Test-Time Efficient Pretrained Model Portfolios for Time Series Forecasting
Authors: Mert Kayaalp
Ali Caner Turkmen
Oleksandr Shchur
Pedro Mercado
Abdul Fatir Ansari
Michael Bohlke-Schneider
Bernie Wang
Venue: ICLR 2026 Poster (2026)
Keywords: pretrained time series models
time series forecasting
foundation model combination
Primary area: learning on time series and dynamical systems

Abstract: Is bigger always better for time series foundation models? With the question in mind, we explore an alternative to training a single, large monolithic model: building a portfolio of smaller, pretrained forecasting models. By applying ensembling or model selection over these portfolios, we achieve competitive performance on large-scale benchmarks using much fewer parameters. We explore strategies for designing such portfolios and find that collections of specialist models consistently outperform portfolios of independently trained generalists. Remarkably, we demonstrate that post-training a base model is a compute-effective approach for creating sufficiently diverse specialists, and provide evidences that ensembling and model selection are more compute-efficient than test-time fine-tuning.

Published as a conference paper at ICLR 2026
TEST-TIME EFFICIENT PRETRAINED MODEL PORTFOLIOS FOR TIME SERIES FORECASTING
Mert Kayaalp's Caner Türkmen', Oleksandr Shchur?, Pedro Mercado?, Abdul Fatir Ansari'
Michael Bohlke-Schneider", Bernie Wang*
Dalle Molle Institute for Artificial Intelligence (IDSIA) USI/SUPSI, Switzerland Amazon Web Services
ABSTRACT
Is bigger always better for time series foundation models? With the question in mind, we explore an alternative to training single, large monolithic model: build- ing portfolio of smaller. pretrained forecasting models. By applying ensembling or model selection over these portfolios, we achieve competitive performance on large-scale benchmarks using much fewer parameters. We explore strategies for designing such portfolios and find that collections of specialist models consistently outperform portfolios of independently trained generalists. Remarkably, we demon- strate that post-training base model is compute-effective approach for creating sufficiently diverse specialists, and provide evidences that ensembling and model selection are more compute- -efppttetefiiinntt than test-time fine-tuning.
INTRODUCTION
The dominant paradigm in pretrained time series models is the "bigger is better" view, supported by evidence that larger models improve forecast accuracy. Building on this observation, recent work has focused on scaling up both model size and training data to build better zero-shot forecasters
(Das et al., 2024 Woo al. 2024 Ansari et al. 2024a Shi et al.) 2024a Edwards al. (025). However, these large, monolithic models come with high training and inference costs, which limits their practical use. To offset the need for ever-larger models, promising alternative has emerged: allocating extra compute at test time. In NL.P and computer vision, this includes generating multiple outputs for a single input and aggregating them either by sampling from a single model (Sun et al., 2024) or using multiple models (Mavromatis et al., 2024). These examples, which partially inspire our work, belong to broader class of methods that continue to achieve increasingly successful results across domains. Nevertheless, such strategies have not yet found their way to time series forecasting.
Inspired by these developments, we explore an alternative path. Instead of scaling up single model, we propose to build portfolio of smaller pretrained time series models and combine them at test time. This strategy offers greater flexibility and efficiency, but introduces two key challenges. First, how can we train a diverse set of models without incurring the full cost of training each one from scratch? Second, how should we combine the models at inference time to make accurate and efficient predictions? We address both challenges and show that this approach match or outperform larger monolithic models, while reducing inference costs and enabling modular design. More specifically, we make the following contributions.
1. We show that a portfolio of small pretrained models (based on encoder-decoder Chronos- Bolt|Ansari et al. (2024b) architecture), each specializing on subset of the training corpus, can match the accuracy of large monolithic models, while being more compute-efficient, By combining models test time using ensembling or model selection, our approach achieves accuracy comparable to state-of-the thtthe-r pretrained forecasters, while dramatically reducing inference cost. We further observe that portfolios follow similar compute-performant scaling trends as monolithic models.
WWrk done during an internship AWS.

---

Published as a conference paper at ICLR 2026
Training Time
Test Time
(a) Classical pretreining
setup
Large Pretrained Medel
Pretrained Model
(I)
[U]]
Generaliat
Genernlist
Protreining partfolio specialist modets from scratch
Goneralat
Forecests
Specialiat
Specialist
Specialist
Model setection
inference with solectedimoden
Speciall ist
Specialist
Generalist
Ceneralise
Goneralist
Forecasts
Specialist
Specialist
Specialist
(oursj: Protaiiin portoolo by post-training the generalist
Specialist
Model
selection
Specialist
Inforence withh selected models
Figure 1: A diagram depicting the approaches explored in this work. (a) Typical setup for pretraining, where single large model is trained large corpus composed of many different datasets. At test time, the model be invoked zero-shot well with finetuning at test-time and used to obtain forecasts. (b) Building model portfolio by training an array of smaller models, where each specialist is trained on single smaller corpus representing different modality or domain. At test time, these models are combined via fitting an ensemble by selecting the best model, and the selected combination (or, single model) is used for forecasting. (c) Specialists can also be trained in two-stages, by post-training the generalist model. We demonstrate that this approach for inducing diversity into the model portfolio results comparable accuracy. while reducing training- time compute by an order of magnitude. The approach also leads to the same forecast accuracy as (a), in return for much smaller number of total parameters used actively for inference.
2. We introduce an efficient strategy for constructing diverse model portfolios without training each model from scratch. The approach starts by pretraining generalist model the full data distribution, followed by targeted fine-tuning on data partitions defined by metadata such as frequency and domain, obtaining specialists (see, Figure I. This reduces overall training time by 10x and yields a portfolio that maintains diversity and delivers accurate forecasts.
3. We explore different methods of combining pretrained portfolios at test time. Unsurprisingly, find that the performance of our portfolios depends on the ability quickly identify the best sparse combination among them in new test task. We find that performing model selection or greedy ensemble selection among specialists is an efficient approach when compared to other methods of utilizing compute at test time (e.g., with fine tuning).
BACKGROUND AND RELATED WORK
2. TIME SERIES FORECASTING
The objective time series forecasting is to predict the future H values of a time series æ, given the previous C observations. Formally, consider time series [2i...., #c) where E RR denotes the value of the time series z at time step t. The goal of probabilistic forecasting to approximate the distribution of future time series values 3C 1.++11 given the history 31.0 with
2

---

Published as a conference paper at ICLR 2026
model po
po(Fc+ic+n(aie) p(sc+1c-ulmie))
Forecasting approaches fall into three broad categories (Benidis et al. 2022: Januschowski et al. 2020). Local models, such as ETS and ARIMA (Hyndman & Athanasopoulos 2018), train separate models, often fully determined by several parameters (~10), for each individual time series. Global
models, including deep learning methods (Salinas et al. 2020 Rangapuram et al. 2018 Oreshkin ct al.] |020] Nie et al.; 2023), learn shared parameters (~ (00K) across a single dataset of multiple
related time series (e.g., energy consumption of different households). More recently, pretrained
models such as Chronos (Ansari et al.) 2024a), TimesFM (Das et al., (024). and Moirai (Woo et al.
2024) have adapted ideas from large language models time series forecasting. These models are trained on large and diverse collections of time series data and aim to generalize to new forecasting tasks without task-specific fine-tuning. With parameter counts typically exceeding 10M, they serve as universal forecasters capable of making accurate zero-shot predictions across wide range of domains. To reduce inference costs associated with such large models, Ekambaram al. (2024) proposed Tiny Time Mixers (TTMMs)lightweite for specific context and forecast settings. Our approach shares the same goal of reducing the test-time cost, but instead of specializing single small model per setting, we train portfolio of small models and select or combine them at test time.
models with around 1M parameters, specialized
2.2 MODEL AND FORECAST COMBINATION
Traditional model combination methods, such as bagging (Breiman, 1996a) and boosting (Freund & Schapire T997). improve prediction accuracy by aggregating the outputs of multiple complementary models. Similar ideas have been applied in time series forecasting under the name forecast combina- tion (Wang et al. 2023b). Forecasts are often combined using weighted averages of individual model predictions. These weights are typically shared across all time series in the dataset and are either set uniformly or in proportion to each model's validation performance (Pawlikowski & Chorowska 2020). More flexible strategies learn these weights directly from data, for example using greedy ensemble selection (Caruana et al. 2004 [Shchur et al. 2023).
In the context of pretrained models, some works have explored combining multiple LLMs, such as Mavromatis et al.) (2024), which studies independently trained base models and their ensembles at test time. Here, the authors compare methods including weighted ensembling, uniform ensembling, and model selection. While their work is closely related to ours in the context of foundation model portfolios, it focuses exclusively on language tasks. In contrast, we focus on probabilistic time series forecasting, where the use of model combination and more generally test-time computation remains largely unexplored. Furthermore. beyond comparing model combination strategies as in Mavromatis et al. 2024), we also investigate how to build diverse portfolio of test-time efficient models.
Another related approach for pretrained models is the mixture-of-er of-eeeerts (MoE) framework (Jaobb et al., [991] (Shazeer et al., 2017), where component models and combination rules are jointly trained. This approach shown to be effective time-series forecasting tasks as well (Shi al. [024b] Liu et al. 2024). Although these works showed that MoE can help reducing inference time and cost, it limits flexibility. In contrast, our approach trains the component models in the portfolio independently, and allows the use of different fusion strategies at test-time. Notably, this flexibility enables the fusion rule to be adapted according to available computational resources during inference. Additionally, approach enhances interpretability during the prediction by clarifying the contribution of each portfolio member, whose training procedures are transparent.
3 CHROMA: A PORTFOLIO OF SMALL PRETRAINED FORECASTERS
In this section, we introduce our method for building portfolios of small pretrained forecasting models. Our approach consists of two stages. In the training stage, our goal is to build a diverse collection of small forecasting models. At test time, we combine their predictions to maximize accuracy for the given forecasting task. Applying this methodology to the publicly available pretrained models Chronos-Bol (Ansari et al. 2024b), we construct our model portfolio, Chroma, which we describe in detail below.
3

---

Published as a conference paper at ICLR 2026
3.1 MODEL ARCHITECTURE
We focus on portfolios composed of models with the same architecture, each pretrained independently on a different subset of the target distribution. The key difference is that we pretrain the full portfolio once, ahead of time, and reuse it across tasks-just just as one would with a single large pretrained model.
We build Chroma using the training setup and architectures of Chronos-Bolt (Ansari et al. 2024b), publicly available pretrained time series model based the T5 encoder-decoder architecture (Raffel et al. 2020). Both pretraining and fine-tuning use average quantile loss, consistent with the original Chronos-Boll implementation. Chronos-Bolt generates multi-step-ahead quantile forecasts and is trained on a diverse collection of time series datasets. We select this model due to its strong benchmark performance, open-source training pipeline and data, and support for a range of model sizes (c.g., t tiny, mini), which enables flexible experimentation under different compute budgets. We train multiple versions of the model on different training datasets to induce diversity. We focus on smaller models ranging from 1M to 9M parameters, corresponding roughly to the tiny (9m) and below scale. Further implementation and training details are provided in Appendix B
3.2 BUILDING DIVERSE PORTFOLIOS
We now turn to the question of how to construct model portfolios that can be effectively combined at test time. It is well established that ensembles benefit from diversity among their members, provided each model is reasonably accurate (Zhou 2012). However, diversity and individual model accuracy often trade off against each other (Wood al. 2023)
In the context of pretrained models, diversity must be introduced during training, as weights are typically frozen at test time. The standard Chronos-Bolt training strategy aggregates all available time series datasets into a single, large corpus and trains one generalist model to perform well on average. In contrast, we aim to exploit structure in the data by explicitly training specialist (see, (Hinton et al. (2155) models on disjoint subsets of the training corpus. Note that these models are not trained at once in cettuures-offxxeertt or on the same data distribution but with perturbed samples as in bagging. Instead, we train each specialist model independently on distinct partition of the corpus representing a different subproblem within the broader target distribution.
Specifically, we partition the data along metadata dimensions that reflect distinct characteristics of the time series. We focus on two such dimensions: frequency and application domain. Each specialist model is trained on a smaller subset of data defined by one of these attributes. For example, a domain- based portfolio may include an "energy" specialist trained only on energy-related datasets, while a frequency- -baeed portfolio may contain an "hourly" specialist trained solely on hourly-resolutien data. In each portfolio, we also include generalist model of identical size trained on the full corpus for comparison. The training corpus of Chroma includes Chronos training data sets, plus some new ones for underrepresented frequencies and domains. The partitions and statistics of the datasets used are specified in Appendix C
3.3 REDUCING TRAINING COST THROUGH POST-TRAINING
While forming portfolios, training all specialists from scratch on large datasets can result in significant computational cost. To mitigate this, we introduce diversity into the portfolio through post-training: we first train generalist model of a given size, then fine-tune it briefly on different subsets of the training data to produce specialist models. This avoids the overhead of training each specialist model from random initialization.
This strategy is particularly important in our setup, where some data partitions may contain far fewer examples than others-fir example, datasets with yearly resolution typically have far fewer observations than those with minutely resolution. In such cases, training from scratch may be inefficient or impractical. By leveraging post-training on a shared base model, we reduce overall training time and compute requirements while still matching the accuracy of portfolios trained independently. The benefits of post-training become even more critical in domains with larger models. While the largest time series models today are around I billion parameters, NLP models can reach up to 1 trillion. In such regimes, training each specialist from scratch is often infeasible, making post-training practical and scalable alternative.
4

