Paper: Incentivized Truthful Communication for Federated Bandits (ICLR-2024)

Reviewer comment (Novelty):
The model in the paper is very complex, combining many different aspects (linear bandits, multiple clients, communication cost, incentivizing payments, truthfulness), and the paper does not provide any motivating application for studying such a complex model.

Author response:
**[Q1]**: The paper did not provide any motivating application for studying this complex model. 

**[A1]**: The line of research on federated bandit learning has been motivated by many real-world sequential decision applications where recommender systems is a representative one. For instance, a recommendation platform (server) wants its mobile app users (clients) to opt in its new recommendation service, which switches previous on-device local bandit algorithm to a federated bandit algorithm. Although the new service is expected to improve the overall recommendation quality for all clients (minimizing the overall regret), particular clients may not be willing to participate in this collaborative learning, as the expected gain for them might not compensate their locally increased cost (e.g., communication bandwidth, added computation, lost control of their data, and etc). 

Therefore, a proper incentive mechanism is required to facilitate the federated learning process, as explained in [CA3]. Following this direction, our work focuses on a more realistic scenario that aims to design a truthful incentive mechanism, ensuring clients won't exploit the server’s commitment by misreporting their data sharing cost.

Reviewer comment (Evaluation):
There is confusion about how the reward/regret of the bandit algorithm can be compared to the incentive payment paid out by the server, and relatedly, what incentive the server has to pay the clients (out of pocket) to get them to participate.

Author response:
**[Q2]**: How can the reward/regret of the bandit algorithm be compared to the incentive payment paid out by the server. Relatedly, what incentive does the server have, to pay the clients (out of pocket) to get them to participate? What benefit does the server get when the agent participates? What if the true “communication costs” are exorbitant (e.g. $1M per communication)? The algorithm in this paper still makes the server pay, regardless of the scale of these costs. 

**[A2]**: As explained in [A1], the server is obligated to reduce regret for all clients. **Therefore, the server’s goal is not to trade off between regret and incentive payment, but to motivate client’s participation for nearly optimal regret**. Note that this obligated server is practical in many real-world scenarios (e.g., distributed users/edge devices on a shared learning platform, different departments within the same company, etc.), where the server/platform/company's best interest is to improve the overall learning across all clients.

With that being said, since Wei et al. (2023) recently opened up an interesting direction in this line of research, we absolutely agree that exploring the non-obligated server setting is a timely and important future work. The interactions between the server and clients can be modeled variously according to real-world applications. For instance, in addition to incentivizing clients for participation, the server can even charge some clients for receiving data from the server.

Reviewer comment (Theory):
From the reviewer's understanding, it is each client performing some action, so it is in the client’s best interest that the bandit algorithm chooses the best actions. Since federated learning improves the learning algorithm by collecting more data, it is in the agent’s best interest to participate; if the “communication cost” is higher than the benefit, the agent can choose not to participate. In this paper, however, the server will pay such a client to participate—yet it is unclear what benefit the server gains from this participation.

Author response:
**[Q2]**: How can the reward/regret of the bandit algorithm be compared to the incentive payment paid out by the server. Relatedly, what incentive does the server have, to pay the clients (out of pocket) to get them to participate? What benefit does the server get when the agent participates? What if the true “communication costs” are exorbitant (e.g. $1M per communication)? The algorithm in this paper still makes the server pay, regardless of the scale of these costs. 

**[A2]**: As explained in [A1], the server is obligated to reduce regret for all clients. **Therefore, the server’s goal is not to trade off between regret and incentive payment, but to motivate client’s participation for nearly optimal regret**. Note that this obligated server is practical in many real-world scenarios (e.g., distributed users/edge devices on a shared learning platform, different departments within the same company, etc.), where the server/platform/company's best interest is to improve the overall learning across all clients.

With that being said, since Wei et al. (2023) recently opened up an interesting direction in this line of research, we absolutely agree that exploring the non-obligated server setting is a timely and important future work. The interactions between the server and clients can be modeled variously according to real-world applications. For instance, in addition to incentivizing clients for participation, the server can even charge some clients for receiving data from the server.

