Flattening Sharpness for Dynamic Gradient Projection Memory Benefits Continual Learning
Authors: Danruo DENG
Guangyong Chen
Jianye HAO
Qiong Wang
Pheng-Ann Heng
Venue: NeurIPS 2021 Poster (2021)
Keywords: continual learning
weight loss landscape
dynamic gradient projection memory
sharpness flatten

Abstract: The backpropagation networks are notably susceptible to catastrophic forgetting, where networks tend to forget previously learned skills upon learning new ones. To address such the 'sensitivity-stability' dilemma, most previous efforts have been contributed to minimizing the empirical risk with different parameter regularization terms and episodic memory, but rarely exploring the usages of the weight loss landscape. In this paper, we investigate the relationship between the weight loss landscape and sensitivity-stability in the continual learning scenario, based on which, we propose a novel method, Flattening Sharpness for Dynamic Gradient Projection Memory (FS-DGPM). In particular, we introduce a soft weight to represent the importance of each basis representing past tasks in GPM, which can be adaptively learned during the learning process, so that less important bases can be dynamically released to improve the sensitivity of new skill learning. We further introduce Flattening Sharpness (FS) to reduce the generalization gap by explicitly regulating the flatness of the weight loss landscape of all seen tasks. As demonstrated empirically, our proposed method consistently outperforms baselines with the superior ability to learn new skills while alleviating forgetting effectively.

Flattening Sharpness for Dynamic Gradient Projection Memory Benefits Continual Learning
Danruo Deng', Guangyong Chen?; Jianye Hao Qiong Wang", Pheng-Ann Heng 'The Chinese University of Hong Kong, *Guangdong-Hong Kong-Macao Joint Laboratory of Human-Machine Intelligence-Syeeet Synergy Systems, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, *College of Intelligence and Computing, Tianjin University, "Huawei Noah's Ark Lab [drdeng, Bardeng.pheng)@eeee pheng)@cse........ppppp..rp. cuhk, , edu.hk, {gy.chy chen, vangqiong) cn, jianye.hao@tjj hao@tju edu.
Abstract
The backpropagation networks are notably susceptible to catastrophic forgetting, where networks tend to forget previously learned skills upon learning new ones. To address such the 'sensitivity-stabiiity dilemma, most previous efforts have been contributed to minimizing the empirical risk with different parameter regularization terms and episodic memory, but rarely exploring the usages of the weight loss landscape. In this paper, we investigate the relationship between the weight loss landscape and sensitivity-stabiiiyy in the continual learning scenario, based on which, we propose a novel method, Flattening Sharpness for Dynamic Gradient Projection Memory (FS-DGPM) In particular, we introduce a soft weight represent the importance of each basis representing past tasks in GPM, which can be adaptively learned during the learning process, so that less important bases can be dynamically released to improve the sensitivity of new skill learning. We further introduce Flattening Sharpness (FS) to reduce the generalization gap by explicitly regulating the flatness of the weight loss landscape of all seen tasks. As demonstrated empirically, our proposed method consistently outperforms baselines with the superior ability to learn new skills while alleviating forgetting effectively
1 Introduction
Humans have the ability to continually learn new knowledge without forgetting their previously learned ones through mediating rich set of neurocognitive mechanisms [41 15 39). This ability, often known as continual learning or lifelong learning (29), is crucial for computational systems, such as deep neural networks (DNNs), which are required to sequentially learn and deal with multiple tasks when implemented in the dynamically changing environment. Continual learning remains long-standing challenge for DNNs since these networks are typically trained with stationary training batches by stochastic gradient descent methods 19), which generally leads to abrupt performance decrease on previously learned tasks as new tasks are learned. To address such catastrophic forgetting, can brutally retrain oracle network the entire dataset containing all tasks capture dynamic changes in the data distribution, but this methodology is obviously too inefficient to hinder the learning of novel data in real time.
During the last few years, lots of research efforts have been devoted improving the stability of DNNs on old tasks while keeping sensitive to new information. The first intuitive idea is to introduce
"ooreeppnning author: Guangyong Chen. <gy.chen@ siat...eesannncm 'The code is available at: httpps://github. htt.:///w...oooooood//rrss//...s
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

---

an independent branch for each new task while freezing the old task parameters to preserve the However, in this way, the network will inevitably become redundant as the task number continually increases. As presented in the neurocognitive works 15]. the reactivation of neuronal activity patterns, representing old memories, plays an important role in the continual learning of humans [9]] Thus, forgetness can be effectively mitigated by training single network for new tasks by considering diverse information stored in the memory, including the original training samples of old tasks [31 5 [3, the gradients induced from old tasks [2] and the feature subspace representing old tasks [4] However, their continual learning performance still limited because DNNs can easily overfit the limited information stored in the small-size memory.
old knowledge 33
25 35 22].
The overfitting problem of DNNs is often attributed to the complex loss landscape containing multiple local optima, and the sharpness of the loss landscape has been widely used to characterize the generalization gap in standard training scenarios from both theoretical and empirical perspectives 21 42 T0 G] While this characterization has inspired new approaches for model training with better generalization, practical algorithms that especially seek out flatter minima to effectively address the "sensitivity stabiiiity dilemma for continual learning have thus far been elusive. In this paper, our first contribution is to characterize the weight loss landscape for the continual learning scenario and identify that a flatter loss landscape with lower loss value often leads to better continual learning performance, as shown in Figure D and Figure 3
27
Further, based on our characterization of the weight loss landscape, we find that the recently proposed Gradient Projection Memory (GPM) method [34] maintains the lowest loss value on old tasks among the previously proposed methods by taking gradient steps orthogonal to the subspace representing old tasks. However, its loss landscape on newly learned tasks is the sharpest due to the lack of sufficient subspace left for new task learning. To improve the network's 's sensitivity, our second contribution is to predict the importance of bases spanning the subspace for old tasks, so that less important bases can be dynamically released. In particular, we introduce a soft weight to indicate the bases importance, which can be dynamically adjusted by combining the Flattening Sharpness (FS) to minimize the loss value and loss sharpness simultaneously. Intuitively, a basis will be regarded as important for preserving old knowledge if the gradients induced by new tasks and old ones are aligned in the opposite direction that basis. As demonstrated through extensive experiments,
proposed method can consistently outperform the state-of-the, the-rttr methods [7 30 24 31 by notable margin across range of widely used benchmark datasets.
2 Related Work
In this section, we briefly survey the representative works of continual learning by highlighting their contributions. To simplify our presentation, this section is organized by dividing the representative works into three categories, parameter isolation, regularization- -basd, memory- methods.
Parameter isolation methods address forgetting by assigning different subset of network parame- ters to each task. Without restrictions on network architecture. new neurons or layers or modules can be added for new tasks, while the previous task parameters can be frozen or copied to preserve old knowledge. For instance, Progressive Neural Network (PGN) 3 freezes the parameters trained with previous knowledge while expands the architecture by allocating new sub-networks with fixed capacity for new tasks. Dynamically Expandable Networks (DEN) 45| selectively retrains or ex- pands network capacity by splitting/dullicating important units on new tasks. Reinforced Continual Learning (RCL) [43] uses reinforcement learning strategy to adaptively expand the network each layer, while 22 use neural architecture search to find optimal network structures for each sequential task. Alternatively, with the architecture remaining static, a fixed part is allocated to each task. During the training of new task, previous task parts are masked out to prevent interference. The mask sets are imposed at parameter level 225), or unit level (35] PackNet (25] uses iterative pruning to fully restrict gradient updates on important weights via binary mask whereas HAT 33 limits the update of important units recognized by the hard attention mask through gradient backpropagation.
Regularization-baat methods introduce an additional regularization term in the loss function to consolidate previous knowledge without using replay. This involves using knowledge distillation
23 14 or penalizing changes in weights deemed important for previous tasks 17 46 28 2 3 to
reduce forgetting. There are many ways to measure the importance. Elastic Weight Consolidation (EWC) [1] identifies important weights based on the diagonal values of Fisher information matrix
2

