Federated Orthogonal Training: Mitigating Global Catastrophic Forgetting in Continual Federated Learning
Authors: Yavuz Faruk Bakman
Duygu Nur Yaldiz
Yahya H. Ezzeldin
Salman Avestimehr
Venue: ICLR 2024 poster (2024)
Keywords: Federated Learning
Continual Learning
Catastrophic Forgetting
Primary area: transfer learning, meta learning, and lifelong learning

Abstract: Federated Learning (FL) has gained significant attraction due to its ability to enable privacy-preserving training over decentralized data. Current literature in FL mostly focuses on single-task learning. However, over time, new tasks may appear in the clients and the global model should learn these tasks without forgetting previous tasks. This real-world scenario is known as Continual Federated Learning (CFL). The main challenge of CFL is \textit{Global Catastrophic Forgetting}, which corresponds to the fact that when the global model is trained on new tasks, its performance on old tasks decreases. There have been a few recent works on CFL to propose methods that aim to address the global catastrophic forgetting problem. However, these works either have unrealistic assumptions on the availability of past data samples or violate the privacy principles of FL. We propose a novel method, Federated Orthogonal Training (FOT), to overcome these drawbacks and address the global catastrophic forgetting in CFL. Our algorithm extracts the global input subspace of each layer for old tasks and modifies the aggregated updates of new tasks such that they are orthogonal to the global principal subspace of old tasks for each layer. This decreases the interference between tasks, which is the main cause for forgetting. Our method is almost computation-free on the client side and has negligible communication cost. We empirically show that FOT outperforms state-of-the-art continual learning methods in the CFL setting, achieving an average accuracy gain of up to 15% with 27% lower forgetting while only incurring a minimal computation and communication cost. Code can be found [here ](https://github.com/duygunuryldz/Federated_Orthogonal_Training)

Published as a conference paper at ICLR 2024
FEDERATED ORTHOGONAL TRAINING: MITIGATING GLOBAL CATASTROPHIC FORGETTING IN CONTINUAL FEDERATED LEARNING
Yavuz Faruk Bakman University of Southern California ybakman@usc.eda
Duygu Nur Yaldiz*
University of Southern California yaldizéusc. edu
Yahya H. Ezzeldin
Salman Avestimehr
University of Southern Californa yessa@usc. edu
University of Southern California avest ime @usc. edu
ABSTRACT
Federated Learning (FL) has gained significant attraction due to its ability en- training over decentralized data. Current literature in FL mostly focuses on single-task learning. However, over time, new tasks may appear in the clients and the global model should learn these tasks without forgetting pre- vious tasks. This real-world scenario known Continual Federated Learning (CFL.. The main challenge of CFL is Global Catastrophic Forgetting, which cor- responds the fact that when the global model trained on new tasks, its perfor- mance old tasks decreases. There have been few recent works on CFL to pro- pose methods that aim to address the global catastrophic forgetting problem. How- ever, these works either have unrealistic assumptions data samples or violate the privacy principles of FL. We propose a novel method, Federated Orthogonal Training (FOT), to overcome these drawbacks and address the global catastrophic forgetting in CFL. Our algorithm extracts the global input subspace of each layer for old tasks and modifies the aggregated updates of new tasks such that they are orthogonal to the global principal subspace of old tasks for each layer. This decreases the interference between tasks, which is the main cause for forgetting. We empirically show that FOT outperforms ttate-of-the-are contin- ual learning methods in the CFL setting, achieving an average accuracy gain of up to 15% with 27% lower forgetting while only incurring a minimal computation and communication cost. Code can be found here
able
the availability of past
INTRODUCTION
Federated learning (FL) is decentralized training solution born from the need of keeping the local data of clients private to train a global model (McMahan et al. (017)) Most of the FL works focus on the global learning of single task (Hard et al. 2018 Yang et al. 2021 Augenstein et al. 2020). However, in real life, new tasks might arrive to the clients over time while previous data disappear due storage limitations. For instance, assume malware classifier is trained over multiple FL clients. The emergence of new malware families (new tasks) is inevitable, making the update of the classifier necessity. Another real-life scenario can be the emergence of new viruses in some clients due to epidemics. The global model also has to learn to classify these new viruses (new tasks) YYon al. 2021). In both of these scenarios, the model should not forget its prior knowledge.
Continual Learning (CL) addresses this issue in centralized machine learning (ML), the problem of learning sequentially arrived tasks without forgetting (Kirkpatrick al. 2017). Learning global
model while new tasks appear in the clients in an online manner is a problem of Continual Federated
Learning (CFL) Ma et al. 2022). An ideal CFL algorithm solving global catastrophic forgetting
Equal Contribution

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Train Task
Train Task t+1
Global Principal Subspace Extraction
مسe S onne Pagestoon
Projecoo
W- 2
W-
5-0 skon S
Figure I: Overview of Federated Orthogonal Training (FOT). Clients do regular SGD training. The server projects the aggregated updates into the orthogonal subspace of previous tasks' activation subspace. At the end of each task. one communication round is reserved to extract global principal activation subspace for each layer.
should not require storage of old task data and an unrealistic amount of computation on the client side because they are already limited in the edge devices. Morcover, it should be compatible with secure aggregation (Bonawitz et al., 2017). The latest research has shown that local data can be extracted from the local updates if there is no protection such as secure aggregation (Boenisch al.)
2021 Wang et al. [022 Fowl et al. 2022). Lastly, it should be resistant to data heterogeneity across clients which mostly the case in weal-world scenarios (Wang et al. 202I).
The goal of centralized continual learning algorithms to prevent the disruption of knowledge acquired from previous tasks while learning new tasks. When the network capacity is fixed, this is accomplished by transferring the information of old tasks to the current learning process and training the model accordingly. Carrying global information of old tasks is non-trivial problem in Continual Federated Learning due to decentralization. One possible solution is that the clients can maintain old tasks' information according to their local data. However, maintaining local information the client side requires extra storage. Considering edge device limitations, this approach is not feasible in real life. Moreover, even if it is pursued, mitigation of global forgetting is not guaranteed because clients do not have global information of old tasks. Clients update the model to prevent forgetting according to their local data. However, aggregation of these local updates might not result in preventing global forgetting. This becomes much major problem in non-IID settings. This effect is visualized in Figure 2 and demonstrated in Appendix A
Continual Federated Learning Challenges:
Our Contributions: In this work, we propose CFL framework named Federated Orthogonal Training (FOT) to address the Global Catastrophic Forgetting problem. Our framework (visualized in Figure I modifies the global updates of new tasks so that they are orthogonal to previous tasks' activation principal subspace. It decreases the interference between tasks therefore learning new tasks does not disrupt model performance on old tasks. Within FOT, we introduce a novel aggrega- tion method, named FedProject, which guarantees the orthogonality in global manner. FedProject requires the global principal subspace information of old tasks for each layer on the server. There- fore, FOT has Global Principal Subspace Extraction (GPSE) to obtain that information. Our method protects the privacy of the clients without any requirement of trusted third parties or representative data in the server. Also, clients do not store anything and there is extra computation local training. Furthermore, our method is robust under data heterogeneity. We perform an exten- sive comparison of our method with state-of-the-ard baselines. Our method outperforms all other methods with significant margin.
the
2 RELATED WORK
Aiming to train a global model in federated learning for multiple tasks with online data arrivals, the
primary challenge is mitigating forgetting while preserving privacy (Ma et al. 2022). Limited prior efforts address this challenge. One approach involves clients storing and sharing perturbed subsets
of previous task samples (Dong et al. 2022). but this strains storage and privacy. Knowledge distil-
lation, another solution, relies server's access to task-specific datasets (Ma et al. 2022), which may not be practical in certain scenarios. Recent proposals (Babakniya et al. 2023] Qi et al. 2023]
Zhang al. 2023 suggest using generative models to generate synthetic samples prevent forget-
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Global Low Loss Region task
Global Low Lass Region for task
Local Low Loss Regions for both task and 1+1 method
updates
Cantralized CL method Update
W.
