Addressing the Stability-Plasticity Dilemma via Knowledge-Aware Continual Learning
Authors: Ghada Sokar
Decebal Constantin Mocanu
Mykola Pechenizkiy
Venue: ICLR 2022 Submitted (2022)
Keywords: Continual learning
Class incremental learning
Stability-plasticity dilemma
Sparse neural networks
Knowledge-Awareness

Abstract: Continual learning agents should incrementally learn a sequence of tasks while satisfying two main desiderata: accumulating on previous knowledge without forgetting and transferring previous relevant knowledge to help in future learning. Existing research largely focuses on alleviating the catastrophic forgetting problem. There, an agent is altered to prevent forgetting based solely on previous tasks. This hinders the balance between preventing forgetting and maximizing the forward transfer. In response to this, we investigate the stability-plasticity dilemma to determine which model components are eligible to be reused, added, fixed, or updated to achieve this balance. We address the class incremental learning scenario where the agent is prone to ambiguities between old and new classes. With our proposed Knowledge-Aware contiNual learner (KAN), we demonstrate that considering the semantic similarity between old and new classes helps in achieving this balance. We show that being aware of existing knowledge helps in: (1) increasing the forward transfer from similar knowledge, (2) reducing the required capacity by leveraging existing knowledge, (3) protecting dissimilar knowledge, and (4) increasing robustness to the class order in the sequence. We evaluated sequences of similar tasks, dissimilar tasks, and a mix of both constructed from the two commonly used benchmarks for class-incremental learning; CIFAR-10 and CIFAR-100.

Under review as a conference paper at ICLR 2022
ADDRESSING THE STABILITY-PLASTICIT DILEMMA VIA KNOWLEDGE-AWARE CONTINUAL LEARNING
Anonymous authors
Paper under double-blind review
ABSTRACT
Continual learning agents should incrementally learn sequence of tasks while satisfying two main desiderata: accumulating on previous knowledge without for- getting and transferring previous relevant knowledge to help in future learning. Existing research largely focuses on alleviating the catastrophic forgetting prob- lem. There, an agent is altered to prevent forgetting based solely on previous tasks. This hinders the balance between preventing forgetting and maximizing the for- ward transfer. In response to this, we investigate the stability-plasticity dilemma to determine which model components are eligible to be reused, added, fixed, or updated achieve this balance. We address the class incremental learning sce- nario where the agent is prone to ambiguities between old and new classes. With our proposed Knowledge-Aware contiNual learner (KAN) we demonstrate that considering the semantic similarity between old and new classes helps in achiev- ing this balance. We show that being aware of existing knowledge helps in: (I) increasing the forward transfer from similar knowledge, (2) reducing the required capacity by leveraging existing knowledge, (3) protecting dissimilar knowledge, and (4) increasing robustness to the class order in the sequence. We evaluated se- quenees of similar tasks, dissimilar tasks, and mix of both constructed from the two commonly used benchmarks for class-incremental learning; CIFAR-10 and CIFAR-100.
INTRODUCTION
Continual learning (CL) aims to build intelligent agents based on deep neural networks that can learn sequence of tasks, use previous knowledge in future learning, and accumulate on it without forgetting. The main challenge in this paradigm is the stability-plasttici dilemma (Mermillod al.) 2013). Optimizing all model weights on a new task, highest plasticity, causes catastrophic forgetting of previous tasks (McCloskey & Cohen 1989). While fixing all weights, highest stability, hinders learning new tasks. Finding the right balance between stability and plasticity is challenging. This sharpens the community's focus on the forgetting problem. The excessive focus on one aspect im- pedes building agents that balance between mitigating forgetting and exploiting relevant knowledge future learning while considering the capacity constraints
et al., 2018).
Task- specffpeiific components methods (Section address the stability-plasticel dilemma by allocating different connections for each task. The newly added connections for new task are flexible to learn while the connections of previously learned tasks are fixed. Despite that these methods are quite successful in mitigating forgetting, some limitations remain to be tackled to achieve the balance between the above-mentioned CL desiderata. First, new components (comnections/neurons are allocated in each layer for every new task. However, the new components might capture knowledge that already exists in the learned components of previous similar tasks. In this case, adding new components would be redundant and resource inefficient. Second, the topology of the new task is allocated using the unimportant components of previous tasks. The design choice of the new topology is made to protect previous knowledge but does not take into consideration the usefulness of this topology for the current task. This limits the forward transfer of useful knowledge for the new task and its speed of learning. Further discussions of these limitations are in Appendix A
'The code will be made publicly available after the publication of this paper and now can be found in the supplementary material in the system.

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In response to these limitations, we study the core of the cubiility-plastiii dilemma in the CL paradigm. In particular, we address the following question: Which components are eligible to be reused, added, updated, or fixed when CL agent faces a new task to achieve the balance be- tween the CL. desiderata? With our proposed Knowledge-Aware contiuNal learner (KAN), we find that considering the semantic similarity between old and new classes is crucial in addressing this question. With the awareness of existing knowledge, the CL agent could identify similar previous knowledge that could be reusable in learning a new task (forward transfer) and add the necessary components only to capture the specific knowledge that cannot be explained with the existing knowl- edge (resource efficiency). Moreover, the agent could protect the dissimilar previous knowledge and limit its transfer to the new task (mitigating forgetting).
To the best of our knowledge, the recent work, CAT (Ke et al. 2020). the only one that attempted to consider the task similarities to balance between forward transfer and mitigating forgetting. CAT was proposed for the maskmencremntal Learning (task-IL.) scenario, where each task is a separate classification problem. In this work, we aim to tackle the more challenging class-Incrementar Learn- ing (class-IL) scenario where a unified classifier is used for all seen classes. The inaccessibility of the task identity inference in the latter scenario arises the following challenges: (1) ambiguities between old and new tasks which affect the balance between maintaining forgetting and forward transfer. (2) Inability to select the corresponding components to task inference; all the existing components in the model are involved in making predictions. To fully analyze the stability-plasticity dilemma in class-IL, -L. given the aforementioned challenges, we focus on the rehearsal-free strategy (Section 2 in which there is no reliance on past data to address forgetting and fixed-capacity model used. We also report the performance of the model in Task-IL to further demonstrate the challenges in class-IL. Our contributions are:
We demonstrate that considering the semantic relation between old and new classes is cru- cial in addressing the stability-plastiitty dilemma and the balance between CL desiderata.
We show that considering both previous and current tasks in allocating the initial topology of new task allows for: selective transfer for the relevant (similar) knowledge, protecting the irrelevant (dissimilar) knowledge, and identifying the reusable previous components to avoid allocating unnecessary resources.
We demonstrate that the standard softmax-classifiatte ambiguities between old and new tasks in class-IL. We study different setups for the output layer which significantly improve the performance of rehearsal-free methods. Our analyses reveal the limitations of the softmax in class-IL and shed light on the necessity of paying more attention to these limitations to narrow the gap between class-IL II and task-IL. II.
layer is source of constituting the
We propose Knowledge-Awee coNtinual leaner (KAN); resource-efficient task specifpei components method to explain these findings.
2 RELATED WORK
We divide the CL methods into two main categories: Rehearsal-free and Rehearsal- methods.
Rehearsal-free methods. In this category, previous data is inaccessible during future learning. There are two strategies to address forgetting. (1) Task-specific components. Specific components are assigned to each task. The components of previous tasks are stable during future learning. While the components that have not been allocated to any task are flexible for future learning. These meth- ods either extend the model size when facing a new task (Rusu et al., 2016) Yoon et al. (018) or uses a fixed-capacity and each task is trained either using sparse web-network within the model (Mallya & Lazebnik] 2018: Mallya et al. 2018: Yoon et al. 2019). These methods are limited to task-Il IL as they require the task identity to specify the components of each task at inference. Recently, few methods have been proposed for class-IL using fixed-capacin model. SupSup (Wortsman et al.) 2020) learns a mask for each task over randomly initialized fixed network. FSLL (Mazumder et al., 2 addressed the few-shot lifelong learning setup. It relies on the availability of large data for the first task to train a dense model. Then, few unimportant parameters are used to learn each task. SpaceNet (Sokar et al. 2021c) learns sparse
for each task from scratch using dynamic sparse training (Mocanu et al. 2018 Hoefler et al.) (02T)) where the weights and the sparse topol- ogy are optimized simultaneously. (2) Regularization-based.. A fixed-capacity model is used and
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all parameters are involved in optimizing each task. Forgetting is addressed either by constraining
changes in the important weights of previous tasks ZZnke et al. 2017: Kirkpatrick et al., 2017
Aljundi al. (2018) via distillation loss (Li & Horem 2017: Dhar et al. 2019). Previous stud-
ies (Kemker et al. (018)] Hsu et al., 2018 Farquhar & Gal 2019 van de Ven & Tolias 2018) showed
the big performance gap of most of these methods between class-Il. and task-ll..
