Title: PROTOTYPE RECALLS FOR CONTINUAL LEARNING

Abstract: Continual learning is a critical ability of continually acquiring and transferring knowledge without catastrophically forgetting previously learned knowledge. However, enabling continual learning for AI remains a long-standing challenge. In this work, we propose a novel method, Prototype Recalls, that efficiently embeds and recalls previously learnt knowledge to tackle catastrophic forgetting issue. In particular, we consider continual learning in classification tasks. For each classification task, our method learns a metric space containing a set of prototypes where embedding of the samples from the same class cluster around prototypes and class-representative prototypes are separated apart. To alleviate catastrophic forgetting, our method preserves the embedding function from the samples to the previous metric space, through our proposed prototype recalls from previous tasks. Specifically, the recalling process is implemented by replaying a small number of samples from previous tasks and correspondingly matching their embedding to their nearest class-representative prototypes. Compared with recent continual learning methods, our contributions are fourfold: first, our method achieves the best memory retention capability while adapting quickly to new tasks. Second, our method uses metric learning for classification, and does not require adding in new neurons given new object classes. Third, our method is more memory efficient since only class-representative prototypes need to be recalled. Fourth, our method suggests a promising solution for few-shot continual learning. Without tampering with the performance on initial tasks, our method learns novel concepts given a few training examples of each class in new tasks.

1. INTRODUCTION:

Continual learning, also known as lifelong learning, is the crucial ability for humans to continually acquire and transfer new knowledge across their lifespans while retaining previously learnt experiences Hassabis et al. (2017) . This ability is also critical for artificial intelligence (AI) systems to interact with the real world and process continuous streams of information Thrun & Mitchell (1995) . However, the continual acquisition of incrementally available data from non-stationary data distributions generally leads to catastrophic forgetting in the system McCloskey & Cohen (1989) ; Ratcliff (1990); French (1999) . Continual learning remains a long-standing challenge for deep neural network models since these models typically learn representations from stationary batches of training data and tend to fail to retain good performances in previous tasks when data become incrementally available over tasks Kemker et al. (2018) ; Maltoni & Lomonaco (2019) .

Numerous methods for alleviating catastrophic forgetting have been currently proposed. The most pragmatical way is to jointly train deep neural network models on both old and new tasks, which however demands a large amount of resources to store previous training data and hinders the learning of novel data in real time. Another option is to complement the training data for each new task with "pseudo-data" of the previous tasks Shin et al. (2017) ; Robins (1995) . Besides the main model for task performance, a separate generative model is trained to generate fake historical data used for pseudo-rehearsal. Deep Generative Replay (DGR) Shin et al. (2017) replaces the storage of the previous training data with a Generative Adversarial Network to synthesize training data on all previously learnt tasks. These generative approaches have succeeded over very simple and artificial inputs but they cannot tackle more complicated inputs Atkinson et al. (2018) . Moreover, to synthesize the historical data reasonably well, the size of the generative model is usually huge that costs much memory Wen et al. (2018) . An alternative method is to store the weights of the model trained on previous tasks, and impose constraints of weight updates on new tasks He & Jaeger (2018) ; Kirkpatrick et al. (2017) ; Zenke et al. (2017) ; Lee et al. (2017) ; Lopez-Paz et al. (2017) . For example, Elastic Weight Consolidation (EWC) Kirkpatrick et al. (2017) and Learning Without Forgetting (LwF) Li & Hoiem (2018) store all the model parameters on previously learnt tasks, estimate their importance on previous tasks and penalize future changes to the weights on new tasks. However, selecting the "important" parameters for previous tasks complicates the implementation by exhaustive hyper-parameter tuning. In addition, state-of-the-art neural network models often involve millions of parameters and storing all network parameters from previous tasks does not necessarily reduce the memory cost Wen et al. (2018) . In contrast with these methods, storing a small subset of examples from previous tasks and replaying the "exact subset" substantially boost performance Kemker & Kanan (2017); Rebuffi et al. (2017); Nguyen et al. (2017) . To achieve the desired network behavior on previous tasks, incremental Classifier and Representation Learner (iCARL) Rebuffi et al. (2017) and Few-shot Self-Reminder (FSR) Wen et al. (2018) follow the idea of logit matching or knowledge distillation in model compression Ba & Caruana (2014) ; Bucilua et al. (2006) ; Hinton et al. (2015) . However, such approaches ignore the topological relations among clusters in the embedding space and rely too much on a small amount of individual data, which may result in overfitting as shown in our experiments (Section 4.2). In contrast with them, without tampering the performance in memory retention, our method learns embedding functions and compares the feature similarities represented by class prototypes in the embedding space which improves generalization, especially in the few-shot settings, as also been verified in works Hoffer & Ailon (2015) ; Snell et al. (2017) .

