Lightweight Online Adaption for Time Series Foundation Model Forecasts
Authors: Thomas L Lee
William Toner
Rajkarn Singh
Artjom Joosen
Martin Asenov
Venue: ICML 2025 poster (2025)
Keywords: Time Series
Foundation Models
Deep Learning
Continual Learning
Online Learning
Primary area: deep_learning->sequential_models_time_series
TL;DR: We propose ELF a model-agnostic and efficient method to improve the forecasts of time series foundation models at deployment by using online feedback.

Abstract: Foundation models (FMs) have emerged as a promising approach for time series forecasting. While effective, FMs typically remain fixed during deployment due to the high computational costs of learning them online. Consequently, deployed FMs fail to adapt their forecasts to current data characteristics, despite the availability of online feedback from newly arriving data. This raises the question of whether FM performance can be enhanced by the *efficient* usage of this feedback. We propose *ELF* to answer this question. ELF is a lightweight mechanism for the online adaption of FM forecasts in response to online feedback. ELF consists of two parts: **a)** the *ELF-Forecaster* which is used to learn the current data distribution; and **b)** the *ELF-Weighter* which is used to combine the forecasts of the FM and the ELF-Forecaster. We evaluate the performance of ELF in conjunction with several recent FMs across a suite of standard time series datasets. In *all* of our experiments we find that using ELF improves performance. This work demonstrates how efficient usage of online feedback can be used to improve FM forecasts.

Lightweight Online Adaption for Time Series Foundation Model Forecasts
Thomas L. Lee * 12 William Toner"
Rajkarn Singh"
Artjom Joosen
Martin Asenov
Abstract
Standard approachi
Foundation models (FMs) have emerged as a promising approach for time series forecasting. While effective, FMs typically remain fixed dur- ing deployment due to the high computational costs of learning them online. Consequently, de- ployed FMs fail to adapt their forecasts to cur- rent data characteristics, despite the availability of online feedback from newly arriving data. This raises the question of whether FM performance can be enhanced by the efficient usage of this feed- back. We propose ELF to answer this question. ELF is a lightweight mechanism for the online adaption of FM forecasts in response to online feedback. ELF consists of two parts: a) the ELF- Forecaster which is used learn the current data distribution; and b) the E..F-Weightt which is used to combine the forecasts of the FM and the We evaluate the performance of ELF in conjunction with several recent FMs across a suite of standard time series datasets. In all our experiments we find that using ELF im- proves performance. This work demonstrates how efficient usage of online feedback be used improve FM forecasts.
Foundation model
Efficient online
+ELF
training
ELF Forecaster
ELF Weighter
Figure I. Exploitation of online information for forecast adap- tion in deployment: Time series foundation models (FMs) are typically seen as fixed when deployed. propose method, ELF, to efficiently improve their forccasts online by leveraging the feedback available in realistic deployment scenarios. ELF consists of two components: the ELF-Forecasterr lightweight forecaster, learnt online; and the ELF- LLFF-Wighteer a dynamic method to adapt the FM forecasts by combining it with the EL.-Forrcasterrs forecasts.
computational resources, machine learning expertise, and ample data (Mercier et al., 2021). These practical hurdles have spurred interest in foundation models (FMs) for time series forecasting (Miller et al., 2024; Ansari et al., 2024), which diverge from the standard paradigm of training on a specific dataset. Instead, FMs large datasets (Woo et al., 2024), enabling them gener- alise to new, unseen time series and deliver strong zero-shot forecasting performance (Ansari et al., 2024; Rasul et al., 2023). This avoids retraining new model for each task, al- lowing FMS to be used out-of-the-box even with no available training data, ML expertise, or computational resources.
trained across multiple
1. Introduction
Over the last decade there has been rapid development in machine learning models for time series forecasting (Darlow al., 2023; Lim & Zohren, 2021; Nie et al., 2022). This has prompted practitioners in fields requiring accurate forecast- ing to adopt and deploy increasingly advanced models to remain competitive. Although these deep models perform well, training them can be prohibitive, requiring substantial
While effective, time series FMs remain fixed during deploy- ment, making them non-adaptive to shifts in the underlying distribution. In time series deployment scenarios however, new data arrives continuously, providing feedback on fore- cast accuracy and changes in the underlying behaviour of the time series (Zhang et al., 2024). This raises the possibil- ity of using online feedback to enhance FM performance. However, this can be challenging as any online adaptation mechanism must be computational efficient to be practical in real-world time series settings (Joosen et al., 2024). Po- tential naive approaches to online adaption, such as regular retraining or continual finetuning on available data, are at present prohibitively expensive as demonstrated in Sec- tion 5.3 and Verwimp al. (2023). Currently, there is no
'Equal contribution 'School of Informatics, University of Ed- inburgh, UK "Work done while intern at Huawei 'Huawei SIR Lab, Edinburgh Research Centre, UK. Correspondence to: Thomas Lee <T..L.ee- wd.aneddd.......... William Toner wwilliam.toner2.6 @ huawei.com?..
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
general, efficient method for leveraging online feedback in time series FMs at deployment.
ing sets of time series, consisting of billions of training points, and whose backhones consist of LLM-based trans- former architectures--ucch as Chronos, which uses the T5 architecture (Roberts et al.. 2019). Also, recently there have been new time series FMs which have gone against this trend by not using LLM architectures as their back- bones. For instance, 7TM (Ekambaram et al., 2024) uses the TSMixer architecture (Ekambaram et al., 2023) as its backbone, which is specific to time series, resulting in much smaller model size when compared to methods using L.M backbones. While, VisionTS (Chen et al., 2024) uses a masked autoencoder (He et al., 2022) as its backbone and does not use time series data for pretraining, instead using images from the ImageNet dataset.
In this work, we propose ELF, an efficient approach for online adaption of FM forecasts at deployment. ELF does not alter the parameters of an FM, which remain fixed during deployment, instead adapting the forecasts output by the FM. In effect, we take a non-online FM and turn it into an efficient online model, FM+ELF (see Figure 1). This approach is lightweight, running on single CPU, making implementation cheap. Importantly, ELF can be used out- gf-the-box-i.e. without human supervision- to enhance the performance of any FM.
ELF consists of two components: the FLF-Forecaster, a lightweight forecast model which is trained online, and the ELF-Weighter which combines the forecasts of the FM and the ELF-Forecaster using an online weighting policy, effec- tively ensembling the two models. To ensure that ELF is computationally efficient we design a complex linear model to use as the trix identity to fit it efficiently online. To construct the Eei-hhigh we build upon exponential weighting, an ap- proach with longstanding pedigree in the online learning community (Cesa- Bianchi & Lugosi, 2006). We construct weighter made out of fast adapting component to extract local information and slow component to extract global information. This allows the weighter to quickly adapt to distribution shift. We experimentally demonstrate consis- tent performance improvements when using ELF, across multiple state-of-the-at foundation models and across the standard time-series benchmark datasets.
A major focus of this work is the exploitation of online feedback available at the deployment stage of time se- ries forecaster. The idea of leveraging online feedback in deployment to improve performance of an ML system has a long history (Hoi et al., 2021; Polikar et al., 2001; Bot- tou & Cun, 2003). Currently, this concept falls within the domain of continual or lifelong learning for deep learning methods (De Lange al., 2021; Wang et al., 2024). Some of these works, for instance, investigate how to update a model online given the newly available data, which is either from the same distribution or from a shifting distribution (Aljundi et al., 2019; Lee & Storkey, 2024b). Importantly, much of the research on continual learning has focused on vision or text tasks (Qu et al., 2021; Wu et al., 2024), with comparatively little attention given to the time series domain (Besnard & Ragot, 2024). The studies that have explored continual learning for time series forecasting concentrate on methods for updating the weights of deep learning models (Ao & Fayek. 2023; Pham et al., 2022; Zhang et al., 2024). This has two problems in the context of time series: a) up- dating the weights of a deep learning model, especially of the size of a FM, the frequency often required for time series data and with the resources usually available (c.g. CPUs) can often make such methods infeasible to use in practice (Diao et al., 2024; Ekambaram et al., 2024). And b) online updating of deep models suffers from the problems of catastrophic forgetting and plasticity loss (Kirkpatrick et al., 2017; Dohare et al., 2023; De Lange et al., 2021). Solu- tions to this currently require retraining on large amounts of historic data and complex, model-specific learning routines (Yang et al., 2024). This is in contrast to the focus of our work, which looks at the efficient online adaption of FM forecasts, so that it can be widely used in the real world.
exploiting the Woodbury ma-
Our work consists of three main contributions:
1. We propose ELF, an efficient way to exploit the online feedback available in the deployment stage of time series foundation model to improve performance.
2. Design a lightweight linear forecaster applied in the Fourier domain the -oreFFFotrr--wwe efficiently fit online via the Woodbury matrix identity.
can be
3. Design a dynamic weighter-the the ELF Weighter--t combine the ELF-Forecaster and FM forecasts. This is constructed out of fast and slow components to quickly adapt the weighting to shifting data distributions.
2. Related Work
Currently, the main paradigm construct (zero-shot) time series FMs is to collect a large pretraining set of time se- ries data and then use it to learn an LLM-like transformer model (Rasul et al., 2023; Chen et al., 2024; Liang et al., 2024). For example, Chronos (Ansari et al., 2024), Moirai (Woo et al., 2024) and TimesFM (Das et al., 2024) are all time series FMs which are trained on large curated pretrain-
3. Rolling Window Forecasting
When deploying time series forecast model in practice, new data becomes available over time. To model this realis- tic scenario, practitioners typically adopt rolling window
2

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
approach in which the time series is processed one time step at time (Nie al., 2022). At each time step, a context window containing the previous L time-series values is con- structed, from which the model produces a forecast for the next H steps. H is referred to as the forecast horizon. By progressing through the time series step-by-step, previous model forecasts can be evaluated against the actual ground truth values of the forecast that subsequently occur, called the target. This online feedback of past forecasts be used to improve future forecasts.
1. The ELF-Forecaster: a lightweight forecast model, trained online on the most recently observed time series data.
2. The ELF- -Li-eeett:::: an online weighting mechanism which adjusts the forecasts of the FM by combining it with the forecasts of the ELF-Forecaster, dynamically ensembling the forecasters.
We display the components of ELF in Figure 2 and describe them in turn below.
In this work we look at the rolling window setting, where the parameters of ELF are altered every M time steps using online feedback; a process we refer to as updating. We keep track of each time ELF is updated by using a counter referred to as the update step which we denote by т, to differentiate it from our notation for time step t. For every M time steps + 1. + M, there is a single update step + 7 + 1.
4.1. ELF-Forecaster
The ELF-Forecaster is a linear forecast model used to pro- vide forecasts based on the data seen online from the time series. We use a linear forecaster for three reasons: a) they have good performance in time series forecasting (Zeng et al., 2023); b) as required by our setting, they can be ef- ficiently updated online; and c) online updating of linear models does not suffer from the same problems of catas- trophic forgetting as the online updating of neural networks (De Lange et al., 2021). In this section we first describe how the ELF-Forecaster generates a forecast and then how it is fit efficiently online. Algorithmic descriptions of these processes are presented in Appendix A.1.3.
3.1. Notation
We list the notation used in this paper below. For notational simplicity, here and in Section 4, we only describe the uni- variate case. This is when each value in the time series is a scalar. This simplification is without loss of generality, as ELF forecasts each dimension of the time series, called channels, separately-i.c is channel independent.
Forecasting The linear -ore-ooreerter forecasts in the Fourier domain, being inspired by the FITS forecaster pre- sented in Xu et al. (2023) (as discussed in Appendix A. 1.1.)), To produce forecast the ELF-Forecaster takes in a context vector æ E 12 and applies the Discrete Fourier Transform (DFT)
L: Context length (number of time steps in the input sequence).
H: Forecast horizon (number of future time steps to predict).
L-1
c: Number of channels (distinet time series).
DFT(x)+ = VL
Σ n=
INE
Bickn
(I)
x: Context window, æ E R-
After this, to improve updating efficiency, the highest fre- quencies are discarded. Concretely, given a E [0, 1] we crop out those components DFT(x)) for which satisfies (5) < k < (1 5) Removing these frequency compo- nents reduces the dimensionality from L to 2
M: The number of time steps between subsequent updating of ELF
t: Current time step.
:: Update Step, number of times forecaster has been updated using online feedback. Given by 1
[ = aL. We multiply the resulting low-dimensional vector by a com-
plex weight matrix W of shape aL x all (a + x WW). The output of this operation is zero-padded up to dimensionality (1 + H), Finally, we apply the inverse Real Fourier Trans- form (iRFT) to generate the forecast ÿ E R The only parameters of the ELF-Forecaster is the complex weight matrix W.
y. ÿ: Target and forecast respectively, y, 2 E H.
2
X, Y: Design matrices containing all observed con- text and target vectors respectively up to the current time step. X, Y Fourier domain representations of the design matrices.
Online Fitting The ELF-Forecaster is refit once every up- date step 7 (every M time steps). We fit the ELF-Forecaster by optimizing the mean squared error (MSE) between its forecasts and the ground truth across all past data up to
4. ELF: Online Adaption of Forecasts
We propose a method for Ensembling with online Linear Forecaster (ELF), to improve the performance of time series FMs during deployment. ELF is a lightweight approach consisting of two parts:
'The RFT of vector is obtained by taking the first I.2 components of the (FT())).
3

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Most Recent
FM Losses
Foundation model
FM Forecast
+Wy fast
Weighter
Most Recent ELF- FLF-Forcastter Losses
Losses
B.
Wy
fast
W+
www.mm past values and prodictions
Exp Weighter
+By me
Wy
Losses using
context
Woodbury update
Weighter
1-0r
Combined Forecast
flo Wr
Losovo
Exp Weightar
Wy
ELF-Foricastor
ELF Weighter
Losses
ELF
Forecaster
ELF Forecast
The left side of the figure shows how ELF adjusts the forecast of the FM. This is achieved by combining the FM forecast with the forecasts of the online (Woodbury) updated ELF- -LF-Foreeerttr using a weighted average. The weights w, in the weighted average determined by the ELF--Wiighter based previous performance of the FM and ELF-Forecaster. The schematic in the blue box to the right, zooms into the components of the ELF-Weighter: the fast, slow and merge weighters. Where the fast weighter gives a weighting based on recent performance; the slow weighter gives a weighting based on performance across the whole history. The merge weighter is used to combine the weights given by the fast and slow weighters. Each weighter uses exponential weighting to construct their weights; the difference being losses used as input. While the slow weigher can seen as using all previous losses to compute its weight, we note that only the last B losses need to stored by ELF.
Figure 2. Forecast construction for FM+ELF:
the current time step 1 M. Importantly, since the ELF-Forecaster is linear model it can be fit in closed- form, avoiding the need for gradient-based based optimisation (Toner & Darlow, 2024), and without storing all the data. We note here that we initialise W to be the weights of a naive seasonal forecaster. Then, when there enough data (rM 2 L + H), we fit the W for the first time.
efficiently update this matrix at each update step via
X; X
where X,, Yy denote the matrices formed from the M most recently observed contexts/targets respectively. We addition- ally speed up refitting in another two ways: a) storing and incrementally updating the matrix (X X + A/) using the Woodbury Matrix Identity (Woodbury, 1950). This, in conjunction with storing X*Y, can be used to update W using Equation 2, without needing to store all previously observed data. b) Removing high frequency components of X, Y. that is removing some specific columns of the matrices. We describe both approaches below:
To refit the ELF-forecaster each update step т, we proceed as follows. Let (I), (2,,22,.... Mr) be the val- ues of univariate time series up to the current time step TM. Let Is =
ys == (Io+L, 1ostL,&++LLD Fs+L-1: ".oo-l......,,+++=-- (1) denote the context and its respective target starting at time step 8. Let X E N and Y NNH, denote matrices formed from all contexts and targets up the current time step that the th row of X is æ; (Note that N == TM L H; this choice of N ensures we do not fit on future time steps). Now, let X denote the matrix obtained by taking the DFT of each row of X and let denote the matrix obtained by applying the RFT to each row of V. The (122-reuularised) Ordinary Least- Squares (OLS) solution for the complex weight matrix of the #LLF-Forecastrr can be expressed as
The Woodbury Matrix Identity is a method for efficiently recomputing the inverse of matrix after a low-rank update (Woodbury, 1950). We apply this in our setting to enable efficient recomputation (X* X + AT) upon receiving new data. Specifically, letting
A =
the Woodbury update is defined as
A A B,
W (X
XT)
where
where À is the regularisation coefficient and denotes the conjugate transpose.