Rehearsal-based methods. In this category, forgetting is addressed by replaying: (1) a subset of old samples (Rebuffi et al., 2017; Lopez-Paz Ranzato 2017 Riemer et al. 2018 Chaudhry al.,
2018). (2) pseudo-samples from a generative model (Mocanu et al. 2016 Shin et al.
2017 Sokaa 2020p. They
al. 2021a), or (3) generative high-level features (Liu al. 2020 van de Ven al.
use classification model and buffer to store old samples or a generative model to generate them.
Other related studies. Recent work by (Ramasesh al., 2020) evaluates forgetting in hidden representations in Task-IL and shows how semantic similarities influence it. They showed that higher layers are more prone forgetting and intermediate similarity results in maximal forgetting.
SAM (Sokar et al.] (02 studies the importance of selective transfer in dense networks by meta-
training (Finn et al. 2017) a self-attention mechanism (Hu et al., 2018). CAT (Ke et al.) 2020) addresses the relation between task similarities and forward/backwand transfer in task-IL where they rely on the task identity to find the previous similar knowledge in dense networks. In (Chen et al. 2020), the lottery ticket hypothesis (Frankle & Carbin 2018) is studied for the CL paradigm.
3 PRELIMINARIES
3.1 PROBLEM FORMULATION
We consider the problem of learning sequence of T tasks. Each task brings new set of C classes. Each task has its own dataset D' = {D'H where D is the data of class c in task t. Once we have trained task its data is discarded. The goal is sequentially learn the model f with a unified classifier that can map any input to its corresponding target for all seen classes up to time step t.
3.2 REHEARSAL-FRES STRATEGY
In this work, we focus to analyze how the model components should be altered for each task in tpsk-specific components methods. In addition, we compare the behavior of these methods against regularization-but based methods which use all model components for each task.
3.2. SPACENET
We analyze the recent task task-specific components method, SpaceNet (Sokar et al., 2021c The mo- tivations for choosing SpaceNet are: (1) unlike most task-specifie component methods, it does not rely on task identity; making it applicable for class-Il., (2) Instead of relying on a pre-trained model, it trains sparse subnetworks from scratch. These are favorable for real-world applications in which we cannot assume the availability of task identity at inference and large datasets for pretraining. To the best of our knowledge, it is only existing work that fulfills these objectives.
The main idea of SpaceNet is to learn sparse representation for each task to reduce the interference between tasks. SpaceNet consists of three main steps. (1) Sparse connections allocation: sparse sub-network W. is randomly allocated for each task in each layer / between subset of the unim- portant neurons badh and hidd where hedd are randomly selected from the free neurons h free that have not been reserved for any previous task. (2) Dynamic sparse training: the sparse weights and -oop--nd-grow" cycles to produce sparse representation. (3) Neu-
topology optimized through ron reservation: after learning, a fraction of the most important neurons for the current task h,pec
fixed and removed from the free list of neurons h free, The full details are provided in Appendix I SpaceNet considers only previous tasks during the allocation of new connections for the new tasks and randomly allocates new sub-network between the free neurons for each task. In this work, we propose novel criteria which consider both past and current tasks in allocation and allow reusing some of the past components instead of allocating new ones.
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Reserved
Hlogurocck
Ireuse
Canditht
Fladd Taak
Model after training Task
(1) Detect itreeereeeewee
(2) Connections aloccatin
Figure 1: An overview of our proposed method KAN. The most left panel shows the network after training on Task 1. When the model faces Task 2, KAN reuses the existing similar knowledge up to layer freuse and adds new components in the subsequent layers. For / reuses (I) the candidate similar neurons for each new class are detected. (2) New sparse connections are allocated between the candidate neurons and free neurons in layer / - 1 and the free neurons in layer
4 PROPOSED KNOWLEDGE--WARE CONTINUAL LEARNER (KAN)
In this work, we aim study how the model components should be altered, when CL agent faces a new task, to balance between the CL desiderata (i.e. forward transfer, reducing forgetting, and resource efficiency). To address this goal, we propose Knowledge-Awar (KAN), a new task-specific components model that exploits the semantic similarity between old and new classes in altering a CL model. The novel contributions of KAN are: (1) the allocation of new topology considers both previous and current tasks: (2) reusing some of the existing learned components from previous similar tasks instead of allocating new components in each layer.
contiNual learner
Figure I shows an overview of KAN. When the agent faces new task t, new connections are al- located for this task. The new connections are trained with stochastic gradient descent (SGD). All previous connections are fixed. Instead of allocating new connections in each layer, KAN starts allo- cation from layer Lyeuse E [1, L - 1] which is hyper-parameter that controls the trade-off between allocating components and reusing old components based existing knowledge. L is the num- ber of layers. The topology from layer 1 up to but excluding layer breuse remains unchanged. In KAN, we operate the class level instead of tasks for connections allocation. Starting from (rease onwards, KAN allocates new connections for each new class. This requires two steps.
The first step is the detection of similar knowledge. Typically, some of the previously learned classes are similar to the new classes and some are dissimilar. Classes that have semantic similarity most likely to share similar representation (Appendix [A] Figures (5d and 6). For each new class KAN finds the candidate neurons hn in each layer 7, vI > treuses that could contain relevant representation learned by a previous similar class. To achieve this, we feed the data of class c, D on the trained model at time step t- ft-1, and calculate the average activation af in each layer We estimate the relevance by the activation value. The higher the activation is, the more relevant is the neuron (i.e. the neurons that fire are the most important the current class). We select a fraction self cand of the most relevant neurons be the candidate neurons hf cand for class c. Please note that by exploiting the representation learned in the candidate neurons h he-eand in layer Ireuses we are reusing previous learned components connected to these neurons in preceding layers (1 (lrewve). Therefore, these components are stable but reusable.
The second step is to exploit the relevant candidate neurons in each layer I, VI 2 freuse, in allo- cating the new connections for each new class c. Adding new connections in layer / between the candidate neurons h he-cond and h he-aand could maximize the forward transfer from previous knowl- edge. However, this would lead to forgetting previously similar knowledge the representation learned by the important neurons for previous classes would be updated. To solve this dilemma, we allow reusing the candidate neurons by allocating new connections out of these neurons. However, we do not allow new connections to be added into the important neurons for previous tasks h (fully filled circles in Figure T). In addition, block the gradient flow through all important
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Algorithm KAN Connections Allocation
1: Require: self, self-rmd freuse, sparsity level él
for each class e task do
Get candidate neurons each layer for each new class
3: añ <- forward pass D on /
4: ncc sorDesscending (af)
5: bf t- select the first self-cund from
end each
7: (h "..bb"") b²")) randomly select seli-1 and self neurons from h and h mee
8: for each class in task do
9: for 4 Lreuse to
do
10:
hn (h U br 11: halloe
<- h
12:
randomly allocate parameters Wi with sparsity et between he and ha Wi Wi U W
14: end for each
15:
end for each
rons for previous tasks h spec even it is a candidate neuron for the current class, to protect previous knowledge. Thus, hapec is used to mask the gradient gi through the neurons in layer I follows: Si = @ (1 h sped) where he is one-hot encoding vector which the entities contain value of represent important neuron in layer 1. This allows not to forget the previously learned knowledge while performing selective transfer for the useful knowledge. Let ha and hafioe be the list of neurons for allocating new sparse connections W for class c in layer 7 with a sparsity level f;; where e is syper-pprammetr that controls the number of sparse connections for each task and O' is the total number of classes. The neurons used for allocation are selected as follows:
h I[1 (hrard U hes and halioe hidd where hedd h free Vi (1, 1}, (1)
where h is subset, sel;, of the free neurons in layer that are not reserved for any previous task. Since the important neurons for the dissimilar classes are not involved in the topology allocation, are protecting their knowledge. Thus, the selective transfer KAN addresses the forward transfer while maintaining previous similar and dissimilar knowledge. Full details are in Algorithm [
After allocating the topology for task t, we follow steps and of SpaceNet, dynamic sparse training and neuron reservation, to train the task and fix its important neurons h afterwards (Section 3.2.
EXPERIMENTS AND RESULTS
We now use KAN to investigate how the model should deal with sequences of similar, dissimilar tasks, or mix of both to achieve the balance between the competing CL desiderata. We compare against other rehearsal-free methods. We analyze the accuracy, forward and backward transfer, model capacity, and class-ordet sensitivity. Each experiment is repeated times with different seeds.
