Cognitively Inspired Learning of Incremental Drifting Concepts
Authors: Mohammad Rostami
Aram Galstyan
Venue: ICLR 2022 Submitted (2022)
Keywords: Complementary Learning Systems
continual learning
Parallel Distributed Processing

Abstract:  Humans continually expand their learned knowledge to new domains and learn new concepts without any interference with past learned experiences. In contrast,   machine learning models perform poorly in a continual learning setting, where input data distribution changes over time. Inspired by the nervous system learning mechanisms, we develop a computational model that enables a deep neural network to learn new concepts and expand its learned knowledge to new  domains incrementally in a continual learning setting. We rely on the Parallel Distributed Processing theory to encode abstract concepts in an embedding space in terms of a multimodal distribution. This embedding space is modeled by  internal data representations in a hidden network layer. We also leverage the Complementary Learning Systems theory to equip the model with a memory mechanism to overcome catastrophic forgetting through implementing pseudo-rehearsal. Our   model   can generate pseudo-data points for experience replay  and  accumulate new experiences to past learned experiences without causing cross-task interference.

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COGNITIVELY INSPIRED LEARNING OF INCREMENTAL DRIFTING CONCEPTS
Anonymous authors Paper under double-blind review
ABSTRACT
Humans continually expand their learned knowledge to new domains and learn new concepts without any interference with past learned experiences. In contrast, machine learning models perform poorly in continual learning setting, where input data distribution changes over time. Inspired by the nervous system learn- ing mechanisms, we develop computational model that enables deep neural network to learn new concepts and expand its learned knowledge to new domains incrementally in continual learning setting. We rely on the Parallel Distributed Processing theory to encode abstract concepts in an embedding space in terms of multimodal distribution. This embedding space is modeled by internal data representations in hidden network layer. We also leverage the Complementary Learning Systems theory to equip the model with a memory mechanism to over- come catastrophic forgetting through implementing pseudo-rehearsal. Our model can generate pseudo-date points for experience replay and accumulate new expe- riences to past learned experiences without causing tross-task interference.
INTRODUCTION
Humans continually abstract concept classes from their input sensory data to build semantic descrip- tions, and then update and expand these concepts as more experiences are accumulated Widmer & Kubat 1996), and use them to express their ideas and communicate with each other Gennari et al. (1989) Lake et al. (2015). For example, "cat" and "dog" are one of the first concept classes that many children learn to identify. Most humans expand these concepts as concept drift occurs, e.g., incorporating many atypical dog breeds into the "dog" concept, and also incrementally learn new concept classes, e.g. "horse" and "sheep." as they acquire more experiences. Although this concept learning procedure occurs continually in humans, continual and incremental learning of concept classes remains a major challenge artificial intelligence (AI). AI models are usually trained on a fixed number of classes and the data distribution is assumed to be stationary during model execution. Hence, when an AI model is trained or updated on sequentially observed tasks with diverse distri- butions or is trained new classes, it tends forget what has been learned before due cross-task interference, known as the phenomenon of catastrophic forgetting in the literature French (1911).
Inspired by the Parallel Distributed Processing (PDP) paradigm McClelland et al. (986);; Mc-
Clelland & Rogers 2003). our goal is to enable deep neural network to learn drifting concept classes Gama al. (014)) incrementally and continually in sequential learning setting. PDP hy- pothesizes that abstract concepts are encoded in higher layers of the nervous system McClelland & Rogers (2003); Saxe et al. (2019). Similarly, and based on behavioral similarities between artificial deep neural networks and the nervous system Morgenstern et al. (2014). we can assume that the data representations in hidden layers of deep network encode semantic concepts with different levels of abstractions. We model these representations similarities between input data points are encoded in terms of geometric distances Jiang & Conrath 1197), i.e., data points that belong to the same concept class are mapped into separable clusters in the embedding space. When a new concept is abstracted, a new distinct cluster should be formed in the embedding space to encode that class. Incremental concepts learning is feasible by track- ing and remembering the representation clusters that are formed in the embedding space and by considering their dynamics more experiences are accumulated in new unexplored domains.
an embedding space in which semantic

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We benefit from the Complementary Learning Systems (CLS) theory McClelland et al. (1995) to mitigate catastrophic forgetting. CLS is based on empirical evidences that suggest experience re- play of recently observed patterns during sleeping and waking periods in the human brain helps to accumulate the new experiences to the past learned experiences without causing interference Mc- Clelland al. (1995); Robins (1995). According to this theory, hippocampus plays the role of short-term memory buffer that stores samples of recent experiences and catastrophic forgetting is prevented by replaying samples from the hippocampal storage to implement pseudo-rchearsal in the neocortex during sleeping periods through enhancing past learned knowledge. Unlike AI memory buffers that store raw input data point, e.g., samples of raw images, hippocampal storage can only store encoded representations after some level of abstraction which suggests a generative nature.
Inspired by the above two theories, we expand a base deep neural classifier with a decoder net- work, which is amended from hidden layer, to form autoencoder with the hidden layer as its bottleneck. The bottleneck is used to model the discriminative embedding space. As result of supervised learning, the embedding space becomes discriminative, i.e. a data cluster is formed for each concept class in the embedding McCCeelanll & Rogers (2003). These clusters can be con- sidered analogous to neocortical representations in the brain, where the learned abstract concepts encoded McClelland et al. (1986). We use a multi-modal distribution to estimate this distribution. We update this parametric distribution to accumulate new experiences past learned experiences consistently. Since our model is generative, we can implement the offline memory replay process in the sleeping brain to prevent catastrophic forgetting McClelland et al. (1995); Rasch & Born (2013). When new task arrives, we draw random samples from the multi-mode modal distribution and feed them into the decoder network to generate representative pseudo-data points. These pseudo-data points are then used to implement pseudo-rehearsal for experience replay Robins (1995). We demonstrate that the neural network can learn drifting conceptsinerementally while mitigating forgetting.
RELATED WORK
The problem of continual learning of incremental drifting concepts lies in the intersection of lifelong learning to encode drifting concept classes and incremental learning to incorporate new concept classes. In lifelong learning setting, the number of classes are usually assumed to be fixed, but the distribution of sequential tasks is non-stationary. In an incremental learning setting, concept classes are learned sequentially while the conditional distribution for each concept class stationary.
the major challenge of continual learning is tackling catastrophic forgetting. Previous works in the literature mainly rely on experience replay [ & Hoiem (2018). The core idea of experience replay is to implement pseudo-rehearsal by replaying representative samples of past tasks along with the current task data to retain the learned distributions. Since storing these samples requires memory buffer, the challenge is selecting the representative samples to meet the buffer size limit, For example, selecting uncommon samples that led to maximum effect past experiences has been found to be effective Schaul et al. (0016). However,
Continual learning:
more tasks are leamed, selecting the effective samples becomes more complex. The alternative approach is to use generative models that behave more similar humans French| (1999). Shin et al. (Shin et al. (2017) use generative adversarial structure to mix the distributions of all tasks. It is also feasible to couple the distributions of all tasks in the bottleneck of an autoencoder. The shared distribution then can be used generate pseudo-samples Rannen et al. (2017). Weight consolidation using structural plasticity Lamprech & LeDoux (2004); Zenke al. (2017); Kirkpatrick et al. (017)) is another approach to approximate experience replay. The idea is to identify important weights that retain knowledge about a task and then consolidate them according to their relative importance for past tasks in the future.
