Response to Question 1

In this paper, we tackle the domain generalization (DG) scenario in which multiple source domains (i.e., $K \geq 2$) are utilized to make predictions for a distinct target domain. This contrasts with single domain generalization (SDG), which involves only one source domain. In this context, N-BEATS as presented in [46] lies on the SDG paradigm from a meta-learning standpoint. Hence, their experimental setup is not directly transferable to our DG context. Nevertheless, we have adapted the datasets from the N-BEATS study and its corresponding results have been added in Appendix G.4 and Table 11. The results demonstrate that Feature-aligned N-BEATS consistently outperform the original N-BEATS.

Response to Question 2

Thanks for pointing this out. According to your suggestion, we have plotted the forecasting and alignment losses during both training and validation steps in Figure 5 in Appendix D. This illustrates stable training process, indicating that the gradients are well-behaved during training.

Feature-aligned N-BEATS is a complicated architecture based on the doubly residual stacking principle with an additional feature-aligning procedure. To investigate the stability of training, we analyze the training and validation loss plots. Figure 5 indicates that the gradients are well-behaved during training. This stable optimization is regarded to the Lemma 3.4 presented in Section 3.1.

Response to Question 3

In Figure 1 (which is Figure 2 in revised version), we demonstrate 1) the effects of FA on the embedding space and 2) its sensitivity to the hyperparameter $\lambda$, with a focus on the alignment aspect (We typically used $\lambda$ values of 1 and 3 for N-BEATS-G due to their superior performance in varied experimental settings.). N-BEATS operates as a recurrent framework, with blocks in stacks capturing distinct temporal features in time series data, such as trends and seasonality. Our goal was to assess whether FA successfully narrows the gap between source domains concerning these diverse features. This involved analyzing all embedding spaces from every block across all stacks. In this regard, it becomes evident that the domains are closely aligned for most features. Although the difference between $\lambda = 1$ and $\lambda = 3$ appears subtle, indicating that N-BEATS-G with FA has low sensitivity to $\lambda$ variations}, we observed that $\lambda$ can be a critical factor in specific cases, such as with N-BEATS-I, as detailed in Appendix F (Figure 8).

Response to Weakness 1

We are grateful for the insightful suggestions, especially those aimed at improving our presentation. For ease of understanding, before starting the main section (Sections 2 and 3), we have signposted how the training objective with the corresponding algorithm will be presented,

The remainder of the article is organized as follows. In Section 2, we set the domain generalization problem in the context of time series forecasting, review the doubly residual stacking architecture of N-BEATS, and introduce the error analysis for domain generalization models. Section 3 is devoted to defining the marginal feature measures inspiring the stack-wise alignment, introducing the Sinkhorn divergence together with the corresponding representation learning bound, and presenting the training objective with the corresponding algorithm. In particular, Figure 1 therein illustrates the overall architecture of Feature-aligned N-BEATS. In Section 4, comprehensive experimental evaluations are provided. Section 5 concludes the paper. Other technical descriptions, visualized results, and ablation studies are given in Appendix.

In addition, we have contextualized $\alpha$, $\beta$, $\gamma$, $K$, $M$, and $L$ in 'Experimental details' in Section 4.

The lookback horizon, forecast horizon, and the number of source domains are set to be $\alpha=50$, $\beta=10$, and $K=3$, respectively (noting that it depends on the characteristics of source domains' datasets). Furthermore, the number of stacks and blocks, and the dimension of feature space are set to be $M=3$, $L=4$, and $\gamma=512$, respectively (noting that it is consistent with N-BEATS [45]).

Response to Weakness 2

We appreciate the suggestion for evaluating the potential increase in computational cost due to the introduction of FA. We have added a new row to Table 1, which presents the training time per iteration.

The updated table reveals that stack-wise alignment does increase the training time compared to N-BEATS. However, Table 2 demonstrates that SD-based alignment, the chosen regularizer, exhibits the most enhanced generalizability while incurring marginal computational cost. Please note that there is no difference in runtime between Feature-aligned N-BEATS and original N-BEATS during inference, since alignment is not applied during this stage.

Response to Weakness 3

We greatly appreciate the suggestion for future work and concur with its significance. We earnestly hope that our empirical findings regarding the potential of feature alignment adaptation in time series forecasting models will contribute valuable insights for future research.

Response to Comment 1

Thanks for this careful advice. We have revised this sentence to 'In particular, the proposed model is built upon a deep learning model which is N-BEATS [45, 46]'.

Response to Comment 2

Thanks for point out our mistake. We have replaced the notation for the expected risk to $R^T, R^k$. Furthermore, we denoted the notations before Proposition 2.1.