In this paper, we propose the method, Prototype Recalls, for continual learning in classification tasks. Similar as Snell et al. (2017) , we use a neural network to learn class-representative prototypes in an embedding space and classify embedded test data by finding their nearest class prototype. To tackle the problem of catastrophic forgetting, we impose additional constraints on the network by classifying the embedded test data based on prototypes from previous tasks, which promotes the preservation of initial embedding function. For example (Figure 1 ), in the first task (Subfigure 1a), the network learns color prototypes to classify blue and yellow circles and in the second task (Subfigure 1b), the network learns shape prototypes to classify green circles and triangles. With catastrophically forgetting color features, the network extracts circle features on the first task and fails to classify blue and yellow circles. To alleviate catastrophic forgetting, our method replays the embeded previous samples (blue and yellow circles) and match them with previous color prototypes (blue and yellow) which reminds the network of extracting both color and shape features in both classification tasks.

We evaluate our method under two typical experimental protocols, incremental domain and incremental class, for continual learning across three benchmark datasets, MNIST Deng (2012) , CIFAR10 Krizhevsky & Hinton (2009) and miniImageNet Deng et al. (2009) . Compared with the state-of-the-arts, our method significantly boosts the performance of continual learning in terms of memory retention capability while being able to adapt to new tasks. Unlike parameter regularization methods or iCARL or FSR, our approach further reduces the memory storage by replacing logits of each data or network parameters with one prototype of each class in the episodic memory. Moreover, in contrast to these methods where the last layer in traditional classification networks often structurally depends on the number of classes, our method leverages on metric learning, maintains the same network architecture and does not require adding new neurons or layers for new object classes. Additionally, without sacrificing classification accuracy on initial tasks, our method can generalize to learn new concepts given a few training examples in new tasks due to the advantage of metric learning, commonly used in few-shot settings Snell et al. (2017) ; Hoffer & Ailon (2015) .

2. PROPOSED METHOD:

We propose the method, Prototype Recalls, for continual learning. For a sequence of datasets D 1 , D 2 , ..., D t , ..., given D t in any task t where t ∈ {1, 2, ..., T }, the goal for the model f T is to retain the good classification performance on all T datasets after being sequentially trained over T tasks. The value of T is not pre-determined. The model f T with learnable parameters φ is only allowed to carry over a limited amount of information from the previous T -1 tasks. This constraint eliminates the naive solution of combining all previous datasets to form one big training set for fine-tuning the model f T at task T . Each dataset

D t consists of N t labeled examples D t = {X t , Y t } = {(x 1t , y 1t ), ..., (x N t , y N t )} where each x it ∈ R D is the D-dimensional feature 𝑇𝑎𝑠𝑘 = 𝑡 -1 𝑓 𝑡-1 𝑐 1,𝑡-1 𝑐 2,𝑡-1 ? 𝐷 𝑡-1 (a) Classification 𝑇𝑎𝑠𝑘 = 𝑡 -1 𝑇𝑎𝑠𝑘 = 𝑡 𝑓 𝑡-1 𝑐 3,𝑡 𝑐 4,𝑡 𝑓 𝑡 ? 𝑐 1,𝑡-1 𝑐 2,𝑡-1 𝑐 2,𝑡 𝑐 1,𝑡 𝐷 𝑡 𝐷 𝑡-1 (b) Eliminating Catastrophic Forgetting

Figure 1 : Illustration of classification and catastrophic forgetting alleviation in our proposed method. In (a), dataset D t-1 contains two classes. Prototypes c 1,t-1 and c 2,t-1 are obtained by averaging example embeddings of the same class. White square with question mark (embedded test data) can be classified by finding its nearest prototype. In (b), dataset D t containing two new classes is introduced in task t. The prototypes c i,t , i = 1, . . . , 4 on task t are computed from D t Dt-1 where Dt-1 is a randomly sampled subset from D t-1 . To eliminate catastrophic forgetting, our method constantly recalls metric space at task t -1 by making example embeddings on Dt-1 , as denoted by dashed color shapes, close to prototypes c t-1 .

vector of an example and y it ∈ {1, ..., K t } is the corresponding class label. S kt denotes the set of examples labeled with class k t .

