# selfless_sequential_learning__c88788be.pdf

Published as a conference paper at ICLR 2019

SELFLESS SEQUENTIAL LEARNING

Rahaf Aljundi KU Leuven ESAT-PSI, Belgium rahaf.aljundi@gmail.com

Marcus Rohrbach Facebook AI Research mrf@fb.com

Tinne Tuytelaars KU Leuven ESAT-PSI, Belgium tinne.tuytelaars@esat.kuleuven.be

Sequential learning, also called lifelong learning, studies the problem of learning tasks in a sequence with access restricted to only the data of the current task. In this paper we look at a scenario with fixed model capacity, and postulate that the learning process should not be selfish, i.e. it should account for future tasks to be added and thus leave enough capacity for them. To achieve Selfless Sequential Learning we study different regularization strategies and activation functions. We find that imposing sparsity at the level of the representation (i.e. neuron activations) is more beneficial for sequential learning than encouraging parameter sparsity. In particular, we propose a novel regularizer, that encourages representation sparsity by means of neural inhibition. It results in few active neurons which in turn leaves more free neurons to be utilized by upcoming tasks. As neural inhibition over an entire layer can be too drastic, especially for complex tasks requiring strong representations, our regularizer only inhibits other neurons in a local neighbourhood, inspired by lateral inhibition processes in the brain. We combine our novel regularizer with state-of-the-art lifelong learning methods that penalize changes to important previously learned parts of the network. We show that our new regularizer leads to increased sparsity which translates in consistent performance improvement on diverse datasets.

1 INTRODUCTION

Sequential learning, also referred to as continual, incremental, or lifelong learning (LLL), studies the problem of learning a sequence of tasks, one at a time, without access to the training data of previous or future tasks. When learning a new task, a key challenge in this context is how to avoid catastrophic interference with the tasks learned previously (French, 1999; Li & Hoiem, 2016). Some methods exploit an additional episodic memory to store a small amount of previous tasks data to regularize future task learning (e.g. Lopez-Paz et al. (2017)). Others store previous tasks models and at test time, select one model or merge the models (Rusu et al., 2016; Aljundi et al., 2016; Lee et al., 2017). In contrast, in this work we are interested in the challenging situation of learning a sequence of tasks without access to any previous or future task data and restricted to a fixed model capacity, as also studied in Kirkpatrick et al. (2016); Aljundi et al. (2017); Fernando et al. (2017); Mallya & Lazebnik (2017); Serrà et al. (2018). This scenario not only has many practical benefits, including privacy and scalability, but also resembles more closely how the mammalian brain learns tasks over time.

The mammalian brain is composed of billions of neurons. Yet at any given time, information is represented by only a few active neurons resulting in a sparsity of 90-95% (Lennie, 2003). In neural biology, lateral inhibition describes the process where an activated neuron reduces the activity of its weaker neighbors. This creates a powerful decorrelated and compact representation with minimum interference between different input patterns in the brain (Yu et al., 2014). This is in stark contrast with artificial neural networks, which typically learn dense representations that are highly entangled (Bengio et al., 2009). Such an entangled representation is quite sensitive to changes in the

Published as a conference paper at ICLR 2019

(a) Parameter sparsity (b) Representation sparsity

T1 T2 T1 T2

Figure 1: The difference between parameter sparsity (a) and representation sparsity (b) in a simple two tasks case. First layer indicates input patterns. Learning the first task utilizes parts indicated in red. Task 2 has different input patterns and uses parts shown in green. Orange indicates changed neurons activations as a result of the second task. In (a), when an example from the first task is encountered again, the activations of the first layer will not be affected by the changes, however, the second and later layer activations are changed. Such interference is largely reduced when imposing sparsity on the representation (b).

input patterns, in that it responds differently to input patterns with only small variations. French (1999) suggests that an overlapped internal representation plays a crucial role in catastrophic forgetting and reducing this overlap would result in a reduced interference. Cogswell et al. (2015) show that when the amount of overfitting in a neural network is reduced, the representation correlation is also reduced. As such, learning a disentangled representation is more powerful and less vulnerable to catastrophic interference. However, if the learned disentangled representation at a given task is not sparse, only little capacity is left for the learning of new tasks. This would in turn result in either an underfitting to the new tasks or again a forgetting of previous tasks. In contrast, a sparse and decorrelated representation would lead to a powerful representation and at the same time enough free neurons that can be changed without interference with the neural activations learned for the previous tasks.

In general, sparsity in neural networks can be thought of either in terms of the network parameters or in terms of the representation (i.e., the activations). In this paper we postulate, and confirm experimentally, that a sparse and decorrelated representation is preferable over parameter sparsity in a sequential learning scenario. There are two arguments for this: first, a sparse representation is less sensitive to new and different patterns (such as data from new tasks) and second, the training procedure of the new tasks can use the free neurons leading to less interference with the previous tasks, hence reducing forgetting. In contrast, when the effective parameters are spread among different neurons, changing the ineffective ones would change the function of their corresponding neurons and hence interfere with previous tasks (see also Figure 1). Based on these observations, we propose a new regularizer that exhibits a behavior similar to the lateral inhibition in biological neurons. The main idea of our regularizer is to penalize neurons that are active at the same time. This leads to more sparsity and a decorrelated representation. However, complex tasks may actually require multiple active neurons in a layer at the same time to learn a strong representation. Therefore, our regularizer, Sparse coding through Local Neural Inhibition and Discounting (SLNID), only penalizes neurons locally. Furthermore, we don t want inhibition to affect previously learned tasks, even if later tasks use neurons from earlier tasks. An important component of SLNID is thus to discount inhibition from/to neurons which have high neuron importance a new concept that we introduce in analogy to parameter importance (Kirkpatrick et al., 2016; Zenke et al., 2017; Aljundi et al., 2017). When combined with a state-of-the-art important parameters preservation method (Aljundi et al., 2017; Kirkpatrick et al., 2016), our proposed regularizer leads to sparse and decorrelated representations which improves the lifelong learning performance.

Our contribution is threefold. First, we direct attention to Selfless Sequential Learning and study a diverse set of representation based regularizers, parameter based regularizers, as well as sparsity inducing activation functions to this end. These have not been studied extensively in the lifelong learning literature before. Second, we propose a novel regularizer, SLNID, which is inspired by lateral inhibition in the brain. Third, we show that our proposed regularizer consistently outperforms alternatives on three diverse datasets (Permuted MNIST, CIFAR, Tiny Imagenet) and we compare to and outperform state-of-the-art LLL approaches on an 8-task object classification challenge. SLNID can be applied to different regularization based LLL approaches, and we show experiments with MAS (Aljundi et al., 2017) and EWC (Kirkpatrick et al., 2016).

