SlowGuess/NeurIPS_2023_OCR huggingface dataset:
For all benchmarks, we set the look-back window length at 60 and vary the forecast horizon from  $H = 1,24,48$ . We split the data into two phases: warm-up and online training, with a ratio of 25:75. We follow the optimization details outlined in [Zhou et al., 2021] and utilize the AdamW optimizer [Loshchilov and Hutter, 2017] to minimize the mean squared error (MSE) loss. To ensure a fair comparison, we set the epoch and batch sizes to one, which is consistent with the online learning setting. We make sure that all baseline models based on the TCN backbone use the same total memory budget as FSNet, which includes three times the network sizes: one working model and two exponential moving averages (EMAs) of its gradient. For ER, MIR, and DER++, we allocate an episodic memory to store previous samples to meet this budget. For transformer backbones, we find that a large number of parameters do not benefit the generalization results and always select the hyperparameters such that the number of parameters for transformer baselines is fewer than that for FSNet. In the warm-up phase, we calculate the mean and standard deviation to normalize the online training samples and perform hyperparameter cross-validation. For different structures, we use the optimal hyperparameters that are reported in the corresponding paper.
License. All the assets (i.e., datasets and the codes for baselines) we use include an MIT license containing a copyright notice and this permission notice shall be included in all copies or substantial portions of the software.
Environment. We conduct all the experiments on a machine with an Intel R Xeon (R) Platinum 8163 CPU @ 2.50GHZ, 32G RAM, and four Tesla-V100 (32G) instances. All experiments are repeated 3 times with different seeds.
Metrics Because learning occurs over a sequence of rounds. At each round, the model receives a look-back window and predicts the forecast window. All models are commonly evaluated by their accumulated mean-squared errors (MSE) and mean-absolute errors (MAE), namely the model is evaluated based on its accumulated errors over the entire learning process.
C.3 Baseline details
We present a brief overview of the baselines employed in our experiments.
First, OnlineTCN adopts a conventional TCN backbone [Zinkevich, 2003] consisting of ten hidden layers, each layer containing two stacks of residual convolution filters.
Secondly, ER [Chaudhry et al., 2019] expands on the OnlineTCN baseline by adding an episodic memory that stores previous samples and interleaves them during the learning process with newer ones.
Third, MIR [Aljundi et al., 2019a] replaces the random sampling technique of ER with its MIR sampling approach, which selects the samples in the memory that cause the highest forgetting and applies ER to them.
Fourthly, DER++ [Buzzega et al., 2020] enhances the standard ER method by incorporating a knowledge distillation loss on the previous logits.
Finally, TFCL [Aljundi et al., 2019b] is a task-free, online continual learning method that starts with an ER process and includes a task-free MAS-styled [Aljundi et al., 2018] regularization.
All the ER-based techniques utilize a reservoir sampling buffer, which is identical to that used in [Pham et al., 2023].
C.4 Hyper-parameters
For the hyper-parameters of FSNet and the baselines mentioned in Section C.3, we follow the setting in [Pham et al., 2023]. Besides, we cross validate the hyper-parameters on the ETTh2 dataset and use them for the remaining ones. In particular, we use the following configuration:
<sup>5</sup>https://www.ncei.noaa.gov/data/local-climatological-data/
Table 6: Standard deviations of the metrics in Table. 2 and Table. 3.