At task T , if we simply train a model by only minimizing the classification loss L classi,D T on dataset D T , the model will forget how to perform classification on previous datasets D t , t < T which is described as catastrophic forgetting problem McCloskey & Cohen (1989) ; Ratcliff (1990); French (1999) . Here we show how the model trained in our method retains the good performance on all previous tasks while adaptively learning new tasks. The loss for all the previous datasets is denoted by

L T (f ) = T t=1 E Dt [L(f (X t ), Y t )].

Our objective is to learn f T defined as follows:

f T = arg min f L T (f ) = arg min f { T -1 t=1 L classi,Dt (f t ) + L classi,D T (f ) + T -1 t=1 δ Dt (f, f t )} (1)

where L classi,D T (f ) defines the classification loss of f on dataset D T and δ Dt (f, f t ) measures the differences in the network behaviors in the embedding space learnt by f and f t on D t , as introduced later in Equ 7. Given f 1 , ..., f T -1 that are learnt from the previous tasks, at task T , learning f T requires minimizing both terms L classi,D T (f ) and

T -1 t=1 δ Dt (f, f t ).

In the subsections below and Figure 1 , we describe how to optimize these two terms.

2.1. CLASSIFICATION:

To perform classification on dataset D t , our method learns an embedding space in which points cluster around a single prototype representation for each class and classification is performed by finding the nearest class prototype Snell et al. (2017) (Figure 1a ). Compared to traditional classification networks with a specific classification layer attached in the end, such as iCARL and FSR, our method keeps the network architecture unchanged while finding the nearest neighbour in the embedding space, which would lead to more efficient memory usage. For example, in one of the continual learning protocols Snell et al. (2017) where the models are asked to classify incremental classes (also see Section 3.1), traditional classification networks have to expand their architectures by accommodating more output units in the last classification layer based on the number of incremental classes and consequently, additional network parameters have to be added into the memory.

Without loss of generality, here we show how our method performs classification on D T . First, the model learns an embedding function f : R D → R M and computes an M -dimensional prototype c kT ∈ R M which is the mean of the embeddings from examples S kT :

c kT = 1 |S kT | (x iT ,y iT )∈S kT f (x iT ).

(2)

The pairwise distance of one embedding and one prototype within the same class should be smaller than the intra-class ones. Our method introduces a distance function d : R M × R M → [0, ∞).

For each example x T , it estimates a distance distribution based on a softmax over distances to the prototypes of K T classes in the embedding space:

p φ (y T = k T |x T ) = exp(-d(f (x T ), c kT )) K T k exp(-d(f (x T ), c k T ))

.

(3)

The objective function L classi,D T (f ) is to minimize the negative log-probability -log p φ (y T = k T |x T ) of the ground truth class label k T via Stochastic Gradient Descent Bottou (2010) :

EQUATION

) In practice, when N T is large, computing c kT is costly and memory inefficient during training. Thus, at each training iteration, we randomly sample two complement subsets from S kT over all K T classes: one for computing prototypes and the other for estimating distance distribution. Our primary choice of the distance function d(•) is squared Euclidean distance which has been verified to be effective in Snell et al. (2017) . In addition, we include temperature hyperparameter τ in d(•) as introduced in network distillation literature Hinton et al. (2015) and set its value empirically based on the validation sets. A higher value for τ produces a softer probability distribution over classes.

2.2. PROTOTYPE RECALL:

Regardless of the changes of the network parameters from φ t to φ T at task t and T respectively, the primary goal of f T is to learn the embedding function which results in the similar metric space as f t on dataset D t in task t (Figure 1b ). Given a limited amount of memory, a direct approach is to randomly sample a small subset Dt = {(x Intuitively, if the number of data samples in Dt is very large, the network could re-produce the metric space at task t by replaying Dt , which is our desired goal. However, this does not hold in practice given limited memory capacity. With the simple inductive bias that the metric space at task t can be underlined by class-representative prototypes, we introduce another loss that embedded data sample in Dt should still be closest to their corresponding class prototype among all prototypes at task t. This ensures the metric space represented by a set of prototypes learnt from Dt by f T provides good approximation to the one in task t.

(t) i , y (t) i )|i = 1, ..

