Event-Driven Online Vertical Federated Learning
Authors: Ganyu Wang
Boyu Wang
Bin Gu
Charles Ling
Venue: ICLR 2025 Poster (2025)
Keywords: Vertical Federated Learning
Online Learning
Event Driven
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: Applying online learning to VFL is not straightforward due to its inherent nature.

Abstract: Online learning is more adaptable to real-world scenarios in Vertical Federated Learning (VFL) compared to offline learning. 
However, integrating online learning into VFL presents challenges due to the unique nature of VFL, where clients possess non-intersecting feature sets for the same sample. 
In real-world scenarios, the clients may not receive data streaming for the disjoint features for the same entity synchronously. Instead, the data are typically generated by an *event* relevant to only a subset of clients.
We are the first to identify these challenges in online VFL, which have been overlooked by previous research. To address these challenges, we proposed an event-driven online VFL framework. In this framework, only a subset of clients were activated during each event, while the remaining clients passively collaborated in the learning process. 
Furthermore, we incorporated *dynamic local regret (DLR)* into VFL to address the challenges posed by online learning problems with non-convex models within a non-stationary environment.
We conducted a comprehensive regret analysis of our proposed framework, specifically examining the DLR under non-convex conditions with event-driven online VFL. 
Extensive experiments demonstrated that our proposed framework was more stable than the existing online VFL framework under non-stationary data conditions while also significantly reducing communication and computation costs.

Published as a conference paper at ICLR 2025
EVENT-DRIVEN ONLINE LEARNING
VERTICAL
FEDERATED
Ganyu Wang'
Boyu Wang! Bin Gu Charles Ling 1.2
Western University
Vector Institute
Jilin University
gwang382$uwo ca bwanglcsd. uwo.ca jsgubin@gmail.com
charles. ling@uwo.co
ABSTRACT
Online learning is more adaptable to real-world scenarios in Vertical Federated Learning (VFL) compared to offline learning. However, integrating online learn- ing into VFL presents challenges due to the unique nature of VFL, where clients possess non-intersecting feature sets for the same sample. In real-world scenarios, the clients may not receive data streaming for the disjoint features for the same tity synchronously. Instead, the data are typically generated by an event relevant to only a subset of clients. We the first to identify these challenges in online VFL, which have been overlooked by previous research. To address these challenges, we proposed an event-driven online VFL framework. In this framework, only a subset of clients were activated during each event, while the remaining clients pas- sively collaborated in the learning process. Furthermore, we incorporated dynamic local regret (DLR) into VFL. to address the challenges posed by online learning problems with non-convex models within a non-stationary environment. We con- ducted comprehensive regret analysis of our proposed framework, specifically examining the DLR under non-convex conditions with event-driven online VFL. Extensive experiments demonstrated that proposed framework more sta- ble than the existing online VFL framework under non -tatiooot data conditions while also significantly reducing communication and computation costs.
INTRODUCTION
Vertical Federated I Learning (VFL.) Vepakomma et al., 2018 Yang et al. 2019 Liu et al. 2019
Chen al. 2020 Gu al.) 2020 Zhang et al.) 2021b a Wang et al. 2023 Qi al. 2022; Wang
et al. 2024) is a privacy-preserving machine learning paradigm wherein multiple entities collabo- rate construct model without sharing their raw data. In VFL, each participant possesses non- intersecting features for the same set of samples, which is significantly different from the Horizontal Federated Learning (HFL) (McMahan et al., 2017 Karimireddy et al. [2000] Li et al. 20200] 202T]
Marfoq et al.) 2022: Mishchenko et al. 2019) where each client possesses the non-overlap samples of the same features
Current research on VFL primarily focuses the offline scenario, characterized by pre-established dataset. However, the limitations of offline learning become obvious when building real-world ap- plications of VFL. First, offline learning is unsuitable for scenarios where the dataset undergoes continual updates, which is typical real-world applications. For example, in the application sce- nario of VFL scenarios involving companies clients (Wei et al. 2022 Vepakomma et al. 2018), new data is constantly generated as new customers engage with the companies, or as existing cus- tomers update their records through ongoing activities. Similarly, in VFL scenarios involving edge devices (Wang& & Xuj 2023 Liu et al. 2022), the sensors, acting as the clients, continuously receive data streams from the environment rather than maintaining static dataset. Second, the dynamic nature of real-world environments leads to data distribution drift, which is particularly evident in edge devices. In response, the offline learning paradigm requires retraining the model from scratch to accommodate shifts in data distribution. While this retraining process may not present significant
"Corresponding authors.
'The term Federated I Learning commonly refers HFT. However, this is not the setting of this study.

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challenges in centralized learning, it becomes prohibitively expensive in distributed learning scenar- ios, as the training process imposes substantial communication and computation costs in distributed learning. Consequently, online learning may provide greater adaptability in VFL by allowing models to update continuously as new data arrives and handle dynamic environments.
However, applying online learning to VFL is not straightforward due to its inherent nature. First, in online VFL, clients receive non-intersecting features the data from the environment. In real-world scenarios, it is rare for all clients receive all these features of sample simultaneously. Instead, is more common for only a subset of clients to obtain the relevant features in response to a specific event. For example, in VFL implementations within large companies (Wei al.) 2022 Hu et al, 2019 Vepakomma et al. (2118), when a customer takes an action such as making payment or purchase, this action typically involves only one company of the VFL. while the data from the other companies remain unchanged. Similarly, in VFI. involving sensor networks (Wang & Xu 2023 Liu et al. 2022), only the sensor triggered by an event will be activated, while others remain inac- live (Suh 2007 Heemels et al. 2012 Trimpe & D' Andrea 2014: Beuchert et al. 2020). The above scenario has brought about the demand for an event-driven online VFL framework, wherein certain participants
activated by events during each round, thereby dominating the learning process, while the rest of the participants remain inactive or passively cooperate with the learning.
The second challenge in online VFL lies in addressing non-convex models and dynamic environ- ments, which are prevalent in practical applications. Current research in online VFL. still pri- marily focuses online convex optimization, assuming both convex models and stationary data streams (Wang& & Xu 2023). While convex models are easier to optimize, they fail to capture the complex patterns necessary for tackling more challenging tasks. Additionally, the assumption of stationary data streams is unrealistic in dynamic real-world environments, where data distributions can shift over time. For example, the environmental sensor that monitors the air quality may ex- perience dynamic change due to natural phenomena. These are challenges that current online VFL frameworks have yet to resolve.
To address the aforementioned challenges, we propose a novel event-driven online VFL framework, which is well-suited for the on- line learning scenario in non-convex cases and non-stationary environments. Figure [ depicts a schematic graph of our framework. In our framework, subset of the clients are acti- vated by the event at each round, while others passively contribute to the training. This ap- proach substantially reduces communication- computation costs and facilitates the client model in learning relevant content. Moreover, we adapt the dynamic local regret approach proposed by Aydore et al. (2019) to our event- driven online VFL. framework to effectively handle online learning in non-convex cases and non-stationary environments.
Server
Activated Client
Passive Client
Event
Activated Client
Figure 1: Event-driven online VFL
In summary, the contributions of our paper are:
We identify the unrealistic assumption of synchronous data reception online VFL research and propose novel event-driven online VFL paradigm that is better suited to real-world scenarios.
We adapt the dynamic local regret approach to our event-driven online VFL. to effectively handle non-convex models in non-stationary data streaming scenarios, and we theoretically prove the dynamic local regret bound for this framework, which incorporates partial activation of the client.
Our experiments demonstrate that our event-driven online VFL exhibits greater stability com- pared to existing methods when confronted with non-stationary data conditions. Additionally, significantly reduces communication and computation costs.
Notation We use a square bracket with multiple items to denote concatenation for convenience.
For instance, given W. E IRdA, wD E Rda, we define ( wn) A [w] wB]" T. The superscript /.
2