Reviewer comment (Theory):
If the problem is considered purely for theoretical interest, the complexity of the model makes it difficult to identify how the results contribute to the theory of multi-armed bandits. Most of the paper focuses on mechanism design, but it takes a long time to understand the underlying federated linear bandit model, and it is unclear which parts of the bandit model are crucial and which are not.

Author response:
**[Q3]**: If one is considering the problem in this paper purely for theoretical interest, the fact that the model was so complex makes it difficult to identify how the results contribute to the theory of multi-armed bandits. Most of the paper was about mechanism design, but it took a long time for me to understand the underlying federated linear bandit model - and it wasn’t clear to me which parts of the underlying bandit model were crucial and which were not.

**[A3]**: Thanks for pointing out the place that could lead to unnecessary confusion. We have provided a detailed discussion on the basics objectives and challenges of federated bandits learning in [CA2].

Typically, the communication scheme (i.e., arm selection strategy, communication event trigger) is a crucial part for federated bandit learning. **However, in the context of incentivized federated bandits, the main interest is on designing the incentive mechanism**, and both Wei et al (2023) [R1] and our work adopt the standard communication scheme, as we explained in [CA3].

For a detailed discussion of our contribution to the theory of bandit learning, please refer to our response in [CA4].

Reviewer comment (Writing):
A client’s utility, \(u_{it}\), is not defined, so truthfulness (definition 1) is not well defined. It is unclear whether a client’s utility involves both the regret and the payment, and if so, how these are combined.

Author response:
**[Q4]**: A client’s utility $u_{i,t}$,  was not defined, so truthfulness (definition 1) is not well defined. Does a client’s utility involve both the regret as well as the payment? If so, how are these combined?

**[A4]**: Following Wei et al. (2023) [R1], our problem considers “individual rational” clients, meaning that client $i$ only participates in data sharing if the incentive mechanism provides a non-negative utility (see Definition 1 of Individual Rationality in [R1]), where utility is defined as the difference between incentive and true cost of client $i$ , i.e., $u_{i,t} = \mathcal I_{i,t} - D_{i,t}$, as is standard in economic analysis [R2]. We have made this notation clearer in our revised manuscript (Section 3).

Nonetheless, we agree that it is also an interesting alternative to include the regret-related term as a factor in the client’s utility design. And we’d like to share some thoughts on it:
- Under the obligated server setting, the server will help every client regardless of whether the clients share their local data or not. As the server always guarantees a nearly optimal regret in each communication round, thus there is no significant difference in such a regret-related utility term across all clients, as everyone can utilize the same amount of aggregated data shared by the server for regret reduction. Therefore, it is less meaningful to take the regret-related term in the utility design. 
- In contrast, for the non-obligated server setting, each client may receive different subsets of the aggregated data and thus obtain different levels of regret reduction. In this case, it becomes more reasonable to include the regret-related term in the utility design, which we believe is an interesting direction to explore combined with the non-obligated setting, as we discussed in [A2].

**References**

[R1]. Zhepei Wei, Chuanhao Li, Haifeng Xu, and Hongning Wang. Incentivized Communication for Federated Bandits. NeurIPS 2023.

[R2]. Roger B Myerson and Mark A Satterthwaite. Efficient mechanisms for bilateral trading. Journal of economic theory, 29(2):265–281, 1983.

Reviewer comment (Novelty):
I am a bit unsure of the contribution and the motivation for the developed solution of this work, particularly considering how this work is built (as an extension) on Wei et al. (2023); hence, the claim for making this work a first attempt in mechanism design for federated bandit learning seems incorrect? Also, for the method developed, several recent works on federated settings, such as FL, tackle such issues for truthful and fair participation, employing economic tools. Contract Theory-based methods or Auction-based methods have been "extensively" employed in mechanism design for truthful interaction between the clients and the server. For instance, [R1, R2].  

