022
0.001
0.006
0.001
0.001
2014
0013
0.007
DSOF SD
2.244
5.109
6.665
1.925
4,433
0.108
0,582
0.805
0,011
0,103
0.213
0.432
0.454
0.361
0.877
1.308
PatchTST
1.773
6.272
5,385
0.528
1.036
0.902
0.015
0.119
0.279
0.015
0.111
0.335
0.022
0.011
0.052
0.038
0.387
0.425
SD
Batch
Learn.
4,770
1.70+1
1.7041
0.915
5.213
6.566
0.155
1.170
2.103
0,024
0.443
1.107
0.290
0.573
0.621
0.423
1.404
2.053
0.085
0.039
0.027
0,016
0.015
0.066
DIM22
0,010
0.019
0.000
0.005
0.005
0.000
0.001
0.001
0.003
0.003
DSOF SD
2,430
5.155
6.015
0.384
1.869
3,465
0,152
0.618
0.856
0,010
0,110
0,218
0.422
0,477
0.336
1,019
1.455
fTrass.
0.036
0.014
0.070
0.118
0.152
0.027
0,106
0.096
0.001
0.004
0.012
0.006
0.026
0.011
0.036
0.218
0.065
SD
Batch
Learn.
1.976
4.119
4.936
0.872
2.688
3.789
0.179
1.049
1.520
0.011
0.117
0.209
0.243
0.459
0.517
0.474
1.492
1,740
0.746
0.157
0.0%0
0.048
0.087
0.008
0.035
0.024
0.000
0,00
0.036
0.002
0.003
0.003
0,10
0,034
0017
DSOF SD
4.733
5,943
2,363
4,03
0,096
0.418
0,554
0.009
0,152
0,289
0.264
0.327
0,369
0
0.671
0,909
OmeNot
1.6es)
9.61
1.8e+1
0.966
1.577
0.019
0.056
0.120
0.044
0.079
0.182
0.011
0.008
0.125
0.133
0.334
SD
Bach
Learn.
3,3c+1
8,5c+1
1.5e+2
2.531
7.017
9.790
0.156
1.094
1.588
0.024
0.487
0.815
0.265
0.576
0.683
0.478
1.377
2,899
0.056
0.133
0.012
0.089
0.175
0.008
0.011
0.017
0.000
0,002
0.004
0.001
0,004
0.003
0.009
0.018
DSOF SD
2.330
5.475
7.000
0.431
3.114
0.121
0,6019
0.843
0.010
0.120
0.270
0.228
0.366
0.388
1,000
1.455
PSNer
1.5e+1
2,816
3.927
1,722
1.693
2.381
0,019
0,163
0.268
0,088
0.200
0. 140
0.005
0012
0.019
0.342
0,334
0.574
SD
Learn.
3.8c+2
4.7622
4.8
1.9c+1
2.3e1
0.190
1.500
2.279
0.034
0.754
1,366
0.599
0.761
0.728
1.760
2.600
0.008
0.009
0.012
0.004
0.010
0-010
0,000
0:001
0.002
0.001
0.000
0.005
0.000
0.001
0.000
0:003
0.008
0.002
DSOF SD
2,235
4,597
5.433
0.375
1.757
0.111
0.542
0,716
0,011
0,093
0.176
0.607
0,680
0.341
1,06
1,344
FITS
0.006
0.009
0.031
0,019
0.003
0.009
0.009
0,000
0.000
0,000
0.001
0.001
0.006
0,005
0.022
SD
Batch
Learn.
2,870
4,509
5.257
0,522
2.189
3.275
0,123
0,732
0,000
0,12
0.025
0.778
0.342
0617
0,707
0,359
1,236
1,64
0.004
0.018
0.021
0.004
0.005
0.027
0,000
0,000
0.001
0.000
0.000
0.012
0.000
0.000
0.000
0.002
0.002
DSOF SD
2,065
4.737
6.181
0.365
1.701
3,082
0.105
OL525
0.695
0.009
0.095
0.192
0.608
0.681
0.337
1,03
1.447
DL incar
0.436
7.014
0.011
0.075
0.097
0.003
0.016
0.022
0,000
0.005
0.014
0.003
0.005
0.001
0.018
0.032
0.082
SD
3
Batch
Lear.
