Unified Probabilistic Deep Continual Learning through Generative Replay and Open Set Recognition
Authors: Martin Mundt
Sagnik Majumder
Iuliia Pliushch
Visvanathan Ramesh
Venue:  (2020)
Keywords: Continual Learning
Open Set Recognition
Probabilistic Deep Learning
Variational Inference

Abstract: We introduce a unified probabilistic approach for deep continual learning based on variational Bayesian inference with open set recognition. Our model combines a joint probabilistic encoder with a generative model and a linear classifier that get shared across tasks. The open set recognition bounds the approximate posterior by fitting regions of high density on the basis of correctly classified data points and balances open set detection with recognition errors. Catastrophic forgetting is significantly alleviated through generative replay, where the open set recognition is used to sample from high density areas of the class specific posterior and reject statistical outliers. Our approach naturally allows for forward and backward transfer while maintaining past knowledge without the necessity of storing old data, regularization or inferring task labels. We demonstrate compelling results in the challenging scenario of incrementally expanding the single-head classifier for both class incremental visual and audio classification tasks, as well as incremental learning of datasets across modalities.

Under review as a conference paper at ICLR 2020
UNIFIED PROBABILISTIC DEEP CONTINUAL LEARN- ING THROUGH GENERATIVE REPLAY AND OPEN SET RECOGNITION
Anonymous authors Paper under double-blind review
ABSTRACT
We introduce unified probabilistic approach for deep continual learning with open set recognition, based on variational Bayesian inference. Our model com- bines joint probabilistic encoder with generative model and linear classifier that get shared across tasks. The proposed open set recognition bounds the approx- imate posterior by fitting regions of high density the basis of correctly classified data points and balances open set detection with recognition errors. Catastrophic forgetting is significantly alleviated through generative replay, where the open set recognition is used to reject statistical outliers from low density areas of the class specific posterior. Our approach naturally allows for forward and backward trans- fer while maintaining past knowledge without the necessity of storing old data, regularization or inferring task labels. We demonstrate compelling results in the challenging continual learning scenario of incrementally expanding the single- head classifier for both class incremental visual and audio classification tasks, as well as incremental learning of datasets across modalities. We further quantíta- tively demonstrate our models ability to successfully distinguish unseen unknown data from trained known tasks and subsequently prevent misclassification
INTRODUCTION
Most machine learning systems make the closed world assumption and are predominantly trained according to the isolated learning paradigm, where data is available at all times and is indepen- dently and identically distributed. However, in the context of continual learning, where tasks and data arrive in sequence, neither of these two principles is desirable. A neural network that is trained exclusively on a new task's data forgets past knowledge and suffers from an early identified phe- nomenon commonly referred to catastrophic forgetting (McCloskey & Cohen 1989). Moreover, to overcome the closed world assumption, inclusion of a "background" class is veritably insufficient as it is impossible to include all unseen concepts and classes explicitly in the loss function before- hand. Likewise, commonly applied thresholding of prediction values doesn't 't prevent resulting large confidences for unseen classes if the data is far away from any known data (Matan al. 1990).
Most of the existing continual learning literature concentrates efforts on either alleviating catas- trophic forgetting, maximizing knowledge transfer addressing ways in which to efficiently store subsets of past data. These works have identified weight regularization (McCloskey & Cohen 1989
Zenke al. 2017 Kirkpatrick al. 2017 Li & Hoiem 2016 Nguyen et al.) 2018)) and rehearsal
techniques (Ratcliff 1990
L9pe- Lopez-Paz & Ranzato 2017: Rebufh et al. 2017] Bachem et al. 2015)
or have postulated methods based on complementary learning systems theory (O'Reilly & Norman 2003) through dual-model with generative memory approaches (Gepperth & Karaoguz 2016 Shin et al. 2017; Wu et al.) [2018, Farquhar Gal. 2018 Achille et al., 2018) as mechanisms against catastrophic inference. On the one hand, regularization techniques can work well in principle, but come with the caveat of relying on a new task's proximity to previous knowledge. On the other hand, training and storing separate models, including generative models for generative rehearsal, comes at increased memory cost and doesn't 't allow for full knowledge sharing, particularly to already stored models. Specifically, the transfer of already attained knowledge to benefit new tasks, known for- ward transfer, as well as the potential positive impact of learning new concepts to aid in existing tasks, known backward transfer, are crucial to any continual learning system. Generally speak-

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ing, most current approaches include set of simplifications, such considering separate classifiers for each new task, referred to as multi-head classifiers. This scenario prevents ccross-talk between classifier units by not sharing them, which would otherwise rapidly decay the accuracy (Zenke et al., 2017 Kirkpatrick et al., 2017 7, |usu et al. 2016, [Shin et al. 2017] Gepperth & Karaoguz 20166 Re- buffi et al. 2017; Achille al.) 2018 Nguyen al. 2018) as newly introduced classes directly impact and confuse existing concepts. In the multi-head scenario task ids thus need to be encoded or are often assumed be given by humans in order know which classifier to use for prediction. Correspondingly, in generative replay, generative and discriminative models are taken to be separate models (Shin et al., |017] Farquhar & Gal 2018 Nguyen et al. 2018). Similar to regularization of a classifier, generative model can suffer from the learned approximate posterior distribution devi- ating further from the true posterior with each further task increment. In order to avoid catastrophic forgetting induced by learning to generate on previously generated data, previous works even store a separate generative model per task (Farquhar & Gal 2018), in analogy to the multi-heat head classifier. An extended review of recent continual learning methods is provided DaPPaisi et al. (2019).
A parallel thread pursues a complementary component of identifying out-of-distribution and open set examples. While current continual learning approaches typically do not include this thread yet, it can be considered crucial to any system and a necessity in order to avoid encoding task labels and distinguishing seen from unknown data. Here, multiple methods rely using confidence values as means of rejection through calibration (Liang et al.] 2018:: [ee et al., 201 8h a). Arguably this also includes Bayesian approaches using variational methods (Farquhar & Gal 2018: Achille al. 2018) or Monte-Carlo dropout (Gal & Ghahramani, 2015) to estimate uncertainties. Since the closed world assumption also holds true for Bayesian methods as the approximated posterior probability cannot be computed for unknown classes, misclassification still occurs, as the open space risk is unbounded (Boult et al.) 2019). Recently TThoma et al. (2014): Bendale & Boult (2016); Dhamija et al. (2018) have proposed extreme value theory (EVT) based open set recognition to bound the open-space risk and balance with recognition errors deep neural networks.
In this work we propose a probabilistic approach to unify open set recognition with continual learn- ing in a single deep model in order to remove or alleviate above mentioned common simplifications. Specifically, our contributions are:
We introduce a single model for continual learning that combines a joint probabilistic en- coder with generative model and linear classifier. Inspired by EVT based open set recognition (Bendale & Boult (2016) for Softmax prediction layers, this model architecture gives rise to natural way of open set recognition with statistical outlier rejection on the basis of the approximate posterior in Bayesian inference. The latter requires no upfront knowledge of open set data or corresponding modifications loss or training procedure and can successfully prevent nonsensical predictions for unseen unknown data, robust- ness feature that is currently not present in closed world continual learning systems.
We show how this EVT bound to the posterior can be used for both identification and rejection of statistically outlying unseen unknown data instances, as well as exclusion of generated samples from areas of low probability density. When used in generative replay this leads to significantly reduced catastrophic forgetting without storing real data.
We share our model across tasks and automatically expand a single linear classifier head with units for new classes, thus not requiring explicit task labels during inference.