In the following, we first discuss related approaches to LLL and different regularization criteria from a LLL perspective (Section 2). We proceed by introducing Selfless Sequential Learning and detailing our novel regularizer (Section 3). Section 4 describes our experimental evaluation, while Section 5 concludes the paper.

Published as a conference paper at ICLR 2019

2 RELATED WORK

The goal in lifelong learning is to learn a sequence of tasks without catastrophic forgetting of previously learned ones (Thrun & Mitchell, 1995). One can identify different approaches to introducing lifelong learning in neural networks. Here, we focus on learning a sequence of tasks using a fixed model capacity, i.e. with a fixed architecture and fixed number of parameters. Under this setting, methods either follow a pseudo rehearsal approach, i.e. using the new task data to approximate the performance of the previous task (Li & Hoiem, 2016; Triki et al., 2017), or aim at identifying the important parameters used by the current set of tasks and penalizing changes to those parameters by new tasks (Kirkpatrick et al., 2016; Zenke et al., 2017; Aljundi et al., 2017; Chaudhry et al., 2018; Liu et al., 2018). To identify the important parameters for a given task, Elastic Weight Consolidation (Kirkpatrick et al., 2016) uses an approximation of the Fisher information matrix computed after training a given task. Liu et al. (2018) suggest a network reparameterization to obtain a better diagonal approximation of the Fisher Information matrix of the network parameters. Path Integral (Zenke et al., 2017) estimates the importance of the network parameters while learning a given task by accumulating the contribution of each parameter to the change in the loss. Chaudhry et al. (2018) suggest a KL-divergence based generalization of Elastic Weight Consolidation and Path Integral. Memory Aware Synapses (Aljundi et al., 2017) estimates the importance of the parameters in an online manner without supervision by measuring the sensitivity of the learned function to small perturbations on the parameters. This method is less sensitive to the data distribution shift, and a local version proposed by the authors resembles applying Hebb rule (Hebb, 2002) to consolidate the important parameters, making it more biologically plausible.

A common drawback of all the above methods is that learning a task could utilize a good portion of the network capacity, leaving few "free" neurons to be adapted by the new task. This in turn leads to inferior performance on the newly learned tasks or forgetting the previously learned ones, as we will show in the experiments. Hence, we study the role of sparsity and representation decorrelation in sequential learning. This aspect has not received much attention in the literature yet. Very recently, (Serrà et al., 2018) proposed to overcome catastrophic forgetting through learned hard attention masks for each task with L1 regularization imposed on the accumulated hard attention masks. This comes closer to our approach although we study and propose a regularization scheme on the learned representation.

The concept of reducing the representation overlap has been suggested before in early attempts towards overcoming catastrophic forgetting in neural networks (French, 1999). This has led to several methods with the goal of orthogonalizing the activations (French, 1992; 1994; Kruschke, 1992; 1993; Sloman & Rumelhart, 1992). However, these approaches are mainly designed for specific architectures and activation functions, which makes it hard to integrate them in recent neural network structures.

The sparsification of neural networks has mostly been studied for compression. SVD decomposition can be applied to reduce the number of effective parameters (Xue et al., 2013). However, there is no guarantee that the training procedure converges to a low rank weight matrix. Other works iterate between pruning and retraining of a neural network as a post processing step (Liu et al., 2015; Sun et al., 2016; Aghasi et al., 2017; Louizos et al., 2017). While compressing a neural network by removing parameters leads to a sparser neural network, this does not necessarily lead to a sparser representation. Indeed, a weight vector can be highly sparse but spread among the different neurons. This reduces the effective size of a neural network, from a compression point of view, but it would not be beneficial for later tasks as most of the neurons are already occupied by the current set of tasks. In our experiments, we show the difference between using a sparse penalty on the representation versus applying it to the weights.

3 SELFLESS SEQUENTIAL LEARNING

One of the main challenges in single model sequential learning is to have capacity to learn new tasks and at the same time avoid catastrophic forgetting of previous tasks as a result of learning new tasks. In order to prevent catastrophic forgetting, importance weight based methods such as EWC (Kirkpatrick et al., 2016) or MAS (Aljundi et al., 2017) introduce an importance weight Ωk for each parameter θk in the network. While these methods differ in how to estimate the important parameters,

Published as a conference paper at ICLR 2019

all of them penalize changes to important parameters when learning a new task Tn using L2 penalty:

Tn : min θ 1 M

m=1 L(ym, f(xm, θn)) + λΩ X

k Ωk(θn k θn 1 k )2 (1)

where θn 1 = {θn 1 k } are the optimal parameters learned so far, i.e. before the current task. {xm} is the set of M training inputs, with {f(xm, θn)} and {ym} the corresponding predicted and desired outputs, respectively. λΩis a trade-off parameter between the new task objective L and the changes on the important parameters, i.e. the amount of forgetting.

In this work we introduce an additional regularizer RSSL which encourages sparsity in the activations Hl = {hm i } for each layer l.

Tn : min θ 1 M

m=1 L(ym, f(xm, θn)) + λΩ X

k Ωk(θn k θn 1 k )2 + λSSL X

l RSSL(Hl) (2)

λSSL and λΩare trade-off parameters that control the contribution of each term. When training the first task (n = 1), Ωk = 0.

3.1 SPARSE CODING THROUGH NEURAL INHIBITION (SNI)

Now we describe how we obtain a sparse and decorrelated representation. In the literature sparsity has been proposed by Glorot et al. (2011) to be combined with the rectifier activation function (Re LU) to control unbounded activations and to increase sparsity. They minimize the L1 norm of the activations (since minimizing the L0 norm is an NP hard problem). However, L1 norm imposes an equal penalty on all the active neurons leading to small activation magnitude across the network. Learning a decorrelated representation has been explored before with the goal of reducing overfitting. This is usually done by minimizing the Frobenius norm of the covariance matrix corrected by the diagonal, as in Cogswell et al. (2015) or Xiong et al. (2016). Such a penalty results in a decorrelated representation but with activations that are mostly close to a non zero mean value. We merge the two objectives of sparse and decorrelated representation resulting in the following objective:

RSNI(Hl) = 1

m hm i hm j , i = j (3)

where we consider a hidden layer l with activations Hl = {hm i } for a set of inputs X = {xm} and i, j 1, .., N running over all N neurons in the hidden layer. This formula differs from minimizing the Frobenius norm of the covariance matrix in two simple yet important aspects: (1) In the case of a Re LU activation function, used in most modern architectures, a neuron is active if its output is larger than zero, and zero otherwise. By assuming a close to zero mean of the activations, µi 0 i 1, .., N, we minimize the correlation between any two active neurons. (2) By evaluating the derivative of the presented regularizer w.r.t. the activation, we get: RSNI(Hl)

j =i hm j (4)

i.e., each active neuron receives a penalty from every other active neuron that corresponds to that other neuron s activation magnitude. In other words, if a neuron fires, with a high activation value, for a given example, it will suppress firing of other neurons for that same example. Hence, this results in a decorrelated sparse representation.