Aggregated Update
Figure 2: Dashed lines show each client maintains its local information of old tasks and apply continual learning method locally. The local updates try to reach the loss region where old and new task loss is low locally (red regions). However, some portion of local low-loss regions might not overlap with the global low-loss regions of both tasks (intersection of gray and yellow regions). Therefore, aggregated update (shown as solid black arrow) may converge to the point where old task loss is high (outside of the gray region)
ting, but these add significant computational and communication overhead. Another method (Halbe et al. 2023) employs foundation models and prompt tuning but requires individual client updates and an available foundation model at the server, increasing costs substantially. Our approach, in contrast, combats forgetting without sharing data samples assuming server-side datasets. It in- curs minimal additional computation and communication costs. While related to GPM (Saha et al., 202T) in centralized continual learning, our adaptation introduces privacy- peeeeyypressriinggg Tramework to extract global principal subspaces at each layer, ensuring orthogonality with a novel aggregation method. We also account for data distribution heterogeneity across clients. Other works leverage layer orthogonality in various contexts, such as one-shot federated learning (Su et al.) (203)) and unlearning (Li al., 2023). but differ in their approaches and goals. extended version of the related work can be found in Appendix
3 PROBLEM FORMULATION
In a traditional federated learning setup, there is single task T and there are C' clients to train global model with parameters W in R communication rounds. At each round, clients receive the global model from the server and perform local training on their local private data. Then clients send the updated models to the server and the server aggregates the locally updated models (McMahan n4
et al. 2017 The optimization objective minw Σi= miotal
L(D;; W) where Dj is the local private dataset of client i, n, is the number of samples in D, and Alotall = Σ, ny is the total number of samples across clients.
In Continual Federated Learning (CFL), there are K tasks ((71, T2, TK sequentially arriving the C clients. Each client has a labelled private dataset for each task Te denoted as Dr, (xx), Vl.s). (x², v²,s); .... (x nt.. V/.5 data sample for task t, and n4,; is the number of data samples on client i for task T.. The aim is to train W in a federated manner to learn the sequentially arrived K tasks. More formally, in CFL we
Pr.")) ")}, where (x/; vt) are the feature and label pair of the i-th
aim to solve the following optimization problem
K
min
(Dr.s W),
(1)
1==1 T4,b
The objective aims minimize the loss of all tasks among all samples, where each task is eval- uated separately (i.e. inckeennremnntal learning). We consider that each task Tú appears for R; communication rounds, during which only data from Ti is available at the clients.
The challenge in solving (T) stems from the nature of continual learning: Dr, is available only during the learning of Tr. Neither clients nor the server can access the old tasks data. Furthermore, we consider that there is no representative data for any of the tasks at the server. Due to the non- availability of old tasks, the model forgets what it learns from previous tasks. Our central goal is to solve this global catastrophic forgetting phenomenon while still preserving the privacy of each of the clients' local datasets. To study this problem, following the existing literature (van de Ven & Tolias 2019: Saha et al. 1021 Augenstein et al. 2020; Dong et al.) 2022; Kirkpatrick et al.) 2017). we focus on the extreme forgetting case where the tasks are clearly separated, L..., there is no intersection between rounds where tasks T; and T, Vi + j.
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4 PROPOSED SOLUTION
Let W be composed of layers, W = 1/ where W is the parameters of layer and X be the input matrix for layer at task Te. Each column corresponds to one input for the layer and these inputs are distributed among the clients x Notice that for the first layer, the columns of X are the samples distributed among clients. Now, applying the model all inputs layer across the clients can be written as:
x,1 n. xx,2...,,,,, nr,2. ....., X,,C G...
H W
(2)
where H is the output of layer at task T. before applying non-linearity. Our aim is change HA as much as possible for each layer while training new tasks ((T++1, Te-2: Tx.).
4.1 FEDERATED ORTHOGONAL TRAINING (FOT)
To present the central idea of FOT, let us focus on the training of the first two tasks Ti and Tz- With no previous tasks to care about, the training of Ti follows the vanilla FedAvg (McMahan al.) 2017 Let W. = {Wi)__ be the global model parameters for layer after concluding the training of T. While training T2, W is updated to W + AW AÊ E [1 L]. Now consider the layer outputs at layer l when applying the new global model parameters on data from T1:
H (Wi + AW
where X1 is the inputs to layer l assuming samples from task Ti. Our goal for task Ti is to achieve that H Hf. Vl E [1: L), such that the layer outputs and therefore the model mapping on data from task Ti is unchanged due to subsequent tasks. From (3), we can see that this is achieved when A AW'X X = as in this case there no interference from other tasks affecting Tı. More generally, we would like to remove interference to task T, by any of its subsequent tasks Tg. with > t, i.e.,
for any task f, we have that:
AW O, e 1t: T7, vL E L]:
(4)
Note, however, that making A AW q = 0 requires the row space of AW; to be orthogonal to the column space of X When there are enough training samples for task T. the column space of X is full rank, which enforces AW, = 0, effectively blocking learning for any subsequent tasks after Tr (no training). Therefore, instead of making A O, we will enforce relaxed condition by requiring AW = O, where P is the low-rank Global Principal Subspace (GPS) of X This subspace includes most of the X/. We use the term "Global" since this subspace is extracted from the whole data distributed among clients instead of the local data of individual clients.
4.1.1 FEDERATED PROJECTED AVERAGE (FEDPROJECT)
Extracting global principal subspace for each layer in a trivacy-preservee manner is a challenging problem which we address next in Section 1.2. For now, let us assume that we have Of which denotes the set of orthogonal vectors covering the GPS of layer l after training for Ti. While training T2, aggregated updates are projected onto the orthogonal subspace of Of for each layer l as follows:
1 c
E Global
5
(5)
C
8C+ ge globat
global 0f
gt gllobal
(6)
We W pu global
where 5 denotes local update client for layer and Of a matrix whose columns correspond
to basis vectors of Of- ( is done with secure aggregation (Bonawitz et al. 2017). By using
the projection in (6) remove the component of the updates that disrupt the previous task knowledge of the model, and make the model learn the new task in a subspace that is not effectively used in previous tasks.
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The local training of clients is not modified. They perform regular training their local data. After T2 is finished, Of is expanded such that also covers the global principal subspace of X2 for each layer & and it is denoted as O2. For upcoming tasks, aggregated updates are projected into to the orthogonal subspace of previous tasks' GPS for each layer. After the training of each task is finished, the orthogonal set is expanded again. The orthogonal set expansion and Extraction of Global Principal Subspace are covered in Section Overall FOT is summarized in the Appendix
and visualized in Figure [
Remark 1. Note that FedProject is guaranteed to converge for each task under the assumptions stated in (Li al., 2020b). This due two key aspects. First, the projection step in FedProject projects the update onto a set defined by O,. As result, applying FedProject with gradient updates equivalent to applying ProximalSGD (Cohen et al. 2017) for constrained convex optimization problem, which is guaranteed to converge if the projection set is convex (we prove that the projection set in FedProject is convex in Appendix C). Secondly, clients perform a number of local steps locally before aggregation, if we consider small enough learning rate, the local updates can be approximated by a single gradient step with a larger learning
shown in (Li et al.) 2020b). even when
rate. This coupled with our equivalence to ProximalSGD, guarantees the convergence even when projection is performed after a number of local epochs.
4.1.2 GLOBAL PRINCIPAL SUBSPACE EXTRACTION (GPSE)
After the training for task T, done, the server and clients go through an additional communication round in order to extract the global principal subspace information for each layer in the model. In this additional round, denoted the GPSE round, the server broadcasts to all clients the global model parameters We and orthogonal vector set O,-1 = (0,-1." 0 } that covers the global principal subspace (GPS) of each model layer to the clients. Each client applies the global model W, on its local dataset, then for each layer É, it projects the layer inputs onto the subspace orthogonal to 0 as follows:
X - X 0 t-1
X
Where X is the input matrix of layer E of client at task t. With B). X across the different clients span a subspace that was explored by the tasks up T--1- 1-
for each layer in order to add it to 0 To do this each, will invoke results from randomized SVD to estimate the global principal subspace without violating privacy. First, each client multiplies projected input vectors with random row vector and sums them up:
The goal now is to estimate the global principal subspace of x
gj
where gj is standard normal random vector of length sf (sampling dimension) sampled from \O I). Note that A is matrix of size d' sf where d' = dim(x(). Each client sends (4 14 to the server, which then sums all A matrices for each layer e:
A
Σ AC.
(10)
The summation in (T) is done with Secure Aggregation (Bonawitz al. 2017 thus ensuring that the server only has access to the aggregate A, (See the discussion in Section 4.2) Note that the resulting A can be written as:
Me,s
A
8 = X G¹
(11)
1=1==1
where X = [ X 'I is the projected X to the subspace orthogonal to 0 and &o is standard normal Gaussian matrix with N₂ x sf dimension; N; represents the total number of samples among all clients task t (i.e. N₂ -i= R2,)).
With (IT) in mind, we can now use an important result in randomized linear algebra, which we state informally in the following theorem for brevity.
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Adaiionn (Secure Apprrggation
Layer
Layer
O 0
Figure 3: Summary of GPSE. At the end of each task, clients multiply layer inputs with a random gaussian matrix. These matrices are securely summed in the server. SVD is applied to the resulting matrix to get the principal subspace.