Formally, for any f after task t, we formulate the regularization of network behaviors δ Dt (f, f t ) in the metric space of task t by satisfying two criteria: first, f learns a metric space to classify Dt by minimizing the classification loss L classi, Dt (f ), as introduced in Sec. 2.1 above; second, to preserve the similar topological structure L regu, Dt,Dt (f, f t ) among clusters on dataset D t , the embeddings f ( xt ) predicted by f based on Dt should produce the similar distance distribution based on a softmax over the distance to prototypes c kt computed using f t on dataset D t :

p φ ( ỹt = kt | xt ) = exp(-d(f ( xt ), c kt )) Kt k exp(-d(f ( xt ), c k t )) , c kt = 1 |S kt | (xit,yit)∈S kt f t (x it ).

(5)

Concretely, L regu, Dt,Dt (f, f t ) is to minimize the negative log-probability p φ ( ỹt = kt | xt ) of the ground truth class label kt conditioned on prototypes c kt , which is pre-computed using f t in Eq 5 at task t and stored in the episodic memory until task T :

EQUATION

Overall, we define δ Dt (f, f t ) in Eq 1 as below: 

EQUATION

2.3. DYNAMIC EPISODIC MEMORY ALLOCATION:

Given a limited amount of memory with capacity C, our proposed method has to store a small subset Dt with m examples randomly sampled from D t and prototypes c kt , k ∈ {1, ..., K t } computed using embedding function f t : R D → R M on D t where t < T . The following constraint has to be satisfied:

EQUATION

When the number of tasks T is small, m can be large and the episodic memory stores more examples in Dt . Dynamic memory allocation of enabling more example replays in earlier tasks puts more emphasis on reviewing earlier tasks which are easier to forget, and introduces more varieties in data distributions when matching with prototypes. Pseudocode to our proposed algorithm in continual learning for a training episode is provided in Algorithm 1.

3. EXPERIMENTAL DETAILS:

We introduce two task protocols for evaluating continual learning algorithms with different memory usage over three benchmark datasets. Source codes will be public available upon acceptance.

3.1. TASK PROTOCOLS:

Permuted MNIST in incremental domain task is a benchmark task protocol in continual learning Lee et al. (2017) 2a ). In each task, a fixed permutation sequence is randomly generated and is applied to input images in MNIST Deng (2012) . Though the input distribution always changes across tasks, models are trained to classify 10 digits in each task and the model structure is always the same. There are 20 tasks in total. During testing, the task identity is not available to models. The models have to classify input images into 1 out of 10 digits.

Split CIFAR10 and split MiniImageNet in incremental class task is a more challenging task protocol where models need to infer the task identity and meanwhile solve each task. The input data is also more complex which includes classification on natural images in CIFAR10 Krizhevsky & Hinton (2009) and miniImageNet Deng et al. (2009) . The former contains 10 classes and the latter consists of 100 classes. In CIFAR10, the model is first trained with 2 classes and later with 1 more class in each subsequent task. There are 9 tasks in total and 5,000 images per class in the training set. In miniImageNet, models are trained with 10 classes in each task. There are 10 tasks in total and 480 images per class in the training set.

Few-shot Continual Learning Humans can learn novel concepts given a few examples without sacrificing classification accuracy on initial tasks Gidaris & Komodakis (2018) . However, typical continual learning schemes assume that a large amount of training data over all tasks is always available for fine-tuning networks to adapt to new data distributions, which does not always hold in practice. We revise task protocols to more challenging ones: networks are trained with a few examples per class in sequential tasks except for the first task. For example, on CIFAR10/miniImageNet, we train the models with 5,000/480 example images per class in the first task and 50/100 images per class in subsequent tasks.

3.2. BASELINES:

We include the following categories of continual learning methods for comparing with our method.

To eliminate the effect of network structures in performance, we introduce control conditions with the same architecture complexity for all the methods in the same task across all the experiments.

Parameter Regularization Methods: Elastic Weight Consolidation (EWC) Kirkpatrick et al. (2017) , Synaptic Intelligence (SI) Zenke et al. (2017) and Memory Aware Synapses (MAS) Aljundi et al. (2018) where regularization terms are added in the loss function; online EWC Schwarz et al. (2018) which is an extension of EWC in scalability to a large number of tasks; L2 distance indicating parameter changes between tasks is added in the loss Kirkpatrick et al. (2017) ; SGD, which is a naive baseline without any regularization terms, is optimized with Stochastic Gradient Descent Bottou (2010) sequentially over all tasks.