---

after training, while Synaptic Intelligence (SI) [46 calculates them online and over the entire learning
trajectory in parameter space. Memory Aware Synapses (MAS) [ estimates importance weights in
an unsupervised manner, while Variational Continual Learning (VCL) 28 introduces a variational
framework that spawned some Bayesian-based works 32
g 7. For example, [32 recursively
uses Gaussian Laplace approximation of the Hessian to approximate the posterior after every task, [ adjusts the learning rate according to the uncertainty defined by the probability distribution of the network weights. [ introduces an interpretation of node-wise uncertainty on the Kullback-Leibler (KL) divergence term the variational lower bound for Gaussian mean-field approximation.
Our method mainly follows memory-baser methods, which mitigate forgetting based on information extracted from old tasks or based on a generative model to generate pseudo samples. For example, iCaRL 30 selects and stores samples closest to the feature mean of each class. ER S 31 suggests reservoir sampling under the limited and fixed budget for replay buffer. Deep Generative Replay (DGR) [6 trains deep generative model in the Generative Adversarial Network (GAN) framework 11 to simulate past data. These previous task samples are mainly reused as model inputs for replay in the above methods. However, replay might be prone to overfitting the subset of stored samples. Alternatively, samples stored memory can also be used to constrain the optimization of the new task loss to prevent previous task interference, thereby leaving more leeway for backward and forward transfer. Gradient Episodic Memory (GEM) [24] projects the estimated gradients in the feasible region, which is outlined by previous task gradients calculated from the episodic memory samples. Averaged-GEM (A-GEM) [4] relaxes the projection to a direction that is estimated from samples randomly selected from memory. [12] proposes unified view of episodic memory- -based continual learning methods, including GEM and A-GEM, and improves performance over these methods by using loss-balancing update scheme. A few other works have utilized gradient information protect previous knowledge. [31 [3] adopt pptimization-basse
meta-learning to enforce gradient alignment between samples from different tasks. GPM (34] minimizes interference between sequential tasks by ensuring that gradient updates only occur in directions orthogonal to the input of previous tasks.
3 The Weight Loss Landscape of Continual Learning
In this section, we first introduce our formulation of continual learning, and then characterize the weight loss landscape for the continual learning scenario from stability and sensitivity. Finally, some insights combining the weight loss landscape and continual learning are provided.
3.1 Problem Formulation
Throughout the paper, we denote scalars as a, vectors as a. matrices as A, and sets A. We consider supervised learning setup where T tasks are sequentially learned from their training data. Each task has an identical task descriptor, т E {1, ,,,,, T), with its dataset D, = {ar.,r, Vi.r p. containing n, samples randomly generated from latent distribution D,. At any time-step during the learning process, we minimize the empirical risk of the model on all f tasks seen so far, with just limited size of memory A summarize the training data of previous tasks (D+)==)- To simplify the notation, we denote LA(w) = L ((aw)) [e (Soo (æ:) : v)] the average empirical loss for the set A. where .(.,)) is an arbitrary loss function (e.g. the cross-entropy (CE) loss), |A| is the sample size of the set A, and fw is the DNN with weight w. Our final goal is to find an optimal parameter w, which minimizes the overall risk Σ Lg. (w) for all tasks.
"a....,T,
3.2 Connection of Weight Loss Landscape and Continual Learning
After learning new task, we visualize the weight loss landscape of each task seen so far by plotting changes in its training loss when moving the weights w in a random direction with magnitude a following (21:
(a) = (w + ad) = iDr
1
Σ
((fo+oo .)),
(a.,))D:
where De is the training set for the t-th task previously learned. To eliminate the scaling invariance of d
DNNs, d is sampled from a Gaussian distribution and further normalized by dr,j + [c. 1r | |
3