Theorem 1. Papadimitriou al. (1988 (Informally Stated): Let A be x matrix and G be a random Gaussian matrix of size n. x where p 2 Let Y AG. Then, for sufficiently large the principal column space of Y recovers the low-rank column space of A up rank with negligible error. The error decreases as p increases.
From Theorem 1 the server can obtain the principal subspace of X approximately by using Aç- To find the subspace where most of the inputs on, the server applies SVD on A for each layer Uf, 2, SVD(AŞ). Then server finds the sufficient rank for each layer t:
PP AilE
X LE 1X
min o' PEN s.t.
+
> the
(12)
|| ||
where P < 0,10 : and the is threshold value. ||x TXX
measures how much of the input matrix
is already covered by 0 and |IPP" Aille measure how much of the remaining input for each |AlI| layer is approximated. Therefore, the threshold value determines how much portion of the input in total for each layer is covered by the subspace. How to find
||X;") I|r is explained in the Appendix |X(||>
After determining the minimum sufficient rank rf, the server extends the orthogonal vector set for each layer as follows O + o U r. Finally, O + GramSchhidde dt (O,)) is applied in order to keep all the vectors orthogonal to each other since the vectors in 0 and [t 1: r'1 are not necessarily orthogonal to each other. The different steps of GPSE are summarized in Appendix and visualized in Figure 3
4.2 DISCUSSION OF FEDERATED ORTHOGONAL TRAINING
Our algorithm is regular training on their local devices and send only the updates the server. Our algorithm al- lows the use of secure aggregation (Bonawitz et al.) (017) at these rounds because FedProject is applied over aggregated updates. More importantly, GPSE rounds are also compatible with secure aggregation (Bonawitz et al. 2017). Each client sends random Gaussian matrix for each layer and the server sums these random Gaussian matrices as (10) with secure aggregation. Secure aggrega- tion is well-studied primitive in federated learning that gives information-theoretti guarantees that the server learns nothing about individual A, except their sum from their encoded values Z;, i.e., [(fA,))1 M EE A,) 0. Recently it has been shown in Elkordy et al. (0022) that the leakage through the sum is limited and is upper-bounded bounded by decreasing function in the number of clients N. These guarantees through secure aggregation ensure the privacy of the proposed ap- proach towards leakage about any individual clients. Furthermore, from the server's perspective, GPSE results in distributed application of the been shown to provide differentiall privacy guarantees (Blocki et al. 2012)).
Privacy:
During the FL training rounds, the clients perform
(JL) transform, which has
Communication Cost: When performing training rounds for task t with FOT, there is no additional communication overhead since only local model updates are aggregated at the server using secure aggregation, similar to FedAvg. The server then projects the aggregated update onto the set O,- When transitioning from task to task t + 1, an additional communication cost is incurred to update the orthogonal set from O to O,+1, by performing an additional secure aggregation round in order to

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compute O1+1 (See Section 4.1.2). Since, the size of A is equal to d' x s then the communication overhead in the GPSE step is OTd x 8 smax), where d = E. d' and 8 s = max(s").
The computation cost of the algorithm the client-side is negligible. During the training of a task, regular training is performed on the clients. At the end of tasks, clients extract internal activations and multiply with them random computationally heavy. Assume there are at most N data points per task, then the computation complexity is ONN x d x smax), where
Computation Cost:
vector in (9). This operation is not
Ex df and 5 max(s"). This is close one local
epoch forward pass complexity which is O(N x d x ddax), where dmax max(d"). The difference comes from the ratio of guus which is a constant in our experiments.
donax
Data Heterogeneity: Our algorithm is not negatively affected by data heterogeneity. The server obtains the Global Principal Subspace of layer inputs by applying SVD on A for Tr. A4 the output of (T), is independent of the data distribution because of the commutative property of addition. That means, for any possible data distribution, the extracted global principal subspace information remains the same. This makes our algorithm resilient to data heterogeneity.
5 EXPERIMENTS
5.1 EXPERIMENTAL SETUP
Here provide summary of the experimental setup with additional details architectures, hy- perparameters, and further details delegated to Appendix [F]
Benchmarks: We evaluate the performance of algorithm four different CL benchmarks.
Permuted MNIST (Ebrahimi et al. 2020) is variant of MNIST (Lecun et al. 1998) containing 10 tasks. Each task is a randomly pixel-wise permuted version of the original MNIST inputs. Split- CIFAR100 (Krizhevsky] 2009) is created by randomly dividing 100 classes into 10 tasks of 10 classes. Split Mini--magenet (Saha al.) [221]: Chaudhry al.] 2019a) is created by randomly dividing 100 classes into 20 tasks of 5 classes. Lastly, we use a sequence of 5-Datasets (Saha et al., 2021, where each dataset is considered a different task. These datasets are CIFAR-10 (Krizhevsky
2009
MNIST, SVHN (Netzer et al. 2011), notMNIST (Bulatov] 2011 and Fashion MNIST (Xiao
ct al.
2017). and all of them contain 10 classes
We compare our method with the Federated Learning adaptations of state-of-the-ard Con- tinual Learning solutions, which are EWC (Kirkpatrick et al. 2017). ER (Chaudhry al. 2019b), a direct CFL approach,
Baselines:
RGO (Lu & Liu 2022) and GPM (Saha et al. 2021). 1) As
compare
to GLFC (Dong et al. 2022), TARGET (Zhang et al. 2023) and FedCIL (Qi et al., 2023). How- ever, since GLFC and FedCIL are proposed for class-incremental setup, adapted them to task- incremental setting with slight changes. We explain the details of the adaptations in the Appendix
F2
Lastly, we only use FedAvg (FL) demonstrate the worst-case baseline.
We distribute each task's data to the clients in both i.i.d, and non-i.i.d manner. To simulate data heterogeneity, we follow the method of (McMahan al. (017). The data is sorted according to the labels, then it is divided into equal-sized shards, where the number of shards is twice the number of clients. Each client is randomly given shards in non-replicative manner. In the end, each client has data from at most two classes.
Data Distribution:
We use two metrics to evaluate the performance of the algorithm. First, we measure the average accuracy (ACC) (Lopez-Paz & Ranzato 2017) of all tasks at the end of the whole CFL process. Second, we use average forgetting (FGT) (Chaudhry et al., 2018) evaluate global catastrophic forgetting. Let a;,: be the global model accuracy of task T, after the training of task T., FGT and ACC are defined as:
Metrics:
K-1
ACC
O
FGT
ai, 0.KK
(13)
K
t=1
5.2 EXPERIMENTAL RESULTS
Table provides average accuracy and forgetting results and Table compares baselines in terms of storage at the edge and the extra computation on local training. Federated Orthogonal Training, in
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Table 1: Performance results of different methods various datasets. ACC (higher is better) stands for average classification accuracy of each task, while FGT (lower is better) denotes average forget- ting in (3).
We run each experiment& 3 times and provide mean and standard deviation.
PMNIST
CIFARIO0
5-Datasets
Mini-Imagenet
Method
ACC(S)
(G)
ACC(S))
FGT(S)
ACC(S)
FGT (8)
ACC(S)
FGT(S)
FL
85.6 85.6850557
8.29:002
63.12+0A8
13.57+1.57
77.95
13.46=1.26
50,43:1997
33.08 08..1.
EWC+FL
86,74-097
8.84-140
63.1340.65
13.48+1.77
77.68-0.55
13.76-1.74
47.00-1 46
36.68+1.57
ER+FL
88.61=0.50
6.62:0066
65,42:0.49
11.72:0.39
80.01a1.17
10.1
55.26:2.95
27.80:3.25
RGO+FL
29.81=020
4.8420.55
63.9140.53
14.3920.54
1.
3.4720.52
51.00:1.88
31.7611.87
GPM-FL
88.27=0.75
6.88:0
59, 49,43:0.48
13.13:
80. 1.01.0
10.7:0.61
50.26:1 90
30.04a1
GLFC
83.76.1005
14.70a1.13
65. 1610.48
11.74:0.59
81 64...2222
10.0740.61
59.26.1 1:2
23.73 23.73
TARGET
89.23=0.77
3.13 20.36
65 28.2888
9.89
82. 1
6.6420.57
58.45a1.3
25, 12:1.92
FedCIL
87 87.43=045
10.01=00.99
60.12++ 12
17.03+1.34
82.433+.33
7.65+0.36
55.98=1 677
28, 90+ 1.84
FOT(Ouur)
90.35-0.06
1.75+000
71.90+0.06
0.87+0.10
85.21-0.93
1.11-031
69.07+0 73
0.19+0.11
FL
80.06=0
9.21
62.56+0.17
10.49+0.40
67.71
21.55=9.73
41 10.00:044
32,92+1.25 1.25
EWC+FL
81.14-1.15
10.78+0..0
62.57-047
10.41+0.83
68.41+3.14
21.42-4.19
41 09.392233
32,23+1.20
ER+FL
82.77.1.56
9.32:2.57
58.5721.59
14.78:2.18
70.66 70.6882.46 46
18.48:3.27
49.23:2.09
24. 40.0.005
nea-HID
RGO+FI.