+=
In a naive implementation of the above one could store all observed data and recompute Equation 2 every update step. However, this approach is expensive. Instead of storing Y we can store X². This speeds up refitting as we can
If model refitting occurs regularly (so that A < L) this approach more efficient than the alternative of stor- ing/updating (X+ + A/) and recomputing the inverse each update step. This because rather than inverting a

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
large matrix of dimension L L (as in Equation 2), we invert smaller matrix of dimension M M.
Exponential Weighting is method for combining K forecasters by computing weighted average of their fore- casts, where the weights are learned online based on the past losses of each forecaster. We describe first the general case, where for K' forecasters wr,k denotes the weight for the hthh forecaster at update step т. The weights are updated at update step r by the learning rule:
Removing High Frequencies Fitting the ELF- Forecasrrr in Fourier space provides convenient mechanism for di- mensionality reduction of the context and target vectors via the removal of high frequency components. This allows us reduce the number of model parameters and further speed up fitting. Specifically, given some a E [0, I] (typi- cally = 0.9) and the Fourier domain representation of a context (or target) vector, we retain the lowest a proportion of frequencies. discarding the rest. In practice this means removing some of the columns of the matrices denoted X, X.Y in Eqn 2. This filtering technique, reduces the dimen- sionality of the weight vector being learned from I. x H to (aL ) speeding up the fitting, with minimal im- pact performance demonstrated in our ablations (see Appendix C.8).
Wr- 1.ke -Ur-IMM-VLoser 1,Los-.
Wr,k
K k==1 W glasss
where Loss,, denotes the loss incurred by the h forecaster on update step т, 1) denotes a predefined learning rate and where we assume wo,k = 1 for all k. By recursively ex- panding the update rule it is possible to give closed form solution the weights learnt at update step T,
Loss,',
Wr, k
LK -i Loss,',M
4.2. ELF-Weighter
Hence the weights learnt by exponential weighting a softmax of the cumulative losses incurred so far by the forecasters. This update rule has two desirable properties: a) the worse a forecaster performs the smaller its weight, and b) the weighter is affected less and less by individual updates as time progresses. Additionally, there is a well known theoretical result which bounds the cumulative loss when using exponential weighting; theoretically motivating its use:
The ELF-Weighter combines the forecasts of the FM and the FFFoorreastte; adapting the FM's forecast by incorporat- ing the knowledge learnt online by the ELF-Forecaster. We combine the forecasts by taking a weighted average of them. In other words, we construct a dynamic weighted ensemble of the FM and EI Fl.e-foracettr. Denoting the forecasts of the FM and the ELF-Forecaster at time step & by FM and Dr, EF, respectively, the combined forecast is given by
ÉLF W. D. + (1 w, )..
(3)
Theorem 4.1. (Cesa-Bianchi & Lugosi, 2006; Rakhlin & Kleiner, 2008) Given convex loss function, define the loss of the weighted-average forecaster at some update step T as and define regret at time T as
The weight w, E [o, 1] is adjusted online by the ELF- Weighter to reflect the changes in the relative performance between the FM's and ELF-Forecaster's forecasts. We use weighted average to combine forecasts because the general idea has been shown theoretically to improve performance (e.g.. Theorem 4.1). We also note that the simpler approach of using an unweighted average (w, = 0.5) leads to poor performance, as demonstrated in Appendix 10
T
RT
Σ 7'=1
Loss-r,weightt
min kE(I....,
Loss
7'=1
Then for a maximum incurred loss of Trmax and a learning =Imms
rute of 1) = "mmax
v
T we have that
SlnK
The basic building block of the ELF- Weighter exponen- tial weighting which is celebrated method from the line learning community (Cesa-Bianchi & Lugosi, 2006; Rakhlin& & Kleiner, 2008) and can be used to weight fore- casts. However. the standard way to perform exponential weighting has the drawback of being quite slow to adapt to changes in the relative performance differences between the forecasters. To be more adaptive we are inspired by Complementary Learning System theory (McClelland et al., 1995; Kumaran et al., 2016; Ba et al., 2016) to combi- nation of slow exponential weighter, which learns weights based on the whole history, and a fast weighter, which only uses the recent history. Below, we first describe exponential weighting in general and then the fast and slow weighers used by ELF- Weightrr to produce wi--
T 2
Rr S Timax
InK.
Slow and Fast Weighters While exponential weighting is an effective approach it has the drawback that it does not quickly adapt to distribution shift (Jadbabaie et al., 2015; Cesa-Biande et al., 2012; Zhao al., 2020). This is be- cause the losses incurred over the most recent time steps are given the same importance as those in the far past when constructing the weights. Hence, it takes several update steps for there to be enough losses from a new data dis- tribution to outweigh older losses to correctly adjust the weights. To remedy this drawback we look at using a com- bination of two weighters (as shown in Figure 2): a) a slow
5

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
weighter which is an exponential weighter, between the FM and ELF-Forecaseer. The update for the slow weight for the FM forecast w wnlow at update step is
Foundation Models (FMs) Our experiments explore using ELF in combination with several of the most recent and well know (zero-shot) FMs for time series: Chronos (tiny) (Ansari et al., 2024). TTM (revision 2) (Ekambaram et al., 2024), TimesFM (Das et al., 2024), Moirai (small) (Woo et al., 2024) and VisionTS (Chen et al., 2024). We compare the performance of each FM with and without using ELF.
w e -L.085,11
slow Wy
uolowe e LLoss,,, + (1 (looo) ) LL088..2
b) A fast weighter, identical to the slow weighter but only using losses from the last B update steps. The fast weight for the FM forecast passt at update step T is
Datasets The datasets we evaluate on are ETThI, ETTh2, ETTm1, ETTm2, Weather, Traffic, ECL. Solar and US Weather (given in Zhou et al. (2021); Wu et al. (2021); Liu et al. (2022); Darlow et al. (2024)). These are standard datasets used widely in time series work (Ekambaram et al., 2024) and importantly, none of the FMs used these datasets for pretraining. This is with the exception of TimesFM which uses Traffic and ECL for training, hence we do not report results for it for those two datasets. Furthermore, we look at these datasets for three different prediction hori- zons: H = 30, 96, 336, which were chosen to be compara- ble to previous work (Woo et al., 2024; Ekambaram et al., 2024). Throughout all experiments we use a context length of L = 520, which is a standard context length for the FMs used in our experiments (Ansari al., 2024). Addi- tional details about our experiments, including additional hyperparameter settings, are given in Appendix B.
-' 1,5
w.f fast
E e 9 - B '
By only using the last B updates for the fast weighter we aim to make it more adaptive than the slow weighter to the current relative performance difference between the FM and ELF-Forecaster. Additionally, note that in our setting we always have K 2, this means we effectively have single weight for each weighter where, for instance, the slow weight for the ELF-Forecase is 1 - wloon.
Merging Fast and Slow Weights The final part of the ELF-Weighter is a mechanism for synthesising a single weight wy from the fast and slow weights (wfoor, wolow respectively). W, is used for combining the forecasts of the FM and EL.-Forecaster as in Equation 3. We use exponential weighter, the merge weighter, to produce this final weight. Specifically, we record the losses Loss,./ obtained when combining the FM and EL.-Forecaster using the fast weight (i.e. setting wi == =wffat wifast and the losses Loss,,. obtained when using slow weight (w, == welow). We then use exponential weighting to fit a weight Smergg:
Metrics We evaluate our results using Mean Absolute Scaled Error (MASE) (Hyndman & Koehler, 2006). This is defined for a given forecast D. context æ and target y as
L-S Σ 1==1 1
Vi
MASE(Ú. y. = H E 24
Ii+
Smerge e .
where S represents the seasonality. MASE measures the absolute error between the forecast and target, normalised by the error of a naive seasonal forecaster on the context. This allows for comparisons across time series with varying scales over time. This is important in the rolling window setting, as it is unrealistic to assume that each channel is scaled to unit variance.
3 gmerge
3 e nloss, + (1 B ye nlass,,
gmerge
This merge weight is then used to combine the fast and slow weights via
Wr 3 w fast + (1 - Jun
gmerge
to generate the final weight given in Equation 3. An al- gorithmic description of the ELF-heighter is given in Ap- pendix A.2.
We also report results using the Root Mean Squared Scaled Error (RMSSE) in Appendix C. This metric is defined similarly to MASE but uses RMSE instead of MAE (Hynd- man & Koehler, 2006). The conclusions drawn from both metrics are the same.
5. Experiments
5.1. Experimental Setup
5.2. Main Results
To evaluate the performance of ELF we simulate how it would perform in a deployment scenario. Specifically, we
Table I presents the results of our experiments applying ELF to FMs, showing that ELF improves performance in all cases. The table displays the MASES of each FM and the difference in MASE when using the FM with ELF (the + +ELF columns). For the difference, a negative number means that FM+ performs better than using the FM on its own, which we colour green. As shown by the table
adopt the rolling window setting described in Section 3, moving through the time series and producing forecasts one time step at a time. We update ELF every M = 200 time steps using the online feedback provided by the newest 200 data points.
6

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Table MASE time series foundation models with and without using ELF: lower MASE better we present results for each dataset over multiple forecast horizon lengths denoted as H in the table. The results show that by using ELF we improve performance across all datasets and forecast lengths tested.
Time Series FMs
Dataset H
TTM
TimesFM
VisionTSS
Chronos
Moirai
+ELF(L)
+ELF(L)
+EIF(L)
+ELF(L)
+FIFF()
30
0.930
-0.019
0.913
-0.022
0.967
-0.063
0.936
-0.047
1.010
-0.089
ETTh1
96
1.081
-0.014
1.107
-0.049
1.084
-0.028
1.120
-0.063
1.168
-0.086
336
1,286
-0.006
1.350
-0.062
1.306
-0.022
1.361
-0.074
1,417
-0.101
30
1,472
-0.018
1.470
-0.024
1.542
-0,071
1,455
-0,021
1,536
-0.058
ETTh2
96
2.786
-0.016
2.799
-0.040
2.787
-0.029
2.801
-0.047
2.863
-0.062
336
6.802
-0.011
6.816
-0.040
6.763
-0.008
6,850
-0.092
6.913
-0.105
30 0.802 -0.048
0.833
-0.079
1.021
-0.252
0.891
-0.139
1.078
-0.311
ETTml
96 0.973
-0.053
1,016
-0.097
1.050
-0.130
1.164
-0.243
1.239
-0.310
336
1,205 -0.063
1,276
-0.131
1.216
-0.077
1.489
-0.341
1.479
-0.327
30 0.799
-0.036
0.814
-0.057
1.040
-0.251
0.843
-0.085
0.946
-0.164
ETTm2
96
0.991
-0.038
1.029
-0.077
1.088
-0.124
1.107
-0.150
1,151
-0.180
336
1.320
-0.040
1.429
-0.128
1,351
-0.072
1.478
-0.193
1.522
-0.219
30
0.893
-0.036
0.868
-0.041
1.027
-0.172
0.939
-0.091
0.896
-0.068
US
96
1.123
-0.040
1.164
-0.102
1.155
-0.085
1,204
-0.121
1.143
-0.088
Weather
336
1,296
-0.044
1.364
-0.123
1,286
-0.048
1.423
-0.163
1.334
-0.111
30
0.887
-0.033
0.798
-0.020
1.342
-0.449
1.077
-0,230
1.161
-0.261
Weather
96
1,205
-0.043
1.269
-0.220
1,436
-0.253
1.697
-0.538
1.720
-0.503
336
1.576
-0.040
3.084
-1.591
1,691
-0.140
2.099
-0.571
2.073
-0.494
30
1.091
-0.031
1.097
-0.058
1.004
-0.020
0.984
-0.030
1.222
-0.132
Solar
96
1.129
-0.031
1,201
-0.097
1.079
-0.021
1.080
-0.046
1.308
-0.164
336
1.166
-0.033
1.248
-0.100
1.236
-0.087
1.107
-0.028
1.292
-0.119
30
1.003
-0.086
0.982
-0.107
0.874
-0.034
1.225
-0.288
ECL
96
1.106
-0.094
1.090
-0.104
1.015
-0.054
1.303
-0.275
336
1,279
-0.086
1.368
-0.177
1.236
-0.079
1.476
-0.264
30
0.887
-0.067
0.962
-0.140
0.659
-0.015
0.770
-0.029
Traffic
96
0.920
-0.077
0.943
-0.112
0.746
-0.033
0.752
-0.017
336 0.965
-0.084
1.012
-0.127
0.923
-0.100
0.801
-0.019
we improve the performance in all cases as we have green numbers for FMs across all of the datasets looked at. In some cases the improvement is quite large. For instance, get
mance. To do this, we benchmark ELF against four different continual learning methods: a) TAFAS (Kim al., 2025), which, like ELF, is method adapt the forecasts of an FM using online feedback. TAFAS differs to ELF in the fact that it adjusts the context not only the forecast, its mod- ule to adjust forecasts is conditioned on the FM forecasts not context and uses gating (Chen et al., 2018) not ex- ponential weighting to adjust the strength of the forecast adaption. b) FSNet (Pham et al., 2023), which is a modified temporal convolutional network (Bai al., 2018), such that it has improved performance when trained continually. OneNet (Zhang et al., 2023), which is ensemble of two forecasters fit online, where one forecaster leverages cross- channel information and the other cross-time information.
by using ELF to adapt the forecasts of Vision'SS average improvement of over 10%. These results illustrate that efficiently and effectively exploiting the online feedback in the rolling window setting allows ELF to improve the forecasts of FMs.
5.3. Comparison to Continual Learning Methods
While we have shown that ELF improves the forecasts of FMs, it is also important to see how well it compares to other methods which use online feedback improve perfor-
7

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Table Results of time series online adaption methods: We report for each method the rel. MASE w.r. the naive seasonal forecaster, averaged over the forecast horizon lengths {30, 96, 336} and the seconds per update step, measured for the ETTh1 dataset with a prediction length of 96. compute seconds per update step using 2 Intel(R) Xeon(R) Platinum 8168 CPUs and put in brackets the proportional speedup of updating using ELF versus the other online adaption methods. The results show that ELF performs the best in terms of both forecast accuracy (MASE) and compute efficiency (sec. per update step).
Dataset
TTM+ELF
TTM+TAFAS
OneNet-TTM
OneNet
FSNet
ETThl
0.882
0.893
1.084
1.123
1.204
ETTh2
0.947
0.954
1.183
1.334
1.130
ETTml
0.810
0.843
1.177
1.146
1.235
ETTm2
0.814
0.834
1.213
1.373
1.363
US- US-Weather
0.817
0.838
1.109
1.105
1.198
Weather
0.770
0.790
3.551
2.043
2.510
Solar
0.936
0.962
1,250
1.300
1.810
ECL.