5.1 DATASETS AND BASELINES
Datasets. We construct sequences with varying similarities from the standard benchmarks for class-IL, CIFAR-10 and CIFAR-100 (Zeke et al., (017), by altering the original class der. From CIFAR-10 10, we construct two-tasks and five-tasks sequences. For the two-tasks se- quences, we construct: (1) Similar sequence (sim (sim_seq 2Tasks): Taskl (ca,cat,, Task2(dog.ttuck) where the two tasks share semantic similarity. (2) Dissimilar sequence (dissim seq 2Taasks): Task Task1 (car,truck). Task2 (cat,dog), where the semantic similarity is limited. For the five-tasks sequence Saan__se_5Taassee der: (atr..aattooreeettto..o hww.aaaaa..oor.rucccoooogoooo htr.ggeeee.rrirlaabbbb....pppppp classes. With this order, we study the case where there is past knowledge that has common mantic similarity with the new task. From CIFAR- we construct a longer sequence of 8 tasks (sim_sed seq 8Tasks). Each task contain two classes coming from two different categories and each two consecutive tasks shares semantic similarity. The classes has the following order: (apples.girl, hwgg:ebboo...uuee....c..ee.....e motooylle
we arranged the 10 classes of (IFAR-10 in the following or-
Each task consists of two consecutive
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Table 1: Forward transfer. Test accuracy [%] of Task in two cases: similar and dissimilar tasks.
Strategy
Method
sim seq 2Tasks
dissim seq 2Tasks
Parameters allocated
Use whole capacity
Grey 0.1
98.19 + 0.240
76.05 3.95
(x(844576
Use whole capacity
SI Grog = 0.01
98.30 +: 0.127
81.91 + 1.60
I (x(884576)
Use netwwek
Scratch
97.73
80.21 0.02
(0.065
Add new sub network per task
Space
98.17 + 0.002
81.27 4: 0.02
(0.131 x)
Reuse relevant knowledge
KAN (ours) lyease
96.29 0.006
66. 0.02
(0.065
Reuse random knowledge
Random reuse beese
95.43 + 0.002
64.60 0.01
(0.065
Architecture. We followed the architecture used by (Zenke et al. 2017 Sokar et al. 2021c) for a
direct comparison. It also allows us to study the challenging case where the available capacity is limited. It consists of 4 convolutional layers (32-32-64-64 feature maps) and 2 feed-forward layers (512-N neurons), where N is the number of classes. More details are in Appendix
Baselines. We compare with the following baselines: (1) Scratch: learning a single task from scratch using sparse neural network with the same sparsity level used for one task in the task specific components baselines, (2) SpaceNet (Sokar et al.] (0211a, (3) SI (Zenke et al. 2017). (4) Random reuse: modified version of KAN which the neurons used for allocating new connections are subset of the free neurons (i.e. no candidate neurons), and (5) Irrelevant reuse: a modified version of KAN in which the candidate neurons are the ones with the lowest activation. The last two baselines allow us to study the role of the initial sparse topology in the performance. In our experiments, we report the performance of our modified improved version of SI where a multi- output layer is used instead of Lingleheaden (Appendix (..2). A comparison with other baselines (PackNet (Mallya
& Lazebnik 2018). EWC (Serra et al.] [2188) p. MAS (Aljundi et al. 2018). and LWF (Li & Hoiem
2017)) and analysis of forgetting can be found in Appendix G
5.2
SELECTIVE KNOWLEDGE TRANSFER FROM PREVIOUSLY LEARNED KNOWLEDGE
5.2.1
ATTENTION TO RELEVANT KNOWLEDGE
In this paragraph, we address the question: Is all previous knowledge always useful for the current task? To answer this question, we begin by investigating the extreme case where new connections for a new task are added in the output layer only (i.e. brease = L - 1). In this case, the minimum new capacity is allocated, and the maximum reusability and highest stability of existing knowledge occur. We performed this analysis two-tasks sequences (sim_seq_2) fetmsse_TTakks, 2Takks, dissim_seq, (etns_sssssssssss___________ To estimate the forward transfer, we focus on the performance of Task assuming the availability of the task identity at inference. Table T shows the accuracy of Task 2 using different methods.
In the similar case, SI and SpaceNet achieve a better performance than learning from scratch which indicates that the learned knowledge by Task is useful for Task 2. Interestingly, despite that Task 2 is trained by learning the output layer only in KAN, it achieves a close performance to SI and SpaceNet. This performance is achieved using only 6.5 % and 50% of the capacity used by SI and SpaceNet respectively. KAN outperforms the Random reuse baseline which reveals the importance of selective transfer. In the dissimilar case, using SI with Greg of 0.1 limits the performance of Task 2 while reducing it to 0.01 gives the connections the flexibility to learn at the expense of forgetting Task 1 as we will discuss next. This shows that solving the stability-plastiiity dilemma by regularization strategies is challenging when tasks are dissimilar. This is because the changes in the model weights after learning a dissimilar task are larger than learning similar one (Appendix A. In KAN, allocating new connections in the last layer only is not sufficient to learn Task 2 since the existing knowledge is irrelevant. This experiment reveals that adding new components in earlier layers is essential when existing knowledge is irrelevant. However, if there is semantic similarity between classes, with a selective transfer, we could learn new classes with minimum components.
To further analyze the importance of the selective transfer, we performed another experiment on him_seq__Taakee in which we fool KAN by making the he.aand of class dog in Task 2 equals to the candidate neurons of the dissimilar class in Task 1 (car). We named this baseline as Inattentive learner. Figure shows the test accuracy of Task along training. As illustrated, attention to irrel- evant knowledge leads to lower initial performance, slower learning speed, and lower final accuracy.

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100
KAN (ou's)
PoNHAR 48 (10-115)
0.900
80
Inatiention leamer
Accurssy dog
Accurtry (16]
0.000
60
60
.800
40
20
KAN tou
Sin KAN (ouns)
0.940
Instiratinn Iuamer
kim, and Iraftendie learTer
disain 550
Taak Taec Aceeage
Task Averaae
Traiiin
Class-IL
T
(a) Learning speed
(b) Class order sensitivity
Backward transfer
Figure 2: (a) Using irrelevant knowledge decreases the learning speed. (b) Unawareness of existing
knowledge decreases robustness to the class order. (c) Selective transfer reduces forgetting.
Previous experiments reveal the sensitivity of the learner to the class order when the relation between old and new classes is not considered or the irrelevant knowledge is used. Figure 26 shows the test accuracy of class dog which appears in Task 2 in three different settings: (1) Similar tasks with KAN, (2) Similar tasks with Inattentive learner (relevant knowledge exists but we reuse the dissimilar one), and (3) Dissimilar tasks with KAN (existing knowledge has limited relevance). Despite that the same class appears at the same time step in the three scenarios, its performance varies.
5.2.2 ADDING NEW COMPONENTS FOR A NEW TASK
We showed that new tasks could be learned using previous similar knowledge in Task-IL. Does this hold in class-IL? Next, we will discuss the ambignities that arise in class-IL between old and new classes when they are similar. We performed this experiment on the iim_seq_2Taske benchmark.
Class ambiguities. In class-Il., a unified classifier is used for all seen classes. Reusing all relevant knowledge from previous similar classes up to the highest representation level (closest layer to the output; freuse L 1) in learning new class maximizes the forward transfer with minimum additional capacity. Nevertheless, it misleads the classifier and leads to ambiguities between old and new classes. The first row in T illustrates this. We report the accuracy of each task after learning the two. We find that in task-IL, we can easily fully utilize previous knowledge up to the last layer in learning a new task; achieving high performance on old and new tasks. However, in class-lL, the model tends to bias toward one of the tasks due to the ambiguity between classes. To address this problem, we propose two modifications for KAN: (1) usage of the free neurons only in allocating the output connections (i.e. no candidate neurons are used; h = se! (2) Enforcing
the re-usage of previous components to be till earlier layers than the highest one (Treuse < L- 1) and adding new components in higher layers to capture the specific representations of the new classes. Next, we will study the effect of these proposed constraints. Ablation study is in Appendix (D]
Adding new components in earlier layer. To reduce the ambiguities, we need to satisfy the condi- tion Treuse < T. T--1, 1. Here, we study the effect of breuse in the performance. As illustrated in Table 2. adding new components in the last three layers, lreuse = T, 3. achieves the highest balance between the two tasks for the two scenarios. Adding new components for Task 2 in earlier layers Greuse < L-33 3) does not lead to a significant performance gain for Task 2 in Task-IL. Nevertheless, consumes additional resources and hinders the balance between forgetting and forward transfer in class-IL. IL. The leamer becomes biased towards the second task and starts forgetting the previous one.
Figure shows the performance on sim sim_seq_20 -etn-seeeTTaasss using different methods. We compare the best two settings for KAN (freuse = {L - 2, L - 3}) against other basclines. SpaceNet achieves higher performance on Task at the expense of forgetting Task 1. KAN achieves a balance between the two tasks and bit higher average accuracy than SpaceNet in class-IL. with less computational cost (Appendix D Random re-usage of previous knowledge or paying attention to the least relevant one (irrelevant reuse) hinders learning the second task. The regularization-ben in balancing between the two tasks in the class-IL. Using all the network parameters in learning each task and relying the regularization cocfficient Creg to address the stability and plasticity trade-off is challenging. The learner either maintains the performance of Task I using a high value of Creg or forgets it and is biased towards the second task by decreasing the value of Greg-
method, SI, struggles
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Table 2:
The accuracy [%] of each task in sim seq 2Tasks after learning the tasks in sequence
along with their average accuracy using different values of reuse in class-IL and task-IL.