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attached to the parameter w indicates the number of rounds (time step). A square bracket enclosing a single integer represents the set of natural numbers from to that particular number. For instance, [M] ..,
2 RELATED WORK: ONLINE HFL AND ONLINE VFL
Online HFL Most existing online federated learning research focuses on HFL because extending the HFL. framework to online learning relatively straightforward. In HFL., each participant pos- sesses the complete feature set of the local sample. Therefore, online HFL. can be easily achieved by assigning each client a unique data stream containing non-overlapping samples. The current re- search on online HFL is focused
speeding up optimization (Mitra et al. [222]] Eshraghi & Liang, 2020), reducing communication cost (Hong Chac 202T) and dealing with concept drift (Ganguly & Aggarwal 2023). Mitra et al. (2021) | applied online mirror descent within the Federated Learning framework, demonstrating sub- liner regret in convex scenarios. Hong & Chae (2021) introduced a randomized multi-kernel algorithm for online federated learning, which maintains the performance of the multi-kernel algorithm while mitigating the linearly increasing communication cost. Wwon al. (2023) incorporated client sampling and quantization into online federated learning. Ganguly & Aggarwal (2023) proposed non-stationary detection and restart algorithm for online federated learning, addressing the concept of drift during online learning.
Online VFL In VFL, each client possesses non-overlapping feature set, which presents chal- lenges when integrating online learning into this framework. The existing approach Online VFL is proposed by Wang & Xu (2023), which applies online convex optimization to synchronous VFL. However, they naively assume that all clients receive a synchronous data stream, which does not align with real-world applications. Apart from this study, no other research has explored online VFL and the characteristics dataset streaming in this context. Through the exploration of event- driven mechanisms, open up new possibilities for real-time data streaming processing across distributed nodes within the VFL framework.
METHOD
3.1 PROBLEM DEFINITION
In the VFL framework, there is a single server and M clients. The server produces the label y' at round 4, while each client may receive non-intersecting features s from the environment during the same round. The model for client m, denoted hm (Wm - parameterized by wm Rdm and takes the local feature xt as input to produce an embedding. We denote w as the concatenation of the parameter from all clients, i.c. embeddings hm (.) from all clients and then calculates the losses with the label y². We define the VFL framework in composite form.
[w...... WM). The server, parameterized by wo. recieves
(wo, w, w'r':"") = (wo, (wwwaa)) (un
where f(-) denotes the model on the server. For brevity, we denote (wo, w, w, by I fw w) throughout all following sections.
Following the work from Aydore et al. 2019), we employ dynamic local regret analysis in online non-convex optimization. To reformulate the dynamic local regret under the context of the VFL framework, we begin by introducing the concept of exponentially weighted sliding-window window average as the basis for its computation.
Definition 1. Exponential weighted sliding window average: Let w6 E Rdo denote the server's parameter at time t, w E Rddm be the client m 's parameter at time w [[1, W2 ... wM) Let l denote the length of the sliding window. Then the exponential weighted sliding-window average can be defined as follows:
1 I-1
(t w²)
Σ
wi
(2)
W
i=0
Extra discussion the related work of VFT. and online learning is in Appendix
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where 0 < < 1 and the superscript of the a' indicates the exponent. W = E a² serves
the normalization parameter for the exponential average, ensuring that W Ei= a² = 1. is worth noting that this window gives more weight to recent values, with the weight decaying exponentially, is computed on the past parameter at round
and the loss f fw" f- .w'=i)
Then, the DLR can be formally defined based on the accumulated square norm of the gradient of the exponentially weighted blding-windoww average.
Definition 2. Dynamic I-local regret: Let Se, w') ) be the sliding-windon defined above, wo be the server's parameter and be the aggregated clients" parameter. The Dynamic I-Local Regret can be defined as:
(TR(T)
w')l?
(3)
3.2 ADAPT DLR TO ONLINE VFL
We integrate the dynamic exponentially time-smoothed online gradient descent method introduced by Aydore et al. (2019) into the VFL framework by incorporating buffer to store the past gradients.
Server update Based on the special characteristic of the dynamic local regret, the server is re- quired to maintain buffer of the past intermediate derivative values of length 1. At each time step, the server computes the gradient and updates the buffer by enqueuing the latest gradient and dequeuing the oldest. Subsequently, the server utilizes the buffer to compute the dynamic exponen- tially time-smoothed gradient. Specifically, the partial derivative w.r.t. the server is shown in Eq. and the buffer stores past gradients.
L-1 Σ a wo. " w-
Van Seta w")
(4)
W
2=0 Server Buffer, 0, 1,...1-1
Finally, the server updated its model with stochastic gradient descent, i.e. wo (+1 + w V Vug Se, (wgh w'), where no is the learning rate for the server.
1j0
Client update The client cannot calculate the partial derivative w.r.t. its model by themselves because they do not hold the label. Consequently, they depend on the server transmit the partial (="(w) derivative w/t 2hm Jwi. w.r.t. the client's model output hm to facilitate model updates. The partial derivative w.r.t. the client m's model computed through chain rules:
Vun S.,,a w')
a'V Wm. j fu= ww'-)
W
Oft ( wi
Ihm (wi-i;;
(5)
W
Ohm (wir x.
Ow
Ruffer,
O,
After receiving vf from the server, the clients update their buffer by enqueuing the vf, Finally, the client is updated with wf+1 + wit 7mm V Vur., St, (wh. w').
Shem (w
Owt
3.3 EVENT-DRIVEN ONLINE VFL FRAMEWORK
The event-driven online VFL framework is designed, and the procedures for both clients and servers are formalized in the algorithm When event occurs at round f, the activated client m will receive the data - Subsequently, it sends the embedding hn .ef,)) the server. The server then requests embeddings from the passive clients m E A, C [M]. After gathering the embedding,
A synchronous version of algorithm provided in the Appendix C.1
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the server calculates the partial derivative of (-) w.r.t its local model wi and w.r.t the output of the activated clients hm (wn't zm)) Subsequently, the server updates the buffer for the sequence of partial derivatives in DLR. Following this, the server updates its local model with the partial derivative w.r.t. its model we (Fq. 4 and sends the partial derivative w.r.t. the client's output of the activated client. After receiving from the server, each activated client partial derivative w.r.t their parameter via chain rule (Eq. 5). and then they update their parameter accordingly. At each round, we denote the set of the activated client as Ar C [M], where [M]]
Ar calculates the
represents the set of all clients. The set of passive clients is denoted by A = [M] \ A
Algorithm Event-driven online VFL
Input: window length coefficient a, learning rate {1m } M Output: server model wo. client models Wm E [M]]
m==
0: initialize model wm for all participants m {0, 1, M.
1: Client procedure:
2: if activated by an event then:
3: sampling the environment to obtain w/n
4: send Amm (wh =m to the server
5: receive "tt, from the server
enqueue of alim ( ( iid into the client's buffer
6:
Or
7: calculate Vwn. S.1.a w²) with client's 's buffer
8: update the parameter w +- w 7m Vun, Se,t (wo, w²)
9: else if the server's query f received then:
10: sampling the environment to obtain ztn
send hn (wf =m the server enqueue 0 to the client's 's buffer Server procedure: when server receives hm (w xtth) from activated client m E Ar, do: send query all passive client m A,
12:
14:
calculate V V f'o w'
updates its model w/+1 <- w
17 V Vwg Std, (wh: w")
0S.,1.a (W/.. w(" all activated clients m A
18:
sends Um
REGRET ANALYSIS
4.1 ASSUMPTION
Assumption Too are the assumptions for analysis of the dynamic local regret bound under the non- convex Specifically, Assumption [ is used for modeling the smoothness of the loss function f()), with which we can link the difference of the gradients with the difference of the input in the definition domain. These the basic assumptions for solving the non-convex optimization problem VFL (Liu et al.) 2019 Zhang et af.] 2021a Chen et al. 2020; Castiglia et al., 2022). Assumption 2 and assumption are the common assumptions convex optimization (Aydore et al. 2019: Hazan et al. 2017). The unbiased gradient assumption means that the expected value the stochastic gradient equals the true gradient for the underlying distribution of the sample. The bounded variance assumption ensures that the variability in the stochastic gradient estimates is limited. Assumption 4 is used for bounding the magnitude of gradient for all participant's models. This is common assumption for the non-convex optimization of VFL (Gu et al.) 2020 [Castiglia et al. 2022 Wang et al. 2023 Zhang et al.) 2021a) and the regret analysis for online learning in VFL Wangg & Xu 2023 This assumption is specifically employed bound the difference between the gradient with missing elements (due to the event- driven framework) and the ideal gradient without such omissions. Assumption 5 assumes that the loss value is bounded, which is mainly used to bound the values the difference between the sliding window average and simplify the theoretical result. This is a common assumption in the regret analysis in online learning under non-convex case (ydore et al. 2019 Hazan et al. 2017).
the analysis of stochastic non-
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Assumption 1. Lipschitz Gradient: V f is L-Lipschitz continuous w.E.f all the parameter, i.e., there exists a constant for V [wo, w|. wgg w'] such that
V "wwooww w (wo, w) [wo.w/] wow:ww/'wuii (w/y,, w'lll I[[wo: w] (woy w'll
(6)
Assumption 2. Unbiased gradient: The stochastic gradient V f w) obtained at each iteration an unbiased estimator of the true gradient the function ('(w,, w). i.e., for all [wo, w), we have
f w) =V VV W); E
where (wo. w) is the true gradient of the global objective function.
Assumption 3. Bounded variance: The variance of the stochastic gradient V (wo, w) obtained at each iteration is bounded, i.e., there exists constant o² such that for all [Wo, w). we have
| w) - ( w|l S
(8)
This ensures that the noise in the gradient estimation is controlled. Assumption 4. Bounded gradient: The gradient of the objective function ( w) bounded, Lee there exist positive constants G such that the following inequalities hold.
||V (mo.w]] f (wo, w)ll S G
(9)
Assumption 5. Bounded loss: For all wo Rdd and [w]. ... wMl. where um Rd for m E [M]. the loss f (wo, w) is bounded:
[ [[(oo, <
(10)
4.2 THEOREM
Theorem 1. Dynamic local regret bound: Under Assumption 7
Assumption 5
solving the event-
driven online vertical federated learning problem with Algorithm 7 Pmin = min Pmo Pmax =
Select constant fee 7, define
maxx Pmo let
7 1
(T)
T
8D
+
Lm
2T
Laje
Pmax
G
(11)
Wpmin
1Jc
Pmin
Remark The last term in Eq. II arises from bounding the error between the gradient with missing elements in the event-driven framework and the ideal gradient without missing elements.
Corollary 1. Select 1 T and = T- Let a - 1-
(LR O(T
(12)
Remark Corollary I suggests a sub-linear growth rate compared the linear regret bound O(T), which demonstrates an effective regret minimization.
The proof of the Theorem is in Appendix A For the completeness of our analysis, we further pro- vide the prevalent regret analysis of the event-driven online VFL in the convex case in Appendix
5
EXPERIMENT
5.1 EXPERIMENT SETUP
Supplementary experimental details, including the algorithms of the baselines specifically adapted for VFL, are provided in Appendix C Furthermore, supplementary experiments addressing sec- ondary aspects, such as ablation studies on the DLR parameter (I, a) and experiments on other datasets (SUSY, HIGGS), can be found in Appendix D
Dataset We leveraged the Infinite MNIST (i-MNIST) dataset (Loosli et al., 2007 to assess the performance of the proposed methodologies in the context of VFL. The i-MNIST dataset extends the MNIST dataset by providing an endless stream of handwritten digit images along with their corresponding labels. To convert the i-MNIST dataset into a distributed dataset, each image was first flattened into a one-dimensional vector. This vector was then divided into four equal segments to ensure distribution of features across the clients. Each client assigned one of the four feature partitions, while the server was assigned the label. Experiments on other practical online learning datasets, including the SUSY and HIGGS datasets, provided in Appendix D.2