Memory Distillation and Replay Methods: incremental Classifier and Representation Learner (iCARL) Rebuffi et al. ( 2017) and Few-shot Self-Reminder (FSR) Wen et al. (2018) propose to regularize network behaviors by exact pseudo replay. Specifically, in FSR, there are two variants: FSR-KLD for logits matching via KullbackLeibler Divergence loss and FSR-MSE for logits distillation via L2 distance loss.

Performance is reported in terms of both mean and standard deviation after 10 runs per protocol. Since generative model-based approaches van de Ven & Tolias (2018); Shin et al. (2017) greatly alter architecture of the classification networks, we do not compare with them.

3.3. MEMORY COMPARISON:

For fair comparison, we use the same feed-forward architecture for all the methods and allocate a comparable amount of memory as EWC Kirkpatrick et al. (2017) and other parameter regularization methods, for storing example images per class and their prototypes. In EWC, the model often allocates a memory size twice as the number of network parameters for computing Fisher information matrix which can be used for regularizing changes of network parameters Kirkpatrick et al. (2017) . In more challenging classification tasks, the network size tends to be larger and hence, these methods require much more memory. In Table 1, we show an example of memory allocation on split CIFAR10 in incremental class tasks with full memory and little memory respectively. The feed-forward classification network contains around 16.3 × 10 5 parameters. Weight regularization methods require memory allocation twice as that, which takes about 32.63 × 10 5 parameters. The input RGB images are of size 3 × 32 × 32. Via Equ. 8, our method can allocate episodic memory with full capacity C = 16.3 × 10 5 and calculate m which is equivalent to storing 16.3 × 10 5 /(3 × 32 × 32) = 530 example images per class. In experiments with little training data as described in Section 3.1, we reduce m to 10 example images per class.

4. EXPERIMENTAL RESULTS:



4.1. ALLEVIATING FORGETTING:

Figure 3 reports the results of continual learning methods with full memory under the two task protocols. All compared continual learning methods outperform SGD (cyan) which is a baseline without preventing catastrophic forgetting. Our method (red) achieves the highest average classification accuracy among all the compared methods, including both parameter regularization methods and memory-based methods, with minimum forgetting.

A good continual learning method should not only show good memory retention but also be able to adapt to new tasks. In Figure 3a , although our method (red) performs on par with EWC (brown) and FSR (date) in retaining the classification accuracy on dataset D 1 in the first task along with 20 sequential tasks, the average classification accuracy of our method is far higher than EWC (brown) and FSR (date) as shown in Figure 3b , indicating both of these methods are able to retain good memory but fail to learn new tasks. After the 13th task, the average classification performance of EWC is even worse than SGD. Across total 20 tasks, our method leads FSR (date) by 3% more accurate on average. Similar reasoning can be applied to comparison with SI (green): although our method performs comparably well as SI in terms of average classification accuracy, SI fails to retain the classification accuracy on D 1 , which is 6% lower than ours in the 20th task.

Figure 3c and 3d show the average task classification accuracy over sequential tasks in incremental class protocol. Incremental class protocol is more challenging than incremental domain protocol, since the models have to infer both the task identity and class labels in the task. Our method (red) performs slightly better than iCARL (date) and has the hightest average classification accuracy in continual learning. Compared with third best method, FSR (green), our method yields constantly around 5% higher on average across all tasks on CIFAR10 and miniImageNet respectively. Note that most weight regularization methods, such as EWC (brown), perform as badly as SGD. It is possible that EWC computes Fisher matrix to maintain local information and does not consider the scenarios when data distributions across tasks are too far apart. On the contrary, our method maintains remarkably better performance than EWC, because ours focuses primarily on the behaviors of network outputs, which indirectly relaxes the constraint about the change of network parameters.

4.2. FEW-SHOT CONTINUAL LEARNING:

We evaluate continual learning methods with little memory under two task protocols with few training data in the second tasks and onwards except for the first tasks. Figure 4 reports their performance. Our method (red) has the highest average classification accuracy over all sequential tasks among state-of-the-art methods with , which is the second best, at the 9th and 10th tasks on CIFAR10 and miniImageNet respectively. Weight regularization methods, such as EWConline (blue) and MAS (brown), perform as badly as SGD (cyan), worse than logits matching methods, such as FSR (green and yellow) or iCARL (purple). Similar observations have been made as Figure 3 with full training data.