---

Published as a conference paper at ICLR 2026
3.4 FORECAST COMBINATION AT TEST-TIME
Given a new time series dataset that was not seen by the portfolio members during training, we can utilize the models through one of strategies: model selection or ensembling. In the first approach, we simply select the single best-perforning model based on its accuracy on a validation set. In the second approach, we combine the predictions from multiple models using weighted averaging
Jens 1I Wm Im. where the weights are determined using the ensemble selection algorithm 2004). For completeness, we provide the full description of the ensemble selection algorithm in Appendix [ Both strategies depend on validation sets produced through time series cross- volssvaali (Hyndman & Athanasopoulos 2018 to rank or weight the predictions in the model portfolio.
.M
(Caruana et al.
4 EXPERIMENTS AND RESULTS
Setup. We extend the range of Chronos-Bolt model sizes to smaller variants and train portfolios of model sizes 1m, 2m, 4m, and tyy (9m). Details of these model architectures are provided in Appendix B We extend the training corpus of Chronos and partition this new corpus across two dimensions: frequency and domain.
For each model size, we first train a generalist using the original Chronos- Bolt corpus and a 200K- gradient-step training procedure following the reference implementations/ Ansari et al. (022b), and incorporating mixup and synthetic data as specified in Ansari et al. (2024a). We then construct a portfolio of specialists by further fine-tuning (i.e., post-training) the generalist weights on different subsets of the training corpus for IK gradient steps. In exploratory experiments, we also tested 3K, 10K, and 20K post-training steps, but observed no significant performance differences. As result, we did not tune this variable further or include it as an experimental factor, and we use IK gradient steps in our main experiments. This post-training phase amounts to only 0.5% of the generalist's training time, making our method of constructing specialists extremely lightweight. Each portfolio includes the generalist and its associated specialists.
To evaluate a portfolio on a target task, we hold out the last H observations from the training set to create a validation window. All models in the portfolio generate forecasts for this window. We then either select the model with the lowest loss on the validation set, or fit weighted ensemble using the greedy ensemble selection algorithm (Caruana et al.) 2004) implemented in AutoGluon (Shchur et al., 2023). These are based rolling-window evaluations (also referred to as backtesting or out-of-time folds) and can be performed even with a single time series. It can also be performed by increasing the number of evaluation windows. Namely, instead of using single window for selecting which specialists provide the best forecasts, one could rely on multiple. The selected model or ensemble is then applied to the test set to produce final forecasts. Note that we use the same methodology throughout all experiments.
Most of our results are reported on Chronos Benchmark I (BM2). Importantly, all datasets in the benchmark are zero-shot examples for our models (these datasets, including their training partitions, were not used in training of Chroma portfolios). Following the benchmark, we report the probabilistic forecasting results using the weighted quantile loss (WQL) and point forecasting results using the mean absolute scaled error (MASE). Finall,, we also evaluate the larger Chroma variants on GIFT- Eva(3 (Aksu et al. 2024). For both benchmarks, we report aggregate performance using the geometric mean of relative errors. Specifically, for each dataset, we divide the model's 's error by that of baseline (Seasonal Naive unless stated otherwise), and then compute the geometric mean across datasets (Fleming & Wallace 1986). Further details on the experiment setup are given in Appendix B
4.1 PERFORMANCE AND TEST-TIME EFFICIENCY
Benchmark performance compared to the state of the art. We report overall results for Chroma on BM2 in Figure 2 (left), comparing the performance of 4m and tiny (9m) Chroma portfolios of frequency specialists to the state of the art in pretrained time series models. We find that, despite
rd
5

---

Published as a conference paper at ICLR 2026
Chronos
GIFT-Eval
Chrenos-Bon Bolt Base
TimesFM 2.0 500M
0.465
Chroma[4m
Chroma[4m Ensemble]
0.631
Chronos-Bolt Base
0.485
Moirai
0.6.11
Chronos-Belt Small
0.487
Chroma[Tiny Ensemble]
0.536
Chroma[Tiny Ensemble]
0.499
Coronos-Bolt Small
Chroma]Tiny Best)
0.505
Moiral
Chrono Mini
0.506
Chronos-Bolt Mini
0.644
Chroma[4m Ensemble)
0.512
Chroma(Tiry Best]
0.645
Chrones-Bolt Tiny
0.516
Chronos Large
0.656
Chromaj4m Best]
0.5:2
Chrones Bolt Tiny
Chrones Large
0.547
TTM
ARIMA
0.741
Auto
0.770
Auto Theta
0.793
Seasonal Naive
Auto ETS
0.824
Auto Theta
1.041
Seasonal Nalve
Auto ETS
1.202
0.4
1.0
0.4
Agg. Relative WQL
Agg. Relative WQL
Others
Chroma (ours)
Chrones
Figure 2: Results on Chronos Benchmark II (BM2) and GIFT-Eval. Results reported for probabilistic forecasting, with weighted quantile losses (WQL), scaled relative Scasonal Naive model, aggregated across all data sets using geometric mean. For Chroma, Best refers to performing model selection on a validation set while Ensemble le refers to the ensemble selection algorithm.
having as little as 4M active parameters test time, the Chroma portfolio under best model selection performs comparably much larger monoliths such Moirai- Large (311M parameters) (Woo cet al., 2024), TimesFM-2.0 (500M parameters) (Das et al. 2024 or Chronos-Bol. Base (205M parameters).
We also report results on the GIFT-Eval benchmark, in Figure (right), including only pretrained architectures and statistical baselines for comparison. In rolling window evaluation tasks, perform model selection or ensembling on the Chroma model portfolio on the first evaluation window alone, and infer with this fixed selection across the subsequent windows. That is, the ensemble fixed in the training window of the first rolling window, and was not refit subsequent windows. this benchmark, too, Chroma's aggregated probabilistic forecasting performance over 97 tasks close some much larger pretrained architectures, including TabPFN-TS (Hoo et al. (2025) which leverages significant test-time computation for in-context learning with tabular foundation model. Note that we report results for TTM (Ekambaram et al.) 2024) in zero-shot mode only. Moreover, in order not to misrepresent the performance TTM, which specializes high-frequency time series, we omit TTM'ss | results on BM2 which primarily contains low-frequency time series datasets. Further details on benchmark evaluations, including point forecasting performance, are given in Appendix
Portfolio design and combination method. We now investigate the impact of key design choices in portfolio construction and test-time combination across different model sizes. To better isolate the effect of these choices, we report results on BM2 using different reference point: instead of comparing the Seasonal Naive baseline, compute relative errors with respect to single generalist model of size 1m. We also include similarly sized portfolio of generalists (models trained with the same large corpus of data but with different random seeds).
We observe that building time series model portfolios by partitioning datasets by frequency is preferable to partitioning them by application domain, as this results in 5% decrease in overall MASE and ~ 4% in WQL, across all model sizes considered (Table I. This is consistent with earlier work such as TTM (Ekambaram et al.) 2024) who used frequency to partition datasets. When comparing the use of ensemble vs. model selection, although ensemble selection leads to slight improvement across model sizes and metrics, we find the evidence for this is not as strong previous conclusion. Therefore, while using ensembling to combine Chroma models can result in improvements in accuracy, this difference may not be strong enough to justify the added computational costs of using multiple models for inference.

---

Published as a conference paper at ICLR 2026
Table 1: Aggregated relative WQL. and MASE on BM2, scaled against single generalist model of size 1m. The best two models for each metric and model size are given in bold.
WQL
MASE
Im
2m
4m
tiny
Im
2m
4m
tiny
Domain
Ensemble (Greedy)
0.957
0.936
0.932
0.915
0.964
0,932
0,928
0.918
Model Selection
0.963
0,950
0.939
0.923
0.973
0.948
0.933
0.930
Frequency
Ensemble (Greedy)
0,926
0.898
0.890
0.896
0.910
0.884
0.878
0,887
Model Selection
0.918
0.916
0.880
0,909
0.920
0.899
0.887
0,893
Generalists
Ensemble (Greedy)
0.987
0.951
0.939
0.919
0.974
0.944
0.931
0.928
Model Selection
0.990
0.966
0.942
0.944
0.979
0,961
0.935
0.941
Single Generalist
None
1,000
0,977
0,960
0.958
1,000
0.971
0.962
0.961
Finally, we observe that simply combining multiple generalists trained using the same large data set does not result in significant performance gains over just using single generalist. We hypothesize this is due variance being a negligible factor in the test error committed by models pretrained on large corpora-an observation that can be supported by Lin et al. (2024), who argues that current LLMs remain underfitted owing to the relatively small number of training epochs. We discuss this observation further in Appendix F Specifically, we show that in the regime where Chroma models are trained, bias component of the generalization error dominates the error due to variance. Since specialists reduce bias within a specific part of the overall data distribution, intelligent model combination or ensembling at test-time can reduce the overall bias of the portfolio. This is as opposed to the traditional ensembling approaches like bagging that tries to reduce error by decreasing variance.
X
Single Generalist alist (a=-0 0.05 Frequency 5p. 0.0-3.013)** Domain Sp. (a- -e-00.0200--
X
Single Generalist (am-0 0.000)
Frequency 0,077ax
0.72
X
Domain Sp. (a- 0.021
0.70
XXX
XOX
0.68
MASE
X
0.64
X
X
X
0.62
0.80
5
8
2 Number Active Parameters (millions)
5
Number Active Parameters (millions)
Figure 3: Scaling behavior of Chroma portfolios. Results are presented for performing model selection from a Chroma portfolio of domain or frequency specialists. Each individual run is for an independently trained generalist in the portfolio, with four trials reported per experiment setting. refers to the slope of the scaling fit, and asterisks denote statistical significance of the fitted coefficient at the 5% level 0.05). Reported results are aggregated across BM2.
Scaling behavior. Having observed that our conclusions hold across different model sizes, we turn our attention to quantifying how Chroma scales. Similar to recent works (Hoffmann et al.) 2022 Kaplan et al., 2020 Shi et al., 2024a Edwards et al. (025)), we carry out ordinary least squares fits to the test error of Chroma (on BM2) and the number of active parameters at inference time. We present results for model selection on the Chroma portfolio in Figure B We observe that despite individually training on much smaller datasets, the overall model portfolio's 's performance scales at rates comparable to the individual generalist models. In further experiments, we found that our training setup of post-training on generalists to obtain the portfolio is essential to this conclusion
7

---

Published as a conference paper at ICLR 2026
We therefore conclude that, provided datasets can be proportionally scaled, our approach can be scaled up to larger model sizes to obtain even better performance.
(see, Appendix G).
Test-time compute efficiency. Chroma delivers strong forecast accuracy, but the cost of additional test-time computation. Since we must either select the best model from the portfolio or combine multiple models via greedy ensembling, each model in the portfolio must run inference both the validation and test sets. This raises a natural question: how does this compute overhead compare to alternative strategies for adapting pretrained models at test time?
The simplest option is to use single generalist model in zero-shot manner- nn adaptation, and only one forward pass. Our approach sits in the middle: it avoids fine-tuning but requires multiple inference passes and some lightweight model selection or combination logic. At the high end of the compute spectrum test-time fine-tuning, where a single generalist model is adapted to the target task using gradient updates.
To quantify these trade-offs,, we compare all three strategies-zerr--hho generalist, Chroma (with model selection ensembling), and fine-tuned generalist-by measuring total test-time floating point operations (FLOPs) and reporting accuracy across the board. Fine-tuning is done with 1K gradient steps for reference. This lets position Chroma in terms of both accuracy and compute efficiency relative to common adaptation strategies. Our results are presented in Figure where different model sizes and combination methods are shown along the efficient frontier of accuracy vs. the total test-time compute required. Fine-tuning a generalist model for 1K gradient steps yields strong accuracy but requires nearly 10x more compute than simple forward passes. Chroma offers a middle ground: achieves competitive accuracy using a fixed portfolio of small models, with only lightweight model selection or ensembling at test time. This places it near the center of the efficiency frontier, offering a favorable trade-of off between accuracy and computational cost. Further fine-tund tenetttnigg may improve accuracy, but it shifts the operating point toward significantly higher compute requirements.
WQL
MASE
Jm
1m
X
Single Generalist
0.72
Test-time Fine-tuned
Proquency Sp. (Best)
Domain Sp. (Best)
0.70
Frequency Sp. (Ens.)
4m Al miny
Domain (Ens.)
1m g.
0.68
WOL
2m
MASE
4m
1m
0.66
1m
3m
Dry
Ainy
Ony
Liny
3m
12m
2m
0.64
Jny
4m
12m
0.80
0.62
109
10A
10a
109
10a
10a
GFLOPs
GFLOPs
Figure 4: Efficient frontier of Chroma portfolios, compared to using a single generalist model without any test- time computation (i.c., zero-shot, x) and fine-tuning generalist model (l). Test computation is computed as the estimated FLOPs for performing model fitting, fine tuning, model selection in addition to single forward pass for inference, per time series the test set. Reported results are aggregated across BM2.
4.2 FURTHER EXPERIMENTS
Ablation studies. We perform three ablation studies summarized in Table 2 A key element in the portfolios we consider is the inclusion a generalist (see, e.g., .inton et al. (2015)). We estimate the effect of excluding the generalist on MASE and WQL errors averaged across different model sizes, specialist types (domain vs. freq) and datasets. In both cases, we find average error increases of ~ 2%, while these regressions in performance are not measured to be statistically significant.
8

---

Published as a conference paper at ICLR 2026
Table 2: Summary results of ablation studies. Results given in the estimated increase in normalized error (where seasonal naive error is 1) on BM2 for the ablation performed, and the p-value associated with the t-test of the regression coefficient.
MASE
WQL
Increase in Error
p-value
Increase Error
p-value
No Generalists Portfolio
0.016
0.107
0.017
0.118
Ensemble Performance- weighted
0.186
0.000
0.142
0.000
Ensemble Simple Average
0.242
0.000
0.180
0.000
Specialists Trained From Scratch
0.002
0.798
-0.001
0.918
Second, we observe that the benefit of specialist portfolios comes primarily from the diversity they introduce. However, this diversity is only useful if we combine models intelligently at test time. In particular, our ensembles not work simply by reducing variance through averaging, as might be expected in settings where randomness in training dominates model differences. We confirm this in ablation studies: both simple averaging and performance-weitn weighted averaging (cf. Pawlikowski & Chorowska 2020) lead to significant performance drops when applied to pretrained portfolios. This contrasts with much the prior forecasting literature, where ensembles of task-specific models with naive combination strategies (Wang al. 2023b).
often effective
An essential part of our methodology training a diverse portfolio of models to pretrain a single generalist before inducing diversity by post-training with different data subsets. In final ablation study, we compare this to training all specialists from random initializations (see, Figure (b)). On average, this does lead to any regressions performance. However, scaling studies we find that post-trained models scale better (see, Appendix Gb.
Credit assignment and interpretability the portfolio. We also investigate which specialists activated during ensembling model selection for given test task. Figure illustrates this activation pattern. The heatmaps are structured as matrix plots, where rows represent tasks and columns represent specialists (categorized by domain frequency). We observe that all specialists are activated for some tasks, indicating no "mode collapse" where only single model dominates. Notably, when the test task shares domain frequency with particular specialist, that specialist tends to be highly activated. This suggests a practical efficiency gain: in many real-world applications, the nature of the test-time task is known advance. This knowledge can be leveraged manually select relevant specialists, thereby reducing test-time computational cost. Additionally, Figure also offers layer of interpretabilityy For each task, the specialist weights provide meaningful insights that can aid practitioners in understanding the model's 's behavior.
Frequency
Domain
buouuurly
energy
0.6
finance
weight
hourly
healthcare
0.4
daily
Tasks
retail
-02
twerage
weekly
transport
-0.1
supramonthly
merg
weathel
retail
tranpporc
Specialists
Figure 5: Distribution ensemble weights for specialist portfolios across distinct tasks, grouped with respect to the their domain or frequency information.