Method / H	MSE
ETTH2	ETTm1	WTH	ECL
1	24	48	1	24	48	1	24	48	1	24	48
Informer	1.370	2.254	2.088	0.088	0.035	0.020	0.005	0.003	0.009
OnlineTCN	0.011	0.017	0.148	0.003	0.002	0.002	0.001	0.001	0.001	0.019	0.077	0.122
TFCL	0.030	0.005	0.279	0.004	0.006	0.010	0.002	0.001	0.004	0.047	0.338	0.253
ER	0.018	0.007	0.141	0.005	0.003	0.004	0.001	0.001	0.009	0.034	0.236	0.320
MIR	0.019	0.017	0.130	0.005	0.005	0.006	0.002	0.001	0.009	0.037	0.261	0.143
DER++	0.022	0.024	0.143	0.003	0.002	0.003	0.001	0.001	0.011	0.027	0.072	0.146
FSNet	0.018	0.014	0.128	0.003	0.002	0.003	0.001	0.001	0.001	0.021	0.096	0.105
Time-TCN	0.020	0.010	0.189	0.004	0.004	0.005	0.001	0.001	0.001	0.033	0.130	0.232
PatchTST	0.022	0.010	0.183	0.005	0.005	0.007	0.002	0.001	0.007	0.039	0.167	0.239
OneNet-TCN	0.015	0.012	0.104	0.003	0.003	0.003	0.001	0.001	0.007	0.025	0.114	0.152
OneNet	0.015	0.014	0.100	0.003	0.002	0.003	0.001	0.001	0.005	0.021	0.086	0.099
Method / H	MAE
ETTH2	ETTml	WTH	ECL
1	24	48	1	24	48	1	24	48	1	24	48
Informer	0.043	0.102	0.091	0.060	0.023	0.014	0.005	0.003	0.008
OnlineTCN	0.007	0.002	0.016	0.002	0.002	0.003	0.002	0.077	0.122	0.002	0.009	0.011
TFCL	0.003	0.003	0.024	0.008	0.005	0.008	0.002	0.001	0.006	0.011	0.019	0.008
ER	0.017	0.006	0.013	0.009	0.002	0.004	0.002	0.001	0.005	0.011	0.017	0.014
MIR	0.018	0.005	0.012	0.009	0.004	0.005	0.002	0.001	0.005	0.013	0.013	0.012
DER++	0.015	0.004	0.015	0.007	0.002	0.002	0.002	0.001	0.007	0.002	0.013	0.014
FSNet	0.009	0.005	0.012	0.004	0.002	0.002	0.001	0.001	0.001	0.001	0.011	0.011
Time-TCN	0.009	0.004	0.018	0.006	0.003	0.005	0.002	0.026	0.044	0.008	0.009	0.011
PatchTST	0.013	0.005	0.016	0.009	0.004	0.006	0.002	0.001	0.005	0.012	0.010	0.011
OneNet-TCN	0.017	0.005	0.013	0.008	0.003	0.004	0.002	0.001	0.006	0.009	0.009	0.013
OneNet	0.014	0.005	0.013	0.007	0.003	0.003	0.002	0.001	0.004	0.005	0.007	0.012
- Learning rate  $3e - 3$  on Traffic and ECL and  $1e - 3$  for other datasets. Learning rate  $1e - 2$  for the EGD algorithm and  $1e - 3$  for the offline reinforcement learning block, where the selection scope is  $\{1e - 3, 3e - 3, 1e - 2, 3e - 2\}$ .
- Number of hidden layers 10 for both cross-time and cross-variable branches, where the selection scope is  $\{6,8,10,12\}$ .
- Adapter's EMA coefficient 0.9, Gradient EMA for triggering the memory interaction 0.3, where the selection scope is  $\{0.1, 0.2, \dots, 1.0\}$ .
- Memory triggering threshold 0.75, where the selection scope is  $\{0.6, 0.65, 0.7 \dots, 0.9\}$ .
- Episodic memory size: 5000 (for ER, MIR, and DER++) 50 (for TFCL).
C.5 Additional Numerical Results
Additional forecasting results. In this section, we analyze the performance of different forecasting methods on four datasets, ECL, WTH, ETTh2, and ETTm1, with various starting points, as shown in Figure 8, Figure 9, Figure 10, and Figure 11, respectively. For the last three datasets, all methods produce similar results that can capture the underlying time series patterns, and the performance differences are not significant. However, when it comes to the ECL dataset, we observe that almost all baselines exhibit poor forecasting results at the onset of the concept shift (time step 2500). As we provide more instances, the performance of these methods improves, as evidenced by the cumulative loss curves in Figure 7 and Figure 6.