Class-IL Task
Task-IL
peuss
Task
Average
Task
Task 2
Average
# &Parameters allocated
69.66
34.77
52.21
95.34
96.29
95.81
Ix(58145)
64.34
61.11
62.72
95.28
97.16
96.21
(1.85x)
64.90
63.97
64.43
94,41
97.90
96.15
(1.93x)
63.36
66.85
65.10
94.43
97.68
96.05
(1.97x)
58.99
69.82
64.41
93.37
98.19
95.78
(1.99x)
59.69
70.81
65.25
95.38
98.29
96.84
(2x)
KAN (ours) frause =
Acuunty (.
Accursa 10
KAN (ours) freuse =
SpaceNet
Irrelevant reuse
Random reuse
30
SI Cing = 0.1
20
Greg 0.01
Average
Tank
(a) Class-IL
(b) Task-IL
Figure 3: Test accuracy of each task in sim 2 2Tasks and their average in class-IL and task- II.
5.3 SELECTIVE BACKWARD TRANSFER FROM NEW TASKS
Constraining the capacity of the learner using a fixed-capacity model one of the desiderata of CL. However, sooner or later this capacity would reach its limit and we have to use the existing connections to learn new tasks. In this experiment, we demonstrate that performing selective back- ward transfer based the relevant knowledge reduces forgetting even if the previous weights are updated without regularization. We performed this experiment on sim_seq 2 We study the extreme case where new connections update the connections in all preceding layers which connected to the candidate neurons h he-cond for 1: 2 < 1 S Irease- We compare KAN with the Inattentive learner described in Section 5.2.1 Figure 2clshows the accuracy of each task in the sequence after learning the two and their average. With the selective transfer KAN, we managed to maintain higher performance for Task in class-IL and Task-IL. More interestingly, KAN outperforms the Inattentive learner by 10.4 % in class-IL. This reveals the importance of selective transfer for both forward transfer and selective backward update.
added in the last layer only (i.e. lreuse = 1). We
5.4 PERFORMANCE ON LONG SEQUENCE OF TASKS
Next, we study the performance of KAN on long sequences. For CIFAR-10 sim_seq wet.seee_.aaaaaa.. STasks, we use freuse L 3. For CIFAR-100 Ammsse_8Tasss, since every two consecutive tasks are dissimilar to previous ones, we allocate new sparse connections in all layers to capture the new knowledge for every odd time step (t 3. In other time steps, we use lreuee L 3. For SI, we use Greg of 0.01. As illustrated in Figure 4 KAN achieves higher performance than SpaceNet on the two benchmarks. The topology of the new connections matters; paying attention to relevant knowledge is better than using random neurons or irrelevant ones. SI is more prone forgetting in long sequences of tasks. Further discussion on scalability and resource efficiency is in Appendix C
5.5 SOFTMAX CLASSIFIER FOR CLASS-INCREMENTAL LEARNING
In this paper, we have observed the performance gap between class-IL and task-IL which was also reported by previous works (Maltoni & Lomonaco 2019 Hsu al. 2018). In our experiments, the learned representations by the base network both scenarios are the same, this suggests investi- gating the output layer in class-IL. Current CL methods use the standard softmax. A sindle-head output layer is usually used and extended when new classes arrive. We named this as Extendable layer. The softmax has some limitations for this setup. It is based on the closed-world assumption (a
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(a) CIFAR-10 sim seq STasks
(b) CIFAR-I00 sim seq 8Tasks
(c) Different output setups
Figure 4: The performance of different methods in class-IL on long sequences of tasks (CIFAR-10 (a) and CIFAR-IO (b)). (c) The performance of Split CIFAR-10 using different output layers.
fixed number of classes). divides the feature space between known classes (Geng et al., 2020a b): normalizing the output probabilities to be summed to one, leaving no space for future classes. In addition, the model is biased towards new classes due to the unavailability of old data (Wu et al.] 2019). Few setups were proposed in the literature to alleviate these limitations: (1) Cosine nor- malization (Hou et al., 2019) which enforces balanced magnitudes of the output weights across all classes, (2) Multi-headed layer (Mittal et al. 2021) where each task has its own output layer, (3) Pre-defined layer (Sokar et al.) (021c) where large number of output neurons is defined at t = 0. At each time step, the output weights of the current task are trainable and all other output weights are set to zero The first two were evaluated for rehearsal based methods. More details in Appendix H
In this work, we evaluate these setups for the rehearsal-free strategy. We analyze the task-specific components method SpaceNet (Sokar et al., (222c) on Split CIFAR-10 benchmark with the typical class order. As illustrated Figure [4c] the commonly used extendable layer is the most prone to forgetting previous tasks. There is no significant gain by using cosine normalization with the ex- tendable layer. Both the multi- headed and predefined single-headed, which address the closed-world assumption, succeed in mitigating forgetting while the latter achieves the best average performance. Analysis on the other rehearsal-free strategy, regularization based, is provided in Appendix H.2
5.6 CLASS SIMILARITY DETECTION: LEARNING CLASS RELATIONSHIP
In the previous analyses, assume that we know whether the model previously learned a similar class to the new one. Automatic detection of class relationships is required. We evaluated the ap- proach by (You et al.
2020) which averages the predictions of the model ft over all samples
of each new class; p(y 1 P. Σ (( ft (z). We observe that it works well in short se-
quences but does not scale for long sequences. Another approach by (Ke et al. 2020) is to compare the performance of task when it learns from scratch against learning from the model ft-1. How- ever, it is computationally expensive. In this work, we focus on analyzing the effect of considering task similarities. We leave designing sophisticated methods for similarity detection for future work.
6 CONCLUSION
In this paper, we study the stability-plastiiity dilemma in continual learning where the model learns incrementally a sequence of tasks. We focus on the rehearsal-free methods using a fixed-capacity model in class-IL. With our proposed Knowledge-Awee how should the model be altered under different tasks similarity. Our analysis reveals that taking into consideration the relation between old and new classes is crucial to balance the CL desiderata.
contiNual learner (KAN), we analyzed
We show that it helps in: (1) identifying the existing similar knowledge which could be exploited to increase the forward transfer and reduce the allocated capacity for a new task; (2) protecting dissimilar knowledge via selective update. We demonstrate that achieving a balance between CL desiderata is more challenging in class-IL than in task-IL. Moreover, we show that the conventional softmax is cause of this challenge. Our findings suggest that more efforts should be dedicated to (1) detecting the class similarities, (2) designing attention mechanisms for selective transfer, and (3) addressing the limitations of the softmax classifier to close the gap between class-II. and task-II..

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dog
Inuck
Tank
Task 2 2
0.28
0.96
Toak
34
cot
0.99
0,5
Siiiar Tasks
(a) New components/task
(b) Representational similarity
(c) Sparse representation
Figure 5: (a) New components are allocated per task in each layer. (b) Cosine similarity between
the last hidden representation of each class in Task 1 and Task 2 produced by the model trained on Task Task 2 has representational similarity with Task that could be exploited its learning. (c) Activation of the last hidden layer of one class in Task 2 using the dense model trained Task 1. The relevant representation from previous knowledge is sparse.
A OBSERVATIONS THE CONTINUAL LEARNING PARADIGM
In this appendix, we will diseuss some the observations in the continual learning paradigm and provide further discussion the limitations stated in Section
New components of each task. Figure 5a show an illustration of connections allocation in the task-specific components method SpaceNet (Sokar et al. 2021c). New connections are added in each layer for each new task. The new connections are randomly allocated between the free neurons that could be shared between tasks. The full-filled colored neurons represent the neurons that important for task and reserved (fixed). The criteria for adding new connections consider only protecting previous knowledge.
Representational similarities between tasks. Figure 5b shows the representational similarities be- tween classes in similar tasks. This experiment is performed on the aim_seq__Taake benchmark using the same experimental setup for KAN described in Section 5. and Appendix B The repre- sentations are obtained from the last hidden layer (closest the output). We average the activation, of the model trained on Task 1 f over all samples of each class in Task 1 and Task 2. The cosine similarity between the two representations a1, ag is calculated as follows: < 21, a2 >= aj aa; where | denotes the L22normalized vector. As illustrated in the figure, when the classes share semantic similarity, as the case in this benchmark, there would be representational sim- ilarity between the classes. In KAN, we exploit this in learning Task 2 by maximizing the forward transfer from Task I instead of allocating new components for learning these representation for the new task.