---

Published as a conference paper at ICLR 2026
5 CONCLUSION
In this work, we introduced Chroma: a portfolio of small, pretrained time series forecasters. Chroma consists of specialist models, each fine-tuned on disjoint subsets of the training data from generalist pretrained model. When model selection or ensembling is applied at test time, Chroma achieves competitive performance on forecasting benchmarks from (Ansaii al. (2024a) and Aksu al. (2024), matching the accuracy of much larger monolithic models. Our experiments demonstrate the test-time efficiency of Chroma, as well as scalability. We believe that the methodology behind Chroma opens a promising new direction for enhancing the accuracy of pretrained forecasting models, especially under limited compute budgets. Furthermore, while our focus on time series forecasting, the underlying principles may also extend to other domains such as natural language processing and computer vision. Our strategy offers an alternative to approaches like Best-of-N which samples from single base model. Instead, it proposes inference-aware portfolio formation strategy: train single generalist base model, fine-tune it into a portfolio of specialized experts, and apply model selection time. Moreover, focused on test-time efficiency because inference often dominates the computational budget in deployed forecasting systems.
Limitations and future work. Although we did not study the impact of the pretraining phase in detail, our design choices follow standard practice: T5-architecture based Chronos-Bot Bolt model with straightforward modifications, see Appendix B Also, our theoretical observations Appendix provides a bias-variance perspective that helps explain the observed gains and suggests implications for the role of pretraining. Understanding how the number training iterations and other pretraining hy perparameterss interact with our conclusions would be valuable direction for future research. Similarly, our experiments were conducted with encoder-decoder transformer architectures. Future work can investigate portfolios of transformers based other types of architectures. Moreover, while we hand-crafted the specialist portfolios using known dataset features (e.g., domain or frequency), future work could explore more principled approaches. For instance, foundation model specialists can be formed by incorporating boosting-style objectives (Freund & Schapire, 1997). Finally, this work focused on weighted ensembling; future research could explore more sophisticated methods such as stacked ensembling (Breiman, 1996b) that can lead to higher accuracy.
10

---

Published as a conference paper at ICLR 2026
REFERENCES
Taha Aksu, Gerald Woo, Juncheng Liu, Xu Liu, Chenghao Liu, Silvio Savarese, Caiming Xiong, and Doyen Sahoo. Gift-eval: A benchmark for general time series forecasting model evaluation. vrXiv:2410.10393 2024.
Abdul Fatir Ansari, Lorenzo Stella, Ali Caner Turkmen, Xiyuan Zhang, Pedro Mercado, Huibin Shen, Oleksandr Shchur, Syama Sundar Rangapuram, Sebastian Pineda Arango, Shubham Kapoor, Jasper Zschiegner, Danielle C. Maddix, Hao Wang. Michael W. Mahoney, Kari Torkkola, Andrew Gordon and Bernie Wang. Chronos: Learning the language of time series. Transactions on Machine Learming Research, 2024a. ISSN 2835-8856. URL
Wilson, Michael
net. F
Abdul Fatir Ansari, Caner Turkmen, Oleksandr Shehur, and Lorenzo Stella. Fast
and accurate zero-shot forecasting with Chronos- Bolt and AutoGluon. Dec.
2024b.
URL https: aww.acoa.cc///booos/maaeeeeeeee. learningg shot orw.::/ww...ot...........a..oana
amazon
Arthur Asuncion, David Newman, et al. Uci machine learning repository, 2007.
Konstantinos Benidis, Syama Sundar Rangapuram, Valentin Flunkert, Yuyang Wang, Danielle Maddix, Caner Turkmen, Jan Gasthaus, Michael Bohlke-Schneide David Salinas, Lorenzo Stella, et al. Deep learning for time series forecasting: Tutorial and literature survey, ACM Computing Surveys, 55(6):1-36, 2022.
Leo Breiman. Bagging predictors. Machine Learning, 24:123-140, 1996a.
Leo Breiman. Stacked regressions. Machine Learning. 24:49-64 1996b.
Rich Caruana, Alexandru Niculescu-Mizil, Geoff Crew, and Alex Ksikes. Ensemble selection from libraries of models. In ICML, pp. 18, 2004.
Luke Darlow, Qiwen Deng. Ahmed Hassan, Martin Asenov, Rajkam Singh, Artjom Joosen, Adam Barker, and Amos Storkey. Dam: Towards foundation model for time series forecasting. arXiv:2407. / 17880, 2024.
Abhimanyu Das, Weihao Kong, Rajat Sen, and Yichen Zhou, decoder-only foundation model for time-series forecasting. In /CML, 2024.
Difan Deng, Florian Karl, Frank Hutter, Bernd Bischl, and Marius Lindauer. Efficient automated deep learning for time series forecasting. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 664 680. Springer, 2022.
Thomas D. P. Edwards, James Alvey, Justin Alsing, Nam H. Nguyen, and Benjamin D. Wandelt. Scaling-laws for large time-series models, 2025. URL httpt::t//aaatt owg.orgg///.o.oo abs hltgg//b/22409 13867
Vijay Ekambaram, Arindam Jati, Pankaj Dayama, Sumanta Mukherjee, Nam Nguyen, Wesley M Gifford, Chandra Reddy, and Jayant Kalagnanam. Tiny time mixers (TTMs): Fast pre-trained models for enhanced zero/few-shot forecasting of multivariate time series. NeurIPS, 37:74147- 74181, 2024.
Philip J Fleming and John J Wallace. How lie with statistics: the correct way to summarize benchmark results. Communications of the ACM, 29(3):218-22) 1986.
Stanislav Fort, Huiyi Hu, and Balaji Lakshminarayanan. Deep ensembles: A loss landscape perspec- tive. arXiv: 19l..9972..2..... 2019.
Yoav Freund and Robert E Schapire. A
generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences, 15(1):119-139) 1997. ISSN 0022-0000. doi: https://doi.org// htt.///iioor/1111000//sssss 1006/ccss.199..
Rakshitha Godahewa, Christoph Bergmeir, Geoffrey I. Webb, Rob J. Hyndman, and Pablo Montero- Manso. Monash time series forecasting archive. In Neural Information Processing Systems Track
Datasets and Benchmarks, 2021.
11

---

Published as a conference paper at ICLR 2026
Geoffrey Hinton, Oriol Vinyals, and Jeffrey Dean. Distilling the knowledge in neural network. In NIPS Deep Learning and Representation Learning Workshop. 2015. URL http:h/arxiii: erg/abs/ 1503. 02531
Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, Tom Hennigan, Eric Noland, Katie Millican, George van den Driessche, Bogdan Damoc, Aurelia Guy, Simon Osindero, Karen Simonyan, Erich Elsen, Jack W. Rae, Oriol Vinyals, and Laurent Sifre. Training compute-optimal large language models. arXiv:2203. 15556, 2022.
Shi Bin Hoo, Samuel Müller, David Salinas, and Frank Hutter. The tabular foundation model tabpfn outperforms specialized time series forecasting models based on simple features. arXiv:2501.02945 2025.
Rob J Hyndman and George Athanasopoulos. Forecasting: principles and practice. OTexts, 2018.
Robert A. Jacobs, Michael Jordan, Steven J. Nowlan, and Geoffrey E. Hinton. Adaptive mixtures of local experts. Neural Computation, 3(1):79-87, 1991.
Tim Januschowski, Jan Gasthaus, Yuyang Wang, David Salinas, Valentin Flunkert, Michael Bohlke- Schneider, and Laurent Callot. Criteria for classifying forecasting methods. International Journal of Forecasting, 36(1):167-177, 2020.
Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B. Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neural language models. arXiv, abs/2001.082 108361, 2020.
08361
Minjung Kim, Yusuke Hioka, and Michael Witbrock. Neural Fourier modelling: A highly compact approach time-series analysis. W0X1:2240.004033, 2024.
Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. NewrlPS, 30, 2017.
Stefan Lee, Senthil Purushwalkam, Michael Cogswell, David Crandall, and Dhruv Batra. Why M heads are better than one: Training diverse ensemble of deep networks. arXiv:1511.06314 2015,
Licong Lin, Jingfeng Wu, Sham M. Kakade, Peter Bartlett, and Jason D. Lee. Scaling laws in linear regression: Compute, parameters, and data. In NeurlPS, 2024.
Xu Liu, Juncheng Liu, Gerald Woo, Taha Aksu, Yuxuan Liang, Roger Zimmermann, Chenghao Liu, Silvio Savarese, Caiming Xiong, and Doyen Sahoo. Moirai-MoF: Empowering time series foundation models with sparse mixture of experts. arXiv preprint arXiv:2410. 100,669,, 2024.
Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. In ICLR, 2019. URL https::/openreviie
Costas Mavromatis, Petros Karypis, and George Karypis. Pack of LLMs: Model fusion at test- time via perplexity optimization. In Conference on Language Modeling, 2024. URL https?
Pablo Montero-Manso, George Athanasopoulos, Rob J. Hyndman, and Thiyanga S. Talagala. Fforma: Feature-based forecast model averaging. International Journal of Forecasting, 36:86-92, 2020.
Yuqi Nie, Nam H. Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. A time series is worth 64 words: Long-term forecasting with transformers. In ICLR, 2023.
Boris N. Oreshkin, Dmitri Carpov, Nicolas Chapados, and Yoshua Bengio. N-beats: Neural basis
expansion analysis for interpretable time series forecasting. In ICLR, 2020. URL https
/oopeneeviww.oo wwt.
Maciej Pawlikowski and Agata Chorowska. Weighted ensemble of statistical models. International
Journal of Forecasting, 36(1):93-97, 2020.
Fotios Petropoulos and Ivan Svetunkov. A simple combination of univariate models. International
Journal of Forecasting, 36:1100111 2020.
12

---

Published as a conference paper at ICLR 2026
Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. JMLR, 21(110):1 2020.
Syama Sundar Rangapuram, Matthias W Seeger, Jan Gasthaus, Lorenzo Stella, Yuyang Wang, and Tim Januschowski. Deep state space models for time series forecasting. In NeurlPS, volume 31, 2018.
David Salinas, Valentin Flunkert, Jan Gasthaus, and Tim Januschowski. Deepar: Probabilistic forecasting with autoregressive recurrent networks. International Journal of Forecasting, 36(3); 1181-1191, 2020.
Noam Shazeer, "Azalaa Mirhoseini, *Krzysztof Maziarz, Andy Davis, Quoc Le, Geoffrey Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated In ICLR, 2017. URL httttttts///ppreeeeeee
layer.
Oleksandr Shchur, Ali Caner Turkmen, Nick Erickson, Huibin Shen, Alexander Shirkov, Tony Hu, and Bernie Wang. Autogluon-timeseriee timeseries: Automl for probabilistic time series forecasting. In International Conference on Automated Machine Learning. volume 224, pp. 9/1-21, 12-15 Nov 2023.
Jingzhe Shi, Qinwei Ma, Huan Ma, and Lei Li. Scaling law for time series forecasting. 2511::4405.15124 2024a.
Xiaoming Shi, Shiyu Wang, Yuqi Nie, Dianqi Li, Zhou Ye, Qingsong Wen, and Ming Jin. Time- MoE: Billion-scale time series foundation models with mixture of experts. arXiv preprint arXiv:2409. 16040, 2024b.
Hanshi Sun, Momin Haider, Ruiqi Zhang, Huitao Yang, Jiahao Qiu, Ming Yin, Mengdi Wang, Peter Bartlett, and Andrea Zanette. Fast best-of-n decoding via speculative rejection. In Newral Information Processing Systems, 2024. URL httpp:://penteeview. nwt./orumffffddcc 348hfcprUs
Meiqi Sun, Wilson Yan, Pieter Abbeel, and Igor Mordatch. Quantifying uncertainty in foundation models via ensembles. In NeurlPS Workshop on Robustness in Sequence Modeling, 2022.
Vladimir Vapnik. The Nature of Statistical Learning Theory. Springer science & business media, 1999.
Xi Wang, Laurence Aitchison, and Maja Rudolph. Ensemble of low-rank adapters for large language model fine-tuning, In NeurIPS Workshop Efficient Natural Language and Speech Processing, 2023a.
Xiaoqian Wang, Rob J Hyndman, Feng Li, and Yanfei Kang. Forecast combinations: An over 50-year review. International Journal of Forecasting, 3914)1518-1144, 2023b.
Gerald Woo, Chenghao Liu, Akshat Kumar, Caiming Xiong. Silvio Savarese, and Doyen Sahoo. Unified training of universal time series forecasting transformers. In ICML, pp. 53140-53164, 2024.
Danny Wood, Tingting Mu, Andrew M Webb, Henry WJ Reeve, Mikel Lujan, and Gavin Brown. A unified theory of diversity in ensemble learning. JMLR, 24(359):1-49, 2023.
Sheheryar Zaidi, Arber Zela, Thomas Elsken, Christopher C. Holmes, Frank Hutter, and Yee Whye Teh. Neural ensemble search for uncertainty estimation and dataset shift. In NeurlPS, 2021.
Zhi- thi-Hua Zhou. Ensemble Methods: Foundations and Algorithms. Chapman & Hall/CRC, 1st edition, 2012.
13

---

Published as a conference paper at ICLR 2026
APPENDIX
A ADDITIONAL LITERATURE
Prior work shows that ensembles of neural networks outperform single networks and provide better uncertainty quantification. Specifically, it has been reported that training the same model architecture from different random initializations is an effective strategy for sufficiently diverse portfolios Lee et al. (2015), and deep ensembles utilize this idea Lakshminarayanan al. (217)) : Fort et al. (2019); Zaidi et al. 2021). p. At the foundation-mont modell scale, however, such work is less common. Notably, [Sun et al. (2022) builds LLM ensembles by varying random seeds, while Wang et al. (2023a) uses low-rank adapters for resource efficiency. We adopt similar approach while forming our generalist portfolios and likewise observe gains over single models. However, we further show that larger improvements can be achieved with specialist portfolios, where models are fine-tuned on targeted subsets of the data distribution. As discussed in Appendix[F] in the foundation- model regime, variance reduction from ensembling is less decisive than improved coverage of heterogeneous data.
In time-series context, we also find that partitioning by frequency to induce specialization across portfolio members performs particularly well. This aligns with broader line of work that exploits frequency-baset representation and basis expansions for time-series forecasting. Namely, Darlow Namely,Daroow al. 2024) proposes foundation model that predicts coefficients of sinusoidal bases for functional decomposition, and Kim et al. (2024) represents time series directly in the frequency space. As opposed to those methods aiming to learn single, monolithic model, our methodology is focused on specialization, i.e., training multiple experts on frequency- clusters and aggregating their predictions. In classical statistical forecasting, beyond our baselines such as Auto-ETS and Auto- ARIMA, there are also recent methods such as FFORMA (Montero-Manso et al. (022), which learns predict ensemble weights over a pool of univariate forecasters using time-series features, and SCUM (Petropoulos & Svetunkov 2020), which aggregates established statistical methods via median- based fusion.
B DETAILS ON MODEL ARCHITECTURES AND HYPERPARAMETERS
We adopt Chronos-Bolt the architecture for Chroma Ansari et al. 2024b). Chronos-Bolt is based on the T5 encoder-decoder architecture (Raffel et al.) (020)_ It chunks the historical time series context into patches of multiple observations, in PatchTST (Nie et al. 2023). which then input into the encoder. The decoder then uses the representations generated by the encoder to directly generate quantile forecasts across multiple future steps method known direct multi- step forecasting. This differs from the original Chronos (Ansari et al.] (024a) models that rely on autoregressive decoding. The Chronos-Boll architecture following standard pattern namely, patching the inputs, using transformer blocks, and multi-step-ahead quantile prediction - makes it a representative choice for time series foundation models, this pattern similar to other recent SOTA series models.
We also adopt the training methodology of Chronos-Bol Boltt which was trained on nearly 100B time series observations from varying frequencies and application domains. The training objective is the mean weighted quantile loss (WQL) over 9 equally spaced quantiles between 0.1 and 0.9. For simplicity, we keep the same training corpus as Chronos-Boll when training generalist models. When training specialists, we added new datasets to the training corpus to balance data for underrepresented frequency and domain classes (see, Appendix C Our generalist models not trained on these new datasets to remain consistent with the Chronos-Boo Bolt setup. Although we do not directly compare generalists and their corresponding specialists in main experiments, explored training generalists on the full specialist dataset. We have observed that this had little effect. The performance of individual generalists, test-time fine-tuned generalists, and generalist portfolios considered in the main paper remained largely unchanged.
also
Note that the models we train are smaller in size ranging from million parameters to 9 million parameters. The 9 million parameter model architecture exists in the released Chronos-Bol and T5-efficient line-ups as the tiny model size The rest of the models were shrunk, roughly keeping
htttss:/huugiini wtt.aacc..c.coaaaaeeeee wttpoooggg///tffffeeee cnnt-tiin
14