We demonstrate that our approach can incrementally learn the classes of two image and one audio dataset, well cross-dataset scenarios across modalities, while allowing for forward and backward transfer due to weight-sharing. When presented with novel data our model is able to distinguish between unseen data from various datasets and data belong- ing to known tasks. We further show that our approach readily profits from recent model advances such as variational lossy auto-encoders (Gulrajani et al. 2017 Chen al. 2017)
UNIFYING CONTINUAL LEARNING WITH OPEN SET RECOGNITION
In isolated supervised machine learning the core assumption is the presence of i..d. data at all times and training is conducted using a dataset D = {(x(n)) (m()) h==r consisting of N pairs of data instances (:(n) and their corresponding labels (n)) E {1 (1...C, CJ for C classes. In contrast, in
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N: {<) m6( ) with & I...., arrives sequentially for T n=1 disjoint datasets, each with number of classes C.. In our work, we consider this class incremental scenario from a perspective of variational Bayesian inference in deep neural networks (Kingma & Welling 2013) that consist of shared encoder with variational parameters 9, decoder and linear classifier with respective parameters & and &. The joint probabilistic encoder learns an encoding to latent variable 2, over which a prior is placed, say unit Gaussian. Using variational inference, its purpose is to approximate the true posterior to both Ps. (z. 2) and Pe (y, 2). The probabilistic decoder Pa(x) (x) and probabilistic linear classifier Pe(y] (z) then return the conditional probability density of the input æ and target under the respective generative model given a sample x from the approximate posterior (z) This yields generative model p(æ: y: 2), for which we assume a factorization and generative process of the form p(æz y, 2) = p(=[ ((zzz))((/) (z))))(z))) For variational
continual learning task data D. = D
inference with our model the following continual learning loss function thus needs to be optimized:
UB 1 +)
LE 40 (= Poo. (z, + log lo ("
-BKI(q0 x (nd) |l p(2))
Here, have added weight term 3 to the KL divergence to balance the individual loss terms, similar to the work of Zhou et al.] (2012) and Higgins et al. (2017). This factor regulates the trade- off between the additional constraint imposed by the classifier, needing be able linearly separate the classes given 2, and the quality of the approximation to the training data. To balance this trade- off irrespective of input and latent dimensionality and number of classes, the losses are normalized according to dimensions. Note that in practice this changes the relative scale of the losses and thus the interpretation of specific 3 values with respect to the original authors Higgins provide a more detailed discussion with empirical examples for the role of [3 in the supplementary section. However, equation | requires the presence of all data for all tasks and is thus generally not feasible for continual learning where only the most recent task's data is assumed to be available. In context of variational inference, two potential approaches offer solutions to this challenge: a prior-based approach using the former approximate posterior -0.--1 the new task's 's prior (Nguyen itt al. (2018) or estimating the likelihood of former data through generative replay or other forms of rehearsal (Farqubar& & Gal 2018 Achille et al. 2018). For proposed model, we follow the latter line of work and let the prior remain the same at all times. Making use of the generative nature of model, we let the above upper-bound to task incremental continual learning become:
al. (2017). We
C, (0, ..
E
v0 (x)((()) (0) logg.s Pos (z) (x("²) + (T 2]]
3K (( II [[)()
No
IE (= [log (x. (x("") (=1= + log 1 [z)] LKL P(=))
(2)
Here, Tz ~ P=.1-((((22) and Jt ~ py)1-1(y(=) with = ~ p(z) is a sample from the generative model with the corresponding label obtained from the classifier. Ñ, is the number of total data instances of all previously seen tasks or alternatively a hyper-parameter. This way the expectation of the log- likeliool for all previously seen tasks is estimated and the dataset at any point in time D. = (x, U ãt, U ())) is a combination of generations from seen past data distributions and the current task's 's real data. For each newly arriving task with novel labels, the classifier is expanded with newly initialized units. We note that whereas the loss function with generative replay in equation 2 is used for continual training, equation I and thus real data is always used for testing.
The model is further trained in denoising fashion, where noise is added to each input x to avoid over-fitting This is preferable to weight regularization as it doesn't 'ttt entail unrecoverable units that are needed encode later stage concepts. We have accordingly coined our model Classifying De- noising Variational Encoder (CDVAE). We optionally enhance the probabilistic decoder with an autoregressive variant, where generation of pixel's value is spatially conditioned on previous pixels (van den Oord et al. 2016 Gulrajani et al. 2017 Chen et al. 2017). Here, the denoising plays additional crucial role of deeqquatization.
Nonetheless, similar to existing moul- approaches (Shin et al. 2017: Wu et al. 2018 Farquhar
& Gal 2018), by itself both CDVAR and PixCDVAR models accumul late errors with each iteration
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5
Petela
5
(OLD
5 5
Chinndee
(a) (OCCDAA/PIXO)CDAAL model
(b) 2-D latent space
(c) Generation
Figure I: (a) Joint continual learning model consisting of a shared probabilistic encoder with vari- ational approximation go (=|zz) probabilistic decoder P z) and probabilistic classifier pe Pe(v, z). The dashed (purple) line denotes an optional pixel decoder with parameters Φp. For open set recog- nition and generative replay with outlier rejection, EVT based bounds for the variational approxima- tion are established. (b) 2-D latent space visualization for continually learned incremental MNIST. (c) Generated MNIST images p (a:))) with z p(z) and their corresponding class ob- tained from the classifier Pe.+ for e = 5, together with their open set outlier probability we,t-
of generative replay deviations of the approximate from the true posterior get amplified. However, in our joint model, the linear classifier directly affects the partitioning of the latent space by influ- encing the joint probabilistic encoder's s weights, resulting class specific areas of large probability density. This is particularly noticeable for lossy VAEs (Gulrajani et al.) 2017 Chen et al. (017) that leave the encoding of local structure to autoregressive layers and hence in our case attribute more influence on the latent space to the classifier. We note that these class specific areas in latent space are not necessarily encouraged for deeper classifiers, however would argue that with a sufficiently expressive probabilistic encoder such a classifier is not necessary. For visualization purposes, have trained a CDVAE following the details of section with a two-dimensional latent space on the class-incremental MNIST (LeCun et al., 1998) upper-bound and show the latent space embedding for the validation dataset at the end of continual learning in figure [b] Corresponding intermediate visualizations for each task increment and PixCDVAE can be found in the supplementary material. We take advantage of the classifier's ' impact on the latent space as the foundation for posterior based open set recognition and complementary generative replay with statistical outlier rejection. We re- (OCDVAE)
fer to this extended model as Open-set Classifying Denoising Variational and PiXOCDVAE respectively. An illustration of our joint probabilistic model is shown in figure la
2.1 OPEN SET RECOGNITION WITH BOUNDS TO THE CLASS SPECIFIC POSTERIOR
We leverage the single-headed linear classifier's presence and the resulting formation of class spe- eific high density regions in latent space as the basis for open set recognition. Specifically, draw inspiration from the EVT based OpenMax approach (Bendale & Boult 2016), that uses knowledge about extreme distance values modify Softmax prediction's confidence, but propose to instead
Algorithm Open set recognition calibration for deep variational neural networks. At the end of task t, a Weibull model fit of tail-size is conducted to bound the per class approximate posterior. Per class Weibull models Pc,,. with their respective shift Te,,, shape Ke,t and scale A,, parameters are returned. The CDVAE model can now be referred to as OCDVAF.
Require: CDVAE with probabilistic encoder qo (2 (z)z)> and probabilistic classifier (vle(
Require: Classifier probabilitics (y.:(yl z) and samples from the approximate posterior = (x)(
90 ()) for each training dataset example x in dataset D.
Require: For each class c, let ((() = z =(x:(()) for each correctly classified training example x!
1: for l.... C do
2: Compute per class latent mean Sa,, = mean ((())
3: Weibull model Pa. (Te,, Ke,t, Ao,t) = Fit Weibull (I]S. - -Scell:))
4: Return means 5, and Weibull models Pa
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Algorithm 2 Open set probability estimation for unknown and uncertain inputs. At the end of any task I, novel data points are considered statistical outliers a Weibull model's cumulative distribution function's 's (CDF) outlier probability value exceeds prior Lz.
Require: OCDVAE with probabilistic encoder (0,x(2/z))
Require: Per class latent mean St,, and Weibull model Pe,t, each with parameters (Te,ts Ric,ts Ac,t)) For a novel input example å sample 2 ~ (2
2:
Compute distances to Se,e de, ||5c,t = for I...C do Compute Weibull CDF (c,e(de,t)
exp (- Iiddla--ean Heat Aeit
4:
Reject input if (c,,) >> Q4 for any class
bound the open-space risk by employing statistical outlier rejection on the basis of the approximate posterior in Bayesian inference. Considering trained model at the end of task t, the EVT based open set recognition fits a Weibull distribution on the distances of each correctly classified training example's sample from the approximate posterior 2 (xx) (0.1(2(<:) to the respective per class sam- ple mean. In other words, regions of high density of the approximate posterior for each class are identified for the subset of correctly identified data points, with the tail of the Weibull distribution bounding the open-space as well as regions of low-density. This procedure is described in algo- established, for any novel input, the Weibull models' cumulative
rithm T Once these bounds distribution function can be used to estimate the statistical outlier probability, based on the unknown example's sample(s) from the posterior and their distance to the class' region of highest density. If the outlier probability is larger than a prior rejection probability, the input can be considered as unknown and false overconfident classifier prediction avoided, or conversely is classified into the already existing classes across all known tasks as detailed in algorithm 2
2.2 GENERATIVE REPLAY WITH STATISTICAL OUTLIER REJECTION
As the obtained open set recognition models provide bounds between the posterior's regions of high and low density, we can extend the use from rejection of statistical outliers for novel input examples to rejection of samples drawn directly from the prior for the purpose of generative replay. Consider generation of data point æ Po. (ar: [2| z). It is common practice to assume that the approximated posterior is close to the true posterior. If a sample from the prior ~ p(z) stems from an area of low density, one further inherently relies In periodic generative rehearsal, these factors can entail accumulation errors through increasing deviations between approximated and true posterior, as well classifier confusion due ambiguous examples. To inhibit the latter and as a result implicitly the former, our obtained bounds can be
the generative model's capability for interpolation.