3.2 SPARSE CODING THROUGH LOCAL NEURAL INHIBITION (SLNI)

The loss imposed by the SNI objective will only be zero when there is at most one active neuron per example. This seems to be too harsh for complex tasks that need a richer representation. Thus, we suggest to relax the objective by imposing a spatial weighting to the correlation penalty. In other words, an active neuron penalizes mostly its close neighbours and this effect vanishes for neurons further away. Instead of uniformly penalizing all the correlated neurons, we weight the correlation penalty between two neurons with locations i and j using a Gaussian weighting. This gives

RSLNI(Hl) = 1

i,j e (i j)2

m hm i hm j , i = j (5)

Published as a conference paper at ICLR 2019

As such, each active neuron inhibits its neighbours, introducing a locality in the network inspired by biological neurons. While the notion of neighbouring neurons is not well established in a fully connected network, our aim is to allow few neurons to be active and not only one, thus those few activations don t have to be small to compensate for the penalty. σ2 is a hyper parameter representing the scale at which neurons can affect each other. Note that this is somewhat more flexible than decorrelating neurons in fixed groups as used in Xiong et al. (2016). Our regularizer inhibits locally the active neurons leading to a sparse coding through local neural inhibition.

3.3 NEURON IMPORTANCE FOR DISCOUNTING INHIBITION

Our regularizer is to be applied for each task in the learning sequence. In the case of tasks with completely different input patterns, the active neurons of the previous tasks will not be activated given the new tasks input patterns. However, when the new tasks are of similar or shared patterns, neurons used for previous tasks will be active. In that case, our penalty would discourage other neurons from being active and encourage the new task to adapt the already active neurons instead. This would interfere with the previous tasks and could increase forgetting which is exactly what we want to overcome. To avoid such interference, we add a weight factor taking into account the importance of the neurons with respect to the previous tasks. To estimate the importance of the neurons, we use as a measure the sensitivity of the loss at the end of the training to their changes. This is approximated by the gradients of the loss w.r.t. the neurons outputs (before the activation function) evaluated at each data point. To get an importance value, we then accumulate the absolute value of the gradients over the given data points obtaining importance weight αi for neuron ni:

m=1 | gi(xm) | , gi(xm) = (L(ym, f(xm, θn)))

where nm i is the output of neuron ni for a given input example xm, and θn are the parameters after learning task n. This is in line with the estimation of the parameters importance in Kirkpatrick et al. (2016) but considering the derivation variables to be the neurons outputs instead of the parameters. Instead of relying on the gradient of the loss, we can also use the gradient of the learned function, i.e. the output layer, as done in Aljundi et al. (2017) for estimating the parameters importance. During the early phases of this work, we experimented with both and observed a similar behaviour. For sake of consistency and computational efficiency we utilize the gradient of the function when using Aljundi et al. (2017) as LLL method and the gradient of the loss when experimenting with EWC (Kirkpatrick et al., 2016). Then, we can weight our regularizer as follows:

RSLNID(Hl) = 1

i,j e (αi+αj)e (i j)2

m hm i hm j , i = j (7)

which can be read as: if an important neuron for a previous task is active given an input pattern from the current task, it will not suppress the other neurons from being active neither be affected by other active neurons. For all other active neurons, local inhibition is deployed. The final objective for training is given in Eq. 2, setting RSSL := RSLNID and λSSL := λSLNID. We refer to our full method as Sparse coding through Local Neural Inhibition and Discounting (SLNID).

4 EXPERIMENTS

In this section we study the role of standard regularization techniques with a focus on sparsity and decorrelation of the representation in a sequential learning scenario. We first compare different activation functions and regularization techniques, including our proposed SLNID, on permuted MNIST (Sec. 4.1). Then, we compare the top competing techniques and our proposed method in the case of sequentially learning CIFAR-100 classes and Tiny Imagenet classes (Sec. 4.2). Our SLNID regularizer can be integrated in any importance weight-based lifelong learning approach such as (Kirkpatrick et al., 2016; Zenke et al., 2017; Aljundi et al., 2017). Here we focus on Memory Aware Synapses (Aljundi et al., 2017) (MAS), which is easy to integrate and experiment with and has shown superior performance (Aljundi et al., 2017). However, we also show results with Elastic weight consolidation (Kirkpatrick et al., 2016)(EWC) in Sec. 4.3. Further, we ablate the components of our regularizer, both in the standard setting (Sec. 4.4) as in a setting without hard task boundaries (Sec. 4.5). Finally, we show how our regularizer improves the state-of-the-art performance on a sequence of object recognition tasks (Sec. 4.6).

Published as a conference paper at ICLR 2019

Figure 2: Comparison of different regularization techniques on 5 permuted MNIST sequence. Representation based regularizers are solid bars, bars with lines represent parameters regularizers, dotted bars represent activation functions. Average test accuracy over all tasks is given in the legend. Representation based regularizers achieve higher performance than other compared methods including parameters based regularizers. Our regularizer, SLNID, performs the best on the last two tasks indicating that more capacity is left to learn these tasks.

4.1 AN IN-DEPTH COMPARISON OF REGULARIZERS AND ACTIVATION FUNCTIONS FOR SELFLESS SEQUENTIAL LEARNING

We study possible regularization techniques that could lead to less interference between the different tasks in a sequential learning scenario either by enforcing sparsity or decorrelation. Additionally, we examine the use of activation functions that are inspired by lateral inhibition in biological neurons that could be advantageous in sequential learning. MAS Aljundi et al. (2017) is used in all cases as LLL method. Representation Based methods: - L1-Rep: To promote representational sparsity, an L1 penalty on the activations is used. - Decov (Cogswell et al., 2015) aims at reducing overfitting by decorrelating neuron activations. To do so, it minimizes the Frobenius norm of the covariance matrix computed on the activations of the current batch after subtracting the diagonal to avoid penalizing independent neuron activations. Activation functions: - Maxout network (Goodfellow et al., 2013b) utilizes the maxout activation function. For each group of neurons, based on a fixed window size, only the maximum activation is forwarded to the next layer. The activation function guarantees a minimum sparsity rate defined by the window size. - LWTA (Srivastava et al., 2013): similar idea to the Maxout network except that the non-maximum activations are set to zero while maintaining their connections. In contrast to Maxout, LWTA keeps the connections of the inactive neurons which can be occupied later once they are activated without changing the previously active neuron connections. - Re LU (Glorot et al., 2011) The rectifier activation function (Re LU) used as a baseline here and indicated in later experiments as No-Reg as it represents the standard setting of sequential learning on networks with Re LU. All the studied regularizers use Re LU as activation function. Parameters based regularizers: - Orth Reg (Rodríguez et al., 2016): Regularizing CNNs with locally constrained decorrelations. It aims at decorrelating the feature detectors by minimizing the cosine of the angle between the weight vectors resulting eventually in orthogonal weight vectors. - L2-WD: Weight decay with L2 norm (Krogh & Hertz, 1992) controls the complexity of the learned function by minimizing the magnitude of the weights. - L1-Param: L1 penalty on the parameters to encourage a solution with sparse parameters.