---

Published as a conference paper at ICLR 2026
the same aspect ratio the Chronos models. These hyperparameter choices given in Table Importantly, we set the patch size and patch stride parameters to 16. Across all models, the number of heads is set to 4, the dropout rate to 0.1, decay the learning rate using triangular schedule initialized at 10-3, and use a batch size of 256. We use the A optimizer (Lohchilovv & Hutter 2019) with weight decay set 0.1, gradient clipping 1.0 and 81 0.98, 32 0.9. All models pretrained for 200K iterations, as outlined in (Ansari et al., 2024a). For post-training fine-tuning, as discussed in the main paper, we perform 1K gradient update steps. For portfolio of N specialists, this yields speedup of x00K N/(200K + 1KxN) z 10x for typical portfolio sizes. This speedup applies equally to wall-clock time, FLOPs, and GPU-hours.
Table 3: Architectural hyperparameters for the model sizes considered in the paper. du. dav, and refer the feedforward model hidden dimension, key-value dimension, and the model (embedding) dimension respectively. L. and La denote the number of layers in the encoder and decoder.
dn
dyv
d
Ld
L
tiny
1024
64
256
4m
768
64
192
3
3
2m
640
32
160
2
2
1m
512
32
128
1
2
We trained and evaluated all models on NVIDIA A10G Tensor Core GPUs, used through AWS EC2 G5 family of instances. For reference, our 1m, 2m, 4m, and 9m (tiny) model sizes take 1.4, 1.7, 3.3, and 4.8 hours on average to train from scratch for 200K iterations, on a single A10G GPU. Post-training runs typically take several minutes.
C DETAILS ON SPECIALISTS
In order train the models of Chroma portfolios, datasets inside the general training corpus as well as additional datasets from Monash (Godahewa et al., 2021) and UCI (Asuncion et al., 2007) repositories, are partitioned in two ways, into six partitions of domain and five partitions of frequency. Details of these assignments, as well as some summary statistics of the data sets are provided in Table
Models trained from scratch follow the exact same training procedure as the generalist models, including 200K training iterations, albeit with access only a specific part of the training corpus.
For Chroma specialists, which were post-trained, we randomly selected generalist model as the initial weights and fine-tuned each specialist via 1K gradient steps, using the same training setup as mentioned above. Due to computational constraints, we trained only one set of specialists for each model size and partitioning scheme, however during experiments, we evaluated each portfolio times varying the generalist in the portfolio and reported their average. Similarly, for all other results reported on single generalists, we give an average independently trained models.
D DETAILS ON MODEL SELECTION
Ensemble selection algorithm Caruana et al. 20044) is a standard approach for forecast combination
in both point and probabilistic forecasting (Deng et al., 2022] Shhhur al. (023). Given set of models producing predictions .I,.... 9M. where each y/ is the tensor of predictions for all item, time steps and quantile levels, the goal to find optimal weights w1,..., wM for the ensemble prediction Jeus= = EM Wm Dm.
The ensemble selection algorithm optimizes validation loss by greedily adding models to equally- weighted ensemble with replacement. Since an eeiglly- ensemble with replacement is equivalent weighted average with fractional weights, this can be interpreted optimizing weights wh...,, WM via coordinate-wise ascent.
https::/huggiiggaacee http:////oooggggoee////////////r
15

---

Published as a conference paper at ICLR 2026
Table 4: Datasets used in training, summary statistics and and their assignments domain and frequency partitions.
Nr. Time Series
Nr. Observatioss
Missing Rate
Asg. Length
Frequency
Domaln
Indian weriinnppweetpeee poveetgeneeattto
13,858
0.0%
1065
daily
energy
vint_tamm_aaaly
119,549
daily
euergy
847
1, 1,501, 263
0.0%%
2129
daily
retai
online_retadi__"
1,742
893
daily
oclinn_retaal_.1
4,772
2,369,710
0.0%
497
daily
recal
nonash/covid inbbiliyyy
150
duily
nonash/oseld mobiliny wichout nissing values
180
0.08
duily
Ingger
262
47,422
0.03
181
daily
ni_dally
4,227
10,023 535
0.0%
2371
daily
nonash/bitssia
18
$2,458
9.88
4581
daily
varous
nonash/bstecln without without
intesing values
T5,164
dally
ronssh/ux_.bree
7,305
0.07
7305
duly
nonash/sunspes
73924
daily
weather
nonwsh/ consst/s:meeooo.tttee withoot wwthhott_ntestngg intestnngue Tuee
73,934
0.00
doily
wrather
ponwsh//erpezaaturee
422
305,950
0.9%
725
daily
weather
usherddaily
1,218
47,060.115
7.7%
38654
dally
weuther
321
5,403,584
0.08
26304
hourty
energy
bourly
energy
337
2. 2,862.144
17.08
8515
hoaey
enargy
resien__sty__biee
38.6870044
bouly
Inneppet
conssh/podisttiion. couuttt
66
3,132,366
0.08
47460
houryy
tnnopr
nonash/ rideshare
44.159
bourty
tmnegert
2,428
1,794,292
0.0%
houty
trzoppo
ober_tlo_houzz)
262
1,138,128
0.08
4544
hourty
trnepor
ot_hourly
973,992
boarly
270
2,942, 364
17.18
10598
boady
weather
nosssh/kdiiccp_p000
2,942,364
houly
weuter
elestzicity_
370
41,93,455
113342
subbour
energy
nonash/aus eclar Tinmn
449
0.00
093169
sabbour
0.00
vabiuur
erargy
nonssh//eeecdemann
17,520
0.0%
17520
vubbour
eneray
5,560
166 518
29951
enargy
nonashh/ccaes_ Paaintee
137
7,200,720
0.0%
52560
salbour
erangy
no0shh/sllas___cccoaa
7,307,222
7397222
eneray
colar
5,105
543 930
0.08
105120
subbour
energy
taxi__
2,425
580 796
0.08
1478
Innque
heart__rae
5,500
0.0%
1375
valiuu
1,397,147
0.0%
7397147
sabioou
weuther
of_xontbly
11,246,413
nonash/m33 other
174
13,325
0.08
77
128,471
waskh
enangy
nonsst/electriotty veekly
321
50,026
0.08
136
weekly
energy
nonseh//olxx
137
7,124
0.08
wirkly
energy
ni_waski
371,570
0.0%
weekh
vaiiou
145,063
16,537,182
0.0%
114
weekdy
web
veathezbench_com daily
225,250
19399199519000
0.88
3500,9
dull
weathe
1000000
274, 100,0000000
0.08
2741
doly
wib
Algorithm: Greedy Ensemble Weighting
1. Initialize weights (0) = 0 E T
2. For iterations j = ,,...., S:
(a) Select model that minimizes validation loss:
m () = arg min L(geus Bval);
where Dens uses the updated weights
(j 1)w((--) + em
j
and em is the m-th canonical basis vector. (b) Update weights:
( (j - 1)w(i-11 1) + emi)
j
3. Return final weights (S)
In our evaluation use S 100 steps of ensemble selection and use the last window of H observations in the training portion of each series as the validation set yvat- As in the rest of the paper, use the weighted quantile loss (WQL) as the loss function L..
16

---

Published as a conference paper at ICLR 2026
E FLOPS COMPUTATION
In our test-time compute we use approximations to the total number of floating point operations used test time. For this, we use approximations which define here for completeness. As in Appendix B let Le, La denote the number of layers in the encoder and decoder, and d the model dimension respectively. We also denote T as the effective sequence length, which is the number of tokens that are given to the encoder as the context. For our work, T = 2018 128 where 2048 is the 16
original sequence length and 16 is the patch size, for all models considered. We compute
47 )
FLOP Storward R (Le La) x T de x (24
When model fine-tuning using backward passes are required at test time, we calculate
FL.OP
FI.OP Soowwrdd
since the backward pass usually incurs about twice the cost of the forward pass, plus some overhead.
Note we compute two numbers for test-time compute, namely, total test-time compute and amortized compute. By the former, we refer to the total effort of performing model selection ensembling with a given portfolio of N models, and performing one inference pass with the resulting selection. For example, total test time compute for performing model selection is given as (N + 1) x FLOPP where the N term results from the necessity to infer with each model once during selection, and the 1 term to the single inference pass with the selected model. Similarly, for ensembling, this factor is = (N + 2.5) where 2.5 is the average number of models selected for the ensemble. Finally, when report amortized test-time compute, omit the N factor the initial cost of performing the model selection or fine-tuning will be amortized over all future inference passes using the resulting ensemble model.
F
BIAS- VARIANCE IN FOUNDATION MODEL PORTFOLIOS
As have seen in Table portfolios composed of generalist models, each trained the full training distribution, yield only small performance improvements over a single generalist model. In contrast, specialist portfolios in Chroma, whether based frequency domain, offer substantial gains. This may seem counterintuitive, since each generalist model is trained on the entire training distribution rather than specific subset, and on average, a model from the generalist portfolio outperforms model from the specialist portfolio in direct, head-to-head comparison. In this section, we offer an explanation for this observation using the variance decomposition of the expected generalization error.
The generalization error of model is known be decomposable as (Vapnik 1999)
Average Test Error Irreducible noise Bias + Variance.
Here, irreducible noise is inherent to the task and cannot be reduced. Typically, model combination methods target either the bias or variance components to lower the overall error.
In generalist portfolio, all models are trained on the same training corpus and therefore share same biases. As a result, ensembles' overall bias remains unchanged from that the individual models in the portfolio. However, such ensembles reduce variance. Indeed, if the models are independently sampled, the variance decreases by factor of N, where the number of models the ensemble. Note that this setup is analogous classical bagging in ensemble learning (Breiman) 1996a Zhou 2012), where multiple models trained on bootstrapped subsets of the full data set are averaged to reduce variance without affecting bias.
We argue that in the regime where pretrained time series models are trained, generalization error due to bias dominates the error due to variance. As result, having a generalist portfolio does not result in significant performance gains over a single model.
To confirm this argument, estimate the bias and variance components of the error of single model. To do so, we train ten distinct generalists using different random seeds following the same training procedure outlined in Appendix B In order to discard the "irreducible noise" component of
17

---

Published as a conference paper at ICLR 2026
the error, test our models synthetic signal inputs without noise component. Specifically, draw thousand time series from a mixture of periodic Gaussian process kernels and offset these with random trends and constants. On this dataset, we compute
1 Variance (1) M E ((())-(()))
(ias(x) (f(x) v)²
where fi(x) denotes the point forecast produced by model realization i for test input æ, f(x) is the average forecast across all . realizations, and y represents the true value of the signal corresponding to x over the prediction horizon. The results are provided in Table[5] Observe that over all model size spectrum consider in this work, the bias component of the error dominates the variance component across all generalist models.
Table 5:
Bias and variance components of the error. It can be seen that bias component dominates the variance component across all model sizes.
Bias
Variance
1m
65.305
10.067
2m
49.031
12.298
4m
22.009
6.423
tiny
20.063
8.367
Our observation for time-series foundation models aligns with the argument of Lin et al. (2024). which states that, in current large language models, the variance component of the error becomes strictly smaller than the bias because these models are typically trained for only one or few epochs on their massive training corpora, leaving them underfit.
These observations also help us understand why Chroma works so well. While building pretrained model portfolios, Chroma trains specialist models by fine-tuning generalist base model. Each of these specialists reduces bias within a specific part of the overall data distribution. As a result, specialist portfolio consists of models with distinct biases.
Crucially, we leverage this diversity in the biases effectively through an intelligent model combination test time. In contrast traditional ensembling approaches, such as bagging, which reduce variance by averaging out randomness across models, Chroma's 's advantage does not rely on variance reduction. Instead, operates through targeted bias reduction. For instance, when applying model selection, selects the most appropriate specialist for each test input, and the overall bias of the ensemble effectively becomes equal to the minimum bias among the models in the portfolio.
Table 2 supports this perspective, as well. A simple, uniform averaging over specialist portfolios fail to deliver performance gains comparable to model selection greedy ensembling. This finding stands in contrast to much of the forecast combination literature, where heterogencous ensembles of task-specific models typically benefit from averaging. In our case, however, the models are used in zero-shot manner, and require test time input specific selection to be effective. This is consistent with our earlier observation that, the pretrained regime, generalization error is dominated by bias rather than variance. Since variance is already low, reducing bias through targeted specialist selection offers a more effective way of improving performance.
G ADDITIONAL EXPERIMENT RESULTS
This section complements Section 4 by providing additional results and plots.
G. SCALING BEHAVIOR
As discussed in Section 4 perform ordinary least squares fits the test error of BM2 analyze the scaling behavior of Chroma. Each run corresponds to an independently trained generalist within the portfolio, with four trials reported per experimental setting. The parameter denotes the slope of
18

---

Published as a conference paper at ICLR 2026
the scaling fit, and asterisks indicate statistical significance of the fitted coefficient at the 5% level (p < 0.00).
Figures 6 and show the test emror plotted against the number of active parameters at inference time for model selection and ensembling on the Chroma portfolio. respectively. Indeed, the overall portfolio performance scales rates comparable to those of the individual generalist models.
In contrast, Figures and present results when the Chroma portfolio is trained from scratch, rather than traininiiing. Here, mentioned in the main paper, the behavior differs significantly. Therefore, we conclude that post-training generalist models to form the portfolio is essential for Chroma's scaling behavior.
We also provide scaling plots in terms of amortized test-time compute in Figures 10 calculated according to Appendix which provide the same conclusions.
Number of Active Parameters Model Selection Post-trained
Single Centralist ac=-0.055)** Frequency Sp. 00-00.133)=
X
Single Generalist (a=-0.022 0.020)
Frequency Sp. (a=00017)
Domain Sp. (a--0. 0.000
Sp. (t-l:l---oo D.01)_02
0.70
XXX
WOL
0.68
MASE
0.64
X
0.62
0.80
5
8
2
Number of Active Parameters (millions)
Number Active Parameters (millions)
Figure 6: Scaling behavior of Chroma portfolios. Results are presented for performing model selection from a Chroma portfolio of domain frequency specialists, where each of these specialists are formed with post-training.
19