2.842
1.5e+1
2.7c+1
2,269
3,389
0.112
0,628
0.818
0.009
0.098
0. 194
0.302
0.649
0.769
0.357
1.182
1,702
48
24
48
1
24
48
48
24
48
26

---

Published as a conference paper at ICLR 2025
Ours
Batch Learning
Ground Truth
Forecand Value
0.0
-2.75
3.200-
Vallu
-15
-.00
-1.0
111
-100
-6.25
-4.00
2093
4022
4060
4560
4590
5000
3020
2240
7000
1220
7241
7285
7200
7393
7522
1300
3000
5220
2040
2000
9220
2108
5120
9043
Timestamps
Timestarpss
Timestam
(a) DLincar at step 4900.
(b) DLinear at step 7200.
(c) DLinear at step 9000.
Farecast Valtee
Forocast Velue
Foreec aist Villee
-1.0
25
-2.0
0920
1623
5940
0965
1680
Sado
5623
5:40
7700
7220
7352
1200
7340
7300
1300
7352
1000
6220
6003
9260
6000
9100
w120
6353
Timestamps
Timestamps
Timestamps
(d) FSNet at step 4900.
(e) FSNet at step 7200.
(l) FSNet at step 9000.
Forecast Value
-1.0
-20
0203 4222 4340
4580 4083 Timestamps
5002
5220
7230 1240 7200 7500
1220
7260
2000 5020
5040
2200 2200
9042
Timestamps
Timestamps
gg) PatchTST at step 4900.
(h) PatchTST at step 7200.
ii PatchTST step 9000.
Figure 12: Visualization of the online learning process for ETTh2 at steps 4900, 7200 and 9000.
E DETAILED PERFORMANCE COMPARISON RESULTS
E. GENERAL FORECASTING RESULTS
This section provides a detailed performance comparison between our framework and batch learning across various datasets and backbones, outlined in Section 4. While Table Sidee2hight highlights average
results across seeds, the standard deviation also included here Table [6 The results show performance stability and robustness against variability from random weight initialization, with most standard deviations being less than 0.1.
27


Reviews:
[{"review_id": "aYi0Vat8PN", "reviewer": "Reviewer_NzDP", "summary": "The paper spots a problem in the training-evaluation pipeline of existing online time series forecasting studies. The problem is that  parts of the future sequence has already been used as labels for updating the model at previous time-steps. This results in information leakage in training. This information leakage results in an overly optimistic training loss and consequently a gap between training and causal evaluation (the latter is what we really care about). The main contribution of the paper is identifying this shortcoming of the previous works. The paper also presents a fast-slow dual network algorithm for OTSF in a setting with no information leakage. They also present comprehensive experiments to showcase the performance of their algorithm.", "questions": "Please see the comments under the Weakness section.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The identified information leakage problem is important, and I think would be influential for the field.", "weaknesses": "The fast learning part (Section 3.3.2) motivates by the necessity of fast updates to stay up-to-date with the recent data. For this purpose they update the student model using labels generated by the teacher. The teacher model is however out-dated. This seems like a contradiction, and it is not clear how it may resolve the problem of being up-to-date, except for the fact that the immediate next label, x_{t+1}, is used in updating the student. This seems very small updates \n\nAlternatively, student updates could leverage k next real data for some k>1, at the cost of a delay of k time-steps. The student of the current work uses k=1. Alternatively, one could consider a whole spectrum (or hierarchy) of models with different values of k (for example powers of 2). \n\nComputational complexity: the DSOF algorithm involves two networks. Moreover, the teacher model requires to generate a long sequence of samples at each time-step, which can be very computationally costly in certain architectures like decoder transformers. I wonder if the reported outperformance could be achieved under the same computational budget as the baselines?\n\n\nComments on the presentation:\n\nThe presentation could be much better. The text involves quite long sentences that reduce fluency, resulting in a tiring read. Fig. 1 does not help much in clarifying the information leakage problem. Despite of extensive use of indices, the arguments are vague in several places; for example in line 45 “the whole sequence” could be replaced by the indices you defined in the same sentence. \nAppendices are missing.\n\nThe claimed new OTSF framework is not rigorously formalized. For example what data can or cannot be used at each time of the training to prevent information leakage. \n\nThe connection to RL and TD learning in section 3.3 seems very loose. It would be better not to use TD terminology. \n\nThe Appendices are missing from the pdf.", "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": "tdJP1I0eOG", "reviewer": "Reviewer_Qe4w", "summary": "This paper discusses a dual-stream framework for online forecasting (DSOF). Two modules separate the learning process into two streams: a “slow stream” which updates with complete data, and a “fast stream” which adapts to the most recent data. Using a teacher-student framework， the proposed model aims to improve forecasting performance by balancing adaptation to new data with robustness against noise. Numerical experiments are provided.", "questions": "Please see the weakness part.\n\nAt the current stage, the novelty is insufficient for acceptance at a top-tier machine learning conference, and I tend to recommend rejection. However, I am open to reevaluating this work based further discussion.