Revisiting Gradient Episodic Memory for Continual Learning
Authors: Zhiyi Chen
Tong Lin*
Venue:  (2020)

Abstract: Gradient Episodic Memory (GEM) is an effective model for continual learning, where each gradient update for the current task is formulated as a quadratic program problem with inequality constraints that alleviate catastrophic forgetting of previous tasks. However, practical use of GEM is impeded by several limitations: (1) the data examples stored in the episodic memory may not be representative of past tasks; (2)  the inequality constraints appear to be rather restrictive for competing or conflicting tasks; (3) the inequality constraints can only avoid catastrophic forgetting but can not assure positive backward transfer. To address these issues, in this paper we aim at improving the original GEM model via three handy techniques without extra computational cost. Experiments on MNIST Permutations and incremental CIFAR100 datasets demonstrate that our techniques enhance the performance of GEM remarkably. On CIFAR100 the average accuracy is improved from 66.48% to 68.76%, along with the backward (knowledge) transfer growing from 1.38% to 4.03%.

Under review as a conference paper at ICLR 2020
REVISITING GRADIENT EPISODIC MEMORY FOR CON- TINUAL LEARNING
Anonymous authors Paper under double-blind review
ABSTRACT
Gradient Episodic Memory (GEM) is an effective model for continual learning, where each gradient update for the current task is formulated as a quadratic pro- gram problem with inequality constraints that alleviate catastrophic forgetting of previous tasks. However, practical use of GEM ís impeded by several limitations: (1) the data examples stored in the episodic memory may not be representative of past tasks; (2) the inequality constraints appear to be rather restrictive for compet- ing or conflicting tasks; (3) the inequality constraints can only avoid catastrophic forgetting but can not assure positive backward transfer. To address these issues, in this paper we aim at improving the original GEM model via three handy tech- niques without extra computational cost. Experiments on MNIST Permutations and incremental CIFAR 100 datasets demonstrate that our techniques enhance the performance of GEM remarkably. On CIFAR100 the average accuracy is im- proved from 66.48% to 68.76%, along with the backward (knowledge) transfer growing from 1.38% to 4.03%.
INTRODUCTION
Catastrophic forgetting (McCloskey & Cohen, 1989) is a common phenomenon in deep learning that the model performance over past tasks can be harmed by the training process of current task when we employ one single neural network to learn consecutive tasks. Continual learning (Ring. 1994), also known lifelong learning (Thrun, 1994), is a specific research field in Al that focuses on avoiding or alleviating catastrophic forgetting. It is well known that humans and large primates can continually learn new skills and accumulate knowledge throughout their lifetime (Fagot & Cook, 2006).
Gradient Episodic Memory (GEM) (Lopez-Paz & Ranzato, 2017) is significant method for contin- ual learning. The basic idea is to modify each gradient of the current task such that the losses at the memories (data examples) from previously solved tasks will not increase, hence allowing positive backward knowledge transfer. The problem is formulated as a Quadratic Program with inequality constraints, and then solving its dual problem provides an efficient computation for a small num- ber of dual variables. GEM outperforms two state-of-the-an methods, Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017) and iCARL (Rebuffi et al., 2017) on variants of the MNIST and CIFAR100 datasets.
Despite the elegance of its mathematical formulation and the computational efficiency in solving the dual problem, the GEM model has following drawbacks for practical use.
1. The episodic memory is simply populated with the last their current implementation, where M is the total budget size of memory locations for totally T tasks. The authors of GEM mentioned that better memory update strategies could be employed to obtain a representative set of examples for past tasks, such as building a coreset per task (Lucic al., 2018).
M/ examples from each task in
2.
The inequality constraints of gradients appear to be quite restrictive, requiring that the gradient update can not increase any loss of all previously solved tasks. When two tasks competing or conflicting in the gradient update directions, especially with a large number of past tasks, these constraints have to be violated and the feasible solution might be null.

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3. The gradient constraints only assure to avoid catastrophic forgetting, but might not promote pos- itive backward knowledge transfer in some scenarios. For instance, we may obtain a gradient vector orthogonal to all gradients of previous tasks in the constrained optimization problem. In this case, GEM is greedy in minimizing the loss of the current task, overlooking the need of back- ward transfer to previous tasks. An ideal model for continual learning should consider better trade-off between optimizing performance of the current task and backward transfer to previously solved tasks.
To address these issues, in this paper we propose three techniques that improve the performance of GEM. These techniques come from successful practice in Support Vector Machines (SVMs) and deep learning approaches. In concrete, we use the notion of margins select representative data points as episodic memories. We also leverage slack variables to allow violations of the constraints, and aggressively encourage backward knowledge transfer to previous tasks. Experiments on MNIST and CIFAR 100 datasets corroborate the improvements of our techniques over the original GEM model. Also our methods increase little overhead in computational efforts.
The paper is organized as follows. In Section 2, we first survey several recent methods on continual learning and explain why we focus on improving GEM. Section offers a brief review of the GEM algorithm. In Section 4, we elaborate our three improvements and discuss the underlying motiva- tions. Section 5 presents the experimental results and comparison analysis, and Scction 6 concludes the paper.
2 RELATED WORK
According to Xu & Zhu (2018), recent methods for continual learning can be roughly classified into two large categories, distinguished by whether the network architecture changes through a contin- uum of tasks.