---

Published as a conference paper at ICLR 2026
Number of Active Parameters Ensemble Post-trained
X
Single Generalist (a=-0.025)***
X
X Single Generalist (a- 0.0
Frequency Sp. (0m-0.117)
Frequency Sp. tor-0,0 (003)aa
Domain Sp. 0.022)
Sp. (a- 0.023
0.90
0.70
WOL
0.68
MASE
X
*
0.66
90²
90²
Number of Active Parameters (millions)
Number Active Parameters (millions)
Figure 7: Scaling behavior of Chroma portfolios. Results are presented for performing ensembling on a Chroma portfolio of domain frequency specialists, where each of these specialists formed with post-training.
Number Active Parameters Model Selection From Scratch
Single (am-0 025 Frequency Sp. la=-0.005)
X
Single (am-0.0 0.020)**
0.72
X
Domain Sp. (am-0.025jtas
Domain Sp. (am-0. 018/aaa
Frequency Sp. 0,044014p
XX
WOL
XX
MASE
0.66
2999
0.64
X
0.62
X
Number of Active Parameters (millions)
Number of Active Parameters (millions)
Figure 8: Scaling behavior of portfolios, when each of the specialists trained from scratch. Results are presented for performing model selection from a portfolio of domain or frequency specialists.
20

---

Published as a conference paper at ICLR 2026
Number of Active Parameters Ensemble- From Scratch
X
Single Generalist t==50.0
X
Single Generalist (x= 0.020) 0
0.72
X
Frequency Sp. (a=-0.002)
Domain Sp. (o-
Frequency Sp. -014p.0
Sp. 000000
0.90
0.70
WOL
0.68
MASE
X
0.66
0.64
0.80
90²
30
90²
Number of Active Parameters (millions)
Number of Active Parameters (millions)
Figure 9: Scaling behavior of portfolios, when each of the specialists are trained from scratch. Results are presented for performing ensembling on portfolio domain or frequency specialists.
Amortized TTC Model Selection Post-trained
X
Single Generalist (am-0. -0040a0 Frequency Sp. (am-0.011)
X
X Single Generalist (a=-0,00 0.019
Frequency Sp. (a=00.06)
Domain (a=00.020)***
X
Sp. (a- 0.000
0.90
0.70
XXX
X
X
WOL
0.68
MASE
X
XOBK
X
0.64
X
XXX
0.62
6x10-
10a
10"
Amortized TTC (GFLOPs)
Amortized TTC (GFLOPs)
Figure 10: Scaling behavior of Chroma portfolios with respect to the amortized test time compute, when the specialists are formed with post-training and model selection is performed during test-time.
21

---

Published as a conference paper at ICLR 2026
Amortized TTC Ensemble Post-trained
Single 0.024)m*
Single Generalist 0.019)
0.72
X
Domain Sp. (am-0.022)4as
Frequency Sp. (a=-0.016)m*
X
Domain Sp. (am-0. (am-0.022/)
Frequency Sp. (x=00.012)**
0.90
X
0.70
X
MASE
X
0.66
0.64
0.80
10"
10
Amertized TTC (GFLOPS)
Amortized TTC (GFLOPS)
Figure 11: Scaling behavior of Chroma portfolios with respect to the amortized test time compute, when the specialists are formed with post-trane post-training and greedy ensembling is performed during test- time.
Amortized TTC Model Selection From Scratch
X
Single Generalist (am-0.024p-x 024 Frequency Sp. (a-00.002)
X
Single Generalist (am-0. 019) Frequency Sp. 0,03..13
0.72
X
Domain (a=40.024)
Domain (a=-0.018)
WOL
0.68
MASE
0.66
0.62
X
10
10
6x
10a
3 10 4 10
6 10
10"
(GFLOPs)
(GFLOPs)
Figure 12: Scaling behavior of portfolios with respect to the amortized test time compute, when the specialists trained from scratch and model selection is performed during test-time.
22

---

Published as a conference paper at ICLR 2026
Amortized TTC Ensemble From Scratch
X
Single Generalist (am-0. (am-0.024)14s
X
Single Generalist (a=-000 -0000
Fraquency Sp. (a=-0.001)
Prequency Sp. (g=0,014
0.72
Domain 5p. (am-0.021)44
X
Domain Sp. (am-0.020)ss
0.90
0.70
WOL
MASE
X
X
0.66
0.64
0.80
10"
10a
Amortized TTC (GFLOPS)
Amortized TTC (GFLOPS)
Figure 13: Scaling behavior of portfolios with respect to the amortized test time compute, when the specialists are trained from scratch and greedy ensembling performed during test-time.
23

---

Published as a conference paper at ICLR 2026
G.2 TEST-TIME COMPUTE EFFICIENCY
Figures 14 and 15 demonstrate the tradeoff between total test-time compute (in terms of FLOPs) and performance. FLOPs estimated as described in Appendix E and incorporate the costs of model fitring, fine-tuning, or model selection, in addition to a single forward pass for inference per item in the test set.
Figure 14 presents results from Chroma portfolios formed through bost-training, while Figure [5 shows portfolios composed of models trained from scratch. Both are compared against two baselines: single generalist model evaluated in zero-shot setting (i.e., without any test-time computation, x), and a generalist model that has been fine-tuned All results are aggregated over the BM2 benchmark.
The conclusion remains the same in both figures. Chroma portfolios with frequency specialists lie at the test-time compute-efficieng frontier, whether the specialists are formed with post-training or trained from scratch.
Total TTC (GFLOPs) Post-trained
WQL
MASE
Im
1m
Single Generalist
0.72
*
Test-time Ane-ttued
Frequency Sp. (Best)
0.90
Domain Sp. (Best)
0.70
A miny
Frequency Sp. (Ens.)
Domain Sp, (Ens..
WOL
MASE
tiyy
Im
0.66
1m
Ary
Ony
0.64
12m
Awy
liy
12m
0.80
10a
10A GFLOPs
10a
10a
10a
10a
GFLOPs
Figure 14: Total test-time compute (GFLOPs) and performance tradeoff, presented for Chroma portfolios built through post-traine post-training.
Total TTC (GFLOPs) From Scratch
WQL
MASE
X
Single Generalist
0.72
Test-time Fine-tuned
2m
1m
0.90
2m
Proquency Sp. (Best)
Domain Sp. (Best)
4m
3m
tiny
Frequency Sp. (Ens.)
Domain Sp. (Ens.)
1m
0.68
MASE
2
1m
2
4m
tiy
tiyy
2m
2m
4mmm
Enty
0.62
0.80
10a
10a GFLOPs
GFLOPs
Figure 15: Total test-time compute (GFLOPs) and performance tradeoff, presented for portfolios when the specialist models are trained from scratch.
24

---

Published as a conference paper at ICLR 2026
G.3 CREDIT ASSIGNMENT AND INTERPRETABILITE OF THE PORTFOLIO
In Figures [6 17 18 and 19 we extend the analysis from Figure 5 by applying the same methodology
portfolios composed of model architectures 1m, 2m, 4m, and t tiny, depicted in Appendix B
We examine which specialists are activated during ensembling for each test task. Like in Figure 5
the heatmaps are presented as matrix plots, with rows representing tasks and columns representing
specialists (categorized by domain or frequency). Tasks are grouped according to their domain or
frequency, and the weights are aggregated across tasks and normalized per group. We can see that our findings from Figure remain consistent across portfolios of different model sizes.
Frequency
Domain
subhourly
energy
finance
0.7
Aunn Bo wiights
hourly
0.6
healthcare
0.5
cally
nature
Tasks
weekly
0.2
transport
0.1
supramorthly
various
subhourly
haurly-
dally"
weakly
suppe supramontti monthty
energy
weather
wab-
retal-
transport
gerveralion
Specialists
Figure Distribution of ensemble weights for specialist portfolios across distinct tasks.
Frequency
Domain
subhourty
energy
0.0
0.7
Sremee em Ceembble weig
hourly
0.6
health
0.5
cally
-0.4
Tasks
retail
-..3
transport
0.1
supr aypramonthly
various
0.0
subhouuy
hourty
energy
weather
web.
trareporrt
Specialists
Figure 17: Distribution of ensemble weights for 2m specialist portfolios across distinct tasks.
25

---

Published as a conference paper at ICLR 2026
Frequency
Domain
subhourly
energy
finance
0.6
So000 meeern wrightss
hourly
0.3
healthrare
0.4
cally
nature
0.3
Tasks
retail
0.2
weekly-
transport
0.1
supramonthly
various
0.0
Aumou
Aypp
Appaa
Aupuouuunns
weater
retaa
Specialists
Figure 18: Distribution of ensemble weights for 4m specialist portfolios across distinct tasks.
Frequency
Domain
subhourly
energy
0.7
finance
0.8
menee Aomemn weig hts
hourly
health
0.5
daily
nature
0.3
Tasks
retall
weekly-
0.2
transport
0.1
supramonthly
various
0.0
Appean
15//9
mottrer
retali
prrspoo
cemeralles
Specialists
Figure 19: Distribution of ensemble weights for tiny specialist portfolios across distinct tasks.
26

---

Published as a conference paper at ICLR 2026
G.4 ABLATION STUDY
We perform three sets of ablations in this study, quantifying the effects of dropping generalists from ensembles, using simple averaging as the ensembling strategy, and for training Chroma portfolio models from scratch instead of post-training, For each study, we fit mixed linear models controlling for the evaluation dataset, and quantify the effect of the ablation and its statistical significance. While overall results for ablations are given in the main text, we also provide detailed results per portfolio type and model size as well.
Table 6: Ablation study, fixed effects and p-values for removing the generalist from model selection ensembles for all model sizes and portfolio types.
Group
WQL Fixed Effect
WQL- p-value
MASE Fixed Effect
MASE p-value
Portfolio Type
domain
0.025
0.000
0.024
0.000
Portfolio Type
freq
0.009
0,015
0.007
0.0536
Model Size
0.017
0.015
0.012
0.184
Model Size
2m
0.017
0.015
0,018
0.0238
Model Size
0.018
0.01225
0.018
0.014
Model Size
tiny
0.016
0.0023
0.014
0.014
Overall
N/A
0.017
0.000
0.016
0.000
Table 7: Ablation study, fixed effects and p-values for replacing greedy weighted averages with simple and performance weighted ensembles.
Group
Type
WQL Fived
WQL-p-valur
MASE Fixed -p-vauue
Type
Perf
Portfolio Type
Simple
0.264
Type
freq
Perf
Portfolio Type
freq
Simple
0.167
0,221
Model Size
Im
Perf
0.142
0.000
0.179
0.000
Model Size
Im
Simple
0.177
0.000
0,231
0.000
Model Size
2m
Porf
0.137
0.000
0.179
0.000
Model Size
2m
Simple
0,172
0.000
0.232
0.000
Model Size
Am
Perf
0.152
0.000
0.199
0.000
Model Size
4m
Simple
0.196
0.000
0.265
0.000
Model Size
timy
Perf
0.138
0.000
0.186
0.000
Model Size
tiny
Simple
0.174
0.000
0.241
0.000
Overall
N/A
Perf
0.142
0.000
0.186
0.000
Overall N/A
Simple
0.180
0.000
0.242
0.000
Table 8: Ablation study, fixed effects and p-values for replacing post-training with training each specialist from scratch.
Group
WQL Fixed Effeet
WQL p-value
MASE Fixed Effect
MASE p-value
Portfolio Type
domain freq
-0.008
0.08575
-0.005
0.20533
Portfolio Type
0.007
0.08575
0.008
0.0144
Model Size
-0.002
0.791
-0.003
0.554
Model Size
2m
-0.009
0.08575
0.004
0.4732
Model Size
-0.001
0.791
0.009
0.1925
Model Size
tiny
0.010
0.08575
-0.004
0.392
Overall
N/A -0.001
0.791
0.002
0.554
G.4.1 PORTFOLIOS TRAINED FROM SCRATCH
In the result Table 2 of our ablation study, we show that there is no statistically significant difference in accuracy between portfolios trained from scratch and portfolios trained with tratinraining. In Table we support this result by a rework of Table that also includes portfolios of models that are trained from scratch, opposed post-training.
27

---

Published as a conference paper at ICLR 2026
Table 9: Aggregated relative WQL. and MASE BM2 including portfolios trained from scratch, scaled against a single generalist model of size 1m.
WQL
MASE
2m
4m
tiny
1m
tiny
Domain
Ensemble (Greedy)
0.957
0.936
0.932
0.915
0.964
0.932
0.928
0.918
Model Selection
0.950
0.939
0.923
0.973
0.948
0.933
0,930
Frequency
Ensemble (Greedy)
0.926
0.898
0.890
0.896
0.910
0.884
0.878
0.887
Model Selection
0,918
0.916
0.880
0.909
0,920
0.899
0.887
0.893
Domain (scratch)
Ensemble (Greedy)
0,953
0.927
0.923
0.911
0,925
0,905
Frequency (scratch)
Ensemble (Greedy)
0.907
0.905
0.893
0.910
0.906
0.895
0.882
0.880
Model Selection
0.935
0.914
0.885
0.940
0,920
0.913
0.895
0.896
Generalists
Ensemble (Greedy)
0.987
0.951
0.939
0.919
0,974
0.944
0.931
0,928
0.942
0.944
0.941
Single Generalist
None
1.000
0.977
0.960
0.958
1,000
0.971
0.962
0.961
can be seen that there is no significant difference between these two type of portfolios in terms of accuracy. Similarly, from the individual tasks point of view, also were not able draw direct conclusion about how from-scratch and post-trained models compare. Despite this performance similarity, the portfolios trained from scratch required 200x more compute for each specialist.
G.5 DETAILED BENCHMARK RESULTS
While we provide the aggregated results in the main text, in this section, we provide detailed results for each individual dataset included the benchmarks. Specifically, for BM2, we report the results for WQL and MASE in Tables I0 and [1] respectively. For GIFT-Eval, WQL results are shown in Tables 12 and while MASE results are presented in Tables and 15
H USE OF LLMs
Large language models are only used for few word choices in the text. They are not used for the implementation of this research.
28