79.31-3.48
15.72+3.6
40.83+2.86
31.57+1.61
77.92-0.80
6.89-1.12
40,43+0.23
33.24-0.80
GPM+FL
72.17a1.22
22.612.54
34.15
34.60:2.87
72.21:0.9
16.8422.15
29.63:3.12
40,73+4.20
GLFC
84.06.024
12.154
59.8911.15
15 15.3621.69
77.8020.49
11.0621.15
50.41:1.85
23.4612.21
TARGET
84.78:0011
3,70e0.44
63.8741.71
8.76:148
78.16s90.4
7.65:0.36
53.67:1999
26.31+2.12
FedCI. 83.23-0.54
8.65+0.0
56.23+1.93
21.32-2.01
77 41.93+
8.01+0.31
51.12+2.11
25.66+2235
FOT(Our))
85.21:0.13
1,97:0
66.31:0:25
0.60:0.43
79.23s1.02
0.65:0.27
62.06:0.59
0.17:0.27
Table 2: Comparison of base- lines in terms of storage on the edge devices and extra computation on local training.
Table 3: Percentage of used subspace at the end of whole training for the average of all layers.
Table 4: Running time of GPSE round and 1 Local Epoch on the client side.
1 Local Epoch(s)
GPSE
Storage Extra Edge
Computation
Average (IID)
Average (non-llI)
(6)
FL.
X
PMNIST
0.067
0.024
EWC+FL
PMNIST
62.18%
61.36%
CIFAR100
0.074
0.172
ER+FL.
RGO+FL
5-Datasets
53.97%
45 64.64%%
Mini-Imagenet
0.194
0.215
GPM-FL
CIFAR100
47.59%
32.96%
GLFC
FedCII
Mini-Imagenet
89.94%
86.44%
FOT(O)r))
terms of average accuracy, has considerably better performance than the FL adaptations of state-of- the-art CL methods in continual federated learning setup. Also, FOT outperforms GLFC and FedCII. in all the benchmarks without any client storage and extra computation. Besides, FOT achieves very small forgetting in all datasets in both i.i.d and non-i.i.d settings. These results validate that FOT successfully transfers old tasks' GPS information and does orthogonal training accordingly. The higher accuracy margin in the non-i.i.d setting is mainly because of extraction of the global sub- space information which independent of distribution of the data (TT) and FedProject guarantees the orthogonality in a global manner. On the other hand, when we apply CL methods locally, in- formation stored for old tasks does not reflect their global distribution, which particularly true when data distributions are heterogeneous. This makes the global model perform worse in hetero- geneous settings. In addition to bata-hhcerogeneitty across clients per task, we explore the effect of task heskhhetrrogeneity across clients (i.e each client does not necessarily train on all tasks) in Appendix G.I More detailed results of Table T can be found in Appendix G.2, Lastly, we provide the running time and computation cost of GPSE in Table 4 and 5 GPSE is performed only once at the end of each task. The computation cost is close to one epoch training and the communication cost is less than the model size. Therefore, FOT has a negligible effect on the costs of client side.
5.2.1 ANALYZING THE SUBSPACE OF LAYERS
In this section, we analyze how much portion of the subspace at layers is used by FOT at the end of each task (Table 3). It is expected that the remaining space might not be sufficient for the proper learning of new tasks. However, the learning capability is limited by the network capacity layer subspace is scaled with network size. Besides, even with 20 tasks in Mini Imai-ne,gene,, our method achieves superior performance while there is still space for new tasks, which shows that our method
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Table 5: Size (in MB) of the model and the matrices used by FOT.
Table 6: Split Mini-Imagener results of Gaussian
Mechanism in GPSE round with (e = 5, 5 = 10 90- DP and freezing the first layer after first task (FRZ).
Model Size
(A
{0')==
non-III
Resnet-18
2.56
0,64
0.08
ACC(%)
FOT(%)
ACC(%)
FGT(%)
Alexnet
1.42
0.24
0.048
DP
66.77
0.31
60.81
0.13
FRZ
69.30
0.15
62.72
0.12
can scale with high number of tasks. The behavior of subspace expansion is controlled by the threshold value in (12). When the threshold is bigger, there is less forgetting but it also restricts the available space for new tasks. If it is smaller, it will allow the network to learn more task, but by
forgetting old tasks more. The threshold value creates trade-off between the amount of forgetting and the number of tasks to learn. We investigate the effect of threshold value in Appendix G.3
5.2.2 ADDITIONAL MECHANISMS TO ENHANCE PRIVACY
As discussed in Section 4.2 applying our GPSE mechanism does provide differential privacy since it (JI.) transform. However concrete privacy
a distributed application of the guarantees in this case dependent the properties of the data matrix X. To further formalize our privacy guarantees, in this subsection, we apply the Gaussian mechanism to the output of the GPSE mechanism (See additional details in Appendix H. As shown in Table O applying the Gaussian mechanism causes a decline in performance. However, our FOT still outperforms other baselines both in the IID and the non-IID settings. Furthermore, freezing the first layer of the model after the first task. In that way, input of the first layer (samples) does not attend in GPSE rounds and only the internal layer's input (activations) is processed in GPSE. This modification does not hurt the performance of FOT as shown in Table 6 Different versions of FOT can be employed depending on the application's 's required privacy.
make additional experiments that
6 LIMITATIONS
Our work is limited to solving the forgetting problem under the assumption of knowing task bound- aries in the task incremental setting. The problem of not knowing task boundaries is an ongo- ing research area in Continual Learning which is known Task-Free Continual Learning (Aljundi)
et al. (2019). Typical continual learning and federated continual learning papers (Ma et al. 2022]
Chaudhry et al. 2019a Saha et al. 2021 van de Ven & Tolias 2019 Qi et al. 2023 Babakniya et al.) 2023) Zhang et al. 2023) assume the boundaries are known and are well-separated. We fol- low the literature yet acknowledge that finding task boundaries is an important problem and can be a topic of further research. Lastly, we investigate task- musk-hetrroenneity cases where each client sees different subsets of tasks. These experiments are provided in Appendix G. We observe that once we know the task boundaries, task heterogeneity is not problem for FOT.
7 CONCLUSION
In this work, we propose Federated Orthogonal Training to mitigate the Global Catastrophic For- getting problem in Continual Federated Learning, where FL clients receive new tasks over time and the global model performance on old tasks diminishes. Alleviating global forgetting is a challenging problem due to the distributed and privacy-preserye our approach is the first solution for CFL that mitigates global forgetting without requiring extra storage and major additional computation at edge devices. Moreover, we protect the client's pri- vacy since FOT is compatible with secure aggregation. We evaluate our algorithm on different CFL benchmarks with various network architectures and compare it with the FL adaptations of of the-art CL solutions. Experimental results show that our method mitigates global forgetting while achieving high accuracy performance. We also present that the extra communication and computa- tion costs of our algorithm negligible.
nature of FL. To the best of our knowledge,

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APPENDIX
A ANALYZING THE EFFECT OF DECENTRALIZAATION ON FORGETTING
In this section, we demonstrate the main challenge of CFL which is carrying global information of old tasks. To demonstrate this, we compare the performance of ER (Chaudhry al., 2019b) on Split- Split-Cifar100 (Krizhevsky) 2009) in centralized and federated settings. Federated ER stores some samples from past tasks the client-side while the centralized ER does the storage and training in a centralized manner. In the federated setup, 125 samples are stored in total among all clients. In the centralized setting, 100 samples are stored in total. We compute the average forgetting difference between federated ER over FedAvg and ER over standard SGD. Table 7 shows that there is more improvement in centralized ER than federated ER in terms of average forgetting with less storage. The performance gap between federated and centralized ER is due to the fundamental challenge in CFL. In the federated setting, old tasks' information only carried locally and aggregated at the server. This may not preserve old tasks' information globally compared to the centralized approach.