Ablation studies of hyper-parameters. We conduct detailed ablation studies about model layers, learning rate, and model dimension here. Taking into account the learning rate for the two-branch framework, the learning rate for the long-term weight, and the learning rate for the short-term weight:  $lr, lr_{\mathbf{w}}, lr_{\mathbf{b}}$ , as shown in Table 10 (left), the impact of the learning rate on dual-stream
Table 7: MSE and MAE of various adaptation methods. H: forecast horizon. OneNet-TCN+Patch is the mixture of TCN, Time-TCN, and PatchTST.
Metric	Method	ETTH2	ETTm1	WTH	ECL	Avg
1	24	48	1	24	48	1	24	48	1	24	48
MSE	TCN	0.502	0.830	1.183	0.214	0.258	0.283	0.206	0.308	0.302	3.309	11.339	11.534	2.522
Time-TCN	0.491	0.779	1.307	0.093	0.281	0.308	0.158	0.311	0.308	4.060	5.260	5.230	1.549
PatchTST	0.362	1.622	2.716	0.083	0.427	0.553	0.162	0.372	0.465	2.022	4.325	5.030	1.512
OneNet-TCN	0.411	0.772	0.806	0.082	0.212	0.223	0.171	0.293	0.310	2.470	4.713	4.567	1.253
OneNet-TCN+Patch	0.355	0.844	1.120	0.079	0.239	0.255	0.163	0.298	0.314	2.172	4.142	4.149	1.178
MAE	TCN	0.436	0.547	0.589	0.085	0.381	0.403	0.276	0.367	0.362	0.635	1.196	1.235	0.543
Time-TCN	0.425	0.544	0.636	0.211	0.395	0.421	0.204	0.378	0.378	0.332	0.420	0.438	0.399
PatchTST	0.341	0.577	0.672	0.186	0.471	0.549	0.200	0.393	0.459	0.224	0.341	0.375	0.399
OneNet-TCN	0.374	0.511	0.543	0.191	0.319	0.371	0.221	0.345	0.356	0.411	0.513	0.534	0.391
OneNet-TCN+Patch	0.338	0.513	0.552	0.184	0.360	0.381	0.217	0.351	0.381	0.297	0.423	0.457	0.371
networks is quite significant. The optimal learning rate varies for each dataset, but we can see that for each dataset, the optimal learning rate is generally within the range of  $[1e - 4,1e - 2]$ .  $lr_{\mathbf{w}}$  has a relatively small impact on the final performance of the model. On the contrary, the offline-RL module determines whether the weights can quickly adapt to the new distribution, which has a greater impact on the final performance. In terms of model parameters, # Layers,  $d_m$ , and  $d_{head}$ , all three have a significant impact on the performance of the model. A small model may not be able to fit the training data, but a model that is too large increases the risk of overfitting, so each dataset has an optimal model size. However, in this paper, we use the same hyperparameters for all datasets to simplify the complexity of training and model selection. Specifically, we set  $lr = 1e - 3, lr_{\mathbf{w}} = 1e - 2, lr_b = 1e - 3, \# layers = 10, d_m = 64, d_{head} = 320$ .
C.6 Ensembling more than two networks.
In the main paper, we verify the effectiveness of ensembling two branches with different model biases. Here, we show that the proposed OneNet framework enables us to incorporate more branches and the OCP block can fully utilize the benefit of each branch. As shown in Table 7, incorporating PatchTST to OneNet-TCN will further reduce the forecasting results during online forecasting.
C.7 More visualization results and Convergence Analysis of Different Structures
As shown in Figure 6 and Figure 7, ETTh2 and ECL datasets pose the greatest challenge to all models due to the sharp peaks in their loss curves. When the forecasting window is short, OneNet outperforms all baselines by a significant margin on all datasets. When the forecasting window is extended to  $H = 48$ , FSNet is comparable to OneNet in the first three datasets. However, when concept drift occurs in the ECL dataset, all baselines experience a drastic increase in their cumulative MSE, except OneNet, which maintains a low MSE. Furthermore, the initialized MSE error of OneNet is consistently lower than that of all baselines, thanks to the two-stream structure of OneNet. For instance, in Figure 6(b) and Figure 6(f), OneNet demonstrates a significantly lower MSE than baselines when the number of instances is less than 100.