---

Published as a conference paper at ICLR 2026
Table 10: Weighted quantile losses (WQL) on Chronos Benchmark II
Sensonnl
1.600
0.015
0.
0.037
0.122
0.141
0.122
0.073
0.191
0.209
0. 0.109
0.101
0.119
0,661
1.024
0.425
0.123
0.:04
0.119
0.209
0.362
0.217
ETS
1.340
0.039
0
0.118
0.13
0.085
0.033
0.033
0.165
0.13
0.093
0
0,12
0.079
0.111
0.628
0,088
0,100
0.01
0.558
0.215
ARIMA
Auto
1.333
0.033
0.029
0.041
0.09
0
0.005
0.699
0.54
0
0.098
0.033
0.156
0.079
0.125
0661
0,044
0.091
0.100
0.354
0.215
TimesMI
1,632
0,053
0.215
0,021
0.085
0082
0,13
0002
0.050
0,130
0.145
0.089
0.075
0,022
0.091
0.557
0.079
0.085
0.070
0,212
0.133
Moiral
0,075
0.014
0.038
0.017
0.0%0
0.073
0
0,052
0,033
0.167
0.13
0.097
0,070
0.131
0,076
0,112
0,522
0,080
0.104
0.082
0.234
0.133
Chrisos
0.004
0.014
0.088
0.017
0.076
0
0,027
0.055
0.130
0,885
0.095
0.033
0,002
0.332
0.584
0,000
0.005
0.067
0.253
0.
Chronos
1.004
0,188
0.061
0,024
0,077
0,051
0.009
0.033
0.058
0,133
0.095
0.102
0.095
0.078
0.157
0,078
0.129
0.598
0,085
0.098
0.065
0.244
0.130
Chronus
0,995
0.016
0.047
0,021
0471
0.052
0.012
0,042
0.057
0.139
0,151
0.092
0.076
0.077
0.121
0.502
0,884
0,090
0,065
0.23
0.134
Chromn
000
0.177
0052
0018
0.079
0067
0012
0.044
0.39
61
0,093
0.72
0075
0.122
0.61
0085
0.055
0.071
02500
01.32
Chroma
1,005
0.016
0.056
0.08
0.079
0.072
0.088
0.54
0.140
0083
0.161
0.093
0,072
0.148
0.076
0,123
0,562
0.085
0.096
0.011
0,255
0.138
Chroma 099.)
1,033
0.017
0.058
0,077
0475
0.099
0,010
0,047
0.054
0.140
0,133
0.094
0.075
0,077
0,114
0.561
0,084
0,100
0
0,254
0.134
Chromn
1010
0.017
0.066
0017
007S
0.62
0.099
0.036
0.054
0.44
0.33
0.094
0.088
0.126
0.799
0.118
0.53
0.066
0.000
0.076
02257
CL134
tourisms montbby
touriam quarterly
tourbm searly
parts
of 2016
covid sleaths
fred md
hoopital
m1 monttbyy
m quarterly
yeary
m1 mombly
m1 quarterly
yourly
quarteriy
md yearly
weekky
waather
reod
eth
ettm
ms
traffic
29

---

Published as a conference paper at ICLR 2026
Table 11: Mean absolute scaled errors (MASE) on Chronos Benchmark II
Scasonal
1.000
0015
0.133
0037
0112
0.41
0.22
0073
0.91
0,209
01:49
0.101
0.19
0.111
1.024
0.425
0.23
0.104
0.119
0,302
0217
Auto ETS
1.340
0.039
0
0.18
0,132
0.085
0.033
0.53
0.165
0.085
0,13
0.093
0,070
0.12
0,079
0.111
0.628
0,088
0.100
0.071
0,558
0,215
Auto
1.333
0.033
0
0.041
0,089
0.105
0,035
0.089
0.154
0.133
0.098
0,071
0.556
0,079
0.125
0.611
0,084
0.091
0.000
0.354
0,215
TimeFM
1,633
0.053
0.215
0,021
0.085
0,022
0.013
0.027
0.050
0.130
0.145
0.089
0.075
0.062
0.091
0.557
0.079
0.088
0.070
0,2
0.138
Moiral
0,975
0,14
0.038
0.017
0.090
0.073
0.012
0,052
0,053
0.162
0.126
0.097
0
0,131
0.076
0,113
0,572
0,090
0.10c
0.082
0,234
0.132
Chroson
1.004
0.014
0.058
0,017
0,076
0.011
0,027
0.055
0.130
0.185
0.095
0.073
0,082
0,132
0.544
0,000
0.095
0.007
0.253
0.138
Chronos
L.004
0.018
0.061
0.350
0.024
0.077
0,051
0.083
0.058
0.133
0.062
0.095
0.078
0.078
0.129
0.366
0.065
0.098
0.065
0.244
0.130
Chrumos
0.995
0.016
0.047
0,021
0.071
0.052
0,012
0.057
0.139
0,151
0.093
0.076
0,077
0,121
0,582
0,084
0.090
0.065
0,231
0,134
Chroma (tiny,
1.000
0.017
0.022
0,018
0.079
0.067
0,012
0.054
0.139
0.161
0,093
0,072
0,075
0,122
0.561
0,085
0,095
0,011
0,250
0.132
Chroma
1,005
0.016
0.056
0.360
0,08
0.079
0.072
0.011
0.088
0.054
0.440
0
0.661
0.093
0.072
0,076
0.123
0.562
0.065
0.096
0.091
0.255
0.338
Chroma (4m,
1.013
0.017
0.058
0,017
0,075
0.059
0,047
0.054
0.140
0.133
0.094
0.075
0,077
0,114
0
0,084
0.100
0.076
0,254
0.134
Chromn best)
1.010
0.017
0.056
0,017
0,075
0.062
0,0236
0.054
0.142
0.133
0.094
0,078
0.079
0,118
0.563
0.086
0,100
0,776
0.257
0.134
Nourian, moothlly
tourbsen quarterty
tourben. yearly
parts
of 2016
covid deatiss
exchonge
fred md
hospital
ml mmonthly
ml quarterly
yearly m3 monthly
m3 quarterly
m3 yearly
quarterly
md. yearly
mnS. weedy
wonther
eroit
etb
ettm
ms
traffic
30

---

Published as a conference paper at ICLR 2026
Table 12: Weighted quantile losses (WQL) on GIFT-Eval (Part 1)
Seaconal Naive
0.081
0.123
0.137
0,131
0.108
0.206
0,245
0.
0.281
0.579
0.552
0.300
0.554
0,229
1.310
1-270
1,290
1.000
1.100
1.260
1.290
1.300
0
0.973
0,262
0.530
0.536
1.400
1.830
0,040
0,049
0,056
1.730
0.165
0.124
0.129
0. 0.109
0.156
0.190
0.099
0,241
0.352
0.396
0.515
0.250
Auto
0.088
0,240
0.038
0..57
0,330
0.436
0.323
0.242
0.48
0..08
0.291
1,600
1.210
1,290
1.100
1,290
1.300
0,088
0,289
0,075
0,385
0.494
1.050
1.310
0,224
0,373
1.300
0,033
0,375
0.121
0.129
0,099
0.155
0.098
0,383
0.867
1.050
0.334
0.40
ARIMA
Autt
0.081
0.137
0.078
0.108
0.154
0,193
0.135
0.281
0.255
0,270
0.309
0.170
1,210
1.270
1,200
0.844
1.100
1,250
0,874
0,035
0,973
0.282
0.500
0.547
0.813
0.757
0,049
0.056
1.290
0.165
0.124
0.129
0.053
0,109
0.156
0.100
0.241
0.352
0.386
0.279
0.223
Thuss
2.0
(500m))
0,051
0.114
0.041
0.039
0.067
0.0N6
0.040
0.1
0.127
0.127
0.199
0.227
0.135
0.445
0.301
0,900
0.6086
0.461
0.706
0.644
0.014
0.033
0.057
0.084
0.539
0.345
0.640
0.728
0.008
0.062
LO06
0.079
0.000
0.083
0.054
0.073
0.009
0.168
0.378
0.283
0.193
TabPEN-
6
0044
60KS
0UST
0154
01:50
0.164
0215
0.40
Q.144
0.456
0,735
0.760
0,591
0.455
0720
0.637
0031
0.0%0
0.088
0.099
0.345
0.511
0.290
0,380
0.440
0.011
0,067
0.991
0,55
0.000
0.101
0.092
0.089
0,000
0.072
0.91
0112
0.511
0.183
0.44
0,260
0
0.19
TTM-R Zerodoot
0.088
0.125 125
0.101
0.995
0.097
0.151
0.225
0.213
0.222
0.208
0.260
0.183
0.596
03006
0.950
0,926
0.605
0,804
0:779
1.000
0,058
0,144
0
0.540
0,509
0.782
0.751
0.053
0.117
0,140
2.285
0,552
0,142
0,143
0.093
0.097
0.109
0.128
0.159
0.235
0.333
0,352
0.416
0.250
Chronos Large
0.770
0.145
0.045
0.066
0.094
0:081
0077
01.37
01
0.339
0.236
0.267
0265
0.148
0-66
0.799
0.725
0.633
0,500
1,0700
0.661
0,022
0.095
0,084
0.0064
0.496
0
0.802
0,757
0.027
0,072
0.09:
1.070
0.02
0.092
0.09%
0.099
0.065
0.0866
0.007
0.:0
0.196
0.376
0.367
0-044
0.19
Chrions Bolt Smalll
0.055
0. 125
0.045
0,0K2
0
0,072
0,133
11134
0. 1414
0,203
0.346
01.45
0.157
0.435
0.807
0.753
01.599
0.453
0,792
0,756
0.623
0.085
0.085
0.092
0.073
0.462
0.204
0,285
0.395
0.052
0.082
0.099
1.007
0.043
0.082
0.087
0.086
0.058
0.067
0.094
1.002
0.048
0. 100
0,288
0.29%
0,283
01 109
Charonos Bolt Bare
0.055
0.129
0,044
0.065
0,076
0,076
0.069
0.125
0.157
0.170
0.302
0,344
0,24N
0.136
0-544
0,755
0,748
0.714
0.238
0.605
0,756
0,624
0,044
0,109
0,074
0.445
0,738
0,189
0.254
0.278
0,0846
0,113
0.095
0,082
0,683
0
0,055
0,064
0
0.098
0,047
0.158
0.281
0.298
0.181
Chroana (d1t) freg.
0.085 (88.)
0.1 25
0.049
0,072
0.033
0.02
010N7
01142
0. -44
0.173
0.3004
0:241
01349
0.155
1..35
0.733
0.71
0.335
01310
0,685
0U021
0.082
0.093
0,077
0.417
0.649
0.210
0.278
01300
0,19
0.075
0.0955
1.000
0.052
0,090
0.096
0.087
0.056
0,073
0.088
0.10P
0.096
0,669
0.376
0.599
01277
0.192
Chroana (tins: Ereg. bead
0.077
033
0.049
0.072
0.082
0.083
0.007
0.445
041
0.154
0.2M
0.25
03338
0.435
0,525
0L686
0.40
0.002
0.706
0.001
0.094
0.098
0.426
0.309
0,277
0,308
0.025
0.074
0.097
1.005
0.006
0.001
0.088
0.092
0.073
0.051
100
0.144
0.71
0.375
1030
0 198
Chronaa (4m. freq.
0,082
0.131
0.052
0.076
0.044
0.083
01788
0.149
0.166
0.305
0.240
0.342
0.137
0.441
0.292
0.735
00099
0.441
0.730
0.364
0.026
0.095
0.110
00079
0.419
0.215
0.290
0.310
0,023
0.081
0.006
1.13
0.058
0.092
0.088
0.089
0.058
0.074
0.091
0.108
0.70
0.202
0.384
095
(4m. freq.
Chrona
besk))
0.06
0.13 19
0052
0075
0.086
0,885
0082
0.149
0.166
0,205
0.250
0.244
0.138
0-448
0.801
0,734
07731
0-441
0.311
0.704
0.796
00137
0.119
0.779
0.667
0,221
0.288
0.314
0.091
0.155
1010
0.056
0.093
0.092
0092
0.588
0.075
0099
0.110
0.73
0.90
0,289
0,311
0.197
-tt--ttrrgg------ 1--------7--1 1--221
SOOPSEEATLE AETTL------
SEAUTL-- 4-57--800
LOOP- LL---LE---1721
ROO--SEATTEE----E 3D
LOOP SEATTLE-H--L
SEAUTLE-H 1--1--00
LOOP LOOP-SEATTEE SEATTLE-H-724
www.rrattoo-aa wtw.rgggoo---- 51-440
bithraat netrratioo alto Fast wwnt-stttrrggg--- 11--1---
hokttohh app.ttsapppplttt applicatioon0
bizztohe aopplctttoonnee 600
beztthbe wwwpletttoon-.oo 9000
car garrrartsso wwt.ppttttttt-hmiseee. mith-iiiinn-12222 12
M-DEANK-D-S -1-30
A-DENSY-HE -E-AN
A.DENSE-HE JI
M-DENST-HH-TL
SZ T--A----5---
SZ 1AXI-15--720
SZ TAXI-T--4
bithraitas anddd S1--T--4 ST-48
hithral tes mnd. mnd-1--- 11-4--4
bizttoheee the 5T-499
hizztthhe De ST-720
bizttbbe Lee 12-4-- 48-8
bizitohs--ee Re 450
bitttohre 22-1--1
bizitobb wewvieeeeeiieeoooeee
bizttobseerreee d00
bisttehe Sereeee---- 9000
osidd deathd 30
dlccrrict--15 1ST-48
Mectrictt--STT- 1ST-450
dectridit-1111 LST-720
dectricity-D..
dectricct--E-- -I-44
dectriiity-H--e 1E-480
decctrictty- Mectriiit----7720 11-720
dlectriitty- WW-8 WW-
151-155-48
401-155-400
601-151-720
(11---30
4111-T--48
31