Incremental learning::
forgetting in incremental learningstems from updating the model when new classes are incorporated, rather concept drifts in a fixed number of learned classes. Hence, the goal is to learn new classes such that knowledge about the past learned classes is not overwritten. A simple approach is to expand the base network as new classes are observed. Tree-CNN Roy et al. (2200) proposes a hierarchical structure that grows like tree when new classes are observed. The idea is to group new classes into feature-driven super-classes and find the exact label by limiting the search space. As the network grows, the new data can be used to train the expanded network. Sarwar et al. Sarwar et al. (2019) add new convolutional filters in all layers to learn the new classes through new parameters. The alternative approach is to retain the knowledge about old classes in
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an embedding feature space. Rebuffi et al. Rebuffi al. (2017) proposed iCarl which maps images into a feature space that remains discriminative as more classes are learned incrementally. A fixed memory buffer is used to store exemplar images for each observed class. Each time observed, these images are used to learn a class-level representative vector in the feature space such that the testing images can be classified using nearest neighbor with respect to these vectors.
class is
Contributions: We develop a unified framework that addresses challenges of both incremental learning and lifelong learning for the first time. Our idea is based on tracking and consolidating the multimodal distribution that is formed by the internal data representations of sequential tasks in a base neural network model hidden layers. We model this distribution a Gaussian mixture model (GMM) with time-dependent number of components. Concept drifts are learned by updating the corresponding GMM component for a particular class and new concepts are learned by adding new GMM components. We also make the model generative to implement experience replay. We provide both theoretical and experimental results to justify why our algorithm is effective.
PROBLEM STATEMENT
Consider a learning agent which observes sequence of observed tasks {2(1)7 (2016) and after learning each task moves forward to learn the next task. Each task is a classification problem in particular domain and each class represents concept. The classes for each task can be new unobserved classes, i.e., necessitating incremental learning Rebuft et al. (2017). or drifted
Jt=1 Chen & Liu
forms of the past learned classes, i.e., necessitating lifelong learning Chen & Liu (2016), or poten- tially mixture of both cases. Formally, a task is characterized by a dataset ())) (x(1), Y
where X [aq..... 211 E idxa, and ( g are the data points and one-hot labels, respectively. The goal is to train a time-dependent classifier function (+))(-)= Rd Cc R where ky is the number of classes for the t-th task and is fixed for each task- such that the classifier contin- ually generalizes on the tasks seen so far. The data points x; ~ (xx))(x) are assumed to be drawn i.idd from an unknown task distribution g (x)) Figure 1 visualizes high-level Block-diagram of this continual and dynamic learning procedure. The agent needs to expand its knowledge about all the observed concepts such that it can perform well on all the previous learned domains.
Learning each task in isolation is standard supervised learning problem. After selecting a suitable parameterized family of functions f66) Rdd + R with learnable parameters 0, e.g. a deep neural network with learnable weight paramters 6, we can solve for the optimal parameters using the empirical risk minimization (ERM): g(t) = arg ming ef() = arg min& E, La(fe (()) y y)),, where La(-) is proper loss function. If n4 is large enough, the empirical risk expectation would be good approximation of the real expected risk function e(0) (0) (en) (La (æ), (x))) As a result, if the hase parametric family is rich and complex enough for learning the task function, then the ERM optimal model generalizes well on unseen test samples that are drawn from (x).
For the rest of the paper, we consider the base model fece to be deep neural network with an increasing output size to encode incrementally observed classes. As stated, we rely on the PDP paradigm. Hence, we decompose the deep network into an encoder sub-network (.)+) Rdd - 2 C R with learnable parameter v, e.g., convolutional layers of a CNN, and a classifier sub-network hw (-)* : IR/ + _ with learnable parameters w, e.g., fully connected layers of a CNN, where Z denotes the embedding space in which the concepts will be be formed as separable clusters.
The concepts for each task are known a priori and hence new nodes are added the classifier sub- network output to incorporate the new classes at time I. We use a softmax layer as the last layer of the classifier subnetwork. Hence, we can consider the classifier be maximum a posteriori (MAP) estimator after training. This means that the encoder network transforms the input data distribution into an internal multi-modal distribution with h; modes in the embedding space because the embedding space Z should be concept- disrreiitiieriiinaiivee for good generalization. Each concept class represented by a single mode of this distribution. We use a Gaussian mixture model (GMM) to model and approximate this distribution (see Figure [ middle panel). Catastrophic forgetting is the result of changes in this internal distribution when changes in the input distribution leads to updating the internal distribution heuristically. Our idea is to track changes in the data distribution and update and consolidate the internal distribution such that the acquired knowledge from past experiences overwritten when new experiences encountered and learned in the future.
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The main challenge is to adapt the network f (-) and the standard ERM training loss such that we can track the internal distribution continually and accumulate the new acquired knowledge consistently to the past learned knowledge with minimum interference. For this purpose, we form generative model by amending the base model with a decoder vu 2 IR. with learnable parameters u. This decoder maps back the internal representations to reconstruct the input data point in the in- put space such that the pair (øu, vuu) forms an autoencoder. According to our previous dis- cussion, multi-modal distribution would be formed in the bottleneck of the autoencoder upon learning each task. This distribution en- codes the learned knowledge about the con- cepts that have been learned from past expe- riences so far. If we approximate this distri- bution with GMM, we can generate pseudo- data points that represent the previously learned concepts and use them for pseudo-rehearsal. For this purpose, we can simply draw samples from all modes of the GMM and feed these samples into the decoder subnetwork generate a pseudo-datasel (see Figure D. After learning each task, we can update the GMM estimate such that the new knowledge acquired is accumulated to the past gained knowledge consistenly to avoid interference. By doing this procedure continually, our model is able to learn drifting concepts incre- mentally. Figure visualizes this repetitive procedure in this lifelong learning setting.
Sample Domsins:
FOREEY
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New Domaias arrive sequentially
pitii
4. The new harned
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apdate the internal
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mutihingg its distribution in the embbediling
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1. Pscudo-data puints concepts geaerated and fed into the encuder implement psendo-
off
Supervised
Commolaateed
Pathway
Decoder
rehearsal
C.(X g-(1)
Pseudo-data points
Incremental Concept Learning Process
Figure 1: Block Block-diagram visualization of the pro- posed Incremental Learning System including the learning procedure steps. (Based viewed enlarged on screen and in color. Enlarged version is in- cluded in the Appendix)
4 PROPOSED ALGORITHM
When the first task is learned, there no prior experience and hence learning reduces the following:
min (x(()) y(u) 0,00,
La (a vt + 7Cr
min o,
(1)
where La is the discrimination loss, e.g., cross-entropy loss, L, is the reconstruction loss for the autoencoder, e.g., fz-norm, L. is the combined loss, and y is a trade-off parameter between the two loss terms. When the first task is learned, also any future task, according the PDP hypothesis, a multi-modal distribution (z))
Σ N Z |1j, E,) with k components is formed in the embedding space. We assume that this distribution can be modeled with GMM. Since the labels for the input task data samples are known, we use MAP estimation to recover the GMM parameters (see Appendix for details of this process). Let (z)) denotes the estimated distribution.
As subsequent tasks are learned, the internal distribution should be updated continually accumu- late the new acquired knowledge. Let K2 Kad + kiews where k denotes the number of the previously learned concepts that exist in the current task and khew denotes the number of the new observed classes. Hence, the total number of learned concepts until t = T is k = ET= k Also, let the index set Nf.or = {l,..., kooor denotes an order on the classes C,, with j E Nf... that are observed until = T. Let Ny = N U No = (i,,..., C NN contains the kT indices of the existing concepts in (7)), To update the internal distribution after learning Z(T), the number of distribution modes should be updated to kfar- Additionally, catastrophic forgetting must be mitigated using experience replay. We can draw random samples from the GMM distribu- tion Zj ~ p(T-1) and then pass each sample through the decoder th (z,)) to generate pseudo-des data points for pseudo-reheamee rehearsal. Since each particular concept is represented by exactly one mode of the
internal GMM distribution, the corresponding pseudo-label labels for the generated pseudo-data points are known. Moreover, the confidence levels for these labels are also known from the classifier softmax
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layer. To generate clean pseudo-datasers points for which the model confidence level is more than т. We also generate a balanced pseudo- dataset with respect to the learned classes. Doing so, ensure suitability of GMM with k components to estimate the empirical distribution accurately after learning the subsequent tasks.
set threshold and only pick the pseudo-date
Let D(t) = (eb (1),, Y denotes the pseudo-dataset, generated at time after learning the tasks (z(*))= ==1 We form the following objective to learn the task z(,, Vt 2 2:
min L. (()) (x Y()) + Σ 1,30,
D( (x ) nld
(C)))
(2)
where ((,)) is probabiliy distribution metric and is a trade-off parameter.
