Scalable Multi-Modal Continual Meta-Learning
Authors: Bin Wu
Shangsong Liang
Qiang Zhang
Venue: Submitted to ICLR 2023 (2023)
Keywords: Continual meta-learning
Indian Buffet Process
Evidential Sparcification

Abstract: This paper focuses on continual meta-learning, where few-shot tasks are sequentially available and sampled from a non-stationary distribution. Motivated by this challenging setting, many works have been developed with a mixture of meta-knowledge to cope with the heterogeneity and a dynamically changing number of components to capture incremental information. However, the underlying assumption of mutual exclusiveness among mixture components prevents sharing meta-knowledge across different clusters of tasks. Moreover, the existing incremental methods only rely on the prior to determine whether to increase meta-knowledge, where the unlimited increase would lead to parameter inefficiency. In our work, we propose a Scalable Multi-Modal Continual Meta-Learning (SMM-CML) algorithm. It employs a multi-modal premise that not only encourages different clusters of tasks to share meta-knowledge but also maintains their diversity. Moreover, to capture the incremental information, our algorithm uses Indian Buffet Process (IBP) as a prior number of components and proposes a sparsity method based on evidential theory to filter out the components without receiving support information directly from tasks. Thus we can learn the posterior number of components to avoid parameter inefficiency and reduce computational consumption. Experiments show SMM-CML outperforms SOTA baselines, which illustrates the effectiveness of our multi-modal meta-knowledge, and confirms that our algorithm can learn the really need meta-knowledge from tasks.

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SCALABLE MULTI-MODALL CONTINUAL
Anonymous authors Paper under double-blind review
ABSTRACT
This paper focuses on continual meta-lcarning, where few-shot tasks from a non- stationary distribution sequentially available. Recent works maintain mixture distribution of meta-knowledge to cope with the heterogeneity and dynamically changing number of components in the mixture distribution capture incremen- tal information. However, the underlying assumption of mutual exclusiveness among mixture components hinders sharing meta knowledge across different tasks. Another issue is that they only use a prior to determine whether to increase meta- knowledge components, leading to parameter inefficiency. In this paper, we propose Scalable Multi-Modal Continual Meta-L Learning (SMM-CML)) algorithm, which employs a multi-modal premise to encourage different clusters of tasks to share meta-knowledge. Specifically, every task cluster is associated with a subset of mixture components, which is achieved by an Indian Buffet Process prior. Besides, to avoid parameter inefficiency caused by the unlimited increase, propose com- ponent sparsity method based on evidential theory to learn the posterior number components, filtering out those meta-ki meta-knowledge without receiving support directly from tasks. Experiments show SMM-CML outperforms strong baselines, which illustrates the effectiveness our multi-modal meta- peota-leoweedee.. and confirms that our algorithm can learn parameter-efficient meta-knowlede
INTRODUCTION
Meta-learning (Vanschoren) 2018 Hospedales et al.) 2020) is widely used in the low-resource setting. The key idea is to transfer meta-knowledge (i.e., the experience about how to learn) to improve data efficiency and enhance model generalization. In contrast to the conventional assumption that data are homogeneous and available at once (Finn et al., 2017 2018), continual meta-learning faces more practical setting where data are heterogeneous and sequentially available (Finn et al. 2019 Denevi et al. 2019). That is, tasks from non-stationary distributions arrive sequentially. There two challenges to consider in this setting: (1) to avoid forgetting the learned meta-knowledge when training on tasks sampled from the heterogeneous distribution, also called as catastrophic forgetting (Kirkpatrick et al./ 10: (2) to capture the incremental meta- knta--knowlede when encountering the newer tasks (Lee et 2017).
For the first challenge, existing works (Jerfel et al. 2019 Yao et al. 2019 Zhang et al. 2021 use
mixture model, associating cluster of similar tasks with single component. One major concern is that they implicitly assume different meta- mneta-knowledge components are mutually exclusive, This assumption impedes the sharing of nnta-knoowedde among clusters of tasks, which could lead to suboptimal performance and bias toward one type of meta-knowledge. For example, in the research of user profiling, a user (i.e., task) can belong to multiple preference groups (i.e., components), so modeling by single meta-knowledge component, the algorithm might focus more one preference and result in biased profiling.
