Rebuttal to Reviewer 3GMe
(1) **Adding background knowledge in Appendix:** Preliminaries on state-space models, Kalman Filter, Granger Causality and how they are all interlinked to motivate our research problem is now mentioned in **Appendix A.1** and cross-referenced in line 71-72 of the main text.
(2) **Pseudocode:** We added pseudocode with detailed steps for implementation of the algorithm as **Appendix A.3** and cross-referenced it in line 156 of main text.
(3) **Beyond Kalman filter based models, Defining functions $f_c, f_a, f_{ML}$, Meaning of Models:** In **Appendix A.2**, we define the functions $f_c(.), f_a(.), f_{ML}(.)$ to clarify their roles in our framework. This is also cross-referenced in **line 154-55** for clarity.
In this context, a "model" refers broadly to any predictive structure that can be used for learning at client and server levels. Therefore advanced machine learning models like neural networks can substitute our choice of $f_c$, $f_{ML}$, $f_a$. However, this is the first work on federated granger causality. Hence, for the purpose of mathematical tractability and theoretical insights, we fixed our clients to be state space models and the ML function to be linear regression. Please refer to our "**Limitations and Future work**'" section in **Appendix A.9** for our vision on extension of this approach to advanced machine learning models.
(4) **Defining the centralized model and $C_{mm}$ before Table 1:** Centralized model is defined now from **line 190-194**, and $C_{mm}$ is defined in **line 181** in the revised paper. Table 1 has also moved to **line 195**.
(5) **Why are two losses considered separately instead of a generic finite-sum objective function?** In our paper, we adopt a vertical federated learning (VFL) framework (**line 101-104**). Unlike most VFL studies, which focus on a classification problem with categorical ground-truth labels, our work addresses a one-step-ahead prediction task for time-series measurements. Here, the "labels" are the future measurements at each client, which differ across clients.
In typical VFL settings, a single finite-sum objective function is feasible because the (1) ground-truth labels are either located at the server, or (2) shared across all clients. This allows for joint optimization. However, our problem is fundamentally different. The client-specific measurements (labels) are high-dimensional and cannot be shared due to data logistics and privacy constraints. Sharing measurements (which are raw data) would violate our goal of reducing communication overhead.
Thus, we use two separate loss functions: one for the client model and another for the server’s ML function. This iterative optimization approach allows each client to maintain its high-dimensional data locally, yet learning the cross-client interdependencies.
(6) **At line 250, The server model is a ML tool whose predictions and labels are... I don't understand what this sentence means:** We acknowledge this sentence in the previous version of the paper might create confusion. The entire paragraph (now starting from **line 256**) is updated in the revised version of the paper.
(7) **Interpreting Table 4 and/or Explaining ``At line 466, Our method performed better than all three baselines mentioned in table 4.'':** Again we acknowledge that the paragraph might not be comprehensible. We addressed it as follows:
**(a)** Line 455-472 is now updated. It contains two discussion sections -- **Interpreting Robustness Results**, and **Baselines**.
**(b)** We provided more details on how the baseline architecture compares to our framework in **line 466-472**.
**(c)** Due to space limitations, a detailed discussion on the performance of baselines is done in **Appendix A.8** (cross-referenced in line 472).
(8) **Some theorems are stated too informally, such as Theorem 6.2:** In the revised version of the paper it is **Theorem 6.1**. The statement of that theorem updated to include $\lim {k\to\infty}$. We discuss the difference between $t$ and $k$ below:
In our work, $t$ refers to the time stamp in the time-series data, and $k$ refers to the iteration index of the gradient descent optimization process. Each iteration $k$ processes one data point (one time step), and $k$ increments with every gradient update. Only the model parameters $\theta$ and $\hat{A}_{mn}$ are indexed with $k$ as they are learned using gradient update. The data is indexed by time $t$.
**E.g.:** Consider a dataset 100 time units ($t = 1, 2, \dots, 100$):
Timestamp ($t$): Represents the sequence of observations, e.g., $y_1, y_2, \dots, y_{100}$.
Total epochs: 2 (the model sees the entire dataset 2 times).
Number of gradient descent iterations ($k$) per epoch: 100 (one iteration per time step).
Epoch 1: At $k = 1$: Use $y_1$ (time stamp $t=100$) for gradient update, ..., $k = 100$: Use $y_{100}$ (time stamp $t=100$).
Epoch 2: $k = 101$: Use $y_1$ (time stamp $t=1$), ..., $k = 200$: Use $y_{100}$ (time stamp $t=100$)