Compared with logits matching methods, our method has the highest average task classification accuracy. It reveals that our method performs classification via metric learning in an effective few-shot manner. It is also because our network architecture is not dependent on the number of output classes and the knowledge in previous tasks can be well preserved and transferred to new tasks. It is superior to traditional networks with new parameters added in the last classification layer, which easily leads to overfitting. As a side benefit, given the same number of example inputs in the episodic memory, our method is more efficient in memory usage since it stores one prototype per class instead of the logits for each example input as verified in Table 1 .

4.3. NETWORK ANALYSIS:

We also study the effects of the following three factors upon performance improvement. Figure 5 reports the average classification accuracy of these ablated methods. (1) Intuitively, limited memory capacity restricts number of example inputs to re-play and leads to performance drop. On permuted MNIST in incremental domain, with full memory capacity reduced by 2.5 times (from 5,000 example inputs to 2,000), our method shows a moderate decrease of average classification accuracy by 1% in the 20th task. (2) We also compare our method with memory replay optimized by cross-entropy loss at full memory conditions. A performance drop around 1.5% is observed which validates classifying example inputs based on initial prototypes results in better performance in memory retention. (3) Given fixed C, our method adopts the strategy of decreasing m numbers of example inputs in memory, with the increasing number of tasks. The performance drop of 1.5% using uniform memory allocation demonstrates the usefulness of dynamic memory allocation which enforces more examples to be replayed in earlier tasks, and therefore promotes memory retention.

In Figure 6 , we provide visualizations of class embeddings by projecting these latent representations of classes into 2D space. It can be seen that our method is capable of clustering latent representations belonging to the same class and meanwhile accommodating new class embeddings across sequential tasks. Interestingly, the clusters are topologically organized based on feature similarities among classes and the topological structure from the same classes is preserved across tasks. For example, the cluster of "bird" (black) is close to that of "plane" (orange) in Task 3 and the same two clusters are still close in Task 9. This again validates that classifying example inputs from previous tasks based on initial prototypes promotes preservation of topological structure in the initial metric space.

5. CONCLUSION:

We address the problem of catastrophic forgetting by proposing prototype recalls in classification tasks. In addition to significantly alleviating catastrophic forgetting on benchmark datasets, our method is superior to others in terms of making the memory usage efficient, and being generalizable to learning novel concepts given only a few training examples in new tasks.

However, given a finite memory capacity and a high number of tasks, we recognize that our method, just like other memory-based continual learning algorithms, have limitations in number of prototypes stored. The memory requirement of our method increases linearly with the number of continuous tasks. In practice, there is always a trade-off between memory usage and retention. We believe that our method is one of the most efficient continual learning methods in eliminating catastrophic forgetting with a decent amount of memory usage. Moreover, we restrict ourselves in classification tasks with discrete prototypes. In the future work, to apply our algorithm in more complex and challenging problems, such as regression and reinforcement learning (RL), one possible solution is to quantize the continuous space in regression or formulate RL in discrete state-action pairs.

:



Figures and Tables:

., m} from D t and replay these examples on task T . There have been some attempts Chen et al. (2012); Koh & Liang (2017); Brahma & Othon (2018) selecting representative examples for Dt based on different scoring functions. However, the recent work Wen et al. (2018) has shown that random sampling uniformly across classes has already yielded outstanding performance in continual learning tasks. Hence, we adopt the same random sampling strategy to form Dt .

Figure 2: Schematics of two task protocols in our continual learning experiments: (a) learning with incremental domain on permuted MNIST; and (b) learning with incremental classes on split CIFAR10.

Figure 3: First and average task classification accuracies from 10 random runs on permuted MNIST, split CIFAR10 and split miniImagetNet in incremental domain (ID) and incremental class (IC) tasks.

Figure 4: Few-shot average task classification accuracies on split CIFAR10 and miniImagetNet in incremental class task with few training data in the second tasks and onwards.

Figure 6: 2D visualization of embedding clusters learnt by our method using t-sne Van Der Maaten (2014) on split CIFAR10 in incremental class task. The first task (Task = 1) is a binary classification problem. In subsequent tasks, one new class is added.