C.8 Online forecasting results with delayed feedback
As illustrated in Section 2, this paper adopts the same setting as FSNet [Pham et al., 2023], where the true values of each time step are revealed to improve the performance of the model in subsequent rounds. However, in real-world applications, the true values of the forecast horizon  $H$  may not be available until  $H$  rounds later, which is known as online forecasting with delayed feedback. This setting is more challenging because the model cannot be retrained at each round and we can only train the model per  $H$  round. Tables 8 and 9 show the cumulative performance considering MSE and MAE, respectively. As expected, all methods perform worse with delayed feedback than under the traditional online forecasting setting. Notably, the state-of-the-art method FSNet is shown to be sensitive to delayed feedback, particularly when  $H = 48$ , where it is even inferior to a simple TCN baseline on some datasets. In contrast, our proposed method OneNet significantly outperforms all continual learning baselines across different datasets and delayed forecast horizons.
Table 8: MSE of various adaptation methods with delayed feedback. H: forecast horizon. OneNet-TCN is the mixture of TCN and Time-TCN, and OneNet is the mixture of FSNet and Time-FSNet.
Method / H	ETTH2	ETTm1	WTH	ECL	Avg
1	24	48	1	24	48	1	24	48	1	24	48
OnlineTCN	0.502	5.871	11.074	0.214	0.410	0.535	0.206	0.429	0.504	3.309	9.621	24.159	4.736
ER	0.508	5.461	17.329	0.086	0.367	0.498	0.180	0.373	0.435	2.579	8.091	17.700	4.467
DER++	0.508	5.387	17.334	0.083	0.347	0.465	0.174	0.369	0.431	2.657	7.878	17.692	4.444
FSNet	0.466	5.765	11.907	0.085	0.383	0.502	0.162	0.335	0.411	3.143	8.722	27.150	4.919
OneNet-TCN	0.411	2.639	4.995	0.082	0.287	0.382	0.171	0.341	0.433	2.470	4.809	6.252	1.939
OneNet	0.380	2.064	4.952	0.082	0.332	0.351	0.156	0.323	0.394	2.351	4.984	6.226	1.883
Table 9: MAE of various adaptation methods with delayed feedback. H: forecast horizon. OneNet-TCN is the mixture of TCN and Time-TCN, and OneNet is the mixture of FSNet and Time-FSNet.
Method / H	ETTH2	ETTm1	WTH	ECL
1	24	48	1	24	48	1	24	48	1	24	48
OnlineTCN	0.436	1.109	1.348	0.085	0.511	0.548	0.276	0.459	0.508	0.635	0.783	1.076
ER	0.376	0.976	1.651	0.197	0.456	0.525	0.244	0.421	0.459	0.506	0.595	0.772
DER++	0.375	0.967	1.644	0.192	0.443	0.508	0.235	0.415	0.456	0.421	0.591	0.758
FSNet	0.368	0.983	1.494	0.191	0.468	0.502	0.216	0.394	0.453	0.472	0.827	1.391
OneNet-TCN	0.374	0.772	0.951	0.191	0.387	0.417	0.221	0.389	0.461	0.411	0.381	0.451
OneNet	0.348	0.684	0.916	0.187	0.428	0.430	0.201	0.381	0.436	0.254	0.387	0.444
Table 10: Results of different OneNet's hyper-parameter configurations on the benchmarks  $(H = 48)$ .  $lr, lr_{\mathbf{w}}, lr_{\mathbf{b}}$  are the learning rate for the two-branch framework, the learning rate for the long-term weight, and the learning rate for the short-term weight. # Layers is the number of layers of the two branches of OneNet.  $d_m, d_{head}$  is the hidden dimension and the output dimension of the encoders, respectively.