The first category attempts to modify the network architecture to alleviate catastrophic forgetting. Fernando et al. (2017) proposed a neural network model called PathNet, in which there are ten or twenty modules in each layer and for one task only three or four modules are dynamically chosen in each layer using an evolutionary approach. Progressive neural networks (Rusu et al., 2016) aims at accommodate the new coming task by dynamically expand the architecture with fixed size of nodes or layers, leading to an extremely large network structure with high redundancy. In Dynami- cally Expandable Network (DEN) ( (Yoon et al., 2017) this issue is slightly alleviated by employing group sparsity regularization when expanding the networks. However many hyperparameters are involved, including various regularization and thresholds, and careful tuning is challenging. Xu & Zhu (2018) proposed Reinforced Continual Learning (RCL) approach that searches for the best neu- ral architecture for each coming task via reinforcement learning strategies. Like other expandable networks, this approach still suffers from higher model complexity than non-expandable networks. Also reducing its training time is major concern.
Another category maintain a fixed network architecture with large capacity through learning. Oquab et al. (2014) proposed weight transfer method by freezing early layers while cloning and fine- tuning later layers on the new task. In "synaptic" memory methods (Kirkpatrick et al., 2017: Zenke al., 2017), learning rates are adjusted tasks. Lee et al. (2017) proposed to incrementally match the moment of the posterior distribution of the neural network trained on the first and the second task, respectively. In Zeng et al. (2018) an Orthogonal Weights Modification (OWM) was proposed to ensure that new learning processes will not interfere with previously learned tasks, but can not enable backward knowledge transfer. In Shin et al. (2017); Kemker & Kanan (2018) previously acquired knowledge can be acquired by replaying of pseudo-data generated by Generative Adversarial Networks (GANs) or generative autoencoders, without revisiting actual past data for the coming task. Several "episodic" memory approaches (Jung et al., 2016; Li & Hoiem, 2016; Triki et al., 2017; Rebuffi et al., 2017) aim at storing and replaying examples from past tasks to keep predictions invariant via distillation (Hinton et al., 2015). GEM (Lopez-Paz & Ranzato, 2017) belongs to this class of episodic memory approaches, but unlike prior work, allows for backward transfer.
minimize updates in weights important for previous
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Comparing to these recent approaches continual learning, GEM is relatively lightweight, compu- tationally efficient and capable of achieving high accuracies. However, there still room to improve GEM with respect to the limitations stated in Section 1.
A BRIEF REVIEW OF GEM
Most of the literature on continual learning (Rusu et al., 2016; Fernando et al., 2017; Kirkpatrick
al., 2017; Rebuffi al., 2017) focused on traditional setting: (1) the number of tasks is small; (2) the number of examples per task is large; (3) the learner allows for several passes over the examples per task; (4) the average performance across all tasks serves as the only metric. In contrast, a "more human-like" setting was considered in GEM: (I) the number of tasks is large; (2) the number of training examples is small for each task; (3) the learner observes the examples of per task only once; (4) knowledge transfer in backward and forward directions is assessed. Therefore following this new setting, the learner will observe the continuum data in a manner of example by example
(I1, 11, 31),................... (a,, t4, Bi),..., (Ins tnx Vn))
where each triplct (I,, ta V/)) contains input vector, task index, and target vector. Note that any order on the tasks can be allowed.
The optimizing problem of GEM (Lopez-Paz & Ranzato, 2017) is
min w
s.t. (w, gk) 2 0, Ak <t
where g is the gradient vector calculated on the batch data of the current task, and Ii here is the gradient on the episodic memory M2 of the past task k. Note that the episodic memory Mr stores a subset of the observed examples from task k. the gradient update g of the current batch fulfills all the constraints, the learner directly uses it to update like the usual gradient descent method. Otherwise, should find a projection of g Lo-norm distance satisfy any constraint related each previous task, and then use the solution w to update the network weights.
We obtain a standard form of Quadratic Program (QP) by expanding the squared objective and discarding the constant term g²
1
min
w
w g"
2
8.1. Gw 2 0,
(01,...99-1)).. The primal QP problem can be translated into its dual problem (Dorn, where G= ........
1960), which is the final optimizing problem:
min a GGG + (Gg) 2
0.
We can recover the projected gradient update as n' = GTa" where a" the solution to the dual problem. Note that solving the dual problem is computationally efficient since there only t variables a, much smaller than the number of weights in a neural network.
OUR IMPROVEMENTS
We propose three improvements to GEM: a technique to choose examples which can represent the past tasks better support examples; a technique to tolerant some slight violation in gradient con- straints on past tasks soft gradient constraints; technique to aggressively encourage gradient descend over previous tasks positive backward transfer.
4.1 SUPPORT EXAMPLES
The first improvement is inspired from the idea of margins in Support Vector Machines (SVMs) (Williams, 2003). Due to the KKT theorem, small set of support vectors obtained from Quadratic
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Program determines the decision boundary of SVM for binary classification. In the context of con- tinual learning, the decision is crucial to identify what samples should be chosen in the episodic memory, since we can only store limited budget of representative samples from previous task. In SVMs support vectors arise naturally as a solution from the QP problem of SVM; however such optimization problem does not apply to neural networks. In GEM simple method is adopted, that is. the last m examples of task are stored into the episodic memory M₁. Also the examples are shuffled randomly, which means that the storage essentially random and ad hoc.