For the second challenge, these works (Jerfel al. 2019 Yao et al. 2020 Zhang al.) 202 l)
incrementally update neeal-nno where new meta- meta- knowlede component is added to the mixture model for tasks. However, all of them just leverage prior (Jerfel et al.) 2019 Zhang ct al. 202T) or make a simple judgment before the update of meta- nota-knowledge (Yao et al. 2019 2020) whether to add new meta-knowledge components but cannot make posterior decision from task

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New Component
New Component
Component
Meell-nooieedge
Mett-KKowlledge/
ketalKnooleedge
Meta-Knowledge
Metz-Train
Meta-Train
Tasks
Tasks
Tasks
Tasks
Tasks
New
Tasks
New
(a) Existing Works
(h) Our Work
Figure 1: The difference in incremental meta-knowledge between the existing works and ours. Previous methods only make prior decision on whether to add new meta-knowledge (the red dashed grid), which might reproduce the redundant components. Our algorithm considers making postenoo decision from tasks after the meta-train to filter out the meta netta-nnowledge without receiving support.
data. Also, they can only increase but is not able decrease the number of mixture components needed, leading to parameter inefficiency when meeting a large number of tasks (seen Fig. I
To solve these problems, propose a Scalable Multi-Modal Continual Meta-Learning algorithm, abbreviated as SMM-CML. The proposed SMM-CML associates a task cluster with subset of components of the knowledee mixture model, where the provided meta- veet--nnowledge is multi- modal (i.e., a statistical distribution of values with multiple peaks) with each mode being a related neta-knowledge component. The multi-men modal meta- eetta-nowwldge relaxes the constraint of a single component, so that it allows different clusters of tasks to share the meta-knowledge via the overlapped components. This achieved by employing the Indian Buffet Process (IBP) prior on the number of new
components when meeting new tasks. To correct the prior after the update of meta- meta-knowledde tasks, we propose an evidential sparsification method decide the posterior number components, filtering the meta-kong meta-knowledde which does not receive support information directly from task data. Our contributions are summarized as:
We propose multi-modal meta-knowledge, where task is associated with subset of components of meta-knowledge mixture model instead of a single one. Our multi-modal premise allows sharing meta-knowledge via the overlapped components among different clusters of tasks so as to avoid bias towards one type of eestllknoweedge
We employ IBP prior allow number mixture components to increase with newer task arriving, and propose evidential sparsification method to learn the posterior number of components from tasks, filtering out the meta-know ketta-knowwegee which does not receive support information directly from all occurring tasks. The combination of IBP and evidential sparsification helps to maintain the scalable knowl-kn to cope with the online non- stationary setting.
We conduct extensive experiments and the results show that our SMM-CML outperforms the-state-of-art baselines under the online non-stationary setting. And also confirms the effectiveness of multi-modal meta-knowledge and that our algorithm can learn the parameter-efficient meta-knowledge from tasks.
RELATED WORK
Meta-Learning. Meta-learning (Vanschoren) 2018 Hospedales et al.) 2020) focuses on a few-
shot setting. It assumes that source tasks can be used help with the learning in the target tasks. Recent works include metric-based(Snelll et al. 2017 Oreshkin et al. 20188) model-based et al.) 2016 Munkhdalai
Yu, 2017). optimization based methods(Finn et al. 2017 2018) and their Bayesian variants (Ravi & Beatson 2018 Gordon et al. 2019 Iakovleva et al. 2020). respectively. However, most of them propose to construct a globally-shared
mneta--nowwedge. which can not the heterogeneous data distribution in the real world (Jerfel et al.) 2019). To solve this problem, some works (Terfel et al. 2019 Zhang et al., (02T) maintain a mixture
nneal-knowlede, where cluster of similar tasks is associated with single component of the meta-knowledge This impedes the sharing of meta- knoweeeewwedgge between different clusters tasks. Different from the existing works, we take into account both sharing and diversity of meta-knowledge simultaneously.
Continual Learning. Conventional continual learning (Delange et al. 202T) concentrates on the large-scale data setting. Existing models prevent the catastrophic forgetting issue via replay (Hu et al.