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "This paper tries to address an important topic in time series forecasting via online model adjusting to hedge the distribution drifting issue.", "weaknesses": "1. The presentation of this paper could be improved. According to the current draft, the novelty appears limited. The way to utilize teacher-student style models and experience replay seems quite standard. I suggest authors include some toy examples or a theoretical analysis to motivate the necessity proposed method from a more principled way.\n\n2. Sample code is not provided to assist the reviewer in verifying reproducibility.\n\n3. Can authors elaborate more on how the proposed model addresses the information leakage? It seems not being well discussed in methodology section (section 3).", "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": "0EnamTZSKq", "reviewer": "Reviewer_7cqR", "summary": "The paper addresses challenges in online time series forecasting (OTSF), particularly the issue of information leakage, where models use historical data for both prediction and evaluation, leading to biased performance assessments. The authors redefine OTSF to focus exclusively on predicting future unknown time steps without using backpropagated data for evaluation, aligning the framework more closely with real-world forecasting tasks.", "questions": ".see weakness", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The paper presents a novel redefinition of online time series forecasting (OTSF), specifically addressing the issue of information leakage, which has not been adequately tackled in prior research. By focusing on predicting future time steps without evaluating on previously backpropagated data, the authors set a new standard for OTSF, ensuring a more realistic and reliable forecasting setting. The introduction of the dual-stream framework, DSOF, which combines experience replay (ER) and temporal difference (TD) learning, is an innovative approach that allows the model to adapt effectively to temporal shifts and generalize better to real-world scenarios.\n\n\nThe study is well-executed, with thorough experimentation that covers multiple datasets and a variety of model architectures. The results clearly demonstrate the advantages of the proposed DSOF framework over traditional batch learning methods, showing significant improvements across diverse real-world scenarios. The experiments also include ablation studies, highlighting the importance of core components such as ER and TD loss, thus providing a detailed understanding of how different design choices contribute to the model’s performance. The methodological framework is backed by rigorous empirical evidence, lending credibility to the findings.", "weaknesses": "The results of the baselines, especially the comparison between OneNet and FSNet, are unexpected and raise concerns about the experimental setup. Specifically, OneNet, which is theoretically more robust and effective, shows worse performance compared to FSNet, especially on datasets like ECL, which have many variables. This discrepancy is puzzling because OneNet should typically outperform FSNet due to its more comprehensive handling of cross-variable and cross-time dependencies. The authors should clarify the hyperparameter settings used for baseline models and ensure that they have been adequately optimized for each baseline. A detailed explanation of why OneNet underperforms would be valuable, as it could indicate either a misconfiguration or potential issues in the experimental protocol. Additionally, including a sensitivity analysis for hyperparameters could help identify whether the results are consistent or if specific settings unduly favor certain models\n\n\nWhile the paper argues that existing OTSF settings introduce information leakage, it overlooks that many practical use cases update the model immediately with each new data point rather than accumulating a sequence. Although the authors highlight the risks of information leakage, real-world scenarios often prioritize fast adaptation to incoming data, even if it means some overlap in training and evaluation periods. This practice, while not perfect, remains a common approach because it ensures the model quickly adapts to recent changes. The authors should address why the proposed method of avoiding information leakage offers a practical advantage, despite differing from common real-world practices. It would also help to show more use cases where avoiding this overlap leads to clear benefits, thus making a stronger case for why users should adopt the new setting. \n\nThe paper does not provide a thorough analysis of the computational cost associated with running DSOF in real-time environments. Given the dual-stream approach and reliance on replay buffers, the method could potentially introduce higher computational demands, especially when dealing with large datasets or long lookback windows. Include a detailed assessment of the computational overhead introduced by the dual-stream approach. Specifically, measure the latency, memory usage, and training time, comparing them with baseline batch learning and other online learning frameworks. This would provide a clearer understanding of the trade-offs involved and help identify scenarios where DSOF is more efficient or where further optimizations are needed.\n\nIn Figure 3, the combined use of ER and TD loss leads to a significant performance boost, but the individual contributions of each are less apparent. This pattern is confusing because it suggests that the benefits of each component alone are minimal, raising questions about their necessity if they do not independently improve performance. Without a clearer understanding of how these components interact, it is difficult to assess their true value within the framework.   