Algorithm Generative replay with outlier rejection. For generative replay after training task samples ~ p(2) are rejected if the Weibull CDF's probability value exceeds the prior Q,
Require: OCDVAE with probabilistic encoder qo (z(2)) and probabilistic classifier (vl:()) Require: Per class latent mean Sa, and Weibull model Pe.,, each with parameters (Te,ss Ric,t, Ac,1)
Require: Number of samples per class M. V I,..., C. with Ö, seen classes up task t Initialize: m. < 04 1,..., C, X, and Y.
while Σ Me < - Me do
>/
in parallel
3:
Sample from prior 2 ~ p(z)
Compute label argmax (log pe, (v]z))
Calculate distance da, = S 2|1
exp (- (-Ildey IIdey-Taull -aull Kea, de,
6: Compute Weibull CDF Wa, (d2,t)
if Wa,t < Q2 and m, < M, then Calculate decoder ã po. (x/22
9:
Append dataset X, < X, U U and Y. + and ma + ma + else reject
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exploited by rejecting samples from low density regions and replacing them with statistically inlying samples. Hence, we extend generative replay for the OCDVAE with such a rejection mechanism. We now first sample from the prior until desired amount of statistical inliers per class is reached, whereas the label is obtained using the linear classifier and is accepted if it is in correspondence with the respective class' Weibull model. We then proceed generate the dataset with the probabilistic decoder. This bounded version of generative replay with statistical outlier rejection is detailed in algorithm 3 An example of MNIST images with outlier probabilities based on their sample from the prior are shown in figure Ic to illustrate the rejection of ambiguous and misclassified instances, with additional images the supplementary material. The reason we use sampling with rejection is because our Weibull models are based on scalar distances and thus samples from the Weibull distributions cannot inverted to high-dimensional vectors. While this may sound detrimental to our method, we argue that both sampling from the prior z p(z) in large parallelized batches and likewise computation of a single layer classifier, even in high dimensions, is computationally negligible in contrast to the much more computationally heavy deep probabilistic decoder. The latter further only needs to be processed for accepted samples and thus does not add any computational complexity with respect to conventional generative replay. Note that the amount of samples that has to be drawn before a sample is accepted scales with the dimensionality of the latent space and could in principle be regarded
limitation for very high dimensional latent spaces. However, practical VAEs typically do not tend to profit from very high dimensional latent spaces.
3 EXPERIMENTS
Similar to recent literature Zenke et al. 2017 Kirkpatrick et al. 2017 Farquhar & Gal 2018 Shin jet al. 2017; Parisi et al.] 2019), we consider the incremental MNIST LLCunn et al.] 1998) dataset, where classes arrive in groups of two, and corresponding versions of the FashionMNiST (Xiao al.] 2017) and AudioMNIST dataset (Becker et al. (2018). For the latter we follow the authors procedure of converting the audio recordings into spectrograms, resized 32 32. In addition to this class incremental setting, we evaluate cross-dataset scenarios with all inputs resized to 32 x 32, where dataset are sequentially added with all of their classes and the model has to learn across modalities.
We compare our proposed OCDVAE model with its counterpart CDVAE to highlight the improve- ment induced by algorithm 3 We further contrast these improvements with the dual model variant, consisting of a VAE for generative replay and a separate deep model for classification (Shin et al.) 2017). Further, autoregressive pixel model variants are reported to demonstrate how approaches benefit from more recent advances in model architecture. We evaluate elastic weight consolidation (EWC) (Kirkpatrick et al. 2017) on the classification task without a decoder to show that approaches based on regularization fail at maintaining previous knowledge in head classifier scenario. Although the latter has already been shown in a recent review by [Parisi et al.] (2019) and even for multi-head classifier scenarios (Kemker et al.) 2018), nevertheless provide these results for emphasis. We do not report episodic memory approaches like coresets (Bachem et al. 2015) that explicitly store real data additional regularization as suggested by Nguyen al. (2018). These methods are complementary to our proposed approach, reporting them separately might mislead the reader and an evaluation that additionally includes these methods on top is left for future work. To provide a frame of reference for achievable performance, we further consider upper- and lower- bounds for our joint model. The CDVAE lower-bound is obtained when only the current task's data is available and provides the worst case performance where absolute catastrophic forgetting occurs. Conversely, the upper-bound is obtained with equation [ when a task's data is added to all previous tasks' real data and yields model's maximum achievable performance if trained in an incremental fashion. The isolated learning baseline corresponds to the typical machine learning practice outside of continual learning where all tasks' data is always present. All models can be trained on a single GTX 1080 GPU and we will make our code publicly available.
3.1 METRICS
Our metrics are inspired by previously proposed continual learning classification measures (opezz Paz & Ranzato 2017 Kemker et al. 2018). In addition to overall accuracy, these metrics monitor forgetting by computing a base accuracy on the initial task, while also gauging the amount of new knowledge that can be encoded by monitoring the accuracy for the most recent increment. In the

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multi-head classification scenario, both the overall and base accuracy is then divided with ideal accuracy. As our single-head classifier scenario implies a natural decay of a task's base accuracy with increasing amount of classes, instead report the raw base and new accuracies and compare them with the upper-bound and isolated performance. This is a particularly important distinction to the multi-head scenario reported in many previous works, where the amount of forgetting can easily be determined by the amount that the accuracy of each binary classifier decays over time as, introduction of new classes does not directly affect the other tasks' weights. We extend these concepts to the probabilistic decoder's 's reconstruction loss. Our metrics are thus:
Classification accuracy: base accuracy Ott,base of initial task at increment 4. New accuracy Olt,new for the freshly added task. Accuracy Ort,lll over all classes of all tasks seen so far.
Reconstruction negative-log-lkkelioon (NLL): base NLL "t,s of initial task at task increment L. New NLL Yt,new for the added task. NLL Ttloll for all tasks seen so far.
Kullback-Leeilee Divergence: between the approximate posterior (=|xr) and the prior p(z) distribution and thus always evaluated for all data up to and including task t.
3.2 TRAINING HYPER-PAPEE PARAMETERS
We base our encoder and decoder architecture on 14-layer wide residual networks (He et al. 2016 Zagoruyko& & Komodakis (0166) as used in lossy auto-encoders (Gulrajani et al. (017: Chen et al., 2017]. with latent dimensionality of 60 to demonstrate scalability to high-dimensions and deep net- works. Our classifier always consists of single linear layer. The optional autoregressive decoder adds three additional layers. For a common frame of reference, all methods' share the same WRN architecture. We use hyper-parameters consistent with the literature (Gulrajani et al., 2017 Chen jet al., 2017). Accordingly, all models are optimized using stochastic gradient descent with a mini- batch size of 128 and Adam (Kingma & Ba, 2015) with a learning rate of 0.001, batch-normalization IIoff & Szegedy 2015) in all hidden layers with a value of 10 55 and ReLU activations. We add noise sampled from (0, 0.25) to the input avoid over-fitting. Due the inevitable data aug- mentation effect, we train all approaches in this denoising fashion. No further data augmentation or preprocessing is applied. We initialize all weights according to He al.] (2015). For our single- head expanding classifier this ensures that all units are always initialized from the same distribution when they get added each task increment, the initialization depends only on the layer's input dimensionality. This is not necessarily the case for other weight initialization techniques where a dependency on the amount of classifier units exists. We train all class incremental models for 120 epochs per task on MNIST and FaSSionMNIST and 150 epochs on AudiOMNIST. Complementary incremental cross-dataset models are trained for 200 epochs per task. While our proposed model exhibits forward transfer due to weight sharing and need not necessarily be trained for the entire amount of epochs for each subsequent task, this guarantees convergence and a fair comparison of results. Isolated models are trained for 200 and 300 epochs until convergence respectively. For the generative replay with statistical outlier rejection, we use an aggressive rejection rate of Ω4 = 0.01 (with analogous results with 0.05) and dynamically set tail-sizes to 5% of seen examples per class. The used open set distance measure is the cosine distance. We provide detailed description of ar- chitectures and EWC baree--arameters in the appendix. Results are averaged over five experimental repetitions, apart from the isolated, lower- and ppuer-bound that show negligible deviations.
3.3 INCREMENTAL CONTINUAL LEARNING RESULTS AND DISCUSSION
Results for the class incremental scenarios for all models, the upper- and lower-bounds and the isolated setting are shown in table [ Corresponding results for the two directions of incremental cross-dataset experiments are summarized in table 2 In general the upper-bound values are almost identical to isolated learning. Similarly, the new task's metrics are negligibly close, as the WRN ar- chitecture ensures enough capacity to encode new knowledge. In contrast to EWC that is universally unable to maintain its old knowledge, CDVAE and PixCDVAE are able to partially retain previous information. Yet they accumulate errors due to samples generated from low density regions. While the dual model approaches do not exhibit this behavior for MNIST, they displays similar forgetting for other experiments, particularly for Audio data. However, proposed OCDVAE and PixOCD- VAE generative replay overcomes this issue to a consideaable degree. For the class incremental scenarios the best models feature less than 10% drop in accuracy all datasets even with repeated
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Results for class incremental continual learning approaches averaged over 5 runs, baselines and the reference isolated learning scenario for the three datasets. orr and "rr indicate the respective accuracy and NLL reconstruction metries the end of the last task increment T = L denotes the corresponding KL divergence. Arrows indicate whether lower or larger values are better.