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Model architecture In our online VFL framework, there were four clients and one server. On the client side, the models consisted of a one-layer perceptron that took flattened features as input and generated 64- 14-dimensional embeddings using ReLU activation. On the server side, a two-layer multi- layer perceptron was employed. The first layer took the concatenated client embeddings as input, produced an output of 256 units, and applied ReLU activation. The subsequent layer generated class logits using softmax activation. The loss function used by the server was cross-entropy.
Baselines The baselines comprised three different online VFL frameworks and three add-ons to those frameworks concerning the activation of participants, resulting in total of nine baselines. The first online VFL framework applied online gradient descent in VFL (referred to as "OGD" for brevity). The second online VFL framework adapted Static Local Regret (Hazan et al. 2017) to online VFL (referred to "SLR"). The third online VFL framework was our VFL framework incorporating dynamic local regret (referred to as "DLR").
The three activation add-ons included "Full" activation, "Random" activation, and "Event" activa- tion. The "Full" activation represented the most common scenario where all clients were activated and received the synchronous data stream. The "Random" activation sets the activation probability for all clients to a constant value, i.e. Pm. = p. which directly interpreted the theoretical result from activation mechanism that designed to simulate the
theorem The "Event" activation tivation in the sensor network. Many sensors are characterized by activation in response to detecting peaks (Suh) (007)) or stimuli (Heemels al. 2012). Inspired by this, we implemented the "Event" activation in which a client is activated when the average of its input features exceeds a threshold, denoted as r. which mimics the stimuli from the surroundings.
To be more specific, the OGD-Full framework was adapted from (Wang & Xu 2023). customized to suit our general VFL setting. The DGD-Random and OGD-Even were oblained by incorpo- rating partial activation into the incorporating Local Regret (Hazan et al., 2017) into the VFL with different activation schemes. DLR-Full was the model obtained by directly adapting the DLR to the synchronous VFL frame- work. DLR- DLR-Random Random and D were the main contributions of our work, which incorporate partial client activation ("Random" or "Event") into VFL.
online VFL. We also designed the SLK baselines by
Training procedure We followed the standard online learning setting, wherein at each round t, client m received only the corresponding feature of a single sample, rather than a batch. Each trial comprised total of 2,000,000 non-repetitive samples. The learning rate v) was tuned from (1, 0,1, 0.001,...). The length of the exponential weighted sliding window for the DLR was tuned from {10, 50, 100, 150). The activation probability p for the "Ran- dom" activation was selected from (0.25, 0.5, 0.75J. The activation threshold I' was tuned from (-0.2, O, 0.2, 0.4, 0.8).
We conducted experiments on both stationary data stream and synthetic non-stationary data stream. In section [5.2] we used the original i-MNIST dataset as the stationary data stream, where each class had an equal probability of being sampled (1)) at each round. In section 5.3 we synthe- sized a non-stationary data stream by altering the class sampling probabilities every 50 rounds. At each stage, the sampling probability for each class was drawn from a uniform distribution (O, 1) and then normalized, Data was then sampled based on the updated probabilities.
Metrics To evaluate the algorithm's performance, we employed the run- run-time error rate (prequen- tial) during online learning. This error rate was averaged and reported every 20,000 samples, pro- viding insights into performance at each stage of the training. Additionally, the accumulated error rate, representing the overall error for the entire training process, served metric for evaluating overall performance. In terms of computational efficiency, we reported the total computation time for all clients, including both active and passive clients at each round. In terms of communication efficiency, we recorded the total communication cost for the VFL framework throughout the entire training process, specifically measuring the size of all communication messages exchanged between the server and the clients.
Algorittm for OGD-Full are provided in Algorithm 4 in Appendix
A Algorithm for OGD-Event are provided in Algorithm 2 in Appendix
" Algorithm for SI.R-Event are provided in Algorithm Appendix
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000
SUR
OLR
Oc0
catn Evert
5/0 Purel
6:1
D18 Pueet
Cree Creettee
Groer Rave
02
04
81
0.1
60
1500n0
160
R600000
166
140
Figure Run-time error rate under stationary data stream
Table 1: Performance metries for online VFL under stationary data stream
OGD
SL.
DL.R
Metric
Full
Random
Event
Full
Random
Event
Full
Random
Event
Accum. Error Rate
0,0575
0.0696
0,1027
0,0846
0.1083
0.0945
0.0618
0.0649
0.0718
Client Comp. (8)
4339,20
2652,38
3182.46
4565,42
2682.47
2992.73
4565.42
2682.47
2992.73
Client Comm. (MB)
1953.13
1465.04
1686.86
78125.02
58599.36
61565.37
1953.13
1464.73
1539.13
5.2 RESULT ON STATIONARY DATA STREAM
Figure illlstrates comparison of the run-time rates across all frameworks within stationary data stream scenario. The x-axis shows the number of observed instances, while the y-axis represents the corresponding run-time error rates. From the analysis of the results, it is evident that in the "Full" activation framework, all models demonstrate stable convergence behavior. However, under the partial activation scheme, both SLR and DLR show superior convergence curves compared to OGD. Notably, DLR converges more rapidly than SLR, entering the convergence phase earlier.
Table presents the performance metries for the entire training process, including the accumulated error rate, the total computational time for the client, and the total communication cost for the entire VFI framework. The accumulated error further illustrates that the DLR exhibits greater stability under the partial activation scheme. Specifically, the accumulated error rate for the DLR remains approximately 0.06 when employing partial activation, whereas the accumulated error rate for OGD undergoes significant reduction from 0.0575 to 0.1027 when implementing the partial activation scheme. By comparing DLR and SLR, we observed that DLR converges more quickly, resulting in a lower average eтror rate. Specifically, due the buffer design, SLR required the communication of the entire buffer between the server and the client in each round" Consequently, the communi- cation volume of SLR "Full" activation, the partial activation approach results in a slightly higher error rate; however, it significantly reduces client computation and overall communication costs.
order of magnitude higher than that of DLR and OGD. Compared to
5.3 RESULT ON NON-STATIONARY DATA STREAM
Figure compares the run-time rates across all the baselines under the non-stationary data stream case. The DLR and SLR frameworks generally outperformed OGD, demonstrating greater stability under the partial activation scheme for clients. Towards the end of the training process, OGD became increasingly unstable, particularly when partial activation add-ons were used. In con- trast, both DLR and SLR maintained stability across all activation schemes. Although SLR was able to converge in non-stationary environments, it was less adaptable to dynamic environments compared to DLR, its convergence was slower at the beginning of training.
Table 2 presents the performance metrics of the frameworks for comparison. Overall, the DLR method demonstrated a lower accumulated eиror rate compared OGD and SLR, primarily due to its stability against non-stationary data streams and partial activation. The communication cost of SLR was significantly higher than that of OGD and DIR, consistent with previous experimental results. When comparing full activation with the partial activation scheme, it was observed that
Thhe step that incurs high communication cost in SLR is highlighted in Algorithm in Appendix
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0GD
SLA
06.
14 Fume
6,1
Crne Teee
10
04
61
61
125
2.5
60
Intance
285
Insoncc
105
Figure Run-time error rate under non-stationary data stream
Table 2: Performance metrics for online VFL under non-stationary data stream
OGD
SLR
DLR
Metric
Full
Random
Event
Full
Random
Event
Full
Random
Event
Accumn Error Rate
0.0592
0,0788
0.1191
0.0481
0.11.99
0.1118
0.0553
0.0561
0.0681
Client Comp. (8)
4406.14
2822.29
3168
3497.49
2172.44
2103.50
4558.20
2872.79
2698
Total Comm. (MB)
1953.12
1464.71
1543.32
78124.61
58589.34
61748.22
1953.12
1465.02
1413.52
the partial activation approach typically resulted in slightly higher total error rate during training. However, significantly reduced both computational and communication costs, making a more practical and efficient solution for real-world applications.
5.4 ENHANCING COMPUTATION-COMMUNNET
EFFICIENCY WITH PARTIAL ACTIVATION
In the partial activation approach ("Random" and "Event"), fewer clients participated in the train- ing of each epoch, therefore reducing both computation and communication costs compared to the framework with "Full" activation.
Activation probability In the "Random" activation framework, the activation probability p de- termines the likelihood of client being activated in each round. A higher value of p increases the probability of activation, while lower value decreases it. We conducted the study on the activa- tion probability within the DLR-Random Framework under the non-stationary data stream, using window length of I = 10 and an attenuation coefficient a = 0.95. We selected p from the range (0.25, 0.50, 0.75 0.75, 1.00).
This observation indicates that the run-time error rate among the DLRRRandom models remains consistent across varying activation probabilities. Table presents the corresponding performance metrics of those trials. This indicates that a small activation probability proportionally decreases the computational cost for the client, albeit with a slight increase in the accumulated error Additionally, both client computation time and communication decrease, as passive clients do not participate in the backward process.
p= 0.25
Table 3: Performance metrics for different activation
p 0.50
p=7.
probability
p- 1.00
Activation Probability
Accum. Error Rate
Client Comp. (s)
Total
Comm. (MB)
0.25
0.0622
1903,91
1220.75
0.50
0.0561
2872.79
1465.02
0.75
0,0560
3788.07
1709.04
2.0 le6
Instance
1.00
0.0553
4558.20
1953.12
Figure 4: Activation probability p

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Published as a conference paper at ICLR 2025
Event activation threshold r Using the DLR-Event framework with window length of I 10 and an attenuation coefficient of a = 0.95, we examined the impact of varying the activation threshold. Table 4 presents the performance metrics for the DLR-Event with different r. As the threshold increased, fewer clients were activated per round, leading to an overall increase in the framework's s accumulated error rate while reducing its computational cost and communication costs. We further provided an examination of the activation rate for each client across various activation thresholds. As the threshold surpasses 0.6, few clients were activated in each round, the framework will mostly rely on the server for learning. with clients primarily offering a nearly invariant mapping of the input features. Conversely, the threshold decreases, clients can be activated more frequently.
Table 4: Performance metrics for activation threshold
Threshold
Accum. Error Rate
Client Comp. (s)
Total Comm. (MB)
0.6
0,1057
928.67
1017.76
0.2
0.0738 0.0595
1852.15
1211,55
-0.2
3752.07
1689.42
LIMITATIONS
Due to the inherent nature of VFL, wherein the clients possess non-intersecting feature sets, the involvement of passive clients each round remains unavoidable. Omitting passive participants leads to missing input at the server's model, which is known the "incomplete view" problem in VFL or the mnon-overlapping sample" problem within the offline VFL setting. Although some research on VFL has attempted address the incomplete view problem through knowledge distil- lation (KD) (Ren et al. 2022), self-supervised learning (SSL. (He et al., 2022] Li et al. 2022 Kang et al. 2022) and sumi-super learning (Li et al. 2024), these approaches entail more complex models and higher
cawntaatioo-oomminctaiiiiio costs. For instance, Ren et al. (2022) train student models for each case with an incomplete view via KD, which imposes a significant computational cost, especially when the number of clients is large. Another example is the work by He et al. (2022), which implements SSL within the VFL framework. This approach requires communication between participants during the computation-intensii pretraining stage. Consequently, in scenarios where participants have limited computational resources, there is no perfect solution to this problem.
The future direction of event-driven online VFI. focuses examining the complete independent data streams for each client by addressing the incomplete view problem within the VFL frame- work. Resolving this problem will eliminate the need for passive client participation in each event, thereby reducing computation and communication costs and enabling the implementation of fully asynchronous online VFL.
7 CONCLUSION
Adapting online learning to vertical federated learning poses challenges due to the unique nature of VFL. The clients may receive data streaming synchronously, as data generated by an "event" typically pertains to only a subset of the clients. To address these challenges, we proposed the Event-Driven Online Vertical Federated Learning framework. Within this framework, only a subset of clients is activated by the event in each round, with others passively participating in the learn- ing process. Moreover, we incorporate dynamic local regret to address non-convex models a non-stationary environment, which further enhances the adaptability of our framework for real- world applications. Through comprehensive regret analysis, we have derived the regret bound of O for dynamic local regret under the non-convex case. Through extensive experiments, we costs
have demonstrated the effectiveness of our approach in reducing conmunication.compuutiec and achieving strong performance in non-stationary environments. Overall, our contributions pave the way for more practical and adaptable implementations of VFL in roal-world scenarios.
10

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Published as a conference paper at ICLR 2025
ACKNOWLEDGMENTS
This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), Discovery Grants program.
REFERENCES
Jacob D Abernethy, Elad Hazan, and Alexander Rakhlin. Competing in the dark: An efficient algorithm for bandit linear optimization. 2009.
Naman Agarwal, Alon Gonen, and Elad Hazan. Learning in non-convex games with an optimization oracle. In Conference Learning Theory, pp. 18 29. PMLR, 2019.
Sergul Aydore, Tianhao Zhu, and Dean P Foster. Dynamic local regret for non-convex online fore- casting. Advances in neural information processing systems, 32, 2019.
Jonas Beuchert, Friedrich Solowjow, Sebastian Trimpe, and Thomas Seel. Overcoming band-
width limitations wireless sensor networks by exploitation of cyclic signal patterns: An event- triggered learning approach. Sensors, 20(1):260, 2020.
Timothy J Castiglia, Anirban Das, Shiqiang Wang, and Stacy Patterson. Compressed-vfl: Communication--ffciiee learning with vertically partitioned data. In International Conference on Machine Learning, pp. 2738-2766. PMLR, 2022.
Tianyi Chen, Xiao Jin, Yuejiao Sun, and Wotao Yin. Vafl: a method of vertical asynchronous federated learning. arXiv preprint arXiv:2007 2000:2007.06081,, 2020.
Nima Eshraghi and Ben Liang. Distributed online optimization over heterogeneous network with any-batch mirror descent. In International Conference on Machine Learning, pp. 2933-2942 PMLR, 2020.
Wenjing Fang, Derun Zhao, Jin Tan, Chaochao Chen, Chaofan Yu, Li Wang, Lei Wang, Jun Zhou, and Benyu Zhang. Large-scale secure xgb for vertical federated learning. In Proceedings of the 30th ACM International Conference on Information & Knowledge Management, pp. 443-452, 2021.
Fangcheng Fu, Xupeng Miao, Jiawei Jiang, Huanran Xue, and Bin Cui. Towards communication- efficient vertical federated learning training via cache-enabled local updates. arXiv preprint arXiv:2207. /4628, 2022.
Bhargav Ganguly and Vaneet Aggarwal. Online federated learning via non-stationary detection and adaptation amidst concept drift. IEEE/ACM Transactions on Networking, 2023.
Xiand Gao, Xiaobo Li, and Shuzhong Zhang. Online learning with non-convex losses and non- stationary regret. In International Conference Artificial Intelligence and Statistics, pp. 235- 243. PMLR, 2018
Bin Gu, An Xu, and Cheng Deng. heng huang. 2020. privacy-preserving asynchronous fed- erated learning algorithms for multi-party vertically collaborative learning. arXiv preprint vrXiv:2008.0625: 2020.
Stephen Hardy, Wilko Henecka, Hamish Ivey-Law, Richard Nock, Giorgio Patrini, Guillaume Smith, and Brian Thorne. Private federated learning on vertically partitioned data via entity resolution and additively homomorphic encryption. arXiv preprint
2017.
Elad Hazan and Satyen Kale. Extracting certainty from uncertainty: Regret bounded by variation in
costs. Machine learning, 80:165-188, 2010.
Flad Hazan, Alexander Rakhlin, and Peter Bartlett. Adaptive online gradient descent. Advances in neural information processing systems, 20, 2007.
Elad Hazan, Karan Singh, and Cyril Zhang. Efficient regret minimization non-convex games. In International Conference Machine Learning, pp. 1433-144) 1441, PMLR, 2017.