---

193.
2.50
2.59
200
1.35
21 133
2 1.59
1.25
31.25
875
0.75
0.75
2.35
3030
0.2
0.28
1at Namber tooko
90
Figure I: The connection between the weight loss landscape and continual learning is investigated on four methods. (a)-( c) shows the stability of the first task. (a) is the test accuracy change curve of the first task; (b) and (c) are the weight loss landscape of the first task after learning the fifth task and all ten tasks. (d)-(e) shows the sensitivity of the fifth task, which are the test accuracy and the weight loss landscape of the fifth task after just learning the fifth task. The shape of the weight loss landscape obtained using ten different random filter-normalized directions. ("Ti" is abbr. of the i-th task)
where di represents the j-th filter at I-th layer of d. and |I-I denotes the Frobenius norm. Compared with our visualization, (26) only consider one task and plot the loss landscape along the directions of its Hessian eigenvectors, which only reflects some of the relationship between forgetting and landscape. Considering d is randomly selected, we repeat the visualization 10 times with different d.
We first study the stability of the network by plotting changes of the weight loss landscape for the first task after new task learning. In particular, We use the previously proposed ER (5). La-MAML [3), and GPM [4 to train MLP network with two hidden layers on the Permuted MNIST (PMNIST) 20 dataset that contains 10 tasks. We also retrain the network on the entire dataset contain all passed tasks as an Oracle network. Early stopping used to halt the training with up to 10 epochs for each task based on the validation loss as proposed in [35). As shown in Figure [a), a)) all three continual learning methods lose their stability as learning new tasks. can be observed from Figure (egure[[)-cc (c) that the weight loss landscape becomes sharper and loss value increases simultaneously when the testing accuracy of the first task continually decreases. We further evaluate the sensitivity of the network by observing the performance of the fifth task just after it has just been learned. As shown in Figure (e)) ER shows the best learning capability compared with other methods with the lowest loss value and the flattest loss landscape. Thus, based on these empirical findings, we assume that lower loss value with a flatter neighbor may lead to better continual learning performance.
3.3
A Case Study of Flattening Sharpness for Vanilla ER
In this part, we further validate our above assump- tion by Flattening Sharpness for vanilla ER (FS-ER). Compared with ER that looks for a solution that jointly minimizes the training loss of current task data and memory data, FS-ER seeks out solution with both low loss and flat neighbor by minimizing the maximal loss in the neighbor around the parameter value. The schematic of the Flattening Sharpness (FS) is shown in Figure 2 We introduce the adversarial weight perturbation (orange dashed line) to explic- itly flatten the weight loss landscape via injecting the worst-case weight perturbation, which is calculated from the current task data and past task data sampled from the replay buffer (Refer to Appendix C. and the next section for more details). Figure 3 respetiveely shows the weight loss landscapes of the fifth task af- ter just learning the fifth task and all tasks. We find that FS-ER successfully gets a solution with a lower loss value and flatter landscape, either after the fifth task has just been learned or all ten tasks have been learned. The average testing accuracy of all tasks using FS-ER is 90.44%. significantly higher than ER (86.16% ),,, which means that flattening sharpness does benefit continual learning.
my PL (wz)
CSER
+o (ow:)((w
4aj VI((w))
of
Figure 2: Schematic of FS-ER update. The dashed line and the solid line indicate the gra- dient ascent and descent, respectively. Orange denotes the actual update of the parameter w.
V
1.75
175
1t 1.50
1.50
1.00
1.00
0.75
0.50
0.25
025
-54
-0.2
0.0
0.2
5.
-04
-02
0.0
02
54
Figure 3: Landscape of the fifth task after just
learning (a) the fifth task and (b) all ten tasks.
4

---

Flattening Sharpness for Dynamic Gradient Projection Memory
As shown in Figure I GPM achieves the highest testing accuracy on old tasks among all three practical continual learning methods, but shows less sensitivity to new task learning. To address this issue, we propose Flattening Sharpness for Dynamic GPM (FS-DGPM), which dynamically adjusts the gradient subspace representing the past tasks to improve the sensitivity to the new task, while ensuring stability of the previous tasks. In particular, we let M [u1, Ug.... uu, denote the bases matrix that spans the gradient subspace of the previous task, A = diag do ..., Au be the diagonal matrix with i-th diagonal element X; E [0, 1] indicating the importance of each basis, and & the number of bases.
4.1 Sharpness Evaluation
we focus on characterizing the weight loss landscape on the new task with respect to the important subspace representing old task. The important subspace can be effectively calculated based on the examples sampled from replay buffer M after each task training. Then, we can find the worst case by maximizing the training loss of the network on the new task in this subspace. Formally, the sharpness of the loss landscape around the solution ur in the old parameter space can be predicted as,
Comparing with the classical strategy that perturbs weight in the entire space 40 16
max VEV
(w vv)
where V denotes the subspace spanned by M and A, and D. denotes the batch samples of the current t-th task. As shown in Eq. (T) the high value be obtained when the network fails learn the new task (sensitiviry) and the new task learning seriously interferes with the past tasks learning (stability). Based on the gradient method, the adversarial weight perturbation can be solved as,
レ U 7/1 MAM (((wo)
(w + +w))
where 7/2 is the update step size. Note that e is initialized as O and layer- updated. As shown in Appendix B. two-step for (default settings) are enough get good improvements.
4.2 Dynamic Gradient Projection Memory
After obtaining the adversarial weight perturbation v, we can further update the bases importance matrix A = diag[Ar A2 ..., Au] by jointly considering the current task batch D, and the batch M sampled from the replay buffer M as following.
A - A; 1/2 (Vx, I.D.Uxs + +)))
where the sigmoid function is used at the end of gradient update to constrain the importance value A; between and 1. In addition, the second term the right side in Eq. (3) be approximated by the first-order Taylor expansion as,
V L. (w + v) ~m (VuLlig, (uo) u.n. (Vw.L.,,un
The above equation characterizes the relationship between the gradients induced by the current task and the old tasks based on the basis u, As illus- trated in the Figure 4 this equation implies that when the projections of two gradients on the basis u, are aligned in the same direction, the gradient of X; will be positive, and when there is interfer- ence, the gradient will be negative. The positive (negative) gradient will decrease (increase) the importance X; thereby releasing (tightening) the update limit of the new task on the corresponding
(a) Transfer
(b) Interference
V (((W)
(w)
VL
VL((w)
Figure 4: depiction of transfer (a) and interfer-
ence (b) in the basis u of gradient space.
basis u,, We provide the full derivation in the Appendix A.2 Note that the initial value of all importance is set to I and dynamically adjusted from the second task.
5