0.831
0.897
1.278
1.299
3.916
Traffic
0.676
0.737
0.756
0.803
1.124
Sec. Per Update Step (ELF Speedup)
0.38
3.76 (10x)
30.66 (81x)
43.12 (113x)
23.69 (62x)
The ensembled forecasters used are typically FSNet models, to mitigate the problems of continual learning. However, here we also look at using TTM as the cross-time model (OneNet-TTMM to explore variant that is more like ELF. We note that all of these methods are fit with gradient de- cent which can be computationally costly and often leads to problems when continually learning (De Lange et al., 2021). I). Additionally, they require an offline training set to initialise their parameters, unlike ELF. Further details of these meth- ods are presented Appendix B. The results of using these methods in the rolling window setting are presented in Ta- ble 2, where we use TTM the FM for TAFAS (results for other FMs are presented in Appendix C.2) and compare to TTM+ELF, as TTM is the best performing FM in our experiments. The results show that TTM+ELF performs better than all of the other methods across all datasets tested. This demonstrates the effectiveness of how ELF uses online feedback to adjust forecasts.
5.4. Why Does ELF Improve Performance?
While Table demonstrates that ELF improves the perfor- mance of FMs, it is important to analyse why this the case. We believe that ELF provides benefit in two main ways: a) the BLF--eighter can identify shifts in data distribution, ensuring that the combined forecast is well suited to the current features of the time series. b) By leveraging online feedback the ELF-Forecaster fits to the specific dynamic features of the time series, enabling the combined forecast to more accurately model these features. We evidence both points below.
The ELF-Weighter makes ELF sensitive to the changing data distribution of the time series. This is achieved by the ELF-Weighter quickly adjusting the weighting between the ELF-Forecaster and FM when distribution shifts occur. An example of the weights adjusting to a shift in distribu- tion is shown in Figure 3, which depicts the evolution of the FM weights (w,) for Chronos on ETTh1. Specifically, the weight given to the FM for the purple channel rapidly increases in the cream shaded region. Additionally, in Fig- ure 4 we plot moving average of the MASEs over time of the Moirai FM, the ELF-Forecaster and the combined Moirai+ELF during a snippet of the ETTh1 dataset. The Figure shows how the ELF- Weighter dynamically weights the forecasts of the FLF-forecaster and the FM based on their performance on the local data distribution. This allows ELF to generally generate forecasts which superior to either forecaster individually. Notably, Figure 4 also shows that utilising the ELF usually provides a benefit regardless of whether, for the current data distribution, the FM outper- forms the ELF-Forecaster (cream shaded regions) or vice versa (light-blue shaded regions). This all provides evidence
Computational Cost A core characteristic of ELF is that it is computationally inexpensive. For example, when deploy- ing on two CPUs, ELF adds only an additional 0.38 seconds per update relative to employing the FM without online adaption. This speed is crucial since, during deployment, online updating must occur faster than new data arrives. To contextualise the speed of ELF, in Table 2 we show the computational cost of other online learning methods. We find that ELF is 10x faster than any other method. We also in Appendix C.3 run an experiment where we continually finetune TTM and find that ELF is 2506x faster. Also, ELF also outperforms this approach, achieving an average 5 5.89% gain in predictive performance on the ETT datasets. These results demonstrate the computational efficiency of ELF.
8

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
ELF Weights (w,) Over Time: ETTh1
MASE over Time: ETTh1
Foundation Model
ELF-Forecaster
166
Moiral
0.8
Moirai+ELF
Moirai+ +LF < Moirai
FM Weight (wr)
0.6
MASE
1.2
0.4
1.0
0.2
0.8
ELF-Forecaster
2000
4000
6000
8000
10000 Time
12000
14000 16000
6200
6400
6600
6800
7000
7200
7400
Time
Figure 3. ELF weight (w,) over time for the Chronos FM for each channel of the ETTh1 dataset: Each line in the plot shows the ELF weight for a channel. A weight of means that the ELF- Forecaster has no contribution to the final forecast, similarly, weight of 0 means that the FM not contributing to the final forecast. The plot shows that for 5 out of 7 channels the FM weights gradually decrease; this reflects the fact that as the ELF- Forecaster observes more data its performance improves and it is upweighted by the ELF-Weighter. Occasionally, the weights adapt quickly changes in the underlying time series, for example the purple channel shows a rapid weighting change between steps 100-10,000 000 (shaded in eream).
Figure 4. MASEs of the Moirai FM, Moirai+ELF and ELF- Forecaster over time on a snippet from channel 6 of the ETTh1 dataset: The plot shows during certain intervals (shaded in cream) Moirai outperforms the Forecasses, on other occasions the reverse is true (shaded in light blue). While the ELF--Weihhe ensures that the performance the combined forecast (the green dashed line) generally outperforms either individual forecast. This demonstrates a reason for why using ELF improves performance, where the gain in using it is given by the green shaded region.
than replacing with other types of forecasters, for exam- ple more complex forecasters like FSNet or forecasters that predict the residual of the FM forecast. Additionally, establish that filtering high frequencies and using the Wood- bury matrix identity leads to significant speed up while only having a negligible impact on forecasting performance. c) In Appendix C.6 we analyse the role of the updating frequency M. We observe that by updating ELF more frequently one may improve its forecasting performance albeit the cost of compute performance.
to the fact that ELF both adjusts to shifts in the data distri- bution and by doing increases the accuracy of forecasts.
Figure also shows that for most channels, the FM weight tends to decrease over time. This phenomenon suggests that as the FL.-Foreeeseer observes more dataset-specific data generally becomes better fitted to the given time series and is consequently up- up-weighted. This has the follow-on effect that the FM's forecast can be increasingly adapted to the characteristics of the time series it has been deployed on. An additional discussion of how the ELF-Forecaster identi- fies time series specific features to improve performance is presented in Appendix C.13. We also remark that, there are some channels in Figure 3 where the FM consistently has a larger weight than the Forecccstt (e-g. the brown line) and hence is performing better. This justifies the decision to make ELF channel-indepennent, if were not, it would not be able to weight these channels differently from the others.
6. Conclusions
In this work we look at how to efficiently improve the fore- casts of a deployed time series foundation model (FM). We propose ELF, method which exploits the fact in deploy- ment there is online feedback on previous forecasts as new data arrives. ELF leverages this feedback in two ways: a) to train a lightweight linear forecaster (ELF- Foreccsre))) on the newly arriving data to learn the up-to-date data distribution; and b) to adapt the FMs forecasts by combining them with the forecasts generated by the ELF-Forecaster using a learnt weighting mechanism the ELF -eeehheeeerr We demonstrate experimentally that by using ELF we improve performance consistently across all datasets and FMs looked at. This indicates that exploiting the online feedback given in de- ployment is an effective way to boost the performance of FMs. Crucially, this online adaption of forecasts is achieved in a FM-agnostic manner and with a small enough overhead that can be widely used in the real world.
5.5. A Ablations
To analyse the respective contribution of different parts of ELF we perform several ablations: a) in Appendix 10 we look at the respective performance of the two main compo- nents of the ELF-Weighter, the fast and slow weighters. We find that using both leads to a gain compared to using either individually. b) In Appendices C.7, C.8 and C.9 we ablate the ELF-Forecaster. Demonstrating that performs better

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Acknowledgements
Chaudhry, A., Rohrbach, M., Elhoseiny, M., Ajanthan, T. Dokania, P. K., Torr, P. H., and Ranzato, M. On Tiny Episodic Memories in Continual Learning, arXiv preprint arXiv:1002. wew.:/902.004400 10486, 2019.
would like to kindly thank Luke Darlow for his help
and Thomas L. Lee would like thank his PhD supervisor, Amos Storkey.
Chen, M., Pennington, J., and Schoenholz, S. Dynamical Isometry and Mean Field Theory of RNNs: Gating En- ables Signal Propagation Recurrent Neural Networks. Proceedings of the 35th International Conference on Machine Learning. 2018.
Impact Statement
This paper looks at the general area of time series forecast- ing. While there are many potential societal consequences of work time series forecasting, due to the general scope of our work we do not feel there are any specific societal consequences note here.
Chen, M., Shen, L., Li, Z., Wang, X. J., Sun, J., and Liu, C. Visionts: Visual Masked Autoencoders Are Free- lunch Zero-shot Time Series Forecasters. arXiv preprint arXiv:2408. /7253, 2024.
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A. Additional Methodological Details for ELF
A.I. Implementation Details of the ELF-Forecaster
Initialisation Initially there is no data to fit ELF-Forecaster since we assume that we are deploying zero-shot without access a training dataset. Consequently, we are required to wait small amount of time before the ELF-Foreeaster can first be fit to data, at least L + H time steps. To handle this period we initialise the linear model to the naive seasonal forecaster. Then after seeing at least L. H time steps we fit the EL.--Forecaster all the available data at the nest update step
Instance Norm The ELF-Forecaster utilises variant of instance norm at inference time. Given a context vector and a target vector and forecast model f, this involves normalising x by its mean fı(z)) applying a model f on the normalised x', and adding this mean back onto the prediction ý. Formally:
x' x p(x).
y f(x') ) you y + u(x))
A.1.I. NUMERICAL ISSUES AND RESOLUTIONS
The naive approach for handling the repeated refitting of the ELF-Forecaster is to store X and X As one observes new contexts and targets these quantities can be updated via;
X ~ X X M
XM,
X + + + YM. YM.
After which one may recompute (XT X to produce the updated weight matrix. While this approach is effective and requires only the storage of L x I. and H matrices, one must take care that the cumulative sums X X, X don't grow too large. For example, in the context of serverless computing practitioners requests. The number of these requests Consequently, in this setting, over the course of a year the values of x X will grow to be in the order of 10 Using, say, bit float the large growth in the magnitude of the elements of x X can result in a loss of numerical precision. To address this issue we adopt subtly different approach, instead storing x X where N is the number of data instances "N
often interested in predicting function sometimes be in the millions per hour (Joosen al., 2023; Diao et al., 2024).
which have been observed thus far. This ensures that the magnitude of the values in these matrices remains approximately constant. Specifically, given M new data instances XM, Yo one updates N
x in the following way:
X
N
X
X7 X XM
+
N
N +
M
N
N
+
The update is similar for X We then observe that
X
A
-1
X N
(x + -1) =
N
N
Thus can compute the OLS solution using the scaled quantities by scaling by x we see more data.
x and x²y N
annealing the regularisation parameter À
N
Data Scaling Typically, when training deep models on time series, practitioners adopt the approach of normalising the data. The standard approach is to compute the mean and the standard deviation of each series on some training set and then scale so that the data is zero mean and unit variance (Nie et al., 2022). The validation and test data are scaled using these same parameters derived from the training set. This technique handles the radically different scales between different time series, and mirrors data standardisation practices in other areas of machine learning such as computer vision. In online setting however we assume that there is no training dataset from which these metrics can be computed. Thus we cannot adopt this approach. We resolve this by computing running standard deviation for each channel which are updated online as more data is observed using Welford's 's online algorithm (Welford, 1962). These values are used to scale the data before fitting the linear model. Note that, at inference time, no data scaling is needed before applying the linear model since aA (#) = Ax for any a.
13

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
A.1.2. How DOES THE ELF-FORECASTER DIFFER FROM FITS?
Our FLF-Forecaster architecture inspired by the FITS linear forecast model; an effective lightweight approach for time series forecasting (Xu et al., 2023). FITS applies the Real Fourier Transform (RFT) context vector, removes high frequency components using low-pass filter (LPF) and then takes the inverse RFT map back into the time domain. This structure is similar to that of the ELre-oore,s, however there are some important differences between our EL.--rorerettrr and FITS, some of which we summarise below:
1. Model Fitting: While FITS is fit using gradient descent, ELF uses closed-for fooed-form solution to find the complex weight matrix.
2. Model Output: FITS outputs a forecast of the target and a reconstruction of the context vector, whereas ELF ELFonly generates target.
3. Loss Function: FTTS is trained to minimise both the reconstruction error and forecast error where we only consider forecast error.
4. Target Compression: FITS uses low-pass filter to compress the context vector. ELF applies an LPF to the context and target.
5. Online Learning: Our method is designed for online learning whereas FITS considers exclusively the non-online setting.
Relation To Standard Linear Regression When FITS does not use LPF and the context length exceeds the prediction horizon length, is known that is equivalent to ordinary least-squares linear regression (Toner Darlow, 2024). Similarly, in the case where our model applies no compression the target the context it will also be equivalent to ordinary least-squares linear regression.
A.1.3. FITTING AND FORECASTING ALGORITTMS
We present here the algorithms which form the ELF-Forecaster. Algorithm describes how the ELF-Forecaster generates forecast given an input context vector æ E RR_. Algorithm describes how the EL.-Forecaster is refit at update step r (i.e. after TM time steps).
Algorithm 1 Predicting with F.FeFarre
Input: Context vector x (I1, ....., ........... .z), forecast horizon H, frequency retention proportion a E [0, 1]. complex weight matrix W E cix.l where L = [ab] and = [aH/22].
Compute mean of input: μ ;== 0i=0 Tz
Normalise data: x < x ------1 1
Apply discrete Fourier transform (DFT): DFT(#)
Discard high-frequency components by removing the central portion of æ;:
T Fal ....... xL)
Remove L- au) components
Apply linear map: ÿ wTW Pad forecast with zeros to length H/2 1:
+ (01 ..... Uū 0 0......) 0) E C//411
Apply inverse real Fourier transform (iRFT): ÿ <- iRFT (
Restore mean: ÿ +- 9 + / Return Forecast vector ÿ E R
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Algorithm 2 Fitting of E..--Forecastr
Input: Update step T, refit interval M 200; context length L; forecast horizon H; the last + L + H time steps of a 1- (= 1-11-11------- -- frequency retention proportion a E [0, 1]; regularisation parameter À = 20; number of datapoints observed far NumSeen; boolean flag FIRSTFIT (TRUE if the model has not yet been fit)
time series
Denote
= (T Patli f - ys == (I Lot4 (441+H-1)
Gather the last M context and target vectors = (æ yo) J.J-M---A
Form matrices X E R&xx and Y E RAxxH from D. Normalise X + X mean(X, axis = 1) if FIRSTFIT then a + stdev di, dim = 1) else
I/ Normalise rows to have zero mean
Calculate initial standard deviation per feature
(ELFORD(X,O)
I/ Update standard deviation online
end if
X + X/o
// Scale data using standard deviation for numerical stability
/1 Transform to frequency domain
X + (X)
II Apply row-wise discrete Fourier transform
Y t RFT(Y)
I/ Apply row-wise real Fourier transform
Reduce dimensionality of X, y
Remove the last
aH columns of Y 2
// Low-pass RFT output (ordered low-to-high) Low-pass on DFT output (high freqs in middle)
Remove the middle [aL] columns of X if FIRSTFIT then
A-
&
B+
1 X
M
FIRSTFIT + FALSE
else
// Apply complex-values Woodbury update rule A=1
A-1 <- WOODBURYUPPATE NumSeer X)
// Update A-1 efficiently; see Alg. 3 // Rescale updated A-1 // Update B via weighted average
A-1 < 1-1 (NumSeen + M)
B + kumSeen+MM
(NumSeen- B + X
end
Update counter: NumSeen + NumSeen + M Set the complex weight matrix W t- A-1B
Algorithm 3 Complex- Complex-Valued Valued Woodbury Update (Woodbury, 1950)
Function: WOODBURYUPDATE(A-!
Input: --1 Cuxd (inverse of d matrix), X E Mxd
Compute the intermediate matrix:
B B-A-XX (I XX-1X*) XA-1
Update inverse:
A + -11-B -B
Return updated inverse matrix A-1
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
A.2. ELF-Weighter Algorithms
We present here the algorithms which form the ELF-Weighter. Algorithm 4, shows how ELF-Weighter ELF the FM forecast by combining with the ELF-Forecaster forecast using a weighted average. While Algorithm shows how the ELF-Weighter exponential weighting to update the weights of the fast, slow and merge weighters which construct the weights w, used to combine the FM and FLFe-aarer forecasts.