---

Published as a conference paper at ICLR 2025
Elad Hazan et al. Introduction to online convex optimization. Foundations and Trends@ in Opti- mization, 2(3-4):155-3333 2016.
Yuanqin He, Yan Kang, Xinyuan Zhao, Jiahuan Luo, Lixin Fan, Yuxing Han, and Qiang Yang. A hybrid self-supervised learning framework for vertical federated learning. arXiv preprint arXiv:2208. 08934, 2022.
WPM Heemels Heemels, MCF Donkers, and Andrew Teel. Periodic event-triggered control for linear systems. IEEE Transactions on automatic control, 58(4):847-861 2012.
Amélie Héliou, Matthieu Martin, Panayotis Mertikopoulos, and Thibaud Rahier. Online non-convex optimization with imperfect feedback. Advances in Neural Information Processing Systems, 33: 17224-17235, 2020.
SC Hoi, D Sahoo, Lu, and P Zhao. Online learning: A comprehensive survey, arxiv. arXiv preprint 00w.:0882002271, 2018.
Songnam Hong and Jeongmin Chae. Communication-effitie randomized algorithm for multi- kernel online federated learning. IEEE transactions on pattern analysis and machine intelligence, 44(12):9872-9886 2021.
Yaochen Hu, Di Niu, Jianming Yang, and Shengping Zhou. Fdml: A collaborative machine learn- ing framework for distributed features. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery Data Mining, pp. 2232-2240. 2019.
Zhenqi Huang, Sayan Mitra, and Nitin Vaidya. Differentially private distributed optimization. In Proceedings of the 2015 international conference on distributed computing and networking, pp- 1-10, 2015.
Yan Kang, Yang Liu, and Xinle Liang. Fedevt: Semi-supervised vertical federated learning with cross-view training. ACM Transactions on Intelligent Systems and Technology (TIST), 13(4): 1-16, 2022.
Sai Praneeth Karimireddy, Satyen Kale, Mehryar Mohri, Sashank Reddi, Sebastian Stich, and Ananda Theertha Suresh. Scaffold: Stochastic controlled averaging for federated learning. In International conference machine learning, pp. 1132-5143. PMLR, 2020.
Dohyeok Kwon, Jonghwan Park, and Songnam Hong. Tighter regret analysis and optimization of online federated learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2023.
Tian Li, Anit Kumar Sahu, Manzil Zaheer, Maziar Sanjabi, Ameet Talwalkar, and Virginia Smith. Federated optimization in heterogeneous networks. Proceedings of Machine learning and sys- tems,
2020.
Wenguo Li, Xinling Guo, Xu Jiao, Tiancheng Huang, Xiaoran Yan, and Yao Yang. Vertical federated learning hybrid local pre-training. arXiv preprint arXiv:2405. 118.24845.888, 1884, 2024.
Wenjie Li. Qiaolin Xia, Hao Cheng. Kouyin Xue, and Shu-Tao Xia. Vertical semi-federated learning for efficient online advertising. arXiv preprint arXiv:2209. 15635, 2022.
Xiaoxiao Li, Meirui Jiang, Xiaofei Zhang, Michael Kamp, and Qi Dou. Fedbn: Federated learning on non- iid features via local batch normalization. arXiv preprint arXiv:2102.07623, 2021.
Peixi Liu, Guangxu Zhu, Wei Jiang, Wu Luo, Jie Xu, and Shuguang Cui. Vertical federated edge learning with distributed integrated sensing and communication. IEEE Communications Letters, 2609):2091-2955, 2022.
Yang Liu, Yan Kang, Xinwei Zhang, Liping Li, Yong Cheng, Tianjian Chen, Mingyi Hong. and Qiang Yang. A communication efficient collaborative learning framework for distributed features. arXiv preprint arXiv: /87, 2019.
Yang Liu, Zhuo Ma, Ximeng Liu, Siqi Ma, Surya Nepal. Robert H Deng, and Kui Ren. Boosting privately: Federated extreme gradient boosting for mobile crowdsensing. In 2020 IEEE 40th International Conference Distributed Computing Systems (ICDCS), pp. 1-11. IEFF, 2020.
12

---

Published as a conference paper at ICLR 2025
Gaëlle Loosli, Stéphane Canu, and Léon Bottou. Training invariant support vector machines using selective sampling. In Léon Bottou, Olivier Chapelle, Dennis DeCoste, and Jason Weston (eds.), Large Scale Kernel Machines, pp. 301-320. MIT Press, Cambridge, MA., 2007. URL http:
//leon. bottou. wwwp://pppppp.oaao
Othmane Marfoq, Giovanni Neglia, Richard Vidal, and Laetitia Kameni. Personalized federated learning through local memorization. In International Conference on Machine Learning, pp- 15070-15092. PMLR, 2022.
Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas. coamunncation-eeficcenn learning of deep networks from decentralized data. In Artificial intelli- gence and statisties, pp. 1282. PMLR, 2017.
H Brendan McMahan, Gary Holt, David Sculley, Michael Young, Dietmar Ebner, Julian Grady, Lan Nie, Todd Phillips, Eugene Davydov, Daniel Golovin, et al. Ad click prediction: view from the trenches. In Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 1222 1230, 2013.
Konstantin Mishchenko, Eduard Gorbunov, Martin Takáč, and Peter Richtárik, Distributed learning with compressed gradient differences. arXiv preprint arXiv:1901.0900 2019.
Aritra Mitra, Hamed Hassani, and George Pappas. Online federated learning. In 2021 60th IEEE Conference on Decision and Control (CDC), pp. 4083-4090. IEEE, 2021.
Vaikkunth Mugumthan, Antigoni Polychroniadou, David Byrd, and Tucker Hybinette Balch. Sm- pai: Secure multi-party computation for federated learning. In Proceedings of the NeurIPS 2019 Workshop on Robust Al in Financial Services. 2019.
Tao Qi, Fangzhao Wu, Chuhan Wu, Lingjuan Lyu, Tong Xu, Hao Liao, Zhongliang Yang, Yongfeng Huang, and Xing Xie. Fairvfl: A fair vertical federated learning framework with contrastive adversarial learning. Advances Neural Information Processing Systems, 35:7852-7865, 2022.
Thilina Ranbaduge and Ming Ding. Differentially private vertical federated learning. arXiv preprint arXiv:2211.06785 2022.
Zhenghang Ren, Liu Yang, and Kai Chen. Improving availability of vertical federated learning: Relaxing inference non-overlapping data. ACM Transactions Intelligent Systems and Tech- nology (TIST), 13(4):1-20, 2022.
Doyen Sahoo, Quang Pham, Jing Lu, and Steven C. H. Hoi. Online deep learning: Learning deep neural networks on the fly. In Proceedings of the Teeeth-Seventh International Joint Conference on Artificial Intelligence, Jun 2018. doi: org/ 2ttp:/0..4493// /ttp:////caaaa..11//..
20188/////. URL http: //xx doi.
Shai Shalev-Shwartz and Yoram Singer. A primal-dual perspective of online learning algorithms. Machine Learning, 69:115-142, 2007.
Arun Sai Suggala and Praneeth Netrapalli, Online non-convex learning: Following the perturbed leader is optimal. In Algorithmic Learning Theory, pp. 441- PMLR, 2020.
Young Soo Suh. Send-on-delta sensor data transmission with linear predictor. Sensors, 7(4): 537-547, 2007.
Sebastian Trimpe and Raffaello D' Andrea. Event-based state estimation with variance-based trig- gering. IEEE Transactions on Automatic Control, 59612))3266-3281.22 2014.
Praneeth Vepakomma, Otkrist Gupta, Tristan Swedish, and Ramesh Raskar. Split learning for health: Distributed deep learning without sharing raw patient data. arXiv preprint arXiv:1812.0001 2018.
Ganyu Wang. Bin Gu, Qingsong Zhang, Xiang Li, Boyu Wang, and Charles Ling. A unified lution for privacy and communication efficiency in vertical federated learning. In shirth- Conference Newral Information Processing Systems, 2023.
13

---

Published as a conference paper at ICLR 2025
Ganyu Wang, Qingsong Zhang, Xiang Li, Boyu Wang, Bin Gu, and Charles X Ling. Secure and fast asynchronous vertical federated learning via cascaded hybrid optimization. Machine Learning, 113(9):6413-645) 6451,, 2024.
Heqiang Wang and Jie Xu. Online vertical federated learning for cooperative spectrum sensing. arXiv preprini arXiv:2312.11360 2023.
Yujia Wang, Lu Lin, and Jinghui Chen. Communication-effiti adaptive federated learning. arXiv preprint arXiv:2205.0277 2022.
Kang Wei, Jun Li, Ming Ding. Chuan Ma, Howard H Yang, Farhad Farokhi, Shi Jin, Tony QS Quek, and H Vincent Poor. Federated learning with differential privacy: Algorithms and performance analysis. IEEE Transactions on Information Forensics and Security, 15:3454-3469 2020.
Kang Wei, Jun Li, Chuan Ma, Ming Ding, Sha Wei, Fan Wu, Guihai Chen, and Thilina Ran- baduge. Vertical federated learning: Challenges, methodologies and experiments. arXiv preprint arXiv:2202.08 04300, 2022.
Daniel Whiteson. HIGGS.
UCI
Machine
Learning Repository, 2014a.
DOI:
http:://bo.org/11.244/////
Daniel Whiteson.
SUSY.
UCI Machine
Learning
Repository, 2014b.
DOI:
https///bb.orr/10.4452/////
Kai Yang, Tao Fan, Tianjian Chen, Yuanming Shi, and Qiang Yang. quasi-newton method based vertical federated learning framework for logistic regression. arXiv preprint arXiv:1912.00513, 2019.
Ke Zhang, Ganyu Wang, Han Li, Yulong Wang, Hong Chen, and Bin Gu. Asynchronous vertical federated learning for kemelized auc maximization. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 4244-4255, 2024.
Qingsong Zhang, Bin Gu, Zhiyuan Dang, Cheng Deng, and Heng Huang. Desirable companion for vertical federated learning: New zeroth-order gradient based algorithm. In Proceedings of the 30th ACM International Conference on Information & Knowledge Management, pp. 2598-2607, 2021a.
Qingsong Zhang, Bin Gu, Cheng Deng, Songxiang Gu, Liefeng Bo, Jian Pei, and Heng Huang. Asysqn: Faster vertical federated learning algorithms with better computation resource utilization. In Proceedings of the 27th ACM SIGKDD conference on knowledge discovery data mining, pp. 3917-3927, 2021b.
Qingsong Zhang, Bin Gu, Cheng Deng, and Heng Huang. Secure bilevel asynchronous vertical federated learning with backward updating. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pp. 10896-10904, 2021c.
Jun Zhou, Chaochao Chen, Longfei Zheng, Huiwen Wu, Jia Wu, Xiaolin Zheng, Bingzhe Wu, Ziqi Liu, and Li Wang- Vertically federated graph neural network for privacy-preserving node classification, arXiv preprint arXiv: //903, 2020.
Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th international conference machine learning (icml-03), pp. 928-936. 2003.
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A REGRET ANALYSIS: NON-CONVEX CASE WITH DYNAMIC LOCAL REGRET
Proof of Theorem
!
Taking Expectation w.r.t
Exg [S., (wb+) w w St. (ur6 w')]
EEz, ( 1. (ugt,, w - -ww w')) + Bxs I1luttt w 17---ig -Iwi, w'lll
2
Ber (VSria( w'), WegSee (w w¹)) Bas
V w²):
E
V a. (wos w/)
mEA.
+ 2
Life Bos - Suiiaa w²)l + 2
Lif
Σ
Exe
4 wm Suiaawo w²)l
meA.
(VSLaaau w'), V Seia w"))
Lufe
V
uny Se w²) /
Ex,.
Life
V.L.Lawu w'),
Σ mEd:
V Sa (wo) w²)
+
Σ
Ess
V Wm Se,r, (wo w³)
2
mEAi
(13)
where 1) applies the Lipschitz Continuous of St,.a (w), w²). 2) applies the update in round of the event-driven online VFL.
For a)
-,EE V (wg. w'), Vwo SeLa(u w")) + Lof E., | no Seo w') /
1) B.E.r
VSSLaa (wo: w²
Tab
Ex, Auo St,1. (aea(uuff (wo w²) - wp (u wt) + V Vwo St wi)
+ 2
VSe.aa w² 1
+ LoN | Sti.a (w), w') - Vay Su. wt)ll + LIj | wo St. w²)l
= (11 Lnp) Ev,
w²)l
+ Lur 4 St (wh, w²) Van Su w²)l"
(14)
where 1) applies assumption] 2 (unbiased gradient). 2) applies ||a + 6112 < 2 ||a]]" + 2b6]
For b)
Ln
-1,E..
V VS,Aa w!).
E
V Vun (wg w²))
Σ
Exe
4 wm Seaaaw w!))
2
meA,
meA.
=- --nEE
V.S, w'),
Σ
Wm Sesaa w²)
Σ
um St,,a w²) +
Σ
V Se. w')
mEA.
mEAr
mEA.
+ Luf
Σ
Exs
V Wm
(eh w²) V Se.aa w') + V
St,1,o (wof wt))
meA.
15