GJsoos the activation of subset of the neurons in the last hidden layer in cases: similar tasks (sim_seq 2Tasks) and dissimilar tasks (dissim_seq (askss, As illustrated in Figure 66a| there are high representational similarities between each class in Task I and the corresponding class that shares some semantic similarity in Task 2. On the other hand, if the tasks are dissimilar (Figure 6b). the new task has a different representation than the previous one.
Relevant representation. As illustrated in Figure 6 the relevant representation from previous knowledge sparse. This representation is obtained from sparse neural network. It would be interesting to study whether the same case holds in dense neural networks. We performed this ex- periment on the sim_seq_2Tase benchmark using a dense model with the architecture described in Section
5.1 |nd and the experimental setting in Appendix B Figure Figure5c/showw shows the 2D visualization of the average the activation of the last hidden layer over samples the class dog in Task 2 using the model trained on Task fl, As illustrated in the figure, even if the network is dense, the relevant representation for this new class is sparse. This is also discussed in (Sokar et al. 2021b Javed & Whitel 2019).
Backward transfer. The main cause of catastrophic forgetting arises from updating the previously learned weights when the model is optimized for one. Next, will analyze the weight change
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car
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cat
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Figure 6: Visualization of the representation of a subset of the neurons in the last hidden layer of each class the sim_seq.2Taae 2Tasks (a) and dissim_seq.27as 2Tasks (b) benchmarks.
dense neural networks two cases: similar and dissimilar sequences tasks. We performed this experiment sim_seq_2Taas 2aakks and dissim_seq -eq benchmarks. We dense model with the
architecture described Section [.I and the experimental settings provided in Appendix B We estimated the average absolute change in the parameters in each layer between the models f and f2. This change is calculated as follows:
A
N 2
(2)
where Wi, is the weight in layer | after learning task t and Nj is the number of weights in layer Tigure/]shove the average absolute change in each layer in the network. As illustrated, the change higher when the two tasks are dissimilar. This experiment demonstrates selective update based on the relation between the classes is essential. Updating the dissimilar knowledge will cause the weights to move far away from the values learned in the previous time step. This causes forgetting of previous tasks.
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0.08
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dissim_seq 2Tasks
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0.05
0.04
0.03
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Layer
Layer 3
Layer 4 Layer
Figure 7: The average absolute difference between the weights of the model at 1 (f)) and the weights at / = 2 (f2). The change is higher when the tasks in the sequence are dissimilar.
B EXPERIMENTAL SETUP
In this appendix, we will describe the experimental settings used in the experiments provided in Section
Datasets. CIFAR-I0 and CIFAR-100 Krizhevsky et al. (2009) are well-known benchmarks for classification tasks. They contain tiny natural images of size (32 x 32). CIFAR-IO consists of 10 classes. contains 6000 images per class (6000 training + 1000 test). While CIFAR-IO0 contains 100 classes, with 600 images per class (500 train 100 test). We split these classes into tasks with different levels of semantic similarity. The network is trained using stochastic gradient descent with batch size of 64 and learning rate of 0.1. Each task is trained for 40 epochs. The hyperparameters are selected using a random search.
Architecture. The model consists of 4 convolutional layers (32-32-64- feature maps). The kernel size is 3 x 3. Max pooling layer is added after every 2 convolutional layers. Two feed-forward lay- ers follow the convolutional layers (512-N neurons), where N is the total number of classes from all tasks. Similar as in (Sokar et al. 2021c). we replace the dropout layers with batch normaliza- tion [Toffe Szegedy 2015).
slyperparameters.. Next, we will report the byperparameters used in each experiment in Section 5 For the SpaceNet baseline, the sparsity levels E (1-density) used in each layer is: {0.84, 0.86, 0.87, 0.87, 0.93, 0.90} for CIFAR-10 and {0.88, 0.91, 0.90, 0.97, 0.97, 0.90} for CIFAR-100. For both datasets, the number of reserved neurons for each task each layer is: {0, 2, 2, 5, 30). Following the original paper of SpaceNet, the number of selected neurons for connections allocation self equals the number of free neurons that layer. For KAN, on CIFAR-10, 10, the same sparsity levels used for SpaceNet are used for the first task. For subsequent tasks, use the above- anntenmet sparsity levels starting from breuee. We will discuss the sparsity levels for KAN on CIFAR-101 in Appendix
For Selective knowledge transfer analysis (Section 5.2 Table shows the number candidate neurons selected for each class self-cand and the selected number of free neurons for each task self in each layer. We report these values for every value evaluated for 4reuse-
For selective buckward transfer analysis (Section 5.3 the number of candidate neurons self-cond used in each layer from 2 to - 1 is as follows: (6, 6, 15, 10.
For longer sequence of tasks (Section 5.4), the number of candidate neurons self-rond used in layers 6-3 3 and L 2 are 15 and 10 respectively. While the number of selected neurons from the free list self in these two layers are 10 and 20 respectively.
C RESOURCE EFFICIENCY AND SCALABILITY
For CIFAR- 100 benchmark, sim seq 8Tasks, we had to increase the sparsity level for each sub- network for SpaceNet to be able to fit the 8 tasks in the fixed-capacity model. The sparsity levels
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Table 3: The number of the candidate neurons for each class self and the number of selected
neurons from the free list in each layer self. is the number of free neurons in layer L-1 (hh (-1
sel
crand
seli
reuse 1#
2
4
5
6
5
L
0
0
0
0
0
50
0
0
0
0
0
0
L
0
0
15
0
0
0
10
Ø
I,
0
15
10
0
0
0
0
10
20
I,
0
8
10
10
0
0
0
6
20
20
L
8
10
10
0
12
20
20
L - 6
6
6
10
10
0
3
10
12
20
20
Ø
Table 4: The accuracy [%] of each task in sim seq 2Tasks after learning the two tasks in sequence along with their average accuracy. The reported performance is in class-II using different strategies for allocating the new connections in the output layer.
Strategy for connections allocation in the last layer
(rease
Task
Task 2
Average
Use candidate neurons only Use candidate neurons and free neurons Use free neurons only
L-1
69.66
34.77
52.21
L-2
68.41
52.71
60.5
L-2
64.34
61.11
62.72
the other hand, since KAN utilizes the existing capacity for similar tasks, have a free capacity that allows to lower sparsity level for allocating the connections in all layers for every odd time step (t 1.).... We use the same sparsity level used for CIFAR-10. For even time steps (t 2, 4....... ..., where utilize the existing knowledge, follow the same sparsity level of SpaceNet for CIFAR- 100 for 2 Lreues-
are mentioned in Appendix B
KAN allows utilizing the resources efficiently. When the new task shares some similarities with previous existing knowledge, KAN exploits this knowledge instead of allocating unnecessary re- sources. The saved capacity is utilized to learn knowledge that the agent has not encountered before. Thus, being aware the existing knowledge helps in utilizing the available capacity effi- ciently and allowing for scaling to a large number of tasks in the CL sequence.
D CLASS AMBIGUITIES
In Section 5.2.2 we demonstrated that reusing the highest level representation (reuse = L- 1) from the similar previous class in learning a new one causes ambiguities between the two classes in class- IL. Therefore, we suggested to: (1) use the free neurons only in allocating the output connections for the new class and (2) add new components for the new class in higher layers (Treuse L 1). In Section 5.2.2 we analyzed point (2) by studying different values for lreuse while satisfying point (1). In this appendix, we analyze the effect of the point (1).
We performed this analysis on the sim seq 2Tasks benchmark. We use breuse 2 candidate neurons in the output layer along with the free neurons to allocate the output connections. The number of candidate neurons self_cand in layers L 2 and L are 15 and 40 respectively. The number of selected free neurons in these two layers are 10 and 55 respectively. We compared The number of
2. We use the
this baseline with the KAN model presented in Section 5.2.2 with frenee candidate neurons in layer L 2 is 15 and no candidate neurons are used in layer L-1 (Table B). We also compare against the KAN model with lyeuee = L 1 presented in Section 5.2. where the connections allocated in the last layer only using the candidate neurons in layer 7.
Table 4 shows the performance these models. Adding new connections in earlier layers (Trexse < L 1) improves the performance of Task 2. However, using the free neurons only in allocating the connections in the last layer reduces the ambiguities between similar classes since each class learns its own specific representation. Thus, satisfying the two suggestions discussed above (drexse L-1 and h gallec
1 hgd helps in achieving the balance in performance between the tasks.
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Table 5: FLOPs required to learn the sim seq 2Tasks benchmark using different methods.