Algorithm 4 ELF-Weise Weighter Combined Forecasts at Update Step T
Input: FM and EF. which are the forecasts of the FM and ELreetrrere for time t, respectively, and let the last update step update step r
Compute weight: gmeerge wast + (1 Binerge samergeeweton woo
Compute and return ELF's forecasts:
Return wy FM + (1 W,y EF
Algorithm 5 ELF-Weighter Update at Update Step r
Input: Fr.PM and Yr, the M x H matrices containing as rows the rolling forecasts the FM and ELF--Forecstee for update step т, respectively; and, X,, Y, the M x and M H matrices containing the true values for the contexts and forecasts for update step т, respectively.
Compute losses for update step (we average MASE over the last M time steps for the loss function): Loss,,1
Aaeaaee_eAvaraaMMA((.... Loss,,2 SompuueeAverrge_MMASEEE EE,, Y, X7)
Loss Averute_Average
Average MASE wel P FM + (1- (1-wwlor) wollow y EFF K,,X)) fast
Loss,, Compute_Average_MAASE (Oat Y + (1- /asl
T.FF Y,, X+)
Update slow weighter:
welow plom,
wstoo
welor plaay, + -eelore nlo0,, 2
Update fast weighter:
w fast
Update merge weighter: guerge oloa
gmerge
gmerge qlin +(1 -. 00 gunnee
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
B. Hyperparameeeess and Additional Experimental Details
There are few additional details to mention about our experiments. First, there are the hyperparameter values used for ELF. These are chosen a priori and are the same across all of our experiments. This is due to the fact in the rolling window setting it is not possible to fix hyperparameters before evaluating/deploinng the forecaster. For the ELF-Forecaster the parameter values À 20 and 0.9. For the ELF- Weighter the hyperparameters B 5 update steps. Second, at the start of deployment the FLee-aoreee has insufficient data to give accurate forecasts therefore we have warm-up period of 5 update steps where it is not used in the combined forecast. Last, in Table 3 we present the seasonalities used to calculate the MASE and RMSSE scores for each dataset along with some general dataset statistic--nune number of channels and time steps.
7) 0.5 for all weighters and
Table 3. Dataset statistics and the seasonalities used for each datasct in the computation of MASE
Dataset
Seasonality
#Channels
#Time Steps
ETTh1
24
7
17420
ETTh2
24
7
17420
ETTml
96
7
69680
ETTm2
96
7
69680
US Weather
24
12
35064
Weather
144
21
52696
Solar
24
88
12840
ECL
24
321
26304
Traffic
24
862
17544
Wind-PerSec
60
86400
Solar-PerSec
60
86400
Cloud-PerSec
240
11
86400
Details of FMs For each of the FMs looked at, we have aimed to use the same configurations in the original works. However, there some necessary changes we needed to make. First, both TTM and TimesFM are trained using context length of 512 and so in our experiments we remove the first 8 values of each 520-long context before giving it to TTM or TimesFM. Also, the currently released models for TTM only predict to maximum horizon length of 96. Hence to forecast with TTM using a horizon length of 336 we auto-regressively feed-in the constructed forecast back into TTM to generate longer forecasts. This is the same technique as done the paper proposing TTM (Ekambaram et al., 2024).
Details of Online Adaption Methods: TAFAS, OneNet and FSNet For the online adaptation methods we look at, as for the FMs, have aimed to keep their setup the same as in the original works. Specifically, this means that for each method we have training step, unlike ELF, which happens after seeing 2000 time steps. Before this training step each method is uninitialised, so we use the naive seasonal forecaster to give forecasts in this region. Then at the training step we do batch training on the first 2000 time steps of the dataset to initialise each method. After this the methods are trained online like ELF, where for TAFAS we set M = 200 to make it as comparable to ELF as possible. While, given how different OneNet, OneNet-TTWe TTM and FSNet are to ELF we set M = for these three methods to maximise their performance, which the same setting as in their original works (Zhang al., 2023; Pham al., 2023). Additionally, as we are the rolling window setting we cannot normalise the data ahead of time, which is done in the original works proposing TAFAS, OneNet and FSNet. Therefore, instead use RevIn (Kim et al., 2021) with each method to standardise the data in experiments. Finally, there are few more specific details we need to mention for TAFAS. One being that we do not use Prediction Adjustment. This is because in our setting we assume the forecasts cannot be modified after they have been given, as in the real-world they are often used immediately for decision making. While the other is that for TTM we use the both gated calibration modules for all datasets bar Solar and Traffic. For the other FMs and for Solar and Traffic for TTM we only use the output gated calibration module due to GPU Memory constraints we had while performing the experiments.
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Table MASE of time series foundation models with and without using ELF on per-second time series: A lower MASE better and we present results for each dataset over multiple forecast horizon lengths denoted as H in the table. The results show that by using ELF we improve performance across per-second datasets and forecast lengths tested.
Time Series FMs
Dataset
H
TTM +ELF(U)
TimesFM
VisionTS +EI.(()
Chronos
Moirai
+ELF(L)
+EL.F()
+EL.F()
30
0.728
-0.041
0.774
-0.089
1.492
-0.794
1.096
-0.398
0.757
-0.070
Wind-PerSec
1.770
-0.036
1.851
-0.111
2.299
-0.552
2.136
-0.386
1.821
-0.082
336
4.759
-0.058
4.923
-0.208
4.975
-0,260
5.219
-0.482
4.786
-0.079
30
0.286
-0.050
0.246
-0.016
0.738
-0.499
0.275
-0,041
0.346
-0.109
Solar-PerSec
96
0.581
-0.103
0.508
-0.030
0.907
-0.417
0.536
-0.051
0.666
-0.183
336
1.543
-0.165
1,528
-0.165
1.647
-0,236
1.429
-0.045
1.684
-0,229
30
0.940
-0.128
0.879
-0,079
0.980
-0.161
0.920
-0.110
0.947
-0.132
Cloud-PerSec
96
1.060
-0.119
1.010
-0.085
1.075
-0.026
1.067
-0.127
1.082
-0.137
336
1.285
-0.126
1.234
-0.094
1.240
-0.082
1.334
-0.172
1.379
-0.212
C. Additional Experimental Results
C.1. Results Per-Second Datasets
In the main text we look standard times series datasets used frequently in the literature (Darlow et al., 2024), here look additionally datasets which have per-second frequency. The reason do this is to explore settings where speed of updating and computing forecast is required to be measured in seconds and so where there is the most need for the compute efficiency of ELF. We perform experiments with three per-second datasets: Wind-PerSec (Godahewa et al., 2021), Volar-PerSec (Godahewa et al., 2021) and Cloud-PerSec (Joosen et al., 2023). The rest the experimental setup the same as for the experiments in the main text. Results for these experiments are presented Table 4 and show that for all the per-second datasets using ELF leads to better performance. Hence, on high-frequency datasets where computational efficiency is major factor, we find that the lightweight ELF improves forecasts.
C.2. Comparison of ELF TAFAS
In the main text we compare ELF with TAFAS using TTM as the FM. Here, we also provide results comparing ELF with TAFAS using the rest of the FMs used in our experiments, except for Chronos. The reason do report results for Chronos is due to its large computational cost. The results presented in Table and show that, with TTM, ELF performs better than TAFAS on FMs tested for all datasets look at. This demonstrates the benefit of using ELF over TAFAS for the online adaptation of FM forecasts. However, we note that given the computational efficiency of both methods. it maybe more beneficial to use both methods at the same time to improve FM performance, which we leave to future work to explore.
C.3. Comparison of ELF to Continual Finetuning
An alternative method to ELF for using online feedback in the rolling window setting is finetuning the FM at each update step. As explained in Section this has three main problems: a) there is no default model-agnostic way to repeatedly finetune time series FMs; continually finetuning FMs lead to problems found continual learning like catastrophic forgetting which are hard deal with (De Lange al., 2021; Lee Storkey, 2024a); and e) finetuning large FMs is computationally expensive and cannot be done in many real world deployment settings (Ekambaram et al., 2024). These reasons why we do not compare the efficient, model-agnostic and unforgetting ELF to finetuning in the main text. However, it is still useful to see how well ELF compares to finetuning approaches.
To look at the performance of finetuning in the rolling window setting we perform an experiment where we finetune TTM at each update step. We use TTM as is the only time series FM which we know of that proposes an efficient finetuning scheme but we note that this scheme specific TTM and cannot be used for other FMs, unlike ELF (Ekambaram et al.,
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Table MASE time series foundation models using ELF TAFAS: lower MASE is better and we present results each dataset over multiple forecast horizon lengths denoted as H in the table. The results show that ELF performs better than TAFAS across all datasets and forecast lengths tested.
Time Series FMs
TTM
FM
VisionTS
Moirai
Dataset
H
+ELF
+TAFAS
+ELF
+7AFAS
+ELF
+TAFAS
+ELF
+TAFAS
30
0.911
0.927
0.891
0,910
0.904
0.960
0.922
1.009
ETTh1
96
1.067
1.080
1.058
1.090
1.055
1.078
1.082
1.166
336
1.280
1.289
1.288
1.317
1.283
1.300
1.316
1.418
30
1.455
1,469
1.447
1.466
1.471
1.538
1.477
1.540
ETTh2
96
2.770
2.788
2.759
2.789
2.759
2.786
2.801
2.872
336
6.791
6.825
6.776
6.797
6.755
6.767
6.808
6.924
30
0.753
0.781
0.754
0.828
0.768
1.008
0.767
1.072
ETTml
96
0.919
0.957
0.920
1.000
0.919
1.021
0.929
1.225
336
1.143
1.193
1.145
1.230
1.139
1.199
1.152
1,459
30
0.763
0.784
0.757
0.812
0.789
1.031
0.782
0.946
ETTm2
96
0.953
0.979
0.951
1,015
0.964
1.065
0.971
1,151
336
1.279
1.306
1.301
1.348
1.278
1.335
1.303
1.521
30
0.857
0.893
0.827
0.857
0.855
1.008
0.828
0.896
US
96
1.083
1.107
1.061
1.115
1.070
1.139
1.055
1.144
Weather
336
1.252
1.269
1.241
1.303
1.238
1.274
1.224
1.336
30
0.855
0.880
0.777
0.795
0.893
1.322
0.900
1.138
Weather
96
1.162
1.197
1.049
1.051
1.183
1.414
1.217
1.652
336
1.536
1.567
1.493
1.613
1.550
1.643
1.579
2.021
30
1.060
1.091
1.038
1.096
0.984
1.004
1.091
1.222
Solar
96
1.098
1.129
1.103
1,200
1.058
1.079
1.144
1.308
336
1.134
1.166
1.149
1.248
1.149
1.236
1.173
1.292
30
0.917
1.000
0.875
0.977
0.937
1.223
ELC
96
1.012
1.100
0.987
1.077
1.028
1.301
336
1.193
1.267
1.191
1.298
1.212
1.474
30
0.820
0.887
0.822
0.962
0.741
0.770
Traffic
96
0.843
0.921
0.832
0.943
0.735
0.752
336
0.881
0.965
0.885
1.011
0.782
0.801
2024). To address the continual learning problems encountered by incrementally finetuning TTM at each update step, finetune using experience replay, well known and well performing continual learning method (Ostapenko et al., 2022; Chaudhry et al., 2019; Wang et al., 2024). More specifically, we set MM = 200 as in our main experiments, maintain a memory buffer 400 previously-seen instances-unin uniformly sampled from seen instances- allow the method to see the last 400 time series instances. Then we finetune using TTMs scheme where the data in the memory buffer and the first 200 of the last 400 time series instances as training data and the last 200 being used validation data for early stopping. We run the experiment on Intel(R) Xeon(R) Platinum 8168 CPUs, to model a realistic deployment scenario, and measure both the MASE forecasting performance and the mean time taken to perform an update step for finetuning TTM and for ELF. The results of the experiments for the ETT datasets are presented in Table 6. The table shows that using ELF improves both forecasting accuracy and computational efficiency. For example, ELF has a 5.89% better forecasting performance and is 2506x faster. This means that while finetuning may not be able to be used in a real-world deployment setting due to computational expense (Joosen et al., 2023; Diao et al., 2024). ELF very likely can be as it incurs only a very small computational overhead.
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Table Computational and forecasting performance of using ELF with TTM compared incrementally finetuning TTM (TTM- Finetune): In the table we present the average results for all the ETT datasets over the forecast horizons of {30, 96, 336]. The results show that using ELF improves both the forecasting performance and computational efficiency when compared to finetuning.
Method
Seconds-Per-Upeee Updaee (CPU)
MASE
TTM-Finetune
911.52
1.746
TTM+ELF
0.38
1.673
Avg. ELF improvement
2506x (f)
5.89% % (t)
Table MASE results of finetuning TTM we finetune TTM on the first 2000 time steps and the use it to forecast the rest of the time series. We compare this method to (zero-shot) TTM and when using TTM with ELF (TTM+ELFF). Furthermore, bold methods which perform the best for each dataset and forecast horizon combination. The table shows that finetuning TTM once damages its performance, as this performs worse than TTM without finetuning.
the ETT datasets Finnetenge-Finnttuee): For each dataset and forecast horizon,
Dataset
H
TTM TTM--Sigt--Finetune
TTM+ELF
30
0.930
0.965
0.911
ETTh
96
1.081
1.172
1.067
336
1,286
1.537
1.280
30
L,472
1,579
1.455
ETTh2
96
2.786
2.927
2.770
336
6.802
7.044
6.791
30
0.802
0.856
0.753
ETTml
96
0.973
1.055
0.919
336
1.205
1,346
1.143
30
0.799
0.885
0.763
ETTm2
96
0.991
1.116
0.953
336
1.320
1.564
1.279
C.4. Comparison of ELF to Single Finetuning Step
In the previous section we explored how well ELF compared finetuning TTM every update step. In this section explore how well it compares to finetuning TTM once, commonly explored setup in previous work (Ekambaram et al., 2024; Ansari et al., 2024). More specifically, we use TTM zero-shot for 2000 time steps and then use the data seen to finetune TTM. For the ETTh datasets this corresponds to finetuning on s 10% of the data. We use the same method as in Appendix C.3 to perform the finetuning with the newest 400 data points used for validation and the rest for training. After this we use the finetuned TTM forecast the rest of the time series. The results these experiments for the ETT datasets are presented in Table 7. The table shows that by only finetuning once we perform worse than using TTM zero-shot and additionally worse than using TTM with ELF. This indicates that while a single finetuning step might help improve forecasts in the short term, in the long term it can damage TTMs ability to generalise to the changing data distribution of the time series. Therefore, these results suggest the need for online updating to improve forecasts of FMs at deployment.
C.5. Continually Finetuning TTM with ELF
Up this point have assumed that we keep the FM fixed while using ELF. This is mainly due the fact, as discussed before, that it is computationally expensive to continually finetune an FM. Additionally, the continual finetuning of FMs is not straightforward, coming with numerous complications as demonstrated by work in continual learning (CL) (De Lange et al., 2021). However, it is still interesting to explore the performance using ELF while continuing finetuning the FM. To do this we perform an experiment where we continually finetune TTM and use ELF (TTM-Finetunes) datasets. We use TTM in this experiment
on the ETT the only FM, to our knowledge, with a efficient finetuning scheme. The
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Table MASE results of finetuning TTM and using ELF (FTM-Finetune+ELF) on the ETT datasets: report results where we continually finetune TTM alongside using ELF to adapt its forecasts. We bold methods which perform the best for each dataset and forecast horizon combination. The table shows that finetuning TTM ELF in general performs worse than keeping TTM frozen.