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Published as a conference paper at ICLR 2025
Σ
V wm (w (g'
mEAt
- -1
VSr,a, w'):
Σ
Vum (u w²)
Σ
V
um Saaa w²)
mEd.
mEde
m.
+ 2
Lak
Σ Exi
Wm Sehaawg w²) V Viw Sedaauy w²) + V wm St,1,a (wgg w//l|
mEA
2 St.la wo)
T/p
Bez
mEA+
- -2,
VSSLaa w'). Σ
Vun Sudalub.
Σ
Vun. um Suaa(csh w')
mEA.
mEd.
+ Luh Σ Exz,
Mm (1. w*) - um Se, w'))
mEAt
V 10m Se.la w')l|
+ Li
Ezg
mEA,
1/ Bur
V am (wo w²)
mEA:
VSI,, w²).
E
Vun Setalwh. w²)
Σ
V Vun Seca(wgg w²)
men.
men.
V Wm (e. w²) - V Wm Se, (wo: wi)l l1?
+ Lan
Σ Raz,
mEA.
+ Lif
Ex,
Vum (e..a wi)l?2 + Lin
Σ Ex,
Vwn. Selaauoo w') V Vwn (ess w
med,
meA,
=- (n - Ink Exz,
Σ
Var Sa w()
imA.
V.S.aaau w
V um (e,,a
w²)
Σ
V
um Sela w³)
meA.
med,
+ Tan Σ Eizz
4 Wm St,,a wh) - V um Se,t (wo: w³)
meA.
+ Lin meA,
Σ
Exi
V Wm Se. w') Vun Sa, w²)l
(15)
where "applies assumption 2 (unbiased gradient),
2) applies ||a + 6112 < ||a| 2 1bbl
3) E(x) E(X)² (X)
Plug in a) and b) and taking expectation w.r.t. the activated client m E Ad, where the activation probability of client m Pm. we derive:
Em, Ez, [Se,,a(wttt w St.,aawff w()]
< (n1 Tnf)) B.,, V St. w²)l1
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| Sed.a (wo; w') V un Sena (wo; wty//
+ Enf
- -(n Lnn)
Σ
Pmllia: |/ wm Staa (wo w²)l
mE(M]
Σ
Pmn V uim St.tia (wos w'), amm St w') - V Fwom (a w²))
mE[M]
+ Lif
Σ
Pr.Ex
V Vee (e w²) V wm (wi w')l²
mEiM
+ Lin Σ
Pin PunBar
V um (w w') V Ven St.era ( w')l|
MEM
1)
3 (n: Li²) PuninEEr || V (wb, w²)l|
+ Lui Pmax Exi, | IVSuLa w') VS..a (wh, w²)l l|?
Σ
Pm V um Se,t (wg w'), V am Setia (wo, w') - V Vm Ser w'))
mE[M]
Pmn Baz, A (w)) w²) V Vwn Ser wo)/:::)/// wt)/?
+ Laf Σ
mE M
2)
$ - (n Lip
Puin Eg,
V.S.La.a w²)l?
0²
a
+ Lak Pmax
W22
1- a²
Σ
Van Se.lia w'), Van. Sela(wh w') V Vwe Se w'))
Pm
ME(M]
E Br. A um (w w') &un (2 w')l "o'':w")
+ Lin
ME
(ne Liã) Pmiin Ear, V. w²)l|
σ²
--a22 25
+ Lni Pmax
W²2 1 1- a²
-npBe Σ Pm V um St,1,0 (wor w'), Van., Se, (wir w²) - V. Sero w²))
mE(M]
+ Tay Pmax 2
(16)
where 1) note that m is in different dimension,
2) applies assumption g (bounded variance), and is weighted average of independently, therefore Ex, 3) plug in e).
w") VS..a w³)l I WT
1-aa1
T-ar.
For c)
Σ Pm Eur mE(M]
4 um Su, (wg, w²) - 4 wm. Se,laauf w()
1-1
Σ
Pm Bay
V w...
fz (10/ ww'-)). (1 -
W
mE(M]
1=0
17

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Published as a conference paper at ICLR 2025
1-1
Σ
Σ
(wn( w) (1 -1)
W2
mE(M]
Σ
Σ
a' (1-1-mm1 ))
4 p w-
W2
Pin
mE|M
i=0
[-1
2
Σ
Σ
a²
fu , 1|
W/2
ME
1=0
mm Srmiini 1-i w ( |
Pm
mE[M]
3)
PIna V (im (w/ , wt- ||?
Σ
ME[M]
4) Pmax G2
(17)
where 1) by triangle inequality
E/=o a wm Vwnn (wb w/'1), . t)
E a Wm (wn t-i w w (1 -1- s Ei a' )) | Vwm (w) i-j w w/)
2) W E.- at
3) denote maxm {Pm) Pmax.
4) applies assumption 4 (bounded gradient).
Taking expectation w.r.t. time step 1 Unif((T)), and use E to denote the expectation
E,E,E We can eliminate ( V Vwn Su,ta (w). w'), V Van.. Std (whr w') Van. Seaaaus w')).
because E, V A.S. (wi, w') = E, V V.S.,,. w²).
E (w w(+1)= - Se,a (woi w¹]] S (n LTT) Pmin V w')ll
o² 1 - 21
+ Loi Punax
1 a-a + Ln Paaxx G2
W2
(18)
Rearrange the above equation we have:
(m - -n²) Pmin E
|| V w/)l1
E S (w/, w') St,ia (wo+)
w((]]]
21
+ Pmxxxxx Pmax
W2 1 a. + Inn Pmax 22
( w') ES wtth
+ E. ESSL 11 w(t) - (. w't)
d)
c)
0²
21
+ Lij Pmax
+ LLn LnpPpax 2²
W2
2D a1-a')
D
(1+a'1)) + (1 (1-ah a1-at-1)(100)
<
(1
W
1
W
21 1 a
+ Lnzpmax Pmax
+ Lnn Pmax 2²
(19)
W
The remaining follow the same procedure as Aydore et al. (019)) for brevity, we use the lemma on their paper. For d) we apply the (Aydore et al. 2019 Lemma 3.3) and derive E.S.,,, (w6. w*)
18