---

4.3 Weight Updating
Finally, we update the model parameters by minimizing the worst performance of fu+v with respect to while adjusting the update magnitude of w on each basis based on its importance to alleviate forgetting. More concretely, the parameter w will be updated to:
(A-MMM) (
ITS
W.LDD +w)
(5)
Note that the optimization is performed over the model parameters w, whereas the objective is computed using perturbed model fw+e- In addition, we update the replay buffer M with reservoir sampling in (3T], and then use Singular Value Decomposition (SVD) to recalculate M based on the sampling data in the replay buffer after learning one task following GPM [34]. Comparing with [4], calculate the important bases in the entire gradient space and use them replace the bases
calculated last time. Besides that, our method degenerates to GPM when 7) and 7/2 are set to O. The complete pseudo-code of FS-DGPM outlined in the Algorithm
Algorithm 1 FS-DGPM (Flattening Sharpness for Dynamic Gradient Projection Memory)
Input: Network weight w, loss function É, learning rate TJa, FS step size 1/1. FS steps K, Soft weight step size 7/2, batch size 6.
Initializing .M <- {). M t 1,
for t = 1,2, ... T do
for ep = .. mumeeoocs do
for batch D. A Dj do
M 2 MM
for l = 1, ..... K do
V - v + mn MAM (o(wto)Lglw + +0)
P Sharpness Evaluation
end for
if 2 2 then
A A A
112 V VALLLLLN (w +v)
D Dynamic Gradient Projection Memory
end if
w +- w 1/3 (I (I- TAMM V Vu (w++)
b. Weight updating
Push Dr to M with reservior sampling
end for end for
M < UpdateGPM (M)
> see Appendix Alg.
end for
4.4 Theoretical Understanding
We further provide theoretical view on why landscape can characterize the continual learning performance and why our proposed FS-DGPM works. To simplify our explanation, we only consider two tasks, which contains the training sets Dj and D: sampled from the distributions 21 and 22, respectively. Based on the previous works on PAC-Bayes bound [27 42 10]. given "prior" distribution P (a common assumption is zero mean, o² variance Gaussian distribution) over the weights, the expected error of the classifier for the continual learning scenario can be bounded with probability least 1 over the draw of n training data:
min E, Le, (w Aw + v)] < Aw
min E. [L (w Aw + v)] + LDD(W Aw)
AWEDe
2n max LD, (w + Aw + +) ( Aw) +41 [K[w Aw + wllP) In DEV
8
where Aw is the update based on the previously optimal solution ur learned on the old task D, when learning the new one D2, and v is often chosen as zero mean spherical Gaussian perturbation with variance x² every direction. Let Aur E Vc. then Aw lies in the complementary space of the important space representing the old task D1. that E. [L, (w + Aw v)] does not increase

---

too much compared with the previously minimized E, [L (w + b)]. The second term denotes the empirical loss on the second task and the third term represents the sharpness of the weight loss landscape around the w + Am. Since we have constrained Aw E ye, then it is natural to assume V. so that Am + e will cover the full space. Thus, our FS-DGPM exactly optimizes the worst- case of the flatness of weight loss landscape to control the PAC-Bayes bound, which theoretically justifies both lower loss value and flatter landscape lead to better continual learning performance, and why our proposed FS-DGPM works.
5 Experiments
In this section, we conduct extensive experiments to compare the performance of our proposed FS-DGPM model with the state-of-the-art methods on widely used continual learning benchmark datasets. Additional results and more details about the datasets, experiment setup, baselines, and model architectures are presented in the Appendix Daand E
5.1 Experimental Setup
Datasets: We evaluate our algorithm on four image classification datasets: Permuted MNIST (PMNIST) 20). CIFAR-100 Split (8). C1FAR-100 Superclass 44 and TinylmageNet (37. The PMNIST dataset is a variant of the MNIST dataset, which each task applics a fixed random pixel permutation to the original dataset. The PMNIST benchmark dataset consists of 20 tasks, and each contains only 1000 samples from 10 different classes [13]. The CIFAR-100 Split is constructed by randomly dividing 100 classes of CIFAR- 100 into 10 tasks with 10 classes per task. The CIFAR-100 Superclass is divided into 20 tasks according to the 20 superclasses of the CIFAR-100 dataset, and each superclass contains 5 different but semantically related classes. Whereas, TinyImageNet is constructed by splitting 200 classes into 40 5-way classification tasks. In our experiments, we do not use any data augmentation. The dataset statistics are given in Appendix D.I
Network Architecture: For PMNIST, we use a fully connected network with two hidden layers of 100 units each following (24). For experiments of CIFAR- 1-00 Split and CIFAR- 100 Superclass, we use a 5-layer AlexNet and LeNet architecture similar to 34] respectively. For TinylmageNet, we use the same network architecture as [13, which consists of 4 conv layers and 3 fully connected layers. In PMNIST, all tasks share the final classifier layer, while other experiments use a multi-head incremental setting, that is, each task has separate classifier. Refer Appendix D.2 for more details.
Baselines: We compare our method against multiple methods described below. (1) EWC [
regularization-basse method that uses the diagonal of Fisher information to identify important weights; (2) ICARL [30]. a memory-basee method that uses know snowledge-distillati
and episodic memory to reduce forgetting; (3) GEM (24, another memory-based method that uses the gradient of episodic memory to constrain optimization to prevent forgetting: (4) ER (5), a simple and competitive
method based on reservoir sampling: (5) La-MAML [13] and (6) GPM [4]] are memory-based
methods inspired by optimization-base meta-lcarning and based on gradient orthogonal constraints, respectively; (7) Multitask is an oracle baseline that all tasks are learned jointly using the entire dataset at once in a single network. Multitask is not a continual learning strategy but serves as an upper bound on average test accuracy on all tasks.
Training Details: All baselines and our method use stochastic gradient descent (SGD) for training. For each task in PMNIST and TinylmageNet, we train the network in and 10 epochs, respectively, with the batch size as 10. These experimental settings are the same as La-MAML [3]. so that we directly compare with their reported results. In the CIFAR- 100 Split and CIFAR- Superclass experiments, we use the early termination strategy to train up to 50 epochs for each task, which is based on the validation loss as proposed in (35). For both datasets, the batch size is set to 64. The replay buffer size of PMNIST, CIFAR-100 Split, CIFAR-100 Superclass, and TinyImageNet are 200, 1000, 1000, and 400, respectively. Details about the experimental setting and the hyperparameters considered for each baseline are provided in Appendix D.5 and D.6
7