Dataset
H
TTM+ELF TTM-Finetune+ELF
30
0.911
0.940
ETTh1
96
1.067
1,081
336
1.280
1.300
30
1.455
1.595
BTTh2
96
2.770
2.784
336
6.791
6.784
30
0.753
0.784
ETTml
96
0.919
0.918
336
1.143
1.140
30
0.763
0.799
ETTm2
96
0.953
0.959
336
1.279
1.282
results of this experiment are presented in Table 8 and show that keeping TTM fixed performs best in most cases. However, for a few dataset and forecast horizon combinations finetuning TTM alongside ELF helps, though more computationally costly. Therefore, this experiments shows that keeping the FM fixed a sensible design decision, performing in general better. But, there maybe cases where computational cost and continual learning are not as big issues where is beneficial to finetune the FM as well.
C.6. ELF Performance versus Update Frequency
In our experiments we refit the ELF-Forecaster every M = 200 time steps. This number is fixed across our experiments and has been tuned. In this section we look how performance impacted by varying this update parameter. Figure plots the MASE of our approach as increase the update frequency of ELF The results shown for the ETTHh (right) and ETTml (left) datasets for the TTM FM (the FM attaining the best results this dataset). These graphs demonstrate how more regular updating generally improves performance of our approach. This supports our core hypothesis that for time series it is crucial to utilise the most up-to-date data.
Performance Retraining Frequency (ETTh2)
Performance vs Retraining Frequency (ETTm1)
0.97
2.890
0.96
MASE
2.885
FM+ELF MASE
0.95
MASE
FM MASE
FM MASE
0.94
2.880
0.93
2.875
50
100
150
200
250
300
60
Times Retrained During Evaluation
Times Retrained During Evaluation
Figure ELF performance as a function of update frequency used during online evaluation, indicating that more frequent updating boosts performance: The left figure shows results for the ETTh2 dataset, while the right figure shows results for the ETTm! dataset.
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
ELF Total Time Batch
ELF Time Taken vs Times Retrained
16
0.15
14
Wondbury
Total (seconds)
Time Taken (secends)
0.13
10
0.12
0.10
0.09
60 Percentage of Frequencies Discarded (%)
60
100
40 Number Retrains
100
Figure 6. The impact on time taken for fitting and inference as function of percentage of frequencies discarded by ELF; We measure the time ELF takes for litting and inference on batch size of 200 creases. As more frequencies are discarded, the total time decreases. Fitting and inference are performed using two CPU cores. Due to the presence of noise, the curve is not perfectly monotonic, as might be expected.
Figure 7. The impact of retraining frequency on ELF fit- ting+inference speed: We plot the total time taken for fitting and inference in ELF as we vary the number of times the model is re- fitted during evaluation on the ETTh1 dataset with horizon 336. We compare the performance of the Woodbury update rule (blue) with the more basic OLS ('linear') ) approach. frequency components are removed in this experiment. As retraining becomes more frequent, the benefit of using the Woodbury identity increases; when retraining occurs 84 times, the speed-up is approximately 25% over the naive 'linear' implementation. Minor fluctuations in the curve reflect the noise inherent in measuring computation speed.
percentage of discarded frequency components in-
C.7. ELF-Forecaster Computational Efficiency Ablation
When training online in real-time is it vital that model fitting be fast and computationally low-cost. This necessity motivates our decision to use a linear model which can be in closed-form for the ELF-Forecaster. Additionally, as outlined in Section 4, we take advantage of two ways to speed up our method: 1) we use the Woodbury matrix identity and 2) we discard high frequency components when fitting the ELF-Forecaster. In this section we evaluate how these techniques impact the amount fitting time CPU. This ablation complements Section C.8 which explores how removing high frequencies impacts the performance of ELF, demonstrating that the decline in performance is slight. Thus together this section and Section C.8 validate our decision to remove high frequencies to improve speed.
C.7.1. IMPACT OF % OF DISCARDED FREQUENCIES ON EXECUTION SPEED
As detailed in Section 4 the ELF includes a feature allowing the model to discard a percentage of the high frequencies of the context and targets when fitting in order to improve inference and fitting time. In this section we explore how varying the percentage of high frequencies which are discarded impacts the total execution time for fitting and inference. Specifically, vary the proportion of frequencies discarded in steps of 5% and record the time taken by ELF, for fitting and inference on single data batch of size 200. For these experiments we do not the Woodbury identity to investigate the role of frequencies on execution time in isolation. To ensure consistent resource allocation, we bound the execution to two CPU cores. This allowed us to isolate the computation to specific cores, mitigating potential variability caused by the operating system's 's task scheduler.
Figure shows the results of these experiments, plotting fitting inference time against percentage of discarded frequencies. The plot suggest that execution speed improves by discarding higher proportion of frequency components. For example, discarding 40% of components results in 25% speed-up compared to using 100% of the components.
C.7.2. SPEED-UP FROM THE WOODBURY MATRIX IDENTITY
The FLr--orrcasttr, detailed in Section 4, uses the Woodbury matrix identity. In this section we record how this design decision impacts the time taken for fitting and inference of ELF. We vary the number of times that we refit the ELF-Forecaster during evaluation on the ETThl dataset and record the total time taken to fit and predict using the ELF model. We repeat twice, once using the Woodbury identity and once not using the identity which we call linear. All fitting and prediction
22

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Lightweight Online Adaption of Time Series Foundation Model Forecasts
CPUs. For these experiments we do not throw away any high frequency components that study the impact of the woodbury identity in isolation. We prediction horizon of 336. The results of these experiments plotted in Figure 7, where we plot the number of retrains on the x-axis aginst total time in the y-axis. 'Woodbury' is plotted in blue and 'linear' in red.
By inspection of Figure 7 we see that, when one retrains rarely, is preferable not to use the Woodbury update rule. This because the Woodbury update rule is suited low-rank updates and when refitting occurs infrequently, the rank of the update matches the rank of the matrix being updated. However, when retraining occurs regularly the Woodbury grants a speed-up over the naive implementation. Moreover, the gain increases as the regularity of retraining increases, In our main experiments refit every 200 times steps. For the ETThi dataset this corresponds to refitting the ELF-Forecaster 84 times. At this retraining frequency the Woodbury matrix identity grants roughly 25% speed-up. The graph features certain bumps which reflect the noise inherent in measuring execution time. Both graphs show a bump at 50 retrains. On the ETTh dataset 50 retrains corresponds to retraining every 336 time steps, this value is equal to the prediction horizon length. Consequently, as the number of retrains increases from 49 to 50 prediction length exceeds the batch size for the first time impacting the speed of the matrix computations and explaining the apparent discontinuity.
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Performance vs % Discarded Frequencies (Weather Dataset)
Performance vs % Discarded Frequencies (ETTm1 Dataset)
FM+ELF
0.96
FM+ELF
0.95
MASE
1.19
MASE
1.18
0.93
1.17
0.92
Percentage of Frequencies Discarded (%)
Percentage of Frequencies Discarded (%)
Performance vs % Discarded Frequencies
Performance vs % Discarded Frequencies
(ECL Dataset)
(Solar Dataset)
1.15
1.14
EM
FM+ELE
MASE
MASE
1.09
1.12
1.11
1.08
1.10
1.07
1.08
100
Percentage of Frequencies Discarded (%)
Percentage of Frequencies Discarded (%)
Figure ELF performance a function of the percentage of high-frequency components discarded before model fitting: We plot the MASE of FM+ELF as the percentage of high-frequency components removed by the EL.-Forecaster increases, for the Weather (top left plot), ETTml (top right), Electricity (bottom left), and Solar (bottom right) datasets. The foundation model (FM) used TTM. The MASE of the FM (black dashed line) is shown for comparison. As more frequencies are discarded, performance begins to decline; however, initially, this decline minor (e.g. ETTml dataset, removing 20% of frequency components results only a roughly 0.1 % increase in MASE).
C.8. ELF Performance as Function of %
Discarded
The -or-ccccar removes high frequencies from the context and target to reduce dimensionality, thereby speeding-up model fitting and inference. In Figure we look at how performance is impacted by this design choice. We plot the performance on the y-axis and the percentage of frequency components which are discarded on the x-axis. The plot shows the Weather (top left plot), ETTml (top right), Electricity (bottom left), and Solar (bottom right) datasets, each using a forecast horizon of 96 and a TTM base forecaster. The performance the TTM FM without online adaption is given by the horizontal black dashed line. While removing high frequencies results in a drop performance, removing only 10-20% of frequencies has negligible impact method performance. Taking into account the meaningful speed-up observed in the ablations in Section C.7 validates this decision drop high frequency components.
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
C.9. ELF-Forecaster Ablation
is important to understand the benefit of using the ELF-Forecaster when compared to other potential options. To do this we perform experiment where we replace the ELF-Forecaster with different types of forecasters, keeping the rest of ELF the same. The different types of forecasters we look are: linear model trained with online gradient descent (OGD), FITS, FSNet and linear forecaster trained online to predict the residuals of the FM. The results of this experiment are presented in Table 9. For brevity, we report for each FM and forecast horizon combination the average MASE relative to the the MASE of the naive seasonal forecaster, averaging over the datasets used in the main experiments. The table shows that for all forecast horizons and FMs the ELF-Forecaster outperforms all of the other forecasters tested. This provides validation for the design decisions taken for the Forecaste..
Table 9. Ablation of using different forecasters instead of the ELF- FLF-Forecatter: We replace the ELF-Forecaster with: linear model trained by online gradient descent (OGD), FITS, ESNet and residual predictor approach. In each case we change nothing except for the forecaster; combining the FM and the forecaster using the FI.--Wiihtter Results are given as MASE relative to the the MASE of the naive seasonal forecaster and we average results across all datasets. We also report the standard error as subtext. The results show that using the -Lrecaseer gives better performance than using any of the other alternatives tested across each FM and forecast horizon.
Average Rel. MASE w.r.t Naive Seasonal Forecaster over all Datasets
FM
H
ELF
OGD-Forecaster
FITS-Forecaster FSNet-Forecaster
Residual-
30
0. 0.796+0.08
0.840+0.039
0.842+0.010
0,833+0.039
0.827=0.013
TTM
96
0. 0.83140.00
0.872+0.029
0.874+0.030
0.869+0.030
0.871
336
0.867 +0.028
0.904+0.022
0..022
0.905+0.023
0.926=0.014
30
0. 0..7620.052
0.817
0.818+0.051
0,814+0.053
0.81640.054
TimesFM
96
0. 0.853 10.029
0.91010.029
0.912
0.926
0.905
336
0. 0..911
0.958
0.964
1.01440.047
0.956
30
0.791 10.000
0.930_1.009
0.935
0.91810.016
0.893
VisionTS
96
0. 1.825+0.033
0.898+0.029
0.902+0.029
0.897+0.082
0.883+0.028
336
0. 00.084
0.928+0.022
0.9022
0.930+0.024
0.92440.021
30
0. 00.03
0.820
0.822+0.032
10.033
0.816.0.032
Chronos
96
0.8 0.810+0.011
0.915 056
0.917 +0.056
0.914+0.08
0.892
336
0. 0.8561..00
0.960.0.046
0.96610.046
0.968
0.933
30
0..00
0.93940.039
16.000
0.928 20.049
0.920
Moirai
96
0.836
0.980
0.98210.062
0.976
0.947 10.000
336
0.870 10.037
0.990
0.997 10.054
0.996
1.001 10.019
C.10. ELF-Weighter Ablation
The BLF-Weighter consists of combination separate weighers- fast and show-therefore it is useful to understand the respective contribution of these two parts. To perform this ablation we an experiment where we only used the slow weighter (LLFF-SooWWiigher) or fast weighter FaFFFhaatWeeigher) on their own. The results of this experiment is recorded in Table 10, where we report for each FM and prediction horizon the average MASE across all datasets used in our main experiments (ETTs, -eether Weather, Solar, ECL and Traffic). The table shows, by the bolded values, that using the full ELF-Weighter is better or equal to only using the fast or slow weighter. Additionally, we find that only in few cases (33%) does the performance of the fast weighter or slow weighter match the performance of the combined weighter. We find that in these cases the combined weighter defaults to either fast weighter slow weighter, perhaps due not finding a benefit in terms of performance of mixing them. Hence, we have shown that making the ELF-weighter a combination of a fast and slow weighter generally improves performance and never makes it worse, while also having minimal additional compute overhead.
In Table 10 we also show the results of using the Hedge weighting algorithm HELF-Hedge or an unweighted mean to combine the forecasts of the FM and FLF--oreereter (FL.F-Unweiggtte)) Hedge is similar to exponential weighting
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Table 10. Results of ablation where we compare using the full ELF-Weighter (ELF) with when using only slow weighter component (ELF-Slow Weighter), fast weighter component (ELF-Fast Weiighter,,,, using Hedge (ELF-Hedge Weighter) or using an unweighted mean (ELF-Unweige) :nweighthtt;;: We present for each FM and prediction horizon the average MASE (#: standard error) across all datasets used in our main experiments. We bold the lowest avgerage MASE weighter each FM and prediction horizon. The table shows that using the full ELF- WLiihheth is better than using either of components independently and that using Hedge or an unweighted mean performs poorly.
Average Rel. MASE w.r.t Naive Seasonal Forecaster over all Datasets
FM
H
ELF
ELF-Slow Weighter
ELF- ELF-FastWeighter GLF-HedgeWeiighter
ELF- ELF-Unweighted
30
0. 79640.038
0.797=0.038
0.798
0.809
0.804+0.037
TTM
96
0. 0.8311 20.03
0.832
0.833+4.0332
0.843
0.839
336
0. 0.86770.002
0.868
0.869+0.027
0.877
0.874+0.025
30
0. 0.77640.032
0.782+0.050
0.776
0.799+0.054
0.784+0.048
TimesFM
96
0. 80.39
0.858
0.853
0.87210.030
0.871
336
0. 0.8988..001
00.017
0.898
0.91310.017
0.967
30
0.791 0.79110.0000 10.030
1..00
0.79210.030
0.811 10.034
0.829
VisionTS
96
0. 0.82540.033
0,828
0.82640.032
0.841
0.842+0.031
336
0. 0.86840.022
0.870
0.870
0.88310.027
0.878
30
0. 0.753jmm 10.037
0.758 100037
0.755
0.777 10.039
0.774
Chronos
96
0.810
0,814
0.811
0.830+0.041
0.841+0.012
336
0.85640.0.2
0.858
0.85640.0s
0.874
0.88640.036
30
00.00
0.802+0.011
0.802+0.012
0.818+0.011
0.830+0.037
Moirai
96
0.836.10.044
10.041
0.838 10.043
0.849 0.8491..044 10.043
0.872
336
0.870+0.037
0.870+0.036
0.872+0.037
0..81+0.036
0.901
where the only difference is that at each update step the FM or ELF-Forecaster is chosen to solely give forecasts until the next update step by sampling from the distribution defined by the weights (Cesa-Bianchi & Lugosi, 2006). While, by an unweighted mean we mean that we set w-- = 0.5 for all update steps т. The results in Table 10 show that using either Hedge or an unweighted mean leads to poor performance and they are never better or equivalent using the ELF-Weighter in our experiments. Hence, this provides evidence that using weighted mean combine forecasts is good design decision.