Dropout is not considered as its role contradicts our goal. While dropout can improve each task performance and reduce overfitting, it acts as a model averaging technique. By randomly masking neurons, dropout forces the different neurons to work independently. As such it encourages a redundant representation. As shown by (Goodfellow et al., 2013a) the best network size for classifying MNIST digits when using dropout was about 50% more than without it. Dropout steers the learning of a task towards occupying a good portion of the network capacity, if not all of it, which contradicts the sequential learning needs.

Experimental setup. We use the MNIST dataset (Le Cun et al., 1998) as a first task in a sequence of 5 tasks, where we randomly permute all the input pixels differently for tasks 2 to 5. The goal is to classify MNIST digits from all the different permutations. The complete random permutation of the pixels in each task requires the neural network to instantiate a new neural representation for each pattern. A similar setup has been used by Kirkpatrick et al. (2016); Zenke et al. (2017); Goodfellow et al. (2013a) with different percentage of permutations or different number of tasks. As a base network, we employ a multi layer perceptron with two hidden layers and a Softmax loss.

Published as a conference paper at ICLR 2019

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

Accuracy % after learning all tasks

SLNID:63.3 De Cov:61.19 L1-Rep:55.76 No-Reg:55.31

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 40

Accuracy % after learning all tasks

SLNID:53.96 De Cov:52.47 L1-Rep:52.57 No-Reg:49.56

Figure 3: Comparison of different regularization techniques on a sequence of ten tasks from (a) Cifar split and (b) Tiny Image Net split. The legend shows average test accuracy over all tasks. Simple L1-norm regularizer (L1-Rep) doesn t help in such more complex tasks. Our regularizer SLNID achieves an improvement of 2% over Decov and 4 8% compared to No-Reg.

We experiment with different number of neurons in the hidden layers {128, 64}. For SLNID we evaluate the effect of λSLNID on the performance and the obtained sparsity in Figure 4. In general, the best λSLNID is the minimum value that maintains similar or better accuracy on the first task compared to the unregularized case, and we suggest to use this as a rule-of-thumb to set λSLNID. For λΩ, we have used a high λΩvalue that ensures the least forgetting which allows us to test the effect on the later tasks performance. Note that better average accuracies can be obtained with tuned λΩ. Please refer to Appendix A for hyperparameters and other details.

Results: Figure 2 presents the test accuracy on each task at the end of the sequence, achieved by the different regularizers and activation functions on the network with hidden layer of size 128. Results on a network with hidden layer size 64 are shown in the Appendix B. Clearly, in all the different tasks, the representational regularizers show a superior performance to the other studied techniques. For the regularizers applied to the parameters, L2-WD and L1-Param do not exhibit a clear trend and do not systematically show an improvement over the use of the different activation functions only. While Orth Reg shows a consistently good performance, it is lower than what can be achieved by the representational regularizers. It is worth noting the L1-Rep yields superior performance over L1-Param. This observation is consistent across different sizes of the hidden layers (in Appendix B) and shows the advantage of encouraging sparsity in the activations compared to that in the parameters. Regarding the activation functions, Maxout and LWTA achieve a slightly higher performance than Re LU. We did not observe a significant difference between the two activation functions. However, the improvement over Re LU is only moderate and does not justify the use of a fixed window size and special architecture design. Our proposed regularizer SLNID achieves high if not the highest performance in all the tasks and succeeds in having a stable performance. This indicates the ability of SLNID to direct the learning process towards using minimum amount of neurons and hence more flexibility for upcoming tasks.

Representation sparsity & important parameter sparsity. Here we want to examine the effect of our regularizer on the percentage of parameters that are utilized after each task and hence the

0 1 2 3 4 tasks

Parmeters Sparsity %

SLNID = 0.01 acc:91.94

SLNID = 0.005 acc:94.44

SLNID = 0.001 acc:95.83

SLNID = 0.0005 acc:95.34

SLNID = 0.0001 acc:94.79 No Reg acc:92.52

0.0 0.2 0.4 0.6 0.8 1.0

SLNID De Cov L1-Rep No-Reg L1-Param

Figure 4: On the 5 permuted MNIST sequence, hidden layer=128, Top: percentage of unused parameters in the 1st layer using different λSLNID; Bottom: histogram of neural activations on the first task.

capacity left for the later tasks. On the network with hidden layer size 128, we compute the percentage of parameters with Ωk < 10 2, with Ωk, see Appendix A, the importance weight multiplier estimated and accumulated over tasks. Those parameters can be seen as unimportant and "free" for later tasks. Figure 4(top) shows the percentage of the unimportant (free) parameters in the first layer after each task for different λSLNID values along with the achieved average test accuracy at the end of the sequence. It is clear that the larger λSLNID, i.e., the more neural inhibition, the smaller the percentage of important parameters. Apart from the highest λSLNID where tasks couldn t reach their top performance due to too strong inhibition, improvement over the No-Reg is always observed. The optimal value for lambda seems to be the one that remains close to the optimal performance on the current task, while utilizing the minimum capacity feasible. Next, we compute the average activation per neuron, in the first layer, over all the examples and plot the corresponding histogram for SLNID, De Cov, L1-Rep, L1-Param and No-Reg in Figure 4(bottom) at their setting that yielded the results shown in Figure 2. SLNID has a peak at zero indicating representation sparsity while the other methods values are spread along the line. This seems to hint at the effectiveness of our approach SLNID in learning a sparse yet powerful representation and in turn in a minimal interference between tasks.

Published as a conference paper at ICLR 2019

Permuted mnist Cifar h-layer dim. 128 64 256 128 No-Reg 92.67 90.72 55.06 55.3 SNI 95.79 94.89 55.30 55.75 SNID 95.90 93.82 61.00 60.90 SLNI 95.95 94.87 56.06 55.79 SLNID 95.83 93.89 63.30 61.16 Multi-Task Joint Training* 97.30 96.80 70.99 71.95

Table 1: SLNID ablation. Average test accuracy per task after training the last task in %. * denotes that Multi Task Joint Training violates the LLL scenario as it has access to all tasks at once and thus can be seen as an upper bound.