The first and the second terms in Eq. equa- tion 2 are combined loss terms for the current task training dataset and the generated pseudo- dataset that represent the past tasks, defined similar to Eq. equation I The second term in Eq. equation 2 mitigates catastrophic for- getting through pseudo-rehearsal process. The third term is crucial term to guarantee that our method will work lifelong learning setting. This term enforces that each concept is encoded in one mode the internal distribution across all tasks. This term is computed set of the concept classes that are shared be- tween the current task and the pseudo-dataset. i.e, No to enforce consistent knowledge ac- cumulation. Minimizing the probability met- ric DI,,-) enforces that the internal conditional distribution for the current task du (q (-CC;), conditioned on particular shared concept Cj. to be close to the conditional shared distribution plr (-IC), Hence, both form a single mode of the internal distribution and concept drifting is mitigated. Conditional matching of the two dis- tributions is feasible as we have access to pseudo-labels. Adding this term guarantees that we can continually use GMM with exactly k// components to capture the internal distribution in this life- long learning setting. The remaining task is to select suitable probability distance metric D(: -) for solving Eq. equation 2 Wasserstein Distance (WD) metric has been found to be an effective choice for deep learning due to its applicability for gradient-baad
Algorithm ICLA (A, 7,7)
1: Input: labeled training datasets in sequence
D(1) (fx((), x ) for =2
3: Initial Learning: learn the first task via Eq. equa- tion[
4: Fitting GMM:
5:
estimate (.) using (oo(x) vi pm
6: For I. 2
7: Generate the pseudo dataset:
8:
D(t)
(a (x((1),, ).(")) "
9:
(2(") yy
(.)
the sub-
10:
Task learning:
11:
Icarnable parameters are updated via Eq. equation
12:
Estimating the internal distribution:
13:
update (-) with k/f components via the
14:
combined samples (do(x= oe (a(")))))
15: EndFor
optimization Courty et al.) (2017).
To reduce the computational burden of computing WD, use the Sliced Wasserstein Distance (SWD) Bonneel et al. (2015). (for details on the SWD loss, please refer to the Appendix). Our Incremental Concept Learning Algorithm (ICLA) is summarized in Algorithm
5 THEORETICAL ANALYSIS
We demonstrate that ICLA minimizes an upperbound for the expected risk of the learned concept classes across all the previous tasks for all t. We perform our analysis in the embedding space as an input space and consider the hypothesis class H (-)(
+ R$ w E [R ]. Let e (w) denote the real risk for given function (y (-) E H when used on task ((t) data representations in the embedding space. Similarly, é (ww) denotes the observed risk for the function ha (-) when used on the pscudo-task, generated by sampling the learned GMM distribution p(t-1), Finally, let (w)) denote the risk of the model ()m when used only on the concept classes the set N, C No., for Vs S I, i.e., task specific classes, after learning the task (1)
Theorem Consider two tasks (t) and ( in our framework, where I f. Let hn be an optimal classifier trained for the z using the ICLA algorithm. Then for any > d and Ç < 2, there exists constant number No depending on d' such that for any & > 0 and min(fgp, n.)
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max(5- max(5-(('+22))) (d)+)) with probability at least 1 for (wtt) H. the following holds:
ea(w) (w) + W " + =c(w") + (2log( )/()
(3)
ReIM,
where (..-) denotes the WD metric, Ri denotes the aseudo-task samples that belong to the classes in N.. (-)) denotes the empirical marginal distribution for Z(*) in the embedding, Pa 1 is the conditional empirical shared distribution when the distribution (-t==))-) is conditioned to the classes in N., and cc(w") denotes the optimal model learned for the combined risk of the
tasks on the shared classes N., i.e., w= = arg mitlor ec ec(0)= = arg minw (vr,.(w) + ex(w)). This is a model with the best performance if the tasks could be learned simultaneously.
Proof: included in the Appendix due to page limit.
We then use Theorem 1 to conclude the following lemma:
Consider the ICLA algorithm after learning z Then all tasks / < T and under the conditions of Theorem 1, we can conclude the following inequality:
Lemma 1
T-2 (w) S er (r- W (,((""),, p. ce(ww)) Σ (el pl + (2
( (1)
T
RyjKe (4)
Proof: included in the Appendix due to page limit.
Lemma concludes that when a new task is learned at time t = T. ICLA updates the model parameters conditioned on minimizing the upper bound of et for all < T in Eq. 4 The last term in Eq. is a small constant term when the number of training data points is large. If the network is complex enough so that the PDP hypothesis holds, then the classes would be separable in the embedding space and in the presence of enough labeled samples, the terms eT-1,(ww) would be small because (ur) is minimized using ERM. The term W ((((())) >.P;
p())) would be small
because we deliberately fit the GMM distribution ((t) to the distribution ((g(t)) in the embedding space when learning the task 2(9, Existence of this term indicates that our algorithm requires that internal distribution can be fit with GMM distribution with high accuracy and this limits applicability of our algorithm. Note however, all parametric learning algorithms face this limitation. continually match the distributions in the embedding space
The term ec ee(w") is small because class-conditionally. Hence, if the model is trained on task (t) and the pscudo-task at - T, it will perform well on both tasks. Note that this is not trivial because the wrong classes are matched across the domains in the embedding space, the term e<(w") will not be minimal. Finally, the sum term in Eq. 4 indicates the effect of experience replay. Each term in this sum is minimized at ++1 because we draw random samples from p!!) and then train the autoencoder to enforce p( R (((())) Since all the terms in the upperbound of ey (ur) in Eq. 4are minimized when a new task learned, catastrophic forgetting of the previous tasks will be mitigated. Another important intuition from Eq. is that as more tasks are learned after learning a task, the upperbound becomes looser as more terms are accumulated in the sum which enhances forgetting. This observation accords with our intuition about forgetting as more time passes after initial learning time of a task or concept.
6 EXPERIMENTAL VALIDATION
To the best of knowledge, no prior method has been developed to address challenges of both con- tinual and incremental learning setting at the same time. For this reason, we validate our method on two sequential task learning settings: incremental learning and continual incremental learning. Incremental learning is a special case of our learning setting when each concept class is observed only in one task and concept drift does not exist. We use this special case compare method against existing incremental learning approaches in the literature to demonstrate that our method is comparably effective. Our implementation is available supplementary.
Evaluation Methodology: We use the same network structure for all the methods for fair compar- ison. To visualize the results, generate learning curves by plotting the model performance on

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the testing split of datasets versus the training epochs, ie, model time. We report the average performance of five runs. Visualizing learning curves allows studying temporal aspects of learning. For comparison, we provide learning curves for: (a) full experience replay (FR) which stores the whole training data for all the previous tasks and (b) experience replay using a memory buffer (MB) with a fixed size, similar to Li et. al (Li & Hoiem (2018). At each time-step, the buffer stores an equal number of samples per concept from the previous tasks. When new task is learned, a portion of old stored samples are discarded and replaced with samples from the new task to keep the buffer size fixed. FR serves as best achievable upperbound to measure the effectiveness of our method against the upperbound. For details about the experimental setup, please refer to the Appendix.