In this work, we propose an idea of choosing samples based on the notion of margins. We assume that the problem to solve here is multi-class classification, so the prediction from the network's forward propagation can be available. Suppose the output prediction h(z, t. y) on triplet example (ar, I, y) computed through softmax activation function, with value falling the range [0, 1]. The following gives the definition of margins:
margin == h(,,,,, max mar t, v'#4
(4)
When the classifier h makes a correct prediction, the margin positive. Otherwise when h misclas- sifies the example, we have margin S 0. A margin value indicates the confidence of the prediction of an triplet example: larger the margin is in magnitude, more confidence we have in the prediction.
In order to alleviate catastrophic forgetting, the most important thing is to ensure that the examples of past tasks can be correctly classified to retain the prediction performance. Usually we have a much smaller budget for the episodic memory than the number of examples per task. For instance, in experiment on CIFAR 100, the number of total samples is 2500 per task, while the episodic memory is 256 per task. On the one hand, we can not afford to assign the memory to the examples whose margins are smaller than 0, which means they could not be classified correctly in the past task or even these examples are outliers. On the other hand, for the examples whose margins relatively large, indicating that the network can classify them with a strong confidence, these example should not be put into the memory either. We define the Support Examples those examples being classified correctly but with a limited confidence. In experiments we simply use a confidence interval to select the support examples (we specify the interval hyperparameters per task). In this way, the episodic memory can be used more efficiently, and at the same time, we argue that the chosen support examples can represent the past tasks better.
4.2 SOFT GRADIENT CONSTRAINTS
This idea is inspired by the soft-margin technique in SVMs, which is compromise to some wrongly classified examples by linear classifiers. In continual learning, the difficulty of optimizing the gra- dient to be consistent to every gradient direction of past tasks is increasing along the process of learning new tasks. The constraints in (2) mean that we want the optimized gradient must have a similar direction with all of the gradients of past tasks, which is too restrictive, especially when the number of tasks is large or the tasks are competing or conflicting.
So propose the following soft gradient constraints:
min w w g"w + +CE
(S)
8.l. Gw 2 -E, & 2 Ak <1,
{= (E.......,Et-1 is the slack vector, C is the trade-off parameter (the bigger, the stricter for w to satisfy the constraints). Once any original constraint in Gw 2 is violated, we impose a penalty in the objective. With the KKT theorem, we have the following dual problem as is the Lagrange multiplier vector:
1
min a TGG + (G 2 s.t. 0 < a S C'
(6)
Like in SVMs, the hyperparameter C' needs to be tuned. For the first several tasks, it is relatively easy make the gradient update completely satisfy the constraints on all the past tasks. When the number of past tasks is increasing, is much more difficult to have the similar direction to all the past tasks' gradient simultaneously. So need to loose the constraint to some extent.
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Here, propose method just like the learning rate decay in neural network training: initialize C' to a relatively big value, which should ensure a very strict constraint for first several tasks, while let C decay along the process of training. In other words, after finishing training certain number of tasks, we set C' smaller to tolerant more freedoms in constraint violations. This trick will maintain good performance, and at the same time make the model more robust.
The authors of GEM claimed that adding a small offset y 2 O to the Lagrangian multipliers in (2) can enable the network to perform better. However, this is ad-hoc practice and lacks rationality: is not faithfully follow their mathematical formulation. In our method, we can just avoid this ad-hoc practice and guarantee that just use the original constraints with slack variables will produce good performance. Our soft constraints offer better explanation to this issue.
4.2.1 POSITIVE BACKWARD TRANSFER
GEM assumes that if the inner product of the proposed gradient update and the gradients of past tasks is no less than zero, the current update will not cause catastrophic forgetting. Some extreme scenarios might occur that the found gradient projection orthogonal to all the gradients of previous tasks, indicating that there is no contribution to backward transfer. Further, the non-negative con- straints in inner products only require the angles between two gradient vectors lying in the interval [-90", ], totally ignoring the cosine similarity in magnitude.
In our opinion, we should specify the magnitude of the inner products of gradient vectors for enhancing positive backward transfer to previously solved tasks. We can modify (2) to impose magnitude penalties:
min 1 w11
s.t. (w, (x) e|| 4k < t, 2.0.
This constraint imposes cosine similarity actually, it represent the angle and the magni- tude simultaneously. Because we do not know |/| in advance, we can use | as proxy since w the projection of g::
min ] 11g wl12
s.l. (w. 9k) 2 el1y| [glll. Ak < 1,
Similarly get the final dual optimizing problem, which includes soft gradient constraints.
1 a GGG + (Gg + ووااا|ااg I
min 2
(8)
s.t. 05 Cl.-
After we solve which has the variables equal to the number of past tasks, we can use w G a + to get the modified gradient and use it to update current network.
Here is hyperparameter. Similarly, the first several tasks are easier for network to enable positive backward transfer. And when the number of tasks increase, we can reuse the idea of 'decay" to weaken the magnitude of inner-product similarity, approaching zero gradually.
