1 Introduction
Recent outbreak of deep learning has led to algorithms with humanlevel performance for many machine learning applications. However, this success is highly limited to single task learning, and retaining learned knowledge in a continual learning setting remains a major challenge. That is, when a deep network is trained on multiple sequential tasks with diverse distributions, the new obtained knowledge usually interferes with past learned knowledge. As a result, the network often is unable to accumulate learned knowledge in a manner consistent with past experience and forgets past learned tasks by the time the new task is learned. This phenomenon is called “catastrophic forgetting” in the literature
[French1999], which is in contrast with continual learning ability of humans over their lifetime.To mitigate catastrophic forgetting, a main approach is to rely on replaying data points from past tasks that are stored selectively in a memory buffer [Robins1995]. This is inspired from the Complementary Learning Systems (CLS) theory [McClelland et al.1995]. CLS theory hypothesizes that a dual long and shortterm memory system, involving the neocortex and the hippocampus, is necessary for continual lifelong learning ability of humans. In particular, the hippocampus rapidly encodes recent experiences as a shortterm memory that is used to consolidate the knowledge in the slower neocortex as longterm memory through experience replays during sleep/conscious recalls [Diekelmann and Born2010]. Similarly, if we selectively store samples from past tasks in a buffer, like in the neocortex, these samples can be replayed to the deep network in parallel with current task samples from recentmemory hipppocampal storage to train the deep network jointly on past and current experiences. In other words, the online sequential learning problem is recast as an offline multitask learning problem that guarantees learning all tasks. A major issue with this approach is that the memory size grows as more tasks are learned and storing tasks’ data points and updating the replaying process becomes more complex. Building upon recent successes of generative models [Goodfellow et al.2014], this challenge has been addressed by amending the network structure such that it can generate pseudodata points for the past learned tasks without storing data points [Shin et al.2017].
In this paper, our goal is to address catastrophic forgetting via coupling sequential tasks in a latent embedding space. We model this space as output of a deep encoder, which is between the input and the output layers of a deep classifier. Representations in this embedding space can be thought of neocortex representations in the brain, which capture learned knowledge. To consolidate knowledge, we minimize the discrepancy between the distributions of all tasks in the embedding space. In order to mimic the offline memory replay process in the sleeping brain
[Rasch and Born2013], we amend the deep encoder with a decoder network to make the classifier network generative. The resulting autoencoding pathways can be thought of neocortical areas, which encodes and remembers past experiences. We learn a parametric distribution in the embedding space that can be used to generate pseudodata points through the decoder network, which can be used for experience replay of the previous tasks towards incorporation of new knowledge. This would enforce the embedding to be invariant with respect to the tasks as more tasks are learned, i.e., the network would retain the past learned knowledge.
2 Related Work
Past works have addressed catastrophic forgetting using two main approaches: model consolidation [Kirkpatrick et al.2017] and experience replay [Robins1995]. Both approaches implement a notion of memory to enable a network to remember the distributions of past learned tasks.
The idea of model consolidation is based upon separating the information pathway for different tasks in the network such that new experiences do not interfere with past learned knowledge. This idea is inspired from the notion of structural plasticity [Lamprecht and LeDoux2004]
. During learning a task, important weight parameters for that task are identified and are consolidated when future tasks are learned. As a result, the new tasks are learned through free pathways in the network; i.e., the weights that are important to retain knowledge about distributions of past tasks mostly remain unchanged. Several methods exist for identifying important weight parameters. Elastic Weight Consolidation (EWC) models posterior distribution of weights of a given network as a Gaussian distribution which is centered around the weight values from last learned past tasks and a precision matrix, defined as the Fisher information matrix of all network weights. The weights then are consolidated according to their importance, i.e., the value of Fisher coefficient