Hyper-Parameter	Value	MSE	Hyper-Parameter	Value	MSE
ETTh2	ETTm1	WTH	ECL	ETTh2	ETTm1	WTH	ECL
lr	1.00E-01	-	-	-	-	# Layers	6	0.632	0.114	0.203	2.402
1.00E-02	0.585	0.152	0.171	3.128	8	0.661	0.101	0.201	2.289
1.00E-03	0.656	0.111	0.196	2.516	10	0.609	0.108	0.200	2.201
1.00E-04	2.994	0.464	0.331	4.949	12	0.652	0.115	0.200	2.328
lrw	1.00E-01	0.619	0.108	0.202	2.177	dm	16	0.679	0.122	0.223	2.201
1.00E-02	0.609	0.108	0.205	2.184	32	0.612	0.116	0.210	2.810
1.00E-03	0.608	0.108	0.201	2.197	64	0.609	0.108	0.200	2.311
1.00E-04	0.607	0.108	0.201	2.197	160	0.619	0.108	0.200	2.141
lrb	1.00E-01	0.899	0.134	0.221	2.499	dhead	80	0.741	0.136	0.219	2.468
1.00E-02	0.876	0.112	0.197	2.372	160	0.600	0.112	0.214	2.364
1.00E-03	0.656	0.111	0.196	2.371	320	0.609	0.108	0.201	2.184
1.00E-04	0.643	0.111	0.196	2.362	500	0.571	0.104	0.182	2.182
Table 11: Ablation studies of the variable independence and frequency domain augmentation, where the metric is MSE. FEDformer-F uses frequency-enhanced blocks with Fourier transform, and FEDformer-W uses frequency-enhanced blocks with Wavelet transform. Time-TCN is the variable independence version of TCN.
Method	Online	ETTH2	ETTm1	WTH	ECL	Avg
1	24	48	1	24	48	1	24	48	1	24	48
FEDformer-F	X	1.922	3.045	4.016	0.922	1.003	1.821	3.544	2.344	1.179	43.852	37.802	37.377	11.569
✓	1.912	3.013	3.951	0.372	0.633	0.586	2.196	0.376	0.562	39.243	35.975	36.092	10.409
FEDformer-W	X	1.816	3.070	3.996	2.275	3.784	2.662	1.220	1.211	1.431	41.791	37.236	37.210	11.475
✓	1.798	2.993	1.623	0.235	0.451	0.516	0.717	0.962	0.372	21.387	24.600	27.640	6.941
TCN	X	27.060	27.760	26.320	2.240	12.170	10.880	0.290	0.480	0.580	538.000	546.000	552.000	145.315
✓	0.530	0.930	0.910	0.130	0.310	0.250	0.300	0.348	0.348	3.010	11.680	10.800	2.462
Time-TCN	X	4.530	7.840	11.017	0.097	0.800	11.017	0.162	0.344	0.429	47.900	48.660	67.150	15.020
✓	0.480	0.780	0.867	0.090	0.280	0.310	0.300	0.310	0.309	4.010	5.220	5.210	1.550
Table 12: Comparison of existing forecasting structures, including TCN [Bai et al., 2018], FEDformer [Zhou et al., 2022b], PatchTST [Nie et al., 2023], Dlinear [Zeng et al., 2023], Nlinear [Zeng et al., 2023], TS-Mixer [Chen et al., 2023], and CrossFormer [Zhang and Yan, 2023].