6.1 INCREMENTAL. LEARNING
The concept classes are encountered only at one task in incremental learning. We design two incre- mental learning experiments using the MNIST and the Pashion-MNIST datasets. Both datasets are classification datasets with ten classes. MNIST dataset consists of gray scale images of handwritten digits and Rashion-MNIST consists of images of common fashion products. We consider an incre- mental learning setting with nine tasks for the MNIST dataset. The first task is binary classification of digits O and and each subsequent task involves learning a new digit. The experimental setup for MNNSST dataset is similar, but we considered four tasks and each task involves learning two fashion classes. We use a memory buffer with the fixed size of 100 for MB. We build an autoencoder by expanding VGG-based classifier by mirroring the layers.
F presents results for the designed experiments. For simplicity, we have provided condensed results for all tasks in a single curve. Each task learned in 100 epochs and at each epoch, the model performance is computed as the average classification rate over all the classes, observed before. We report performance on the standard testing split of each dataset for the observed classes. Figure 2a and present the learning curves for the MNIST experiments. Similarly, Figure 2b present learning experiments. We can see in both figures that FR (dashed blue curves)
curves for the Fashion- leads to superior performance. This is according to expectation but as we discussed, the challenge is the requirement for memory buffer with an unlimited size. The buffer cannot have a fixed size the number of data points grows when more tasks are learned. MB (solid yellow curves) is initially somewhat effective and comparable with ICLA, but as more tasks are learned, forgetting effect becomes more severe. This is because fewer data points per task be stored in the buffer with fixed size as more tasks are learned. As a result, the stored samples would not be sufficiently representative of the past learned tasks. In comparison, we can generate many pseudo-data points.
We can also see in Figure 2a and Figure 2b] that ICLA (dotted green curves) is able to mitigate catastrophic forgetting considerably better than MB and the performance difference between ICLA and MB increases as more tasks are learned. We also observe that ICLA is more effective for MNIST dataset. This is because FMNIST data points are more diverse. As a result, generating pseudo-data points that look more similar to the original data points is easier for the MNIST dataset given that we are using the same network structure for both tasks. Another observation that the major performance degradation for ICLA occurs each time the network starts to learn a new concept class as initial sudden drops. This degradation occurs due to the existing distance between the distributions P.,
(T-1) 1) and ())())) at t = T for s < T. Although ICLA minimizes this distance, the autoencoder is not ideal and this distance is non-zero in practice, leading to forgetting effects.
For comparison against existing works, we have listed our performance and number of methods for incremental learning on MNIST in Table 3 Two sets of tasks for incremental learning setting have been designed using MNIST in the literature: 5 tasks (5T) setting and 2 tasks (2T) setting. In
the 2T setting, two tasks are define involving digits (0- 4) and (5 1 -9). In the 5T setting, five binary classification tasks are defined involving digits (0, 1) to (8, 9). We have compared our performance against several methods, representative of prior works: CAB He & Jaeger (2018), IMM Lee et al.)
(017)) OWM Zeng et al. 2019), GEM Lopez-Paz & Ranzato (2017). iCarl Rebuffi et al. (2017) GSS Aljundi et al. (2019), DGR Shin et al. (001)), and MeRGAN Wu et al. (2018). 8). The CAB, IMM, and OWM methods are based on regularizing the network weights. The GEM, iCarl, and
GSS methods use memory buffer to store selected samples. Finally, DGR and MeRGAN methods are based on generative replay similar to ICLA but use adversarial learning. We have reported the classification accuracy the ten digit classes after learning the last task in Table 3 A memory
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Books
(a) FR vs. MB
(b) FR vs. ICLA
FR
(d) ICLA
Figure 4: Learning curves for the five continual incremental learning tasks, designed using the
permuted MNIST tasks (a) FR (solid) MB (dotted), (b) FR (solid) vs. ICLA (dotted); UMAP visualization of (c) FR and (d) ICLA in the embedding space. (Best viewed in color on screen. See Appendix for enlarged versions.)
buffer with fixed size of 100 is used for GEM, iCarl, and GSS. Following these works, an MLP with two layers is used as the base model for fair comparison.
We observe Table that when the buffer size is small, buffer-based methods perform poorly. Methods based on weight regularization per- form quite well but note that these methods limit the network learning capacity. As a result, when the number of tasks grow, the network cannot be used to learn new tasks. Generative methods, including ICLA, perform better com- pared to buffer-based methods and at the same time do limit the network learning capacity because the network weights can change after generating the pseudo-dataset. Although ICLAA has the state-of-the-are performance for these tasks, there no superior method for all con- ditions, because by changing the experimental setup, e.g., network structure, dataset, hyper- parameters such memory buffer, etc, dif- ferent method may have the best performance result. However, we can conclude that ICLA has a superior performance when the network size small and using memory buffer is not possible.
Eacess
Eaads
(a) MNIST
(b) FMNIST
Figure Learning curves for the incremental learning experiments (a) MNIST and (b) Fashion- MNIST (FMNIST) datasets; (c) MNIST per- formance comparison (Best viewed in color screen. See Appendix for enlarged versions.)
6.2
CONTINUAL INCREMENTAL LEARNING
Permuted MNIST task a common supervised learning benchmark for sequential task learn- ing Kirkpatrick et al. (2017). The sequential tasks are generated using the MNIST dataset. Each task Z is generated by rearranging the pixels of all images in the dataset using fixed random predetermined permutation transform and keeping the labels as their original value. result, we can generate many tasks that are diverse, equally difficult. As result, these tasks are suitable for performing controlled ex- periments. Since prior work has addressed incremental learning of drifting concepts, we should design suitable set tasks.
Method
2T
ST
CABlHe Jagger 2018
2018
94.9:0.3
IMN Les ef al MmL7
1--033
Zong
2019
96.3--0.1
GFM
opez
Kanzate 2017
78.0
(Car Kebuth et
A01
81.0
GSS Auuund
61.0
DGR Shin et al
2017
88.722.6
MeRGAN Wa al
2018
97.0
ICLA
97.2-022
91.020.4
Figure 3: Classification accuracy for MNIST.
We design continual incremental learning tasks that share common concepts using five permuted MNSIT tasks. The first task is a binary classification of digits and 1 for the MNIST dataset. For each subsequent task, we generate a permuted MNIST task but include only the previously digits plus two new digits in the natural number order, e.g., the third task includes permuted versions of digit 0 5. This means that each task, new forms of all the previously learned concepts
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encountered, i.e, we need to learn drifting concepts, in additional to new tasks. Hence, the model needs to expand its knowledge about the previously learned concepts while learning new concepts. We use a memory buffer with size of 30000 for MB. Due to the nature of these tasks, we use a multi- layer perceptron (MLP) network. Figure 4 presents learning curves for the five designed permuted MNIST tasks. In this figure, the learning curve for each task is illustrated with different color and different line styles are used to distinguish the different methods (for enlarged versions, please refer to the Appendix). At each epoch time-step, model performance is computed as the average classification rate on the standard testing split of the current and all the past learned tasks.
Figure 4a presents learning curves for MB (dotted curves) and FR (solid curves). Unsurprisingly FR leads to almost perfect performance. We also observe MB is less effective in this setting and catastrophic forgetting is severe for MB beyond the second task. The reason is that the concepts are more diverse in these tasks. As a result, it is more challenging to estimate the input distribution using a fixed number of stored samples that also decrease due to a fixed buffer size. We can conclude that as tasks become more complex, larger memory buffer will be necessary which poses challenge for MB, Figure 4b presents learning curves for FR (solid curves) and MB (dotted curve). As can be seen, ICLA is able to learn drifting concepts incrementally. Again, major forgetting effect for ICLA occurs as a sudden performance drop when learning a new task starts. This observation demonstrates that an important vulnerability for ICLA is the structure of the autoencoder that we build. This can be deduced from our theoretical result because an important condition for tightness of the provided bound in Lemma I is that we have: & Re o Both our theoretical and experimental results suggest that if can build auto-encoders that can generate pseudo-data points with high quality, incremental learning can be performed using ICLA. In other words, learning quality depends on the generative power the base network structure. Finally, we also observe that as more tasks are learned after learning a particular task, model performance on that particular task degrades more. This observation is compatible with the nervous system memories fade out when time passes.