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Published as a conference paper at ICLR 2025
1, f+1 w(t1) v 2D (1-a'), For e) we apply the (ydore et al. 2019 Lemma 3.2) and
W I-a
derive (L
1+1 w(ll) (w) ++1 w(fl) s w (1 (1+a'-1) + (1 (1+a) 1)
plugs d) and e).
Divide both side by (Me Lag Pmin-
E V (w), w')l
2D (1-a') D
(1 + 1 + (1-aa="(1+a)
d² 1-a Luk Pinax
Lng Prmax G2
W
1-a
W 1-
(11 Lmiin
1) 4D
(1 (1-d')
2D
(1 (1+a'=1)+ 11-1111-11000 1-1a-1)(11 a)
+
1/ WPann
1 a
W Pinin
1 a
Pmax
0²
21
Pmax G²
+ 2Lm,
+ +2L1
Pmin
W2
1 a²
Pmin
2D
2(1
at)
(1 a² a'=1)(1+a))
+ +1+aa'1)
T/ W Pmin
1
1-a
0² 1-a 21
Pmax
Puax G²
+ 2LT,
+ +2L1
(20)
Pmin
W2
1- a²
Pmin
where note that when т)
ZI 1/e
Laj s Z
Follow the proof of (Aydore al. 2019 Theorem 3.4), 1
lim
E VSL. (wo: w²)l|
a=
8D
2Lm
Pmax
o²
1 1 at
2Lm
Pmax G² Pmin
n Pmin
Pmin
W2
1-a2
1
8D
2mpp
+ 2
Pmax G2
(21)
W Pmin
1)t
Piii
Summing from = 0, 1,..7 concludes the proof.
(LR(() = Σ lim E SelV w')ll2
1= a-1
T
18D
+ 2mpppaxeo?
+ 2T 2TLm
Puax G²
(22)
W
Pmin
1J
Pmin
Proof of Corollary 7
Select
T and 17. T 4, note that lim W lim 1-a¹ d.
Σ lim E / St (wh, w'll?
a-1
t=0
11
T
8D
+ 2.
2T
Pmax G² Pmin
Ipmin
n
T
(SDT' + 2LT
imaxx + 2T Lap
Pmax 2 Pmin
VTPmiu
8D
2T 2mp
2I 2T G2
T +
T
T
Pmiiu
Pmin
Puin
=0(T?)
(23)
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Published as a conference paper at ICLR 2025
B REGRET ANALYSIS FOR OGD-EVENT IN CONVEX CASE
In the convex case, OGD already an efficient algorithm to solve the online learning problem with partial client activation.
We start by defining Regret in the VFL framework below.
Definition 3. Regret for online convex optimization in VFL framewokk
RT
Ef(ub w() E(wg
f(w w")
(24)
where [t0g*, w*] = argmin
Et= 1= I (woj (w)
[ug w*]
Following the online convex optimization (Hazan al. 2016). design the event-driven online
VFL with online gradient descent (OGI-Event in Section 5.
The algorithm is provided in the
algorithm Sooritthm@bbllo below.
Algorithm Event-driven online VFL with online gradient descent (OGD-Event)
Input:
Output: model parameter Win for all workers (0, 1, M... 0: Initialize Wm for all participants m E {0, .. for E [T'] do
2: for m A, do
3: client m send hmm (W): Fmm) to the server.
4: end for
5: for m A do
6: Server queries the embeddings from passive client
7: Client m send hm (wmi 2m) to the server.
8:
end for
(f(w.))) Sulg
9: The server updates its model wi + WF 1o0
10: for m E A, do
af(w,,w) to the client m.
11: Server send Um Sh...
Client m update parameter Wm <- Wm 7m Um
ahm. 8w
end for 14: end for
We use different set of assumptions on convexity, the diameter of the space, and the client's 's delay make the regret analysis framework for online convex optimization with Algorithm 2
Assumption 6. Convexity: for any (wo, w] and [w/. w'), wo; w/ IRdo, Win: w'n
Rd...
(' w) satisfy
('wyy w') Z fw(wo, w) + (V f w), (wy) w'] - [wo,, w/)
(25)
Assumption 7. Bounded space diameter: For any (Wo, w] and [wgg, w'), satisfy
[|[wo. w [wh,, w'll S D
(26)
Assumption 8. Independent client activation: The activated client my the global iteration t is independent of ma. Mp-1 and satisfies F(m, = =m) := Pm-
Assumption 9, Uniformly bounded delay: For each client m, the delay at each global iteration is bounded by a constant i.e. T <.
Theorem 2. Under Assumptions ~~ to solve the online VFL problem with partial client partici- pation using Algorithm 2
the following inequality holds.
3G + 4D3 + 2 VT
Rr
(27)
2G min {pm)
where the learning rate is chosen Tt GVt
D
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Remark 3. The regret is sublinear O(VT).
Proof of Theorem
First, bound the update of the participants. For notation brevity, V W., [ w) and V Vun (Wo, w) are the partial derivative w.r.t the corresponding parameter in the space of the pa- rameter of the glohal model, where the position of all other parameters are filled with 0.
[1o c-1 [ [wg, w*ll
(1o:: w/] - 7/e Vwn, S' w/ T/e Σ
V Vun Jwrgs w²)
[wo w-]
mEA.
[[og] w'1 - [[2o,, w']l + i
V Vua f(wo, wr) +
V Σ
Wm (we w²)
mEA,
2mp V woo [wogs w²*) Σ
Wm (wrs w²), [mo w²] [urō= w']
mEA.
[ w'1 - (ww w'1l12 1T G2
2mh V Vaoo Jwe' w²+++ Σ
V Vum (wir w²),, [wc. w'l - -(wrō= w'1
mEA:
-I[uo0. w'1 - [wog w'll v/t 2
2n7 (V we (wo, w'), (wo. w'] Iwo w'l)
2m Σ
Vum f( fwoj (wg, w'J -wr] w"]
men.
= =I[(vó w') - [wo w'1l22 + n/ G2
2mp I w²) - I w') + Vay (wo w²), [wg w"] -wwi w'))
Σ
Wmn (wr. W¹)
Vwn I (wg, w²) +
V Vun ( (wo w²), (1o6- w') - [106 w/)
meAi
mEAe
mEA.
- -I[iug w'1 - [w, w'1l2 G²
+ 2 V "w (wg, w²) - Vio (wij w²), mh (lwg) w'] [ur w'))) 217, (V no [ w'), [") w'] - -rug w'))
+
Σ
Wm I (woj w²)
Σ
V
Vumf [wi w'), ne (lwy w'] - [wo. w'l)
me.A
mEA.
2m
Σ AmAA.
Wim (wor w'), (wr w'J - -[u w")
41
[ w'] - [1og w1]l| 1/2 G2
+ Vnn Vwn w') - Vw, w w²|l + n/2 (wwb. w']-[w w")) ||
27p (V wg fwob w²), (wg wt] (wg w*))
+
V mm (wi) w²)
V Vum fwuy w²r)
+ m/ ([ww) w'1 - -[w w 1)[
imAA
mEA.
2m
Vum (woj, wx), [ww w'] -mw w"]
mEA.
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Published as a conference paper at ICLR 2025
[weg: w"| - -(wr w w*ll n
+ V Jwo w') + Σ
V Vem f(wo; w') - V Wwo((ut w²)
Σ
V Vu (wo w')
mE Ar
m mEAe
+ n/ (1wwg w'] - www- w w1]]11
-2 (V Wuoffuit ("),(ug)) w') - w'l)
+ ni (1wgg. w'] - [w6: w w11)1|² -2n
Σ mmEA,
V Vunf w'), [eog w'1 - 120 w'l
6)
- [[ w'1 - [w06, w'1l + 12262
+ 1 1/w² - -11
+ 2n || (lwhr w²) [wg= w*))
2n (V wa (wog w'), (wg. w'] lwo: w'l) -2m
Σ
V Vun fwub. w'), w'] fwo: w"]
meA.
2 [[ug w"] - [wO w'll + v/6 G² + 22 max }
1E(T]
+ 2n/ || (lwj, w'] - [wij w")) 1
- 2m (V ww (wol, w [uv w'] - wwg. w")) - -2m
Vun f(eeg [w), w/] [wo w"]
me A
S [[:v w') - [wo w w'll G/ GG2
max { + D2² sE/T]
2m W [w), w²] - [wg: w")) - -21je
Σ mmEA A
V Vun. f w²), ['), w²] [-- [wo w")
(28)
where 1) is the partial activation of clients and server optimization step at time step 1, 2) note that (V [wwj w w²)]mee(juus are in the non-intersect dimensions, 3) applies assumption (bounded
gradient), 4) applies (a, b) < 1 ||a| + |bb|.. 5) {V mm f [woj w w') mmm)muu are the non- intersect dimensions, 6) applies assumption assumption [ (Lipschitz Gradient). 7) applies Eq. 29
below, 8) applies assumption 7 (bounded parameter diameter).
wn w111
( ++1-
w(-))
w/ 1-1 w/ 111 2
<T
Wm f(w)/ wr)
- T/
mEAL-
2
Σ
V Wm Fwub w/(-))
111
1=1
Aa-s
ST
1=1
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Published as a conference paper at ICLR 2025
<=² G² max { sE[r]
(29)
where 1) applies assumption (uniformly bounded delay), 2) by Cauchy-Schwarz inequality, EE xi - (Ei 1.1;)
>n xi. 3) applies assumption (bounded gradient), noting that the gradients are in non-intersect dimensions.
Rearrange Eq. 28
21/ (V moo Woo w"), [wo w'] [wg w²)) 27je
Σ
V Wm Jung w'), w'] [wog= w*)
mEA.
- (wo, w"]l 1[ + w - [wug w w'll| < |[uó w'1
+ T/e G² G²²
max {nh- + 2012 D²
[E[r]
(30)
Taking expectation w.r.t. the activation client sets A, from both sides.
2mp (V f w w*), [wog w'] - [wn. w=1) + 2nJe
Σ
Pm V wm Ff f(ug. w'), [wg. w²1 - -iwō w')
m==1
< [w] w"] [wg: w w*]l TE ||( w --ft)][0 [wr. w']l + n/2 + 1,, G? max
{12-1) + 27/ D2
iE(T]
(31)
Taking the minimum pm:
M
21 min {pm } (V wn (wo, w²), [wá. w²] - -wo w*]) + 21/² min {pm)
Σ
V
1mm swn. w²), [wo. w'] - -(wō= w*)
m
<B [wo w"1 - -wo w*1l|2 IE || w [wrg, w*]l + G²
2 G2
max (1i11} + 21 1D²
kE[r]
(32)
Combining the gradient from different dimensions, and rearranging the equation:
V (wo. [wig, w'), [wo: w"] - [wo w w'}>
[E ||[ugr w'1 - -1w w P- EE ||wbt +1 w w -lw6. w w=1l()
21p min {pm}
11 G/2 + G2 max
:E(r] { + 21 2 D22
2m min {pm}
1) 1
Q
+ G2 + G? max (ni-e) + 21, DD2
(33)
2 min {pm)
17t
where 1) denote Q = E [[ w²] - [urg w']l1 E (| w - [wr w w*]l for brevity.
By convexity (assumption
("(wi, w²) - -f f' w*) < (w.. f* [wo w'] - [(w' w'))
(34)
Summing over 1..., and we a diminish learning rate m) GVt with (1 & 0):
( w²) - - (' w")
((V "" w') - p w1))
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Q + G² + G2 max {nh-s} + 2070 DD?
2 min {pm]
t=
1
E Il[wg= w'] - -wug w"]l
[wo+t w
[wg: w'll
[mn + + + -1?r3)3)
2 2in {pm]
N
1
E ||wgg w'] [uog, w']l
(1
(G² + -222+222773))
+
2 min (pm)
1/1
M|++1
tm
41
1
D2 1
(n( + 2D2 - =1
2 2mi [pm}
1
T + (G² + T?, G?)
1
D²
Σ
2 min {pm]
1/7
t=1
1
GDVT+(GD + 2D3
+ + 2
2 2ii (Pm)
G
3G + 4D3 + 2 VT
(35)
2G min (pm)
where 1) applies Eq. 33
2) т diminish. Expand Q
3) e 0, and [[x] w T
(wg. wI| 20 0,
no
4) applies assumption 7 (bounded parameter diameter).
5) EE VT 22T..
The proof Theorem is complete.
C EXTRA DETAILS
C.1 ALGORITHM 1 IN A SYNCHRONOUS MANNER
We also provide synchronous version of the algorithm below.
Algorithm -ivien online VFL on Dynamic Local Regret
Input: hyperparameter a,
Output: server model wo. client models wm E [M]] 0: initialize model wm for participants m E {0, 1, M)
1: for / E [T) do
2: for m Ar do
3: activated client m send Imm
(w km oft)) to the server.
4: end for
5: for Å₁ do
6: server queries the passive client m.
7: passive client vn send hm (w ) to the server.
8: end for
9: server updates its model w
++1
w/
170
Vuy wt)
10: for m A, do
as, c. ( (w)
server send v.. ahm (ari, sn
to the client m.
client m update parameter w++1
w 7m
Um
Dh. Fw
13: end for
14: end for
24