Table 7: Performance improvement of ER over baselines
Centralized
Federated
Total Number of Samples
100
125
ER Improvement over Basclines
2.71 %
0.48%
B RELATED WORK
Centralized Continual Learning. Several methods for continual learning and dealing with catas- trophic forgetting have been recently proposed (De Lange et al., 2019). We can categorize these solutions for the centralized settings into three methods: (1) Rehearsal-based approaches replay previous tasks' data that is stored in limited memory (Lopez-Paz & Ranzato 2017: Chaudhry jet al., 2019a; Rolnick et al. 2019 Guo et al. 2020) or that is generated using generative mod- els (Shin al. (0017) reduce forgetting of older tasks. However, the application of these solutions in FL may violate federated learning's 's data storage and privacy requirements. Li & Hoiem (2016)
follows an experience replay based method, where the model at the end of the previous task is stored and used as teacher, i.e knowledge distillation is applied on the features before the classification layer. The application of this method to CFL overperformed by generative based approaches (Qi eet al. 2023 Babakniya et al. (023). (2) Expansion-based approaches increase the model capacity to solve the problem forgetting. The model a dynamic network that increases in size as tasks arrive (Rusu et al. 2016: Yoon et al.] 2018; Sarwar et al. 2020; Yoon et al., 2020). Distributed application of these methods in FL may result in excessively big models, especially in scenarios (3) Regularization-
approaches modify the direction of gradients for each task by adding regularization such that the optimal parameters of the new task are close the optimal points of previous tasks (Kirkpatrick et al. 2017 Liu & Liu 2022 Serra et al.] 2018 Maalyya & Lazebnik 2018). These approaches (Kirkpatrick et al. 2017 Liu & Liu 2022 Serra 2018 | & Lazebnik 2018) require access to information from previously trained tasks. Another approach for regularization was adopted by GPM (Saha et al., 202I] which modifies the model gradients per layer for the sequential tasks such that the updates are orthogonal to the core gradient subspace of previous tasks. When applied distributively in an FL system, GPM not guaranteed to achieve orthogonalization in the global model as core gradient subspaces might differ across clients. It also requires extra computation and storage the client side.
where client datasets can be highly heterogeneous.
Continual Federated Learning aims to train a global model in federated learning system for multiple tasks as data for different tasks arrives at the clients in an online manner. Similar to the centralized continual learning, the main challenge is how to alleviate forgetting of the previous task by the global model, but while maintaining the privacy requirements of federated learning (Mal et al. 2022). A solution to this problem is not trivial because of the privacy constraints of federated
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learning (Ma et al., 2022). A limited set of recent works have attempted to solve this non-trivial problem. Dong et al. (2022) proposes a solution where clients store a subset of the samples from previous tasks and share them perturbed with a proxy server, which can be limiting factor in terms of storage (particularly for edge devices) and can still leak information through multiple views of the perturbed task samples. Ma et al. (2022) follows
approach based on knowledge-ddsstilatio in order to alleviate forgetting of the global model. This assumes that the server has a representative dataset for each task to be used for knowledge distillation. The availability of these datasets at the server may not be feasible, particularly in scenarios where FL is used to train models in the absence of representative centralized distributions.
Recently, (Qi et al., [023] Babakniya et al., 2023 Zhang et al. 2023) propose using generative models to generate synthetic samples to prevent forgetting. However, these approaches increase the computation and communication overhead significantly because they require training additional generative model which relatively bigger than the classifier model. Besides, the server generates synthetic samples based on the generative model which can cause privacy leakage. (Halbe et al.) 2023) uses foundation model and prompt tuning to prevent forgetting but that approach requires individual updates of clients which cannot preserve privacy, and available foundation model at the server which is not always realistic. It also increases the computation and communication costs lot because of the size of the foundation models. Lastly, Yoon et al.] (2021) addresses fundamentally distinct problem from ours while categorized under CFL It is primarily interested in learning individual local tasks, not learning global model but utilizing the information of other clients. It employs federated learning not to train a global model but to enhance the continual learning process for individual local tasks by leveraging information from other clients.
Our proposed approach for dealing with forgetting in the global model does not require sharing of the data samples with the server or assumes the availability of a dataset subset at the server. Also, our method's 's additional computation and communication cost is negligible. Perhaps, the closest related work to ours is GPM (Saha et al. 202T) centralized continual learning, however unlike in the adaptation of GPM to Fi., we introduce privacy-preservve
framework to extract the global principal subspaces of each layer and guarantee the orthogonality with a new aggregation method, while also being resilient to the heterogeneity of data distributions across clients. There are also
several works using layer's orthogonality in different problems (Li et al., 2023 Su et al.] 2023). shot federated learning work and does not extract the global subspace of the global model. It extracts a local subspace for each client at the client-side and sends it to the server. Also, they do not make global update; instead, they update each model separately regarding the local subspace of each client. (Li et al. 2023) uses layer's orthogonality idea in unlearning setting. They directly send scaled samples the server which violates the privacy and scales the communication cost with number of samples.
(Su et al., 2023) is
C CONVEXITY OF PROJECTION SET IN FEDPROJECT
Throughout this section, we will abuse notation for ease of presentation, by treating all param- eter updates 8 in their vectorized form instead of matrix, i.e., 8 is a vector of size di, instead of a matrix whose number of elements is equal to de.
Let S be the set projected on in (6). i.e. the set where Stot lives in. We want prove that S Rd is convex set where d Ee de and de is the parameter-size of layer f. In other words, we want to show that (1) (2)) S and a E [0, 1], we have that (1) (1 a) (2) e S.
Note that the set S C IRd is the cartesian product of the sets { and as a result, it is convex if the individual sets St are all convex.
Thus, we only need to focus on showing that S; is convex Vf. Recall that, by definition, the set S₁ is the set of all vectors orthogonal to all the columns of Of, i.c.
S = { u E Rd O u -- }).
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(1) (2)
Now let Bi BB Sr, then have that
oeT
age (11 a)g +11
(1)
(2)
= a0 eT (1)
8e + (1 -
a)
gi
(2)
agi
(1)
+ -a) St.
(2)
which proves that St is convex.
D ALGORITHMS
The Federated Orthogonal Training is summarized in Algorithm The Global Principal Subspace Extraction is summarized Algorithm 2
Algorithm Federated Orthogonal Training
Let C' be number of clients, 7' be number of tasks, L be number of layers.
8 represents update for layer !
8 = {6'] ]==1
Input: Total communication rounds per task R, and data of the client for task Deç. where
E [1 : CJ and / E [1:
Initialize: O = (
O {0-)
1=11 for task t = to T do
for round
to R2 do
Server runs:
Send the global model W to clients
Each client runs:
Train model with local data Dr,;
Send update 6, to Server
Server runs:
FedProjecct (W, O, (
end for
// End of task Server extracts new principal subspace O + GPSEG O, (D,
end for
FedProject W. O, {5;);
5atobaal = = - 5, + SecAgg (16:)2mm)
for layer = do
11 Applying 66) O is the orthonormal basis for O 8%a global
8t. end for global
O 0 global
W + W nd u
global
E COMPUTING PROJECTED PORTION OF THE DATA
After each task is finished, clients project their layer inputs as = - 0 X where is the input matrix for client i and layer at task t. After the projection is done, clients send the [|x "e
X
ratio
[X, to the server, which is the information of how much portion of the input is already
o_ the served-side, these rates are summed up E ||X
This summation gives an [X
[X
approximation of [
Note that this summation can be done with secure aggregation.
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Algorithm 2 Global Principal Subspace Extraction (GPSE)
Input: Global model W, orthogonal set O, and set of data of the clients (Diji=1 where is the
number of clients
Output: Updated orthogonal set O
Server runs:
Send global model W and O to the clients
Each client rums:
A, f-Randomi www.letiaaatoooooeeeeeoooatttC..
Send A, to the server
Server runs:
O t Exgganndrtthoogoaaeeee444///1/01
return O
Randomized hatpmiiiidAAiiaaiiinoollccttttoo (W, O)
Feed local data to the model Store inputs of each layer X VE [1: for layer to do
X X , 1-1 X
//Projecting input orthogonal subspace as in (
G
(0
// Normally distributed Gaussian matrix
A X G
// Applying
end for return (A 11=1
ExpandOOthoognaalSet/ (A:) (A (A,) for layer / = to I. do
A E A - SecAgg ((A.).=))
Uº v v + SVD(A*)
re CalculateRank U // Solving the problem in 12 find the minimum rank
Oc - O Oc < (rradcchmidt(")
end for return (0')
F EXPERIMENTAL DETAILS.
F.I DATASETS
Dataset statistics provided in Table
F.2 BASELINE DETAILS
EWC (Kirkpatrick et al. 2017) is regularization-base method and defines important weights as the Fisher Information diagonals. We adapt this approach FL (EWC+FL) that each client calculates and stores its own fisher information matrix and uses it in the local training process. The server performs regular FedAvg (McMahan al. (007)) as FL.