---

65
60
2007
PS 30P0
Averrye ACCUrDIy 196)
ICABL
Nemory (esampte))
2502
Normpe bestrncy
15
2002
FS.CPM
ICAAL
10
15
20
200
400 tytbod)
600
PUNGTT
CRAR-130 3pH Supettions
(4) Number basks
1c1 Danaser
Figure 5: (a) Average accuracy as function of the number of tasks trained on 20-Split CIFAR-100 Superclass. (b) Training time per task 20-Split CIFAR-100 1-00 Superclass. (c) Memory usage four datasets. ("Superclass" is abbr. of CIFAR-100 Superclass).
Table Experimental results on 10-Split CIFAR-100, 20-Split CIFAR-100 Superclass and 40-Split TinyImageNet in 50 epochs. Each experiment is run with seeds. and denotes results reported by and 44
respectively.
CIFAR-100 Split
CIFAR-100 Superclass
ImagenteeNet
Method
ACC(%)
BWT(%)
ACC(%)
BWT(%)
ACC(%)
BWT(%)
EWC
72.77 + 0.45 0.45
-3,59
0.55
50.26 + 1.48
-7.87
1.63
GEM
70.15
-8.61 = 0.42
50.35 + 0.80
-9.50 + 0.85
50.57
0.61
-20.50 + 0.10t
ICARL
53.50 0.81
20.444
49.05 +
-11.24
54.77
0.32
-3.93 0.551
ER
70.07 +: 0.35
-7.70 = 0.59
51.64 + 1.09
-7.86
48.32
1.51
-19.86 1: 0.70
La-MAML
71.37 +: 0.67
-5,39 = 0.53
54,44 +: 1.36
-6.65 0.85
66.90 = 1.65
-9.13 +: 0.90!
GPM
73.18 + 0.52
-1.17 = 0.27
57.33 + 0.37
-0.37 + 0.12
67.39 + 0.47
1.45 0.22
ES-DGPM
74.33 + 0.31
-2.1 + 0.17
58.81 + 0.34
-2.77 + 0.35
70.41 + 1.30
-2.11 + 0.84
Multitask
79.75 0.38
61.00 0.20
1.06
Metrics: We evaluate the continual learning performance by the average accuracy (ACC) and backward transfer (BWT) [24 5), formulated as following.
T ACC T
T T
Σ
RT BWT
RT - R4,55
where T' is the total number of learned sequential tasks, Rú, the classification accuracy of the model on j-th task after learning the last sample from i-th task. ACC is the average test classification accuracy of all tasks, bigger better. BWT is the interference of new learning on the past knowledge. More specifically, negative BWT implies (catastrophic) forgetting whereas positive BWT indicates learning new task increases performance on some preceding tasks.
5.2 Results and Discussion
Table 2: Experimental results (mean + std in runs) on PMNIST in single-epoch.
We first evaluate algorithm for 20 sequential PMNIST tasks with only 1000 samples per task in single-head setting. From the results, as shown in Table 2 we see that our method (FS-DGPM) achieves the best age accuracy (76.96% + 0.77). Moreover, we achieve the least amount of forgetting except GPM, which is essentially a trade-off in accu- racy to minimize forgetting. As shown in Figure 5(c), we only use about 31 % of the final mem- ory size of GPM and achieve 2.5% better accuracy.
PMNIST:
PMNIST
Method
ACC(%)
BWT(%)
EWC
62.25 1.44
-15.22 1.25
GEM
61.82 + 0.85
-15.58 + 1.17
ER
68.31 0.56
-13.91 0.67
La-MAML
75.98 0.60
-10.21
+ 0.90
GPM
74.54 0.36
-7.17
0.51
FS-DGPM
76.96 + 0.51
-7.45 0.30
Multitask
86.54 1.74

---

1.
4.
68
31
OCEN
80
300
0.0
12
Ja Mumbor
The ablation study implemented on CIFAR-100 Split with 10 tasks. (a)-(c) shows the stability of the first task. (a) the test accuracy change curve of the first task; (b) and (c) are the weight loss landscape of the first task after learning the fifth task and all ten tasks. (d)-(e) shows the sensitivity of the fifth task, which are the test accuracy and the weight loss landscape the fifth task after just learning the fifth task. The shape of the weight loss landscape obtained using ten different random firter-normalized directions. ("Ti" abbr. of the i-th task)
Figure 6:
CIFAR- and TinylmageNet: Next, we use multi-head setting to evaluate our algorithm under the more challenging visual classification benchmarks. Table[Treport all results of these experiments. We outperform all basclines on three datasets, with achieving the best average accuracy 74.33 % 58.81 % and 70.41 %. In these experiments, we observe that GPM is a strong baseline with the least forgetting. At the same time, we highlight that our method achieves the highest accuracy on all datasets and the recond-lowest forgetting after GPM. Figure shows the process of performance changing with the number of tasks on the CIFAR- 100 Superclass. We consistently see the superior performance of our method at any stage of model evolution. It is also worth emphasizing that although our method requires more time for training than GPM, it has lower memory usage and better test accuracy (See (C)) As noted by and [8), EWC performs poorly without multiple passes over the datasets, and GEM is not very effective under the single-head variants. These situations have also been observed in our experiments.
5.3 Ablation studies on FS-DGPM
We further investigate our model per- formance with an ablation study and summarize it in Table 3 We respec- tively ablate the effects of flattening sharpness and dynamically adjusting the soft weight for bases. We refer to them as DGPM and FS-GPM. We also construct an ablation referred to as La- DGPM (Look-ahead DGPM), where the adversarial weight perturbation in Eq. (2) is changed to the direction
Table 3: The ablation study results on CIFAR- 100 Split and Superclass. Each experiment is run with 5 seeds.
CIEAR-100 Split
CIFAR-100 Superclass
Method
ACC(%)
BWT(%)
ACC(%)
BWT(%)
FS-DGPM
74.33 : 0.31
-2.71
0.17
58.81
0.34
-2.97 : 0.35
La-DGPM
73.74
-3.05
0.73
58.18
0.41
-2.11 4: 0.39
ES GPM
73.96
11.9640.44 0.44
-3.12 + 0.43
58.61
+ 0.53
-2.79
+ 0.30
DGPM
73.78
1 0.32
-3.67
: 0.42
0.449
-244
+ 0.40
GPM
73.18 0.52
-1.17 + 0.27
57.33
0.37
-0.37 1 0.12
of gradient descent. At the same time, we also change the sign in Eq. (3) to ensure that the soft weight of the basis is reduced when the gradients aligned in the same direction. From the results, shown in Table 3 we observe that flattening sharpness does benefits GPM, with ~ 1.0% improvement over GPM on both datasets. We can further observe through Figure tigure/othat all landscapes of FS-DGPM have lower loss values and flatter neighbors than DGPM and La-DGPM on the CIFAR-100 Split experiments. In addition, we see that DGPM performs well in learning new tasks, but it also leads to forgetting previous tasks. This situation can be efficiently alleviated by flattening sharpness. Hence, FS is indeed a powerful method worthy of being widely adopted for continual learning scenarios.
Conclusion
In this paper, we explore the weight loss landscape to characterize the well-known 'sensitivity- stability" dilemma faced by continual learning algorithms, and find that lower loss value with flatter neighbor often leads to better continual learning performance. Based on this finding, we propose our FS-DGPM algorithm, which introduces soft weight to represent the importance of each basis representing past tasks in GPM, that less important bases can be dynamically released to improve

