Sliced Cramer Synaptic Consolidation for Preserving Deeply Learned Representations
Authors: Soheil Kolouri
Nicholas A. Ketz
Andrea Soltoggio
Praveen K. Pilly
Venue:  (2020)
Keywords: selective plasticity
catastrophic forgetting
intransigence

Abstract: Deep neural networks suffer from the inability to preserve the learned data representation (i.e., catastrophic forgetting) in domains where the input data distribution is non-stationary, and it changes during training.  Various selective synaptic plasticity approaches have been recently proposed to preserve network parameters, which are crucial for previously learned tasks while learning new tasks. We explore such selective synaptic plasticity approaches through a unifying lens of memory replay and show the close relationship between methods like Elastic Weight Consolidation (EWC) and Memory-Aware-Synapses (MAS).  We then propose a fundamentally different class of preservation methods that aim at preserving the distribution of internal neural representations for previous tasks while learning a new one. We propose the sliced Cram\'{e}r distance as a suitable choice for such preservation and evaluate our Sliced Cramer Preservation (SCP) algorithm through extensive empirical investigations on various network architectures in both supervised and unsupervised learning settings. We show that SCP consistently utilizes the learning capacity of the network better than online-EWC and MAS methods on various incremental learning tasks.

Published as a conference paper at ICLR 2020
SLICED CRAMÉR SYNAPTIC CONSOLIDATION FOR PRESERVING DEEPLY LEARNED REPRESENTATIONS
Soheil Kolouri, Nicholas A. Ketz, & Praveen K. Pilly
HRL Laboratories, LLC Malibu, CA, 91301, USA {skolouri, naketz, pkpilly]ehr1.com
Andrea Soltoggio
School of Computer Science, Loughborough University, Leicestershire, UK a. ssolttoggio@@bbro......
ABSTRACT
Deep neural networks suffer from the inability to preserve the learned data repre- sentation (i.e., catastrophic forgetting) in domains where the input data distribu- tion is non-stationary, and it changes during training. Various selective synaptic plasticity approaches have been recently proposed to preserve network param- eters, which are crucial for previously learned tasks while learning new tasks. We explore such selective synaptic plasticity approaches through a unifying lens memory replay and show the close relationship between methods like Elastic Weight Consolidation (EWC) and Memory Memory-Aware-he Swaar--Snapsses (MAS). We then pro- pose fundamentally different class of preservation methods that aim at preserv- ing the distribution of the network's output at arbitrary layer for previous tasks while learning a new one. We propose the sliced Cramér distance as a suitable choice for such preservation and evaluate our Sliced Cramér Preservation (SCP) algorithm through extensive empirical investigations on various network architec- tures both supervised and unsupervised learning settings. We show that SCP consistently utilizes the learning capacity of the network better than online-EWC and MAS methods on various incremental learning tasks.
INTRODUCTION
Incremental learning without catastrophic forgetting is one of the core characteristics of a lifelong learning machine (1.2M) and has recently gained renewed attention from the machine learning com- munity. In real-world applications, the input distribution of the data (e.g., sensory inputs) is prone constant changes due environmental variations (e.g., seasonal changes), exposure new situ- ations (e.g., change in the surface friction), sensory malfunction (e.g., water droplets on camera), among others. therefore desirable continue train the base computational model only on the new data/task and incrementally accumulate knowledge to improve the performance of the system over time, opposed retraining the model on the composition of old and new data.
The existing computational models, for instance deep convolutional neural networks (CNNs), face two fundamental issues regarding incremental learning, 1) catastrophic forgetting (MCCloskey & Cohen 1989), which refers to the forgetting of previously acquired knowledge when learning new tasks as a result interference between the old and new tasks, and 2) intransigence, which refers to the inability to acquire new knowledge while trying to preserve old knowledge (e.g., reducing the learning rate) [Caudhry et al. (2018). Note that we use the term 'knowledge" here to indicate the input/output behavior of the computational model in (Hinton al., 2015). A successful incremental learner should be able to overcome both forgetting and intransigence. The commonly used strategies overcome catastrophic forgetting include:

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1. selective synaptic plasticity to preserve learned knowledge
(Kirkpatrick et al. [017 Zenke et al., 2017, Aljundi et al.) 2018 Chaudhry et al. 2018), which is rooted in the idea of homeostatic plasticity in neuroscience,
2. additional neural resource allocation to learn new knowledge and preserve old knowledge, Rannen et al. 2017, Schwarz et al. 2018 Li et al. 2019), which similar to neurogenesis in the hippocampus,
(Rusu et al. 2016 Lee et al., 2017 Li & Hoiem 2017
3. memory and experience replay (Rebuffi et al., 2017 Shin et al. 2017: Wu et al., 2018 Hu et al. 2019; Rostami et al.) 2019), which is based on the well-established theory of complementary learning system (CLS) (McClelland et al. 1995).
Each framework has its advantages and disadvantages, and et al. (2019) provide excellent survey of these methods. We note that the term "synaptic weights" refers to the strength of a con- nection between two nodes. The term 'plasticity" is used analogously human brain, which refers to the ability of the neurons to change their synaptic weights. The term "selective plasticity" refers to the desired capability of a network to selectively increase or decrease the plasticity of individual synapses throughout the neural architecture.
'neural plasticity" in the
Our focus in this paper on selective synaptic plasticity. The standard deep neural network ar- chitectures are uniformly plastic; hence, all neurons are prone to changes during training, and this powerful capability is also the demise of these networks and leads to catastrophic forgetting. The idea of selective synaptic plasticity is to partially preserve synapses that are critical for previously learned tasks by rigidifying those synapses (i.e., to enforce critical synapses to change less). Rigid- ifying the network over time leads to loss of learning capability for future tasks, which is known as 'intransigence' in the literature. Selective synaptic plasticity, by itself, could not fully overcome intransigence. A combination of strategies like efficient memory replay for reconsolidation, neuro- genesis, and selective synaptic plasticity could lead to superior methods that defeat both catastrophic forgetting and intransigence. [Cauddhry et al. 2018). for instance, provide such a combination of memory replay and selective synaptic plasticity. Also, there have been various efforts toward mak- ing the idea of neural resource allocation scalable, the progress and compress work Schwarz et al. 2018), and the incremental moment matching Lee et al. (017) work fall under this category.