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2019 Titsias al. 2019), regularization (Benjamin et al. 2018 Pan et al. 2020 and incremental model selection (umar et al. 2021; Kessler et al. (021). Recently, many works based on meta-
learning (Finn et al., 2019: Zhuang al.] 2020) focus on the low-resource setting. Inspired by the
incremental model selection, some existing works extend meta-knowledge when encountering new
tasks, via increasing the number of mixture components (Yao al. 2019) or adding novel block to construct the mate-path (Yao et al. 2020). Moreover, the Chinese Restaurant Process (CRP) has been used to determine the prior number of meea-knowledge components (Jerfel et al. 2019
Zhang et al.) 2021). However, these methods only consider how to construct the prior number of components and do not make a posterior decision from tasks. Such a prior determination only allows the increase of knttlknnoledgge, which would lead to parameter inefficiency and large computational consumption. In our work, we learn the posterior number of meta- satta-knwwldgge components from tasks via the combination of IBP prior and the evidential sparsification method.
Sparsification Method In recent years, a number of methods have been proposed to sparse the multi-modal space. Most of them (Martins & Astudillo 2016;] Laha al. 2018) aim propose softmax alternative to sparse the large output space. [[kina et al. (2020) pointed out that the above methods are aggressive, and propose post hoc evidential sparsification for conditional variational auto-encoder. based on the conclusion in (Denceux) 2019) that most existing classifiers can be seen as converting features into mass function and merging them to the final result. Following [ttkina et al. (2020), Chen et al. (2021) presented evidential softmax method. However, these methods operate on mutual exclusiveness, which is in confict with ours. Moreover, our evidential sparsification method provides a novel view of how to apply the evidential theory on continual learning.
3 BACKGROUND
3.1 BAYESIAN ONLINE META-LEARNING
Suppose there are sequentially arriving tasks Tt with a dataset Dr from a non-stationary distribution p(T)- Note that the dataset 'D is split into two sub-datasets, a support set D = {x ys ) for training and query set DP {zi, y/
Mr for validation.
fi
Catastrophic forgetting (Lee et al., 2017 is key issue in continual learning. To overcome such an issue in the non-stationary task flow, some variational methods (Yap et 2021 Zhang et al. 202T) have been developed. They update the mnta-knowledge in an online way following Variational Continual Learning (VCL)(Nguyen et al. 2018):
p(0: x p(D,
where O₂ is the meta-knowledge and used as the initialization following MAML (Finn et al. (017). Note that it assumes that the datasets Di:t independent given 8,- Thus, the mnwwele can be updated in a recursively way. Then, the meta-learning framework can be reformulated as variational way (Gordon et al. 2019 lakovleva et al. 2020):
((D, = / p ( (o)
(2)
where Φe is the task-specific parameter. To learn such intractable posteriors, some inference meth- ods (c.g., variational inference (Kingma & Welling 2013)) are applied to infer the approximate distributions. More details of inference are in Appendix
3.2 EVIDENTIAL THEORY
Evidential theory (Denœuxx 2019) works on discrete set of hypotheses (or equivalently, components of meta-knowledge in this paper). Let Z = {=1, 22, .3...., =x be finite set, the element of which ze a binary variable indicating whether the current task associated with k-th component or not, and the power set of Z, donated by 22. A mass function on Z is a mapping m: 2 - [0, 1] and satisfies the following constraints:
m0) 0,
Σ m(A) = 2
(3)
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The Update of Meta-Knowledge
Non Statiiononny
Knowlodge
Posterior
Meta Knowledge
Data Stream
Prior
Postarior
Pr
Meta- Train
Filter
Puss
(P
(9(P
m(
Evidential Sparsification
Meta Knowl-nooweegee
Task spee--pecffic Parameter
Di
Task-specific Adaption
Figure 2: The framework of SMM-CML. The top the updating of meta-knowledge, where the IBP prior determines whether to add new meta- knowledge components (the red box), and the evidential sparsity method used to filter out the component without receiving support directly from tasks based on the posterior of beta distribution from the current time (the solid line) and the previous time (the dashed line). The bottom is decided and then adapted to the task-specific parameter based on the support set D (the dashed line), and then the fask-specific parameter is used to make the prediction on the data (the solid line).
task-specific adaption, where the multi-modal
-noweeeeee
The mass function m(.) represents the support each potential subset of components provided by piece of evidence, and any subset is called focal set if m(A) > 0. As a particular case, the vacuous function (i.e., m(2)
indicates that the evidence not provide any information. One mass function is said simple when:
m(A) = m(Z) = -1- s, w = - -n s),
(4)
where A is single strict subset A C 7, E [0. represents the support degree of A, and w donates the evidential weight A.