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end for Published as a conference paper at ICLR 2025
C.2 ALGORITHM: OGD-FULL
Below we introduce the algorithm for the OGD-Full baseline in Algorithm adapted from Wang & Xu (,023), and transformed into a partial gradient-baser approach similar the split neural network method proposed by Vepakomma et al. (2018). It is important to highlight that the utilization of multiple local updates Wang Xuj (2023) is not applicable within the context our study on the general VFL framework because the multiple local update approach requires that clients possess
labeled data, which does not align with the setting in our paper.
Algorithm 4 Online VFL with Online Gradient Descent (OGD-Full)
Input: Learning rate 17
Output: Model parameter Wm for all workers m E {0, 1, L...M. 0: Initialize W., for all participants E {0, 1..
1: for t E [T] do
2: for m [M] do
3: Client m send 1tm (Wii Fim) to the server.
4: end for
5: The server updates its model wo + 170
(f..
Chon
6: for m [M] do
7: Server send U/m 8kmm
af(w,.)) the client m.
8: Client update parameter Wm <- Wm 7m Um
Ok.. Them
9: end for
10:
C.3 ALGORITHM: EVENT-DRIVEN ONLINE VFL USING STATIC LOCAL REGRET
We also provide a Static Time-Smoothed Stochastic Gradient Descent algorithm (Aydore et al,, 2019 Hazan et al. 2017) which adapted Event-Driven Online VFL. Note that the static slide window (wo w') = El (w- w*), where the time step for the parameter is L.
Algorithm Event-driven online VFL on Static Local Regret
Input: Hyperparameter a, 17, fixed model w"
Output: Server model wo, client models wm E [M] 0: Initialize model wm for all participants m E {0, 10.
1: for E [T] do
2: for m E A do
3: Activated client m sends window of embeddings { { (w ( + to the server.
4: end for
5: for A. do
6: Server queries the passive client m, for window of embeddings length /
7: Passive clients sends the window of embeddings {lmm (w., ) to the server.
8:
end for
9: Server updates its model f-1 +- w No Vung Fe,((g w*)
10:
# Fe, Fr,( (wf, w²) = + V Wwo -wir w*)
11:
for m A, do
Server sends window of gradient V Vmm fr (w-i w w²) 1/1 to client m.
13:
Client m updates parameter ua++1 < wi 7m V w.n (wg w*)
14:
#
V Wm (i. w²)
Σ 0= wm ft (wi, w')
+E V ft (wo w') Vun hym (w z
15:
end for
16:
end for
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D SUPPLEMENT EXPERIMENTS
D.1 ABLATION STUDY OF DLR PARAMETERS
In order to achieve thorough comprehension of applying DLR in the VFL framework, conduct an ablation study on several key parameters, including the sliding window length 4, and attenuation coefficient a.
Sliding window length 1 The sliding window length / determines the length of the exponential average window of the DLR. A larger window implies that greater number of past gradients influence the update at the current time. In the DIR-Full framework, employed under the con- cept drift data stream and with an attenuation coefficient of 0.95, we vary the window length I = 10, 50, 100, 150, and conduct the training. It's worth noting that 0.95 0.60, 0.95 s 0.08, and 0.95 200 0.006 represent three different shapes the exponential sliding average window.
Figure 5 presents the run-time error rate of different window lengths in the DLR-Full framework. As observed in the figure, the DLR is stable across different window lengths, which consistent with the result reported by Aydore al. 2019).
0.30
10
0.25
= 50
1 100
0.20
Error Rate
0.15
0.10
0.05
0.00
0.0
0.5
1.0
1.5
2.0
Instance
1e6
Figure 5: Ablation study on window length /
Attenuation coefficient a The attenuation coefficient of dynamic local regret influences the weighting of past gradients: a larger a indicates that the influence of past gradients decreases more slowly, while a smaller a leads to quicker attenuation of the effect of old parameters. For our experiment, we utilize a sliding window length of 100 and vary a across 0.99, 0.95, and 0.9 in DLR-Full framework. Notably, 0.99100 0.37, 0.95 ~ 0.01, and 0.944 as 0.01 represent three distinct shapes of the exponential averaging window. Figure Bollusttill the run-time error rate for different values of a. We observed that better results were obtained for 0.95 and 0.9. Therefore, we recommend using exponential average windows where the tail approximates 0.
D.2 EXPERIMENT ON OTHER ONLINE LEARNING DATASET
a physics dataset from the UCI repository. is clas- sification problem to distinguish between a signal process that produces supersymmetric particles and a background process that does not. Eighteen features are used for each sample to determine whether it belongs to the signal or background class. The first 8 features are kinematic properties measured by the particle detectors in the accelerator. The last ten features are functions of the first 8 features; these are high-level features derived by physicists to help discriminate between the two classes. The preprocess method follows the same process in Sahoo et al. (0188). In the experiment, two clients are employed, and each holds half of the feature set. Each client models composed of four fully-connected layers with the following neuron configurations: 32, 64, 98, and an output layer containing 128 neurons. After each hidden layer, Rectified Linear Unit (ReLU) activation function is applied to introduce non-linean non-linearity. On the server side, a five-layer MLP is employed. This model integrates the concatenated outputs from the client models and processes them through
SUSY dataset SUSY WWiiesson 2014b)
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0.30
0.99
0.25
M
Q=
0.95
0.90
0.20
Error Rate
0.15
0.10
0.05
0.00
0.0
0.5
1.0
1.5
2.0
Instance
1e6
Figure 6: Attemuation coefficient a
successive fully-connected layers with neuron counts of 256, 128, 64, and 32, ending with an output layer with neurons. We tested various learning rates within the range [0.01, 0.003, 0.001] to identify the optimal setting for models under different experimental conditions. The exponential weighted sliding window's ' length of the DLR is tuned from {10, 50). and the attenuation coeffi- cient from (0.95, 0.99). The activation probability p for the "Random" activation is selected from (0.25, 0.5, 0.75). The activation threshold I is tuned from {-1, -055, 0, 0.5). The results for both the stationary and non-stationary data streams are depicted in Figures and [] respectively. These figures demonstrate that the DLR model achieves stable convergence in both Full and Partial activa- tion modes, consistently outperforming OGD across varying data conditions. Additionally, DLR is learning faster than SLR in both scenarios.
GCO
SUR
DLR
0.5
Gindal
Biedut
OCD-Evert
SUN-Dvert
Duh-Cvert
0.2
0.2
0.2
6.-
1.50
0.2
2.00
Figure 7: SUSY dataset, stationary case
0.5
0.5
03
DCD Random
SUN Sundorr SURivve!
0.5
6.5
D..NEvent
Enveitaae
50
0.22
0.23
1.00
3.2
2.50
1.70
6/
0.22
0.23
3.00
122
1.50
1.75
2.50
0.0
0.20
0.75
1.00
225
130
1.5
200
11220000
Indarro
Iestanoo
Figure 8: SUSY dataset, non-stationary case
HIGGS dataset HIGGS (Whiteson 2014a) is also physics dataset from UCI repository. It is a classification challenge aimed at distinguishing between a signal process that produces Higgs bosons and background process that does not. Each sample comprises 28 features: 21 low-level
27

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features representing kinematic properties measured by particle detectors in the accelerator and high-level features derived from the first 21. The model architecture employed in this experiment is identical that used in the SUSY study, with the sole difference being the number of input features. The hyperparameter tuning procedure also remains consistent with the previous experiment. The results for both stationary and non-stationary conditions are depicted in Figure | and Figure [T] respectively. These results illustrate that the DLR model achieves comparable performance under stationary conditions and exhibits enhanced stability under non-stationary conditions. Furthermore, DLR demonstrates faster learning compared to SLR, with less initial stagnation at the early phases of training.
SUNPUl
DURPul
0.43
060 hondor
0.43
SU8 Randon
0.43
DUR Funder
0:43
0.42
0.43
0.30
0.25
0.23
0.25
0.20
0.20
0.33
0.2
0.53
8.5
1,51
125
1.53
1.25
2:55
0.10
0.3
3.5
0.58
0.75
1.00
1.20
1.50
2.50
0.33
3355
0.52
0.75
1.30
3.55
1.50
3.75
2.50
Interce
Instance
Inderne
Figure 9: HIGGS dataset, stationary case
SUR
0.50
0.30
0.30
0.4
Rondor
0.43
SUN Randorr
0.43
OE1h Evees
SUN Eveel
0:33
0.37
0.30
0.30
0.25
0.25
0.15
0.33
0.70
1.52
3.25
1.50
0.13
0.20
0.25
0.50
0.15
1.50
1.30
3.75
2.50
0.00
0.20
225
1.50
3.75
2.00
800ance
366
Figure 10: HIGGS dataset, non statioott case
D.3 ADDITIONAL EXPERIMENT ON EVENT-DRIVEN ONLINE VFL.
Scalability of the framework (number of total clients): To evaluate scalability, we increase the total number of clients the experiment described in Section 5.2 on the stationary iMNIST dataset to 8 and 16 clients. Most settings remain consistent with those described in Section 5.I with the number of clients increased the input size of each client's model is adjusted accordingly. While the output size of the clients remains unchanged, the input size of the server model scaled up to accommodate the increased number of clients. The results for 8 clients are presented in Figure [1] while the results for 16 clients are presented in Figure 12 Those figures demonstrate that when scale up the number of clients, the conclusion remain consistent: OGD is less stable under partial client activation, while DLR converges more rapidly than SLR. This consistency supports the robustness and scalability of our proposed framework.
and 16. Features are evenly distributed among the clients, and
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000
SUR
OLR
0c0
catn Evere
5/8 Purel
6:1
D18 Pueet
Cree
04
01
0.1
60
1503nce
160
R60000
166
1550000
140
Figure 11: Run-time error rates on the iMNIST with 16 clients under stationary data stream
DLR
0G0+Fa
Suhdul Sul-Randoon
DUR-uu Buk-tanden
OGD fandon
Doo Tate
0.3
Rate
0.3
0.30
6.35
1.75
0882
0.75
1.00
0.25
Figure 12: Run-time ertor rates the iMNIST with 16 clients under stationary data stream
Number of activated clients: To further examine the impact of partial client activation,
ducted experiments which a fixed number of clients were randomly selected to participate each round. The experimental setup remains consistent with the configuration described in Section [. The experiment involves 4 clients, with the activated clients randomly selected in each round. The number of activated clients varies from suggest that increasing the number of activated clients enhances the stability of learning and reduces the variability in error rate over time. However, even with single activated client, the model even- tually achieves comparable performance, demonstrating robustness under limited client activation.
The results are presented in Figure [3] The results
0.30
#Activated Clients
0.25
#Activated Clients
#Activated Clients =
Error Rate
#Activated Clients
0.15
0.10
0.05
0.00
0.0
0.5
1.0
1.5
2.0
Instance
1e6
Figure 13: Number of activated client.
Client activation statistic: To further analyze and validate the partial client activation mechanism, conducted additional experiments to monitor and record the activation frequency of each client throughout the training process. Table 5 presents the activation frequencies for each client under various event-driven framework settings described in Section 5.4 considering different activation probabilities and activation thresholds Г. Note that the activation frequency for each client is defined as the ratio of the number of iterations in which the client activated to the total number
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of iterations. Additionally, we present results for extreme I values, including I = +o0 (where clients are activated) and I = -oo (where all clients are always activated). This table also provides insight into the selection of I values in Section 5.4 ensuring that the chosen I' values cover wide range of activation frequencies for the clients.
Table 5: Frequency of activation for each client under different settings
Setting
Client 1
Client 2
Client 3
Client
Random
p = 0.25
0.249747
0.250473
0.250277
0.250142
p = 0.5
0.499507
0.499811
0.500266
0.500179
p = 0.75
0.749623
0.750214
0.750155
0.750257
p 1.0
1.0
1.0
1.0
1.0
Event
I +00 (+100)
r = 0.6
0.000006
0.066729
0.101827
0.000204
r 0.2
0.006346
0.461186
0.456317
0.038649
r = -0.2
0.355166
0,982896
0.985216
0.596582
r (-100)
1.0
1.0
1.0
1.0
E RELATED WORKS
E. VERTICAL FEDERATED LEARNING
Federated learning is a decentralized machine learning approach where models are trained collabo- ratively across multiple participants. In terms of how the data is distributed among the participants of federated learning, federated learning can be roughly categorized into Horizontal Federated Learn- ing (HFL) and Vertical Federated Learning (VFL). HFL is centered on collaborative model training among the devices possessing the same feature but with non-overlapping samples (McMahan et al. 2017 Karimireddy et al., 2020 Li et al. 2020: [021] Marfoq et al. 2022; Mishchenko et al.] 2019) .. In contrast, VFL involves collaboration among participants with non-overlapping feature sets
but on the same sample /Vepakomma et al. 2018 Yang et al. 2019 Liu et al. 2019 Chen et al.,
2020
Gu al. 2020 Zhang et al. 2021e b Wang al. 2023 Zhang et al. 2024] Qi et al. 2022)
Research on VFL is facing a wide variety of challenges. The primary concern in VFL lies in privacy and security. To protect the privacy, VFL frameworks commonly employ privacy protection meth- ods such as differential privacy (DP) (Ranbaduge & Ding 2022 Wei et al. 2020 Zhou et al. 2020
Huang et al.
[0155] Zhou al. 2020; Huang al. 2015), homomorphic encryption (HE) (Hardy
et al. 2017 Liu et al. (220)) secure multiparty computation (SMC)(Fang et al.) 202T Mugunthan
cct al. 2019 Gu et al. 2020). The second challenge encountered in VFL pertains to communication costs. This due to the transmission of large intermediate model information through the network during VFL training. To enhance communication efficiency in federated learning. the most common method is to compress the communication of the VFL. (Castiglia et al. 2022 Wang et al. 2022
2023). Besides, another common strategy involves using multiple local updates on each participant reduce the number of communication rounds (Liu et al. 2019 Fu et al., 2022). However, these methods often assume the client possesses additional information, such as data labels, which may align with the general VFL framework. The third challenge of VFL involves adapting data distribution efficiently. The distribution of data may change throughout the life cycle of VFL. Offline learning settings require retraining the VFL model when encountering concept drift, which is neither computationally efficient nor communication efficient within the VFL framework. There- fore, applying online learning to the VFL framework is essential (Wang & Xu 2023). Moreover, online VFL also alleviates storage costs, particularly when clients have limited storage space. In the offline learning paradigm, both the client and the server possess large datasets. Certain types of VFL based the offline learning paradigm to each sample which takes large storage cost (Chen et al. 2020; Wang et al.) 20233 Zhang et al.) 202 In contrast, online learning obviates the need store large datasets. Participants receive
new
required store the intermediate results corresponding
30