[Kirkpatrick et al.2017]. In contrast to EWC, Zenke et al. [Zenke et al.2017] consolidate weights in an online scheme during learning a task. If a network weight contributes considerably to changes in the network loss, it is identified as an important weight. More recently, Aljundi et al. [Aljundi et al.2018] use a semiHebbian learning procedure to compute the importance of the weight parameters in a both unsupervised and online scheme. The issue with the methods based on structural plasticity is that the network learning capacity is compromised to avoid catastrophic forgetting. As a result, the learning ability of the network decreases as more tasks are learned.Methods that use experience replay, use CLS theory to retain the past tasks’ distributions via replaying selected representative samples of past tasks continuously. Prior works mostly have investigated on how to store a subset of past experiences to reduce dependence on memory. These samples can be selected in different ways. Schaul et al. select samples such that the effect of uncommon samples in the experience is maximized [Schaul et al.2016]. Isele and Cosgun explore four potential strategies to select more helpful samples in a buffer for replay [Isele and Cosgun2018]. The downside is that storing samples requires memory and becomes more complex as more tasks are learned. To reduce dependence on a memory buffer, Shin et al. [Shin et al.2017] developed a more efficient alternative by considering a generative model that can produce pseudodata points of past tasks to avoid storing real data points. They use a generative adversarial structures to learn the tasks distributions to allow for generating pseudodata points without storing data. However, adversarial learning is known to require deliberate architecture design and selection of hyperparameters [Roth et al.2017], and can suffer from mode collapse [Srivastava et al.2017]. Alternatively, we demonstrate that a simple autoencoder structure can be used as the base generative model. Our contribution is to match the distributions of the tasks in the middle layer of the autoencoder and learn a shared distribution across the tasks to couple them. The shared distribution is then used to generate samples for experience replay to avoid forgetting. We demonstrate theoretically and empirically effectiveness of our method on benchmark tasks that has been used in the literature.
3 Generative Continual Learning
We consider a lifelong learning setting [Chen and Liu2016], where a learning agent faces multiple, consecutive tasks in a sequence . The agent learns a new task at each time step and proceeds to learn the next task. Each task is learned based upon the experiences, gained from learning past tasks. Additionally, the agent may encounter the learned tasks in future and hence must optimize its performance across all tasks, i.e., not to forget learned tasks when future tasks are learned. The agent also does not know a priori, the total number of tasks, which potentially might not be finite, distributions of the tasks, and the order of tasks.
Let at time , the current task with training dataset
arrives. We consider classification tasks where the training data points are drawn i.i.d. in pairs from the joint probability distribution, i.e.,
which has the marginal distribution over. We assume that the lifelong learning agent trains a deep neural network
with learnable weight parameters to map the data points to the corresponding onehot labels . Learning a single task in isolation is straight forward. The agent can solve for the optimal network weight parameters using standard empirical risk minimization (ERM), , whereis a proper loss function, e.g., cross entropy. Given large enough number of data points
, the model trained on a single task will generalize well on the task test samples as the empirical risk would be a suitable surrogate for the real risk function, [ShalevShwartz and BenDavid2014]. The agent then can advance to learn the next task, but the challenge is that ERM is unable to tackle catastrophic forgetting as the model parameters are learned using solely the current task data which can potentially have very different distribution. Catastrophic forgetting can be considered as the result of considerable deviations of from past values over time as a result of drifts in tasks’ distributions . As a result, the updated can potentially be highly nonoptimal for previous tasks. Our idea is to prevent catastrophic forgetting through mapping all tasks’ data into an embedding space, where tasks share a common distribution. We represent this space by the output of a deep network midlayer and we condition updating to what has been learned before in this embedding. In other words, we want to train the deep network such the tasks are coupled in the embedding space through updating conditioned on .High performance of deep networks stems from learning datadriven and taskdependent high quality features [Krizhevsky et al.2012]. In other words, a deep net maps data points into a discriminative embedding space, captured by network layers, where classification can be performed easily, e.g., classes become separable in the embedding. Following this intuition, we consider the deep net to be combined of of an encoder with learnable parameter , i.e., early layers of the network, and a classifier network with learnable parameters , i.e., higher layers of the network. The encoder subnetwork maps the data points into the embedding space which describes the input in terms of abstract discriminative features. Note that after training, as a deterministic function, the encoder network changes the input task data distribution.