Given mass function, there are two corresponding functions, called belief and plausibility function, respectively, which are defined as follows:
Bel(A) Σ m(B), PI(A) Σ m(B) = Bel(A).
(5)
BCA
BOAYU
(Al can be interpreted as the total support degree A, while the 1 PI(A) can be interpreted as the total doubt degree to A. Besides, when the plausibility function is restricted to singletons, then is called contour function pl 2ς + (0, 1).
Given two mass functions provided by different evidence, the fusion of them follows Dempster's 's rule (Dempster 2008 More details about the computing rules are in Appendix
4 SCALABLE MULTI-MODAL CONTINUAL META-LEARNING
In this section, we present our Scalable Multi-Modal Continual Meta-Learning algorithm (SMM- CML). The total framework of SMM-CML is seen Fig.
4.1 MULTI-MODAL CONTINUAL META- SETA-LEARNNIG
SMM-CML relaxes the constraint that task is associated with only a single component of meta- knowledge. The restriction of one-to-one mapping prevents the sharing of meta-knowledge among
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different clusters of tasks. We employ the multi-mode meta- eeta-knowledge where multiple meta- knowledge components are maintained. It assumes that a cluster of similar tasks is associated with a subset of components of meta knotllnnowedgge:
z))((z)) - / p(Pilos) =z) pl dzz,
p(D,)0))
(6)
where 2t the indicating vector consisting binary elements, each element which indicates whether the current task is relevant to the component of meta-knowledge or not. The multi-modal premise enables the sharing of meta-knowledge among different clusters tasks via the overlapped related components, relaxing the restriction of mutual exclusiveness.
4.2 INDIAN BUFFET PROCESS PRIOR
In the non-stationary regime, one important requirement is to capture incremental information when newer task is encountered. Thus, the fixed meta-knowledde not appropriate. To capture the incremental meta-knowledge and fit the multi-modal premise, we employ the Indian Buffet Process (IBP) (Griffiths Ghahramani 2011) make a prior decision on the number of components 2t IBP(a), where the number of the added components at each time is:
ke ~ Possion
where is the hyperparameter to control the rate of increase. The IBP prior for ze is formulated based on the stick- beek-bra process:
Uk ~ Beta(a, 1), Th =
2, ~ Bern(T)), for ks = I,..,,00
(8)
where (eta(-) and Bern(-) represents the Beta distribution and the Bernoulli distribution, respec- tively. Based the IBP prior, the generative process of SMM-CML follows:
0i,k ~ ((e, ot,)), 0 2i p( 2:),
(9)
where Pr. is the meta- weta-knowledge of the k-thh component, and the task-specific parameters Φe are associated with subset of kota-knowledge components determined by zz. The inference can be seen in Sec. and the probability graph model is shown in Fig. 7 in the Appendix [D
Hereby the IBP provides prior on the number of components when encountering new tasks that can capture the incremental knowledge in the online non-stationary setting. However, the IBP just provides prior and cannot make posterior decision after the updating of meta-knowlese When meeting large number of task distributions, the unlimited increase in the number of components would cause a large computational consumption and lead to parameter inefficiency.
4.3 EVIDENTIAL SPARSIFICATION FOR MODAL META SETA-KNOWLEDDCE
To learn the posterior number of components from tasks, we propose an evidential sparsification method for multi-modal knowledee which post hoc method after the update of meta- knowledge. Since the components in our multi-modal meta-knowledge are mutually independent, there might be redundancy across time. How to merge the information about different components from both previous and current times remains an issue. The evidential theory provides good way to merge independent pieces of evidence and make decision (Dempster) 2008).
After updating the meta-knowledge at one time, the relationship between the current clusters of tasks and each koota-knowwedgee component is built up. The relationship between tasks and one certain component is piece of independent evidence, containing the support and doubt information. Such information from the current and previous times can be merged to illustrate the unified relationship between the occurring tasks and the meta-knowledge components. The components receiving support are cast as redundant and removed. Fig. 6lin Appendix Chshows an intuitive explanation of the combination between evidential theory and multi-modal meta- meta-knowledge.