Strategy Regularization
Method SI 6.69e13
FLOPS
PackNet
1.004e14
-ask-specific components
SpaceNet
1.02e13
KAN (ours)
8.14e12
E COMPUTATIONAL EFFICIENCY
In this appendix, report the number of floating-poin operations (FLOPs) required for training the regularization based and task-specific components models. Regularization-Du -assed models are based on dense networks while tasc-specific components models are based sparse networks either trained from scratch (i.e. SpaceNet (Sokar et al. (021c), KAN (ours)) or pruned after training a dense model (PackNet) (Mallya & (eneebnik]00188
The FLOPs metric is the traditional metric used in the literature to estimate the speed of training algorithms and compare the computation efficiency of sparse model against its dense counterpart (Evci et al. 2020; Yuan et al. [222]]. This is because current sparse neural network models are simulated using binary masks over network weights. Truly sparse implementations with arbitrary sparsity during training is a highly researched approach with no general solution yet (Curci et al. 202T.
We follow the method described in (Evci et al. 2020 to calculate the FLOPs. The FLOPs are calculated with the total number of multiplications and additions layer by layer in the network. We compute this metric for the sim_seq wwt.ss.......ee 2Task benchmark.
Table shows the computed FLOPs for each method. As illustrated the table, SI requires more FLOPSs than SpaceNet because it trains the dense model for each task. KAN, requires fewer FL.OPs than SpaceNet as updates the weights starting from frease- PacKNet requires the highest number of FLOPs because trains the dense model for each task and performs additional fine-tuning epochs after pruning the least important weights to recover the model performance.
F PERFORMANCE ACROSS SEVERAL BENCHMARKS
In this appendix, analyze the performance of the studied task-specific components models on a sequence that contains two different datasets from two domains. We construct this se- quence using two standard datasets for class-IL; MNIST LeCun (9988, Zenke al.) (017)) and Fashion MNIST (Xiao et al.) 2017] Farquhar & Gal 2019). We name the new benchmark as MNIST FashionMNNST 6taskss The benchmark consists of tasks. Every odd task contains two consecutive classes from the MNIST dataset while every even task contains two consecutive classes from Fashion MNIST.
We performed this experiment using the same experimental setting used for the CIFAR-10 bench- mark described in Appendix eexeept for the number of epochs. We train each task for 5 epochs. We allocate new sparse connections in each layer for Task 2 as it contains new knowledge (Fashion MNIST) different from the one learned from Task (MNIST), Starting from the third task on-wards, use 1yeuse= =L-3.
Figure shows the average accuracy on the seen tasks at each time step. KAN succeeds main- taining higher average performance than SpaceNet especially when the number of tasks increases. Consistent with previous findings, selective transfer of KAN is more effective than random reuse of previous components or irrelevant reuse.
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100
KAN (ours)
SpaceNet
Random reuse
Accuracy [%)
Irrelevant reuse
60
50
1
2
3
4
5
6
learned tasks
Figure 8: The performance of different methods in class-IL MNIST FANSTTFFshionMNISST.T... benchmark.
G ADDITIONAL EXPERIMENTS AND ANALYSES
G.1 PERFORMANCE OF REHEARSAL BASED METHODS
In this paragraph, we compare the performance of KAN to more rehearsal-based methods. We ana- lyze the performance of the task-specific component method PackNet (Mallya & Lazebnik] 2018). PackNet originally designed for the task-IL II as discussed in Section 2 To adapt PackNet to the class-IL, we use all the learned connections during inference without masks. The dense model is trained from scratch on the CL tasks. We prune 85% of the network weights after training each task to get approximately the same sparsity level used for the task-specific components baselines. We perform 20 training epochs for fine-tuning the network after pruning to restore its performance. We use the official code from the authors to obtain the results of PackNet on the studied benchmarks.
We also analyze the performance of other well-known regularization-ben EWC (Serra et al.) 2018), MAS (Aljundi et al. 2018), and LWF (Li & Hoiem 2017). We use the public code from (Masana et al. 2020) to get the results of these algorithms.
methods including
100
PackNet
Nocursay [X)
(ours) freuse
SpaceNet
EWC
LWF
MAS
SI Crep 0.01
Tosk
Tosk
Average
Tosk
Task2
Average
(a) Class-IL
(b) Task-IL
Figure Test accuracy of each task in sim_seq 2Tasks and their average in class-IL. and task-II.
100
PackNel
KAN (our)
SpaceNet
Acourany (%)
EWC
MAS
1
2
3
5
learned tasks
Figure 10: The performance of different methods in class-IL II on sen.-see-Taaske
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Table 6: The average accuracy (ACC) [%] and the average backward transfer (BWT) [%] on
sim_seq seq 2Tasks and sim_seq_57las 5Tasks benchmarks in class-IL.
sim_seq ain_seq-Taske
2Tasks
sim Hon_seq_STaake
Strategy
Method
ACC
BWT
ACC
BWT
SI
54.51
-6..6
30.72
-47.66
EWC
49.95
788.60
27.70
-54.90
Regularization
LWF
57.30
-5490
33.50
-57.4
MAS
51.80
60.50
25.40
-42.3
PackNet
57,42
-68885
29.64
-58.01
SpaceNet
64.05
-36.67
41.53
-28.56
Task specific components
KAN (ours)
64.43
-32.58
43.48
-22.46
Random reuse
60.51
-33.17
41.10
-23.87
Irrelevant reuse
59,25
-29.39
39.54
-26.13
Figure 9 and Figure I0 show the performance the studied methods on sim_seq 2Tasks and
sim_seq STaks.. Consistent with our previous findings, the stability-plasticii dilemma is more challenging be addressed using regularization-busee methods especially in class-IL. The methods based on Sparse representation (KAN, SpaceNet) perform better than PackNet. KAN outperforms all studied baselines on both benchmarks.
G.2 NEGATIVE BACKWARD TRANSFER (FORGETTING)
In this paragraph, we estimate the average forgetting that occurred in the studied methods. We use the backward transfer metric (Lopez- Paz & Ranzato 2017). BWT, which measures the influence of learning new tasks the performance of previous tasks. Formally BWT is calculated as follows:
BWT T Σ Rr. - R1,
where Ry. is the accuracy on task i after learning the j-th task in the sequence, and is the total number of tasks.
Thole the average accuracy over all tasks (ACC) after the model learned the whole sequence along with the average backward transfer (BWT) the sim_seq_220 2Tasks and Seq_seeSTaasss bench- marks. As illustrated in the table, all regularization-based methods are more prone to forgetting than most of the task -pse-fpec components methods. PackNet is more prone to forgetting than other task-specific components methods which learn sparse representations. KAN achieves the highest average performance and least forgetting on the two benchmarks.
H SOFTMAX CLASSIFICATION LAYER FOR CONTINUAL LEARNING
H.1 DIFFERENT SETUPS FOR THE OUTPUT LAYER
Next, we will discuss the different setups proposed in the literature for the softmax classification layer in class-Il..
Cosine normalization. In (Hou et al. 2019), the authors proposed to use cosine normalization in the last layer to address the imbalance in the magnitudes of the output weights between old and new classes. Thus, the predicted probability for sample æ is calculated as:
exp(1 < Wi, >) exply Exexp( W,, (1) >)
pi(x)
(4)
where fz-1(x) is the normalized vector the last hidden representation, W, is the normalized weights connected to the output neuron i, < b > is the cosine similarity between the two vectors a and b. and 77 is a learnable scalar to control the peakiness softmax distribution. The setup was evaluated on rehearsal-base method.
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Masked weights
Unmasked output weights
1. norm of wrightts
Figure 11: L2 normalization of the weights of the output layer corresponding each task in the benchmark. The magnitude of the weights of Task 2 is higher than Task I when the Task 1 weights are not masked during learning Task 2.
Multi-headed output layer. The multi headed output setup is the common practice used in Task IIL where each task has its own output head. In (Mittal et al. [022]) the authors showed that this setup improves performance of rehearsal-based methods in class-IL. In the multi-headed setup, the softmax activation function (.) is applied on the classes of the current task only (y E yft C t * C' + CD, where the number of classes each task. The cross-entropy loss is evaluated on this selected part of the vector only.
Pre-defined single-headed. SpaceNet (Sokar et al. 2021c) tries to address the closed-world as- sumption by defining output layer with large number of neurons before start learning the quence of tasks
0. To address the weights imbalance between old and new classes, only the weights of the classes of the current task are active during their learning. During learning task I, a mask M is used to indicate the connections connected to the output neurons for the current task where Mf s C: & C' + CJ equals and 0 in the other indices. During learning task t, the weights of the output layer are multiplied by this mask which blocks the gradient flow from other output neurons:
Wi-1 = WT-1 & Mi-11
(5)
where is the element-wise multiplication operator. Figure II shows the effect of masking the output weights in learning a new task. We calculate the L2 normalization of the output weights of each task the sim seq 2Tasks benchmarks using SpaceNet. As illustrated in the figure, using all output weights in learning new tasks leads to higher weights for the new task. This causes bias in prediction towards the new task. While using a mask over the current task output weight alleviates this problem.