EXPERIMENTS
5.1 DATASETS AND SETTINGS
We do experiments two dataset same to GEM (Lopez-Paz & Ranzato, 2017): MNIST Permuta- tions (Kirkpatrick et al., 2017) and Incremental CIFARI00 (Rebuffi et al., 2017). Besides, we also do experiment MNIST Mix (Xu & Zhu, 2018),
1. The MNIST Permutations dataset is derived from the famous dataset on handwritten digits (Le- Cun et al., 1998), which randomly generates different pattern of pixel permutation for each task exchange the position of the original images of MNIST.
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2. The Incremental CIFARIO0 dataset is derived from CIFARI00 (Krizhevsky. 2009), which aver- agely devides the whole classes in CIFARIO0 into parts, and 'I' here is the number of tasks. So in each task, there is 100/T different classes that are not included in other tasks.
3. The MNIST Mix dataset is also derived from MNIST, which has 10 MNIST Permulations (P1, P2,....... ..., P10) and 10 MNIST Rotations (RI, R2, .... R10) (Lopez-Paz & Ranzato, 2017), whose image rotated by fixed angle from original image of MNIST. The consecutive tasks are arranged in the order P1, R1, P2, R2, P10, R10.
To compare our method to the original GEM, we use the same settings on datasets. The number of tasks is set T 20. On the MNIST Permutations dataset there 1000 examples per task and On the Incremental CIFAR100 dataset there are 2500 examples per task. In the basic comparison, the size of the total budget on episodic memory set as 5120, which means that we can store 256 examples for each task.
The network architectures are also same to the experiments of the original GEM: for MNIST Per- mutations, we use fully-connected neural networks with two hidden layers of 100 ReLU units; for Incremental CIFARIO0 we use a smaller version of ResNet18 (He et al., 2016), which has three times less feature maps for each layer than the orginal ResNet18 and has a final linear classifier for each task.
5.2 COMPETING METHOSS
To solve the optimizing problem of soft gradient constraint, we have to change the toolbox in original code of GEM evxpy. So conduct experiments both on run of the original GEM and the GEM with the cvxpy toolbox.
Besides, we compare with other several methods:
1. Single, single predictor for all the tasks.
Multimodal, multimodal predictor, whose architecture same "single" and has dedicated imput layer for each task (only tested on MNIST Permutations dataset).
3. Independent, network that there is a specific independent predictor for each task which has the same architecture as "single" while number of hidden units are times less than "single".
Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017).
5. iCARL (Rebuffi et al., 2017), the class-incremental learner uses episodic memory and a nearest- exemplar algorithm (tested only Incremental CIFAR
These compared methods parison, we report the results reported in original paper of GEM, as well as our run of the original GEM, the GEM using cvxpy toolbox and our improved method. In other experiments, the method GEM is always the version of using evxpy.
to the settings in (Lopez-Paz & Ranzato, 2017). In Basic Com-
5.3
EVALUATION METRICS
The evaluations we use are average accuracy and positive backward transfer. The backward transfer can be calculated as:
1
Backward Transfer
E(An
AAs)
T
i=1
where Ai the accuracy of task j after learning of task i.
5.4 BASIC COMPARISON
Table shows the basic comparison results. To be succinct, MNIST Permutation denoted by MNIST_P and MNIST Mix is denoted by MNIST.M. We can that method perform better both on average accuracy and positive backward transfer than the original GEM on all three datasets. For example on CIFAR100, method improves from 66.48% (our run of GEM) to

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Table I: Results in basic comparison.
Metrics
single
independent
multimodal
EWC
Average Accuracy
0.5742
0.3700
0.7158
0.6185
MNIST_P
Backward Transfer
-0.2220
0.0000
-0.0683
-0.1653
Metrics
GEM
GEM (rerun)
GEM-cvx
Ours
Average Accuracy
0.8260
0.8232
0.8213
0.8331
MNIST P
Backward Transfer
0.0247
0.0386
0.0230
0.0409
Metrics
single
independent
iCARL
EWC
Average Accuracy
0.4666
0.4415
0.5359
0.4984
CIFARI00
Backward Transfer
-0.1240
0.0000
-0.0405
-0.0799
Metrics
GEM
GEM (rerun)
GEM-cvx
Ours
Average Accuracy
0.6783
0.6648
0.6703
0.6876
CIFAR 100
Backward Transfer
0.0042
0.0138
0.0147
0.0403
Metrics
single
independent
multimodal
EWC
Average Accuracy
0.7206
0.5275
0.7656
0.7285
MNIST_M
Backward Transfer
-0.1107
0.0000
-0.0194
-0.0885
Metrics
GEM (our run)
GEM-cvx
Ours
Average Accuracy
0.8521
0.8446
0.8537
MNIST M
Backward Transfer
0.0246
0.0129
0.0237
Table 2: Hyperparameters of basic comparison
Support examples
Soft gradient constraint
Positive backward transfer
MNIST_P
0.11, 0.8]
oo & #1 per 20 tasks
0.2 & #0.8 per 3 tasks
CIFARION
00.2, 0.8
5 & *0.6 per 3 tasks
0.2 & *0.8 per tasks
MNIST_M
[0.1, 0.8
oo & 11 per 20 tasks
0.25 & #0.8 per 5 tasks
68.76% with backward transfer growing from 1.38% to 4.03%. Table 2 lists the hyperparamters used in experiments.