Method	Online	ETTH2	ETTm1	WTH	ECL	Avg
1	24	48	1	24	48	1	24	48	1	24	48
FEDformer-F	X	1.922	3.045	4.016	0.922	1.003	1.821	3.544	2.344	1.179	43.852	37.802	37.377	11.569
✓	1.912	3.013	3.951	0.372	0.633	0.586	2.196	0.376	0.562	39.243	35.975	36.092	10.409
FEDformer-W	X	1.816	3.070	3.996	2.275	3.784	2.662	1.220	1.211	1.431	41.791	37.236	37.210	11.475
✓	1.798	2.993	1.623	0.235	0.451	0.516	0.717	0.962	0.372	21.387	24.600	27.640	6.941
PatchTST	X	0.427	2.090	3.290	0.083	0.433	0.570	0.163	0.375	0.467	2.030	4.395	5.101	1.619
✓	0.362	1.622	2.716	0.083	0.427	0.553	0.162	0.372	0.465	2.022	4.325	5.030	1.512
Crossformer	X	23.270	28.904	29.218	0.400	1.433	1.691	0.146	0.327	0.426	469.260	475.490	478.270	125.736
✓	9.873	2.856	5.772	0.096	0.356	0.370	0.149	0.317	0.359	68.300	92.500	94.790	22.978
TCN	X	27.060	27.760	26.320	2.240	12.170	10.880	0.290	0.480	0.580	538.000	546.000	552.000	145.315
✓	0.530	0.930	0.910	0.130	0.310	0.250	0.300	0.348	0.348	3.010	11.680	10.800	2.462
Time-TCN	X	4.530	7.840	1.300	0.097	0.800	1.030	0.162	0.344	0.429	47.900	48.660	67.150	15.020
✓	0.480	0.780	1.300	0.090	0.280	0.310	0.300	0.310	0.309	4.010	5.220	5.210	1.550
DLinear	X	2.91	10.25	7.53	0.538	1.461	1.233	0.266	0.462	0.542	12.03	51.28	58.46	12.247
✓	2.44	9.24	6.91	0.46	1.3	1.12	0.262	0.459	0.541	6.69	27.82	31.54	7.399
NLinear	X	0.424	50.15	49.52	0.09	4.02	4.13	0.171	1.07	1.08	2.14	930	929	164.316
✓	0.369	50.24	49.6	0.089	4.035	4.141	0.171	1.053	1.064	2.135	930	930	164.408
TS-Mixer	X	1.968	3.525	4.88	0.335	0.726	0.855	0.255	0.429	0.503	11.16	30.93	44.68	8.354
✓	0.78	2.05	3.060	0.219	0.550	0.660	0.237	0.413	0.482	2.798	4.983	5.764	1.833
C.9 The effect of variable independence and frequency domain augmentation
As shown in Table 11, we observe that frequency-enhanced blocks, which use the wavelet transform, offer greater robustness to the Fourier transform. FEDformer outperforms TCN in terms of generalization, but online adaptation has a limited impact on performance, similar to other transformer-based models. Notably, we find that variable independence is crucial for model robustness. By convolving solely on the time dimension, independent of the feature channel, we significantly reduce MSE error compared to convolving on the feature channel, regardless of whether online adaptation is applied.
C.10 Comparison of existing forecasting structures.
Results are shown in Table 12. Considering the average MSE on all four datasets, all transformer-based models and Dlinear are better than TCN and Time-TCN. However, with online adaptation, the forecasting error of TCN structures is reduced by a large margin and is better than DLinear and FEDformer. Specifically, we show that the current transformer-based model (PatchTST [Nie et al., 2023]) demonstrates superior generalization performance than the TCN models even without any online adaptation, particularly in the challenging ECL task. However, we also noticed that PatchTST remains largely unchanged after online retraining. In contrast, the TCN structure can quickly adapt to the shifted distribution, and the online updated TCN model prefers a better forecasting error than the adapted PatchTST on the first three data sets. Therefore, it is promising to combine the strengths of both structures to create a more robust and adaptable model that can handle shifting data distributions better.
Algorithm 1 Training and inference algorithm of OneNet
1: Input: Historical multivariate time series  $\mathbf{x} \in \mathbb{R}^{M \times L}$  with  $M$  variables and length  $L$ , the forecast target  $\mathbf{y} \in \mathbb{R}^{M \times H}$ , where we omit the variable index and time step index for simplicity.
2: Initialize a cross-time forecaster  $f_{1}$ , a cross-variable forecaster  $f_{2}$  with corresponding prediction head, long-term weight  $\mathbf{w} = [0.5, 0.5]$ , short term learning block  $f_{rl}: \mathbb{R}^{H \times M \times 3} \to \mathbb{R}^2$ , and step size  $\eta$  for long-term weight updating.