Furthermore, denote by $R^k(\cdot):{\cal H}\rightarrow \mathbb{R}$ and $R^T(\cdot):{\cal H}\rightarrow \mathbb{R}$ the expected risk under the source measures $\mathbb{P}^k$, $k=1,\dots,K$, and the target measure $\mathbb{P}^T$, respectively.

Response to Comment 3

Sorry for this mistake. We have revised it. Please refer to Definition 3.1 in Section 3.1.

Response to Comment 4

Following this suggestion, we have added the pointer in the caption of Table 1.

Response to Comment 5

To improve readability, we have bolded the best performance in each row of the all tables. We appreciate the suggestion, which significantly improves the clarity of our paper.

Response to Weakness 4

We appreciate your concern about the interpretability of our model. In this revision, we have provided a visualization (Figure 7) and additional explanation in Appendix F for interpretability analysis,

Figure 7 exhibits the interpretability of the proposed method by presenting the final output of the model and intermediate stack forecasts. N-BEATS-I and N-HiTS presented in Appendix A have interpretability. More specifically, N-BEATS-I explicitly captures trend and seasonality information using polynomial and harmonic basis functions, respectively. N-HiTS employs Fourier decomposition and utilizes its stacks for hierarchical forecasting based on frequencies. Preserving these core architectures during the alignment procedure, Feature-aligned N-BEATS still retains interpretability.

Response to Question 1

We consent on this advice. Conceptually, divergences such as Wasserstein Divergence (WD) [28], Sinkhorn Divergence (SD) [44], Maximum Mean Discrepancy (MMD) [30], and Kullback–Leibler (KL) Divergence [65] can be applied for the alignment loss. Our choice is supported by the experimental advantages of SD, which are 1) computational cost and 2) robustness on performance. It is empirically demonstrated in Table 2 in the main script and Table 6 in Appendix E. On top of that, SD enables us to leverage the theoretical properties of optimal transport problems (Lemma 3.4 and Theorem 3.6).

Response to Question 2

We highlight that under doubly residual stacking architecture, defining feature alignment is nontrivial, which is well-described in the 4-th paragraph of the Introduction in the revised script.

First, N-BEATS does not allow the feature alignment in a 'one-shot' unlike the aforementioned references. This is because it is a hierarchical multi-stacking architecture in which each stack consists of several blocks and is connected to each other by residual operations and feature extractions.

In this regard, we adopted a 'stack-wise' feature alignment strategy rather than 'naive' or 'block-wise' alignment. In particular, we believe that stack-wise alignment ensures wide feature coverage in the context of alignment and promotes efficient gradient descent. The N-BEATS framework, which employs a doubly residual stacking architecture, encodes diverse features across its blocks and stacks [45]. Our goal was to align these features by frequently calculating divergence (either block-wise or stack-wise) and incorporating this into the alignment loss. However, as [47] noted, overly frequent loss propagation can lead to issues of gradient vanishing or exploding. Consequently, we determined that 'stack-wise' alignment offers an optimal frequency for this process. A comparative analysis of block-wise and stack-wise alignment is detailed in Table 8 in Appendix G.1.

Response to Suggestion 2

We already clarified our contributions explicitly in the 7-th paragraph of the Introduction (Section 1),

To provide an informative procedure of stack-wise feature alignment, we introduce a concrete mathematical formulation of N-BEATS (Section 2), which enables to define the pushforward feature measures induced by the intricate residual operations and the feature extractions for each stack (Section 3.1). From this, we make use of theoretical properties of optimal transport problems to show a representation learning bound for the stack-wise feature alignment with the Sinkhorn divergence (Theorem 3.6), which justifies the feasibility of Feature-aligned N-BEATS. To embrace comprehensive domain generalization scenarios, we use real-world data and evaluate the proposed method under three distinct protocols based on the domain shift degree. We show that the model consistently outperforms other forecasting models. In particular, our method exhibits outstanding generalization capabilities under severe domain shift cases (Table 1). We further conduct ablation studies to support the choice of the Sinkhorn divergence in our model (Table 2).

Response to Weakness 1 and Suggestion 1

We appreciate the valuable suggestions for our presentation. In response, we have provided a simplified illustration of Feature-aligned N-BEATS in Figure 1 in Section 3 of the revised script for better accessibility (noting that its detailed version is provided in Figure 3 in Appendix A). Additionally, we have relocated some of the more detailed technical aspects to the appendix for improved readability. Furthermore, for ease of understanding, before starting the main section (Sections 2 and 3), we have signposted how the training objective with the corresponding algorithm will be presented,

The remainder of the article is organized as follows. In Section 2, we set the domain generalization problem in the context of time series forecasting, review the doubly residual stacking architecture of N-BEATS, and introduce the error analysis for domain generalization models. Section 3 is devoted to defining the marginal feature measures inspiring the stack-wise alignment, introducing the Sinkhorn divergence together with the corresponding representation learning bound, and presenting the training objective with the corresponding algorithm. In particular, Figure 1 therein illustrates the overall architecture of Feature-aligned N-BEATS. In Section 4, comprehensive experimental evaluations are provided. Section 5 concludes the paper. Other technical descriptions, visualized results, and ablation studies are given in Appendix.