Method Avg-acc Finetune 32.67 LWF (Li & Hoiem, 2016) 49.49 EBLL (Triki et al., 2017) 50.29 IMM (Lee et al., 2017) 43.4 Path Integral (Zenke et al., 2017) 50.49 EWC (Kirkpatrick et al., 2016) 50.00 MAS (Aljundi et al., 2017) 52.69 SLNID-fc Pretrained (ours) 53.77 SLNID-fc randomly initialized (ours) 54.50

Table 2: 8 tasks object recognition sequence. Average test accuracy per task after training the last task in %.

4.2 10 TASK SEQUENCES ON CIFAR-100 AND TINY IMAGENET

While the previous section focused on learning a sequence of tasks with completely different input patterns and same objective, we now study the case of learning different categories of one dataset. For this we split the CIFAR-100 and the Tiny Image Net (Yao & Miller, 2015) dataset into ten tasks, respectively. We have 10 and 20 categories per task for CIFAR-100 and Tiny Imag Net, respectively. Further details about the experimental setup can be found in appendix A.

We compare the top competing methods from the previous experiments, L1-Rep, De Cov and our SLNID, and No-Reg as a baseline, Re LU in previous experiment. Similarly, MAS Aljundi et al. (2017) is used in all cases as LLL method. Figures 3(a) and 3(b) show the performance on each of the ten tasks at the end of the sequence. For both datasets, we observe that our SLNID performs overall best. L1-Rep and De Cov continue to improve over the non regularized case No-Reg. These results confirm our proposal on the importance of sparsity and decorrelation in sequential learning.

4.3 SLNID WITH EWC (KIRKPATRICK ET AL., 2016)

We have shown that our proposed regularizer SLNID exhibits stable and superior performance on the different tested networks when using MAS as importance weight preservation method. To prove the effectiveness of our regularizer regardless of the used importance weight based method, we have tested SLNID on the 5 tasks permuted MNIST sequence in combination with Elastic Weight Consolidation (EWC,Kirkpatrick et al. (2016)) and obtained a boost in the average performance at the end of the learned sequence equal to 3.1% on the network with hidden layer size 128 and a boost of 2.8% with hidden layer size 64. Detailed accuracies are shown in Appendix B. It is worth noting that with both MAS and EWC our SLNID was able obtain better accuracy using a network with a 64-dimensional hidden size than when training without regularization No-Reg on a network of double that size (128), indicating that SLNID allows to use neurons much more efficiently.

4.4 ABLATION STUDY

Our method can be seen as composed of three components: the neural inhibition, the locality relaxation and the neuron importance integration. To study how these components perform individually, Table 1 reports the average accuracy at the end of the Cifar 100 and permuted MNIST sequences for each variant, namely, SNID without neuron importance (SNI), SNID, SLNID without neuron importance (SLNI) in addition to our full SLNID regularizer. As we explained in Section 3, when tasks have completely different input patterns, the neurons that were activated on the previous task examples will not fire for new task samples and exclusion of important neurons is not mandatory. However, when sharing is present between the different tasks, a term to prevent SLNID from causing any interference is required. This is manifested in the reported results: for permuted MNIST, all the variants work nicely alone, as a result of the simplicity and the disjoint nature of this sequence. However, in the Cifar 100 sequence, the integration of the neuron importance in the SNID and SLNID regularizers exclude important neurons from the inhibition, resulting in a clearly better performance. The locality in SLNID improves the performance in the Cifar sequence, which suggests that a richer representation is needed and multiple active neurons should be tolerated.

Published as a conference paper at ICLR 2019

4.5 SEQUENTIAL LEARNING WITHOUT HARD TASK BOUNDARIES

In the previous experiments, we considered the standard task based scenario as in (Li & Hoiem, 2016; Zenke et al., 2017; Aljundi et al., 2017; Serrà et al., 2018), where at each time step we receive a task along with its training data and a new classification layer is initiated for the new task, if needed. Here, we are interested in a more realistic scenario where the data distribution shifts gradually without hard task boundaries.

Method Avg.acc -tasks models No-Reg w/o MAS 69.20% SLNI w/o MAS 72.14% SLNID w/o MAS 73.03% No-Reg 66.88% SLNI 71.32% SLNID 72.33% Method Avg.acc-last model No-Reg w/o MAS 65.15% SLNI w/o MAS 63.54% SLNID w/o MAS 70.75% No-Reg 66.33% SLNI 64.50% SLNID 70.94%

Table 3: No tasks boundaries test case on Cifar 100. Top block, avg. acc on each group of classes using each group model. Bottom block, avg. acc. on each group at the end of the training.

To test this setting, we use the Cifar 100 dataset. Instead of considering a set of 10 disjoint tasks each composed of 10 classes, as in the previous experiment (Sec. 4.2), we now start by sampling with high probability (2/3) from the first 10 classes and with low probability (1/3) from the rest of the classes. We train the network (same architecture as in Sec. 4.2) for a few epochs and then change the sampling probabilities to be high (2/3) for classes 11 20 and low (1/3) for the remaining classes. This process is repeated until sampling with high probability from the last 10 classes and low from the rest. We use one shared classification layer throughout and estimate the importance weights and the neurons importance after each training step (before changing the sampling probabilities). We consider 6 variants: our SLNID, the ablations SLNI and without regularizer No-Reg, as in Section 4.4, as well each of these three trained without the MAS importance weight regularizer of Aljundi et al. (2017), denoted as w/o MAS. Table 3 presents the accuracy averaged over the ten groups of ten classes, using each group model (i.e. the model trained when this group was sampled with high probability) in the top block and the average accuracy on each of the ten groups at the end of the training (middle and bottom block). We can deduce the following: 1) SLNID improves the performance considerably (by more than 4%) even without importance weight regularizer. 2) In this scenario without hard task boundaries there is less forgetting than in the scenario with hard task boundaries studied in Section 4.2 for Cifar (difference between rows in top block to corresponding rows in middle block). As a result, the improvement obtained by deploying the importance weight regularizer is moderate: at 70.75%, SLNID w/o MAS is already better than No-Reg reaching 66.33%. 3) While SLNI without MAS improves the individual models performance (72.14% compared to 69.20%), it fails to improve the overall performance at the end of the sequence (63.54% compared to 65.15%), as important neurons are not excluded from the penalty and hence they are changed or inhibited leading to tasks interference and performance deterioration.