C.11. Analysis of the Performance of ELF-Forecaster Compared to FMs
To understand the impact of learning from the up-to-date feedback given in the rolling window setting, we present here the performance of using the ELF-Forecaster on it own. We compare this against using (zero-shot) TTM, the best performing FM in our experiments, and when using ELF with TTM (TTM+ELF). The results are displayed in Table 11, from which we can draw three main conclusions: a) for some datasets using TTM zero-shot performs better than learning from the given dataset online using ELF-Forecaster (e.g. ETThl and ETTh2); b) for other datasets by learning online on the given dataset
performs better than using TTM (e.g. ECL and Traffic); and e) using ELF-Forecaster to adapt the forecasts of TTM (TTM+ always improves performance against using either separately. While using summary MASES informative, it also useful look at how the relative performance between the FLF-Forecaster and FMs changes over time. We can look at Figure 3 to do this, as it displays the weight w, used to weight between the ELF-Forecaster and the Chronos at each update step for ETTh1. The figure shows that for most channels that at the start Chronos is preferred and performs better than the ELF-Forecaster. But, as the ELF-Forecaster sees more data and therefore learns the specific time series characteristics it gradually performs better and therefore is weighted more heavily. This all shows that, as expected, using the up-to-date feedback in the rolling window setting (i.e. deployment stage), means we can learn a dataset specific forecaster (EILF Foreccaast)) which steadily improves in performance over time to be comparable to the FM. Therefore, can be used to adapt the FMS forecasts to be more dataset specific, to improve performance, which our results experimentally validate (e.g., see Appendix C.13).
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
Table MASE ELF-Forecaster,
when adapting TTM with ELF (TTM+ELF); results show that Forecaastr. which is trained online on each dataset, and TTM, performing zero-shot forecasting, perform similarly. TTM sometimes performs better than Eore-aoreeeterr and othertimes not. Additionally, TTM+ELF improves upon both TTM using ELF-Forecaster on own for all datasets and forecast horizon lengths.
Dataset
H
ELF-Forecaster
TTM TTM+ELF
30
0.946
0.930
0.911
ETTh1
96
1.113
1.081
1.067
336
1.335
1.286
1.280
30
1.503
1.472
1.455
ETTh2
96
2.819
2.786
2.770
336
6.822
6.802
6.791
30
0.769
0.802
0.753
ETTml
96
0.931
0.973
0.919
336
1.150
1,205
1.143
30
0.793
0.799
0.763
ETTm2
96
0,981
0.991
0.953
336
1.300
1.320
1.279
30
0.877
0.893
0.857
US
96
1.096
1.123
1.083
Weather
336
1.262
1,296
1.252
30
0.907
0.887
0.855
Weather
96
1.205
1.205
1.162
336
1.568
1.576
1.536
30
1.084
1.091
1.060
Solar
96
1.124
1.129
1.098
336
1.172
1.166
1.134
30
0.930
1.003
0.917
ECL
96
1.021
1.106
1.012
336
1.205
1.279
1.193
30
0.871
0.887
0.820
Traffic
96
0.888
0.920
0.843
336
0.922
0.965
0.881
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
C.12. RMSSE Results
Table RMSSE time series foundation models with and without using ELF: lower RMSSE is better and we present results for each dataset over multiple forecast horizon lengths denoted 'I"" in the table. The results show that by using ELF we improve RMSSE scores across all datascts and forecast lengths tested, as when ovaluating with MASE.
Time Series FMs
Dataset H
TTM +ELF(J)
TimesFM
VisionTS
Chronos
Moirai
+ELF(L)
+ELF(L)
+ELF(L)
+ELF(()
30
0.806
-0,016
0.805
-0.023
0.864
-0.068
0.840
-0.052
0.877
-0.076
ETTh1
96
0.954
-0.012
0.994
-0.049
0.975
-0.031
1.017
-0.068
1.032
-0.074
336
1,149
-0.005
1.215
-0.058
1.176
-0.024
1,231
-0.072
1.259
-0.085
30
0.838
-0.015
0.842
-0.021
0.911
-0.072
0.868
-0.034
0.896
-0.056
ETTh2
96
1.149
-0,012
1.178
-0.037
1.166
-0.029
1.209
-0.054
1.219
-0,060
336
1.807
-0.010
1.832
-0.036
1.802
-0.015
1.897
-0.086
1.900
-0.084
30
0.680
-0.039
0.710
-0.067
0.860
-0.208
0.763
-0.122
0.905
-0.253
ElTml
96
0.865
-0.046
0.910
-0.088
0.963
-0.139
1.040
-0.216
1.093
-0.265
336
1.077
-0.052
1.135
-0.106
1.108
-0.080
1.321
-0.287
1,316
-0,282
30 0.668
-0.032
0.693
-0.055
0.868
-02213
0.742
-0.097
0.805
-0.154
mT
96
0.847
-0.034
0.892
-0.075
0.955
-0.129
0.981
-0.156
1.001
-0.173
336
1.123
-0.036
1.187
-0.094
1.163
-0.073
1.285
-0.186
1.299
-0.198
30
0.744
-0.030
0.746
-0.038
0.875
-0.149
0.803
-0.088
0.779
-0.067
US
96
0.971
-0.032
1.035
-0.092
1.015
-0.072
1.063
-0.115
1.030
-0.087
Weather
336
1,148
-0.033
1.230
-0.108
1.159
-0.043
1.278
-0.149
1.232
-0.108
30
0.580
-0.028
0.521
-0.017
0.941
-0.359
0.724
-0.167
0.753
-0.184
Weather
96
0.987
-0.035
0.888
-0.034
1.203
-0.223
1.363
-0.402
1.337
-0.365
336
1.643
-0.031
1.767
-0.214
1.761
-0.131
2.061
-0.439
2.001
-0.369
30 0.790
-0.005
0.831
-0.043
0.836
-0.045
0.846
-0.065
0.922
-0.104
Solar
96
0,866
-0.006
0.957
-0.079
0,919
-0.045
0.997
-0.106
1.031
-0.131
336
0.903
-0.005
1.000
-0.077
0.973
-0.057
1.033
-0.091
1.046
-0.109
30 0.846
-0.066
0.863
-0.102
0.781
-0.040
1.049
-0.250
ECL
96
0.956
-0.070
0.964
-0.092
0,929
-0.063
1.135
-0.235
336
1.115
-0.062
1.191
-0.138
1.130
-0.085
1.285
-0.214
30 0.709
-0.051
0.793
-0.128
0.587
-0.027
0.630
-0.026
Traffic
96 0.783
-0.057
0.811
-0.090
0.723
-0.052
0.678
-0.020
336 0.843 -0.057
0.876
-0.090
0.895
-0.115
0.754
-0.021
C.13. The ELF-Forecaster Learns Dataset-Specific Features
While FMs are designed to produce good forecasts out-of-thee tft-of-the-box, avoiding a dataset-specific training routine can mean that such models are not optimally tuned to the given data distribution. contrast, the ELF-Forecaster is fit to data drawn from the specific time series. This allows it to pick up on dataset-specific features than the FM may miss. This way combining the forecasts of the FM with those of the FLF--oreeaster can boost performance. Two concrete examples of this are given in this section.
Figure 9 shows forecasts the Traffic dataset (channel 620) made by TTM (left, orange) and ELF-Forecaster (right, blue). We see that while the forecasts of TTM are good, capturing daily periodicity, it does not model the decrease in traffic which occurs during the weekend (regions shaded in blue). By contrast, the ELF-Forecaster is able to identify and predict this decrease traffic.
Figure 10 shows forecasts on two cloud time series (Joosen et al., 2024) made by TTM (orange) and TTM+ELF (blue)
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Lightweight Online Adaption of Time Series Foundation Model Forecasts
TTM vs ELF-Forecaster Forecasts Traffic
TTM
ELF-Forecaster
0.20
0.20
Values
0.15
0.15
0.10
0.05
0.00
10000 10200
10400
10600 10800
10000
10200
10400
10600 10800
Time
Time
Figure 9. TTM (left) and ELF- Forecasrrrr (right) forecasts on the Traffic dataset: The TTM forccast (right figure, orange) is good, picking up on the daily periodicity traffic levels. However, TTM fails model the weekly periodicity; specifically the decline in traffic occurring at the weekend (see the blue shaded regions). Conversely, the FLF--orcess,er, which fit to the dataset, predicts this decline in weekend traffic numbers.
TTM and TTM+ELF on Cloud Data
5500
12000
Ground
TTM-
5000
10000
Requests
8000
Requests
4000
Ground Truth
6000
3500
TTM+
3000
TTM
2500
2000
2000
7000 7200
6800
7000
7200
Time
Time
Figure 10. TTM and TTM+ELF forecasts on two cloud time series: In both figures, the TTM forecast (orange) is accurate but omits the large spikes which occurs around time step 7100 on left figure and around step 7170 on the right figure. This despite the regularity of these spikes making them seemingly easy to predict. In contrast, ELF predicts these spikes well that TTM+ELF (blue) generates superior forecast to TTM by itself.
compared against the ground truth (red). We see that while the forecasts of TTM are fairly accurate does not anticipate spikes which occur periodically in either of the datasets. However, the Foreccaas,,, which is fit on drawn from these dataset, is able insert the missing spikes boosting overall performance.
C.14. Note on the Performance Ordering of FMs When Using or Not Using ELF
interesting to see how the relative performance of FM change when using ELF to adjust their forecasts online. The results in Table show that in our experiments TTM is generally the best performing FM without ELF, But, when using ELF the best performing FM is less clear: the performance of each of the FMs becomes more similar and the best model varies across dataset and prediction length. This suggests that in realistic settings where it is possible to use ELF exploit online feedback to improve forecasts, the performance improvement in newer FMs is less stark than in the zero-shot setting. This adds qualifications to the suggested progress made in time series forecasting by successive generations of FMs.
29


Reviews:
[{"review_id": "WB4aee12JN", "reviewer": "Reviewer_26Ko", "summary": "The paper proposes a method to combine foundation model forecasts with forecasts from an online learner. They innovate on 2 components, the online learner, and the algorithm to combine the forecasts. The online learner is a linear model in the frequency domain, learned via efficient closed form updates. The algorithm to combine the forecasts is based on the exponential weighting algorithm, combined with the idea of slow and fast learning. Experiments are performed on some standard datasets found in the literature, and shows  consistent improvement across 5 foundation models.\n\n## update after rebuttal\nI increased my rating as my concerns were addressed.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer, thank you for your comments and constructive feedback. We are especially happy you found that \"the contributions of the paper are novel and present ideas on how to ensemble predictions in an online fashion\". We answer the comments you had below and are thankful that you pointed out issues in notation and framing which we have fixed.\n    \n**Ensembling: You identify that the proposed method is actually an ensembling method - encouraging us to reframe with the terminology of ensembling or exponential weights.**\n\nWe have edited the paper to incorporate this feedback and to frame the method in ensembling terminology. Additionally, to fully take your feedback into account we have renamed our method to **ELF**, standing for **E**nsembled with online **L**inear **F**orecaster. e.g. TTM+AdapTS becomes TTM+ELF. To reduce confusion in the rebuttal we have resorted to using the name AdapTS. \n\n**Details and Formal Notation: You point out that the paper is lacking details and formal notation in places. In particular, the AdapTS-Forecaster description could be made clearer.**\n\nThank you for pointing this out; we feel that addressing these points has improved the quality of the paper. We have now updated the paper to make the explanation of the AdapTS-Forecaster clearer, clarifying the notation and providing a more precise explanation of model fitting. We have added dimensionality information where missing, as well. Additionally, an algorithmic description of fitting the AdapTS-Forecaster is now given in the Appendix.\n\n**Baselines: You identify a lack of comparison with relevant baselines, both for the AdapTS-Forecaster and AdapTS-Weighter**\n\nThank you for this comment, it was mirrored in the feedback of other reviewers as well and has now been addressed. We have compared the AdapTS-Forecaster with FSNet, FITS and a linear model trained by online gradient descent (OGD) and we have compared the AdapTS-Weighter with Hedge as recommended. The results for TTM on the ETT datasets are provided in the table below. Results are given relative to our approach so that values above 1 correspond to worse performance. In each case results are worse than using AdapTS. These results will be extended to all datasets and FMs. \n\n||||Relative MASE to TTM+AdapTS|||\n|-|-|-|-|-|-|\n|**Dataset**|$H$|**Hedge-Weighter**|**OGD-Forecaster**|**FSNet-Forecaster**|**FITS-Forecaster**|\n|ETTh1|30|1.020|1.037|1.029|1.037|\n||96|1.018|1.025|1.023|1.026|\n||336|1.013|1.007|1.027|1.013|\n|ETTh2|30|1.017|1.023|1.021|1.026|\n||96|1.008|1.010|1.008|1.010|\n||336|1.003|1.004|1.009|1.003|\n|ETTm1|30|1.021|1.070|1.068|1.070|\n||96|1.013|1.061|1.060|1.059|\n||336|1.009|1.057|1.057|1.054|\n|ETTm2|30|1.018|1.051|1.051|1.052|\n||96|1.015|1.040|1.042|1.041|\n||336|1.012|1.034|1.034|1.032|\n\n**Results Presentation: You propose that it would be better to normalize the metrics to a baseline, then aggregate over the different datasets and settings.**\n\nThank you for suggesting this way to incorporate model results into the main text. We have updated the paper to normalized MASE (w.r.t. the naive seasonal forecaster) and once all our additional experiments have run we will aggregate in the way you propose to ensure that as many of them can be presented in the main text as possible.\n\n**I want to verify that the loss used in exponential weighting is from previous time steps, i.e. no data leakage?**\n\nYes, the true values used to train both the weighter and the forecaster at some time step _t_ are only those already seen by that time step (i.e. 0 to t).  \n\n**\"Time series foundation models could be replaced with standard deep learning forecasting models, and the paper still reads exactly the same.\"**\n\nYou are correct that this method could in principle be applied to standard deep-learning forecasters. We decided to focus on FMs due to the particular difficulties and computational cost of finetuning these approaches online.\n\n## Alteration Summary\nThank you again for your feedback. Below is a summary of improvements made based upon your feedback. We look forward to hearing your thoughts.\n\n1) Compared Forecaster and Weighter against Baselines\n2) Addressed issues with notation and terminology\n3) Reframed our approach with the terminology of ensembling\n4) Updated to Normalized MASE", "claims_and_evidence": "One major issue I have is that the paper claims to propose a method for adaptation of foundation models in the online/continual learning setting. However, the proposed method is actually an ensembling method - the foundation models are used as is, and are not adapted, instead, their predictions are ensembles with another online learning forecaster. I strongly encourage the authors to rename/reframe the proposed method, with the terminology of ensembling or exponential weights.\n\nUnfortunately, Table 5 in the appendix makes the evidence for the idea of slow and fast weighting to be weakened, as the fast weighting only seems to match the full approach in most cases.", "methods_and_evaluation_criteria": "### Method\nThe paper is lacking details and formal notation. Especially in the methods section, AdaptTS-Forecaster should be described with more precise formal notation. I do not have a clear and exact understanding of how the linear model is learned in Fourier space. Information like what is the dimensionality of the weights, X, and Y, are missing. \n\n### Evaluation\nNo baselines are presented. We cannot judge how well the proposed innovations perform because results are only presented for the proposed method. For the AdaptTS-Forecaster, relevant baselines would be to replace it with a similar linear model but with various forms of online gradient descent, and as well as FSNet (Pham et al, 2022). For the AdaptTS-Weighter, the relevant baselines would be the naive exponential weights algorithm (I suppose this is equivalent to slow weighter only, with results present in the appendix averaged over all models), and other similar methods from the online learning literature, e.g. hedge.\n\nI encourage authors to present these results for each model, unlike table 5 - the results shouldn't really be averaged across models. Instead, it would be better to normalize the metrics to a baseline, then aggregate over the different datasets and settings (see the normalized MAE/MASE/... metrics in the Moirai and Chronos papers). That way, more results across different models and baselines can make it into the main paper, then the appendix contains the full breakdown.", "theoretical_claims": "No theoretical claims were made.", "experimental_designs_or_analyses": "I want to verify that the loss used in exponential weighting is from previous time steps, i.e. no data leakage?", "relation_to_broader_scientific_literature": "The contributions of the paper are novel and present ideas on how to ensemble predictions in an online fashion. I'm not too convinced about the need for time series foundation models for this case - they could be replaced with standard deep learning forecasting models, and the paper still reads exactly the same.", "essential_references_not_discussed": "None that I'm aware of.", "other_strengths_and_weaknesses": "None.", "other_comments_or_suggestions": "None.", "questions_for_authors": "None.", "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": "LSLNon2QnY", "reviewer": "Reviewer_6Y6Z", "summary": "This paper proposes AdapTS, a lightweight method for the online adaptation of time series foundation model forecasts. It consists of an AdapTS-Forecaster and an AdapTS-Weighter. Experiments clearly show that AdapTS can significantly improve prediction performance across multiple models and datasets.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer, thank you for your kind words about our work and constructive comments/questions. We are particularly happy that you found that AdapTS gives \"significant performance improvement with low computational cost\" and is \"applicable to any FM\". We have included additional baseline comparisons against test-time adaptation methods and provide answers to your questions below.