ER (Chaudhry et al.) 2019b) is memory-based approach, where certain number of samples from each task is stored the memory and the loss of that is equally contributed the new tasks. For the FL adaptation of this method (ER+FL), we let each client store certain number of data points from each task and do local training with stored samples and new tasks' samples together. The server performs regular FedAvg (McMahan et al. 2017) FL.
RGO Liu & Liu 2022) updates the optimizer iteratively to modify gradients and decreases the interference of tasks. We adapt this method to FL (RGO+FL) that each client calculates the Hessian matrix on its local data and maintains its own local projection matrix. The server performs regular FedAvg (McMahan al. 2017) FI..
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Table 8: Dataset statistics.
5-Datasets
PMNIST
(CIFARIO, MNIST, SVHN, notMNIST, FMNIST)
Input Size
1228228
3x32x32
# tasks
10
5
# clients
125
150
# training/task
50000
50000, 60000, 73257, 60000. 16853 10000, 10000, 26032, 10000, 1873
# test/task
10000
CIFAR100
Mini- Imagentt
Input Size
3x32x32
3 x4
# tasks
10
20
# clients
50
50
# training/task
5000
2500
# test/task
1000
500
2021) is applied locally for each client, i.e. clients compute orthogonal set locally at the end of each task and apply projection to their local updates in each round. The server performs regular FedAvg (McMahan al. 2017) as FL.
GPM (Saha et al.
GLFC Dong et al. 2022) is designed for dass-incremennal while our setting is task mss-incremental. GLFC proposes method consisting of parts prevent forgetting:
1. They make clients store some old tasks data and during the training of new tasks, old samples also used with normalization factor. This is very similar ER (Rolnick ct al. 2019) with normalization difference.
2. Clients store the previous task's model and calculate KL divergence loss between the out- put of the new model and old model. This output comes from the last layer. As it is class-continuous setting, the output dimension of old models is less than the new model. Therefore, they add new dimensions the old task's model by using the one-hot repre- sentation of the label. This approach is specific to class-incremental settings because in task-incremental there are different heads for different tasks so the old model does not have output for new tasks.
3. They require an additional proxy server for collecting perturbed gradients to generate sam- ples of the clients.
In order to fairly compare with GLFC, we drop 2 and because 2 is not proper for task-continual setting and 3 assumes there an additional server.
F.3 ARCHITECTURES
We use the same architecture for all of the methods we compare and the network size is fixed throughout the whole CFL process.
MLP architecture: For Permute MNIST, we use a 4- MLP as in (Kirkpatrick et al. 2017), where the first fully-connected layers' hidden size 400. The last layer the classification layer with output size of 10. We use ReI.U as non-linearity and dropout of 0.2 after the first layer and 0.5 for the next 2 layers.
AlexNet-like architecture: For Split-CIFAR100 100 dataset, we use an AlexNet-like architecture sim- ilar to GPM (Saha et al., 2021). There are 3 convolutional layer in the network with 64, 128 and 256 filters, and with 4x4, 3x3, and 2x2 kernel sizes, respectively. ReLU is used as non-linearity and 2x2 Max-Poollee Max-Pooling is applied after each Conv layer. Conv layers are followed by two fully-connected layers with hidden size of 2048. Dropout of 0.2 is applied for the first 2 layers, and with 0.5 for the rest of the network. The model is multi-headed,, where separate classification layer for each task exists at the end of the network. The output size is 10 for each of the heads.
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reduced ResNet18 architecture similar to the one in (Lopez-Paz & Ranzato 2017). We subtitute the x 4 Average-Pooling before classifier layer with 2 x Average-Pooling For Split Mini Imagenet,, use convolution with stride 2 the first layer. The model is multi-headed, where a separate classifi- cation layer for each task exists the end of the network. The output size is 5 (Split Mini-Imagenet) and 10 (5-Datasets) for each of the heads.
Reduced ResNet18 architecture: For 5-Datasets and Split Mini- Imagenet,
F.4 HYPERPARAMETERS
Note that we obtained the optimal hyperparameters presented in our work by performing grid search.
Local Training Details: all of the experiments, local number of epochs is with SGD optimizer and learning-rate=0.0 Batch size is set to 64 for Permute MNIST, and 16 for Split-CIFAR100, 5-Datasets and Split Mini-Iman Imageemaeent.
Number of Communication Rounds: For Permute MNIST, we run vanilla FL for 1100 rounds in total, where first task is run for 200 and the others for 100 rounds. All of the other methods' total communication round is set 2000, divided equally to the tasks. For 5-Datasets, again each method is run for the same round configuration. Number of rounds are 1000, 100, 200, 100, 100 in the order of 5-Datasets tasks, making in total 1500 rounds. For Split-CIFAR 100, total number of round is the same for each method with being 10000 and shared equally among 10 tasks. For Split Mini-Imagent, each task trained for 250 rounds, making in total of 5000 rounds, and this is set the same for each method.
Number of Clients: For Permute-MNIST dataset, we set the number of total clients 125 and select 64 of them randomly. In this setup each client has 480 data points per task. For 5-Datasets, we consider 150 clients in total and randomly select 64 of them. Local number of data points per client is as follows in the task order: 333, 400, 488, 400, and 112. For Split-CIFAR 100 and Split Mini- Imagenet, there are 50 clients total and the selection rate in in round is 0.5. Local number of data points per client per task is 100 and 50 for Split-CIFAR 100 and Split Mini-Imagenn respectively.
Method Specific Hyperparameters:
EWC+FL
regularization coefficient: 40 (PMNIST), 20 (CIFAR100), 1 (5-Datasets), 0.01 (Mini-Imagenet)
ER+FL
total number of stored data among all clients: 125 (PMNIST), 50 (CIFAR100 and Mini-Imagennt), 150 (5-Datasets)
RGO+FL
percentage of clients (randomly selected) that store projection matrix per task: 40% (for all datasets)
GLFC
total number of stored data among all clients: 125 (PMNIST), 50 (CIFAR 100 and Mini-Imagenet), 150 (5-Datasets)
FOT
sampling dimension sf is the same as d' for MNSIT and 5d for 5-Datasets, CI-
FAR100 and Mini-Imagener 9
Li et al. 2020a)
Threshold values for Federated Orthogonal Training:
Threshold values for Federated Orthogonal Training (see 12 are provided in Table 9 Threshold value is the same for each layer of the network. We do not tune the threshold value for each task, we solely set the threshold value for the first task and increment it evenly after each task is finished. The initial threshold values are 0.94, 0.95, 0.93 and 0.90 for PMNIST, 5-Datasets, Split-CIFAR100 and Mini-Imann Imagenet, respectively. The increment value for 5-Datasets, Split-CIFAR100 and Mini- Imagenet 0.001 while it is zero for Permute MNIST. Note that there are 5 tasks in 5-Datasets, while Permute MNIST and Split-CIFAR10 have 10 and Mini-Imagenet has 20.
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Table 9: Threshold values for Federated Orthogonal Training (see 12 each layer of the neural network.
Threshold is the same for
Dataset
Task
Task
Task
Task
Task
Task
Tisk
Task
Task
Task 10
PMNIST (idd)
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
PMNIST
5-Datasets
0.95
0.951
0.952
0,953
0,954
CIFAR100
0.871
0.872
0.874
0.875
0.877
0.878
Mini-Inagenet
0
0.902
0.903
0,904
0.905
0.906
0,907
0.908
0.909
Mini-Imagennet
0.91
0.911
0.912
0.913
0.914
0,915
0.916
0.917
0.918
0.919
F.5 GPU DEVICES
We measured local epoch training time and GPSE time (in Table 4 for computation in NVIDIA Quadro RTX 5000 GPU. We run our experiments in parallel on NVIDIA Quadro RTX 5000 GPUs.
G ADDITIONAL RESULTS
G.1 EFFECT OF TASK HETEROGENEITY
We analyze the effect of task heterogeneity across tasks for each client. other words, a client does not necessarily see all the tasks. In experiments, each client 60% of the tasks on average. In global manner, we follow continual learning literature that to separate tasks clearly (i.c. two different tasks do not arrive to clients simultaneously). Number of rounds and hyperparameters set the same as before. However, to simulate the task heterogeneity, we increase the total number of clients so that 60% of that is equal the number of clients set for the main experiments for each dataset. By this way, the number of clients that have data for each task is the same as before. We provide the results for task heterogeneity in Table TO
Table 10: Performance results of different methods Permute MNIST, Split CIFAR-100, 5- Datasets and mini-Imagene' under task-heterogeneity setting. ACC (higher better) stands for average classification accuracy of each task at the end of whole CFL rounds, while FGT (lower is better) denotes average forgetting as in (13).