In this paper. we focus on selective synaptic plasticity to preserve learned representations in a deep neural network. Inspired by (Chaudhry et al.) 2018), we take a geometric view and devise new method for selective synaptic plasticity. The proposed method is fundamentally different from the previous approaches like Elastic Weight Consolidation (EWC) (Kirkpatrick et al. (217)) and Mem- ory Aware Synapses (MAS) (Aljundi et 2018), which we indicate as sample-based approaches. Instead, we focus on identifying synaptic importance parameters that preserve the 'distribution' of the latent representation of task. Focusing on preserving the distribution of the latent represen- arbitrary layer, as opposed to the expected change in network's
tation of neural network response for individual samples, enforces a less restrict regularization on the network, and enables better utilization of the network learning capacity. The primary concept of sample-based versus distribution- regularizations are visualized in Figure T Similar to the MAS framework, our proposed method denoted as Sliced Cramér Preservation (SCP) is also able to preserve task rep- resentation in any layer of neural network, hence, enabling its application to various unsupervised or self- supervisdd learning settings.
Our specific contributions in this paper are:
Introducing a distribution-basee regularization using the sliced Cramér distance (aka, Cramér Wold distance (Tabor et al., 20188) for selective synaptic plasticity that preserves the distributions of the representations of previously seen tasks at an arbitrary layer of deep neural network while learning a new task.
Providing geometric interpretation for the MAS algorithm (Aljundi et al. 2018) that further gives insight into the otherwise heuristic choices made in the approach.
Comparing the proposed method to online-EWC and MAS on the benchmark permuted MNIST dataset, sequential unsupervised learning with auto-encoders, and the more inter- esting problem of semantic segmentation of driving scenes, and demonstrating significant improvements in overcoming catastrophic forgetting and intransigence over these methods.
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Sample-Based
Distribution-Baser
Change Norm
LWC Regularizer
MAS Regularizer
SCP Regulanizrr
Ex- [d(o(x; 0). o(x;
de
Figure 1: Sample-based approaches regularize the learning by the expected change, as measured by dissimilarity measure, of the response of the network for individual samples from Task A, after learning Task A and during learning Task B, Ea=p& [d(p(æ; 0), ((x; 0x))]. where d(:.)) is dissimilarity measure between two K K-ddimensional vectors. Therefore, the regularization is an empirical expected change of the response for samples. EWC and MAS fall under the sample-based category. The proposed distribution- the overall distribution of the network's 's output for input samples from Task A, after learning Task A and during learning Task B, dp (12(-10), P2 (-102)), where p2 is defined in equation 8 and ..) is distance measure between two probability distributions defined on Z C R
approach, on the other hand, regularizes the change in
PROBLEM SET-UP AND PRELIMINARIES
Consider data from stream of tasks X = {ar ~ ) where p'x is the probability density function (PDF) for task t defined on X C Rd. We consider both supervised and unsupervised tasks, where in the supervised case the input sample, æ/,, is accompanied with the corresponding label y E KRKK Let ((:; 0) Rd - RR K denote parametric function (e.g., neural network) that is to be optimized to solve the stream of tasks. In the supervised learning setting, we consider á to be the mapping to the logits prior to applying the softmax layer.
2.1 GEOMETRIC VIEW OF ELASTIC WEIGHT CONSOLIDATION
In their seminal work, Kirkpatrick et al. (2017) considered the problem of overcoming catastrophic forgetting supervised and also reinforcement learning scenarios where supervisory signal y exists, whether in the form of labels/annotations, reiterate the geometric interpretation of the EWC framework following the work of (Chaudhry et al.) 2018), which we will then adapt to define our generic consolidation framework. Let po po (y)
environmental rewards, respectively. Here we
(o]tmax(((<;; 0)), where (pe(y)an æ;)] y is the softmax probability of the j-th class. For simplicity, let consider the case where want to learn only two tasks consecutively, i.e., tasks 'A' and 'B.' Then, EWC ensures that while learning task 'B,' the conditional likelihood po(y (A does not drift far from the optimal conditional likelihood paz, (y) x AA), where 0% the parameters initially optimized for task A:
arg min (B) = aig (B + AB [DKL. (poz (3/|z) po Po [))] 0
0
poz (y/ xx) logg
arg argmin min ( XE x Ey-ros
po (o())))
0
where À is the regularization coefficient. Note that equation essentially optimization with memory replay regularizer. Here, while learning task B, samples from task (i.e., memory buffer from this task) are fed through the network and the conditional likelihood is constantly checked against the optimal conditional likelihood for task A, p0z, (yl A), to ensure a minimal deviation from those parameters.