In our IBP-based knowlegge the relationship between tasks and components at each time is determined by & beta distributions of uy Following the evidential theory (Denceux 2019),
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each beta distribution either the current time or the previous time piece of evidence, that there are t. k pieces of evidence. Intuitively, as all beta distributions of V (i.e, the evidences) are independent, each of them only provides the support doubt for the corresponding component. That is, supports the corresponding component {=e] J or the complementary set of the corresponding component { } And this piece of evidence does not by itself provide 100% certainty, which in evidential theory means that the remaining probability commits to the universal set Z.
In this way. each piece of evidence (i.e., each beta distribution of vx,,)) can provide the evidential weight w conducting two simple mass functions with the focal set {z*} and {zx), respectively. The evidential weight can be defined as:
We,k exp((r.*) - 7 exp(3r.s),
(10)
where the evidential weight will increase with a larger Ogt and decrease with a larger Br Note that the hyperparameter 7 can effectively adjust the sparsity of meta-knowledge. This weight w can deduce other evidential weights wi, and w supporting the singleton of corresponding component } and its complementary set [ respectively. The similar derivation (Itkina et al. 2020) can be used as:
t mar(O, w),+) O, W mar(0, -w;,k) 0.
(11)
For each piece of evidence kt there exist two mass functions supporting {2+} and { respectively:
m ((zx)) }-- = exp( wtk): mi, (Z) expl - m k ((x)) = w m4, (Z) = exp(-
(12)
(13)
And these mass functions provided by different pieces of evidence can be fused using Dempster's rule and get the final result as follows:
m({zx))= = -C exp(-w5) exp(w + II (1-erp(-uup w 11
(14)
where C,C+ and are the normalization terms and can be omitted when computing, and w and WE are the merged evidential weight each component. The computational details can be seen in Appendix C If a component k does not receive support (i.e., m(=)) O), then there must be evidence directly supporting this component (i.e., W, 0) and there is at least one other component receiving doubt directly from the evidence (i.e., w7 0, k))
For the sparsity of the multi modall meta-knowledge, we can apply the mass function developed above to filter out the component without receiving the support information directly. That is, components with zero singleton mass value (i.e., m({2))) = 0) are removed and the construction of meta-
knowledge for the specific task in Eq. 6 are modified as:
p(o, 21) [mm(fx])) 0)1 + Obp(0rul00,s
(15)
4.4 STRUCTURED VARIATIONAL INFERENCE
The exact inference is intractable because of non-conjugacy, thus, the approximation is required. In our work, we employ the variational inference (Blei et al. (2077) to approximate the posterior. The evidence lower bound (ELBO) of the observation at the current time f can be derived as following:
L(,i, Dí) = (g( (D.gp()))o)] +
DkL(a(rux)lp(uuu)
Diulglaadeewwllleeeede.. + Σ Dx (xx(g((u)))p(0...))
(16)

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where DE is the Kullback-Leible divergence and q(-) is the variational distribution for each latent variables, respecively. Note that the expectation of likelihood in Eq. [16 can be computed using Monte Carlo sampling, while all the KL-terms can be computed directly as they have closed-fore expressions via implicit reparameterization gradients (Figurnov et al.] (018). Details of the definition of variational distribution, the sampling gradient computation for the likelihood term and the closed form expression for KL-terms can be seen in Appendix
4.5 DISCUSSION
In contrast to recent works (Jerfel al. 2019 Zhang et al, 202T), our work has two major differences that enhance the performance and confirm contributions. (1) Different from the one-to-one matching between the cluster of tasks and the meta- weta-knowledde component, our algorithm constructs many-to-man to-many matching, where multiple task clusters share meta-knowlede and one task cluster needs multiple getta-knowledge components. This is to avoid bias toward one meta-knowledde component and improve performance on heterogeneous tasks, (2) Secondly, our algorithm combines the IBP prior with the evidential sparsification to learn the posterior number of weta-knowledde components. Compared to the existing works only using CPR as prior, our algorithm makes a posterior decision, which achieves parameter efficiency and reduces computational consumption. The analysis of complexity shown in Appendix E
component
EXPERIMENTS
To examine the effectiveness of our SMM-CML, we design experiments, make comparisons and analyze the results. Specifically, the research problems that guide the remainder of the paper are: (RQ1) Can our proposed SMM-CML achieve a better performance than the state-of-the-arr baselines under the online non-stationary setting? (RQ2) Can the increasing number of components capture the incremental information? (RQ3) What the impact of evidential sparsification on performance?