PMNIST
CIFARIOU
5-Datasets
Mini-Imagenet
Method
ACC(S)
PGT(S)
ACC(%)
FGT(S)
ACC(S)
FGT(S)
ACC(%)
FGT(S)
FL.
86.93
7.15
63.89
13.51
77.86
13.62
48.43
35.34
EWC+FL
87.92
7.92
63.13
14.20
78.38
13.10
47.73
35.92
ER+FL
89.97
5.60
65.68
11.84
79.00
12.23
55.80
27.78
RGO+FL
89.27
6.13
62. 84
16.24
85,29
2.57
50.89
31.98
GLFC+FL
88.11
9.86
65.17
12.44
81.20
58.17
25.32
FOT(Ours)
90.18
1.60
71.68
1.13
85.06
0.88
68.95
0.04
FL
79.66
9.81
63.41
9.63
66.24
24.51
43,49
31.72
BOn-HI
EWC+FL
81.56
10.49
62.41
10.14
63.87
27.90
41.49
33.69
ER+FL
67.14
RGO+FL
76.68
18.31
45.54
26.18
79.90
5.11
42.95
31.22
GLFC+FL
85.36
60.74
14.20
77.65
11.42
49,58
24.93
FOT(Ours)
85.61
2.16
66.03
0.93
79.37
1.99
62.10
0.19
G.2 DETAILED RESULTS
In this section, we provide more detailed results on Split CIFAR-100 and Split mini- Imagenet for each method. In Figure we provide curve of average accuracy measured the end of each task. For both datasets, FOT outperforms other methods throughout the whole training process. This shows that our method has better performance independent of the number of tasks.
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In Figure provide curves of first task accuracy measured at the end of each task for each method. We can deduce that FOT protects first-task accuracy only with a negligible accuracy drop. The performance of other methods the first task degrades significantly when tasks arrive.
In Table III we provide the results for Single Task. Single Task stands for the training of each task separately then calculating the average accuracy of each task, i.e. there is different model for each task.
Aususacy Cilar000-10
Aug Aucurucyy-yy..... minimnpeeedd..
Eatwitt Rooert
9
DVCHI
mart
3MCAFL 301
50
0.50
Figure 4: Average accuracies of the model the end of each task for Split-Cifar and Split-Mini- Imagenet Datasets.
CUCC
10st
ASOHLL
101
Figure 5: Task 1 and Task 11 accuracies of the model at the end of each tasks for Split- Cifar100 and Split Mini- Imagenet Datasets.
Accuracy (%) results of Single Task training on various benchmarks. Please note that FGT is not applicable.
Table 11:
Method
PMNIST
CIFAR100
5-Datasets
Mini-Imagenet
Single Task (IID)
91.10 86.42
67.83
87.74 83,86
67.60 58.71
Single Task (non-IID)
61.93
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G.3 EFFECT OF THRESHOLD IN FOT
We analyze the effect of threshold value in FOT algorithm (22 experimentally. For Split Cifar- 100 dataset and under non-IID data distribution among clients, alter the threshold value of FOT algorithm ranging from 0.80 to 0.96. We provide accuracy curves of all tasks in Figure 6 In Table
12 we provide the numerical results of final accuracy for each task as well as final average accuracy and average forgetting as defined in (I3). Also, we demonstrate how the activation space is utilized for all the layers of the neural network (Figure
Table 12: Accuracy of each task the end of whole training for different threshold values of FOT on Split-Cifar100 benchmark and under non-lll) data distribution. ACC and FGT are as defined in
Task (%)
Task 2 (%)
Task (%)
Task 4 (%)
Task (%)
Task (%)
Task 7 (%)
Task (%)
Task (%)
Task 10 (%)
ACC (%)
FGT
Threshold
(%)
0.80
51.9
57.0
70.9
66.2
69.4
70.2
72.9
70.0
72.6
76.2
67.7
2.2
0.85
53.8
59.2
72.2
68.5
67.9
69.7
67.8
74.2
66.9
0.87
53.8
60.5
71.8
65.0
68.9
65.9
69.3
68.0
68.5
71.5
66.3
1.1
0.90
57.1
61.5
70.6
64.1
66.6
64.7
65.7
65.1
67.2
69.2
65.2
0.6
0.93
61.0
62.7
67.0
59.3
65.1
61.4
63.4
62.5
64.1
66.4
63.3
0.1
0.96
62.4
61.6
64.2
57.9
61.7
59.3
61.2
61.0
60.6
64.5
61.4
-0.1
Take
010
2
160.0
Figure 6: Accuracy of each task measured at the end of training of each task for different threshold values of FOT Split-Cifarl00 benchmark and under non-iid data distribution.
G.4 COMMUNICATION COST COMPARISON
As we claim in the main paper, the communication cost of FOT is negligible. This is because the only additional cost of FOT is GPSE round, which only happens at the end of each task. Besides, the communicated objects in that round are {Ar)/- and {O+) which have smaller size than the model itself we does not scale with the number of data points or the number of tasks. On the other hand, recent Continual FL work Qi al. (2023) adds additional communication cost per round. The server and
from Table 5 This additional cost of FOT per client is fixed. It
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clients has to communicate an additional generative model after the first task. This increases the total communication cost more than 100%. In Figure 8 Sand Table we provide the total communication cost of one client at the end of each task for Cifar 100 and Mim--Imagent datasets. At the first task, the communication cost is equal for all methods because they all apply FedAVG. After the first task, the communication cost of FOT and FedAVG is almost equal but cost of FedCIL. increases more than others because of the generator model communication.
1DWV1
DDW2
CDNV
0.-0.2
2-2215
Und Sp
0-0005
14
RC1
RC3
0.03
6.000
34000
3=0.00
0.00
6.6
3-01.00
(=0000
13
36
10k
Figure 7:
Percentage of used space at the end of each task on Split-Cifar100 (non-llI) data distri- bution) for different threshold levels of FOT.
Table 13:
Cumulative communication cost comparison of FedAvg, FOT and FedCIL measured at the end of each task Split- Sifit-Cifarl000 100 and Split-mini- Imagenet. moilt-mini-mmmgenet.
Tusk1
Task 2
Tusk
Task 4
Task 5
Task 6
Task 7
Task 8
Task 9
Task 10
Cifar100
FedAvg
2840.0
5680.0
8520.0
11360.0
14200.0
17040.0
19880.0
22720.0
25560.0
28400.0
FOT
2840.0
5680.3
8520.6
11360.9
14201.2
17041.4
19881.7
22722.0
25562.3
28402.6
FedCII.
2840.0
8880.0
14920.0
20960.0
27000.0
33040.0
39080.0
45120.0
51160.0
57200.0
FedAvg
1280.0
2560.0
3840.0
5120.0
6400.0
7680.0
8960.0
10240.0
11520.0
12800.0
mini-Imagend
mini- Imagenct
FOT
1280.0
2560.7
3841.4
5122.2
6402.9
7683.6
8964.3
10245.0
11525.8
12806.5
FedCII.
1280.0
4020.0
6760.0
9500.0
12240.0
14980.0
17720.0
20460.0
23200.0
25940.0
Task11
Task 12
Task 13
Task 14
Task 15
Task 16
Task 17
Task 18
Task 19
Task 20
FedAvg
14080.0
15360.0
16640.0
17920.0
19200.0
20480.0
21760.0
23040.0
24320.0
25600.0
FOT
14087.2
15367.9
16648.6
17929.4
19210.1
20490.8
21771.5
23052.2
24333.0
25613.7
FedCIL
28680.0
31420.0
34160.0
36900.0
39640.0
42380.0
45120.0
47860.0
50600.0
53340.0
G.5 CONVERGENCE ANALYSIS OF FOT
In this section, we provide the convergence curves of FOT and FedAVG for different tasks in Figure g As the reader could approve, the accuracy-round curve of FOT and FedAVG exhibits similar convergence behaviour in task training. One difference is that as we do a convex restriction on the model updates in FOT, the convergence point of FOT is less optimal compared to FedAvg- Therefore, the accuracy of FOT is less than FedAVG at the end of the task. However, when the model moves to the next tasks, FOT achieves to keep that accuracy but FedAVG cannot. This is the reason why FOT have better average accuracy and almost 0 forgetting.
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Cfar100
innn--magnntt
FestVG FeeCIL
50000
50000
40000
43000
28402.6
25613
30000
30000
20000
20000
28400.0
25800.0
10000
10
Number Tasits
Number Tasks
Figure 8: Communication Cost Comparison of FedAvg, FOT and FedCIL.