The key question answered by the EWC framework is on how to avoid memory replay and yet achieve similar result, i.e., not forget the knowledge from old task (task A). The answer lies in the second-order Taylor expansion of the regularizer around the parameters optimized for the old task, 07 It is straightforward to show (Chaudhry al. 2018) that the second-order Taylor expansion of

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the regularizer is of the form:
1
R [DL (po((aa) Pe-50 (3)=))
FØó0 500"
1
(2)
2
where Fg the Fisher Information Matrix (FIM) and is defined as:
[
@log(po(slx))
(Dlog(po x))
Fe B Bapp
By-nez
(3)
00
00
see supplementary material for complete derivations. Therefore, when 80 - the KL-divergence regularizer enforces closeness of 0 to 0 Riemmanian coouudo-maifolld induced by the FIM. Given that the number of parameters could easily reach several million in standard deep neural networks, it is practically infeasible to store and use the FIM matrix, Fg. Therefore, Kirkpatrick et al. (017) assume that Fo is diagonal and further approximate the Kl.-divergence with:
M
E [DEL (po((ax) (o)ka(
Σ m=1
Folmm [60]
where M is the total number of parameters in the neural network. This leads to the main equation the EWC framework:
M
Foz m,, arg min _ = arg min (0) + Σ
(0 0
(5)
From our point of view, the critical aspect of these derivations is the connection between memory replay and structural plasticity. Note that we started with equation [ which uses the idea of memory replay regularizer. Then by assuming the KL-divergence around 0% and assuming that FIM is diagonal matrix arrived at equation 5 which provides the idea of synaptic importance parameters and an embodiment of the selective synaptic plasticity framework. Next, (i.e., based on memory replay) for the MAS algorithm.
- O, using the second-order Taylor expansion of
this critical aspect and develop analogous regularizer
2.2 GENERALIZING TO UNSUPERVISED LEARNING
The EWC framework as explained in the previous section preserves the softmax probability of sam- ples from Task A while learning Task B. This limits the applicability of the method to networks with outputs living on K-dimensional simplex, e.g., supervised learning and reinforcement learn- ing where the network outputs a probability over finite set of actions. More recently. Aljundi et al.) (2018) presented their Memory-Aware Saaarsee (MAS) framework, which lifts the requirement for EWC outputs to live even unsupervised learning and also during testing. While the method is exciting and practically very useful, there is no geometric motivation behind the algorithm. Here reverse engineer the importance term used in MAS and show a simple regularizer that leads to the MAS algorithm and more importantly provides geometric interpretation for the algorithm.
simplex, and enables calculation of the synaptic importance parameters
Let the regularizer for Task B, be the expected absolute difference between squared f2 norms of the output of the network for samples from Task A, i.e.:
arg min _ arg min L (0) θ
(0) + AE x
1 0)1| -|o 11od
(6)
0
is straightforward (see Supplementary material) to show that using the second-order Taylor ex- pansion of the above regularizer leads to:
M
arg min CB (0)
arg min (
Σ m.m 0 0%
which is the importance parameter used by Aljundi al. (2018). From geometric perspective, MAS preserves the norms of the samples from
where [9]m.
E Banpg
01 o 0
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Task A while learning Task B. In other words; MAS enforces closeness of 0 to 0% in Riemmanian useudo-manifoll induced by matrix f2
The general idea of using the expected value of a "suitable" distancc/divergenee between samples, Ea-pt d( (x; 0), c(;; x)) and leveraging its second-order Taylor expansion to obtain synaptic importance values is crucial here and could lead to various undiscovered algorithms based on new distances/divergenness We refer these approaches sample-based regularization methods.
2.3 PRESERVING DISTRIBUTION OF AN ARBITRARY LAYER
We approach the problem of overcoming catastrophic forgetting from the angle of preserving Task A's distribution at an arbitrary layer of the neural network, when learning Task B. Let z = o(æ;;; 0) E R be the output of the network, for a sample from Task A, at the target layer (e.g., output logits NN classifier, or reconstructed image of an autoencoder). The distribution of the random variable x in the target layer follows from the Random Variable Transform (RVT) theorem (Gillespie 1983):
(= f (x()(((z 6(æ; 0))
(8)
Then we propose the following general regularization overcome forgetting when learning task B: arg min L (0) arg min (0 + x0)p2((1))
where d(:,.) .) is discrepancy measure between the two probability distributions defined in R Note that equation could be performed with any discrepancy measure or distance, e.g., the Wasserstein distance (Villani 2008 Kolouri al., 2017), using memory replay strategy. In what follows, we describe a 'suitable" distance d(.,-) that: 1) respects the underlying geometry of the space, and 2) enables a similar strategy to that of the EWC framework to provide importance parameters.
SLICED-CRAMÉR DISTANCE FOR STRUCTURAL PLASTICITY
3.1 CRAMÉR DISTANCE
The p-Cramér distance (Cramér) 1928, Székely & Rizzo 2013) between two one-dimensional probability density functions Po and pr is defined the ly -,-norm between their cumulative dis- tribution functions. Note that here we avoid any measure theoretic notations for simplicity. Let (:)))= f"x p:(T)dr denote the cumulative distribution function for pı, then the p-Cramér distance is defined as:
Cp(po.P)) =(/ (10((t)
(1 (t)
(10)
for p 2 1. Similar to the Wasserstein distance and unlike the KL-divergence and its symmetric form Jensen-Shannon distance (i.e., the square root of the Jensen-Shannon divergence). the Cramér distance respects the underlying geometry of the space. Moreover, the Cramér distance provides unbiased sample gradients (Bellemare et al. 2017). and for p = 1 C1 is equivalent to the 1- Wasserstein distance, W,. In addition, and similar to the Wasserstein metric. the dual of the Cramér distance is of the form of an integral probability metric (IPM) (Dedecker & Merlevède 2007).
To further demonstrate the favorable characteristics of this distance, consider the following para- metric distribution matching in one-oonenett die-imennionn where the target distribution, is box distribution defined on and p- (t) p(t-T) ) the shifted version p and the goal is to optimize the distance between p- and (r" = 0). For this simple setting, we calculate the energy landscape (i.e., the distance between p- and p) as function of for the Sensen-Shanne Cp for 2. Figure 2 shows the distributions p and pr on the left and the energy landscape function on the right. can be clearly seen that the Wasserstein and Cramér distances respect the underlying geometry of the problem, while the
minimize
distance, W,. and
(JS) distance fails to do
To extend the Cramér distance to slicing used in various recent publications (Kolouri et al. 2016: 2019 2018). We note that the sliced- Cramer distance (also known as the Cramer- Wold distance) was recently used in Tabor et al. (2018) for generative modeling. We briefly describe the idea in the following section.
distributions, we utilize the idea distribution
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Distributions and
The Normalized Laddscape Ditferent Distances
Pr
-25
25 -19
10
Figure 2: The energy landscape of various distances as function of the translation parameter. It can be seen that both Wasserstein and Cramér distances respect the underlying geometry of the problem while the
distance fails to do so.