Extendable layer. This the typical setup used in class-II. where single-head output layer used. At each time step, the layer is extended to include new output neurons for the new classes in the current task. our analysis in Section 5.5 we implement the masking trick described in the previous paragraph for the extendable layer baseline. As you see from Figure 4c this does not solve the other limitation for the softmax being based on the closed world assumption. Figure 12 illustrates this limitation. We show the learned biases after learning two tasks from the Split CIFAR-10 benchmark in two different setups: pre-defined single-headed and extendable layer. As illustrated in the figure, the learned biases for the first task in the of extendable layer divides the feature space between the two classes in the first task. The biases of Task classes have larger value of the bias than for Task The pre-defined single-head layer overcomes this limitation.
Few previous works (Yu et al. 2020 Li et al. 2021) started to propose alternatives for the standard softmax for continual learning overcome its limitations. More efforts are needed in addressing the limitations of the softmax to improve the performance of the CL agent in class-IL.
H.2 RE-EVALUATION OF REGULARIZATION BASED METHODS CLASS-IL
Previous studies show the huge performance drop of the regularization- baasz method in class- IL (van de Ven & Tolias 2018 Hsu et al. 2018 Maltoni & Lomonaco 2019). In these methods, the agent tends be biased toward the last task while catastrophically forgets the previous ones.
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Extendable singkihead
Pro-defined singlehcad
0.8
0.6
0.4
0.2
Figure 12: The learned biases in two different setups: Extendable single-headed and Pre-defined single-head. Each column represents the value of the bias for each class in the two tasks.
Table 7: Test accuracy on Split CIFAR-10 of the regularization method SI using different setups of classification output layer the class-incremental learning.
Setup
Accuracy
Extendable layer
19.20 = 0.04
Multi-headed layer
36.84 + 0.94
We evaluated one of the regularization-basee methods SI (Zenke et al. 2017) using one of the se- tups discussed before; multiheaded setup. We performed this experiment the Split CIFAR-10 benchmark with 5 tasks.
Table 7 shows the average test accuracy overall tasks after learning the whole sequence. The multi- headed output layer increases the performance of the agent by 17.64%. This experiment suggests to re-consider the regularization methods in class-Il. II. while addressing the limitations of the softmax output layer.
I SPACENET
In this appendix, include the details of the three main steps of the SpaceNet algorithm (Sokar et al. 2021c): (1) Connection Allocation (Algorithm (3, (2) Dynamic Sparse Training (Algorithm and (3) Neuron Reservation 5
The full algorithm of SpaceNet is illustrated in Algorithm 2
Algorithm 2 SpaceNet for Continual Learning
I: Require: loss function C training dataset for each task in the sequence D Require: sparsity level e, rewiring fraction
3: Require: number of selected neurons self, number of specific neurons specf
4: for each layer / do
h free h;
Initialize free neurons with neurons /
6:
h spec <-
7: Wi
+- 0 Wenved
8:
for each
10:
for each available task t do
11: W +- Connection CommectionsAAlloaattoo Allocatione,, self, frree
Perform A Algorithm
12:
W +- TaskKTrininggW.W DD,
Perform Algorithm
13: h free +- NeuronsReseervaiionn
b Perform Algorithm
Wravedd <- W. U Wt
W
> Retain connections of last layer for
15: WL + Ws. \ W..
16: end for each
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Algorithm 3 Connections allocation
1: Require: number of selected neurons self, sparsity level
2: for each layer do
3: (hn-1: hn +- randomly select seli-; and self neurons from free and free
( 4: randomly allocate parameters W, with sparsity between h and h
5: Wi + WF
6: end for each
Algorithm 4 Dynamic sparse training
1: Require: loss function C, training dataset D', rewiring fraction
2: for each training epoch do perform standard forward pass through the network parameters W
4: update parameters W using stochastic gradient descent
5: for each sparse parameter W do
6: Wf + sort W based on the connections importance f2
7:
(W7, ki) drop (Wf.r)
Remove the weights with smallest importance
compute neuron importance a:-1 and
Neurons importance for task
9:
G - -
G t- sorDescending(G))
11:
Gpos select top-k positions in Gj where Wi equals zero
12: Wi < grow(W?. Gpos)
Grow k zero-initialized weights Gpos
13: end for each
14:
end for each
Algorithm 5 Neurons reservation
1: Require: number of specific neurons speci
2: for each layer l do
compute the neuron importance a: for task
4: a: <- sortDescending(a)
I sxpec <- top-spect from di
hopec +- hopec Tayce
h free <- h free tapeo
8: end for each
J SYNAPTIC INTELLIGENCE (SI)
SI Zenke al. 2017 is regularization-bass method proposed for the continual learning paradigm. It uses a fixed- capacity dense model in learning the sequence of tasks. During train- ing each task, it estimates how important each parameter model faces new task, it regularizes (penalizes) changes to parameters according to their impor- tance to previously learned tasks. The importance of each parameter
learning the current task. When the
certain task is estimated by: (1) its contribution to the change of the loss w/ (i.e. how much does a small change to the parameter change the loss function?) and (2) how far moved ( Following the paper notations,
for the current task I, the per-parameter contribution in loss (:()) is calculated as follows:
8C 80,
w/ (0 - 0: 1))
(6)
where Nuw is the total number of training iterations per task, ())) is the value of parameter i after sct performing iteration on task and 60, the gradient of the loss with respect parameter iteration during learning task t.
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The distance travelled by parameter i during learning task is calculated as follows:
A 0g (Niers) 00"-1)(( (("
(Nnes).
(7)
Thus, the estimated importance of each parameter for previous & tasks is calculated as follows:
(t-1)
2 (1111)
(8)
+
where & is a damping parameter to bound the expression in cases where A goes to zero. During learning task regularization term fieg is added to the classification loss L to penalize the change in the weights based on their importance. fize is given by:
Siyg Σ 12(1-1) (0. ((-1(((nens) 2
(9)
The total loss for task is:
L'= C. + Creg Flog
(10)
where Creg is a eiyyper-paramterr to control the trade-off between the classification and regularization terms.
24


Reviews:
[{"review_id": "TFFZPgOOjB", "reviewer": "Reviewer_UadE", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "The paper proposed a class incremental learning method that can exploit the similarity of classes for knowledge transfer. The previous work CAT did this only in the task incremental learning setting. \n\nThe paper started off quite interesting as the proposed idea is new. Limited work has been done to improve class incremental learning by transferring knowledge from previous similar classes. However, the work is incomplete as it assumes that the class similarity is given instead of being detected automatically like the CAT system. This significantly reduces the value of this work. \n\nThe experiments section has many issues. It is more like an analysis rather than showing the superiority of the proposed method. The author acknowledged this in Section 5.6. \n\n1. Comparison with state-of-the-art class incremental learning systems is seriously lacking. Many such systems have been published recently. Practically, only SI and SpaceNet are compared, but SI is fairly old and your system is based on SpaceNet. Although existing methods do not explicitly transfer knowledge in the class-IL setting, but they do have implicit sharing and transfer because the new tasks are learned based on the model built for old tasks. Thus, a comparison with them should be included. \n\n2. Only a fixed sequence of classes and tasks arranged by you is used. It raises the question of how you did the arrangement and what about the random sequence or simply follow the original class order in the original data. For CIFAR 100, you used only 16 classes to make 8 tasks. Why did not you use all classes in the full data to make a sequence of 50 tasks. In section 5.4, you said “every two consecutive tasks are dissimilar”. Did you arrange the order of classes to make this happen? You reported a study in figure 2 about the sensitivity of class order, but that involves only two tasks in the task-IL setting with task identity known. \n\n3. In Section 5.2.1, you wrote “to estimate the forward transfer, we focus on the performance of Task 2 assuming the availability of the task identity at inference. Table 1 shows the accuracy of Task 2 using different methods.” Since you are working on class-IL, your assumption that the task identity is known at inference, which is the task-IL case, makes the result not reflecting the of class-IL knowledge transfer. In fact, knowledge transfer of similar classes may make the classification of the old and new classes more difficult. The question is whether this transfer in the task-IL case is beneficial to Class-IL. Also, in the results, you only reported the result of task 2. What about task 1? \n\n4. In section 5.2.2, you reported the same experiment in the class-IL setting and showed that it does not work for class-IL and proposed two constraints. This makes the writing problematic because the constraints should have been given somewhere in the model sections as it should be part of the technique. Using only two tasks to study this is not interesting because when a large number of tasks are learned, it is very hard to know what happens. \n\n5. Figure 4 shows that the forgetting is very serious. Many existing methods do better. \n\n6. Section 5.2.1 and 5.2.2 are quite hard to follow as it has too many details. \n\n7. The architecture is also very simple. Most of continual learning systems for class-IL uses ResNet. \n", "summary_of_the_paper": "The paper proposed a class incremental learning method that can exploit the similarity of classes for knowledge transfer. The previous work CAT did this only in the task incremental learning setting. The paper did some analysis in the experiment section to show how the transfer can happen in some limited experiments. ", "summary_of_the_review": "This paper studies an interesting problem of knowledge transfer in the class-IL setting. Existing work only did it in the task-IL setting. However, this work is incomplete as it assumes that class similarities are known, which significantly decreases the value of this work at this stage. There are also a lot of issues in the experimental section. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "QGPX0WH6aw", "reviewer": "Reviewer_o6Js", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "Strengths:\n- I really appreciated that the paper explicitly tackles the elementary questions of CL: \"we investigate the stability-plasticity dilemma\nto determine which model components are eligible to be reused, added, fixed, or updated to achieve this balance\". Most existing methods consider a fix-size backbone and introduce new regularization or training mechanisms to avoid forgetting but do not explicitly address these problems.\nWeaknesses: \n- First, the method is very incremental with respect to spaceNet. In my understanding, the main difference with this previous approach lies in the lines 3 and 4 of Alg.1.\n-The contribution of the paper and the difference with spaceNet should be clarified. This is discussed only in 3.2 while this should be discussed from the start in Sec.1.\n-In the introduction, the second contribution is not specific to this paper. It was already in SpaceNEt\n-The authors claim that they address only the class incremental learning problem. However, I don't see anything in the method that prevents the use of this method in the case of task-incremental learning. It would strengthen the experimental conclusion significantly.\n-Many baselines should be added to the experiments: DEN,EWC,LWF,MAD\n- The datasets used in the experiments are simple and very similar: cifar 10 or 100. I would recommend adding at least another dataset (Mnist,svhn, tinyimagenet...)\n-The structure of the paper could be improved:\n- Sec 3.3 should not be here. This is not part of the method. This is only the description of a baseline so I would recommend moving this part to the experimental section.\n-Sec 3.2. For a reader that does not know SpaceNet, it's almost impossible to understand.\nin Table1, the performances of KAN  are much worse than the other methods but it uses much fewer parameters. In my understanding, it would be possible to use different hyperparameters to have a more fair comparison allocating a more similar number of parameters? From the reported performances, the superiority of the method is not clear.\n- No readme is provided with the code. So it's hard to replicate the experiments. Comments in the code are very sparse and not very informative.\n\n\ndetail: in figure 1 block (2), a connection is missing in the bottom-left.", "summary_of_the_paper": "The paper address the the problem of continual learning for image classification. This approach focuses on how to reuse, add, fix or update neurons to learn tasks incrementally. The method is based on an existing method name SpaceNET.", "summary_of_the_review": "The proposed method is based on an existing method and is very incremental. The presentation of the method could be improved. Experiments are not sufficiently convincing.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "ELwJtRn_3ft", "reviewer": "Reviewer_NFc9", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "### Strengths:\n(1) The paper is well-motivated, well-written, and easy to follow.   \n(2) I like the overall experimental designs where different aspects of continual learning with KAN are studied with several experiments.\n\n\n\n### Weaknesses:\n\nAlthough I enjoyed the experimental design, and agree with the intuition behind KAN, the proposed method have major weaknesses:\n\n(1) First, there are several hyper-parameters involved, such as the number of layers to be reused, the sparsity level, etc. \nWhile the paper studies the impact of layer reuse, this is usually an unknown factor in practice for new and unseen data.\n\n(2) Second, given all the interesting effort for developing KAN, the proposed method does not outperform existing baselines.\nFor instance, when the benchmark is challenging, such as Fig 4. b, KAN, SpaceNet, and even Random Reuse achieve the same performance.\nI believe since the paper is an empirical work, the fact that the proposed method is not performing better than baselines hurts its contribution. \n\n(3) Generally, while reporting metrics, it is not clear how much average forgetting for different methods are (see [1, 2] for the definition of forgetting metric). I think it is important to report these metrics as well, which indicates the negative backward transfer.\n\n\nMisc: \n- In Fig. 1, a fixed connection from the last neuron of the first layer is removed from steps 2 to 3. \nI think there was an error in figure generation. \n- It will be interesting to include the run-time of different algorithms to see how much different modifications by KAN impact the speed. \n\n[1] Chaudhry, Arslan, et al. “Efficient Lifelong Learning with A-GEM.” International Conference on Learning Representations, 2018.   \n[2] Mirzadeh, Seyed Iman, et al. “Understanding the Role of Training Regimes in Continual Learning.” Advances in Neural Information Processing Systems, vol. 33, 2020, pp. 7308–7320.\n\n\n\n-----\n***UPDATE**   \nWhile I am not fully convinced regarding (1), the new version of paper addresses (2) and (3) an I now lean towards accepting this work. \n\n\n ", "summary_of_the_paper": "### Summary:\nThe paper studies replay-free continual learning with the focus on the plasticity-stability dilemma.\nMore specifically, the paper proposes the KAN method for class-incremental learning where the forward transfer is achieved by detecting similar knowledge and reusing the first few layers,\nand negative backward transfer can be alleviated sparse connection allocations for different classes of different tasks.\nThe paper also studies a limitation of the softmax layer, which is an interesting contribution.  ", "summary_of_the_review": "I believe the proposed method has its flaws but overall I believe it is an illustrating study and I lean towards acceptance.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "3eOMOXoS-k2", "reviewer": "Reviewer_xDnv", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "Pros:\n+ This work is well motivated. Exploring similarity between tasks could be indeed a promising way to address the stability-plasticity dilemma to simultaneously promote forward transfer and avoid catastrophic forgetting. \n+ Extensive ablation studies are conducted to show the proposed method work to some extent.\n\nCons:\n- The organization of method section could be improved.   \nFor example, the first and second steps could be divided into two separate paragraphs, and selective gradient back-propagation could become another paragraph.   \nIn addition, it is better to summarize the categorization of neurons in a layer.\n- There is no explanation on why activation of a neuron could be a good indicator for relevance between classes. What else do authors also attempt?\n- It is suggested to calculate the statistics about how much portion of neurons are reserved, reusable and free for switching between similar tasks, and how much for the switch between dissimilar ones.\n- The proposed method also shares some similarity with PackNet although the PackNet does not explore similarity between tasks but tries to compress the used neurons for each new task. Authors should compare the proposed method with it to see which method is more effective in address the stability-plasticity dilemma.\n[a] A. Mallya and S. Lazebnik, PackNet: Adding Multiple Tasks to a Single Network by Iterative Pruning, CVPR 2018.\n- More rehearsal-free CL methods should be compared with. In addition, comparison on larger datasets such as tinyImageNet and also the one across several datasets such as Oxford Flowers, Caltech-UCSD Birds and MIT Scenes should be conducted.", "summary_of_the_paper": "This work aims at addressing the stability-plasiticity dilemma in the class incremental learning scenario by exploring which model components shold be reused, added, fixed or updated. Authors proposed to make use of existing knowledge in previous tasks, including identifying and adding similar knowledge that could be reusable for forward transfer and preventing dissimilar knowledge from being transferred to avoid forgetting, hence achieving better balance between catastrophic forgetting and forward transfer. Extensive ablation studies are conducted to demonstrate the effectiveness of the proposed method.", "summary_of_the_review": "Good motivation and extensive ablation study. But the method should be described in more details with more insightful explanation in a better organizing way. In addition, it should be compared with more rehearsal-free methods on larger datasets.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper investigates the stability-plasticity dilemma in the class incremental learning context. It investigates which model components are eligible to be “reused, added, fixed, or updated” to achieve a good balance. Initially the paper had one supporter (xDnv) who liked the motivation and extensiveness of the ablation. NFc9 and UadE also echoed some similar points about motivation and liked that the work was easy to follow. Reviewers expressed concerns such as incrementality w.r.t. spaceNet, too many hyper-parameters, unclear performance benefit, lack of comparison to SOTA, fixed sequence of classes specified by the authors, not clear how much forgetting is happening in each method (echoed by multiple reviewers), and limited datasets used for evaluations. The authors responded to the critical reviews and provided a revised version of the paper with additional comparisons to rehearsal-free methods and with more datasets (MNIST/FashionMNIST). 

Following the author response, NFc9 stated that they thought the paper was in better shape with the revisions and upgraded their score claiming it was “closer to acceptance”. Yet, they still had concerns with the practical implications of having too many hyper-parameters. UadE engaged further with the authors but claimed that they avoided the reviewer’s direct concerns. UadE maintained their concerns with the manual ordering of classes and older baselines. I agree that there are several rehearsal or pseudo-rehearsal methods to which they could have compared. Reviewer o6Js did not engage further. Overall this is a borderline paper. I appreciate the authors engaging in the discussion period, though my assessment is that the key issues still remain. This paper could use further development so my recommendation is reject.
Decision: Reject