In addition to requiring a memory buffer with an unlimited size, we also demonstrate that an is- sue for FR is inability to identify concepts across the tasks in the embedding space. We use the UMAP McInnes et al. (2018) tool to reduce the dimensionality of the data representations in the embedding space to two for 2D data visualization. We illustrated the testing split of data for all the tasks in the embedding space Z in Figure 4e for FR and Figure 4d for ICLA when the final task is learned. In these figures, each color corresponds to one of the digits {0, 1. , 9). As expected from the learning curves, data points for digits form separable clusters for both methods. This result verifies that the PDP hypothesis holds in these experiments and hence the internal distribution can be modeled using GMM. The important distinction between FR and ICLA is that FR has led to the generation of distinct clusters for each concept class per task. This means that each concept class has not been learned internally as one concept and FR learns each concepts as several distinct con- cepts across the domains. This observation also serves as an ablative study for our method because it demonstrates that matching distributions glass-conditionally in the embedding space is necessary for our method to work, as justified by the theoretical analysis.
In figure 4d we observe that ten clusters for the ten observed concepts are formed when ICLA is used. This observation demonstrates that ICLA is able to track modes of the GMM successfully as more tasks are learned. ICLA is also able to build concept classes that are semantically meaningful across all tasks based on the labels. This is the reason that we can learn new classes incrementally in a continual lifelong learning scenario. In other words, opposed to FR, ICLA encodes each cross-task concept in a single mode of the internal GMM distribution. This allows for expanding concepts for cross-domain abstraction similar to humans when new forms of concepts are observed.
7 CONCLUSIONS
Inspired by the CLS theory within the PDP paradigm, we developed an algorithm for continual incremental learning of concepts. Our algorithm is based on modeling the internal distribution of input data as a GMM and then updating the GMM as new experiences are acquired. We track this distribution to accumulate the new learned knowledge to the past learned knowledge consistently. We expand the base classifier model to make a generative model to allow for generating a pseudo- dataset to represent past learned tasks. The aseudo-dataset is used for reeuro-rehearral to implement experience replay. We provided theoretical and empirical result on three benchmark datasets to validate our algorithm. Future works includes extension domains with unlabeled data.

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A
APPENDIX
A.1
COGNITIVE MODELING BACKGROUND
Our work is inspired by the "complementary learning systems" (CLS) theory within the "parallel distributed processing' (PDP) paradigm.
A.2 PARALLEL DISTRIBUTED PROCESSING
Parallel distributed processing (PDP) approach in cognitive science tries to explain mental phenom- ena using structures similar artificial neural networks McClelland al. (1986) which were his- torically inspired by biological neurons and their parallel processing ability in low-level structures of the nervous system. Within this framework, learning process is modeled as adjusting weights in network according to various rules such as Hebbian learning Song et al. (2000). PDP models data representations in the nervous system as distributed representations that are encoded in the neural activation functions Hinton et al. (9844) which is analogous to representing data in semantically meaningful embedding space. Hasson et al. Hasson et al. (2020) argue that although evolution trains the biological neural networks blindly based on behavioral advantage, but the emerging behaviors are similar to behaviors that are observed in the artificial neural networks. They argue that both biological and artificial neural networks learn a meaningful embedding space by optimizing an ob- jective function on densely sampled training data, i.e., empirical risk minimization. As result, the dimensions of the embedding space capture features that help to encode informative variations across the input data points. We have based our work on this hypothesis. This means that when we train an artificial neural networks for classification, data representations encode input data sim- ilarity in terms of belonging to the same class. This means that we model the data representation distribution using a multi- modal distribution.
A.3 COMPLEMENTARY LEARNING SYSTEMS
We rely on the Complementary Learning Systems (CLS) theory McClelland et al. (1995) to prevent catastrophic forgetting both when concepts drift or when new concepts are observed. CLS theory is proposed within the PDP paradigm and hypothesizes that continual lifelong learning ability of the nervous system is result of a dual long- and short-term memory system. The hippocampus acts as short-term memory and encodes recent experiences that are used to consolidate the knowledge in the neocortex long-tern term memory through offline experience replays during sleep minnelmann & Born (2010). The hippocampal experience replay is more of a generative process because the input stimuli is absent during these replays. In our work the internal multimodal distribution mod- els the neocortical consolidated knowledge. When a task is learned, this distribution is updated to incorporate the new learned knowledge update the long-term memory. The hippocampal experi- ence replay is modeled when the pscudo-dataset is generated to prevent catastrophic forgetting using pseudo-rehearse This neudo-dataset more representative of the recent memory, as demonstrated by both our theoretical and empirical results.
A.4 BLOCK DIAGRAM OF THE PROPOSED METHOD
Figure 5 presents an enlarged version of the system block-diagram for more clarity on how the PDP and the CLS theories are reflected in our model. Visualization of the data representation in the embedding space in Figure 5 highlights the PDP hypothesis. An important condition for the proposed method work is that the PDP hypothesis holds. This means that the concepts are formed as clusters in the embedding. As result, the task data in the embedding would follow GMM distribution and the number of components of this GMM is equal to the number of observed classes. As result, the second term in Eq. (3) is the distance between the empirical and the real distributions for a GMM. Hence, the second term is minimized by fitting a GMM on the drawn distribution samples. Similar to all the parametric algorithms, the data distribution is correct. All parametric algorithms are limited in this sense.
method works only if the assumption about
After learning each task, need to update the estimate of the internal GMM distribution in two aspects for generating representative pseudo-dataset in the future. First, the number of components should be updated to k, because new classes may be observed. Second, estimates for parameters
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AVAA 7AAL
Sample Domains:
77778
N99979
Latin digits and
Arabic digits
L-1
[+1
1-2
1. New Domains arrive sequentially
D
Current Domain Data
The new learned knowledge about all concepts is used to update the internal data distribution
Encoder
(x(x (1))
3. The new domain is learned through matching its distribution in the embedding
Embedding Space
L
New Concepts
Classifier
Concept Labels
Pseudo-das data points for old concepts are generated and fed into the encoder to implement pseudo- rehearsal
PJ
(t-1)
Supervised Pathway
Consolidated Concepts
Decoder
L
Lc Cc (t) (1)
Pseudo-data points
Incremental Concept Learning Process
Figure 5: Block-diagram Architecture of the proposed Incremental Learning System.
of each concept cluster, i.e., the mean and the variance of the corresponding Gaussian component, should be updated to incorporate potential concept drifts. Updating this distribution models the process of knowledge consolidation in the nervous system using recent experiences.
A.5 GMM ESTIMATION
Upon learning a task, the internal distribution will be updated according to the input distribu- tion. The empirical version of the internal distribution is encoded by the training data samples (=)o( y (1) ) where with a slight abuse of notation, we use the same notation to denote the sseulo-samples. We consider the distribution (z) to be a GMM with ky components:
Re E,),
(2) Σ (z)
(5)
where a, denotes the mixture weights, i.e., prior probability for each class, μ, and E, denote the mean and oo-variance for each component. Since have labeled data points, can compute the GMM parameters using MAP estimates. Let S, denote the support set for class j in the train-
ing dataset, i.e., S₁ (w (0) E DDl arg max y; (() j]. Then, the MAP estimate for the
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parameters would be:
|
0g Th
A
Σ
Σ
(0o(x)")) - (a)) (oo(x) - pe)).
[S]
(a ()))
(o(9) ( (
jes
We can these estimates to draw samples from (()) to generate representative pseudo-datased before learning the subsequent task.