3.2 SLICED-CRAMÉR DISTANCE
The idea of slicing higher-dimeensional distribution has roots in the Radon transform that is com- monly used in computational tomography. The idea is to represent high-dimensional distribution via the infinite set of its marginal distributions. short, let po and pi be d-dimensional probability density functions defined on X C IR, then their Radon transform is defined as:
(4. pi(x))((-- x .
(11)
for E IR and vg E Sd-1 where gd-1 denotes the d-dimensional unit sphere. Note that Rp .) called slice of Pt, which is one-dimensional marginal distribution of p:- Let (; be the corresponding cumulative distribution function of R):(-;)):
Rg. (1,E) S Rpi(T,
(12)
then the sliced-Cramér distance between and is the expected value of the Cramér distance between their vee-dimensional slices, i.e., (- E) when & Uga-1 for Uga-1 being the uniform distribution on the d-dimensional unit sphere. In other words, the sliced Cramér distance is defined as:
SCp(Po,P1)
( (R) R4i ()
(13)
Now we are ready to propose our method for overcoming representation forgetting.
3.3 OVERCOMING REPRESENTATION FORGETTING
The critical point here is that the sample-based regularizers could over-estimate the importance of synapses, leading to intransigence faster. More importantly, a substantial expected change in the network's output (over individual samples) would not necessarily mean catastrophic forgetting. Changes the network's 's output within a mode (e.g., in supervised classification within the distribu- tion of particular class) harmless, long the representation of the data not significantly changing. Our proposed regularizer tolerates such changes and therefore has the potential for better utilization of the network's 's learning capacity. We propose the following regularization for overcom- ing catastrophic forgetting:
arg min A (0)
arg min (B ASC
(14)
equation 14 requires memory replay from the old task(s) while learning the new one. To transition
from memory replay selective synaptic plasticity, derive the second-order Taylor expansion of the regularizer SC3 around the optimal parameters for the previous tasks, 0% Assuming that
0 50 where 50 0, it is straightforward show that:
10 (50)² To: (50) |500 ||50||roz
(15)

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where Toy is defined as:
/ gá-1
/
d (2 (t.
(d. d0
Top
dtdl
(16)
d0
See supplementary materials for the detailed derivations. Note that similar to Før; Tøz, is also ceattie-seei-defiinee (PSD) and therefore our and 0 in a Riemannian pseudo-manifold induced by the PSD matrix To.
regularizer enforces closeness of 0
While the definition of in equation T6 regardless of its similarity timidating, leads to a straightforward empirical algorithm, which we discuss in the next section. Before that, point that similar to the FIM, calculating I is also practically infeasible, and approximate I' with diagonal matrix that simplifies the regularization into:
equation may in-
arg min (0)
arg min (B())
Σ M=
I m. [80
(17)
Finally, we emphasize that while equation 17 is similar to equation it enforces a very different constraint on the neural network. equation which is sample-based; however, our proposed formulation in equation 17 preserves the distribu- tion of network's 's outputs at a particular layer for old tasks, i.e., it preserves the distribution of previously learned representations.
5 SJenforces conditional class likelihoods to be preserved,
4 PROPOSED ALGORITHM
Here derive the algorithmic steps required to calculate I empirically, shown in Algorithm 1 The empirical distribution at the network's output can be written as, p2 N/ E 8(2 (a 0)), and the slices of this empirical distribution are defined as:
(12(1,
1 N 8 o(
Σ
Let ul-) denote the step function, which is the cumulative distribution of the Dirac delta function. Then the cumulative distribution of Rp E0 can be written as, R4 (1, El ~ 1. E u(l
E ( 0)). Therefore we have:
d Rg (l)
1 N
-d 66 0)
5(t E ((x
(18)
d0
d0
substituting equation 18 into equation 16 and using a Monte-Carlo approximation of the integration of gd-1, with L samples, leads to:
d E: d E:
r
(19)
d0
d0
where Eis are randomly drawn from S -1 birth to proposed algorithm shown in Algorithm practically feasible (due to the large number of parameters of deep neural network), we follow the work of (Kirkpatrick et al. 2017) and approximate be diagonal matrix, which simplifies to:
and
N
===1 @xx,1 03). These derivations give Given that calculation of matrix I is not
I
N
d E Φ(x 02)
2 L
E: 02
2
[To:e,
Σ
L
N n=l
d0,
=
de,
1=1
1=1
where 3y N EN

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Algorithm 1: Sliced Cramer Preservation (SCP)
Input: Data, X = { ~ ] n= and the optimized neural network, o(-; 02). ) for Task A.
Parameters: Number of random projections, L.
Output: Synaptic importance matrix r
Calculate the mean response of network at the targeted layer: 5 = # E (( 07).
Initialize the synaptic importance matrix, [T] =
for [ 1 to L do
Sample & from Sk-1
Slice the mean response: p= E 0%
Calculate Vep using auto-differentiation
Update r: += {(Vo))(Voo)*
LOVE
Stankor
Chies Chentrater Aecaaos
SCPVOast
DAC
MAS
303
125
150
175
200
1/00
1000500
[b]
Figure 3: Comparison among OWW, MAS, and SCP on learning ten permuted MNIST tasks (a), and the histogram of Log(A x x) of the synaptic importances (i.e., the diagonal values of F, Ω and r, for FWC, MAS, and SCP, respectively) for each method (b).