4.6 COMPARISON WITH THE STATE OF THE ART

To compare our proposed approach with the different state-of-the-art sequential learning methods, we use a sequence of 8 different object recognition tasks, introduced in Aljundi et al. (2017). The sequence starts from Alex Net (Krizhevsky et al., 2012) pretrained on Image Net (Russakovsky et al., 2015) as a base network, following the setting of Aljundi et al. (2017). More details are in Appendix A.4. We compare against the following: Learning without Forgetting (Li & Hoiem, 2016) (Lw F), Incremental Moment Matching (Lee et al., 2017) (IMM), Path Integral (Zenke et al., 2017) and sequential finetuning (Fine Tuning), in addition to the case of MAS (Aljundi et al., 2017) alone, i.e. our No-Reg before. Compared methods were run with the exact same setup as in Aljundi et al. (2017). For our regularizer, we disable dropout, since dropout encourages redundant activations which contradicts our regularizer s role. Also, since the network is pretrained, the locality introduced in SLNID may conflict with the already pretrained activations. For this reason, we also test SLNID with randomly initialized fully connected layers. Our regularizer is applied with MAS as a sequential learning method. Table 2 reports the average test accuracy at the end of the sequence achieved by each method. SLNID improves even when starting from a pretrained network and disabling dropout. Surprisingly, even with randomly initialized fully connected layers, SLNID improves 1.8% over the state of the art using a fully pretrained network.

Published as a conference paper at ICLR 2019

5 CONCLUSION

In this paper we study the problem of sequential learning using a network with fixed capacity a prerequisite for a scalable and computationally efficient solution. A key insight of our approach is that in the context of sequential learning (as opposed to other contexts where sparsity is imposed, such as network compression or avoiding overfitting), sparsity should be imposed at the level of the representation rather than at the level of the network parameters. Inspired by lateral inhibition in the mammalian brain, we impose sparsity by means of a new regularizer that decorrelates nearby active neurons. We integrate this in a model which learns selflessly a new task by leaving capacity for future tasks and at the same time avoids forgetting previous tasks by taking into account neurons importance. Acknowledgment: The first author s Ph D is funded by an FWO scholarship.

Alireza Aghasi, Afshin Abdi, Nam Nguyen, and Justin Romberg. Net-trim: Convex pruning of deep neural networks with performance guarantee. In Advances in Neural Information Processing Systems, pp. 3180 3189, 2017.

Rahaf Aljundi, Punarjay Chakravarty, and Tinne Tuytelaars. Expert gate: Lifelong learning with a network of experts. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.

Rahaf Aljundi, Francesca Babiloni, Mohamed Elhoseiny, Marcus Rohrbach, and Tinne Tuytelaars. Memory aware synapses: Learning what (not) to forget. ar Xiv preprint ar Xiv:1711.09601, 2017.

Yoshua Bengio et al. Learning deep architectures for ai. Foundations and trends R in Machine Learning, 2(1): 1 127, 2009.

Arslan Chaudhry, Puneet K Dokania, Thalaiyasingam Ajanthan, and Philip HS Torr. Riemannian walk for incremental learning: Understanding forgetting and intransigence. ar Xiv preprint ar Xiv:1801.10112, 2018.

Michael Cogswell, Faruk Ahmed, Ross Girshick, Larry Zitnick, and Dhruv Batra. Reducing overfitting in deep networks by decorrelating representations. ar Xiv preprint ar Xiv:1511.06068, 2015.

T. E. de Campos, B. R. Babu, and M. Varma. Character recognition in natural images. In Proceedings of the International Conference on Computer Vision Theory and Applications, Lisbon, Portugal, February 2009.

M. Everingham, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman. The PASCAL Visual Object Classes Challenge 2012 (VOC2012) Results. http://www.pascalnetwork.org/challenges/VOC/voc2012/workshop/index.html.

Chrisantha Fernando, Dylan Banarse, Charles Blundell, Yori Zwols, David Ha, Andrei A Rusu, Alexander Pritzel, and Daan Wierstra. Pathnet: Evolution channels gradient descent in super neural networks. ar Xiv preprint ar Xiv:1701.08734, 2017.

Robert M French. Semi-distributed representations and catastrophic forgetting in connectionist networks. Connection Science, 4(3-4):365 377, 1992.

Robert M French. Dynamically constraining connectionist networks to produce distributed, orthogonal representations to reduce catastrophic interference. network, 1111:00001, 1994.

Robert M French. Catastrophic forgetting in connectionist networks. Trends in cognitive sciences, 3(4):128 135, 1999.

Xavier Glorot, Antoine Bordes, and Yoshua Bengio. Deep sparse rectifier neural networks. In Geoffrey Gordon, David Dunson, and Miroslav Dudík (eds.), Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, volume 15 of Proceedings of Machine Learning Research, pp. 315 323, Fort Lauderdale, FL, USA, 11 13 Apr 2011. PMLR. URL http://proceedings.mlr.press/v15/ glorot11a.html.

Ian J Goodfellow, Mehdi Mirza, Da Xiao, Aaron Courville, and Yoshua Bengio. An empirical investigation of catastrophic forgetting in gradient-based neural networks. ar Xiv preprint ar Xiv:1312.6211, 2013a.

Ian J Goodfellow, David Warde-Farley, Mehdi Mirza, Aaron Courville, and Yoshua Bengio. Maxout networks. ar Xiv preprint ar Xiv:1302.4389, 2013b.

DO Hebb. The organization of behavior. 1949. New York Wiely, 2002.

Published as a conference paper at ICLR 2019

James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, et al. Overcoming catastrophic forgetting in neural networks. ar Xiv preprint ar Xiv:1612.00796, 2016.

Jonathan Krause, Michael Stark, Jia Deng, and Li Fei-Fei. 3d object representations for fine-grained categorization. In Proceedings of the IEEE International Conference on Computer Vision Workshops, pp. 554 561, 2013.

Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In F. Pereira, C. J. C. Burges, L. Bottou, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 25, pp. 1097 1105. Curran Associates, Inc., 2012.

Anders Krogh and John A Hertz. A simple weight decay can improve generalization. In Advances in neural information processing systems, pp. 950 957, 1992.

John K Kruschke. Alcove: an exemplar-based connectionist model of category learning. Psychological review, 99(1):22, 1992.

John K Kruschke. Human category learning: Implications for backpropagation models. Connection Science, 5 (1):3 36, 1993.

Yann Le Cun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278 2324, 1998.

Sang-Woo Lee, Jin-Hwa Kim, Jung-Woo Ha, and Byoung-Tak Zhang. Overcoming catastrophic forgetting by incremental moment matching. ar Xiv preprint ar Xiv:1703.08475, 2017.

Peter Lennie. The cost of cortical computation. Current biology, 13(6):493 497, 2003.

Zhizhong Li and Derek Hoiem. Learning without forgetting. In European Conference on Computer Vision, pp. 614 629. Springer, 2016.

Baoyuan Liu, Min Wang, Hassan Foroosh, Marshall Tappen, and Marianna Pensky. Sparse convolutional neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 806 814, 2015.