Ard wag Aecumary of Mastrrparst
Ancluge Accuragy Trals Mesiirggeeet
Ancrage Aecuragy of Taak 2 imoggeeed
022
1300
500
1303
1500
1000
2500
W00
2000
25.0
1000
Averigo Arrurry Bes Pa3gENe
Auttogo Acosory Todk? Crartee
10T
9
04
03
13
2.003
40:00
4.23
-101
Juc upo Accuncc olfccag
Aucrape acurasy of asl (Murt0
815
3003
0000
3000
340
0000
X,C
1000
5000
X008
1000
Figure 9: Accuracy Round Curves of FOT and FedAvg for various tasks in Mini-Imagenet and 10f
G.6 APPLICATION OF FOT TO SUBSET OF LAYERS
FOT can be applied some layers of the model rather than the whole layers. However, if we exclude some of the layers from FOT, we should choose whether they will be frozen in the later tasks or continue be trained. If we freeze these layers, FOT still works well as we show in section 5.2.2. In that experiment, we freeze the first layer and exclude it from FOT and the result is still as good as the original FOT. However, if we choose continue training the excluded layers the later tasks, FOT might not work well. The intuition behind this is that, when we fine-tune the excluded layers, these layers' activations in response to inputs from earlier tasks could change drastically and this change can propagate to the activation response in later layers in the model.
To empirically show the affect of this, we use FOT on only subset of layers. In the first experiment, we apply FOT the last half of the layers and fine-tune the early half of the layers. In the second experiment, we still exclude the first half layers from FOT but we freeze them after the first task. We also do the same experiment where apply FOT only the first half of the layers and fine- tune/freeze the second half. The experiments are done in mini- Imagentt with IID distribution and the same hyperparameters as the main experiments are used. Results shown in Table 14
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When do not freeze the excluded layers, the average accuracy is low which approves hypoth- esis. If we freeze these layers, we still get comparable results but the average accuracy is lower. This is because when freeze some layers, restrict the learning of subsequent tasks which leads to learning those tasks worse and lower average accuracy.
Table 14: Performance of applying FOT to the subset of layers on Mini-Imagenet. (A)-(B) denotes that method (A) is applied to the first half of the model and method (B) is applied to the second half.
FOT-FOT
FOT-frozen
FOT-finetune
frozen-FOO
finetune-FOT
ACC(%)
69.07
67.48
49.79
67.58
47.31
FGT(%)
0.19
0.14
28.15
0.12
33.41
H ADDITIVE GAUSSIAN MECHANISM
Instead of relying on the privacy guarantees coming from JL transform (Blocki et al. 2012). we formalize the privacy guarantees by applying the Gaussian mechanism (Dwork) 2006) the clients and use Secure Aggregation to get central approximate differential privacy guarantees at the server. In particular, before sending the JL transformed matrices in (II. we first clip the transformed activates and then add Gaussian noise as shown in 14 below
A CLIP
Σ
gj e +
(14)
where CL
|x min(I]X]|, L) (typically we use 1) and G sampled from 15/(1.25/6) then using standard arguments we can achieve (c, 5),
o2). If we choose o² central differential privacy, where C is the number of clients in the federated learning system.
22C
25


Reviews:
[{"review_id": "UYTzL0Xg6T", "reviewer": "Reviewer_Zv1b", "summary": "To address the catastrophic forgetting problem in the context of federated continual learning, the paper introduces a novel method that leverages orthogonalization of tasks to mitigate global forgetting in the course of continuous learning. This approach effectively reduces interference between distinct tasks, as demonstrated empirically. FOT exhibits superior performance compared to existing methods, manifesting improvements in accuracy and reduction in forgetting rates, all while incurring minimal additional computational and communication costs.", "questions": "I have a question of how to understand the correctness of the orthogonal process in this paper (intuitively)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The study presents a CFL framework, Federated Orthogonal Training (FOT), which addresses Global Catastrophic Forgetting by modifying global updates for new tasks to reduce interference with previous tasks. FOT also ensures client privacy, eliminates the need for client-side storage, and outperforms other methods in cross-device settings, even though they have additional computation and storage requirements.", "weaknesses": "Communication overhead represents a significant weakness in the methodology presented in the paper. However, the paper's analysis is somewhat superficial, lacking a comparison with baseline experiments. The argument regarding the minimized communication overhead is not sufficiently elaborated upon.\n\nSimilarly, Remark 1 highlights that the convergence analysis in the paper is relatively straightforward and lacks comprehensive theoretical underpinnings.", "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": "y4J9xqr9U8", "reviewer": "Reviewer_vBpD", "summary": "This Paper works on the continual federated learning problem. The author proposed a framework named Federated Orthogonal Training (FOT) to address the Global Catastrophic Forgetting problem.", "questions": "1. On page 6, Theorem 1, the author mentioned that \"for sufficiently large n, the principal column space of Y recovers the low-rank column space of A up to rank k with a negligible error.\". In my understanding, n corresponds to the number of data in every client. Then what is the number of data in your experiments? For the 100 clients' experiments, is there enough data? If not, how do you explain the effectiveness of the FOT?\n2. Authors said that they used the same round number for different methods. How did you decide the round number? Did other methods converge earlier than the round number?\n3. According to the description of the FOT, to me, it seems like that FOT would also work well if we only apply this to several layers in the model. Have the authors tried this?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper clearly describes the problem setting and states their target -- solving the catastrophic forgetting problem. The authors used illustrations, formulas, and well-written paragraphs to explain the FOT framework step by step clearly.\n2. When estimating the global principal, clients need to upload locally extracted principal subspace information to the server. The authors used knowledge from randomized SVD to realize this in a privacy-preserving way.\n3. In the paper, the authors discussed several important aspects, especially the privacy, of the algorithm that people worried about. \n4. Empirical results showed that the FOT provided significant improvement in average forgetting compared with extensive baselines in various benchmarks", "weaknesses": "1. There are still some baselines that are not compared in the paper, for example, the methods in paper [1],[2],[3].\n[1] Zhizhong Li and Derek Hoiem. Learning without forgetting. IEEE transactions on pattern analysis and machine intelligence, 40(12):2935–2947, 2017.\n[2] Jaehong Yoon, Wonyong Jeong, Giwoong Lee, Eunho Yang, and Sung Ju Hwang. Federated continual learning with weighted inter-client transfer. In International Conference on Machine Learning, pages 12073–12086. PMLR, 2021.\n[3] Jie Zhang, Chen Chen, Weiming Zhuang, and Lingjuan Lv. Target: Federated class-continual learning via exemplar-free distillation, 2023.\n2. In FOT, the global update was generated by projecting the aggregated updates onto the orthogonal subspace for each layer in the model. Intuitively, this will make the converge slower, which is directly related to potential extra communication costs. The authors did not provide a discussion about this.", "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": "mkd2Zn3wZQ", "reviewer": "Reviewer_VJN9", "summary": "This paper discusses the concept of Continual Federated Learning (CFL), a real-world scenario where new tasks emerge over time in a decentralized data setting. In CFL, the main challenge is Global Catastrophic Forgetting, where the performance of the global model on old tasks deteriorates as it is trained on new tasks. While previous works have attempted to address this problem, they often rely on unrealistic assumptions about past data availability or violate privacy principles. To overcome these limitations, the authors propose a novel method called Federated Orthogonal Training (FOT). FOT works by extracting the global input subspace of each layer for old tasks and modifying the aggregated updates of new tasks in a way that ensures they are orthogonal to the global principal subspace of old tasks for each layer. This reduces interference between tasks, which is the main cause of forgetting. Empirical evidence shows that FOT outperforms existing continual learning methods in the CFL setting, achieving better average accuracy and lower forgetting while incurring minimal computation and communication costs.", "questions": "The author could have a clear statement about the technical novelty and contribution of this paper compared to previous contralized continual learning.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well-written.\n2. The authors propose a CFL framework named Federated Orthogonal Training (FOT) to address the Global Catastrophic Forgetting problem. \n3. Within FOT, they introduce a novel aggregation method, named FedProject, which guarantees the orthogonality in a global manner without privacy leakage and more communication.", "weaknesses": "My primary point of concern revolves around the differentiation in technical innovation when compared to related works. More precisely, the paper that introduced the GPM (s Gradient Projection Memory) method for centralized continual learning seems to employ a similar approach to address the issue of forgetting. This similarity is evident in the equations provided – Eq 8 and 9 in the GPM paper and Eq 12 in the current paper.\nI acknowledge that this paper introduces an additional layer of aggregation, which generates $A^l$ through secure aggregation in the context of federated learning. However, it's worth noting that secure aggregation inherently lends itself to operations such as summation or averaging. Consequently, I am inclined to question the extent of the technical contribution and novelty offered by this paper. Please correct me if I misunderstand something.", "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)