[R1] Karimireddy, Sai Praneeth, Wenshuo Guo, and Michael I. Jordan. "Mechanisms that incentivize data sharing in federated learning." arXiv preprint arXiv:2207.04557 (2022).  
[R1] T. H. Thi Le et al., "An Incentive Mechanism for Federated Learning in Wireless Cellular Networks: An Auction Approach," in IEEE Transactions on Wireless Communications, vol. 20, no. 8, pp. 4874-4887, Aug. 2021, doi: 10.1109/TWC.2021.3062708  

In that reference, I am not sure relaxing the data sharing cost with a valuation function, commonly used in standard economic analysis, is a sufficient contribution to this work. Can you explain more about the missing guarantees on the social cost of Wei et al. (2023)? Later, after Def. 2, you mentioned the definition of social cost in this work is different than theirs.

Author response:
**[Q1]**: I am a bit unsure of the contribution and the motivation for the developed solution of this work, particularly considering how this work is built (as an extension) on Wei et al. (2023); hence, the claim for making this work a first attempt in mechanism design for federated bandit learning seems incorrect? 

**[A1]**: Thanks for pointing out the place that could lead to unnecessary confusions. The truthfulness guarantee is a crucial property in mechanism design, especially in the context of **federated bandit learning** (see definition in [CA1]). Motivated by this practical demand, our work provides the first solution that simultaneously ensures both truthfulness and nearly optimal learning performance. Please also refer to the motivating example in our response to [Q1 of Reviewer Jfq6].

We acknowledge that Wei et al. (2023) [R6] is the first work to investigate mechanism design in the incentivized federated bandits. Indeed, in our introduction section, we only claim to present "**the first incentive-compatible (i.e., truthful)** communication protocol", signifying that our work is the first solution to achieve nearly optimal performance without relying on the truthfulness assumption in the incentivized federated bandit problem.

Please refer to our general responses for a detailed comparison on the commonality (i.e., using the same communication scheme, see [CA3]) and differences (i.e., our unique contribution, see [CA4]).

Reviewer comment (Writing):
Following my earlier comment, the setup and the interaction procedure is unclear to establish the contribution, again, as compared to Wei et. al., 2023. The preliminary model is not rigorous to that end. For instance, what exactly is pulling the arm characterised in Section 3 in a federated bandit setting? How to interpret y_t following it? what is w in footnote 1? and so on. This can be significantly improved.

Author response:
**[Q4]**: Following my earlier comment, the setup and the interaction procedure is unclear to establish the contribution, again, as compared to Wei et. al., 2023. The preliminary model is not rigorous to that end. For instance, what exactly is pulling the arm characterised in Section 3 in a federated bandit setting? How to interpret y_t following it? what is w in footnote 1? 

**[A4]**: In the context of federated bandits (introduced in Section 3 of our paper), there are $N$ clients interacting with the environment to learn the unknown parameter, coordinated by a central server. At each time step $t$, an arbitrary client becomes active to pull an arm $x_t$, and receives the corresponding reward $y_t$ from the environment. A vivid illustration of the interaction procedure can be found in Figure 1(a) of [R5].

As is standard in federated bandit learning, we use the upper confidence bound (UCB)-based arm selection strategy to guide the client’s interaction with the environment, where the collected data $(x_t, y_t)$ is used for model estimation, as detailed in [CA2]. Regarding the communication scheme between the server and clients under the incentivized setting, please see our response [CA3].

The notation $w$ in footnote 1 represents how much client $i$ weighs its data collection cost $\det(\Delta V_{i,t})$ in its data-sharing cost function $D_{i,t} = f(\Delta V_{i,t})$. However, as we clarified in [A2] and in our response to Q3 of Reviewer 1gFC, the specific form is trivial and will be removed from our final version to avoid further confusion.

Reviewer comment (Evaluation):
Can the authors support the claim that "more frequent communication leads to lower regret"? (and the line that follows in pg. 3) It is understandable in terms of the communication overhead, but this is also the fact that you gain in training efficiency. This leads back to my earlier confusion regarding how "regret" is quantified.