Our experiments conducted under the online non-stationary settings. We compare our algorithm to the following baselines: (1) Train-On-Evert aan-Everythhng (TOE); an intuitive method that re-initializes the meta- netaa-knowledge each time Ê and trains all the arriving data D1: (2) Train- - -Son-Sh (TFS): another intuitive method that also re-initializes the meta-knowledde at each time but trains only the current data Dr; (3) Follow the Meta Leader (FTML)(Fin
al.] 2019):
method utilizing the Follow the Leader algorithm (Kalai & Vempala 2005) to minimize the regret of meta-learner. (4) Online Structured Meta-Learning (OSML): a method via conducting pathway to extract meta-knowledde from a meta- veca--hirarrhhial graph: (5) Dirichlet Process Mixture Model (DPMM): an algorithm that employs CRP conduct a mixture knoweene using point estimation; (6) Bayesian Online Metning with Variational Inference (BOMVT): method that uses Bayesian meta-learning address the catastrophic forgetting issue; (7) Variational Continual Bayesian Meta-L (VC-BML): a state-of-thh-aar method that aims to conduct mixture meta- netal-nnoweede via a Bayesian method.
Following the exiting works Yap et al. 202 Zhang et al. 2021), } we conduct the experiments on Larochelle 2017]. CIFAR-FS(Bertineto et al.) 2018). and Omne et al.) 2011). Tasks sampled from different datasets correspond to different task distribution, so that the online mon-stationary environment can be created via chronologically sampling tasks from different datasets. Specifically, the sampled task is 5-way 5-shot task, and 5 classes are sampled randomly from a dataset for a task. In our experiment, we sequentially meta-train the model on tasks sampled from the meta-training dataset of these four datasets, which means that the model is trained on the tasks sampled from VGG-Flowers dataset, and then proceeds to the next dataset. The showed performances are evaluated on the test set after tuning hyper- param-paraeeters on the validation set. More details about experiment are in Appendix F
four datasets: VGG-Flowers hww.FloowessNisssacee.
Nilsback& & Zisserman 2008), ) minimmagenet((Rai
5.1 RQI: PERFORMANCE UNDER ONLINE NON-STATIONARY SETTING
To examine the effectiveness of our algorithm, we present the mean meta-test accuracy on all the learned datasets at each meta-tras training stage in Tab. [ and the details of performance each training stage are in Appendix F.4.I Our SMM-CML achieves the best performance at each meta-training stage (i.e., VGG-Flowers, minilmagenet, CIFAR-FS and Omniglot), which illustrates

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Table I: Mean meta-test accuracy (%) of the learned dataset at each meta-training stage. The best performance is marked with boldface.
Algorithms
VGG-Flowers
minilmagenet
CIFAR-FS
Omniglot
FTML
76.84 + 1.75
60.74 + 1.85
66.71 +: 1.86
61.89 1.49
OSML
79.61 = 1.50
66.15 + 1.73
68.24 + 1.73
65.65 + 1.40
DPMM
78.97 = 1.52
66.55 + 1.77
67.18 + 1.86
68.26 + 1.47
BOMVI
77.05 = 1.80
60.44 - 1.86
59.57 1.77
69.04 1.54
VC-BML.
83.71 +: 1.58
68.09 +: 1.58
69.87 1.74
69.48 1.51
SMM-CML
85.11 1.46
69.45 1.54
70.72 1.61
71.46 1.39
VGG-Flowers
minilmagenet
CIFAR-FS
Omniglot
1.0
0.8
Posterlor
0.6
0.4
0.2
2
3
i
3
Index of Components in Multi-Modal Meta-knowledge
Figure 3: Each column represents the posterior probability æ of the Bernoulli distribution different component in the multi-modal meta-knowledge on different datasets.