---

Published as a conference paper at ICLR 2026
Table 13: Weighted quantile losses (WQL) on GIFT-Eval (Part
(aart 2)
Seasoull
Naive
0.540
0.616
0.338
0.096
0.143
0.105
0.205
0.094
0.241
0.287
169
2,360
1,030
0.062
0.155
0.277
0.304
0.297
0.173
0.486
0.598
0,888
0.559
0,949
1.250
0.026
0.040
0. 0.126
0.09M
0.073
0.138
0,907
0.754
0.445
0,855
0.860
0.771
0.786
0.757
0.628
1.270
0.236
1.630
00.44
0,017
0.022
Antoo BTS
0.277
0.103
0.325
0.317
0.158
0.103
0.115
0,931
0,642
0,055
0.082
0.310
0
0.074
1,160
0,949
1.250
0.029
0.070
0.000
0.079
0,052
0,111
0.596
0,322
0,896
0,850
2,030
0,281
1.080
2.120
1.720
0.139
0.754
0.074
0,012
0,018
0.1
ARIMA
Auto
0.384
0,430
0,305
0.096
0.143
0.105
0.125
0.089
0.245
0.136
0,735
0.485
0,000
0.155
0.277
0.304
0.080
0.145
0.230
0,393
0,559
0,851
1,080
0.023
0.034
0.098
0,082
0,050
0.130
0,362
0.564
0,326
0,549
0,860
0.771
0,786
0,282
0.628
0,557
0,607
0,152
0,094
0.074
0,010
0,018
TimesfM
(50hh))
0,252
0,310
0,272
0.085
0.105
0.106
0.108
0.006
0,130
0,576
0,330
0,050
0,016
04031
0,035
0,058
0.045
0,006
0.068
0,378
0,376
0,466
0,518
0,021
0,011
0.067
0
0,042
0
0,261
0,408
0,342
0
0,804
0,516
0,408
0,278
0,406
0,376
0,493
0,171
0,586
0,019
0,111
0,13
2,0
TabPFN-
0.276
0,290
0.256
0.073
0.998
0.101
0.129
0.070
0.120
0.593
0,342
0
0.035
0.057
0
0.047
0.043
0.066
0,359
0.437
0.462
0.462
0.024
0.078
0
0,075
0.036
0.297
0.384
0,278
0,380
0.545
0.359
0.352
0.209
0.358
0.324
0.565
0,018
0,013
0011
TS
TTM-R2
Zeroshot
0.339
0,342
0.448
0.099
0.128
0.126
0.119
0.088
0,200
0,792
0,725
0.123
0.845
0.069
0.068
0.124
0.060
0.073
0,084
0.522
0,514
0,532
0.542
0.835
0.040
0.177
0.139
0.000
0.197
0.438
0,589
0,405
0.006
0.785
0.573
0.545
0,396
0.468
0.493
0.512
0.531
0,791
0.104
0,036
0,027
Charanoss Large
0.333
0.350
0.316
0.072
0.126
0.140
0
0,022
0.125
0.139
0,071
0.599
0,365
0.087
0.044
0.075
0,077
0.009
0.046
0.071
0,072
0,502
0.458
0,658
0.636
0.022
0.076
0.103
0.083
0.337
0.135
0,279
0,420
0.415
0471
0.575
0.681
0.747
0.328
0.337
0,355
0.448
0.162
0,009
0,023
0,013
0010
Churomus Bolk Small
0.295
0.337
0,293
0,070
0.119
0.118
0
0
0.118
0,121
0,094
0,582
0,354
0,058
0.037
0.060
0
0.047
0.043
0.058
0,068
0,375
0,267
0,364
0.419
0,021
04020
0.094
0.078
0,038
0,264
0,354
0,293
0,372
0,298
0,453
0.497
0,286
0,303
0.356
0,573
0,136
0,544
0,028
0,016
0,013
Chromus Bolt Base
0,303
0,311
0.29%
0.067
0.110
0.111
0.094
0,663
0,115
0,008
0.576
0,553
0,052
0,033
0.057
0.064
0,045
0.042
0.054
0,062
0,372
0.246
0,301
0,30M
0.21
0.025
0.094
0.077
0.038
0,121
0,338
0.296
0,363
0.511
0.436
0.443
0.287
0.298
0,368
0,405
0,538
0.026
0,019
0,013
Chromut (ting, Freg.
0.343
0,467
0.260
0,044
0.107
0.113
0.104
0.067
0.121
0.130
0.095
0.57
0,511
0,084
0.048
0.05
0.077
0.446
0.04
0.059
0.083
0.375
0,580
0.25
0.571
0.022
0.026
0.093
0.075
0.043
0.12
0.278
0.387
0.599
0.499
0.504
0,350
0,334
0.278
0.316
0.331
0.343
0,147
0,508
0,023
0.020
0,015
Chronu (tiny, freg. hest)
0.298
0.305
0,273
0.064
0.209
0.117
0.096
00067
0.121
0.099
0,580
0.556
0,084
0.046
0,058
0.070
0.016
0.042
0.059
0.088
0.399
0,357
0-23
0.471
0.023
0.027
0.094
0.076
0.443
0,123
0.279
0.388
0.324
0.449
0.478
0.356
0.340
0.281
0.317
0.352
0.338
0.136
0.548
0,022
0.019
0.015
Chromu (4m, freq.
0.318
0.541
0.253
0,088
0.315
0.117
0.099
01066
0.044
0,576
0.500
01054
0.034
0.060
0.099
0D16
0046
0.055
0.368
0
0.443
0.498
0.023
0,029
0.094
0,077
0044
0,114
0
0.47
0,354
0.479
0.532
0,353
03355
0.279
0.322
0.333
0,359
0.383
0,551
0.03
0021
0.16
Chroma (4m. free. best)
309
.99
281
0.0F1
0.119
0.121
0.096
0.003
0.577
0,350
0.084
0,035
05059
0.060
0.016
0.045
0.055
0.375
0.441
0.456
0.404
0,024
0.028
0.095
0,079
0.043
0.118
0.287
0.440
0,334
305
0.658
0.353
0.355
0.284
0.322
0.336
0.341
22227
0.551
0.023
0.1
0.016
kddd-up-22 201 18 with- wwwssiinnnn...... D0300
kdd-ccup 2018 with missing--I---ee -I-4--I-8 I1-48
kdd--cup-a 201 will-miissngg 11 430
kdd cap- 201 with miseems-s-in---- HI-72
temperature-ree rain- caai-mmiiiinng--
hierarchical- saleal-slle--.... D.30 30
Autrarcicaa-ssin soles------ W-8
rene-wootbeee 1ooobe---1---00000 101--490
jena- mnn--watberr-... 10T-720
eHt-II-500
-1I-I--700
152-55T-88
-12-1571-800
en 151-720
eH2----0
012-11-48
e122-11-4 480
122-1--720
etz---8 W-B
hospital-12 12
jenu mou--wotherr--te 101th----10---- 10T-448
jenur aana-watter-- 30
Jenas wwwther--11 H14
jenu wwnther II
jenn 11-2--11-720 11-720
mdd-dail--14
mo4-boorly-40
and-anathly-B 18
ind-quartert---
and 103
md euu-y
restaurant-30
cangenddy--D... D0.30
M--1 M--12
saugeenday-
sangemday- W--
solor- 10T-48
solar-10r-000 10T-480 -80
solnr- 101-720
solar D-50
solar-11--8
soloe-33 31-480 4000
3alor-11--20 7200
WV-
us- ee-hirtha--D- 30
us- Me-hhrths---112
us hi-hirth--W. S
solar-
32

---

Published as a conference paper at ICLR 2026
Table 14: Mean absolute scaled errors (MASE) on GlFl-Eval vval (Part 1)
Scanooaal Naive
0.081
0.123
0.:37
0.:31
0
0,206
0.345
0.294
0
0.479
0.552
0.309
0.454
0.554
0,229
1,210
1.220
1,290
1.000
1.100
1.360
1,290
1,300
0.08S
0.042
0.973
0,202
0.530
0.536
1,430
1.830
0,000
0,000
0,056
1.730
0,065
0.124
0.129
0,009
0.156
0.190
0.099
0,341
0.352
0,396
0.515
0.250
Auto
0.085
0,240
0.058
0.157
0,330
0.426
0.125
0.242
0,458
0.508
0.291
1.620
1.210
1.270
1.290
1.100
1.260
1.290
1.300
0,161
0,269
0,075
0,355
0,494
1.050
1.110
0,224
0,373
1.340
0,175
0.124
0.129
0,099
0.156
0.383
0,887
1
0.354
0.460
ARIMA
0,041
0.125
01
0:078
0.10K
0.154
0,193
0.135
0.281
0,235
0,200
0.509
0.351
0,39%
0,170
1,210
1.270
1,200
03644
1,100
1.260
1,290
0.874
0,028
0.042
0,973
0,262
0.530
0,674
0,547
0.813
0,787
0,049
0,056
1.200
0,165
0,124
0.129
0.083
0,109
0.155
0.190
0,100
0,241
0.352
0.396
0,279
0,223
ThassFFM 2,0
(50000)
0.551
01114
0.04
(LAS9
BL067
IL127
0L.27
0..19
11.27
0L.33
0.447
01.88
0L.000
01.008
0.461
0.640
0.014
0,057
0L084
0.539
0.365
0.640
0.728
0.003
0.062
1.96
0,079
0.000
0.083
0,054
0.073
0.000
0.049
0.168
0.278
01.283
0.193
TabPEN-
0462
0,094
0.044
0,065
0
0,857
0.354
0.159
0.164
0.215
0.245
0.248
0.344
0.456
0.735
0,760
0
0,555
0.750
0.635
0,011
0,000
0.085
0.099
0.345
0,290
0,380
0.440
0.001
0.065
0.091
0.955
0.104
0.092
0.089
0.000
0,072
0.091
0.112
0,051
0.183
0.348
0.260
0
0. 101
TTM-82
Zeroshot
0L0S
0.13 15
0.001
00095
0.104
0.997
01:51
0.255
0213
0,22
0.208
0.200
01133
0.598
0U06
0.99
0392
0,608
0.779
1.000
0005
0.14
0.107
0.40
0.785
0,549
0.782
0.51
0.055
0.177
0.140
2,255
0..52
0.142
01143
0.099
0.997
0.009
0.128
0.55
0.33
0,352
0416
0250
Chroness Large
0.070
0.44
0.045
0.066
0.84
0.8
0X075
0137
0134
0.339
0.236
0.2667
0.265
0.148
0-465
0.299
0.723
0,306
1,070
0,061
0.002
0.0R4
0.004
0,496
0
0.502
0,757
0,072
0.091
1,070
0,042
0,092
0.094
0.0%
0.065
0.006
0
0.996
0.376
0.367
0.404
0.198
Charmos Belt Small
0.055
0. 125
0.045
0.066
0.082
0.062
0,072
0,133
0.134
0. 146
0.203
01245
0.137
0.435
0.807
0.755
01.89
0.453
0.756
0,623
0.08S
0.092
0,733
0.482
0,790
0.204
0,285
0.395
0,082
0.09%
1.007
0.043
0.082
0.067
0.086
0.058
0.067
0.084
100
0. 10
0,288
00.296
0.283
0.009
Chroonos Bolt Bisse
0.055
0.129
0444
0.065
0.076
0,076
0.069
0.125
0.173
0.202
0.244
1:244
0.136
0.454
0.755
0.74N
0C774
0.498
0.605
0,756
0,624
0,054
0,109
0,074
0.445
0,189
0.254
0,278
0,051
0,095
0,113
0
0.047
0,0822
0,083
0
0,055
0,064
0
0.098
0,047
0,158
0,281
0.298
Chroonn (103)) freq. 408.
0.065
125
0.049
0.072
0.083
0.082
01087
01142
14440
0.173
0.2044
0.241
0.289
0.157
01.355
0.800
0.736
0.671
0.435
0.000
0.685
0101
0.082
0,093
0,777
0.417
0.649
0.280
0.278
0,300
0.019
0.075
0.095
1.000
0.090
0.086
0.057
0.056
0.073
0.08%
0.109
0.086
0.169
0.276
0:205
0,275
192
Chroana (Liss feeg.
Bost)
0.017
0,133
0.049
0.072
0.0R2
0,083
0.0K5
0.145
0.341
0.354
0.204
0.255
0.138
0.435
0.758
0.686
0.430
0.802
0.706
0.031
0.094
0.078
0.426
0.299
0,377
0.308
0.074
0.097
1.005
0.056
0.091
0.088
0.092
0.05
0.073
0.091
0.103
0.37
0.375
0.306
0.295
0.193
Chrmmaa (4m) treg-
(01-)
0,082
01:31
0,522
0.07
0004
0083
0U778
01.49
01446
01066
0.305
0L24
0.137
0.441
0.792
0:35
0L6999
0.441
0.730
0.364
00036
0.955
0.110
00099
0.44
0.737
0215
0.250
0.310
0,023
0.81
0.106
1.013
0,058
0.02
0.088
0.000
0.98
0074
0091
0.00
0.770
0.83
0.54
0198
Chronaa (4m. Breg.
best)
0.041
0.139
0.053
0.075
0.85
0,02
0.149
0.166
0.305
0.348
0.13M
0.448
0.301
0.334
0.731
0.411
0.751
0.714
0.796
0.087
0
0.079
0467
0,785
0.221
0.288
0.314
0.002
0.091
0.115
1.010
0,056
0.093
0.092
0.092
0.058
0.075
0.095
0.110
0.173
0.200
0.289
0,311
0.99
bitter wwwII www.rrii-------tttaoonn www.-ttttggg---o... T--48
Anturratts-e -ttttttogg---- 1-400
Witbrutie wtt.sttrrge------ 5 11-730 20
bittrr Wittrratoseeee dast weteetttrggg---- 41 -48
LOOP SEATTLE-ST-AE
L.OP KOOP-SEATHE MAATLE--S--
SOOP-SEATTLE----- ATTTE----3
WATTL-----M
LOOP-ST BAATLE- 1--1--400
LOOP-SE SEATTLE- 8E-H-720
tatttbbe a appl ceplicattionniii 40
bizttbb applicattionn 600
haztmbee cennnbeappe applecatioonn 900
car garts www.rrt-wit---iiiiii mis0-mmiss--uu-1- 12
L.OPPSSAA SEATTLE--SSSE
M-DRANSE-IE D-330
ALDENSE-IE HLA000
MEDENSE-H-FD H-773
SZ--AAI-JFT-SS
SZ-TAM--5Y--80
bithrrat watrrrtttssaaaa and $51-88
bitbral 9:720
bithral hithrales-aa alee and 11 1--48
biztimbee 12 T-445
bizitthee bittmbel22 Le S T1480
11720
bitttbee 23e 1e-1--44 11-48
bizitobe12e- 2e 4890
bizitobo hizettnbee-ee 22-11-720 200
60
bizttbbe 400
bitiibe orer vice- -900
deathsshh333 30
dletriictt-155 15T-48
dlectrictt-1S5 -55- LST-489
plectricity-151 1ST--720
dlectricit--EE TE-48
electricity- Mettrriity----8 HE-880
olectricity - 11-720
electricitt- WW-
A-DENNE--E
SZ. -A--AN-----
bithrattss mnts
electricity-1 -D-
155-555-48
ont -15--00-580
111-15777201
on 1-333
alt 31-40 4M
39