Xialei Liu, Marc Masana, Luis Herranz, Joost Van de Weijer, Antonio M Lopez, and Andrew D Bagdanov. Rotate your networks: Better weight consolidation and less catastrophic forgetting. ar Xiv preprint ar Xiv:1802.02950, 2018.

David Lopez-Paz et al. Gradient episodic memory for continual learning. In Advances in Neural Information Processing Systems, pp. 6470 6479, 2017.

Christos Louizos, Karen Ullrich, and Max Welling. Bayesian compression for deep learning. In Advances in Neural Information Processing Systems, pp. 3290 3300, 2017.

S. Maji, J. Kannala, E. Rahtu, M. Blaschko, and A. Vedaldi. Fine-grained visual classification of aircraft. Technical report, 2013.

Arun Mallya and Svetlana Lazebnik. Packnet: Adding multiple tasks to a single network by iterative pruning. ar Xiv preprint ar Xiv:1711.05769, 1(2):3, 2017.

Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning. 2011.

M-E. Nilsback and A. Zisserman. Automated flower classification over a large number of classes. In Proceedings of the Indian Conference on Computer Vision, Graphics and Image Processing, Dec 2008.

Ariadna Quattoni and Antonio Torralba. Recognizing indoor scenes. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pp. 413 420. IEEE, 2009.

Pau Rodríguez, Jordi Gonzalez, Guillem Cucurull, Josep M Gonfaus, and Xavier Roca. Regularizing cnns with locally constrained decorrelations. ar Xiv preprint ar Xiv:1611.01967, 2016.

Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. Image Net Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 115(3):211 252, 2015. doi: 10.1007/s11263-015-0816-y.

Published as a conference paper at ICLR 2019

Andrei A Rusu, Neil C Rabinowitz, Guillaume Desjardins, Hubert Soyer, James Kirkpatrick, Koray Kavukcuoglu, Razvan Pascanu, and Raia Hadsell. Progressive neural networks. ar Xiv preprint ar Xiv:1606.04671, 2016.

Joan Serrà, Dídac Surís, Marius Miron, and Alexandros Karatzoglou. Overcoming catastrophic forgetting with hard attention to the task. ar Xiv preprint ar Xiv:1801.01423, 2018.

Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. ar Xiv preprint ar Xiv:1409.1556, 2014.

Steven A Sloman and David E Rumelhart. Reducing interference in distributed memories through episodic gating. Essays in honor of WK Estes, 1:227 248, 1992.

Rupesh K Srivastava, Jonathan Masci, Sohrob Kazerounian, Faustino Gomez, and Jürgen Schmidhuber. Compete to compute. In C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 26, pp. 2310 2318. Curran Associates, Inc., 2013. URL http://papers.nips.cc/paper/5059-compete-to-compute.pdf.

Yi Sun, Xiaogang Wang, and Xiaoou Tang. Sparsifying neural network connections for face recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4856 4864, 2016.

Sebastian Thrun and Tom M Mitchell. Lifelong robot learning. Robotics and autonomous systems, 15(1-2): 25 46, 1995.

Amal Rannen Triki, Rahaf Aljundi, Mathew B Blaschko, and Tinne Tuytelaars. Encoder based lifelong learning. ar Xiv preprint ar Xiv:1704.01920, 2017.

P. Welinder, S. Branson, T. Mita, C. Wah, F. Schroff, S. Belongie, and P. Perona. Caltech-UCSD Birds 200. Technical Report CNS-TR-2010-001, California Institute of Technology, 2010.

Wei Xiong, Bo Du, Lefei Zhang, Ruimin Hu, and Dacheng Tao. Regularizing deep convolutional neural networks with a structured decorrelation constraint. In ICDM, pp. 519 528, 2016.

Jian Xue, Jinyu Li, and Yifan Gong. Restructuring of deep neural network acoustic models with singular value decomposition. In Interspeech, pp. 2365 2369, 2013.

Leon Yao and John Miller. Tiny imagenet classification with convolutional neural networks. CS 231N, 2015.

Yuguo Yu, Michele Migliore, Michael L Hines, and Gordon M Shepherd. Sparse coding and lateral inhibition arising from balanced and unbalanced dendrodendritic excitation and inhibition. Journal of Neuroscience, 34 (41):13701 13713, 2014.

Friedemann Zenke, Ben Poole, and Surya Ganguli. Improved multitask learning through synaptic intelligence. In Proceedings of the International Conference on Machine Learning (ICML), 2017.

Published as a conference paper at ICLR 2019

A DETAILS ON THE EXPERIMENTAL SETUP

In all designed experiments, our regularizer is applied to the neurons of the fully connected layers. As a future work, we plan to integrate it in the convolutional layers.

A.1 PERMUTED MNIST

The used network is composed of two fully connected layers. All tasks are trained for 10 epochs with a learning rate 10 2 using SGD optimizer. Re LU is used as an activation function unless mentioned otherwise. Throughout the experiment, we used a scale σ for the Gaussian function used for the local inhibition equal to 1/6 of the hidden layer size. For all competing regularizers, we tested different hyper parameters from 10 2 to 10 9 and report the best one. For λΩ, we have used a high λΩvalue that ensures the least forgetting. This allows us to examine the degradation in the performance on the later tasks compared to those learned previously as a result of lacking capacity. Note that better average accuracies can be obtained with tuned λΩ.

In section 4.1 we estimated the free capacity in the network with the percentage of Ωk < 10 2, with Ωk, the importance weight multiplier estimated and accumulated over tasks. We consider Ωk < 10 2 of negligible importanc as in a network trained without a sparsity regularizer, Ωij < 10 2 covers the first 10 percentiles.

A.2 CIFAR-100

As a base network, we use a network similar to the one used by Zenke et al. (2017) but without dropout. We evaluate two variants with hidden size N = {256, 128}. Throughout the experiment, we again used a scale σ for the Gaussian function equal to 1/6 of the hidden layer size. We train the different tasks for 50 epochs with a learning rate of 10 2 using SGD optimizer.

A.3 TINY IMAGENET

We split the Tiny Image Net dataset (Yao & Miller, 2015) into ten tasks, each containing twenty categories to be learned at once. As a base network, we use a variant of VGG (Simonyan & Zisserman, 2014). For architecture details, please refer to Table 4 below.

Layer # filters/neurons Convolution 64 Max Pooling - Convolution 128 Max Pooling - Convolution 256 Max Pooling - Convolution 256 Max Pooling - Convolution 512 Convolution 512 Fully connected 500 Fully connected 500 Fully connected 20

Table 4: Architecture of the network used in the Tiny Imagenet experiment.

Throughout the experiment, we again used a scale σ for the Gaussian function equal to 1/6 of the hidden layer size.