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Published as a conference paper at ICLR 2025
one sample from the environment at time, making it suitable for VFL scenarios with low-capacity participants, such as sensor networks (Wang & Xu 2023 Liu et al. 2022).
E.2 ONLINE LEARNING
Online learning is a paradigm wherein models are continually updated as fresh data becomes ac- cessible, as opposed to being trained on a static dataset in a batch mode. In the establestbli online convex optimization, the primary objective to minimize regret (Hoi et al., 2018 Hazan et al., 2016), which is the difference between the cumulative loss of the player and the optimal solution. The most straightforward approach to online convex optimization involves directly opti- mizing regret, known as follow the leader (FTL) (Hazan al., 2016). However, FTL can be unstable which motivates the need to stabilize the training through regularization. Therefore the idea of fol- low the regularized leader (F (FTRL) (Shalev-Shwarz, & Singer 2007] Abernethy et al. 2009; Hazan & Kale, 2010; McMahan et al., 2013) was proposed to address this problem. In FTL and FTRL, the learner is required to store all previously seen samples, which is inefficient in terms of memory and computation. Another prevalent algorithm for online convex optimization is online gradient descent (OGD) (Zinkevich, 2003). This iterative optimization algorithm updates the model using the gradient on the incoming data point. In theory, it achieves a sublinear regret of O(VT). Hazan iet al. (2007) further introduced the adaptive OGD algorithm, which offers intermediate regret rates between ((vT) and (log(T))
While traditional online learning predominantly addresses convex cases with shallow models. recent research has shifted its focus towards the non-convex case of online learning. Hazan et al. (2117) in- troduced the concept of local regret as a surrogate for regret analysis non-convex online learning- Unlike the regret utilized in online convex optimization, the fundamental idea is to confine the re- gret within a sliding window, rendering it "local". [Aydore et al. (2019) further explores the concept of local regret, introducing dynamic local regret to address concept drift in the data stream. This approach incorporates an exponential average over the sliding window of local regret. Additionally, they utilize past gradients for the window, enhancing computational efficiency compared to the static 20a
local regret proposed by Hazan et al. (2017). Gao et al. (2018) present an online normalized gradient descent algorithm for scenarios with gradient information available and bandit online normalized gradient descent algorithm when only loss function values are accessible. They demonstrate achiev- ing regret bound of
Sahoo et al. (2018) propose online deep learning tackle online learning within the deep learning paradigm. Aggawal et al. (2019) addresses online learning in non-convex setting by leveraging an offline optimization oracle. Their study demonstrates that by enhancing the oracle model, online and statistical learning models achieve computational equiv- alence. Suggala & Netrapalli (220) demonstrate achieving an (VV)) rate using the Follow the Perturbed Leader (FTPL.) algorithm. [Héliou et al. (2020) address online learning with non-convex losses, introducing mixed-strategy learning policy based on dual averaging under the assumption of inexact model feedback for the loss function.
To facilitate a clear comparison, we present Table summarizes the regret bounds of the most relevant works, including both standalone online learning algorithms and the online VFL. Specif- ically, compared to the theoretical results of standalone DL.R (Aydore et al., (009)) the additional constant term (2WC) G) arises from the missing gradient elements of passive clients due to the dy- namic partial activation of clients in the event-driven online VFL framework.
Table 6: Comparison of the regret bound
Method
Online Convex Learning
Online Non-Convex Learning
Standalone
OGD Hazan al.
2016
OlvT)
SLR
Hazan et al.
2017
SLR. s 7 [83M 02) DLR (T) S Tor (83M + 02)
DLR
(Aydore et al.
2019
Online VFL
Online VFL WWng & Xu
2023
O(VT)
Event-Driven Online VFL
(ours)
O(VT)
DLR, (T) < $
Paaz Pretn
5AM Pmaz
+ +02 +2W
31


Reviews:
[{"review_id": "uB1yBucrjE", "reviewer": "Reviewer_H7dv", "summary": "This paper focuses on online vertical federated learning and proposes an innovative event-driven framework. The main contributions are particularly noteworthy: (1) The authors make a novel observation that in real-world scenarios, clients in vertical federated learning are unlikely to receive different features of the same sample synchronously. This perspective is novelty and addresses a significant gap in current research. (2) The authors then develop an event-driven online vertical federated learning framework. A particularly valuable contribution is the incorporation of Dynamic Local Regret into this framework to handle challenges arising from non-convex models in non-stationary environments. (3) This framework effectively bridges the gap between theoretical VFL models and practical applications, addressing real-world challenges that have been overlooked in previous research.", "questions": "1.\tCould this paper include a convergence comparisons with existing approaches? The current convergence analysis lacks comparative results that would help demonstrate the relative convergence performance and efficiency of the proposed method against other existing frameworks.\n2.\tI am curious about how this paper address privacy in this framework. Could you demonstrate on the acceptable privacy protection mechanisms and safeguards implemented in the proposed approach?\n3.\tHow does the framework's performance scale with different total numbers of clients? It would be valuable to evaluate whether the proposed approach maintains its effectiveness as the number of clients increases or decreases.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1.\tThe paper makes an insightful observation about asynchronous data reception in VFL - a practical issue that has been surprisingly overlooked in previous research but significantly impacts real-world applications.\n2.\tThe integration of Dynamic Local Regret into VFL shows technical sophistication, offering an elegant solution for non-convex and non-stationary scenarios that extends beyond traditional convex-only approaches.\n3.\tThe theoretical analysis is rigorous, with well-constructed proofs of regret bounds that provide solid mathematical foundation for the event-driven framework.\n4.\tThe practical benefits are clear - by activating only relevant clients, the approach naturally reduces communication and computation overhead, making it more feasible for real-world deployment.", "weaknesses": "1.\tUnclear innovation contribution\n\n(a) The paper would benefit from a clearer discussion of the specific challenges encountered when adapting event-driven client participation to VFL, along with the corresponding design considerations and solutions proposed to address these challenges.\n\n(b) Similarly, the paper could better elucidate the specific technical challenges encountered in DLR integration and more clearly demonstrate the novel solutions developed to overcome them.\n\n2.\tLack of comparative analysis\n\nThe comparative analysis could be expanded. The paper does not discuss several relevant works in online VFL, such as \"Online Vertical Federated Learning for Cooperative Spectrum Sensing, Wang et al. \" and \"Vertical Semi-Federated Learning for Efficient Online Advertising, Li et al.\". Including comparisons with these works would help better position this paper's contributions within the existing works.\n\n3.\tLack of comparative analysis\n\nThe experimental evaluation would benefit from including standard VFL baselines, such as Local Model and Vanilla VFL in “Fedcvt: Semi-supervised vertical federated learning with cross-view training, Kang et al.”, “VERTICAL FEDERATED LEARNING HYBRID LOCAL PRE-TRAINING, Li et al.”, to provide a more comprehensive comparison of the proposed approach's performance.\n\n4.\tOnline varying numbers of active clients\n\nGiven that the paper focuses on event-driven client participation, and changing event, network disconnections, or other factors, the experimental section would benefit from exploring scenarios with varying numbers of active clients.", "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": "cTFJt8wgOx", "reviewer": "Reviewer_qrpn", "summary": "This paper discusses vertical federated learning (VFL) in an online setting when all clients receive a synchronous data stream. Instead of reviewing the updates of a model from a client-focus aspect, this paper proposes to review this problem through an event-driven online VFL framework. That is, only a subset of clients were activated during each event, while the remaining clients passively collaborated in the learning process. As this will lead to the non-convex optimisation problem, a dynamic local regret approach is adapted to handle online learning in non-convex cases and non-stationary environments.", "questions": "1. I might miss the result of SLR for SUSY and HIGGS datasets. Where can I find that?\n2. Here, each of the images in iMINST has been divided into 4 segments. However, it didn't specify if segments are overlapped with each other. \n3. If the segments can have overlaps, I would like to know the results of having more than 4 clients, especially how the performance might be influenced when overlaps increase.\n4. If the segments do not have overlaps, please either explain why or add the results for an overlapped case.\n5. Is only one client activated each time? If yes, can multiple clients be activated at the same time?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The idea is novel, and the presentation is clear.", "weaknesses": "The solution is simple. As there are not many available baselines for direct comparison, the experimental results can only demonstrate the proposed solution is a feasible plan. Therefore, although the limitations of the proposed framework have been discussed at the end, it lacks the support from experiments to form a deep understanding.", "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": "MvP7IZnyia", "reviewer": "Reviewer_x1ux", "summary": "This paper proposes an online vertical federated learning framework,  based on the mechanism of exponential weighted sliding-window averaging. The dynamic local regret analysis by Aydore et al. is employed and extended to analyze this framework. Typically, only a subset of clients were activated during each event, and meanwhile the remaining clients can be reached out for passive collaboration during the learning process. Dynamic local regret (DLR) can be derived from the analytical framework, under some reasonable assumptions. Finally, experiments are conducted under various settings to verify the theoretical results.", "questions": "Q1. Is $x$ missing from $f^t(w_0, {\\bf w},)$ in (1)? Should it be something like $f^t(w_0, {\\bf w}, x; y)$?\n\nQ2. What is the biggest challenge when extending the results about DLR in Aydore et al. 2019 to the setting of this paper?\n\nQ3. Is it possible to extend the proposed framework for HFL? If not, why? If yes, how does it compare with previous works on online HFL?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1. A reasonable setting of VFL is considered.\n\nS2. The proposed solution is general (for a large class of learning algorithms) and easy to be implemented.\n\nS3. Some sound theoretical results are derived under reasonable assumptions.\n\nS4. Experiments are conducted under various setting.", "weaknesses": "W1. The theoretical contribution is incremental.\n\nW2. More discussion about the implication of theoretical conclusions about DLR in practice is needed.", "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": "S8DHhQanAF", "reviewer": "Reviewer_MnZ6", "summary": "This work tackles a practical problem with online and vertical federated learning where clients do not receive the streaming data synchronously. In other words, the features of a data sample are assigned to a few of the participating clients in an event-driven manner. The work proves its empirical superiority against traditional online regret minimization and static local regret baselines. A sublinear regret bound for non-convex models is also derived.", "questions": "1. What are the differences between the use of dynamic local regret proposed in the Aydore paper cited and your work? Is it that in your case, half of the model is on server and the other half is at clients? What specific challenges did you encounter in adapting the said dynamic local regret to your framework?\n\n2. For eq. 1, are embeddings from each client concatenated?\n\n3. Referring to lines 162-163, is $W=M$?\n\n4. I acknowledge that the discussion related to the following question is provided in \"Limitations.\" However, I still want to clarify this point: Are all features of a data sample \"covered\" (trained on) by all the activated clients?\n\n5. Do we always need to divide the features of a data sample among all the participating clients? Can there be overlapping features across clients? Can there be multiple clients activated for the same feature? Under the \"Event\" experimental setting, do all four clients participate each round? If not, what happens to the unassigned features? \n\n6. Do all activated clients receive the assigned features at the same time/synchronously?\n\n7. Referring to line 278, is capital $F$ in $\\nabla F^t (w_o, \\mathbf{w})$ a typo? \n\n8. I am curious to know if an even distribution of features across the clients is a practical assumption (with reference to line 322).\n\n9. Is a batch of embeddings from a client sent back to the server? Or the embeddings are also sent in a streaming setup?\n\n10. How do you define a round for the streaming/online FL?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. The writing is very clear. \n2. The work is novel; addressing the practical setting of streaming data for vertical federated learning (for non-convex models) is timely and relevant.\n3. The proposed solution is elegant, and the authors have done a good job at rationalizing each component of their methodology.", "weaknesses": "1. I did not fully get the significance of passive clients. Are passive clients also assigned unique features of a data sample? If the features assigned to the passive clients are not unique, then why do we need the derived embeddings from the passive clients?\n\n2. I acknowledge that the work is based on a cross-silo or vertical FL setting, where a small quantity of clients is the norm. However, experiments with 4 clients and a 3-layer model still make me question how scalable this work is.", "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)