Figure reports the evolution curves of average accuracy for all tasks and the accuracy of the first task as well. The details of numerical values are given Appendix A. The evolution of average accuracy of all tasks indicates the comprehensive performance of the method. Our method perform best comparing all other methods on average accuracies. Also on all three datasets, our method keeps relatively stable performance on the first task and always has improvements through the whole learning process. We think the performance evolution of the first task only provide an small aspect of the comparison; improvements of average accuracy could be more important.
5.5 PERFORMANCE ON DIFFERENT MEMORIES
We also do experiments different budget of memories for the three datasets, with results shown in Table 3. We can see that larger budgets of memories can enhance the accuracy, and method improves the original GEM on all settings. Table 4 shows the hyperprameters for method.
5.6 ABLATION STUDY
We conduct ablation study to figure out the contribution of each improvements, with results shown in Figure 2. The details of numerical results reported in Appendix B.

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0.99
0.55
041
0.25
0.25
GIM
KARL
EVC
Tigh
Iidapendert
Average accuracy C0F
(h) Average accuracy MNIST_P
038
10
11
11
Du't
GCM
TULE moddl
EWC
Angle
Indeperdent
(c) Average accuracy on MNIST M
(d) Accuracy of the first task on CIFAR100
5.
335
245
235
86
235
Task
Task
(e) Accuracy of the first task on MNIST.P
(1) Accuracy of the first task on MNIST.M
Figure 1: Evolution curves along with more tasks.
Table 3: Average accuracy in different memories
Memories
GEM-cvx
Ours
256
0.8213
0.8351
MNIST_P
128
0.8046
0,8096
64
0.7636
0.7762
256
0.6703
0.6876
CIFAR100
128
0.6489
0.6556
64
0.6193
0.6248
256
0.8446
0.8537
MNIST_M
128
0.8165
0.8309
64
0.8072
0.8169
We can see that using the positive backward transfer separately can give an improvement, but the support examples and soft gradient constraint can not independently improve the performance. How- ever, support examples can give an improvement combined with soft gradient constraint or positive backward transfer, while soft gradient constraint can probably produce a further promotion with the combination to the other two methods. So the combination of support examples and positive back- ward transfer can provide stable improvements, whereas the contribution of soft gradient constraint depends the dataset.
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Table 4: Hyperparameters of different memories
Memories
Support examples
Soft gradient constraint
Positive backward transfer
256
(0.1, 0.8]
00 & #1 per 20 tasks
0.2 & +0.8 per 3 tasks
MNIST
128
[0.15, 0.89]
00 & per 20 tasks
0,2 & +0.8 per 3 tasks
00.15, 0.85)
00 per tasks
+0.8 per 3 tasks
256
(0.2,0.8)
& +0.6 per tasks
0.2 +0.8 per 3 tasks
CIFARI00
128
0..2
& +0.6 per 3 tasks
0.2 & #0.8 per 3 tasks
64
0.22.088
10 & +0.6 per tasks
0.2 & +0.8 per 3 tasks
256
00.1,0.8]
oo & #1 per 20 tasks
0.25 & *0.8 per 5 tasks
MNIST.M
128
0.110.8)
00 & #1 per tasks
0.25 & #0.8 per 5 tasks
64
0.15
& 1 per tasks
0.25 & +0.8 per tasks
0.7
3.4
0.68
3.0
Rapport
385
Rispprt
325
137
(a) CIFARI00
(b) MNIST
Figure 2: Left: Ablation study CIFAR100. Right: Ablation study on MNIST.
CONCLUSION
We notice the fact of the precious space of memories in continual learning and the incremental difficulty through progressive learning. The GEM model is relatively lightweight, computationally fast and robust to yield high accuracy, we propose support examples to make good use of memories, formulate soft gradient constraints to accommodate competing tasks, present cosine similarity to encourage positive backward transfer over previously solved tasks. From experiments we can see the performance improvements of our techniques over the original GEM, obtaining better average accuracy and higher backward transfer. Also our method increase little computational burdens, with the cost of several needing to tune. Therefore we argue that our improvements can promote GEM be applied to more practical applications.
propose three techniques improve GEM. Specifically,
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10

---

Under review as a conference paper at ICLR 2020
Guanxiong Zeng, Yang Chen, Bo Cui, and Shan Yu. Continuous learning of context-dependent processing in neural networks. arXiv preprint arXiv/810.01255. 2018.
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A A RESULTS IN ALL TASKS' AVERAGE ACCURACY AND THE ACCURACY OF THE FIRST TASK
Table 5: Results of all tasks' average accuracy in CIFARI00
0.2074
0.2328
0.2368
0.2629
0.294
0.3248
0.3589
0.3777
0.3744
0,4158
Ours
0.4472
0.4759
0.5014
0.5181
0.5555
0.568
0.581
0.6283
0.668
0.6876
0.2074
0.2284
0.2485
0.2748
0.296
0.325
0.3583
0.3819
0.3924
0,4318
GEM
0.4493
0.4733
0.496
0.5049
0.5419
0,5617
0.5788
0.6224
0.6538
0.6703
0.2111
0.2167
0,2236
0.2342
0.2532
0.274
0.2882
0.3076
0.3123
0.3333
iCARL.