---

Published as a conference paper at ICLR 2026
intons (MASE) Borl (Part
Table 15: Mean absolute scaled errors
(aart 2)
Seusonal
Naive
0.540
0.616
0.338
0.096
0.143
0.165
0.205
0.994
0.287
0.169
2,360
1,030
0.062
0.155
0.277
0.314
0.297
0.173
0.486
0.598
0,888
0.559
0.949
1,250
0.026
0.040
0.136
0.099
0.073
0.138
0.907
0.754
0.445
0.855
0.860
0.771
0.786
0.757
0.628
1.220
1.470
0.236
1.630
0.144
0.077
0,022
Auto ETS
0.277
0.108
0.325
0.317
0.158
0.103
0.329
0.115
0,931
0.642
0.053
0
0.210
0,282
0.074
0.1588
1.160
0.949
1,250
0,029
0.070
0.100
0.079
0.052
0.111
0.596
0,322
0.896
0.870
4,590
2,930
0.281
1.080
2,120
1,720
0.139
0.754
0,074
0.012
0.018
ARIMA
Auto
0.384
0:430
0:305
0.096
0.143
0.165
0.125
0.089
0.245
0.222
0.36
0775
0.465
0.000
0.15
0.77
0.304
0.0R0
01443
0.230
0.393
0.550
0.851
1.050
0.023
0.34
0.098
0.082
0,000
0.30
0.362
0.564
0.336
0.540
0.860
0.771
0.766
0
0.628
0.557
0,607
0.52
0.694
0,074
0010
0,188
TimesUM
(500m)
0.282
0,310
0,272
0,065
0,105
0.106
0.108
0,006
0.110
0,125
0,110
0.576
0,330
0,050
0,016
0431
0033
0,088
0.455
0,066
0,068
0,578
0,376
0,466
0,518
6021
6011
0967
0962
0,43
0,991
0,261
0,408
0,342
0,601
0,804
0.516
0:98
0.778
0.466
0.776
0,93
0,71
0,566
0019
6011
0,13
2.0
TabPEN.
0.276
0.290
0.256
0,033
0.09H
0.101
0.129
0.079
0.120
0.593
0.342
0,050
0,035
0,577
0,055
0.047
0.043
0,060
0,066
0,359
0.43
0.462
0.462
0.024
0,288
0,088
0,075
0,036
0.384
0,278
0.389
0.565
0.55
0352
0.309
03388
0.565
0,118
0013
00111
TS
TTM-R2
Zenoboot
0.339
0.342
0.488
0.093
0.128
0.125
0.119
0.088
0.139
0,144
0.300
0,792
0,725
0.123
0.045
0.069
0.068
0.128
0.00
0,075
0,084
0452
0.514
0.522
0.542
0.035
0.440
0.17
0.339
0.099
0.197
0.438
0.589
0.405
0.996
0.785
0.573
0.545
0.396
0.468
0.493
0.12
0.51
0
0,104
0.066
0,02
Chronao Large
0.33
0,350
0.16
0,022
0.126
0.140
0
0,72
0.125
0
0,599
0,365
0,057
0,044
0.7S
0.072
0.019
0.46
0.071
0,502
0.458
0,658
0,636
0,022
0.066
0.108
0.083
0
0.135
0.420
0.415
0.471
0.575
0,681
0.747
0.328
0.337
0,355
0.448
0.162
0,009
0,023
0013
0,010
Chronos Bolt Small
0.295
0,337
0.203
0.070
0.139
0.118
0
0
0,094
0,582
0.354
0,058
0,037
0.050
0.063
0.047
0.045
0.058
0.065
0.373
0,267
0,364
0.419
0,021
0.020
0.094
0.078
0.038
0.128
0,264
0,354
0,293
0,372
0.498
0.453
0.407
0,286
0,303
0.356
0,573
0.136
0.544
0,028
0.016
0.033
Chronos Belt Base
0.303
0.31
0,2086
0.067
0.110
0.111
0.094
0
0.088
0,576
0,353
0,05
0,033
0.05
0.004
0.045
RJM4
0,054
0.372
0,246
0,301
0,300
0,021
0.025
0.094
0.077
0.038
0,121
0.264
0,338
0.2996
0,363
0.511
0.436
0.443
0.287
0,298
0,405
0,538
0,026
0,019
0013
Chroma (tiny, freq.
0.343
0.467
0,260
0.064
0.107
0.113
0.104
0.967
0.121
0,095
0.577
0.351
0.054
0.048
0.05
0.007
0.046
0.044
0.039
0,083
0.375
0,380
0.455
0.471
0.022
0.076
0.093
0.075
0.043
0,387
0,379
0.449
0.504
0,350
0,334
0,278
0.316
0.331
0,343
0.147
0.548
0,023
0,020
0,15
Charoma (diny, freq. best)
298
0.305
0.277
0.064
0.109
0.117
0.996
0.967
0.121
0.129
0,099
0,580
0,356
0.054
0.946
0.058
0.070
0.046
0MM42
0.059
0.08S
0.579
0.357
0,23
0.471
0.023
0.027
0.094
0.076
0.043
0.123
0.279
0,388
0.324
0.499
0.475
0.356
0,340
0,281
0,317
0.332
0,338
0,548
0,022
0,019
0,15
Charama (4m. free.
0,318
0.541
0,253
0.068
0.115
0.117
0.099
0,066
0,094
0,576
0,350
0.084
0.034
0.060
0.059
0.046
0.046
0.055
0.074
0,368
0
0.443
0,298
0,023
0.029
0.094
0.077
0.044
0,114
0,282
0.377
0,334
0479
0,532
0,353
0.355
0.279
0,322
0,333
0.339
0,183
0,551
0,023
0,021
0,016
0.
Charmuu (4m, freq. beed)
0.309
0.592
0
0.071
0.119
0.121
0,996
0,056
0.669
0,093
0.577
0.350
0,084
0.035
0.059
0.060
0.016
0.043
0,055
0,375
0,441
0,456
0.494
0.024
0.028
0.095
0,079
0,043
0,118
0.287
0.450
0,334
0,506
0,658
0.353
0.355
0,284
0.322
0.336
0,341
0,551
0,023
0,021
0,016
kdd kdd- 20 8 with www.iing---iiii. 10:30
kdd 2015 Wiih-withh... with missing-- 1ep--I--- 11-44
Audd-cu-222 20118- with Ses---il--mmsss mes-smiiiiiig--- 1t -000
kdd cup- 2018. with wwwhhwhthsssiii waao-m-iiiiingggg Ht 720
femperature- raim hetnew-ith-mmssttt miih---iiiinng.00
neerrrciical--oo wales-DD 30
hierurchical--e watiicaa-ttee----ee- saales--- VN-88
jema www.werrtrrreeeeeeoo. 1017-48
jenss-weatber-- 107-4000
Jena www.terreeeereeeeeeeeeeeeeeeeeee 10T-720
wwathers 81-20
-II-1-720
112-15T-48
412-151-400
112-1511-200
v122-D-30
411--11488
480
122-1--720
122-W-8
hosqitaal-12
jema wwwtherr-eD---.... 30
Jena www.thhrr-----e
jena weathers
md -1---i11-114
mos-hoorly--i 488
md modmmottllyy 18
and quuarterrly--
m4 woollyyy-11
m& yearly-6
resforrant-300
saugeenday-D... DD-30
saugeenday-M-u M-12
smggrenday-ye W-S
solor solar-10T-48
solar-
molr-1101-720
wobr--11-48
solar-t aN0
molur-11-73) -200
solur-Wys W
us birthts----00
me- ne-birth-M--12 M-12
us birthe W-88
solur-D-a
34


Reviews:
[{"review_id": "TBS3zDUKzh", "reviewer": "Reviewer_QGqS", "summary": "This paper investigates fine-tuning/post-training of time series foundation models. Rather than using pre-trained TSFMs for zero-shot forecasting directly, they propose to specialize a base model into different groups (can be flexibly defined, but the paper experimented with domains and frequency groupings). Given a group of models, a single forecast can be obtained either by model selection, or model ensembling. Experiments are done to identify the best approach. Results show that an ensemble of small models are on par with much larger models.", "questions": "* if model selection/ensembling is selected using the first evaluation window, how are the metrics for this first evaluation window obtained without using the ground truth values? Shouldn't the window before the first evaluation window be used for selecting the model combination method instead?\n* Is the same model combination selection method also used for other experiments, e.g. scaling plot?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "This is a very interesting investigation into the capabilities of time series foundation models. Paper is well written and well positioned in the literature. The paper proposes an innovative method to utilize pre trained time series models, and performs extensive experiments, yielding useful insights.", "weaknesses": "* Scaling results are not strong. There are likely too few data points across the scaling axis to reliably trust the results. For frequency specialists, 4m models are consistently stronger than tiny (9m) models. There is no clear indication we should expect the regression to extrapolate. This is also reflected in the main results figure, leading to some confusion when one is reading about scaling behavior but in the main results, 4m is clearly stronger than 9m.\n* It is unclear what \"active parameters\" refers to in this setting. From my reading I understand active parameters for an ensemble of 4m models to be 4m. However, I would think that for model ensembles, the number of active parameters should be ensemble size * model size\n* Omits more recent models with much stronger results, e.g. Toto, Tirex, TimesFM 2.5", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "eI7xEY7mIw", "reviewer": "Reviewer_Z6Sb", "summary": "The paper investigates whether building portfolios of smaller pretrained models can match or surpass large monolithic time-series foundation models (e.g., Chronos-Bolt) under constrained compute budgets.", "questions": "See the weaknesses", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Elegant and Compute-Efficient Framework. The proposed portfolio of specialists provides a pragmatic alternative to large models, achieving substantial compute savings. The post-training step is extremely lightweight (only 1K updates), making the approach appealing for practitioners with limited resources.\n- Strong Empirical Insights. The work provides valuable empirical findings: ensemble diversity is critical for the forecasting tasks and ensembling yields a favorable accuracy-compute trade-off close to the efficiency frontier.\n- Potential for Broader Impact. By reframing scaling from “bigger models” to “smarter test-time computation,” the study could inform future directions in efficient time-series foundation modeling and resource-adaptive forecasting systems.", "weaknesses": "1. **Validation data at test time is impractical and misaligned with typical deployment.** The method assumes the availability of non-trivial validation data at test time to select or weight experts. In realistic forecasting deployments, only inputs (and possibly a short warm-up context) are available. Labeled validation slices are rare or prohibitively small. This creates a distributional and resource gap between the paper’s evaluation protocol and common practice.\n2. **Zero-shot and OOD forecasting ability are not validated.** Although the paper positions the approach against “zero-shot” foundation models, the core mechanism, model selection or greedy ensembling using validation losses, consumes task labels and thus departs from zero-shot forecasting. **The zero-shot experiments are highly recommended, especially when the test data have a significant domain shift with the validation data.**\n3. **Greedy ensemble selection risks overfitting to the validation window.** The ensemble is built by iteratively adding experts to minimize validation error. Repeated selection against a small, local window is prone to selection bias and variance inflation, particularly with correlated experts. We may need further experiments to show that this method has resistance against overfitting\n4. **Scaling-law claims rest on a narrow 1–9M parameter band.** Figure 3 supports “comparable scaling to single models” only in a small-model regime. Extrapolating this trend to ≥100M/1B parameters is not justified without additional anchor points, thus being unconving.\n5. **Expert construction is largely manual and metadata-dependent.** Experts are partitioned by frequency/sector categories, and those categories are highly dependent on the manual design. Hence, it is unclear how this manually designed expert construction performs on the unseen domain/tasks.\n6. **Experimental gains are modest and sometimes negative.** Across benchmarks, the ensemble often only slightly outperforms the best single expert. On Chronos, the 4M-parameter ensemble even underperforms the top single model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VvOt71uAmO", "reviewer": "Reviewer_2k41", "summary": "The authors proposed chroma to reframes “bigger is better” for time-series FMs by replacing one large monolith with a portfolio of small specialists. Starting from a generalist base model, the authors fine-tune disjoint subsets to create diverse experts, then use model selection or ensembling at inference. Across standard benchmarks, this portfolio matches large models with far fewer parameters and is more compute-efficient than test-time fine-tuning, suggesting a simple, scalable alternative that may generalize beyond forecasting.", "questions": "Questions:\n1. post-training helps to reduce training cost but I wonder if it reduces the diversity. Because if we train the specialist models from scratch, the specialist models may learn more on the unique pattern in different subsets. Particularly, if we train from scratch, could we get better performance? Or only similar performance but with significantly more computational cost. \n2. How to determine the validation sets, especially if we don't have enough data? Additionally, when we only want to make a forecast for a single time series instead of a whole evaluation dataset, how we can do the forecast combination at test-time? \n3. I notice that 1K gradient steps seems sufficient for the fine-tuning (i.e., post-training) and further increases this doesn't help. I wonder if this is because of the data amount used in post-training or any other possible reasons? \n4. I wonder if the scaling behavior holds if we further increase the model size to 200M (the size of chronos-bolt-base).", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Strengths:\n1. The problem is well motivated and of interest in the community.\n2. The empirical study is comprehensive. The authors investigate different portfolio design and the combination methods. One thing I personally like is that, from Table 2, the findings of simply using ensemble cannot help improve performance.\n3. Technical contributions are somewhat limited. But I feel the problems is still worth investigated from this perspective.", "weaknesses": "Weakness:\n1. The proposed ensemble method is a bit hand-wavy. It shows the good performance but doesn't provide insights on why/how it works. Althrough some ablation studies are provided, I still wonder, in general, what kind of ensembling methods can help to improve the performance and any principles behind that. If we have this kind of insights, that would bring this work to a better level. \n2. I am not sure if the conclusion can extend to other model architecture. In this paper, it only uses chronos-bolt which is encoder-decoder architeture. But other time series foundation models use decoder-only (e.g., TimesFM) or maksed-encoder (e.g., Moirai). I think that if only encoder-decoder architecture is considered, maybe it is better to adjust some of the claims and the conclusion to limit the model type.\n3. For time series foundation models with MoE, there are some other works. For examples, time-moe and moirai-moe. May also consider has some discussions on these in related works. \n\nReferences \n\n[1] Time-MoE: Billion-Scale Time Series Foundation Models with Mixture of Experts. ICLR 2025\n\n[2] Moirai-MoE: Empowering Time Series Foundation Models with Sparse Mixture of Experts. ICML 2025", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gxwcIiVRwa", "reviewer": "Reviewer_tn7T", "summary": "This paper presents Chroma and framework for having portfolio of specialized models originated by fine-tuning a general purpose model for time-series forecasting. At inference time, this frameworks employs a subset of the models based a Greedy Ensemble Weighting algorithm. The paper provides a lot of empirical evaluations ranging from performance and ablation to test-time compute (in terms of FLOPs) and scaling. The results (including statistical analysis and bias-variance tradeoff) suggest the utility of the proposed framework in terms of time saving while maintaining performance.", "questions": "- How many times each experiment was repeated? I recognize the paper is quite strong in terms of experiments but providing confidence interval or standard error can make the message stronger specially for key experiments\n\n- Did you perform p-value correction for multiple hypothesis testing? if so, what was the employed method?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper is strong in terms of empirical contributions to support claims \n- I also found the paper strong in terms of reproducibility \n- Overall, I think the paper provides insights for the community that could be interesting therefore, I am in favour of acceptance.", "weaknesses": "- I think the paper lacks technical novelty because training a source model and then fine-tuning it for different target tasks has been round in transfer learning (particularly domain adaptation) for a long time. Similarly, the studied selection strategies are also known. Therefore, it seems the main contribution of Chroma is the fine-tuning step as opposed to training from scratch  (comparing figure 1 b and c) but that step according to table 2 did not have an impact on the performance as opposed to the ensembling which makes the technical contributions very limited. \n\n- This is not a weakness per se but I think it will be beneficial to show that the presented results are not only specific to T5 architecture \n\n- I think the paper can benefit from more discussion around G.5 results. Particularly to identify for what type of problems this approach works better and where we should train from scratch. Currently, these results are buried in the appendix but a discussion around them can clarify the applicability domain of the presented framework better.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)