4.1 ONLINE EXTENSION OF THE ALGORITHM
To extend the framework into sequential learning of multiple tasks and avoid memorizing task- specific Is (or their diagonals), we follow the EWC++ framework proposed by Chaudhry et al. (2018), which is, in essence, identical to the online-EWC proposed by [Schwarz al. (2018). The EWC++ (and online-EWC) methods ameliorate the need for predicting task identities and calculat- ing task-specific FIMs and keep the memory requirement of the method constant (EWC requires linear growth of memory function of number of tasks). In EWC++, given F( ((t-1) at task (t (t-1),
the accumulated FIM after learning task f is calculated as Fx(() = aFø; + (1 - a) where Fo; the task specific FIM for task f, and a E [0. 1) is hyperparameter that indicates the importance of preserving the most recent task over the older ones. Similarly, we use,
r al' o; + - -a)r
(20)
obtain the sliced-Cramér regularizer for task (1 + 1).
NUMERICAL EXPERIMENTS
5.1 PERMUTED MNIST
We first test our proposed algorithm on the benchmark permuted MNIST task and compare the performance with online-EWC and our implementation of the online-MAS algorithm. For this ex- periment, we used a single head model that learns ten tasks, where each task contains permuted version of the original MNIST dataset. We note that the reported results are function of the archi- tecture the underlying network. Meaning that, while all three methods perform well on this task for larger networks, the true competitiveness of the proposed method emerges for smaller networks where "over- eettmmttttaaiio"" of the synaptic importances significantly hinders learning the subse- quent tasks and leads intransigence. For this experiment, we used fully-connected network (i.e.,
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Standard MLP
LSG mantt
Chiine MAS
2003
c0022
2
246
No Consolisation
MAS
Figure 4: Qualitative and quantitative comparison between MAS and SCP on sequential learning of auto-encoders Columns in Panel (a) show the reconstruction of data after learning consequent tasks random permutation of MNIST sequence. Panel (b) shows the average & teroccnsttruction loss for each method over all tasks and over 10 runs.
Input
Ground Truth
No Consolidation
EWC
SCP (Ours)
Learning
1ask
Leaming
Performance after learning T2
EWC
MAS
SCP
Figure 5: A qualitative comparison of the EWC, MAS, and SCP algorithms semantic segmenta- tion of the SYNTHIA Dataset (Ros et al. 2016). Task (TI) semantic segmentation in summer, and Task (T2) is semantic segmentation in winter. The first row shows performance after learning Ti on input from T1. Second row, shows performance after learning T2 on input from T2. The third row shows performance input from TI after learning T2. The last row, magnifies the last three images the third row for the ease of comparison.
a multi-layer perceptron) with the following architecture, 784 - 1024 - 512 + 256 - 10 neu- rons, and for all optimizations used the ADAM optimizer with learning rate, Ir 1c 4. For our proposed method, SCP, we used = 100 slices. We repeated each experiment 10 times (with different permutations), and reported the average accuracy over all tasks in Figure (a). We can see that SCP is capable of utilizing the capacity of the network better, which points to the fact that regularizing the distribution as opposed to the samples provides less restricted regularization for the network and still enables the network to freely move individual samples so long as the overall latent distribution of the data consistent.
The regularization coefficients for each algorithm was cross-validated on the following grid, A E (1e + i i [-3, -2, .... 9]} and the optimal value was used report the results in Figure 3 (a). Moreover, in Figure (b) we show the histogram of the logarithm of the product of the regulariza- tion coefficients with importance parameters for each method (i.e., times the diagonal values of Fo. Ω. and I' for EWC, MAS, and SCP, respectively). One can see that the optimal values of the reg- ularization coefficients provide, more or less, ccale-consistent synaptic importances for all methods and distinguishing factor between the methods is the difference between these distributions. An- other interesting observation is that the distribution of the synaptic importances are more similar for sasple-based methods (i.e., EWC and MAS) compared to the proposed distribution based method.

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5.2 SEQUENTIAL LEARNING OF -UTO-ENCODERS
Next, we consider an experiment consisting of unsuperriieeddsfffspprrvvvvi that end, we learn an auto-encoder on single digits of the MNIST dataset sequentially. The model is chosen to be fully connected auto-encoder, with the following encoder 728 - 1024 + 1024 - 1024 -> 256, mirrored decoder 256 - 1024 - 1024 - 1024 + 784, and Rectified Linear Unit (ReLUU activations. For the loss function, we used cross-entropy plus the f of the reconstruction error. Similar to the previous experiment, we used the ADAM optimizer (Kingma & Ba 2014) for training the network with (r = 14 We perform 50 epochs of learning on each digit, before switching to the next digit. For consolidation, we used L = 100 slices for SCP. Finally, we permute the order of the digits and run our experiments 10 times, and compare 'No Consolidation,' with MAS, and SCP. Due to the unsupervised nature of the experiment, the EWC framework does not apply here.
sequential learning. To
Figure 4 demonstrate the results of this experiment. Panel (a) shows the reconstruction of a sample digit from one of our with the input digit sequence of [2, 7, 4, 0, 9. 1, 1, 5,8,3). It can be that both MAS and SCP can successfully retain the learned knowledge while acquiring new knowl- edge, while without consolidation the auto-encoder suffers from catastrophic forgetting. Panel (b) shows the average fn-rrrm of the reconstruction error for all seen tasks over the 10 runs. We can see that SCP and MAS are qualitatively on par, and SCP provides a modest yet statistically significant improvement over MAS for this task.