A.6 PROOF OF THEOREM I AND LEMMA 1
Our proof is modeled after Redko et Redko al. (017)). The proof by Redko et al. Redko al. (017)) is limited to the problem of domain adaptation in which the same classes exist across two domains. We adapt the proof to work our learning setting, where the two distributions share only subset the classes.
We first review the definition of the optimal transport. Let Ω C Rd be a measurable space and P(12) denote the set of probability distributions that are defined over Q2 Given two distributions q()),())) E P(2)) and the cost function e D² + R+, the optimal transport distance between p(-)
and q()) is defined as:
/ c(x, y) y)dy(( y)
W 4) inf yEll(p.))
(7)
where ..:))) denotes the set of all joint distributions over Ω2 that have marginal distributions p and q. Optimal transport is well-defined for any proper selection of the cost function. In our proof, consider that the cost function has the specif form: c(æ;, y) [|))(>)) nyy)ll. where r RR - Rdd
is an embedding function and || llg denotes the norm function in this space.
We will need the following lemma in our proof.
Lemma 2: Consider two distribution
E P(12) and two functions hw hw H. and the cost function c(x, y) ||m)(x) - (g)) Assume that the hypothesis space H. is Reproducing Kernel Hilbert Space (RKHS) equipped with a kernel, induced by by the feature map v) + Rdd,, Let the loss function ((.,.) to be mathematical metric which is convex and bounded by 1. Additionally, assume that the loss function para metrically depends on /// (x) hw (x)llx. Then the following
inequality holds:
(8) Proof: First note that since the difference lww (x)) hw (w) lies in the hypothesis space, then the loss function is nonlinear function that maps member of the H to positive numbers. Using re-
Exe (La(tea (ww - (Ca (x), hew (x))) S (p.4)
sults from Saithh| (1997). we can deduce scalar RKHS space is formed. Following the above assumptions, we can deduce:
Expp (arnp(aalhie (ha(hrrr (x:), New (x:)) Eeong(Ca(ahha (x),, (x))) Exzp (L, (Tx)) Be (L, ())))))))))))))))))))))))))))(
(T (1(x)) P (z)) L)g S
|| (ml E. Ee (v) (z) =
qdllp S)l(o
/ [[(((x) -
(9)
v)llo
f Jgp [1)((x)) - -(0))
*El((p.)) / [l((x)) n(v))lad.(( y) = ((p.g)
inf
In the first and the second lines, we have used the reproducing property in 9 space. In the third and fourth lines, first used the property of the expectation and then inner-product property. In the
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fifth and sixth lines, have used the property of the joint distribution and then the definition of the optimal transport. We note that this proof is specific to a particular form of cost functions.
We also need the following result on convergence of the empirical distribution the real distribution in the optimal transport norm in our proof.
Theorem 2 (Theorem 1.1 Bolley
P( and Ja exp (a | (2)) < oc for some a > O. Let p(z) = N E, 5(x;) denote the empirical dis- tribution that is built from the samples {x, that are drawn i.d from æ, ~ p(x). Then for any
al. (2077)) consider that p(-)
d' > d and & < V2, there exists No such that for any > 0 and N 2 No e ((d)+))) we have:
5 Ne) 2
P(W(,, 2) & exp(
(10)
We combine the above result and the previous lemma to prove Theorem 1.
Theorem 1: Consider two tasks g and ((s) in our framework, where s S t. Let hygt be an optimal classifier trained for the z using the ICLA algorithm. Then for any > d and ( < 2 and min, ng) 2
there exists constant number No depending on d' such that for any
max(f" - (d'+2), 1) with probability at least 1 -
f for hwn(t) E H. the following holds:
e (w) Sec-1, (w) + ( (V()!""))((())))) + =c(w"))+
(2218(2211 log(
(11)
Der,e|N
1hss
where -::-) denotes the optimal transport distance, Ner,t|N, denotes the subset of samples of the pseudo-task that belong to the classes in N., (-)())()) denotes the empirical marginal distribution for (*) in the embedding space, p denotes the conditional empirical shared distribution when the distribution (-) is conditioned to the classes in N,, and ec(w") denotes the optimal model
for the combined risk of the two tasks on the shared classes in Ng. i.c., w" = arg min cc( =
arg min (w) + ex(w)).
Proof:
(uw) S (ww") + (eonoggo)s((aa(eee (x), how (x)))
(e((w)) ) Ben (Ca(kew-
=- (Ca(hro (x), hew (ar))
E b 1- (hw (x)), Mow (x)))
ex(w") - 2nD (Ca (x), kw) (0
W(p! (1-1) "
e, + 61- (w) + c, -.-
W W (1-1) ,0(g =
(12)
(t-1.e(m) + +cc(w") W (p((-1) 669 (1)) S
e (-"""(w) + +eo(w") W
- + W (p! ;((((")))))))))))))))))))))))))) s
( + W(!! 11 o(((0))) + =e(e")+
W W(Dh- (t-1) -()1)) + (( 2(q s
(w) W ((((())) cc(w")
(210g( //)
n
In the above proof, fifth line is deduced from Lemma 1. In the sixth, have used the triangular inequality on the loss function. In the seventh line, we have used the definition of the joint optimal model. In the lines eighth to tenth, we have used the triangular inequality on the optimal transport. In the last two lines, we have used Theorem 2.
We use Theorem to deduce Lemma 1.
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Lemma 1 Consider the ICLA algorithm after learning Z Then all tasks t < T and under the conditions of Theorem 1, we can conclude the following inequality:
((w) I (r-1i(w)) + (((()())) ).P)) + =c(2"))
(13)
Σ W(t .r + (2log( 1/5)
n + Aupe.
Proof: We consider z with empirical the distribution o(g()) in the embedding space and the pseudo-task with the distribution p(r-11 in Theorem 1. Applying the triangular inequality
on the term ((((g()), P.t ( recursively, i.e., ((((g()), P -) I W(o(f()), p. p +
(2) p (T-1)) for all t < concludes Lemma 1. W W(P
A.7 DETAILS OF EXPERIMENTAL IMPLEMENTATION
A.7.1 DATASETS
We investigate the empirical performance of our proposed method using two commonly used bench- mark datasets: MNIST (M) and MashiT (4) MNIST is a collection of hand written digits in 28 28 pixels with 60000 and 10000 training and testing data points, respectively. Fashion- MNIST has similar properties but the images are more realistic. To generate permuted MNIST tasks, we followed the literature and applied fixed random permutation to all the MNIST data points for generating each sequential task. We used cross entropy loss as the discrimination loss and the Euclidean norm the Reconstruction loss. We used Keras for implementation and ADAM optimizer. We run our code on cluster node equipped with 2 Nvidia Tesla P100-SXM2 GPU's.
A.7.2 EVALUATION METHODOLOGY
All these datasets have their own standard testing splits. For each experiment, we used these testing splits to measure performance of the methods that report in terms of classification accuracy. We used classification rate on the testing set of all the learned tasks to measure performance of the algorithms. At each training epoch, compute the performance to generate the learning curves. We performed 5 learning trials on the training sets and reported the average performance the testing for these trials. We used brute force search cross-validate the parameters for each sequential task.
the testing split of these tasks
A.7.3 NETWORK STRUCTURE
For visual recognition experiments, we used convolutional structure as spatial visual similarity can be captured by convolutional structures. We used VGG16-based encoder. The decoder subnetwork is generated by mirroring this structure. We flatten the last convolutional layer response and used a dense layer to form the embedding space with dimension 64. The classifier subnetwork is single layer with sigmoid.
Following the literature, have used an MLP with two layers for tasks of Table 1. The first layers has 100 nodes and the second layer has nodes equal to the number of learned concepts.