A.4 8 TASK OBJECT RECOGNITION SEQUENCE

The 8 tasks sequence is composed of: 1. Oxford Flowers (Nilsback & Zisserman, 2008), 2. MIT Scenes (Quattoni & Torralba, 2009), 3. Caltech-UCSD Birds (Welinder et al., 2010), 4. Stanford Cars (Krause et al., 2013); 5. FGVC-Aircraft (Maji et al., 2013); 6. VOC Actions (Everingham et al.); 7. Letters (de Campos et al., 2009); and 8. SVHN (Netzer et al., 2011) datasets. We have rerun the different methods and obtain the same reported results as in Aljundi et al. (2017).

Published as a conference paper at ICLR 2019

B EXTRA RESULTS

B.1 PERMUTED MNIST SEQUENCE

In section 4.1, we have studied the performance of different regularizers and activation functions on 5 permuted Mnist tasks in a network with a hidden layer of size 128. Figure 5 shows the average accuracies achieved by each of the studied methods at the end of the learned sequence in a network with a hidden layer of size 64. Similar conclusions can be drawn. Maxout and LWTA perform similarly and improve slightly over Re LU. Regularizers applied to the representation are more powerful for sequential learning than regularizers applied directly to the parameters. Specifically, L1-Rep (orange) is consistently better than L1-Param (pink). Our SLNID is able of maintaining a good performance on all the tasks, achieving among the top average test accuracies. Admittedly, the performances of SLNID is very close to L1-Rep. The difference between these methods stands out more clearly for larger networks and more complex tasks.

T1 T2 T3 T4 T5

Accuracy % after learning all tasks

SLNID:93.89 De Cov:93.58 L1-Rep:93.85 Orth Reg:92.65 L1_Param:91.06 L2_WD:90.99 Maxout:91.52 LWTA:91.22 Rel U:90.72

Figure 5: Comparison of different regularization techniques on 5 permuted MNIST sequence of tasks, hidden size=64. Representation based regularizers are solid bars, bars with lines represent parameters regularizers, dotted bars represent activation functions. See Figure 2 for size 128.

B.2 SLNI WITH EWC

To show that our approach is not limited to MAS (Aljundi et al., 2017), we have also experimented with EWC (Kirkpatrick et al., 2016) as another importance weight based method along with our regularize SLNID on the permuted Mnist sequence. Figure 6 shows the test accuracy of each task at the end of the 5 permuted Mnist sequence achieved by our SLNID combined with EWC and by No-Reg (here indicating EWC without regularization). It is clear that SLNID succeeds to improve the performance on all the learned tasks which validates the utility of our approach with different sequential learning methods.

T1 T2 T3 T4 T5

Accuracy % after learning all tasks

SLNID:96.98 No-Reg:94.12

T1 T2 T3 T4 T5

Accuracy % after learning all tasks

SLNID:95.86 No-Reg:92.26

Figure 6: (a) SLNID with EWC on 5 permuted Mnist sequence of tasks, hidden size=128, (b) hidden size=64.

B.3 CIFAR 100 SEQUENCE

In section 4.2 we have tested SLNID and other representation regularizers on the Cifar 100 sequence. In Figure 3(a) we compare their performance on a network with hidden layer size 256. Figure 7 repeats the same experiment for a network with hidden size 128. While De Cov and SLNID continue to improve over No-Reg, L1-Rep seems to suffer in this case. Our interpretation is that L1-Rep here interferes with the previously learned tasks while penalizing activations and hence suffers from catastophic forgetting. In line with all the previous experiments SLNID achieves the best accuracies and manages here to improve over 6% compared to No-Reg.

Published as a conference paper at ICLR 2019

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 40

Accuracy % after learning all tasks

SLNID:61.16 De Cov:58.43 L1-Rep:52.61 No-Reg:55.06

Figure 7: Comparison of different regularization techniques on a sequence of ten tasks from Cifar split. Hidden size=128. See Figure 3(a) for size 256.

B.4 SPATIAL LOCALITY TEST

To avoid penalizing all the active neurons, our SLNID weights the correlation penalty between each two neurons based on their spatial distance using a Gaussian function. We want to visualize the effect of this spatial locality on the neurons activity. To achieve this, we have used the first 3 tasks of the Permuted Mnist sequence as a test case and visualized the neurons importance after each task. This is done using the network of hidden layer size 64. Figure 8, Figure 9 and Figure 10 show the neurons importance after each task. The left column is without locality, i.e. SLNID, and the right column is SLNID. Blue represents the first task, orange the second task and green the third task. When using SLNID, inhibition is applied in a local manner allowing more active neurons which could potentially improve the representation power. When learning the second task, new neurons become important regardless of their closeness to first task important neurons as those neurons are excluded from the inhibition. As such, new neurons are becoming active as new tasks are learned. For SLNID all neural correlation is penalized in the first task. And for later tasks, very few neurons are able to become active and important for the new task due to the strong global inhibition, where previous neurons that are excluded from the inhibition are easier to be re-used.

Published as a conference paper at ICLR 2019

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

Figure 8: First layer neuron importance after learning the first task (blue). Left: SNID, Right: SLNID. More active neurons are tolerated in SLNID.

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

Figure 9: First layer neuron importance after learning the second task (orange), superimposed on Figure 8. Left: SNID, Right: SLNID. SLNID allows new neurons, especially those that were close neighbours to previous important neurons, to become active and to be used for the new task. SNID penalizes all unimportant neurons equally. As a result, previous neurons are adapted for the new tasks and less new neurons are getting activated.

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

Figure 10: First layer neuron importance after learning the third task (green), superimposed on Figure 9. Left: SNID, Right: SLNID. SLNID allows previous neurons to be re-used for the third task. It avoids changing the previous important neurons by adding new neurons. For SNID, very few neurons are newly deployed. The new task is learned mostly by adapting previous important neurons, causing more interference.

Published as a conference paper at ICLR 2019

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

Figure 11: First layer neuron importance after learning the first task, sorted in descending order according to the first task neuron importance (blue). Left: SNID, Right: SLNID. More active neurons are tolerated in SLNID.

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

Figure 12: First layer neuron importance after learning the second task sorted in descending order according to the first task neuron importance (orange), superimposed on top of figure 11. Left: SNID, Right: SLNID. SLNID allows new neurons to become active and be used for the new task. SNID penalizes all unimportant neurons equally and hence more neurons are re-used then initiated for the first time.

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

0 10 20 30 40 50 60 Neurons Location

Neurons Importance

Figure 13: First layer neuron importance after learning the third task sorted in descending order according to the first task neuron importance (green), superimposed on top of figure 12. Left: SNID, Right: SLNID. SLNID allows previous neurons to be re-used for the third task while activating new neurons to cope with the needs of the new task. For SNID, very few neurons are newly deployed while most previous important neurons for previous tasks are re-adapted to learn the new task.