5.3 SEMANTIC SEGMENTATION OF SYNTHIA DATASET
Lastly, we go beyond the benehmark yet less practical MNIST dataset and address catastrophic for- getting more interesting/critical application of autonomous vehicles. We specifically consider the problem of learning semantic segmentation of road scenes in a sequential manner, where the in- put distribution of the data changes over time. Semantic segmentation is the task of assigning a class label to every pixel of an input image. To that end, we use two sequences of the SYNTHIA dataset SETTHHA-SSSS-O--SSSSSS
(Ros et al.) 2016), namely as Task 1 and Task 2, respectively. There are 13 classes in the dataset namely: Miscellaneous, Sky, Building. Road, Sidewalk, Fence, Vegetation, Pole, Car, Sign, Pedestrian, Cyclist, and Lane Mark- ing. We keep the last 100 frames of each sequence as the testing-set and train deep convolutional U-Net architectrr (Ronneberger et al., 2015) on the tasks mentioned above. For the loss function, we used (1 Dice) (Zou al., 20044. and each task was learned over 100 epochs. For the opti- mizer, we used the ADAM optimizer (Kingma & Ba 2014) with learning rate, lr = le - 4. For consolidation, we used L, 100 slices for SCP.
and
We learn the tasks sequentially (summer first and then winter), and qualitative compari- son of onl, MAS, and SCP on sam- ple test frame is in Figure 5 The first row shows the performance on TT after learning T1 (base model), the second row shows perfor- mance T2 after learning TI and then T2 (indicator of intransigence), and the third row shows performance on TI after learning on T1 and then T2 (indicator of the catastrophic for- getting). We can see that catastrophic forget- ting happens when no synaptic consolidation is leveraged. Moreover, compared to online-EWC and MAS, SCP suffers less from intransigence as fewer artifacts are present in the second row. Lastly, SCP overcomes tittsttrphiccforgettin more successfully compared to EWC and MAS as it is apparent from the lack of artifacts in the last row. Finally, we provide a quantita- tive comparison between the methods in Figure 6, where we report the Dice score (Zou et al.] 2004), averaged over ten runs, for each task and each method during the sequential training. As
Performance Task (Suumer)
Nerformance en Tink (Winter)
0.45
0.43
0.02
0.42
Ma Connal dattio
No Consriitattan
0.33
SWC
INC
(Ounl
5C (Ouns)
Ranch lleration
Rwirth heratinn
Figure 6: Testing Dice score Zou et al. 2004) of online-EWC. EWC, MAS, and SCP on sequential learn ing for semantic segmentation of the summer im- ages (Task 1) and winter images (Task 2) from SYNTHIA dataset (Ros et al. 2016). The blue and red shadings on the plots indicate the dura- tions in which the models were trained on summer and winter data, respectively
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Published as a conference paper at ICLR 2020
can be seen, SCP significantly outperforms online-EW and MAS on this task both in overcoming catastrophic forgetting (left plot) and overcoming intransigence (right plot).
6 CONCLUSION
We introduced a new generic approach towards selective synaptic plasticity for preserving the dis- tribution of the network's output, at an arbitrary layer, for previously learned tasks. We started from ceattory-repay-bbaee work's output and showed that a second-orde Taylor expansion of such regularization would lead selective synaptic plasticity approach that does not require memory of samples from previously seen tasks. Furthermore, we proposed the sliced-Cramér distance as a suitable metric for preserving these distributions, which leads to a straightforward algorithm for selective plasticity. Also, using similar approach, we reverse-enginge engineered the Memory Aware Synapses (MAS) framework and pro- vided geometrically meaningful regularization that leads to this algorithm. We then compared the online-EW EWC, MAS, and SCP methods on a variety of learning tasks, including supervised and unsupervised/self- supprrers learning, and consistently showed competitive performance.
regularization that penalized the change in the distribution of the net-
7
ACKNOWLEDGMENTS
This material is based upon work supported by the United States Air Force and DARPA under Contract No. FA8750- 18-C-0133 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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8 SUPPLEMENTARY MATERIALS
8.1 TAYLOR EXPANSION OF THE KL-DIVERGENCE
For the sake of completion, here derive the second-order Taylor expansion of DKL.
DEL (po, [ho)plpo) == {f { poo (2) log
Poa (z)
de
around 0g where we can write 0 0o 600 The second-order Taylor expansion is:
00" [dDKL (pe))
[d2DLL (pe, (po)
DKL (po (po)) x (po lpp.))
2 000
0. 80
d0
0
d02
Where for the first-order term we have:
dDKL (pe, (pe) - -/- po. (x) d (og(po
dr 1
peo (I) dpo(x)
dr
d0
d0
po(x)
d0
Therefore at 0 = Og we have:
dD (po, (po)
f (po(a)
d (J poo (2)dx
lou
dx d0
d0
d0
2. and for the second-ored order term: 1 po)
d2 D (pe,
(-
po, (x)
dpo(xx) dxr
d0a
d0
Pa (x)
d0
J Peo (z)
(z)
(1)
dx
x P (x)
de
d0
J pe. (x)
de (
da
Po
d0
Therefore at 0 = Oo we have: de DRL. (po, l1po)
/ pao (x)
1
dyo, (2))
1
dpo (1r)
100
dr
d02
(o.
d0
Po. (x)
d0
Jx
dl po (x)
dr
d02
/ poo (x)
d (dlog(po((x))
d Log(po
dr
d0
d0
d (og((po)
Log(pa, (x))
T
F
d0
d0
which is the Fisher Information Matrix (FIM).
Finally, putting everything together we have:
DKL (ko,Po,lppo - -00 - F50 2
1
which concludes the derivation.