Table 1:
Clas-incremental
(rr(%)
"Ir (nats)
(Ly(ta)))
base
new
base
new
all
CDVAE ISO
99.45
78.12
22.12
COVAR UB
99.57
99.10
81.97
CDVAE LB
00.00
99.85
20.16
123.2
92,00
163.7
24.87
MNIST
EWC
00:45 4:0
99.58 was
20.26
Dual Model
97.31
98.59
96.64
75.08
89.32
88,29
16.13
Dual Pix Model
98.04 341
97.31 1o5s
96.52
purs
92.05 1: 1:13
113,4 1:03
106.1 nak
9.296 1:16
COVAE
19.86
99.00
64,
101.6
93.55
107.6 6
30.61
PIRCDVAE
56.53 4-4852
96.77
MUST
83.61
1927
102.4 5:133
118.2 1:172
118.7
5320
16.37 tono
OCDVAE
92.35 4-4485
99.06
9171
93.24
77.16
89.68 a-asss
92.92
6.777
PixOCDVAE
97.44 1:0785
98.63
96.84
0:36
100.5 191
113.3 1.0785
111.9 2.00
12.49
0.005
COVAE ISO
89.54
224.8
23.27
COVAE UB
92.20
97.50
89.24
208.4
246.2
226.2
20.27
FashionMNNIS
CDVAE LB
00.00
99.80
19.97
306.5
242.0
275.1
21.61
EWC
00.17
99.60
20.06
Dual Model
94.26 9:92
93.55 977
63.21 1987
217.7 1:10
242.8 4.9.
230.5
11.45 0.22
Dual Pix
60.04 1:5
72.41 3.911
274.1
305.8 1.0786
289.5
9,781 100a
CDVAE
39.51 177
96.92
58.82
232.8
248.8
242.2
26.68 0059
PixCDVAE
47.83 1541
97.91 1ote
63.05
1ach
241.
1:47
283.2 +:2000
271.7
=2177
22.14
0.77
OCDVAE
60.63
96.51
69.88
222.8
244.0
234.6
20.47
PiXCCDVA
74.45 2.3
98.63 0176
80.85
234.1
283.5 2:24
267.2 0.5
17.93 0700
COVAE
429.7
COVAE UB
98.42
98.67
97.87
418.4
421.3
427.2
15.15
Andio
CDVAE LB
00.00
100.0
20.02
432.9
425.2
440.4
14.52
EWC
00.11
99.41
19.98
Dual Model
61.58 4:434
89.41 mmm
47.42 1.47
425.2
534
422.7 4:07
432.7 = 0385
5,470 Dues
Dual Pix Model
64.60
98.18
75.50
435.
431.9
440.3
6.031
CDVAE
59.36 1:7
84,93 6:201
81.49
1944
422.7
4: 2.19
423.9 1:14
431 n
22,96 0912
PixCDVAE
29.94
97.00
63.44
428.0
436.9
27.14 1.19
OCDVAE
79.73 4:4000
89.52
6.50
87.72
1.594
423.5
2.90
422.9
4:333
430.9
18.52
1133
PixOCDVAE
75.25 a
90.23
432.3
1
437.7
8835
Table 2: Results for incremental cross-dataset continual learning approaches averaged over 5 runs, baselines and the reference isolated learning scenario for the three datasets. oT and "r indicate the respective accuracy and NLL reconstruction metrics at the end of the last increment 7 = 3. KLT denotes the corresponding KL divergence. Arrows indicate whether lower or larger values are better.
Cross-datavet
Tty (nats)
K (.y((nats)]
base
new
all
base
new
all
all
CDVAE ISO CDVAE UB
94.95
269.6
24.97
89.10
97.88
95.00
311.2
434.3
269.7
25.20
CDVAE LB
00.00
98.12
22.70
689.7
341.0
511.7
98.74
Fashion-MNIST--
EWC
22.85
93.31
43.42
Dual Model
81.89 a:s
96.78 sow
91.75
tum4
320.0 + 1215
431.1 1434
273.7 1314
12,80 tueo
Dual Pix Model
82.88 4-1.
97.23
92.16
288.5
437.7
251.6
9,025 1375
CDVAE
$7.70 1:4400
96.73 0.26
81.10 1,71
360.9 20.5
432.1 1:021
296.4
7am6
44,29
4.077
PixCDVAE
56.44
181 97.63
80.76
1:1
1:000
252.6
OCDVAE
80. 4:2222
91.13
345.
430.7
1:000
280.2
25.42
PixOCDVA
81.84 4-0212
97.75
91.76
1221
288.8
437.1
+-0725
251.8
wade
21.07
0244
CDVAE ISO
94,95
269.6
24.97
Andio-MNIST-Faslie
CDVAE UB
97.17
89.16
94.91
428.8 8
311.9
268.2
23.91
CDVAE LB
00.00
506.6
311.0
351.1
EWC
3.420 1.0%%
87.54
45.42 6/77
Model
66.82 4:135/
89.15 4020
87.70 tute
447.3 # 5.70
308.5 1:139
270.9 1:29
12.89 tute
Dual Pix Model
71.58 1.1%%
88.76 n
88.61
1:57
445.8
29014 1.0
255.0 ns5a
9,164 131
CDVAE
79.74 4:24
88,50 10:2
89.46
tuke
448.6
3.17
315.1 1.3
281.6 =
33.38 t208
PIxCDVAE
49.38
88.54
82.18
441.4
287.0
252.5
30.60
OCDVAE
94.53 4:45
0307 89.91
89.53
94.06
nits
433.4
a 5404
311.6 435
271.2 0-044
23.16 0121
PiNOCDVAE
91.90
93.82
438.5
289.4
251.3
20.35
generative replay. Even stronger results can be observed for the cross-dataset scenarios, where for- getting is alleviated to the extent that final accuracy values are close to the upper bound. Likewise improvements noticeable in the reconstruction NLL and KI. divergences. The OCDVAR models
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can consequently achieve reconstruction likelihoods akin to dual model's separate VAE, while fully sharing encoded knowledge and maintaining a classifier. As result of OCDVAE's shared weights, observe backward transfer for some experiments. This is particularly apparent for AUdiOMNIST,, where addition of the second increment first decays and inclusion of later tasks improves the second task's accuracy. Due space constraints, and examples of generated images for all increments in the supplementary material. We note that the pixel decoders are trained for classification and reported NLL values are obtained through sampling from the multinomial distribution. Original losses are provided in the supplementary material.
provide detailed account of all intermediate results
3.4 OUT OF DISTRIBUTION DETECTION RESULTS AND DISCUSSION
In addition to previous results with focus on continual learning accuracy, where the open set recogni- tion approach has been used for generative replay, we proceed to quantitatively analyze the models' ability to distinguish unknown tasks' data from data belonging to known tasks. Here, the challenge consider all unseen test data of already trained tasks as inlying, while successfully recognizing 100 % of unknown datasets as outliers. For this purpose, trained models on each of the three datasets are evaluated on their test set, the remaining datasets and additionally the KMNIST (Clanuwat al.) 2018), SVHN (Netzer et al. 2011 CIFAR1O 10 and CIFARION 100 (Krizhevsky 2009) datasets.
We compare and contrast three criteria that could be used for open set recognition: classifier pre- dictive entropy, reconstruction loss and our proposed latent space based EVT approach. Naively one might expect the Bayesian approach to yield distributions that are sufficiently different for un- known data. We thus evaluate the respective expectation for each criterion with 100 variational samples from the approximate posterior per data point. Figure shows the three criteria and respec- tive percentage of the total dataset being considered outlying for the OCDVAE model trained on FaShionMNISTT. In consistence with recent literature (Nalimicck et al. 2019). we can observe that use of reconstruction loss sometimes distinguish between the known tasks' test data and unknown datasets, but results in failure for others. In the case of classifier predictive entropy, depending on the exact choice of entropy threshold, generally only a partial separation can be achieved. Furthermore, both of these criteria pose the additional challenge of results being highly dependent on the choice of the precise cut-off value. While drawing 2 samples multiple times, as well as repeatedly calculating the classifier, is computationally feasible, we further note that sampling the entire decoder is com- putationally prohibitively expensive in practice. In contrast to the other criteria, the test data from the known tasks is regarded as inlying across wide range of rejection priors Ω4 and the majority of other datasets is consistently regarded as outlying by our proposed open set mechanism.