If the embedding space is discriminative, this distribution can be modeled as a multimodal distribution for a given task, e.g., Gaussian mixture model (GMM). Catastrophic forgetting occurs because this distribution is not stationery with respect to different tasks. The idea that we want to explore is based on training
such that all tasks share a similar distribution in the embedding, i.e., the new tasks are learned such that their distribution in the embedding match the past experience, captured in the shared distribution. Doing so, the embedding space becomes invariant with respect to any learned input task which in turn mitigates catastrophic forgetting.The key questions is how to adapt the standard supervised learning model
such that the embedding space, captured in the deep network, becomes taskinvariant. Following prior discussion, we use experience replay as the main strategy. We expand the base network into a generative model by amending the model with a decoder , with learnable parameters . The encoder maps the data representation back to the input space and effectively make the pair an autoencoder. If implemented properly, we would learn a discriminative data distribution in the embedding space which can be approximated by a GMM. This distribution captures our knowledge about past learned tasks. When a new task arrives, pseudodata points for past tasks can be generated by sampling from this distribution and feeding the samples to the decoder network. These pseudodata points can be used for experience replay in order to tackle catastrophic forgetting. Additionally, we need to learn the new task such that its distribution matches the past shared distribution. As a result, future pseudodata points would represent the current task as well. Figure 1 presents a highlevel blockdiagram visualization of our framework.4 Optimization Method
Following the above framework, learning the first task () reduces to minimizing discrimination loss term for classification and reconstruction loss for the autoencoder to solve for optimal parameters and :
(1) 
where is the reconstruction loss and is a tradeoff parameter between the two loss terms.
Upon learning the first task, as well as subsequent future tasks, we can fit a GMM distribution with components to the empirical distribution represented by data samples in the embedding space. The intuition behind this possibility is that as the embedding space is discriminative, we expect data points of each class form a cluster in the embedding. Let denote this parametric distribution. We update this distribution after learning each task to accumulative what has been learned from the new task to the distribution. As a result, this distribution captures knowledge about past. Upon learning this distribution, experience replay is feasible without saving data points. One can generate pseudodata points in future through random sampling from and then passing the samples through the decoder subnetwork. It is also crucial to learn the current task such that its corresponding distribution in the embedding matches . Doing so, ensures suitability of GMM to model the empirical distribution. The alternative approach is to use a Variational Auto encoder (VAE), but the discriminator loss helps forming the clusters automatically which in turn makes normal autoencoders a feasible solution.
Let denote the pseudodataset, generated at . Following our framework, learning subsequent tasks reduces to solving the following problem:
(2) 
where is a discrepancy measure, i.e., a metric, between two probability distributions and is a tradeoff parameter. The first four terms in Eq. (2) are empirical classification risk and autoencoder reconstruction loss terms for the current task and the generated pseudodataset. The third and the fourth term enforce learning the current task such that the past learned knowledge is not forgotten. The fifth term is added to enforce the learned embedding distribution for the current task to be similar to what has been learned in the past, i.e., taskinvariant. Note that we have conditioned the distance between two distribution on classes to avoid class matching challenge, i.e., when wrong classes across two tasks are matched in the embedding, as well as to prevent mode collapse from happening. Classconditional matching is feasible because we have labels for both distributions. Adding this term guarantees that we can continually use GMM to fit the shared distribution in the embedding.
The main remaining question is selecting such that it fits our problem. Since we are computing the distance between empirical distributions, through drawn samples, we need a metric that can measure distances between distributions using the drawn samples. Additionally, we must select a metric which has nonvanishing gradients as deep learning optimization techniques are gradientbased methods. For these reasons, common distribution distance measures such as KL divergence and Jensen–Shannon divergence are not suitable [Kolouri et al.2018]. We rely on Wasserstein Distance (WD) metric [Bonnotte2013] which has been used extensively in deep learning applications. Since computing WD is computationally expensive, we use Sliced Wasserstein Distance (SWD) [Rabin and Peyré2011] which approximates WD, but can be computed efficiently.