0,3536
0.3606
0.3835
0.4096
0.4332
0.4589
0.4867
0.4805
0,509
0.5359
0.2189
0.2164
0.1968
0.2157
0.2499
0.2486
0.2411
0.2649
0,2936
0.2964
EWC
0,3134
0.2716
0.3318
0.3346
0.3424
0.3718
0.3968
0.4007
0.4314
0,4657
0.2189
0.2152
0,2183
0.2343
0.24
0.2503
0.2449
0.2782
0.2827
0.2953
single
0,3243
0.2816
0,3382
0.3474
0.377
0.3703
0.4043
0.4134
0.4406
0,4666
0.2139
0,2268
0.233
0,2401
0,2595
0.2666
0,2802
0,2894
0,3009
0,3147
independent
0,3356
0.344
0,3599
0.3692
0.38
0.3822
0.397
0.4119
0.4256
0,4415
Table 6: Results of the first task's accuracy in CIFARI00
0.468
0.578
0.55
0.552
0.528
0.558
0.512
0.516
0.508
0.516
Ours
0.544
0,548
0.532
0.544
0.554
0.51
0.464
0,502
0,548
0.468
0.622
0.584
0.56
0.502
0,536
0.544
0.61
0.558
0.598
GEM
0.614
0.62
0.608
0.626
0.62
0.622
0.644
0.572
0.61
0.426
0.468
0.378
0.412
0.414
0.408
0.432
0.43
0.488
0.462
iCARL.
0.436
0.398
0.51
0.518
0.528
0.526
0.528
0.514
0.518
0.416
0.31
0.282
0.232
0.256
0.252
0.24
0.23
0.28
0.29
EWC
0.26
0.32
0.246
0.208
0.198
0.222
0.228
0.218
0.208
0.416
0.302
0.274
0.26
0.286
0.292
0.238
0.276
0,214
0.238
single
0.29
0.216
0.308
0.214
0.23
0.19
0.212
0.222
0.214
0.482
0,482
0.482
0.482
0.482
0.482
0.482
0.482
0,482
0.482
independent
0.482
0.482
0.482
0.482
0.482
0,482
0.482
0.482
0.482
Table 7: Results of all tasks' average accuracy in MNIST Permutations
0.1364
0.1733
0.2165
0.2419
0.2754
0.3197
0.3612
0.3971
0.4302
0.468
Ours
0.5052
0.5396
0.5848
0.6189
0.6557
0.6948
0.7226
0.7645
0.7856
0.8331
0.14
0.1741
0.2147
0.2315
0.2772
0.3187
0.3579
0.3956
0.4157
0.4625
GEM
0.5
0.5323
0.5783
0.6082
0.6498
0.6837
0.7243
0.7567
0.7844
0.8276
0.1256
0.1648
0.21
0.2394
0.2667
0.3081
0.3471
0.3765
0.3958
0.4242
multimodal
0.4382
0.4831
0.5239
0,5373
0.5491
0.6017
0.6358
0.6605
0.6974
0.7158
0.1364
0.1704
0.205
0.2271
0.2689
0.2954
0.3355
0.3495
0.3576
0.3646
EWC
0.4177
0.4234
0.4646
0.4633
0.4843
0.5057
0.526
0.5874
0.5932
0.6213
0.1364
0.171
0.1959
0.2164
0.2599
0.2861
0.3235
0.3518
0.3732
0.3878
single
0.4238
0.4399
0.4767
0.4838
0.493
0.52
0.5089
0.5283
0.534
0,5742
0.1094
0.1176
0.1267
0.1374
0.1502
0.1618
0.1735
0.1873
0.2037
0,2178
independent
0.2317
0.2478
0,2631
0.276
0.2912
0.3083
0.3236
0,341
0.3547
0.37
11

---

Under review as a conference paper at ICLR 2020
Table 8: Results of the first task's ' accuracy in MNIST Permutations
0,691
0.8559
0.8527
0.8109
0.8167
0.8331
0.8321
0.8314
0.8159
0.82
Ours
0.812
0.8036
0.8154
0.8183
0.8179
0.825
0.8161
0.8194
0.8178
0.691
0.831
0.8246
0.7178
0.7946
0.8164
0.8035
0.8226
0.8016
0.8188
GEM
0.8194
0.7867
0.8159
0.7991
0.8116
0.8176
0.8176
0.8086
0.797
0.6274
0.7109
0.7138
0.7227
0.6278
0.6846
0.6899
0.6795
0.688
0.7
multimodal
0.6155
0.6612
0.6409
0.6361
0.6192
0.6525
0.6228
0,5679
0,5712
0.691
0.7576
0.6677
0.6616
6544
0.673
0.7021
0.6164
0.5332
0.4895
EWC
0,5609
0,5634
0,5655
0.5151
0,5049
0,5332
0.5381
0.5394
0,5001
7.767
0.7699
0.6753
0.5782
0.6531
0.6459
0.6201
0.5827
0.5457
0.4566
single
0.4455
0,4732
0,5009
0,4416
0.4323
0.4465
0.4085
0.3767
0,3865
0.1889
0,1889
0.1889
0.1889
0.1889
0.1889
0.1889
0.1889
0,1889
0.1889
independent
0.1889
0.1889
0,1889
0.1889
0,1889
0.1889
0.1889
0.1889
0.1889
Table 9: Results of all tasks' average accuracy in MNIST Mix
0.691
0,8502
0,8584
0,8348
0.8368
0.8532
0.8401
0.8321
0.8403
0.8339
Ours
0.8172
0.8214
0.8134
0.8145
0.8169
0.8184
0.8107
0.8147
0,8172
0.691
0.8019
0.8256
0.828
0.7953
0.8045
0.8019
0.7881
0.8212
0.8073
GEM
0.7886
0.7949
0.8086
0.783
0.8027
0.7938
0.8001
0.801
0.7991
0,6274
0,6871
0.6865
0.6638
0.6638
0.7225
0.7349
0.7214
0.7379
0.6676
multimodal
0.6347
0.6224
0.5992
0.6664
0.6933
0.7262
0.736
0.7064
0.6864
0.691
0.7205
0.7356
0.6811
0.6823
0.6718
0.6831
0.4955
0,6263
0,6342
EWC
0.6134
0.5594
0.5966
0.6658
0.5849
0.5676
0.5547
0.541
0.5528
0.7062
0.7431
0.749
0.6757
0.7315
0.6863
0.6796
0.4702
0.6201
0.5759
single
0.4812
0.5436
0.4643
0.4981
0.4607
0.4785
0.4318
0.4134
0.4387
0.3213
0.3213
0.3213
0.3213
0.3213
0.3213
0.3213
0.3213
0.3213
0.3213
independent
0.3213
0.3213
0.3213
0.3213
0,3213
0.3213
0.3213
0,3213
0.3213
Table 10: Results of the first task's accuracy in MNIST Mix
0.1116
0.342
0.3809
0.4658
0.5388
0.5392
0.589
0.5948
0.6523
0.6481
Ours
0.6897
0.6811
0.7415
0.7238
0.781
0.7725
0.8185
0.8082
0.8593
0.8537
0.1116
0.3403
0.3789
0.4749
0.5228
0.4995
0.5695
0.5829
0.6407
0.6513
GEM
0.6765
0,6873
0.738
0.7303
0.7771
0.7771
0.8145
0.8126
0.8493
0.8446
0.1272
0.1673
0.2084
0.2371
0.2672
0.3094
0,3419
0,3802
0,4011
0.4351
multimodal
0.4688
0.4966
0.5328
0.5598
0.5992
0.6338
0.6651
0.7043
0.7336
0.7656
0.1116
0.3362
0.3688
0.41
0.4221
0.4449
0.4966
0.4494
0.4835
0.5506
EWC
0.5877
0.5494
0.6503
0.5714
0.6278
0.663
0.685
0.6992
0.7335
0.7285
0.1124
0,3329
0.3649
0.4168
0,4317
0.4572
0.5002
0.4642
0.4651
0.5531
single
0.5736
0.5868
0.6336
0.5857