As our latent space based EVT approach for open set recognition and respective algorithms and do not technically require a decoder, we can evaluate similar outlier detection for the variational dual model approach. While we also evaluate this scenario, note that the distinction the generative model makes need not necessarily apply to a separate classifier and vice versa. We report the outlier detection accuracy in table 3 Here, 5% validation split is used to determine the respective value
Percenbog of dottapet outtresuuttess outters
Parcentage of detaeet
Peccettnge
Dalivvet
10
14
0.5
Webbill CDF outlie rejection
Figure 2: Trained MaSSionMNIST OCDVAE evaluated on unknown datasets. All metrics are av- eraged over 100 approximate posterior samples per data point. (Left) Classifier entropy values are insufficient to separate most of unknown from the known task's 's test data. (Center) Reconstruction loss allows for a partial distinction. (Right) Our posterior- based open set recognition considers the large majority of unknown data as statistical outliers across wide range of rejection priors Ωz

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Under review as a conference paper at ICLR 2020
Test accuracies and outlier detection values of the joint OCDVAE and dual model ap- proaches when considering 95 % of known tasks' validation data is inlying. Percentage of detected outliers is reported based on classifier predictive entropy, reconstruction loss and our posterior based EVT approach, averaged over 100 qo () samples per data-point respectively.
Table 3:
Outlier detection at 95% validation inliers (%)
MNIST
Fashion
Audio
KMNIST
CIFARIO
CIFAR100
SVHN
Trained
Model
Test acc.
Criterion
Dual,
99.40
Class entropy
90.43
97.53
95.29
98.54
98.63
95.51
MNIST
CL
Reconstruction
5.522
99.98
99.97
99.98
99.99
99.96
99.98
VAE
Latent EVT
4.362
99.41
99.80
99.86
99.95
99.97
99.52
Joint, OCDVAE
99.53
Class entropy
3,948
95.15
98.55
95.49
99.47
Reconstruction
5,083
99.50
99.90 99.98
99.91
99.97
99.99
99.98
Latent EVT
4.361
99.78
99.67
99.73
99.96
99.93
99.70
Dual, CL+
90.48
Class entropy
74.71
5.461
69.65
77.85
24.91
28.76
36.64 64
FashionM
Recon Latent EVT
5.535
5.340
64.10
31.33
99.55
98.41
97.24
VAE
96.22
5.138
93.00
91.51
71.82
72.08
73.85
Joint. OCDVAE
90.92
Class Entropy
66.91
5.145
61.86
56.14
43,98
46.59
37.85
Reconstruction
0.601
5.483
63 10.00
28.69
99.67
98.91
98.56
Lutent EVT
96.23
5.216
94.76
96.07
96.15
95.94
96.84
AudioMNIST
Dual, CL+
98.53
Class entropy
97.63
$7.64
5,066
95.53
66.49
65.25
54.91
Reconstruction
6,235
46.32
4.433
98.73
98.63
98.63
97.45
VAE
Latent EVT
99.82
78.74
5,038
99.47
93.44
92.76
88.73
Joint, OCDVAE
98.57
Class eatropy
99.23
89.33
5,731
99.15
92.31
91.06
85.77
Reconstruction 0.614
38.50
3,966
36.05
98.62
98.54
96.99
Latent EVT 99.91
99.53
5.089
99.81
100.0
99.99
99.98
at which 95% of the validation data is considered as inlying before using these priors determine outlier counts for the known tasks' test set as well as other datasets. While MNIST seems to be distinct and easy enough dataset for all approaches, we can make two further major observations:
1. The latent based EVT approach outperforms the other criteria, particularly for the OCD- VAE where near perfect open set detection can be achieved.
2. Even though we can apply our proposed open set approach to just classifier, the joint model with shared latent space consistently exhibits higher outlier detection values. While future investigation into this aspect is needed, we hypothesize that this is due to the joint model also optimizing a variational lower bound to the data distribution p(z).
We provide figures analogous to figure 2 for all models reported in table B in the supplementary material. We emphasize that our open set detection does not rely on knowing any open set data examples beforehand as we do not need modification the loss function training. As there exists a body of complementary work (Liang et al. 2018: Lee et al. 2018b] Dhamija al., 2018) that could readily be integrated, we leave such analysis for future work.
other forms of explicit
CONCLUSION AND OUTLOOK
We have proposed unified probabilistic approach to deep continual learning. At the heart lies Bayesian inference with a model combining a shared probabilistic encoder with a generative model and a expanding linear classifier. Weight sharing across tasks allows for forward and backward transfer, while generative replay alleviates catastrophic forgetting. We have then introduced EVT based bounds the approximate posterior enabled through class specific latent space partitioning induced by the classifier. Derived open set recognition and corresponding generative replay with statistical outlier rejection have been shown achieve compelling results in both task incremental as well as cross-dataset continual learning across image and audio modalities, while being able to distinguish seen from unseen data. We have demonstrated that our approach readily benefits from recent model advances such as autoregressive models (Gulrajani et al. 2017]] Chen et al. 2017) and therefore expect to apply our approach to more complicated data such as larger scale color images with further improvements in generative models. As our approach is extendible, we envision future work to encompass dynamical neural network expansion to increase representation capacity when task complexity increases (Yoon et al. 2018 Rusu et al. 2016: Zhou et al. 2012), combination with soft-targets (Hinton et al. 2014 Li & Hoiem (016) or transfer to entirely unsupervised scenarios where the classifier learns only dataset ids instead of distinct classes (Achille al. 2018).
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A CONTINUAL LEARNING 2-D LATENT SPACE VISUALIZATION
natural consequence of our joint model with shared probabilistic encoder is that the classifier encourages the formation of class-specific regions of high density in the latent space. During con- tinual incremental learning, these regions keep shifting with every task increment while maintaining their bass-specifiicty. New regions of high density emerge for newly added classes. As can be observed in figure [] at the end of the first task two regions have been formed around the mean of the (0, 1) prior when training our CDVAE model on the MNIST (LeCun al. 1998) dataset in class-incremental upper-bound fashion. With every addition of the next classes, the latent em- bedding shifts around the mean of the prior to accommodate the new classes with distinct classes separated by regions of low density. Furthermore, it can also be seen in figures Be and 31 that the shape and the location of the high density regions in the latent embedding are model dependent.
dimerdion
dimension
dimension
33
dimension
(a) 2-class
(b) 4-class
(c) 6-class
dimension
-44
dimersion
1
dimension
(d) 8-class
(e) 10-class
(I) 10-class with PixeIVAE
Figure 3: 2-D latent space visualization for continually learned incremental upper-bound MNIST at the end of every task increment for CDVAE (a-e) and at the end of training for all task increments for PixCDVAE (f).
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B BALANCING CLASSIFICATION ACCURACY AND KULLBACK LEIBLER DIVERGENCE THE ROLE OF
In equations 1 and we have added weight term 3 to the KL divergence to balance the individual loss terms, similar to the work of Zhou et al. (2012) and Higgins et al. (2017). In standard B- VAE Higgins al. (2017), B can be seen as hyper-parameter that balances the reconstruction accuracy with the distribution approximation quality the latent space. As mentioned in the main body, our work, this factor also regulates the trade-off between the additional constraint imposed by the classifier, needing to be able to linearly separate the classes given 2, and the quality of the approximation the training data. As reconstruction losses are typically dependent on the inputs' spatial dimensionality and the classification loss on the respective number of classes, we normalize the loss accordingly. This yields loss values whose scale comparable and in practice changes the relative scale of the losses to each other. Whereas typically the reconstruction loss heavily outweighs the Kullback Leibler divergence and classification in non-normalized training for 3 value of 1.0, this is not the case for our approach. Although the benefit is more stable training irrespective of chosen dimensionalities, note that this however changes the role of specific 3 values with respect to the original authors' Higgins al. (2017) interpretation and the contribution of regularization.
To illustrate this we have trained our WRN model on MNIST with 2-D and 60-D latent spaces for multiple values of 3 in tables 4 and respectively. In these tables report both the normalized
Table 4: Losses for the used WRN architecture with 2-D latent space and different 3 values for MNIST. As stated in the main body. losses normalized per dimension for training. Un- normalized values in nats are reported brackets for reference purposes.
In nats per dimension (nats in brackets)
2-D latent
Beta
KLD
Recon loss
Class Loss
Accuracy [%]
train
1.0
1.039 (2.078)
0.237 (185.8)
0.539 (5.39)
79.87
test
1.030 (2.060)
0.235 (184.3)
0.596 (5.96)
78.30
train
0.5
1,406 (2.812)
0.230 (180.4)
0.221 (2.21)
93.88
test
1.382 (2.764)
0.228 (178.8)
0.305 (3.05)
92.07
train
0.1
2.055 (4.110)
0,214 (167.8)
0.042 (0.42)
99.68
test
2.071 (4.142)
0.212 (166.3)
0.116 (1.16)
98.73
train
0.05
2.395 (4.790)
0.208 (163.1)
0.025 (0.25)
99.83
test
2.382 (4.764)
0.206 (161.6)
0.159 (1.59)
98.79
Table 5: Losses for the used WRN architecture with 60-D latent space and different 9 values for MNIST. As stated in the main body, losses are normalized per dimension for training. Un- normalized values in nats reported in brackets for reference purposes.