---

the sensitivity of new skill learning, Flattening Sharpness (FS) is also introduced here to reduce the generalization gap by explicitly regulating the flatness of the weight loss landscape of all tasks seen so far. The evaluation of various image classification tasks with different network architectures and the comparison with some state-of-the-a art algorithms show the effectiveness of our method in achieving high classification performance while alleviating forgetting. Although our method theoretically and empirically demonstrates the advantages of introducing FS into continual learning, whether there exists closer upper bound for the continual learning performance remains an unresolved problem and left for our future exploration. Although our method theoretically and empirically demonstrates the advantages of introducing bases soft weight and FS into continual learning, whether there exists better dynamic adjustment and a closer upper bound for the continual learning performance remains an unresolved problem and left for our future exploration.
Acknowledgments
We would like to thank Dr. Yongzhe Deng and Zhuo Zhang for their helpful discussions, and anonymous reviewers for their valuable comments to improve this work. This work sup- ported by a grant from the National Key Research and Development Program of China (Project No. 2020YFB1313900 Hong Kong Research Grants Council under General Research Fund (Project No. 14201620), the National Natural Science Foundation of China (Project No. 62006219, 62072452) and Guangdong Provincial Basic and Applied Basic Research Fund Regiinal Joint Fund (Project No. 2020B (12001515130004)
References
[1] Hongjoon Ahn, Sungmin Cha, Donggyu Lee, and Taesup Moon. Uncertainty-based continual learning with adaptive regularization. arXiv preprint arXiv: /905 1614, 2019.
[2] Rahaf Aljundi, Francesca Babiloni, Mohamed Elhoseiny, Marcus Rohrbach, and Tinne Tuyte- laars. Memory aware synapses: Learning what (not) to forget. In Proceedings of the European Conference Computer Vision (ECCV), pages 139-154, 2018.
[3] Sungmin Cha, Hsiang Hsu, Taebaek Hwang, Flavio Calmon, and Taesup Moon. Cpr: Classifier-rooeectoon regularization for continual learning. arXiv preprint arXiv:2006.074: 073266 2020.
[4] Arslan Chaudhry, Marc' Aurelio Ranzato, Marcus Rohrbach, and Mohamed Elhoseiny, Efficient lifelong learning with a-gem. arXiv preprint arXiv:1812.0044 2018.
[5] Arslan Chaudhry. Mareus Rohrbach, Mohamed Elhoseiny, Thalaiyasingam Ajanthan, Puneet Dokania, Philip HS Torr, and M Ranzato. Continual learning with tiny episodic memories. 2019.
[6] Pengfei Chen, Guangyong Chen, Junjic Ye, Pheng-Ann Heng, et al. Noise against noise: stochastic label noise helps combat inherent label noise. In International Conference Learning Representations, 2020.
[7] Yu Chen, Tom Diethe, and Neil Lawrence. Facilitating bayesian continual learning by natural gradients and stein gradients. arXiv preprint 1904. 10644, 2019.
[8] Sayna Ebrahimi, Mohamed Elhoseiny, Trevor Darrell, and Marcus Rohrbach. Uncertainty- guided continual learning with bayesian neural networks. arXiv preprint 2019.
[9] Mehrdad Farajtabar, Navid Azizan, Alex Mott, and Ang Li. Orthogonal gradient descent for continual learning. In International Conference on Artificial Intelligence and Statisties, pages 1762-3773 PMLR, 2020.
[10] Pierre Foret, Ariel Kleiner, Hossein Mobahi, and Behnam Neyshabur. Sharpness-aware mini- mization for efficiently improving generalization. arXiv preprini arXiv:2010.01412, 2020.
[11] Ian J Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks. arXiv preprint arXiv:1406.266 2661, 2014.
10

---

[12] Yunhui Guo, Mingrui Liu, Tianbao Yang, and Tajana Rosing. Improved schemes for episodic memory-basee lifelong learning. Advances in Neural Information Processing Systems, 33, 2020.
[13] Gunshi Gupta, Karmesh Yadav, and Liam Paull. La-maml: Look-ahead meta learning for continual learning. arXiv preprint 2tt1:2007.119000 2020.
[14] Wenpeng Hu, Zhou Lin, Bing Liu, Chongyang Tao, Zhengwei Tao, Jinwen Ma. Dongyan Zhao, and Rui Yan. Overcoming catastrophic forgetting for continual learning via model adaptation. In International Conference on Learning Representations, 2018.
[15] Daoyun Ji and Matthew A Wilson. Coordinated memory replay in the visual cortex and hippocampus during sleep. Nature neuroscience, 10(1):100-107, 2007.
[16] Mohammad Khan, Didrik Nielsen, Voot Tangkaratt, Wu Lin, Yarin Gal, and Akash Srivastava. Fast and scalable bayesian deep learning by weight-perturbate in adam. In International Conference on Machine Learning, pages 2611-2620. PMLR, 2018.
[17] James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Overcoming catastrophic forgetting in neural networks. Proceedings of the national academy of sciences, 114(13)3522-3356 2017.
et al.
[18] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.
[19] Quoc Le. Jiquan Ngiam, Adam Coates, Abbik Lahiri, Bobby Prochnow, and Andrew Y Ng. On optimization methods for deep learning. In ICML, 2011.
[20] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11))2278-2333 1998.
[21] Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, and Tom Goldstein. Visualizing the loss landscape of neural nets. In Proceedings of the 32nd International Conference Neural Information Processing Systems, pages 6391-6401, 2018
[22] Xilai Li, Yingbo Zhou, Tianfu Wu, Richard Socher, and Caiming Xiong. Learn to grow: continual structure learning framework for overcoming catastrophic forgetting. In International Conference on Machine Learning, pages 3925-3934. PMLR, 2019.
[23] Zhizhong Li and Derek Hoiem. Learning without forgetting, IEEE transactions on pattern analysis and machine intelligence, 40 (0(11):2935-299) 2017.
[24] David Lopez-Paz Paz and Marc' Aurelio Ranzato. Gradient episodic memory for continual learning, arXiv preprint arXiv: 100. 088400 2017.
(25] Arun Mallya and Svetlana Lazebnik. Packnet: Adding multiple tasks single network by iterative pruning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7765-7773, 2018
[26] Seyed Iman Mirzadeh, Mehrdad Farajtabar, Razvan Pascanu, and Hassan Ghasemzadeh. Under- standing the role of training regimes in continual learning, arXiv preprint urXiv:2006.06955 2020.
[27] Behnam Neyshabur, Srinadh Bhojanapalli, David McAllester, and Nathan Stebro. Exploring generalization in deep learning. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 5949-5958, 2017.
[28] Cuong V Nguyen, Yingzhen Li, Thang D Bui, and Richard E Turner. Variational continual learning. arXiv preprint arXiv: 10628, 2017.
[29] German Parisi, Ronald Kemker, Jose Part, Christopher Kanan, and Stefan Wermter. Continual lifelong learning with neural networks: A review. Neural Networks. 113:54-71, 2019.
[30] Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, Georg Sperl, and Christoph H Lampert. icarl: Incremental classifier and representation learning. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 2001-2010, 2017.
[31] Mathew Riemer, Ignacio Cases, Robert Ajemian, Miao Liu, Irina Rish, Yuhai Tu, and Ger- ald Tesauro. Learning to learn without forgetting by maximizing transfer and minimizing interference. arXiv preprint arXiv:1810.11910 2018.