\n\n**Summary of comparisons against test-time adaptation and gradient-based methods**\n\nIn response to your comments, we have now run experiments with several additional baselines: TAFAS (TTM-TAFAS); AdapTS-forecaster to predict residuals of the FM forecasts (TTM+Residual-Adjustment); a gradient-based online-ensembling approach for time series forecasting (OneNet-TTM) and fine-tuning the FM (TTM-Fine). The experiments are shown in the table below for TTM, the best performing FM in our experiments, on the ETT datasets. We give the results relative to the MASE performance of TTM+AdapTS, for example a score of 1.043 means that TAFAS performs 4.3\\% worse than AdapTS.\n||||Relative MASE to TTM+AdapTS||||\n|-|-|-|-|-|-|-|\n|**Dataset**|$H$|**TTM+TAFAS**|**TTM+Residual-Adjustment**| **OneNet-TTM**|**TTM-Fine**|\n|ETTh1|30|1.043|1.029|1.013|1.072|\n||96| 1.038|1.021|1.277|1.092|\n||336|1.028|1.011|1.284|1.092|\n|ETTh2|30|1.030|1.019|1.012|1.036|\n||96|1.020|1.010|1.167|1.027|\n||336|1.006|1.004|1.333|1.013|\n|ETTm1|30|1.033|1.065|1.905|1.031|\n||96|1.051|1.059|1.285|1.062|\n||336|1.049|1.050|1.576|1.059|\n|ETTm2|30|1.041|1.050|1.764|1.072|\n||96|1.038|1.043|1.502|1.076|\n||336|1.025|1.033|1.555|1.074|\n\nWe find that, in all cases looked at, AdapTS performs better than the baselines and discuss each experiment in more detail in the sections below. We hope these experiments satisfy your comments on comparing to test-time adaptation and gradient-based methods and we believe those additions significantly strengthen the paper. We further note that we are currently running experiments across all the datasets and FMs and will add them to the paper once they complete.\n\n****1. Why is $w_r$ a scalar, rather than being designed as a vector or a tensor?****\n\nIf we understand correctly, when you mention $w_r$ you are referring to the weight used to merge the forecasts of the FM and AdapTS forecaster (in Eq 2.)? If so, making $w_t$ a vector would mean that there would be a different weight applied per time-step (using the element wise product between $w_r$ and $y_{t,FM}$). In the development stage of this work we did look at this, but found that using only per channel weights yielded better performance, due to overfitting. Therefore, we did not discuss it in the paper, however to clarify the text and to address your comment, we will add appropriate mentions of it in the paper. As for making the weights a tensor, we do not know exactly what you mean by this, could you please clarify?\n\n****2. Comparison to test-time adaptation method TAFAS****\n\nWe agree that this work would be greatly strengthened by comparisons with the test-time adaption baseline TAFAS. As mentioned before, results of these experiments and other baselines are given in the table at the start of our comment. We find that in all cases looked at AdapTS performs better than TAFAS. We also note that TAFAS requires backpropagating through the FM, making it computationally slower than AdapTS. This demonstrates the advantage of AdapTS over TAFAS.\n\n****3. Why didn't the gradient-based approach work in experiments? In addition, gradient updates are not just about fine-tuning; for example, adding a small neural network to fit the residuals.****\n\nGradient-based fine-tuning does not work well for the online learning of neural networks due to the problems studied in the continual learning literature. These problems mainly manifest as catastrophic forgetting whereby updating on new data the model forgets large amounts of information about old data. Hence, the gradient-based fine-tuning of TTM did not work for online updating (as shown in the table above). We note here that by using a linear model as the AdapTS-forecaster we can sidestep all the problems of continual learning which only occur for more complex models, while being significantly faster.\n\nAdditionally, we have run an experiment in which we use the AdapTS-forecaster to predict residuals of the FM's forecasts and use this to modify the forecast. We present the results for this method in the table at the start of our comment, and find that it performs worse than AdapTS in all cases looked at.\n\n****\n\nWe thank the reviewer again for your constructive feedback. We hope given our answers and especially the added experiments comparing to TAFAS, fine-tuning, residual predictions and OneNet, we have satisfactorily addressed your concerns and believe that your comments have made our submission stronger.", "claims_and_evidence": "The claims in the paper are strongly supported by experiments. The authors conducted experiments using several foundation models and standard time series datasets. The notable reduction in the MASE when using AdapTS clearly indicates that it can remarkably enhance the prediction ability of foundation models.", "methods_and_evaluation_criteria": "The proposed methods are highly reasonable. The two - part structure effectively addresses the problem of TSFM inability to adapt online. Using standard datasets and the MASE metric is extremely appropriate for evaluating the prediction performance of different time series.", "theoretical_claims": "The paper correctly applies well - established theories such as the Woodbury matrix identity and exponential weighting. These theories firmly provide the basis for AdapTS to achieve efficient online adaptation.", "experimental_designs_or_analyses": "The experimental designs are very sound. The rolling window setting closely mimics real - world deployment. Ablation experiments greatly help analyze the role of each part of AdapTS, and comparisons with other methods (such as fine - tuning) also strongly prove the effectiveness of AdapTS. However, the lack of comparison with Test time adaptation methods is an obvious and significant shortcoming.", "relation_to_broader_scientific_literature": "This paper focuses on TSFMs and clearly differentiates from previous time series continual learning work by avoiding direct gradient - based updates.", "essential_references_not_discussed": "There are no obvious missing essential references.", "other_strengths_and_weaknesses": "- **Strengths**: AdapTS is lightweight, applicable to any FM, and shows significant performance improvement with low computational cost.\n- **Weaknesses**: Hyperparameters are set a priori without tuning, which may not be optimal.", "other_comments_or_suggestions": "None", "questions_for_authors": "- **Q1**: Why is $w_r$ a scalar, rather than being designed as a vector or a tensor?\n- **Q2**: Many Test Time Adaption methods also meet the requirement of parameter freezing, so why don't the authors compare them? (such as https://doi.org/10.48550/arXiv.2501.04970)\n- **Q3**: Why didn't the gradient - based approach work in experiments? In addition, gradient updates are not just about fine - tuning; for example, adding a small neural network to fit the residuals.", "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": "kicSI0CBZ5", "reviewer": "Reviewer_GmRz", "summary": "This paper introduces AdapTS, a lightweight mechanism designed to enhance the adaptability of Foundation Models (FMs) for time series forecasting by incorporating online feedback. Traditional FMs remain fixed after deployment due to the high computational cost of online updates, preventing them from adapting to changing data patterns. To address this limitation, AdapTS consists of two key components: AdapTS-Forecaster, which learns the current data distribution to capture recent trends, and AdapTS-Weighter, which dynamically combines forecasts from both the FM and the AdapTS-Forecaster. The paper evaluates AdapTS across multiple benchmark time series datasets and demonstrates that it consistently improves forecasting accuracy.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your helpful comments and questions! We provide answers to your questions below and hope we have satisfactorily answered them, in particular by performing several additional experiments.\n\n**1. Why fixed FMs?**\n\nThe main reason we keep the FM fixed is due to the computational expense of updating it. This is demonstrated by the fact AdapTS is 2506x faster than online finetuning TTM; the fastest FM to finetune. Additionally, online updating of FMs and neural networks more generally is not straightforward, coming with numerous complications as demonstrated by work in continual learning (CL). Furthermore, current CL solutions for these problems like FSNet are often complex and architecture specific. In contrast, one of the goals of AdapTS is to be simple and FM agnostic. Your comment illustrates that this point needed to be clarified in the paper which has been edited accordingly.\n\nTo justify using a frozen FM instead of learning it online, we run an experiment where we finetune TTM alongside AdapTS. The results of this experiment (TTM-Fine+AdapTS) are given in the table for question 2. and show that it performs generally worse than AdapTS with TTM frozen (results are given relative to TTM+AdapTS). This alongside the vast compute benefits demonstrate why keeping the FM fixed is beneficial. \n\n**2. Comparisons with OneNet, FSNet and TAFAS**\n\nThank you for pointing out the missing comparisons to OneNet, FSNet and TAFAS, which we agree should be compared to AdapTS. We have run these experiments as well and we provide the results for these baseline methods on the ETT datasets in the table below. Additionally, we provide results for OneNet where one of its forecasters is replaced with TTM (OneNet-TTM). We find that in all cases TTM+AdapTS performs better. Importantly, we perform much better than OneNet-TTM, our closest comparator. Importantly, AdapTS is also more computationally efficient than the other methods. We hope now to have better contextualised the effectiveness of AdapTS relative to these more compute-intensive adaptation methods.\n||||Relative MASE to TTM+AdapTS||||\n|-|-|-|-|-|-|-|\n|**Dataset**|$H$|**TTM+TAFAS**|**OneNet**|**FSNet**|**OneNet-TTM**|**TTM-Fine+AdapTS**|\n|ETTh1|30|1.043|1.405|1.555|1.406|1.032|\n||96|1.038|1.281|1.355|1.277|1.013|\n||336|1.028|1.275|1.333|1.284|1.016|\n|ETTh2|30|1.030|1.273|1.359|1.259|1.096|\n||96|1.020|1.169|1.184|1.167|1.005|\n||336|1.006|1.196|1.102|1.333|0.999|\n|ETTm1|30|1.033|1.702|2.176|1.905|1.041|\n||96|1.051|1.295|1.338|1.285|0.999|\n||336|1.049|1.603|1.709|1.576|0.997|\n|ETTm2|30|1.041|1.769|2.025|1.764|1.047|\n||96|1.038|1.421|1.596|1.502|1.006|\n||336|1.025|1.657|2.359|1.555|1.002|\n\nRegarding the novelty of AdapTS compared to OneNet, the main contribution of AdapTS over OneNet is its superior computationally efficiency stemming from our ability to avoid performing online gradient optimization. The table indicates that this approach also provides a performance gain. \n\n**3. Higher frequency datasets and AdapTS performance when using a more advanced forecaster (or FITS)**\n\nWe have run experiments using TTM on 3 per-second datasets: SWind, SSolar[5] and SCloud[6]. The results are given in the table below. As in the rest of our experiments we find that using AdapTS improves performance of FM forecasts. This shows that on high frequency datasets where computationally-fast adaptation is necessary, AdapTS still improves performance.\n\nWe also looked at using a more advanced forecaster, FSNet, and FITS instead of the AdapTS-forecaster. The results are provided in the table given in response to reviewer 26Ko. We find that using FSNet or FITS instead of the AdapTS-Forecaster results in reduced performance. *We hope the results for FITS answers your 5th question.* The reason we believe FSNet performs poorly compared to the AdapTS-Forecaster is that the online updating of complex models like FSNet is to a large degree an unsolved problem as shown by results in continual learning. In contrast, learning the AdapTS-forecaster online is relatively easy, resulting in better online performance.\n|**Dataset**|$H$|**TTM**|*+AdapTS*|\n|-|-|-|-|\n|SWind|30|0.728|-0.041|\n||96|1.77|-0.036|\n||336|4.759|-0.058|\n|SSolar|30|0.286|-0.050|\n||96|0.581|-0.103|\n||336|1.543|-0.165|\n|SCloud|30|0.94|-0.128|\n||96|1.06|-0.119|\n||336|1.285|-0.126|\n\n**4. Why does linearly combining AdapTS-Forecaster and FM forecasts improve performance?**\n\nAdapTS corresponds to ensembling an FM (trained across numerous diverse time series) and a lightweight online forecaster (fit on recent time series data). The difference in training data means the forecasters are less correlated which is known to provide a performance benefit. Also, the weighter is able to optimally tune the ensemble weight online based on past performance leading to better performance than the individual models.\n\n[5] Monash Time Series Forecasting Archive\n\n[6] How does it function? characterizing long-term trends in production serverless workloads", "claims_and_evidence": "Overall, the claims made in the paper are clear, but some aspects require further justification. Specifically, the decision to keep the foundation model fixed despite distribution shifts needs more explanation. Additionally, comparisons with prior adaptation-based forecasting methods should be included to support claims of novelty and contribution.", "methods_and_evaluation_criteria": "The proposed method's contribution—being fast and lightweight—needs additional justification, particularly in relation to the chosen benchmark datasets. Since the datasets used have relatively low sampling frequencies, the necessity of a fast adaptation mechanism is unclear. Further experiments on higher-frequency datasets or with more computationally intensive forecasters would strengthen the evaluation.", "theoretical_claims": "I have checked the correctness of the theoretical claims presented in the paper.", "experimental_designs_or_analyses": "I have reviewed the experimental design and analyses, and they appear to be sound. However, additional experiments comparing AdapTS with adaptation-based forecasting methods such as FSNet, OneNet, and TAFAS would help better position the contribution.", "relation_to_broader_scientific_literature": "The paper contributes to time series foundation models by introducing an adaptation mechanism that leverages online feedback. The proposed approach builds on prior work by incorporating a lightweight linear forecaster and weighting mechanism to improve forecasting performance.", "essential_references_not_discussed": "The paper appropriately discusses related work, and I did not identify any missing essential references.", "other_strengths_and_weaknesses": "Strengths\n- The paper is well-written and clearly structured, making it easy to follow.\n- The proposed framework is general and can be applied to various time series foundation models.\n- Experimental results show consistent performance improvements across multiple datasets.\n\nWeaknesses\n- Some aspects of the method, including the decision to keep the foundation model fixed, require further justification (see Questions for Authors).\n- The novelty and contribution of the proposed method compared to existing adaptation-based time series forecasting approaches need to be better clarified.\n- The choice of benchmark datasets may not fully align with the claimed advantages of the method, particularly regarding computational efficiency.", "other_comments_or_suggestions": "Please see Questions For Authors.", "questions_for_authors": "1. When the data distribution shifts, the performance of the foundation model itself is expected to degrade. Why is the foundation model kept fixed rather than updated? Wouldn't a two-online forecaster approach, such as OneNet [1], be more appropriate in this framework? The justification for using the fixed foundation models in this setup needs to be clarified.\n\n2. Compared to OneNet, which replaces one of its online forecasters with a foundation model, what are the unique contributions and novelty of AdapTS? Additionally, experimental comparisons with adaptation-based time series forecasting methods (e.g., OneNet, FSNet [2], TAFAS [3]) would help contextualize the proposed approach.\n\n3. The paper emphasizes the computational efficiency of AdapTS-Forecaster, yet the datasets used for evaluation have sampling frequencies in the range of minutes or hours. In such cases, rapid adaptation may not be as critical. Have you tested AdapTS on datasets with higher sampling frequencies? Additionally, how does AdapTS perform when using a more computationally expensive but advanced forecaster?\n\n4. Why does linearly combining the forecasts from the foundation model and AdapTS-Forecaster lead to improved forecasting performance? Further theoretical or empirical justification would strengthen this claim.\n\n5. The supplementary material discusses the differences between AdapTS and FITS [4]. However, given the structural similarities, it would be helpful to include an experimental comparison where FITS is used instead of AdapTS-Forecaster.\n\nReferences  \n[1] OneNet: Enhancing Time Series Forecasting Models under Concept Drift by Online Ensembling  \n[2] Learning Fast and Slow for Online Time Series Forecasting  \n[3] Battling the Non-stationarity in Time Series Forecasting via Test-time Adaptation  \n[4] FITS: Modeling Time Series with 10k Parameters", "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": "9bM7VJA3cu", "reviewer": "Reviewer_wUFv", "summary": "This paper identifies that existing foundation models (FMs) fail to fully utilize the large amount of online feedback obtained during the deployment phase. This is due to the high computational cost associated with regular retraining or fine-tuning, which often leads to the neglect of this valuable feedback. To address this issue, the authors propose a lightweight online adaptation mechanism named AdapTS, which enhances the performance of time-series forecasting tasks during the deployment phase of FMs by dynamically adjusting the prediction results of the FMs.