8.2 TAYLOR EXPANSION OF THE MAS REGULARIZER
Here drive the second-order Taylor expansion of the MAS regularizer in equation
1
MASreg = Ex-px
2 (1l(( 0) |0(<;; 0o)
1 / px (1(
00) 2
2
around 0g. The second-order Taylor expansion is:
( (MASSey
ddMAS
MASreg Ri 50
+ 50
d e. 80
d0
10
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Published as a conference paper at ICLR 2020
1. Where for the first-order term we have:
dMAS reg
/ px (az) d 1lo(2;0)1 0)1
(116(1;; 0)1| (1 0o)1
de
d 0
which for 0 0g zero.
2. For the second-order term we have:
dMMASre
/ px(x))
11 0)11
2 d [|o(2; 0)1
dx + [0),
d02
d
0 m0
d
[ Px(xx)
dd2 (1( 12
(1lo(2; 0)1 2 11bba;; 0o) )
where evaluated at 0 = 0g the second term on the right-hand-side vanishes.
Finally, putting everything together we have:
MASreg = _0 Exepx [[dlledee (dllo(az0)11)
[[)((()))11" d0
(d [bb(;; 0)11 27
d
which is the term reported in the paper.
8.3 TAYLOR EXPANSION OF SLICED-CRAMÉR DISTANCE
Here we derive the second-orde, Taylor expansion of the squared 2-Sliced-Cramae distance,
SC (po,, po) = f Ryoo (1, E) Rto d
around Oo, reported in equation 16
The second-order Taylor expansion is:
(po, po) SC2 (pe,, Po.) + 60 (d SC2 (LSC2(po,,,))) po)
(d' (d-SC3
d0
80
2 50000
d02
10:
60
1. Where for the first-order term we have:
d SC2 po) = 22 g4 JR
(t,
(Rgo E) - Rgo, (t, d))
d0
d0
which for 0 Og equal zero.
2. For the second-order term we have:
d² SC2 (pe., Po)
f
d2 Rgo E)
(Rgo(t, E) - -Roo (1, E))
d0a
d
ga-1 /
f
d (go((,))) d0
d (gol1,,))
dtdE
d0
which evaluated at 0 Oo is:
d2 SC2 (po,, Po)
2 ga-1 /
d Rqo, (t,E)
(a Rqo, (t.E) T
didg
d02
d0
d0
21
Finally, putting everything together we have:
SC Po) = 50"
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Published as a conference paper at ICLR 2020
Task I: MNIST
Performance on MNIST
Performance on SVHN
Average Performance
1.00
1.00
:00
8
4
0
/
7
/
No Conselidation
5
0
CWC
>
5
MAS
6
8
SCP (Ours)
0.55
/
0
8
6
7
0
0
Accuracy
No Consolidation
EWC
Task SVNNN
MAS
22
020
(Ours)
0.85
0.60
0.00
0.80
No Consolidatier
MAS
[Ours
0.75
20
0.75
20
0.75
10
(2) (8)
Epochs
Epochs
Epochs
Figure 7: Sample images from MNIST and SVHN dataset (a), and the testing performance of the Online-EWC, MAS, and SCP on sequential learning starting from MNIST (Task 1) and then SVHN (Task 2) (b). The blue and red shadings on the plots indicate the durations in which the models were trained on the MNIST and SVHN datasets, respectively.
VGG like
(activation)= (iIp(a)))
(features): Sequential
[0): Conv2d(1, 64, sixe=__ 3), stride-(1, 1), (2d 2))
(1): JetchNorm2d(64, egs-Ie-05, monentur=0.1, affine-True,, stack_running staniing_tttt--reee
(inplan)))
[3): Conv2d(64, 64, bernel_size-1l 3), stride=(1, 1), padding= 2))
Batchhorm2d(64, eps-ie-05, momentunr0.1, affine-true,
(5): RetU(inglace)
(6): MaxPoo12d(kernal size-2, stride-2, padding=O, dilation=1, ceil node-False)
(7): Conv2d(64, 128, kernel size=((, 3), stride=(1, 1), padding= 2))
[8]1 BatchNorm2d(128, eps-1e-05, nomentum-U.1, affinentrus, baack__unninn_ttttrraee
[9]1 (epu(in)
[10): Conv2á(128, 128, kernel_ size-(3, 3), stride-(1, 1), padding-(2, 2))
[11): Bat.chNorm2d1128, eps=1e-05, nomentum=0.1, affine=True,
[12]: Retl((nplacc)
[13]] MxxPool2((kernel _iize2,, stride=2, padding=0, diletion=l, ceil pode-False)
(14): Conv2d(128, 256, kernel size-(3, 3, 3), stride-(1, 1), padding-(2, 2))
(15): BatchNorm2d(25,, ops-1e-05, momentun=D.1, affine-Trus, track running stats=True)
(16): ReLD[(nglace)
[17)1 Conv2d(256, 256, semnel_iizem3,, 3), stridem(1, 1), padding 2)}
(10): DetchNorm2d epe-le-05, nomentur-0.1, affine-Prue, beackkrunning_com state-fruue)
(19): (etu(ing)acc)
[20): MxxPoo12((kernal size=2, stride=2, padding=0, dilstionsl, ceil mode=Folse)
)
(fc): Sequential(
(0): Linear(ia festures=12544, out featuree=2048, bias-Prus)
(1): 3atchNormid(2048, ops=1e-05, nomentun=0.1, affinesTrus, track wwwnirguntttaaeeeeennneeeeeeeoee ataat=/rrue
(2): 3c1D(inplace)
(3): Dropout(p=0.21
(4): Lineac(in festuree-2048, out features-1024, bias-Prue)
(5): Batchhormid(1024 aps=1e-05, nomentum=,,1, affine-Trus, track ronninggatatseeennn stats=True)
(ctD(inplace)
[7]1 Dropout(p=0.2)
(clasaify): Sequential(
(0): (aaaarriinttttuess110102 out, foatures=10, bias=True)
Figure 8: The model used in the MNIST-to-SVHN experiment.