SWD is computed through slicing a highdimensional distributions. The dimensional distribution is decomposed into onedimensional marginal distributions by projecting the distribution into onedimensional spaces that cover the highdimensional space. For a given distribution , a onedimensional slice of the distribution is defined as:
(3) 
where denotes the Kronecker delta function,
denotes the vector dot product,
is the dimensional unit sphere and is the projection direction. In other words, is a marginal distribution of obtained from integratingover the hyperplanes orthogonal to
.SWD approximates the Wasserstein distance between two distributions and by integrating the Wasserstein distances between the resulting sliced marginal distributions of the two distributions over all :
(4) 
where denotes the Wasserstein distance. The main advantage of using SWD is that it can be computed efficiently as the Wasserstein distance between onedimensional distributions has a closed form solution and is equal to the
distance between the inverse of their cumulative distribution functions. On the other hand, the
distance between cumulative distribution can be approximated as the distance between the empirical cumulative distributions which makes SWD suitable in our framework. Finally, to approximate the integral in Eq. (4), we relay on a Monte Carlo style integration and approximate the SWD between dimensional samples and in the embedding space as the following sum:(5) 
where denote random samples that are drawn from the unit dimensional ball , and and are the sorted indices of for the two onedimensional distributions. We utilize the SWD as the discrepancy measure between the distributions in Eq. (2) to learn each task. We tackle catastrophic forgetting using the proposed procedure. Our algorithm, Generative Autoencoder for Continual Learning (GACL) is summarized in Algorithm 1.
5 Theoretical Justification
We use existing theoretical results about using optimal transport within domain adaptation [Redko et al.2017], to justify why our algorithm can tackle catastrophic forgetting. Note that the hypothesis class in our learning problem is the set of all functions represented by the network parameterized by . For a given model in this class, let denote the observed risk for a particular task and denote the observed risk for learning the network on samples of the distribution . We rely on the following theorem.
Theorem 1 [Redko et al.2017]: Consider two tasks and , and a model trained for , then for any and , there exists a constant number depending on such that for any and with probability at least for all , the following holds:
(6) 
where denotes the Wasserstein distance between empirical distributions of the two tasks and denotes the optimal parameter for training the model on tasks jointly, i.e., .
We observe from Theorem 1 that performance, i.e., real risk, of a model learned for task on another task is upperbounded by four terms: i) model performance on task , ii) the distance between the two distributions, iii) performance of the jointly learned model , and iv) a constant term which depends on the number of data points for each task. Note that we do not have a notion of time in this Theorem and the roles of and can be shuffled and the theorem would still hold. In our framework, we consider the task , to be the pseudotask, i.e., the task derived by drawing samples from and then feeding the samples to the decoder subnetwork. We use this result to conclude the following lemma.
Lemma 1 : Consider GACL algorithm for lifelong learning after is learned at time . Then all tasks and under the conditions of Theorem 1, we can conclude the following inequality:
(7) 
Proof: We consider with empirical distribution and the pseudotask with the distribution in the network input space, in Theorem 1. Using the triangular inequality on the term recursively, i.e., for all , Lemma 1 can be derived.
Lemma 1 explains why our algorithm can tackle catastrophic forgetting. When future tasks are learned, our algorithms updates the model parameters conditioned on minimizing the upper bound of in Eq. 7. Given suitable network structure and in the presence of enough labeled data points, the terms and are minimized using ERM, and the last constant term would be small. The term is minimal because we deliberately fit the distribution to the distribution in the embedding space and ideally learn and such that . This term demonstrates that minimizing the discrimination loss is critical as only then, we can fit a GMM distribution on with high accuracy. Similarly, the sum terms in Eq 7 are minimized because at we draw samples from and enforce indirectly . Since the upper bound of in Eq 7 is minimized and conditioned on tightness of the upper bound, the task will not be forgotten.
6 Experimental Validation
We validate our method on learning two sets of sequential tasks: independent permuted MNIST tasks and related digit classification tasks. Our implementation code and experimental details are available in an online public domain.
6.1 Learning sequential independent tasks
Following the literature, we use permuted MNIST tasks to validate our framework. The sequential tasks involve classification of handwritten images of MNIST () dataset [LeCun et al.1990], where pixel values for each data point are shuffled randomly by a fixed permutation order for each task. As a result, the tasks are independent and quite different from each other. Since knowledge transfer across tasks is less likely to happen, these tasks are a suitable benchmark to investigate the effect of an algorithm on mitigating catastrophic forgetting as past learned tasks are not similar to the current task. We compare our method against: a) normal back propagation (BP) as a lower bound, b) full experience replay (FR) of data for all the previous tasks as an upper bound, and c) EWC as a competing weight consolidation framework. We use the same network structure for all methods for fair comparison.