For permuted MNIST experiments, we used an MLP network. This selection is natural the cepts are related through permutations which can be learned with an MLP structure better. For this reason, the images were normalized and converted to 784 x vectors. The network had three hidden layers with 512, 256, and 32 nodes, respectively. We used ReLu activation between the hidden lay- ers and selected the third hidden layer as the embedding space. This selection is natural because the last hidden layer, supposedly should respond to more abstract concepts. The decoder subnetwork is generated by mirroring the encoder subnetwork and the classifier subnetwork is one layer with 10 nodes and sigmoid activation.
A.8 ENLARGED FIGURES
For possibility of better inspection by readers, enlarged versions of Figure 2 and Figure in the main body of the paper are provided in this section.
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80
Testing Accuracy
Testing Accuracy
40
20
50
7000
350
Epochs
Epochs
(a) MNIST
(b) FMNIST
Figure 6: Learning curves for the incremental learning experiments (a) MNIST and (b) Fashion- MNIST (FMNIST) datasets; (c) MNIST performance comparison (Best viewed in color on screen. Enalarged version included in the Appendix.)
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90
80
Tesling Accuracy
Testing Accuracy
70
60
40
20
30
100
100
200
400
500
Epochs
Epochs
(a) FR vs. MB
(b) FR vs. ICLA
10
-5
-10
-10
-10
10
-10
10
(c) FR
(d) ICLA
Figure 7: Learning curves for the five continual incremental learning tasks, designed using the permuted MNIST tasks (a) FR (solid) vs. MB (dotted), (b) FR (solid) vs. ICLA (dotted); UMAP visualization of (c) FR and (d) ICLA in the embedding space. (Best viewed in color screen)
19


Reviews:
[{"review_id": "p55iNFmSou2", "reviewer": "Reviewer_vhwe", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Generally, I like the strong theoretical background of the paper and approach. The paper builds on important theories that are central in lifelong learning (PDP, CLS) and mathematical approaches GMM, WD. The formalism, both in the paper and in the appendix seems solid although I did not carefully analyse all the steps of the mathematical derivations.\n\nMy main concerns are two.\n\n(1) The narrative of the paper is not easy to follow.\n\nFor example, the abstract does not mention drifting concepts that is present in the title, nor GMM that are mention in the ‘contribution’ section. However, these are central parts of where the paper aims to advance knowledge, and yet are not stated in the abstract, which makes it hard to find the claim and objectives of the paper. As a consequence, the abstract reads to generic and does not address core ideas in the paper. \n\nOne other aspect that makes it difficult to follow the paper is that the terms \"lifelong learning\", \"incremental learning\", \"continual learning”. “continual incremental learning”, and “sequential learning\" are not well defined and are used somehow loosely in the paper. It is conceivable that different researchers in the field might have different understanding how what each term means, and thus I think that the paper needs to explain more clearly and formally each problem. This confusion makes it difficult to  understand what problem is addressed specifically in each section. For example, differently from what appears to transpire from the paper, I assume that lifelong learning is a higher level term that include both incremental and continual learning. In other words, I would argue against the sentence \"In a lifelong learning setting, the number of classes are usually assumed to be fixed”. Similarly, I’m also confused by the use of the terms ‘concept’ ‘concept class’ and ‘class’ as sometimes they are used in the same sentence but it's unclear whether they mean the same thing or not. It would be helpful if the text explains what the authors mean for each of such term. \n\nThe results in Fig 4 are difficult to read and to me not very clear (particularly in the graphs at the left).\n\nMinor observation: Fig 2 and 3 appear after Fig.4 and all figures are difficult to read because too small. \n\n(2) The simulation results appear limited or do not illustrate well the potential of the method.\n\nThe continual learning dynamics in Fig. 2 and 4 for the novel approach (ICLA) are compared only against standard baselines that include the full experience replay (FR) and memory buffer replay (MB). But how do these compare against well established continual learning preservation mechanisms such as EWC or MAS? \n\nThe main result for continual incremental learning in Fig 4 are not compared with the baselines that are instead shown in Fig 3. It's not clear to me why. \n\n ", "summary_of_the_paper": "This paper proposed an algorithm for learning concepts incrementally taking inspiration from Parallel Distributed Processing and Complementary Learning Systems. The core idea is to use Gaussian Mixture Models and update them incrementally plus exploit a generative model to perform pseudo-rehearsal. The paper presents a methodology and follow-up experimental results that are shown to perform well. ", "summary_of_the_review": "The paper addresses in important challenge in lifelong learning, i.e., to be able to generalise across a variety of classes that are learned sequentially through different tasks, meeting both new classes and drift, when the distribution of one class changes. The method appears sound and has a strong theoretical background. However, I feel that the paper could be improved significantly in clarity, readability, and presentation of results. It's not clear to me what are the main claims of the paper, and while some results appear to be good, the final set of results (fig 4) are difficult to interpret. In short, this paper has a strong potential, but requires a better presentation and a better illustration of the results and baselines for the last experiments in Fig. 4.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "YgZUokXpc9a", "reviewer": "Reviewer_QevD", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The submission is interesting and well motivated. The main novelty of the approach is fitting the latent space of the auto-encoder structure with a Gaussian mixture model which is then used to help with both generation of pseudo samples and to prevent concept drift. The final term of Eq. 2 which is used to minimize this concept drift is particularly interesting.\n\nMy main criticism of the paper is that the authors spend very little time talking about the consequences of the unimodality assumption which is the backbone of their work. To be more specific, the main innovation in this paper is the use of a GMM to interpret the latent space of an auto-encoder. This assumes that each concept is unimodal which is a reasonable assumption. However, the authors do not explore the consequences of this assumption in cases were it is not satisfied. As an example when the concepts of a new task have non-trivial relationship with previous tasks (e.g. task 1 is classifying color and task 2 is classifying shape), clearly one modality per class is no longer satisfied. What happens in such cases or in other cases where the semantic relationship between the classes is more complicated than ‘same’ or ‘new’? What are the other failure modes of the approach? \n\nThe authors spend valuable real estate to discuss PDP and CLS to argue that concepts should be encoded in a deep layer of a network and that generative processes can help alleviate catastrophic forgetting. Neither of these ideas are new. The paper can be strengthened by focusing more on what is novel and less on what has come before.", "summary_of_the_paper": "The submission proposes a new unified methodology for continual and incremental learning based on Parallel Distributed Processing theory and Complementary Learning Systems theory. The authors prove that their methodology optimizes an upper bound for the combination of previous tasks and also demonstrate the effectiveness of their approach empirically.", "summary_of_the_review": "The submission is a step forward in implementing approaches that can learn in a continual manner. The analytic demonstration that the loss is an upper bound is a cherry on top. However, the submission can be greatly improved if the authors focus more on the consequences of what is in fact novel in their approach (i.e. the GMM on top of the autoencoder latent), providing more experiments and a deeper discussion of the failure modes.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "Axs8W6DVSgd", "reviewer": "Reviewer_kyrh", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "This is a neatly conceptualized, well-motivated approach that is grounded in cognitive science theory. On one hand, in contrast to models with an explicit memory buffer, ICLA avoids a blow-up in memory complexity with an increasing number of samples/classes encountered. On the other hand, in contrast to models with synaptic constraints, ICLA doesn't restrict the expressivity of models. This also seems to be among initial efforts to jointly tackle the problem of continual and incremental learning.\n\nHowever, several aspects of this manuscript need to be improved before it is publication-ready.\n\n(i) Strength of empirical evaluation:\n(a) Recent papers on continual learning evaluate their method on several benchmarks, including mini-ImageNet, CIFAR-100, CIFAR-10, etc. In comparison, the experiments presented in this paper seem too narrow in scope. It also does not provide an intuition for this algorithm's ability to scale.