8.4 MNIST-TO-SVHH EXPERIMENT
Given the space constraint of the conference, we include our results on the MNIST-to-SWE SVHN exper- iment in the supplementary material. In this experiment, Task 1 is learning the MNIST digits, and Task 2 is learning the SVHN digits. We show sample images from these two datasets in Figure 7h. In this experiment, used VGG-like architecture, for which include the details in g
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Published as a conference paper at ICLR 2020
Average Perfermance
Rowers
0.8
0.6
Consolidation
0.4
MAS
0.3
SCP (Ours)
50
100
130
200
250
Figure 9: Sample super-classes from the CIFAR 100 dataset and visualization of the data augmenta- tion procedure used in this experiment (a), and the average testing accuracy of the Online-EWC. MAS, and SCP on sequential learning of the tasks (b). We note that we were, unfortunately. only able to the experiment once, and will update the Figure with the results from multiple runs as soon as they are available.
We performed incremental learning with consolidation, Online-EWCC MAS, and SCP. The im- portance parameters were cross-validated on coarse grid (due to the computational and time re- strictions) {le + ili E [2, 3, 4, 5, 6]}, where the optimal parameters for the methods where 1e + for SCP, 1e++5 for MAS, and le+44 4 for OnWine-EWW.. We chose 20 epochs per task. Figure P shows the results of the experiment. Note that the first plot shows the testing performance of the methods on Task1 (i.e., MNIST dataset), where the blue shade indicates the first 20 epochs (training on the MNIST dataset), and the red shade indicates the second 20 epochs (training on the SVHN dataset). As can be seen, all methods can address catastrophic forgetting, while Online-EWC and SCP out- perform MAS (However, a one should perform finer grid search on the importance-parametio definitive evaluation). The middle plot shows the testing performance of the model the SVHN dataset (Task 2). The MAS and SCP methods outperform online-EWC. and SCP performs slightly better than MAS (i.e., overcomes intransigence better). The third plot shows the average testing performance of the methods on both tasks. Similarly, we can see that SCP outperforms MAS and OnWine-EWC. We repeated the experiments ten times, and the plots show the average performance.
for
8.5 CIFARI00 EXPERIMENTS
The CIFAR100 (Krizhessky et al. 2009) dataset contains images from 20 super-classes, where each super-class contains five sub-classes. For this dataset, considered an experiment in which the tasks supervised classification of the super-classes. In short, split the data into five sequential tasks, where each task contains a sub-class from all the 20 super-classes. model and utilized the
used a Wide Residual
Network
ZZgoruyko & Komodakis (2016) network
data augmentation suggested by Vries & Taylor 2017). Figure (] shows sample super-classes together with data augmentation. For each method, we used coarse grid search for the importance parameters (1e + E [2, 3.4, 5, 6].. Unfortunately. and due to the lack of time, we were able to only run the experiment for each method once. We report the average classification results in Figure Db. For the final version of the paper, we will complete this experiment with 1) finer grid search the importance parameters and multiple runs of the experiments and reporting the average.
17


Reviews:
[{"review_id": "SJlTfBCMqH", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "[Summary]\nThis paper proposes a new method for overcoming catastrophic forgetting in continual learning, based on distribution-based regularization using the sliced Cramer distance, i.e. Sliced Cramer Preservation (SCP). Unlike previous work on catastrophic forgetting, this paper tackles unsupervised learning scenarios as well as supervised learning. They evaluate the proposed SCP on permutated MNIST, sequential learning in autoencoder task, and sequential learning for segmentation. \n\n[Pros]\n- This paper tackles unsupervised learning scenarios beyond the classification on benchmark datasets.\n- This paper employes sliced Cramer distance with theoretical justification.\n- The analysis on EwC and MAS in terms of geometric view\n- Experimental results look promising.\n\n[Cons]  \n- Even if MAS[1] describes the details on synaptic concept and Hebbian rule, many readers might be not familiar with synaptic or neuro-science terms. So, in prelimiary session, more explanation can help readers to understand.\n- In addition to permute-MNIST, it is required to be evaluated on more conventional tasks such as MNIST->SVHN or CIFAR-10, 100 datasets. Also, more recent work should be compared such as IMM [2] and PGMA [3] for supervised learning scenarios.\n- All graph result figures can be improved for enhancing legibility. In Figure 6, In particular, the subtitle of Summer case confuses me.\n \n[1] Aljundi et al. Memory Aware Synapses: Learning what (not) to forget, ECCV 2018.\n[2] Lee et al. Overcoming Catastrophic Forgetting by Incremental Moment Matching, NIPS 2017.\n[3] Hu et al. Overcoming Catastrophic Forgetting for Continual Learning via Model Adaptation, ICLR 2019.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "BylVcusk5r", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "I think the paper is written quite well, and the approach makes a lot of sense. I think the idea of replacing generalizing the KL to sliced Cramer distance is quite interested. The authors put in some effort in explaining why they propose this alternative distance between distributions. \n\nI think overall this is a great paper, very informative. If I need to nitpick, I think the experimental section relies heavily on MNIST (e.g. permuted MNIST, auto-encoding MNIST), which I think is not a hard enough task. It is though widely used in the continual learning community, though maybe it should not anymore. But I think given the reliance of the field on these datasets it makes sense, plus I think the strength of the paper is not in the empirical evaluation but rather in the derivation of the method. \n\nI think while the authors put quite a bit of effort in explaining the difference between KL and Cramer distance, I would have appreciated a even more detailed exposition. I think the difference between these metrics is not well understood by the majority in the community. ", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper addresses an important problem (preventing catastrophic forgetting in continual learning) through a novel approach based on the sliced Kramer distance. The paper provides a novel and interesting conceptual contribution and is well written. Experiments could have been more extensive but this is very nice work and deserves publication.
Decision: Accept (Spotlight)