We learn permuted MNIST using standard stochastic gradient descent and at each iteration, compute the performance of the network on the testing split of each task data. Figure
2 presents results on five permuted MNIST tasks. Figure 1(a) presents learning curves for BP (dotted curves) and EWC (solid curves) ^{1}^{1}1We have used PyTorch implementation of EWC
[Hataya2018]. . We observe that EWC is able to address catastrophic forgetting quite well. But a close inspection reveals that as more tasks are learned, the asymptotic performance on subsequent tasks is less than the single task learning performance (roughly less for the fifth task). This can be understood as a side effect of weight consolidation which limits the learning capacity of the network. This in an inherent limitation for techniques that regularize network parameters to prevent catastrophic forgetting. Figure 1(b) presents learning curves for our method (solid curves) versus FR (dotted curves). As expected, FR can prevents catastrophic forgetting perfectly but as we discussed the downside is memory requirement. FR result in Figure 1(b) demonstrates that the network learning capacity is sufficient for learning these tasks and if we have a perfect generative model, we can prevent catastrophic forgetting without compromising the network learning capacity. Despite more forgetting in our approach compared to EWC, the asymptotic performance after learning each task, just before advancing to learn the next task, has improved. We also observe that our algorithm suffers an initial drop in performance of previous tasks, when we proceed to learn a new task. Forgetting beyond this initial forgetting is negligible. This can be understood as the existing distance between and at . These results may suggest that catastrophic forgetting may be tackled better if both weight consolidation and experience replay are combined.To provide a better intuitive understating, we have also included the representations of the testing data for all tasks in the embedding space of the neural network in Figures 3. We have used UMAP [McInnes et al.2018] to reduce the dimension for visualization purpose. In these figures, each color corresponds to a specific class of digits. We can see that although FR is able to learn all tasks and form distinct clusters for each digit for each task, but five different clusters are formed for each class in the embedding space. This suggests that FR is unable to learn the concept of the same class across different tasks in the embedding space. In comparison, we observe that GACL is able to match the same class across different tasks, i.e., we have exactly ten clusters for the ten digits. This empirical observation demonstrates that we can model the data distribution in the embedding using a multimodal distribution such as GMM.
6.2 Learning sequential tasks in related domains
We performed a second set of experiments on related tasks to investigate the ability of the algorithm to learn new domains. We consider two digit classification datasets for this purpose: MNIST () and USPS () datasets. Despite being similar, USPS dataset is a more challenging task as the number of training set is smaller, 20,000 compared to 60,000 images. We consider the two possible sequential learning scenarios and . We resized the USPS images to pixels to be able to use the same encoder network for both tasks. The experiments can be considered as a special case of domain adaptation as both tasks are digit recognition tasks but in different domains.
Figure 4 presents learning curves for these two tasks. We observe that the network retains the knowledge about the first domain, after learning the second domain. We also observe that forgetting is less sever for the first task in Figures 5. This stems from the fact that MNIST has more training data points. As a result, the empirical distribution can capture the task distribution more accurately. As expected from the theoretical justification, this empirical result suggests the performance of our algorithm depends on closeness of the distribution to the distributions of previous tasks and improving probability estimation will increase performance of our approach. We have also presented UMAP visualization of all tasks data in the embedding space in Figures 5. We can see that as expected the distributions are matched in the embedding space.
7 Conclusions
Inspired from CLS theory, we addressed the challenge of catastrophic forgetting for sequential learning of multiple tasks using experience replay. We amend a base learning model with a generative pathway that encodes experience meaningfully as a parametric distribution in an embedding space. This idea makes experience replay feasible without requiring a memory buffer to store task data. Our algorithm is able to accumulate knowledge consistently to past learned knowledge as the parametric distribution in the embedding space is enforced to be shared across all tasks stored . Compared to modelbased approaches that regularize the network to consolidate the important weights for past tasks, our approach is able to address catastrophic forgetting without limiting the learning capacity of the network. Future works for our approach may extend to learning new tasks and/or classes with limited labeled data points.
8 Acknowledgment
We thank James McClelland, Amarjot Singh, Charles Martin, Nicholas Ketz, and Jeffrey Krichmar for helpful feedback in the development and analysis of this work and conceptual discussions surrounding the work. This material is based upon work supported by the United States Air Force and DARPA under Contract No. FA875018C0103. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Air Force and DARPA.
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