---

[32] Hippolyt Ritter, Aleksandar Botev, and David Barber. Online structured laplace approximations for overcoming catastrophic forgetting. arXiv preprint arXiv:1805.0788/1 2018.
[33] Andrei A Rusu, Neil C Rabinowitz, Guillaume Desjardins, Hubert Soyer, James Kirkpatrick, Koray Kavukcuoglu, Razvan Pascanu, and Raia Hadsell. Progressive neural networks. arXiv preprint arXiv: (11660666677 2016.
[34] Gobinda Saha, Isha Garg, and Kaushik Roy. Gradient projection memory for continual learning. arXiv preprint 1rXi::2003.09622 2021.
[35] Joan Serra, Didac Suris, Marius Miron, and Alexandros Karatzoglou. Overcoming catastrophic forgetting with hard attention to the task. In International Conference on Machine Learning, pages 4548-4551 PMLR, 2018.
[36] Hanul Shin, Jung Kwon Lee, Jachong Kim, and Jiwon Kim. Continual learning with deep generative replay. arXiv preprint arXiv: 2017.
[37] Stanford. tinyimagenet. http://cs231n htt.:ss22nnnttaforr...oaoo
www.ssaad.....omo ewdutto..eduuutiiny--a....... -mmnnnet--eeee----- wateeett-00..000 200.zip 2017.
[38] Binh Tang and David Matteson. Graph-based continual learning. arXiv preprint arXiv:2007.0488 13, 2020.
[39] Gido M van de Ven, Hava T Siegelmann, and Andreas S Tolias. Brain-inspired replay for continual learning with artificial neural networks. Nature communications, 11(1):1-14, 2020.
[40] Yeming Wen, Paul Vicol, Jimmy Ba, Dustin Tran, and Roger Grosse. Flipout: Efficient iseudo-independent weight perturbations on mini-batches. arXiv preprint 1803v:1803004388 2018.
[41] Matthew A Wilson and Bruce I. McNaughton. Reactivation of hippocampal ensemble memories during sleep. Science, 265 (72))666-666777 1994.
[42] Dongxian Wu, Shu-Tao Xia, and Yisen Wang. Adversarial weight perturbation helps robust generalization. Advances in Neural Information Processing Systems, 33, 2020.
[43] Ju Xu and Zhanxing Zhu, Reinforced continual learning. arXiv preprint arXiv:1805. 12369,
2018.
[44] Jachong Yoon, Saehoon Kim, Eunho Yang, and Sung Ju Hwang. Scalable and order-robust continual learning with additive parameter decomposition, arXiv preprint arXiv: (XXiv:9992009433 2019.
[45] Jachong Yoon, Eunho Yang, Jeongtae Lee, and Sung Ju Hwang. Lifelong learning with dynamically expandable networks. arXiv preprint arXiv:1708. 01547, 2017
[46] Friedemann Zenke, Ben Poole, and Surya Ganguli. Continual learning through synaptic intelligence. In International Conference on Machine Learning, pages 3987-3995. PMLR, 2017.
12