The authors should provide a more detailed analysis explaining why ER and TD loss, when used together, show such a marked improvement over their separate contributions. This might include examining how these methods complement each other and identifying specific scenarios where one is more effective than the other. Additional ablation studies or visualizations showing how each component contributes to different aspects of the model’s learning could offer deeper insights into their interactions.", "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": "YcfP1e1pid", "reviewer": "Reviewer_oqzm", "summary": "The paper corrects a fundamental bug in previous papers for Online Time Series Forecasting, proposing a definition of the OTSF task without data leakage. To solve arising challenges of this new scenario, the authors then propose a dual-stream framework (DSOF) for this task, which updates a teacher-student model through residual learning.\n\nIn detail, there are two data streams: slow and fast.\n- The slow data streams are stored into a buffer in an FIFO basis for updating teacher and student residual model. This requires complete ground truth data for all time steps in data stream. This is called Experience Replay (ER) Update in the paper.\n- The fast data stream, designed to utilize data from timestamps where the ground truth for predictions is not yet fully available, updates the student residual model only, according to discord of (the prediction of the teacher-student model in the previous step) and (the ground truth of this step along with the pseudo-label generated by the prediction of the teacher-student model at this step). This is called Temporal Difference (TD) Update in the paper.", "questions": "- Could you conduct a grid search for different look-back horizons on selected datasets, models, and forecasting horizons to demonstrate robustness with respect to look-back horizon? I'm not suggesting that you repeat all experiments. Conducting grid searches for a subset, particularly with smaller models and datasets (e.g., DLinear, iTransformers on ETTh1, Exchange; H=24), would be computationally feasible and enhance the robustness of your results.\n\n- I noted the experiments on different buffer-size in appendix. Would it be possible that these experiments could partially represent or replace experiments about different look-back horizon?\n\n- Recent work has proposed Foundation Models for Time Series Forecasting, and there are papers discussing Test Time Adaptation for Time Series Forecasting. Would it be possible that the predictions of base models can be improved by the DSOF framework?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "This work demonstrates several notable strengths.\n\n- Most significantly, the authors point out a significant problem of several works for Online Time Series Forecasting related to information leakage. The bug pointed out here is such fundamental that I did not even believe it existed before I checked the code of several previous works for OTSF and verified the claim of the authors. The redefinition of OTSF problem would definitely make it clearer for the time series community to develop online models that are not only good on benchmarks, but also practical and useful.\n\nIn this data-leakage-free setting, there are data points in timestamps whose ground truth for prediction is currently not fully available, which is an arising problem.\n\n- The authors propose the DSOF framework as a logical solution to this emerging problem. This framework will serve as a baseline for future comparisons. The proposed framework provides insights related to differential learning in reinforcement learning, making it more explainable.\n\n\n\n- The authors evaluate their proposed framework with extensive experiments conducted across multiple Models (DLinear, FITS, FSNet, OneNet, iTransformer, PatchTST, NSTransformer, etc) and multiple Datasets (ECL, ETT, Traffic, Weather, Exchange, etc) on multiple settings (different forecasting horizons, different learning rates, etc), demonstrating the effectiveness of their method.\n\n\n\nThe significance of this work is apparent, especially in its correction to the OTSF problem practice, and also in its contribution to point out the arising problem, and propose a method to (partially) deal with this problem.", "weaknesses": "A fixed look-back length of 96 may not be entirely convincing. Some models may show a significant advantage at a certain look-back length, but less advantage or even underperformance at longer look-back lengths. It would be better if the authors could follow the practice in the paper of FITS, PatchTST, etc (which are cited by the authors in their paper) to do grid search for the optimal lookback length for each dataset for each forecasting model, and to compare this optimal value against each other.\n\nThe authors' approach is understandable, given that (1) some previous work accepted at top conferences for time series forecasting uses similar settings of fixed look-back length and (2) this approach would require much larger computation resource. However, I believe it would benefit the time series community if new papers utilized grid search and comparing with baselines across different look-back lengths, though this would not affect the contribution and significance of this work.", "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)