Author response:
**[Q5]**: Can the authors support the claim that "more frequent communication leads to lower regret"? (and the line that follows in pg. 3). It is understandable in terms of the communication overhead, but this is also the fact that you gain in training efficiency. This leads back to my earlier confusion regarding how "regret" is quantified.

**[A5]**: As in line with the established common ground and convention of this line of research, our claim that "more frequent communication leads to lower regret" and the definition of “regret” are the standard consensus/terminology shared by federated bandits literature. Please refer to our response [CA2] for a detailed clarification.

Reviewer comment (Theory):
Is the critical value defined in Def. 4 unique?

Author response:
**[Q6]**: Is the critical value defined in Def. 4 unique?

**[A6]**: Yes. Suppose $c_{i,t}^{1}$ and $c_{i,t}^{2}$ are two distinct critical values for client $i$ at time step $t$. When client $i$ claims its data sharing cost as $\widehat D_{i,t} = (c_{i,t}^{1} + c_{i,t}^{2}) / 2$, then client $i \in S_t$ and $i \notin S_t$ will simultaneously hold true according to Definition 4, which is clearly a contradiction. Therefore, by Definition 4, the critical value for each client $i$ is unique.

Reviewer comment (Theory):
I must admit the proof of monotonicity has not been conveyed clearly in its current form; can you provide a discussion on this?

Author response:
**[Q7]**: I must admit the proof of monotonicity has not been conveyed clearly in its current form; can you provide a discussion on this?

**[A7]**: As defined in Definition 3, Algorithm 1 is monotone if for any selected participating client $i$ with a reported data sharing cost $\widehat D_{i,t}$, it will remain selected by Algorithm 1 whenever it reports a lower value, i.e., $\widehat D^\prime_{i,t} \leq \widehat D_{i,t}$.

Note that Algorithm 1 searches for the participating set $S_t$ by starting from a minimal budget (Line 2) and repeatedly increases the budget (Line 4) until the resulting client set found by Algorithm 2 satisfies the specified condition (Line 3). Therefore, decreasing a client's reported cost $\widehat D_{i,t} \rightarrow \widehat D^\prime_{i,t}$ can cause a different output by Algorithm 2 and, consequently, terminate the search process of Algorithm 1 at a different budget $b \rightarrow b'$ with a potentially different selection of the participant set ${S_t} \rightarrow {S^\prime_t}$.

**As a result, to prove monotonicity, we need to show that given $i \in S_t$ when client $i$ reports $\widehat D_{i,t}$, the proposition $i \in S^\prime_t$ always holds true when $\widehat D^\prime_{i,t} \leq \widehat D_{i,t}$**. This is done by demonstrating that the resulting objective value $g_t({S_t})$ from Algorithm 2 is non-decreasing with respect to its input budget $b$, as we proved in Lemma 12. In other words, it becomes impossible for $b' > b$ because otherwise the search process in Algorithm 1 would have stopped at $b$. Since decreasing client $i$'s claimed cost will increase its chance of getting selected by Algorithm 2 (Line 3), we can show that $i \in S'_t$ always holds true for all $b' \leq b$, as explained in Appendix A. This finishes the proof.

Reviewer comment (Experiments):
The simulations build on Wei et. al. 2023 but didn't use it as a baseline.

Author response:
**[Q8]**: The methods build on Wei et. al. 2023 but didn't use it as a baseline in simulation

**[A8]**: We have to clarify that despite achieving the same results on regret and communication cost, our incentive mechanism is completely different from that of Wei et al. (2023) [R6], and **these two are not comparable as their method builds on the truthfulness assumption, which is absent in our setting**. Please find a detailed comparison in our general response [CA4].

Reviewer comment (Experiments):
There is no time-complexity analysis and no overall learning performance evaluation following the proposed approach; also, it is unclear how well the method scales.

Author response:
**[Q9]**: What about the time-complexity analysis and the overall learning performance following the proposed approach? Also, how well the method scales.