We tried to lighten the exposition up in the main paper by putting some remarks (Remarks C.1 and C.2 in the revised script, which were Remarks 3.7 and 3.8 in the original script) in Appendix C.

Response to Weakness 2

We kindly ask to take a careful look at the case of CDG and ODG with MASE metric in Table 1. Since CDG represents the case where the source domain data are distinctive from one another, we can say that the representation learning through the alignment of distinctive source domains' feature measures can help to improve the original model's forecasting power. Furthermore, since the ODG represents that the source data and target data belong to the different super domains (that is, the source and target data are distinctive from each other), we can say that our FA-NBEATS improves the forecasting power on the (unknown) distinctive target sequential data (when domain shift is significant). For the case of IDG, as the source data and target data belong to the same super domain (that is, the source and target data are relatively not distinctive from each other), the improvement seems to be marginal. But this is quite intuitive because the model would not have much chance to learn domain invariant features through the similar types of the source domain and the model's prediction power at the not distinctive target domain could not be improved through the feature alignment of similar distributions. In this regard, we further emphasize that our experiment performances are verified in diverse domain generalization environments, and the performances' improvement also stands out.

Furthermore, our choice of SD over other divergences is substantiated by Table 2 in Section 4 and Table 6 in Appendix E for the following reasons: 1) While SD yields performance comparable to WD (noting that SD is an approximate of WD.), SD is more computationally efficient. 2) SD outperforms both MMD and KL (in the ODG context) in terms of forecasting, with MMD incurring higher computational costs compared to SD. 3) Despite KL being the most computationally efficient, Table 6 reveals its lack of robustness, as indicated by its frequent inability to produce a result, denoted as 'NaN'.

Response to Weakness 3 and Questions 3-4

We appreciate the suggestion for evaluating the potential increase in computational cost due to the introduction of FA. We have added a new row to Table 1, which presents the training time per iteration.

The updated table reveals that stack-wise alignment does increase the training time compared to N-BEATS. However, Table 2 demonstrates that SD-based alignment, the chosen regularizer, exhibits the most enhanced generalizability while incurring marginal computational cost. Please note that there is no difference in runtime between Feature-aligned N-BEATS and original N-BEATS during inference, since alignment is not applied during this stage.

I thank to authors for their responses and addressing raised concerns. Especially a clarifying response to W2, pointing out FA addresses out-of-domain generalization well in experiments, and adding runtimes to Table 1, brought more confidence in the practical usefulness of the proposed method.

Altogether, after having read remaining responses and revisions suggested, I increase my rating of the paper to 6 (weak accept). Thanks again for your contributions and good luck.

Response to Weakness

We sincerely appreciate this valuable advice. To underscore our specific contributions and further illustrate this point, we would like to emphasize the following:

• Under doubly residual stacking architecture, defining feature alignment is nontrivial, which is well-described in the 4-th paragraph of the Introduction in the revised script.

  First, N-BEATS does not allow the feature alignment in a ‘one-shot’ unlike the aforementioned references. This is because it is a hierarchical multi-stacking architecture in which each stack consists of several blocks and is connected to each other by residual operations and feature extractions.

  In this regard, we have put lots of effort into representing N-BEATS in a mathematical manner (noting that it does not appear in the original N-BEATS paper) in Section 2.2. This mathematical definition has allowed us to establish important theoretical foundations for learning invariance.

  Based on this formal representation, we could achieve Lipschitz continuity of marginal feature maps (Lemma 3.4). Hence, by leveraging this with theoretical properties of optimal transport problems, we could establish representation learning bounds (Theorem 3.6). Therefore, it is not a direct extension of [41] where only simple fully-connected layers are considered.

• We have carefully considered the optimal alignment frequency (Remark 3.3), resulting in the 'stack-wise' approach. This application is not simple but rather a rational design (see Figure 1 in the revised script), addressing the gradient flow challenges in recurrent models as discussed by [47] and leveraging the expressiveness characteristic of N-BEATS, as detailed by [45].

• We have found an effective divergence for stack-wise feature alignment by extensive qualitative analysis with respect to computational cost and robustness performance (Table 2); that is, Sinkhorn divergence. To demonstrate the generalizability and forecasting power of our approach, we have presented results across a wide range of scenarios, including IDG, CDG, and ODG, all with real-world data (Evaluation details, Section 4). These protocols represent novel contributions to the field of forecasting (Table 1).