0.6136
0.6509
0.6388
0.6912
0.7098
0.7206
0.1166
0.1302
0.1465
0.1631
0.1829
0.2068
0.2322
0.2534
0.2752
0.3011
independent
0.3246
0.3498
0.3757
0.3967
0.4191
0.4443
0.465
0.4904
0.5041
0.5275
12

---

Under review as a conference paper at ICLR 2020
B В ACCURACY IN ABLATION
Table II: Average accuracy on MNIST Permutations
Metrics
GEM-cvx 0.6703
transfer
soft
support
transfer &
Accuracy
0.6718
0.6536
0.6535
0.6747
support & soft
support & transfer
0.6564
0.6849
0.6876
13


Reviews:
[{"review_id": "SkeYd-i2FB", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper presents three improvements to the previous GEM algorithm: choosing support examples better (the GEM paper used a random set), incorporating soft gradient constraints, and specifying the magnitude of the dot product in the gradient optimisation problem (in order to increase positive backward transfer). They then compare the original GEM algorithm (and other baselines) with their algorithm on two datasets, consider the case where there is less memory available, and provide ablation studies for their three improvements.\n\nI recommend to reject this paper. I am struggling to see the improvements in the results, as each algorithm is only run once on each dataset, and all the numbers seem very close together (GEM vs new algorithm). There are also a significant number of hyperparameters added by this paper: how are these tuned? I also do not understand the reasoning behind the third idea ('positive backward transfer'). I do, however, like the other two improvements and see the reasoning behind them, however I still have some misgivings. I will now elaborate on these points.\n\nFirstly, there are no standard deviations provided for the experiments. I believe this is especially important to do for the original GEM algorithm and the proposed algorithm because the two provide extremely similar metric values. Comparing values from the original GEM paper and this paper's run of GEM indicates to me that there is a significant chance that any improvements could be within error bars. For example, on CIFAR100, the authors' run of GEM gives 66.48%, the original GEM paper reports 67.83%, and the authors' method is 68.76%.\n\nSecondly, this paper introduces a significant number of hyperparameters over the original GEM algorithm. How are these tuned? Is there a validation set? Does having to tune these hyperparameters slow down computation significantly (how much exactly)? It would be nice to see how much more computationally expensive the new method is compared with GEM: the authors claim \"little computational burdens\".\n\nI like the soft gradient constraint idea introduced in this paper. It is more principled than the hack that the original GEM paper used. The method of choosing the support set also makes sense to me, however there are many hyperparameters in this idea. Additionally, the results (Table 3) for different memory sizes are confusing. It seems like smaller memory sizes lead to less improvement over random memory. Surely cleverly choosing memory should be more beneficial when constrained to smaller memory? This indicates to me that improvements need to still be made to the idea. I also do not follow the explanation given in Section 4.2.1 for the positive backward transfer improvement. Why should the magnitude of the inner products be specified? What do you mean by \"cosine similarity in magnitude\"?\n\nAs a final comment, I will say that there are many typos in this paper.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}, {"review_id": "HJxngHKiKB", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes an extension to gradient episodic memory (GEM) to improve its performance and backwards transfer. Specifically, the proposed method selects \"support examples\" to represent each task (versus the last M examples for GEM); introduces slack variables to ensure the constraints imposed by GEM are not too restrictive; and uses cosine similarity between sample gradients to encourage backwards transfer.\n\nThe ideas proposed are interesting, and the paper is easy to follow.\n\nHowever, I have a number of concerns which I think, unfortunately, preclude publication at this point.\n\nPrimarily, while I believe the ideas are intuitive and novel, the experimental evaluation does not appear to support the claim that the proposed method significantly improves GEM (only a minor improvement is shown at best). This is further compounded by the fact that variances over multiple seeds/runs are not reported, which makes it difficult to gauge any statistically significant performance improvement.\n\nSome claims in the paper also need to be tempered:\n- The abstract suggests performance improves \"remarkably\", but experiments do not support this.