**[A9]**: As detailed in the response to [Q4 of Reviewer 1gFC], the proposed method holds a polynomial time complexity of $O\left(\left\lceil \log_{1+\epsilon}\left(\sum_{i=1}^{N} \frac{\widehat{D}{i,t}}{b_0}\right)\right \rceil \cdot N^2\right)$, where $b_0 = \min_{i \in \widetilde S} \widehat D_{i,t}$, scaling quadratically with the number of clients $N$.

Our method archives nearly-optimal overall learning performance with $O(d\sqrt{T} \log T)$ regret and $O(N^{2} d^3\log T)$ communication cost, which is confirmed by our empirical study in Section 5. For example, as Figure 1(a) shows, the regret curve of our proposed method converges after ~1000 iterations, indicating that the algorithm already learned an accurate estimation of the unknown parameter and can consistently choose the best arm, thus incurring no regret.

Reviewer comment (Theory):
The general setup with "sequential interaction" is limiting in itself, in my understanding.

Author response:
**[Q10]**: The general setup with "sequential interaction" is limiting in itself, in my understanding.

**[A10]** : We should clarify that our studied problem is **federated bandit learning**, which is an important research field with a wide range of real-world applications featuring sequential decision-making processes, as discussed in [CA1]. Therefore, investigating the incentive design with this setup holds significant value both in research and practical applications.

Reviewer comment (Novelty):
The regret analysis seems to follow quite straightforwardly from Wei et al. (2023) with slight modifications in the hyper-parameter values (as indeed noted on page 17 of the supplementary material, in the description trailing Lemma 18). Therefore, it appears that there is not much novelty in the regret analysis, leaving the novelty to the demonstration of truthfulness and near-optimality of social cost.

Author response:
**[Q1]**: It appears that there is not much novelty in the regret analysis, leaving the novelty to the demonstration of truthfulness and near-optimality of social cost. 

**[A1]**: As we discussed in the general response [CA4], our method achieves the **same order of regret** as previous works that count on strong assumptions (e.g., the willingness assumption in DisLinUCB [R2], the truthfulness assumption in Inc-FedUCB [R4] as illustrated in the table presented in [CA4]), while we drop all those assumptions. We truly believe our finding that one can achieve the same level of regret while being free from these strong assumptions is already a non-trivial contribution to the community.

We should emphasize that our focus is on devising the incentive mechanism that ensures truthfulness and near-optimal social cost, without compromising the guarantees in regret and communication cost. Moreover, as the previous work has shown the obtained regret and communication upper bounds are already tight [R5], there is little room to develop new regret analysis techniques to improve the results.

Reviewer comment (Novelty):
In continuation to the last sentence of the previous point, there appears to be no motivation to study the criterion delineated in (1). The authors simply proceed to analyse the objective function in (1), simply because it appears in Wei et al. (2023). No further explanation of the criterion in (1) or of its relevance to FL is provided in the paper.

Author response:
**[Q2]**:  There appears to be no motivation to study the criterion delineated in (1). No further explanation of the criterion in (1) or of its relevance to FL is provided in the paper.


**[A2]**: In our concerned federated bandit problem, optimizing the determinant ratio directly corresponds to the reduction of regret [R4], which is a well-established standard metric in the literature [R1, R2, R3]. Therefore, any feasible solution to Eq. (1) that satisfies the determinant ratio constraint immediately guarantees the near-optimal regret. A more detailed explanation of this criterion is provided in Sections 4.1 and 4.2 in Wei et al. (2023) [R4].

We want to clarify that in our work, using Eq. (1) as our optimization problem is not an assumption but a design choice to achieve our goal. We acknowledge the possibility of other forms of optimization problems that, when solved, can yield similar results with proper proofs.

Reviewer comment (Theory):
On the algorithmic side, the authors prove in Lemma 6 that if the selection rule (a rule for selecting a set \( S_t \) of clients that would participate in the FL task at time \( t \)) can be computed in polynomial time, then so can the critical value associated with the selection rule. However, no explicit result stating how much time is taken by Algorithm 1/2 to compute the selection rule is provided in the paper. While the authors provide an explicit scheme to compute the critical value, the authors do not explicitly prove that their Algorithm 1/2 computes the selection rule in polynomial time.