In nats per dimension (nats in brackets)
60-D latent
Beta
KLD
Recon loss
Class Loss
Accuracy [%]
train
1.0
0.108 (6.480)
0.184 (144.3)
0.0110 (0.110)
99.71
test
0.110 (6.600)
0.181 (142.0)
0.0457 (0.457)
99.03
train
0.5
0.151 (9.060)
0.162 (127.1)
0.0052 (0.052)
99.87
test
0.156 (9.360)
0.159 (124.7)
0.0451 (0.451)
99.14
train
0.1
0.346 (20.76)
0.124 (97.22)
0.0022 (0.022)
99.95
test
0.342 (20.52)
0.126 (98.79)
0.0286 (0.286)
99.38
train
0.05
0.476 (28.56)
0.115 190.16)
0.0018 (0.018)
99.95
test
0.471 (28.26)
0.118 (92.53)
0.0311 (0.311)
99.34
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24 corrorsei
2- morsion
(a) B 1.0
(b) 9 0.5
(c) 8 0.1
(d) / 0.05
Figure 4: 2-D MNIST latent space visualization with different / values for the used WRN architec- ture.
obtained losses during training, as well as the equivalent computed un-normalized losses as typically reported in the literature. While have reported the latter for comparison purposes in main body, the former are used in practice. We can observe that decreasing the value of beta leads to improvement of the classifier and reconstruction accuracy the expense of KL divergence quality. In both 2-D and 60-D cases, a large beta value leads to significant underfit of the classifier. In order enable adequate training of the classifier, corresponding value of thus has be chosen. Empirically we have found that / of 0.1 works well universally. A nuance of the normalized loss that 3 values cannot be compared to ones the un-normalized scenario. To pick a concrete example, the ratio between KLD) and reconstruction loss with a / value of 1.0 for 60-D latent space in table is roughly 22 without normalization, but approximately 1: 1.6 our case. In practice choice of B would translate to much larger values in a conventional 3-VAE.
However, the presence of the classifier, irrespective the value latent space. For the 2-D latent space trained model for MNIST, density regions become more apparent with smaller values of 8, but manifest even for large values the center due the classifier's 's need to disentangle the class representations.
induces low density regions in
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C GENERATIVE REPLAY EXAMPLES WITH CDVAE AND OCDVAE
As stated in section 2 of the main body, as well as exemplified in the previous section, the jointly optimized linear classifier directly affects the emergence of class specific areas of large probability density in the latent space. The effect of sampling from the prior without statistical outlier rejection for low density regions is shown in figure for the MNIST dataset. For CDVAE/PiXCDVAL we ob- serve classifier confusion due to class interpolated examples, mentioned section 2.2. As the gen- erative model needs to learn how to replay old tasks' data based on its own former generations, this confusion and interpolations accumulate rapidly. This is not the case for OCDVAE/PIXOCDVAEE where misclassifications scarce and the generative model is capable of maintaining high visual
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Under review as a conference paper at ICLR 2020
fidelity throughout continual training. As the OCDVAE constrains the sampling regions of high density, in principle the generative replay could reproduce solely the original data without any in- terpolation akin to over-fit. However, for the purpose of generative replay and estimating the log-likelihood of former seen data distributions, this can be desirable in the continual learning sce- nario as long as variety is ensured. Both continual learning results presented in the main paper, as well as the visual examples of this section's 's figures indicate that this is the case, Similar tenden- cies can be observed for the other two datasets FashionMNIST Xiao al. (2017) (figure ( and AudioMNIST (Becker et al., 2018) (figure 77.. We note that we show AudioMNIST for the purpose of completeness as generated examples are difficult to interpret visually.
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Under review as a conference paper at ICLR 2020
(a) CDVAE 2-class gener-
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Figure 7: Generated images for continually learned incremental AudioMNIST at the end of task increments for CDVAE (a-d), OCDVAE (e-h), PiXCDVAE (i-l) and PixOCDVAE (m-p).
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D ILLUSTRATION OF GENERATIVE REPLAY WITH STATISTICAL OUTLIER REJECTION
In figure 8 we show generated images P with p(2) and their corresponding class e obtained from the classifier ()))) for an OCDVAE model trained on the class incremental MNIST, after the last task increment
Based on its sample from the prior, for each image we have further noted the open set statistical outlier probability We,t from the respective class' Weibull model. Images are depicted in rows, whereas each row corresponds ure exemplifies the sampling process for a maini-batch of size 128, where the amount of class samples within the mini-batch is not necessarily balanced. observe how generated images that feature blurring and ambiguity are considered strong statistical outliers, as well as examples with class interpolation and therefore hold a misclassified label. Using the latter examples to create a dataset for continual learning with generative replay hence entails accumulation of errors. In contrast to the conventional version with unconstrained sampling, our generative replay with statistical outlier rejection algorithm shown in algorithm 3 of the main body rejects these examples and prevents such errors to a large degree.
distinet class label. The fig-
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T. From
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Under review as a conference paper at ICLR 2020
E FULL CLASS INCREMENTAL RESULTS
In addition to the comparative analysis provided section 3.4 of the main body, we provide the class-incremental results for each of the three datasets the end of every task increment, averaged over 5 experimental repetitions in tables 7 and (idd88resppp respectively. These tables aid in making some additional observations about the behavior of the different continual learning algorithms across con- secutive task increments.
We once again observe the increased effect of error accumulation due to unconstrained generative sampling from the prior in the CDVAE and the PixCDVAE models in comparison to their open set counterparts. The statistical deviations across experiment repetitions in the base and the overall classification accuracies are higher and are generally decreased by the open set models. For example, table the MNIST base and overall accuracy deviations CDVAE are higher than the respective values for OCDVAE starting from the second task increment. Correspondingly, the accuracy values themselves experience larger decline for CDVAE than for OCDVAE with progressive increments. This difference not pronounced the end of the first task increment because the models haven't 't been trained on any of their own generated data yet. Successful literature approaches such as the variational generative replay proposed by the authors of Farquhar & Gall (20T8) thus avoid repeated learning based on previous generated examples and simply store and retain separate generative model for each task. The strength of our model that, instead of storing trained model for each task increment, we are able to continually keep training our joint model with data generated for all previously seen tasks by filtering out ambiguous samples through statistical outlier rejection. Similar trends can also be observed for the respective pixel models.
E.I BACKWARD TRANSFER
The weight sharing and the presence of generative expanding single-headed classifier open up the scope for both forward and backward transfer of knowledge in the continual learning context. Figure shows an interesting case of the latter for class-incremental learning with our OCDVAE model on the AudioMNIST dataset. The addition of increment leads to an improvement in the classification performance on class two, as indicated by the confusion matrices.
classes (four and five) the end of the second
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Figure 9: AudioMNIST confusion matrices for incrementally learned classes the OCDVAE model. When adding classes two and three the model experiences difficulty in classification, how- ever is able to overcome this challenge by exhibiting backward transfer when later learning classes four and five. is also observable how forgetting of the initial classes is limited.
E.2 PIXEL MODEL BITS PER DIMENSION CLASSIFICATION LOSSES
Although the main body reports PixeIVAE reconstruction ligelikess in nats, these models are practically formulated as a classification problem with a 256-way Softmax. The corresponding loss in bits per dimension. We have converted these values to have better comparison, but in order to do so we need to sample from the pixel decoder's 's multinomial distribution to calculate binary cross-entropy reconstructed images. The bits per dimension classification loss values for PixeIVAE based experiments in the main body are provided for reference here. The PixCDVAE and PiXOCDVAE achieve final losses on all tasks of 1.019 + 0.014 and 1.047 + 0.010 for MNIST,
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Table 6: Results for class incremental continual learning approaches averaged over 5 runs, baselines and the reference isolated learning scenario for MNIST at the end of every task increment. at and y indicate the respective accuracy and NLL reconstruction metries at the end of every task increment t. Le denotes the corresponding KL. divergence.
MNIST
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Under review as a conference paper at ICLR 2020
Table 8: Results for class incremental continual learning approaches averaged over 5 runs, baselines and the reference isolated learning scenario for AudioMNIST at the end of every task increment. or and "/t indicate the respective accuracy and NLL reconstruction metrics at the end of every task increment I. Lr, denotes the corresponding KL divergence.
Audio
CSV
COVAR UR
CDVAF
EWC
Dual Model
Pix Model
COVAR
PRCCDVAF
OCDVAF
PivCCIVAR
99,99
100.01
100.0
102.0
100,0
99.21
99.71
99.27
99.92
98.58
17.88
100.0
00.29
83.25
15
92.44
51.38
95.12
15.
99.02
00.31
55
91.
98.4
99.99
100.0
100.0
100L0
100,0
99.21
09.71
99.95
99,
99.70
71
98.9
25
95.31
25
95.20
97.33
97.03
62.52
U5.33
17.
98.67
98.
100.0
100,0
99.71
99.83
50.00
50.16
99.38
04
99.67
Gallit
99.