\n\nAdapTS consists of two main components: AdapTS-Forecaster and AdapTS-Weighter. The AdapTS-Forecaster employs a linear prediction model, which is updated using the mean squared error (MSE) loss to avoid potential issues associated with gradient optimization. It also leverages Fourier transforms to remove high-frequency components and utilizes the Woodbury matrix identity to achieve efficient updates. The AdapTS-Weighter, on the other hand, combines fast and slow weight mechanisms to dynamically adjust the prediction weights of the FM and AdapTS-Forecaster, thereby adapting to changes in data distribution.\n\n## update after rebuttal\n\nI am leaning towards accepting that even though this paper uses techniques from other fields applied to the time series field, I think it is innovative. I also hope that the authors enrich the related work section.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer, thank you for your detailed review and constructive comments. We are happy that you thoroughly assessed our claims and found that they held up. We provide answers to your question below.\n\n**1. Adding details to ablations section**\n\nThank you for raising this issue. We have updated the paper to expand the ablation section, including adding experimental data presented in a table, which we hope will address your concerns.  \n\n**2. Ablation shown in Figure 8.**\n\nWe agree that our rationale behind the decision to remove high-frequency components would be further enhanced by including more datasets to Figure 8. We have updated Figure 8 to include the Traffic and Solar datasets, and we find that the same conclusions hold. \n\n**3. Definition of online feedback**\n\nThe online feedback we discuss in the paper is given by the dynamically arriving data points from the time series. Each data point gives feedback on the accuracy of the previous forecasts which aimed to predict the value of that data point (and others). This feedback is then used to update the AdapTS-Weighter and AdapTS-Forecaster. Additionally, we do not assume the data points are corrupted in any way other than that modelled by the time series itself (i.e. they are reliable). We tried to explain this on the fourth line of the second paragraph of the introduction but understand from your comment that this need to be more clear, especially in the \"Rolling Window Forecasting\" subsection which formally describes the setting we look at. Therefore, we have updated the paper to fix this issue.     \n\n**5. Prompt and rapid fine-tuning techniques**\n\nWe agree with you that there have been many methods proposed in vision and NLP to perform rapid finetuning of FMs and will add discussion of this to our related work section. There is however one large difference between vision and NLP and time series, which is that in time series linear models are still competitive. We exploit this fact in our construction of AdapTS in that the AdapTS-Forecaster is a lightweight linear model and that we do not require the adding of parameters or the backpropagating through the FM which prompt tuning and typical adaptor based methods require. This makes AdapTS computationally efficient and agnostic to the FM used. Additionally, looking at work on rapid finetuning of FMs, AdapTS would roughly fall into the adaptor based fine-tuning paradigm. As pointed out by the other reviewers there has been a work on using adaptor based continual finetuning in time series, i.e. TAFAS. We have now compared to TAFAS, with the results shown in the table in our comment to reviewer 6Y6Z. Our findings are that AdapTS outperforms TAFAS in all cases looked at.\n\n**6. Use of AdapTS in other TSA tasks**\n\nThank you for suggesting these potential avenues to increase the scope of this work. While we focus on time series forecasting in this work and AdapTS is built for forecasting, it certainly is interesting to think whether it could be extended to perform continual adaption for other TSA tasks. To construct such a method for one of these tasks you would need some lightweight and online updatable method to replace the AdapTS-Forecaster. For example, for imputation you can use FITS. Then for generative tasks you can still use the AdapTS-Weighter. However, for predictive tasks you would need to use a slight modification whereby, as in the Hedge algorithm, you select a class/action using the weights as the probabilities instead of doing a weighted average. We see this direction as future work and want to focus this paper on time series forecasting but, given your comment, have added mention that AdapTS could be extended to these other tasks into the conclusions section.\n\n****\nWe would like to thank you again for your thoughtful review and hope that we have satisfactorily answered your remaining questions about the work.", "claims_and_evidence": "### Claim 1: lightweight and efficient \n\nFirstly, in the design of the AdapTS-Forecaster component, a linear forecasting model is selected and fitted using the mean squared error (MSE) loss during each update. This approach avoids the complexities associated with gradient optimization. Secondly, the use of Fourier transforms to remove high-frequency components and the application of the Woodbury matrix identity enable efficient updates.\n\nTherefore, it can be concluded that the paper's component design is centered around the goals of lightweight and high-efficiency, with efforts made to optimize and improve in these aspects.\n\nMoreover, the experimental results demonstrate that AdapTS has high computational efficiency. Each update requires only an additional 0.38 seconds, which is 2,506 times faster than the existing online fine-tuning method (Online Fine-tuning TTM). This makes AdapTS suitable for use in resource-constrained environments.\n\n### Claim 2: Enhancement of Online Prediction Performance of Time-Series Foundation Models (FMs) Using AdapTS\n\nThe results in Table 1 demonstrate that AdapTS efficiently and effectively utilizes online feedback in the rolling window setting, thereby improving the prediction results of the foundation models. The experimental findings show that AdapTS significantly enhances the forecasting performance of the FMs across multiple standard time-series datasets used as baselines. In some cases (e.g., when AdapTS adjusts the forecastings of VisionTS), the average improvement exceeds 10%.\n\n### Claim 3: Can the AdapTS-Weighter be used to combine the forecasting  of FMs and AdapTS-Forecaster?\n\nThe experimental results in Table 5 of Appendix C.6 show that using the full AdapTS Weighter consistently achieves better results compared to the AdapTS unweighted experimental settings.\n\nHowever, except on the ECL and Traffic datasets, the full AdapTS Weighter yields results that are mostly comparable to those of the fast weight mechanism in most cases. Therefore, the effectiveness of the AdapTS-Weighter component in dynamically adjusting weights to combine the forecasting of AdapTS-Forecaster and the foundation model (FM) is somewhat lacking in persuasiveness.", "methods_and_evaluation_criteria": "The AdapTS-Forecaster component selects a linear forecasting model and achieves efficient updates through the use of Fourier transforms and the Woodbury matrix identity. The AdapTS-Weighter, on the other hand, combines fast and slow weight mechanisms to adapt to changes in data distribution. The design of the components focuses on efficiency, lightweight implementation, and rapid adaptation to changes in data distribution, which is rational.", "theoretical_claims": "### Statement 1: Three Reasons for Choosing Linear Prediction Models\n\nThe paper provides three reasons for selecting linear forecasting models as the AdapTS-Forecaster: a) They perform well in time-series forecasting (Zeng et al., 2023); b) They can be efficiently updated online according to the requirements of our setting; c) Unlike neural networks that are updated online, linear models do not encounter catastrophic forgetting during online updates (De Lange et al., 2021).\n\n**Correctness of the Justifications:**\n\n- **a) Good Performance:** The paper cites the study by Zeng et al. (2023) to support the good performance of linear models in time-series forecasting.\n- **b) Efficient Updates:** Linear models are updated online via closed-form solutions, avoiding the computational burden of gradient descent. Parameter updates can be directly completed through matrix operations, which confirms the validity of this statement.\n- **c) Avoiding Catastrophic Forgetting:** The paper references De Lange et al. (2021), noting that linear models update based on the entire data history and do not forget old data. In contrast, online updates of neural networks may lead to forgetting of old data, resulting in catastrophic forgetting.\n\n### Statement 2: Upper Bound on Cumulative Loss of Exponential Weighting Methods\n\nTheorem 4.1 (Cesa-Bianchi & Lugosi, 2006; Rakhlin & Kleiner, 2008): For convex loss functions, the cumulative regret of a weighted average predictor is given by:\n\n$$R_T=\\sum_{\\tau=1}^T\\mathrm{Loss}_{\\tau,\\text{weighted}}-\\min_{k\\in\\{1,\\ldots,K\\}}\\sum_{\\tau=1}^T\\mathrm{Loss}_{\\tau,k}$$\n\nWhen the maximum loss is *L*max and the learning rate is *η*=*L*max1*T*8ln*K*, the cumulative regret satisfies:\n\n$$R_T \\leq L_{\\max} \\sqrt{T \\ln K}$$\n\n**Correctness of the Proof:**\n\nThe paper cites the classic results from Cesa-Bianchi & Lugosi (2006) and Rakhlin & Kleiner (2008) to prove the upper bound on cumulative loss for exponential weighting methods under convex losses. This statement is correct.\n\n### Statement 3: Adaptability Issues of Exponential Weighting Methods\n\nAlthough exponential weighting methods are effective, they struggle to adapt quickly to distribution shifts (Jadbabaie et al., 2015; Cesa-Bianchi et al., 2012; Zhao et al., 2020).\n\n**Correctness of the Proof:**\n\nThe paper references studies by Jadbabaie et al. (2015), Cesa-Bianchi et al. (2012), and Zhao et al. (2020) to highlight the limitations of exponential weighting methods in adapting to distribution changes. This statement is correct.", "experimental_designs_or_analyses": "### Rationality\n\nTo demonstrate the performance improvement brought by the AdapTS-Weighter component, the paper designed corresponding ablation experiments. In Table 5 of Appendix C.6, the full AdapTS-Weighter setting is compared with the AdapTS Unweighted setting, and the full AdapTS-Weighter consistently achieves better results.\n\n### Questions\n\n- In Section 5.5 Ablations, the paper only presents the conclusions drawn from the ablation experiments, with all experimental results relegated to the appendix. The main text lacks a systematic explanation of the reasons behind these conclusions (even a brief analysis of the specific numerical results would be helpful). As a result, the conclusions presented in Section 5.5 Ablations lack concrete experimental data to support them and are somewhat less convincing.\n- The paper attempts to validate the rationale behind the decision to remove high-frequency components to improve speed by showing the proportion of discarded frequency components and performance on the Weather and ETTm1 datasets through Figure 8, in conjunction with Figure 6. However, there are some issues: Figure 8 only displays results from the Weather and ETTm1 datasets, and experimental results from more datasets are not shown in this paper. Therefore, the design of this ablation experiment is somewhat lacking in persuasiveness.", "relation_to_broader_scientific_literature": "The AdapTS method holds significant importance in the field of time-series forecasting and is closely related to the literature on continual learning, online learning, and efficient computation. First, AdapTS enhances the performance of foundation models through its online adaptation mechanism, thereby avoiding the catastrophic forgetting problem often encountered in continual learning (De Lange et al., 2021; Lee & Storkey, 2024). Second, its design draws on classic methods in online learning, such as exponential weighting and the Woodbury matrix identity (Cesa-Bianchi & Lugosi, 2006; Rakhlin & Kleiner, 2008), and systematically applies them to time-series forecasting for the first time, effectively addressing dynamic changes in data distributions. Additionally, the lightweight design of AdapTS meets the demands of efficient computation, resolving the computational bottleneck associated with online fine-tuning of large-scale models (Ekambaram et al., 2024). Experiments demonstrate that AdapTS has broad adaptability across different datasets and models, aligning with the goals of zero-shot time-series forecasting (Ansari et al., 2024; Woo et al., 2024). In summary, AdapTS not only proposes an innovative method but also offers new perspectives for research in related fields.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "## Strengths\n\n- Appendix C.1 clearly compares AdapTS, proposed in this paper, with the existing time-series FM (TTM) that is the only one to propose an effective fine-tuning scheme, demonstrating the advantages of AdapTS in generalization, efficiency, and performance.\n\n- Section 5.4 provides a detailed explanation and reasoning for the improvement of online forecasting performance of time-series FMs by AdapTS, in conjunction with Figure 3 and Figure 4.\n\n## Weakness\n\nFrom the working mechanism of AdapTS-Forecaster, it can be inferred that its training data consist of the online feedback dynamically generated during the deployment phase. These data reflect the characteristics and changes of the current time series, enabling AdapTS-Forecaster to learn the most up-to-date data distribution. However, the paper lacks clarity and completeness in its design of “training data.” In particular, there are some omissions in the description of data processing and allocation in the experimental settings.\n\nFor more information, please see the Question For Authors section.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- Neither the description of the experimental settings in the main text nor the detailed experimental information in Appendix B specifies what kind of online feedback data AdapTS, proposed in this paper, is based on. Does it assume that the online feedback data is reliable? If there is a specific explanation regarding this in the paper, please provide the exact location.\n\n- ==Compared to the Prompt fine-tuning techniques that have been widely applied in computer vision and natural language processing [1-5], I believe your approach also involves adding lighter, trainable components to the frozen backbone of the model to achieve more efficient fine-tuning and better performance. Given this, I wonder if it is necessary to analyze the differences between your work and the existing rapid fine-tuning techniques in computer vision and natural language processing in the related work section. I am particularly curious about the challenges that might arise when transferring these techniques from the domains of image and natural language processing to time-series analysis, and how you have addressed these challenges. Alternatively, are there already similar works in these two domains that you have adapted to time-series foundation models? Please forgive my concerns because this technology is indeed very mature in other fields.==\n- ==In addition to models used for zero-shot forecasting such as Moriai and Chronos, can the fine-tuning techniques you proposed be applied to pre-trained models for processing the five major tasks of TSA such as SymTime [] and UniTS []? And use this lightweight and efficient fine-tuning method for other tasks such as classification, filling and anomaly detection.==\n\n[1] Lester, Brian, Rami Al-Rfou, and Noah Constant. \"The power of scale for parameter-efficient prompt tuning.\" *arXiv preprint arXiv:2104.08691* (2021).\n\n[2] Jia, Menglin, et al. \"Visual prompt tuning.\" *European conference on computer vision*. Cham: Springer Nature Switzerland, 2022.\n\n[3] Han, Cheng, et al. \"E^ 2vpt: An effective and efficient approach for visual prompt tuning.\" *arXiv preprint arXiv:2307.13770* (2023).\n\n[4] Sohn, Kihyuk, et al. \"Visual prompt tuning for generative transfer learning.\" *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*. 2023.\n\n[5] Yao, Hantao, Rui Zhang, and Changsheng Xu. \"Visual-language prompt tuning with knowledge-guided context optimization.\" *Proceedings of the IEEE/CVF conference on computer vision and pattern recognition*. 2023.\n\n[6] Wang, Wenxuan, et al. \"Mitigating Data Scarcity in Time Series Analysis: A Foundation Model with Series-Symbol Data Generation.\" *arXiv preprint arXiv:2502.15466* (2025).\n\n[7] Gao, Shanghua, et al. \"UniTS: A unified multi-task time series model.\" *Advances in Neural Information Processing Systems* 37 (2025): 140589-140631.", "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:

Meta-review: This paper introduces a simple method to improve time series foundation model forecasts using online feedback. Reviewers found the approach intuitive and lightweight, with clear empirical gains across several models. While initial concerns were raised about framing, baseline comparisons, and clarity, the authors provided strong rebuttals, reframed the method appropriately, and added relevant baselines including test-time adaptation approaches. The method is general, efficient, and shows consistent benefits, justifying acceptance.  The authors are advised to factor in the reviewer feedback when preparing the updated version of the paper, including expanding related work.
Decision: Accept (poster)