Author response:
**[Q4]**: No explicit result stating how much time is taken by Algorithm 1/2 to compute the selection rule is provided in the paper. While the authors provide an explicit scheme to compute the critical value, the authors do not explicitly prove that their Algorithm 1/2 computes the selection rule in polynomial time.

**[A4]**: For Algorithm 2, the worst-case time complexity of the while loop is $O(N)$. The operation inside the while loop involves finding the maximum element in a set, which takes $O(N)$ time. Therefore, the time complexity of Algorithm 2 is $O(N^2)$.

Let $M$ be the number of iterations of the while loop in Algorithm 1. Hence, the time complexity of Algorithm 1 is $O(M \cdot N^2)$. Specifically, the worst case is to consistently increase the budget $b$ until it reaches the sum of $\widehat D_{i,t}, \forall i \in \widetilde{S}$. Therefore, we can upper bound $M$ by considering the loop-breaking case: $b_0 \cdot (1 + \epsilon)^M \geq \sum_{i=1}^{N} \widehat D_{i,t}$, i.e., $M \leq \left\lceil \log_{1+\epsilon}\left(\sum_{i=1}^{N} \frac{\widehat D_{i,t}}{b_0}\right)\right \rceil$, where $b_0 = \min_{i \in  \widetilde S} \widehat{D}_{i,t}$.

As a result, the overall time complexity of Algorithm 1 is $O( \left\lceil \log_{1+\epsilon}\left(\sum_{i=1}^{N} \frac{\widehat D_{i,t}}{b_0}\right)\right \rceil \cdot N^2)$, which is a polynomial time complexity.

Reviewer comment (Theory):
The statement of Lemma 7 appears to be contradictory to one of the statements made in its proof (presented in Appendix D). In the proof, the authors identify that \(\left(1+\frac{t L^2}{\lambda d} \right)^{-d}\) is a lower bound on a certain ratio of determinants, and further note that "if the hyper-parameter \(\beta\) is **greater** than the (preceding) lower bound, it is guaranteed that no client can be essential". However, Lemma 7 states the contrary. Furthermore, the constants \( L \) and \( \lambda \) appearing in the statement of Lemma 7 are not defined in the main text.

Author response:
**[Q5]**: A typo in the proof of Lemma 7 (presented in Appendix D). Furthermore, the constants $L$ and $\lambda$ appearing in the statement of Lemma 7 are not defined in the main text.


**[A5]**: Thanks for pointing out this typo. We have corrected it (and other typos pointed out by the reviewer) in our updated manuscript. To follow the convention in the bandit literature and reduce the barrier of access, we choose the standard notations $L$ and $\lambda$ to represent the regularization parameters, in line with previous works [R1,R2, R3, R4]. Due to the space limit, we left these standard technical details to the appendix, please refer to the definitions in Lemma 15 and Eq. (12).

Reviewer comment (Theory):
There are no comments on the tightness of the bounds on the communication cost and pseudo-regret in relation to the bounds appearing in the work of Wei et al. (2023) (the key piece of work the current paper seems to be based upon). In the process of achieving near-optimal social cost, what is the impact on the communication cost / number of rounds of communication and regret? A formal comparison along these lines with Wei et al. (2023) is missing.

Author response:
**[Q6]**: There are no comments on the tightness of the bounds on the communication cost and pseudo-regret in relation to the bounds appearing in the work of Wei et al. (2023). In the process of achieving near-optimal social cost, what is the impact on the communication cost / number of rounds of communication and regret? A formal comparison along these lines with Wei et al. (2023) is missing.


**[A6]**: As clarified in [A1], our results on the communication costs and regret are essentially in the same order as those in Wei et al. (2023) [R4], which are already tight order wise. Please see a detailed comparison in the general response [CA4].