\n\n(b) Is there a reason why these popular methods weren't included as part of the benchmark? Happy to stand corrected if the authors have a valid explanation.\nSynaptic Intelligence [Continual Learning Through Synaptic Intelligence. Zenke et al (2017) ICML]\nElastic Weight Consolidation [Overcoming catastrophic forgetting in neural networks. Kirkpatrick et al (2017) PNAS]\n\n(c) The improvement over competing methods for the 2T MNIST task is marginal. It is also unclear to me why Table 3 is not complete.\n\n(ii) Literature review:\nThis method is also very related to \"Online Class-Incremental Continual Learning with Adversarial Shapley Value\" [Shim et al (2021) AAAI], where they introduce a scoring method to preserve latent decision boundaries. The authors should clarify similarities/differences. \n\n(iii) Paper clarity:\nThis paper gives the impression that it was hurriedly put together. There are several grammatical and typographical errors, some of which I've tried to list below. A thorough proofread will enhance the quality of this manuscript.\n\nPg. 2 conceptsincrementally \n\nPg. 2 Incremental learning::\n\nPg. 2 incremental learningstems\n\nPg. 4 In Figure 1 caption: Based viewed enlarged\n\nPg. 4 \"reduces the following\" --> reduces to the following\n\nEq 2 (third term) there is a parenthesis mismatch\n\nPg. 7 \"Figure 2a and present the learning curves...\". Incomplete phrase\n\nPg. 7 \"two tasks are define\"\n\nTable 3 is actually Figure 3 in the text.\n\nMoreover, the citation style the authors use makes for a very unpleasant reading experience. In several places, the references directly interfere with the text. This needs to be fixed.\n\n(iv) Other clarifications on the method:\n(a) It is true that with an increasing number of samples/classes ICLA doesn't suffer from a memory bottleneck. However, the same cannot be said about the size of the \"pseudo-dataset\". Can the authors quantify how this will scale? This could prove to be a computational bottleneck on larger-scale tasks.\n\n(b) \"However, we can conclude that ICLA has a superior performance when the network size is small and using a memory buffer is not possible\" --> Model parameter complexity results are not presented in this study. How did the authors arrive at this conclusion?\n\n(c) $\\gamma$ and $\\lambda$ seem to be important hyperparameters of this model (for both learning and stability). Were these fixed apriori or dynamically adapted? The same goes for other hyperparameters such as the $N_{old}^{t}$. These choices need to be explained better.\n\n(d) \"Existence of this term indicates that our algorithm requires that internal distribution can be fit with a GMM distribution with high accuracy and this limits applicability of our algorithm. Note however, all parametric learning algorithms face this limitation\". Have the authors considered directly modeling the density p(\\phi(x)) using something like a flow network? This can circumvent the hacky GMM fitting.\n \n(e) \"This is because FMNIST data points are more diverse. As a result, generating pseudo-data points that look more similar to the original data points is easier for the MNIST dataset given that we are using the same network structure for both tasks\". The way I understood from the paper, only the modes of the latent embedding density matter to \"reinforce\" knowledge from the past. Does this mean that the method is going to have bigger issues when moving to, say, mini-ImageNet?\n\n(f) Can the authors give a sense of how much \"drift\" can be accommodated from time to time? For instance, would a full permutation of labels be amenable? To clarify, say, at time t+1 we change the labels of all 0s to 1s, all 1s to 2s, etc. There are no new classes added, but the mapping has been fully modified. I am just trying to get some intuition for what would happen in this case. ", "summary_of_the_paper": "In this paper, the authors propose ICLA, an approach to tackle the problem of incremental and continual learning. By keeping track of a model's \"internal\" representations of data, they identify and overcome the \"drift\" issue (as in continual learning). Particularly, they adopt an encoder-decoder style architecture to learn an embedding space that serves two purposes. (i) The embeddings from the encoder serve as an \"internal\" representation of data whose distribution is explicitly estimated using a Gaussian Mixture Model. (ii) By sampling embeddings from this GMM and passing them through the decoder, they turn this into a generative framework to support memory replay to overcome forgetting. When new classes are incorporated (as in incremental learning), additional components are added to the parameterization of the GMM. The authors provide some theoretical guarantees for error bounds and conduct empirical evaluations on MNIST and FMNIST.", "summary_of_the_review": "This is a neat and well-thought-out idea. However, as I've expressed in the main review, some of the claims need further justification and the paper clarity needs to be significantly improved. I am willing to update my score if the authors are able to provide a convincing response!", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "4RTGz-IhVnH", "reviewer": "Reviewer_WRij", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The idea of this paper is kind of reasonable but the significance and novelty are rather incremental. There are quite a few prior works that have proposed to apply VAE in continual learning and there are also some works that have proposed to generate pseudo samples for replay.  The discussion with related work is very insufficient. There are very close prior works that should be discussed and compared, such as [1,2,3,4,5,6]. Using the GMM model to simulate a multi-modal distribution is a rather naive way and the relation to PDP is also loose. As in the PDP the main point is the hierarchy of clustering concepts which is not exploited in this paper.  The experiments are insufficient because there are only MNIST and FMNIST tasks without any more realistic dataset such as commonly used CIFAR or TinyImgNet. The comparison with other baselines is insufficient either as mentioned above. And the memory size of memory-based reply methods should be equivalent to the memory cost of the VAE and GMM model, otherwise, it is not a fair comparison. The basic memory-based replay methods such as ER or Gdumb should be considered as a baseline as well. \n\nSome minor issues:\n1. Most of the citations should be in parentheses.\n2. No enough explain of the GMM model, is it isotropic Gaussian? What's the definition of the covariate matrix?\n3. Fig.1 is confusing as the pseudo points from the decoder should be fed to the classifier for replay, but the arrow is pointing to the encoder.\n4. Fig.3 should be Tab.3\n \n\n\n[1]. Rannen, Amal, et al. \"Encoder based lifelong learning.\" Proceedings of the IEEE International Conference on Computer Vision. 2017.\n[2]. Rao, Dushyant, et al. \"Continual Unsupervised Representation Learning.\" Advances in Neural Information Processing Systems 32 (2019): 7647-7657.\n[3]. Jeon, Ik Hwan, and Soo Young Shin. \"Continual Representation Learning for Images with Variational Continual Auto-Encoder.\" ICAART (2). 2019.\n[4]. van de Ven, Gido M., Hava T. Siegelmann, and Andreas S. Tolias. \"Brain-inspired replay for continual learning with artificial neural networks.\" Nature communications 11.1 (2020): 1-14.\n[5]. Pellegrini, Lorenzo, et al. \"Latent replay for real-time continual learning.\" 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2020\n[6]. Silver, Daniel L., and Sazia Mahfuz. \"Generating accurate pseudo examples for continual learning.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops. 2020.", "summary_of_the_paper": "This paper proposes training a VAE in parallel with a classifier in continual learning and learning a Gaussian Mixture Model in the embedding space for sampling pseudo data points in experience replay.  The components in the Gaussian Mixture Model correspond to the classes of the data.   ", "summary_of_the_review": "The contribution of this work is rather incremental and there is not enough discussion with related work, the experiments are not convincing either. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper tackles the challenge of continual learning. It approaches the problem by combining a Gaussian Mixture Model (GMM) to model concepts in a latent space and and a decoder system to generate new data points for pseudo-rehearsal and maintenance of previous information. When new concepts arrive, the GMM can be updated with rehearsal serving to prevent forgetting. The authors show competitive results on incremental learning of MNIST and FMNIST.

The scores were mostly below threshold, with one above threshold (5,3,5,6). The reviewers generally agreed the approach was interesting and they appreciated the theoretical treatments. However, there were a number of concerns, the central ones being the lack of clarity and the lack of convincing empirical demonstrations of scalability. The authors attempted to address the concerns, but they were not able to show good performance on larger datasets. The suggested this was due to the complexity of the encoding model, but they were unable to demonstrate this concretely. The reviewers' scores did not change, though, and the consensus was that this paper was not quite ready for publication. Given these considerations, and an average final score of 4.75, a decision of reject was reached.
Decision: Reject