; Lopez-Paz et al. (2017); Zenke et al. (2017) (Figure

Table content: <table/>

Peer review (iclr2020):
The proposed method addresses continual learning, by learning a mapping from the input space to an embedding space, and employing a loss that encourages clustering the embeddings by class (and task?) around some centroids called prototypes. Catastrophic forgetting is mitigated by adding a penalty term that is proportional to the distance of the embeddings under the current network of some samples from the past tasks, and the centroids previously associated to each of them. The idea might have some merit, but the paper in its current state fails to convey it clearly and is too vague on the experimental settings for the results to be considered in any way. The notation is at times unnecessary heavy (see specific comments below), and many details on the method and experimental setup are missing to the point that it’s difficult to judge it and it would be impossible to reproduce it. In addition, it’s unclear whether the baselines have been properly tuned since the reported performance is incompatible with that of the original paper. The submitted paper is clearly not ready for publication. I can consider modifying my vote if the paper undergoes a major review, fixing the notation, explaining the method and the experimental settings extensively and clearly, with proper comparison in terms of capacity/performance tradeoff w.r.t. the state of the art. Detailed feedback: 1) Notation - I believe the exposition would be more fluent and easier to follow if you described the method under the assumption that all tasks have the same number of classes and a shared dataset, and mentioning that the method can be easily extended to work in the more general setting. Under this assumption, e.g., c_{kT} would simplify to c_{k}. - When you refer to a generic task, please use a lower case letter (e.g., t) instead of T (e.g., first paragraph of page 3, Eqn. 1, Eqn. 2, ..). Use capital letters for constants only (e.g., use T for the total number of tasks). When you need to refer to two tasks that are in a particular relative ordering. For instance, Eqn. 1 should be the argmin of “sum_{s=1}^{t} [...], with s < t”. Similarly, Sec 2.2 should refer to tasks s and t, rather than t and T (unless T is specifically referring to the last task alone, which doesn’t seem the case here as from what I gather the same should hold for all tasks, i.e., for a specific task t and any previous task s, with s < t.). - If I am not mistaken, L_T is defined just before Eqn. 1 and only used there. It is not very informative and it’s based on some undefined loss L. I would suggest to drop it entirely, and simplify Eqn. 1 accordingly. - Please avoid subscript with commas as much as possible, as they are difficult to parse. E.g., L_{classi, Dt} can become L_{classi_t} (or even L_{c_t} or L_{c}^{(t)}), remaining equally informative. All other “D_t” subscripts can be similarly replaced by “t” throughout the text. Generally speaking, I would recommend the ^{(t)} notation to denote task t (which you already adopted when defining 	ilde{D}_t - I believe Eqn. 1 can be simplified by merging the two summations into a single expression. Also, I think it would be clearer to define the argmin to be over “f in mathcal{F}”, since it is a search over the space of all possible functions. Finally, I also find a bit confusing the use of f and f_t to denote an evolving (currently evaluated) function and a fixed (best function from previous search) one respectively, but I am not sure I have a suggestion to improve this notation. - Please avoid unnecessary repetitions. In a number of cases a big part of a formula is defined inline in the text just before the equation. This doesn’t add to the conversation and makes it unnecessarily hard to parse the text (e.g., inline formulas just before Eqns. 4 and 6. Also prefer a textual description instead of the variable when introducing a definition (e.g., before Eqn. 7, “we define the distance as” or “we define the distance between the functions f and f_t embeddings as”) - Please fix all citations according to the ICLR style guidelines: When the authors or the publication are included in the sentence, the citation should not be in parenthesis (as in ``See citet{Hinton06} for more information.**). Otherwise, the citation should be in parenthesis (as in ``Deep learning shows promise to make progress towards AI~citep{Bengio+chapter2007}.**). For instance, all citations in Sec 2.2 should use citep. 2) To my best understanding classification in the embedding space is performed with some variation of k-means, with k being the number of “prototypes” for the task at hand (i.e., the number of classes for the specific task). If this is correct, referring to k-means in the paper and highlighting the similarities and differences to it, would have been a more effective way to describe your method. 3) The authors claim not to use a final classification, as opposed to competing methods, hence not being subject to memory usage increases when the number of classes increase (sec 2.1). Surprisingly though, they then employ a softmax over the prototypes (Eqn. 3) which to the best of my knowledge would require an incoming layer whose dimensionality would need to grow with the increase of the number of prototypes, (i.e., classes). Please update the text to either correct the claim or explain exactly how your method can drive the softmax in the case of an increasing number of classes without increasing the dimensionality of the layer before the softmax. 