As analyzed in our Section 4.3, to achieve a near-optimal social cost, our method needs to slightly relax the determinant ratio constraint by a small constant factor of $(1 - e^{-1})$. Technically, our method guarantees the determinant ratio $\frac{\det(V_{g,t}(S_{t}))}{\det(V_{g,t}(\widetilde{S}))} \geq \beta^{1-e^{-1}}$, while their method guarantees $\frac{\det(V_{g,t}(S_{t}))}{\det(V_{g,t}(\widetilde{S}))} \geq \beta$. Correspondingly, we need to set a lower communication threshold $D_c = \frac{T}{N^2d\log T} - (1-e^{-1}) \sqrt{\frac{T^2}{N^2dR\log T} }\log \beta$ compared to $D_c = \frac{T}{N^2d\log T} - \sqrt{\frac{T^2}{N^2dR\log T} }\log \beta$ in Wei et al. (2023).

Despite these trivial constant gaps, **there is no impact on the order of communication cost and regret**, as we presented in the table of [CA4] and proved in Appendix E.

Reviewer comment (Theory):
Immediately following the statement of Lemma 18 in Appendix E, the authors state that "In each communication round, **all** the clients upload \(O(d^2)\) scalars to the server and then download \(O(d^2)\) scalars." Why do **all** the clients participate in the communication and not just the ones from the selection set \( S_t \) at the communication time instant \( t \)? This is a little confusing, and not discussed elsewhere in the paper.

Author response:
**[Q7]**: Why do all the clients participate in the communication and not just the ones from the selection set $S_t$ at the communication time instant $t$? This is a little confusing, and not discussed elsewhere in the paper.

**[A7]**: To facilitate the data valuation of clients' local updates at time step $t$, all clients need to first upload $\Delta V_{i,t}$ to the server, and only those who decide to participate will subsequently upload the corresponding $\Delta b_{i,t}$. Please refer to [CA3] for a detailed clarification on the communication scheme.

Reviewer comment (Writing):
The *incentive* \(\mathcal{I}_{i,t}\) introduced in (2) does not appear to be used elsewhere in the paper.

Author response:
**[Q8]**: The incentive $\mathcal{I}_{i,t}$ introduced in (2) does not appear to be used elsewhere in the paper. 

**[A8]**: In the incentivized communication setting, any individual rational client only participates in data sharing if the incentive provided by the server is no less than the client's intrinsic data sharing cost, i.e., $\mathcal I_{i,t} \geq D_{i,t}$. This property is formally introduced in Definition 1 of Wei et al. (2023) [R4], and we have made this notation clearer in our revised manuscript (see Section 3).

Reviewer comment (Writing):
The notion of "communication threshold" in Theorem 10 is not explained in the paper.

Author response:
**[Q9]**: The notion of "communication threshold" in Theorem 10 is not explained in the paper.

**[A9]**: We have made this notation clearer in our revised manuscript (see Theorem 10). The communication threshold $D_c$ is a hyperparameter of the standard communication event trigger, which is used to control the communication frequency, as defined in the communication protocol (Line 9 of Algorithm 6).

Reviewer comment (Experiments):
The paper is generally missing the overall feel of an FL paper (no details about the communication protocol, which set of clients participate in communication, what information is communicated from the clients to the server), and seems to only build upon the setting and results of Wei et al. (2023) on the social cost and truthfulness aspects.

Author response:
**[Q10]**: The paper is generally missing the overall feel of an FL paper (no details about the communication protocol, which set of clients participate in communication, what information is communicated from the clients to the server), and seems to only build upon the setting and results of Wei et al. (2023) [R4] on the social cost and truthfulness aspects.

**[A10]**: We should clarify that our focus is on mechanism design for incentive compatibility in **federated bandit learning**, instead of the general FL. Please find our detailed clarifications in [CA1] and [CA2].

Due to space limit, we leave the details of the standard communication scheme to Appendix F, where the communication protocol is presented in Algorithm 6. Please see also our general response [CA3].

As we clarified in the general response [CA4], our incentive mechanism design (Algorithm 1) is completely different from Wei et al. (2023) [R4] (Algorithm 3 in their paper), as their design builds on the strong truthfulness assumption. **Note that despite achieving the same results on regret and communication cost bounds, our method requires fewer assumptions but enjoys richer guarantees**.