\n- The last paragraph of Section 4.2 declares that adding a offset to the Lagrangian, as suggested by the GEM authors, \"is an ad-hoc practice and lacks rationality: it is not faithfully follow their mathematical formulation\". This may come across as confrontational, and I don't feel the point is valid in this case, given that the soft gradient constraints introduce a similar trade-off parameter.\n\nThe language is also imprecise and conversational at times; I would suggest another read-through and careful rewording for clarity. For example, in 4.2.2, \"this new constraint imposes cosine similarity actually...\".\n\nLastly, I'm not sure about this, but does the link to the GitHub repository break anonymity? Given the language difference, I'm not sure what the url refers to; but wonder if it could be a name.\n\nThe ideas have potential, and I suggest the authors explore avenues to further improve the efficacy of the method and expand the experimentation, and improve some of the writing in terms of language and claims made.\n\n(A note on the score: I don't think this is a 1 under the old ICLR system, but unfortunately have to recommend it be rejected in its current state)", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}, {"review_id": "Bylpt08hYS", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Paper proposes three improvements upon the gradient episodic memory (GEM) method [Lopez-Paz,2017]. The three improvements address 1. A way of selecting which exemplars to store (originally random), 2. The strictness of the constraints is loosened by considering slack vectors, 3. The update is improved by promoting positive backward transfer, not only limiting the gradient to the constraints imposed by exemplars of previous tasks, but aiming to improve also for previous tasks by preferring gradients which have high cosine similarity with the gradients on exemplars. Results on CIFAR 100 and MNIST permuted are presented. Noteworthy is the improved backward transfer on CIFAR 100 when compared with GEM.\n\nConclusion: The proposed improvements seem sensible. I liked especially the one which aims at improved backward transfer. However, the paper should have built upon the more recent A-GEM paper. Also, the proposals show strange behavior in the ablation study, and I am not convinced they all contribute to better performance. Finally, the gain with respect to GEM is very small. I, therefore, recommend a weak reject.\n\n1.       The authors somehow missed the more recent paper ‘EFFICIENT LIFELONG LEARNING WITH A-GEM’. Their analysis should compare with this paper. This new paper does not address the points addressed in this paper, but even so, since it obtains generally slightly better results but is much more efficient, it is a better starting point. The relaxing of the constraints (with slacking vectors) might be less efficient in A-GEM which already relaxes the constraints.\n2.       Improved exemplar sampling: The authors say ‘we cannot afford to assign memory to the examples whose margins are negative’, why not ?  I would like to see this ablated.\n3.       Why does the average accuracy start so low ? Are they also averaging over the unseen classes? Does it not make more sense to just average over the seen classes (those considered until the current task)?\n4.       The gain with respect to GEM is very small in Figure 1.\n5.       Could the authors explain their validation protocol (A-GEM also makes a point of that).\n6.       I think there is a typo in the equation, should have a j index on the left side, and should be A_jn-A_nn\n7.        Ablation study should be better written (it takes a while to understand to which of the three proposals the abbreviations link). The results of the ablation do not convincingly show the merits of two of the three proposals. These only work when combined with ‘threshold’. No reason/explanation is provided for this strange behavior. (error in the legend the support+soft should be green).", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes an extension of Gradient Episodic Memory (GEM) namely support examples, soft gradient constraints, and positive backward transfer. The authors argue that experiments on MNIST and CIFAR show that the proposed method consistently improves over the original GEM.

All three reviewers are not convinced with experiments in the paper. R1 and R3 mentioned that the improvements over GEM appear to be small. R2 and R3 also have some concerns without results with multiple runs. R3 has questions about hyperparameter tuning. The authors also appears to be missing recent developments in this area (e.g., A-GEM). The authors did not provide a rebuttal to these concerns.

I agree with the reviewers and recommend rejecting this paper.
Decision: Reject