Reviews:
[{"review_id": "ekh1Pa9V_ws", "reviewer": "Reviewer_zbwu", "summary": "The article proposes a continual learning method that innovatively improves on an existing algorithm. Precisely, the current work takes GPM (Gradient Projection Memory) which they show is good at preserving performance on old tasks (good stability), but it's not that efficient at learning new ones (low sensitivity) and modifies it in two ways: by dynamically adjusting the importance of the various bases such that the less important ones can be dropped to increase sensitivity. In addition it improves the stability by employing a form of adversarial parameter perturbation to flatten the loss landscape.\nThe contributions of the paper are:\n- an analysis that shows the loss landscape and its \"flattness\" / \"sharpness\"; this analysis is used to support the idea that flatter minima tend to forget less in continual learning setups;\n- changing the GPM such that it has an increased sensitivity while also improving the stability by flattening the loss landscape around the current solution\n", "questions": "", "limitations": "", "rating": 7, "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": "The current work is novel and relevant to the continual learning literature. It identifies a weakness in a competitive algorithm from the literature (the low \"sensitivity\" of GPM = low performance on new task) and improves it with a well motivated modification.\n\nThe article is well written, the formulas and the pseudocode are clear, and the experiments support the claims made.\n\nThe algorithms used for comparison are relevant and diverse. The only weakness of the approach is its limitation to the \"known task boundaries\" setup and the lack of large-scale or reinforcement learning experiments. Although they are not the most challenging, the set of tasks are adequate and common in the continual learning literature.\n\nGiven the originality of the analysis of the sharpness of the loss landscape, the well-motivated changes to GPM, and the empirical results, I consider the paper should be accepted.\n\nMinor observation: Humans are animals.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Limitations: not really. The approach is applicable to a subset of continual learning scenarios with assumptions such as known task boundaries, control over the times the data is used, etc.\nSocietal Impact: Not applicable.", "review_text": "", "extra_scores": {"ethical_concerns": "No.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "YoI-x6-Q2nm", "reviewer": "Reviewer_hs57", "summary": "In this paper, the authors propose a method called Flattening Sharpness for Dynamic Gradient Projection Memory (FS-DGPM) for continual learning. The main idea is to investigate the effect of weight loss landscape on overcoming catastrophic forgetting. With the understanding of the weight loss landscape, the authors further propose a method based on the recently proposed Gradient Projection Memory (GPM) method to predict the importance of bases which span the subspace of old tasks. Less important bases can be dynamically released based on the proposed soft weight. Extensive experiments show that the proposed method can consistently outperform the state-of-the-art methods across a range of widely used benchmark datasets.", "questions": "", "limitations": "", "rating": 6, "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": "\nOriginality: The idea of leveraging the flatness of the minima for continual learning is novel and interesting. However, the proposed method seems incremental. \n\nQuality: The paper is technically sound. \n\n\nClarity: The writing of the paper needs improvement. The paper should include a clear introduction to GPM such that it is easier to understand the proposed method. Meanwhile, the proposed method is not clearly explained. There are also several typos in the paper.  \n\n\nSignificance: The improvements are not significant but meaningful which show the effectiveness of utilizing the loss landscape. \n\n\n\nQuestions and suggestions:\n\n\n1. Can the idea of leveraging the loss landscape be used for other continual learning methods other than GPM?\n\n2. To show the effect of loss landscape on continual learning, maybe the authors should conduct a simpler experiment to remove the factor of continual learning methods themselves. For example, just train a vanilla network sequentially on all the tasks, then compare the performance of the network with or without enforcing flat minima. \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3 hours"}, "extra_text": {}}, {"review_id": "Myps7aAm6gL", "reviewer": "Reviewer_sRK3", "summary": "The paper studies catastrophic forgetting and continual learning from the loss landscape perspective.\nThe primary motivation is that from the plasticity/stability view, stable networks tend to have flatter minima.\nWhile this has been observed before in [1], the paper studies the problem in more depth. Also, it proposes a solution to reduce the sharpness of minima for each task based on the gradient projection memory work.  \nOverall, I like the paper, and I think it is an influential continual learning paper. However, I think the paper can be even better if the authors can address a few shortcomings.", "questions": "", "limitations": "", "rating": 7, "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": "\n### Overall\n**Originality**: The significance of minima sharpness has been studied before in CL literature [1], but the work is novel enough regarding the depth and the approach of analysis.   \n**Significance**: The paper is a significant contribution since it gives a better understanding of catastrophic forgetting and continual learning in general.  \n**Quality**: I believe this is a high-quality work both theoretically and empirically.  \n**Clarity**: The paper is well-written, aside from a few problems regarding the appendix (see below).\n\n----\n\n### Strengths\n**1-** I appreciate the fact that this paper aims to increase our understanding of catastrophic forgetting. In my opinion, the CL literature needs more works that try to understand the continual learning challenges.   \n**2-** The proposed method and analysis seem novel and reasonable.  \n**3-** The experiments are very well designed. However, they can be improved to make the contributions of the paper even more significant.  \n\n \n### Weaknesses\n**1-** Since the main objective of the proposed method is to reduce the sharpness of minima, and since the authors seem to be familiar with the impact of training regime on sharpness [1], I wonder why they have chosen to use a single hyper-parameter rather than integrate the findings of [1] regarding the training regime. Can the authors elaborate on this? Aside from that, I think it will be fairer if the results of Tables 1 and 2 were calculated using a grid-search over different hyper-parameters since using only one set of parameters might favor a specific method.\n\n**2-** Figure 6 illustrates the effectiveness of FS-DGPM over other methods and shows that around the local minimizer, the loss landscape of FS-DGPM seems to be flatter than other methods. But, what about the Multitask method? What would the results look like compared to the upper-bound, which is the multitask training? Could the authors report the results in this regard?\n\n**3-** While Figure 5 (c) shows the memory usage, it is unclear what would happen if the memory budget increases/decreases. Could the authors perform an experiment where they compare the results for ER and FS-DGPM when the memory size increases? For instance, an experiment similar to figure 1 in [2] can show the behavior of FS-DGPM over different scenarios. For instance, for Split CIFAR-100 (super), memory sizes of 100, 500, 1000 seem representative enough.\n\n**4-** It seems like the appendix is not well-connected to some parts of the main text. For instance, in Section D.4 (L135), the paper mentions figure 5 (b) while it refers to figure 5 of the main text rather than figure 5 of the paper. Also, in the main text (L283), the paper mentions the training details hyperparameters are reported in Appendix D.4 while they are discussed in Appendix D.5 and D.6. Also, Figure 1 in the appendix does not have a legend.\n\n**5-** Do authors plan to release their code for this paper?\n\n----- Update after discussion period ------  \nThe authors have addressed my comments and I still vote accepting the paper. \n\n\n[1] *Mirzadeh, Seyed Iman, et al. \"Understanding the Role of Training Regimes in Continual Learning.\" Advances in Neural Information Processing Systems, 2020*   \n[2] *Chaudhry, Arslan, et al. Continual Learning with Tiny Episodic Memories. 2019. https://arxiv.org/pdf/1902.10486.pdf*", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "C2xPNA89Pc7", "reviewer": "Reviewer_tT1e", "summary": "The paper proposes a new continual learning method that promotes the loss shapes to be flattened. The main idea is to optimize the worst case perturbed loss function at every gradient step, and the empirical investigation of the loss values around the learned parameter corroborates their assertion. Also, experimentally, they showed promising results compared to other recent baselines. ", "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": "First of all, I think the paper misses an important reference that also had a similar motivation of attaining the flat minima, or the wide local minima (WLL), in the context of continual learning:\n\nCha. et al., CPR: Classifier-projection regularization for continual learning, ICLR 2021. \n\nWhile the approach is different, I think it definitely need to be cited. \n\nHaving said that, the proposed method of the paper seems novel in the context of attaining flat minima in the continual learning context. Followings are what I think about the paper:\n\nPros:\n(+) Framework of minimax optimization and empirically showing that the loss function becomes flat seems to be novel. \n(+) Provided theoretical explanations, although the connection with the proposed method is not very straightforward.\n(+) Experimental results tend to be thorough, providing ablation study and memory usage, time complexity comparisons. \n\nCons:\n(-) Missing the reference mentioned above. \n(-) The computational complexity of the method seems to be a limitation. At every task, one needs to carry out the SVD for obtaining M matrix, which could become quite expensive if large network model is used. \n(-) In Eq.(4), using sigmoid should be more explicitly explained. I believe the sigmoid is applied at the end of the gradient update, but is \\lambda appearing at the right-hand side of (4) also the one with sigmoid, or not? Also, this step seems to be a heuristic, and it's not clear whether this is the best option. \n(-) Experiments on larger datasets would show more compelling results. \n\nOverall, despite some weakness mentioned above, I am leaning toward weak accept, but I will also see what other reviewers think about the paper. \n\n-------\n\nI have read the author's rebuttal and other reviewers' reviews, and I would like to keep my score to 6. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "N/A", "review_text": "", "extra_scores": {"ethical_concerns": "N/A", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1.5 hour"}, "extra_text": {}}]

Author rebuttal:

Meta-review: Echoing what the reviewers highlighted as well. I think the idea of this work is quite novel, and the paper is well executed, where there was due diligence in terms of explaining the hypothesis, and doing ablation studies. As it stands, I think the work is definitely interesting for the community and should be accepted. 
Decision: Accept (Poster)