33:28
78.24
95.32
94.25
90.10
95.70
47.77
24.50
95.41
97.75
97.87
19.98
90.23
433.7
423.2
422.3
434.2
435.2
432.5
426
433,8
422.5
439.4
425.6
434.4
425.9
432.5
425
433.5
Phose
420.7
425.0
434.6
422.7
433,]
419.9
425.4
434.2
422.8
433,0
433.0
(hits/dm
418.4
432.9
425.2
435.1
422.7
432.3
433-7
422.3
434.2
435.2
432.5
433.8
381.3
390.4
382.5
389,4
385
389.4
Themut
438.9
436.7
436.8
144.7
436.3
442.7
436.9 9
442.4
(nats)
485.9
485.5
497.4
486.7
494,4
486.5
694.8
(hirkkim)
422.4
429.7
433.7
423.2
422.3
434.2
435.2
432.5
426
433.8
401
411.8
404.0
412.4
405.2
410.9
403.5
411.5
412.1
423.3
413.6
413.8
(nats)
430.3
433.9
432.4
4326
439.8
(hits/dm)
429.7
427.2
440.4
432.7
$40.3
431.4
436,9
430.
431.7
11.65
11.20
4.639
4.561 2010
11.78
9.293
11.16 2m
11.87
K 7a
13.40
11.78
17.09
5,135
5.427
5,330
5.399
15.13
18.18
20.28
14.00
14,06
13,81
14.41
12.40
(natc)
13.61
14.41
5,243
5.812210
22.91
24.91
17.58
16.00
17.89
15:15
22.96
2.851 + 0.0026 and 2.852 + 0.0047 for FashionMNIST, 4.425 + 0.0010 and 4.451 + 0.0198 for Au-
dioMNIST. For cross-dataset experiments starting with FasHionMNIST first, the corresponding loss
values in bits per dimension for PixCDVAE are 2.260 + 0.0078 and 2.238 + 0.0021 for PixOCD- VAE. In the reverse direction the values are 2.232 = 0.0177 and 2.218 = 0.0014 respectively. In the dual model scenario, the separate Pix VAE achieves final losses on all tasks of 1.040 + 0.0103 for MNIST, 2.242 + 0.0124 for FashionMNIST and 4.406 + 0.0024 for AudioMNIST. Loss values for the cross dataset setting are 2.253 + 0.0047 when FashionMNIST gets trained first and 2.279 + 0.0104 for the reverse direction starting with AudioMNIST.
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Under review as a conference paper at ICLR 2020
Percensag dataver outhers
Perconinge 0--bteee conters
Dataset entropy
Dataset reconstruction ghatst
Webuull CDP outleer rejection prior
(a) OCDVAE classifier entropy based open recognition
(b) OCDVAE reconstruction loss
(c) OCDVAE posterior EVT based open recognition
based open recognition
Patceroap oitattett outtivrs
Percentage ootteet dotter outtees
Perontagge
Zii
Walbull CDF outier rejictnnncttonn
(d) Classifier prediction entropy based open set recognition
(c) Reconstruction loss based
(1) Posterior EVT based open set recognition
open set recognition
Figure 10: Trained MASHI OCDVAE evaluated on unseen datasets. All metrics are reported as the mean over 100 approximate posterior samples per data point. (a) The classifier entropy values by itself are insufficient to separate most of unknown from the known task's test data. (b) Recon- struction loss allows for partial distinction. (c) Our posterior-basee based open set recognition considers the large majority of unknown data as statistical outliers across a wide range rejection priors Ω4. (d-f) Corresponding outlier detection using the two separate models of the dual-model approach.
F ADDITIONAL OPEN SET RECOGNITION VISUALIZATION
As we point out section of the main paper, our posterior based open set recognition considers almost all of the unknown datasets statistical outliers, while the same time regarding unseen test data from the originally trained tasks as distribution inliers across a wide range of rejection priors. In addition in the main body, we also show the full outlier rejection curves for the remaining datasets, as well as all dual model approaches in figures TO 11 and 12] These figures visually support the quantitative findings described in the main body and respective conclusions. In summary, the joint OCDVAE performs better open recognition in direct comparison the dual model setting, particularly when using the EVT based criterion. Apart from the MNIST dataset, where reconstruction loss can be a sufficient metric for open detection, the latent based approach also exhibits less dependency on the outlier rejection prior and consistently improves the ability to discern unknown data.
the outlier rejection curves for FashionMNIST and the quantitative results presented
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Under review as a conference paper at ICLR 2020
Percentage dAtANet outler
Petertage of denoee: ounters
Dotaset entrosy
Dottset reconstructinnneeeo loss (natg)
Weitall CDP cuther rejection prior
(a) OCDVAE classifier entropy
(b) OCDVAE reconstruction loss
OCDVAE posterior EVT based open set recognition
based open set recognition
based open set recognition
Netentape Sataet cudiers
Perensagge of dameet nutters
Datasot entrasy
Dataset reconssiuctio
(d) Classifier prediction entropy based open set recognition
(e) Reconstruction loss based
(1) Posterior EVT based open set recognition
open set recognition
Figure 11: Trained MNIST OCDVAE evaluated on unseen datasets. All metrics are reported as the mean 100 approximate posterior samples per data point. (a) The elassifier entropy values by itself are insufficient to separate most of unknown from the known task's test data. (b) Reconstruc- tion loss allows for distinction the cut-off is chosen correctly. (c) Our posterior-based open set recognition considers the large majority of unknown data as statistical outliers across wide range of rejection priors Q4. (d-f) Corresponding outlier detection using the two separate models of the dual-model approach.
25

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Under review as a conference paper at ICLR 2020
Parcenaage dicauee outters
Percontbig of 0100m outthers
outtees
Dotaset entropy
Dataset resonstruction loss thatst
CDE repection prioe
(a) OCDVAE classifier entropy based open recognition
(b) OCDVAE reconstruction loss
(c) OCDVAE posterior EVT based open recognition
based open recognition
Phrcentag dateet outter
Magg of deraeet outtess
CAROD
Dataset resonstructin
Wabbll CDF outier rejictionniie
(d) Classifier prediction entropy based open set recognition
(c) Reconstruction loss based
(1) Posterior EVT based open set recognition
open set recognition
Trained AudiOMNIST OCDVAE evaluated on unseen datasets. All metrics are reported as the mean over 100 approximate posterior samples per data point. (a) The classifier entropy values by itself are insufficient to separate most of unknown from the known task's test data. (b) Recon- struction loss allows for partial distinction. (c) Our posterior-basee based open set recognition considers the large majority of unknown data as statistical outliers across wide range rejection priors Ω4. (d-f) Corresponding outlier detection using the two separate models of the dual-model approach.
Figure 12:
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Under review as a conference paper at ICLR 2020
G ARCHITECTURE DEFINITIONS AND ADDITIONAL HYPER-PARAMETER
Our previous description the training hyperparameters the main text is extended here by speci- rying the exact encoder and decoder architecture, and the additional hyperparameters for the Adam Kingma & Ba 2015) optimizer used for training in each of our evaluated methods. We also pro- vide the hyperparameter values necessary for evaluating EWC in the glass-incrementee learning and cross-dataset scenarios.
We point the reader to tables and 10 for detailed encoder and decoder configurations. For the autoregressive addition to our joint model, we set the number of output channels of the decoder 60 and append pixel decoder layers, each with a kernel size of hyperparameters for Adam optimization include a ,3 of 0.9, .2 of 0.999 and of 10-S.
and 60 channels. The
For the EWC experiments, the number of Fisher samples is fixed the total number of data points from all the previously seen tasks. A suitable Fisher multiplier (A) value has been determined by conducting a grid search over a set of five values: 50, 100, 500, 1000 and 5000 on held-out validation data for the first two tasks in sequence. We observe exploding gradients if À is too high. However, very small leads to excessive drift in the network weight distribution across subsequent tasks that further results in catastrophic inference. A balance between these phenomena is achieved for value of 500 in the muas-inccrementl scenario and 1000 in the cross-dataset setting.
Table 9: 14-layer WRN encoder with a widen factor of 10. Convolutional layers (conv) are parametrized by a quadratic filter size followed by the amount of filters. p and represent padding and stride respectively. If no padding or stride is specified then 0 and s 1. Skip connections an additional operation at layer, with the layer to be skipped specified brackets. Every con- volutional layer is followed by batch- norralizaaiiiiionnn and ReLU.
Layer type Layer
WRN encoder
conv 3 x 3 48, 4
conv 3 x 3 160, p
conv 1 x 1 160 (skip next layer)
conv 3 x 3 160, p
Block
conv 3 x 3 160, p=
shortcut (skip next layer)
conv 3 x 3 160, p
conv 3 x 3
2, p= conv x 1 120, 2 (skip next layer)
conv 3 x 3 3-0, p
Block
conv 3 x 320,0 p= 1;
shortcut (skip next layer)
conv 3 x 3 3-00, p=
conv 3 x 640, 2, p=1;
conv x 640, (skip next layer)
cony 3 x 3 640, p
Block
conv 3 x 3 640,
shortcut (skip next layer)
conv 3 x 3 640, p
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Under review as a conference paper at ICLR 2020
Table 10: 15-layered WRN decoder with widen factor of 10. Prefers the quadratic input's spatial dimension. Convolutional (conv) and transposed convolutional (conv.t) layers parametrized by quadratic filter size followed by amount of filters. p and represent padding and stride respectively. If padding stride is specified then p 0 and additional operation at layer, with the layer to be skipped specified in brackets. Every convolutional and fully-connected (FC) layer followed by natch-normalization and ReLU, The model ends on a linear transformation with a Sigmoid function.