3: Get prediction results from two forecasters.
4:  $\tilde{\mathbf{y}}_1\in \mathbb{R}^{M\times H} = f_1(\mathbf{x})$  // Prediction result from the cross-time forecaster.
5:  $\tilde{\mathbf{y}}_2\in \mathbb{R}^{M\times H} = f_2(\mathbf{x})$  // Prediction result from the cross-variable forecaster.
6: Get combination weight from the OCP block.
7:  $\mathbf{b} = f_{rl}(w_1\tilde{\mathbf{y}}_1\otimes w_2\tilde{\mathbf{y}}_2\otimes \mathbf{y})$  // Calculate the short term weight.
8:  $\tilde{w}_i = (w_i + b_i) / \left(\sum_{i=1}^d (w_i + b_i)\right) // \text{Calculate the normalized weight.}$
9:  $\tilde{\mathbf{y}} = w_{1}*\tilde{\mathbf{y}}_{1} + w_{2}*\tilde{\mathbf{y}}_{2}$  // The final prediction result.
0: Update the long/short term weight.
11:  $w_{i} = w_{i}\exp (-\eta \parallel \tilde{\mathbf{y}}_{i} - \mathbf{y}\parallel^{2}) / \left(\sum_{i = 1}^{2}w_{i}\exp (-\eta \parallel \tilde{\mathbf{y}}_{i} - \mathbf{y}\parallel^{2})\right)$
12:  $f_{rl}\gets \mathrm{Adam}(f_{rl},\mathcal{L}(w_1*\tilde{\mathbf{y}}_1 + w_2*\tilde{\mathbf{y}}_2,\mathbf{y}))$  //Parameters such as learning rate are omitted.
13: Update the two forecasters.
14:  $f_{1}\gets \mathrm{Adam}(f_{1},\mathcal{L}(\tilde{\mathbf{y}}_{1},\mathbf{y}))$ $f_{2}\gets \mathrm{Adam}(f_{2},\mathcal{L}(\tilde{\mathbf{y}}_{2},\mathbf{y}))$
(d) ECL
(e) ETTh2.
(f) ETTm1.
(g) WTH.
(h) ECL
Figure 6: Evolution of the cumulative MSE loss during training with forecast window  $H = 1$  (a, b, c, d) and  $H = 48$  (e, f, g, h). 27
(f)ETTml.
(h) ECL
Figure 7: Evolution of the cumulative MAE loss during training with forecasting window  $H = 1$  (a,b,c,d) and  $H = 48$  (e,f,g,h). 28
(a) Channel 1.
(b) Channel 2.
(c) Channel 3.
(d) Channel 1.
(e) Channel 2.
(f) Channel 3.
(g) Channel 1.
(h) Channel 2.
(i) Channel 3.
(j) Channel 1.
(k) Channel 2.
(1) Channel 3.
Figure 8: Visualization of the model's prediction throughout the online learning process in the ECL dataset. We focus on a short horizon of 50 time steps and the start prediction time is from 5000 (a,b,c), 7500 (d,e,f), 10000 (g,h,i), and 12500 (j,k,l) respectively.
(1) Channel 3.
Figure 9: Visualization of the model's prediction throughout the online learning process on the WTH dataset. We focus on a short horizon of 50 time steps and the start prediction time is from 5000 (a,b,c), 7500 (d,e,f), 10000 (g,h,I), and 12500 (j,k,l), respectively.
(1) Channel 3.
Figure 10: Visualization of the model's prediction throughout the online learning process in the ETTh2 data set. We focus on a short horizon of 50 time steps and the start prediction time is from 2500 (a,b,c), 5000 (d,e,f), 7500 (g,h,i), and 10000 (j,k,l) respectively.
(1) Channel 3.
Figure 11: Visualization of the model's prediction throughout the online learning process on the ETTm1 dataset. We focus on a short horizon of 50 time steps and the start prediction time is from 2500 (a, b, c), 5000 (d, e, f), 7500 (g, h, i) and 10000 (j, k, l), respectively.