4) Similarly, the authors claim that “Unlike parameter regularization methods or iCARL or FSR, our approach further reduces the memory storage by replacing logits of each data or network parameters with one prototype of each class in the episodic memory”. This is not entirely true, as they also need to store a (unspecified) number of samples from all tasks to prevent forgetting. It is unclear that replacing a snapshot of the network parameters is better (in terms of memory consumption) than storing some data from each task, and a proper discussion on this crucial point of the paper is missing. 5) Apart from the losses, all the details on the architecture (type and number of layers, activation functions, ..) and on the hyperparameters are missing. The paper only mentions that the same feedforward architecture as EWC is used, with similar memory usage, but also mentions that unlike all other methods (including EWC I assume), theirs doesn’t need to vary the architecture if the number of tasks is increased. It is then completely unclear to me how this is possible if the architecture is the same. If it’s not, which parts of the network are shared and which are task specific. I assume most of the network to be shared among tasks and only the last layer (that drives the softmax that determines the probabilities of each class (prototype) for the current task) to be task-specific. This details are crucial to the understanding of the method and to be able to reproduce the results. 6) Crucially, the reported Permuted MNIST performance on EWC is much worse than that on the original paper. This suggests that the baselines have not been properly tuned or there are issues in the implementation. 7) Unless I am mistaken, in Sec 3.3 there seems to be a flaw in the calculation of the number of examples that can be stored by the proposed method in order to use approximately the same memory as the baselines. Indeed, the memory allocation for weight regularization is independent of the number of classes and so should be the equivalent capacity allocated for samples retention, i.e., in order to be comparable with EWC (and similar methods), the proposed model should be allowed to store 530 examples in total, rather than per class. 8) I suggest to drop the definitions of the obvious functions, such as the softmax and the NLL (that is even defined twice!), and use that space to improve the description of the model and algorithm instead. 9) Can you clarify what you mean by “The pairwise distance of one embedding and one prototype within the same class should be smaller than the intra-class ones.”? Should it be inter-class? 10) In my opinion the narrative would be clearer backwards: rather than first defining the loss as a sum of three sub-losses, I’d suggest to define the sub-losses first and finally aggregate them. In its current state, Eqn. 1 is not informative until delta_{D_t} is defined, a full page later. 11) Eqns. 3 and 5 are not “distance distributions” but rather “class probabilities distributions”, if I understood correctly. Can you confirm and amend the text? 12) Eqn. 5 is more confusing than informative, I recommend to remove it and replace it with a better textual description. There is no need to redefine c_{kt}, as it is already defined in Eqn. 2 exactly in the same way. Secondly, there is no need to define the softmax over the distances a second time, as it’s already defined in Eqn. 3 and only the arguments change. 13) Re Sec 2.3, to my best understanding there is no “emphasis on reviewing earlier tasks”, as the number of rehearsal samples is the same for all tasks. In other words, 	ilde{D}_t is sampled uniformly across tasks I believe. Is this right? 14) Sec 3.1, “typical continual learning schemes assume that a large amount of training data over all tasks is always available for fine-tuning”. This is not true, and it’s actually usually not the case (see e.g., generative replay approaches, or those that rely on pruning or other forms of architecture modification.) 15) The introduction fails to mention the category of approaches that try to combat forgetting by being smart about the architecture, e.g., Progressive nets, Progress & compress, etc, .. Please amend this. Minor: - Add more space after Figure 1’s caption, so that it’s clear where the caption ends and the text begins. - Eqn. 2, no need to define yiT since it’s not used in the equation. - The sentence at the end of Sec 2.1 (“In practice … estimating the distance distribution” ..) should go after Eqn. 2. Similarly, the sentence “Our primary choice of the distance function” would be better placed right after the definition of the distance function IMO. - The MNIST citation is wrong. Please correct it: LeCun, Yann and Cortes, Corinna. *MNIST handwritten digit database.* (1998) - The introduction is a bit long and a “related work” section is missing. You might want to split the introduction into two chapters. - Do not use D to denote the datasets and the dimensionality of the datasets. - The sentence after Eqn. 1 “learning f_T requires minimizing both terms ..” is incorrect, as it goes against Eqn. 1 in which there are 3 terms rather than 2. - End of page 3: “without loss of generality”...w.r.t. what? The phrase doesn’t seem to fit the context, consider changing it. - End of page 7, the detailed description of Figure 3 would be better placed in the caption of the figure itself. Typos: - Page 2: state-of-the-arts -> state-of-the-art - “There have been some attempts [citations] selecting”, “of” is missing - End of page 4: which is pre-computed -> are - End of page 5: public -> publicly