Skip connections
Layer type
WRN decoder
Layer
FC 640 x [=] P/4
conv_t3 x
p= 1;
conv_t X 1 320 (skip next layer)
conv 3 x 3- p=1
Block I
conv 3 x 3 -2000
shortcut (skip next layer)
conv 3 x 3 -20,
upsample x 2
conv 3 x 3 160, p= 1;
conv x 1 160 (skip next layer)
conv 3 x 3 1
Block 2
conv 3 x 3 1=
shortcut (skip next layer)
conv 3 x 3 160, p=
upsample
conv x 3 48, p= 1;
conv It 1 48 (skip next layer)
conv 3 x 48, p=
Block 3
conv 3 x 3- 48, p= 1;
shortcut (skip next layer)
conv 3 x 3
Layer 2
conv 3 x 3 -333p=
28


Reviews:
[{"review_id": "S1eM-mFCFr", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "This paper combines replay and openMax approach to help continual learning.  The results shows robustness on different dataset include image and audio in the continual learning condition, where the new come data has a different distribution but the model still able to maintain reasonable quality for the previously and newly come examples. To my understanding, this approach was not ground-breaking but seems a reasonable combinations.\n\nI'm learning to give weak reject for this paper because of it's poorly written. (1) It's very hard to align the contribution claimed by the paper and previous work in the introduction section. I highly suggest the author re-write this part and has a separate section about related work and explicit describe the difference compare to others. (2) The contribution seems over-claimed, it said it's a unified framework, but I don't understand what it has been unified. (3) In the experimental part, it use audioMNIST. Any reason to use this dataset? There are more well-defined audio task such as TIMIT for phoneme classification or Aurora for digital recognition. They are more easy to understand since they have well established benchmark.\n\nGiven my limited knowledge on this the literature of this topic, I'm happy to change the score if the written being improved and the following question being addressed.\n\n(1)  Introduction. I take most of space to describe previous work, but hard to find out what's difference of this paper. My understanding it combines A and B and apply it to C. But it claim it's a unified framework. \n(2) \"We fully share our model across tasks and automatically expand the linear classifier with additional units when encountering new classes, thus not requiring explicit task labels.\" I cannot link \"automatic\" with the proposed method. Is that doable because of the proposed framework? \n(3) Why use AudioMNIST which is an unusual task for audio?\n(4) For the giant Table 1, I suggest link each acronym with the reference paper. So it can easily get how it associate with different approach. Highlight some numbers can also help the reader understand what's going on in this giant table.\n(5) Can the author give me some insights, what these KL loss demonstrated in the table? I feel since you use the beta-vae version, the kl scale is depend on different approach, not really comparable for these different models.", "extra_scores": {"experience_assessment": "I do not know much about 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": "B1x_nF116B", "reviewer": "AnonReviewer3", "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: This paper proposes a unified model for continual learning and aims to address the following problems:\nOut-of-train-domain dataset recognition\nCatastrophic forgetting\nThe out-of-domain or open set recognition model is not only used to detect outliers but also for sampling “representative data” of previous tasks for forward (and backward) transfer. \n\nCOMMENT(S)/QUERIES:\n------------------------------\n \nWhile the experiments do justice to the contributions mentioned in the paper, I believe that certain sections need clarifications and expansion (while some can be dismissed). \n\n1. Given my limited knowledge in the literature on this topic, I would have appreciated a proper related work section. \n\n2. I believe equation (1) could have been moved to supplementary for reference since equation (2) is the only important equation in the paper. \nI don’t believe Equation (1) is really used for testing purposes. It’s just that the data is sent through the probabilistic encoder and then classified. There is no need for reconstruction of the data point.\n\n3. It’s difficult to follow the flow of the method. Shouldn’t generative replay (algorithm 3) be placed before open set recognition of unknown and uncertain inputs (algorithm 2), since the latter is probably just used during the test, while the former affects the training procedure directly (implicit data augmentation)?\n\n4. Major concern: It’s somewhat strange to observe that the VAE model is able to generate multiclass data so fluidly with a simple gaussian prior. This kind of challenges the belief that current generative models are unable to capture all modes perfectly. A small note about why multimodal prior was not used (which intuitively and mathematically makes more sense) and also a statement about the average time to generate multiclass multi-modal data using algorithm 3 would have been nice. \n\n5. In section 3.2, “For our single-head expanding classifier this ensures..”. While having a single-head expanding classifier is listed as one of the important contributions, it hasn’t been given enough justice w.r.t to the implementation details. Is it like adding a completely new classifier during the training of the new task?\nThe entire section on hyper-parameters could’ve been moved to the supplementary if space was a major constraint but compromising on details about an important contribution only weakens the paper. \n\nI especially like figure 2 in the experiments, where the importance of Weibull CDF outlier rejection prior is highlighted.\n\n\nTypo: \nincorrect opening inverted comma for the word background in the introduction section (page 1)\n\nOVERALL COMMENT\n-----------------------------\n\nThe paper combines the generative, and discriminative models into one framework for multiple important tasks. While the contributions are clear in the introduction, the presentation of the paper is somewhat too complicated at a couple of places. It does not do full justice to explaining its more vital components and some parts of the method section feel like a jigsaw puzzle, where readers are heavily expected to “figure out on their own”. \nWith proper presentation though, I can envision this paper contributing positively to unified frameworks in general.\n\nDue to the above reasons. I am giving it a score of 6.\n", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "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_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "B1xEKODCKH", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "This paper tackles the problem of catastrophic forgetting when data is organized in a large number of batches of data (tasks) that are sequentially made available. To avoid catastrophic forgetting, the authors learn a VAE that generates the training data (both inputs and labels) and retrain it using samples from the new task combined with samples generated from the VAE trained in the previous tasks (generative replay). In this way, there's no need to store all past data and even the first learned batch keeps being refreshed and should not be forgotten.\n\nI like that this paper uses a single global probabilistic model instead of separate discriminative and generative ones. Unfortunately, there are several things that left me unconvinced about this paper:\n\n1) Presentation of the paper\n\n- Variables x, y, z are introduced and talked about without explanation. The graphical model or factorization assumptions are not even mentioned until after the loss has been defined. A normal flow is to first describe the model and what the involved variables mean, and then talk about what the loss for learning it should be, not the other way around.\n\n- Text contradicting the equation: \"In order to balance the individual loss terms, we normalize according to dimensions and weight the KL divergence with a constant of 0.1\". But equation (2) shows a loss with no weighting. I'm assuming the text is correct, but then a beta should be added to the equation in front of the KL divergence.\n\n- Tables and figures are inconveniently far from where they are referenced in the text.\n\n2) Theoretical inconsistencies\n\nAlthough the system might work overall, two things seem to be technically incorrect:\n\n- The decoder and classifier are expected to approximate the distribution of training data according to the authors (for valid generative replay). This is not true in a beta-VAE. The weighting of the KL that the authors introduce is going to bias the learned generator towards the high probability regions. This is not a sound mechanism to achieve an as-faithful-as-possible (limited by the expressiveness of the encoder-decoder architectures) approximation to the training data.\n\n- A Weibull distribution is used to model the same data, again, in a different way. I.e., there are two different probabilistic models modeling the same data in inconsistent ways and one or the other is used depending on the part of the system. (As an example, q(z) could be arbitrarily multimodal as far as the encoder is concerned, but the Weibull seems to force one mode per class. But regardless of this, both models are inconsistent.)\n\n- Similarly, the proposed rejection sampling scheme of OCDVAE is not consistent with the theory of VAEs and it's a post-hoc tweak that is not theoretically expected to provide a pdf of data with lower KL divergence to the true data pdf.\n\n3) Experiments\n\nFinally, the experimental results do not look very compelling, it seems to be overall worse than the baselines in the two image datasets and slightly better in the audio dataset, so it's unclear that this approach is superior.", "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": {}}]

Author rebuttal:

Meta-review: This paper presents a unified probabilistic approach for deep continual learning by combining generative and discriminative models into one framework that solves the following problems: catastrophic forgetting, and identifying out of distribution and open set examples. The method termed, OCDVAE in the paper achieves closer or better to SOTA results in different evaluation tasks. 

The reviewers had several concerns about the presentation of the paper and some errors in the equations, all of which seem to have been fixed in the latest upload made by the authors. Blind review #3 was delayed as the original reviewer refused to review the paper and this review was then obtained by someone else after the new upload of the paper, so this review looks at the new version of the paper. I would recommend the authors to incorporate suggestions provided by reviewer #3 in the final version of the paper including expanding on the related work section. 

However, as of now I recommend to reject the paper.
Decision: Reject