Table 2:
Mean meta-test accuracy (%) under the sequential task setting, where the performance represents the average accuracy across the whole training tasks sequence. The best performance is marked with boldface.
Algorithms
VGG-Flowers
minilmagenet
CIFAR-FS
Omniglot
FTML
57.29 2.28
31.92 1 1.58
39.21 1: 1.75
82.03 1.42
OSML
56.07 = 2.10
32.41 + 1,36
40.75
1.85
82.89 1.42
DPMM
64.21 = 2.06
36.68 + 1.46
47.47 + 1.88
88.39 1.48
BOMVI
64.71 = 1.78
38.44 1.41
48.19 + 1.88
90.49 + 1.62
VC-BMI.
65.28 + 2.19
38.65 + 1.83
47.07 + 1.75
89.97 + 1.11
SMM-CML
66.27 + 2.01
40.04 + 1.48
48.97 + 1.71
91.13 + 0.20
that SMM-CML not only maintain the weeta-knowledge learned from previous times but also capture the incremental meta-knowledge from the current tasks. Moreover, the comparison between the performance of SMM-CML and the baselines (i.e., DPMM and VC-CML), which maintain the mutually exclusive meta-knowledge components, confirms that sharing helps to improve performance.
To further illustrate the association between tasks and meta-knowledge, we show the posterior of the Bernoulli distribution of each component on each dataset. As in Fig. 3 the probabilities of Bernoulli distribution of each component are distinct. For example, the VGG-Flowers dataset has strong association with the later three components, while the Omniglor dataset is closely relevant to all the components except the third one. It confirms that in our learned wetu-knowledde. different clusters of tasks share multiple components and maintain their diversity via the other different ones.
Besides, we take into account another more challenging setting, where tasks from different datasets are mixed and randomly arrive one by one. Because of the more non non-stationary task stream, catastrophic forgetting is more serious. Tab. shows the average accuracy results over all times. Our SMM-CML achieves the best performance on all four datasets even in such a challenging setting. It further confirms that our algorithm has the capability to cope with the online non-stationary task streams.
5.2 RQ2: THE IMPACT OF INCREASING NUMBER OF COMPONENTS
To capture the incremental knowledoweeee in continual meta-learning, we employ the Indian Buffet Process to allow the increasing number of components. We conduct the experiment with different numbers of meta-knowledge components to test its effectiveness. The evolution of meta-test accuracy when training on different datasets shown Fig. TOE has the best performance on most stages because it can replay all the available data. With the number of components increasing, our proposed algorithm has better performance in both the learned and the new datasets. It further demonstrates that more components capture the incremental meta-knowledge and alleviate the forgetting issue.
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Omniglot
CIFAR-FE FS
minimmgeenet
VGG-Flowers
Onniglot
90
Meta seas-tett Accuracy (1%)
VCG-Flowers minilmagenet CIFAR-FS
65
55-
TOF
K-1
K-2
K-4
Meta training Iteration
Figure 4: The evolution of meta-test accuracies (%) of SMM-CML with different numbers of components when training on different datasets. TOE and TFS are two baselines for comparison.
Table The meta-test accuracy (%) before sparsification and after sparsification each dataset
Omniglot
CIFAR-FS
minilmagenet
VGG-Flowers
original sparse
99.31 + 0.25 99.43 + 0.21
85.99 + 1.07
76.07 + 1.40
71.41 + 1.48 71.22 + 1.36
86.53 + 1.05
75.91 + 1.38
68.5
Guis (E-CMAAA))
05.25
05.00
86.75
06.50
04.25
57.0
69.0
04.00
83.75
66.5
685
k=2
(a) VGG-Flowers Stage
(b) minilmagenet Stage
(c) CIFAR-FS Stage
(d) Omniglot Stage
Figure 5: The comparison between SMM-CML and VC-BML with different numbers of components on each training stage.
5.3 RQ3: THE EFFECTIVENESS OF EVIDENTIAL SPECIFICATION
To reduce computational consumption, we propose an evidential sparsification method. To examine the impact of methods, we compare performance before and after sparsification. The meta-test accuracy at each meta-training stage shown in Tab. 3 and more results are shown in Appendix F.4.2) Compared to the original meta-knowledge. the sparse meta-knowledge comparative performance. This confirms that our method can reduce redundancy and computational consumption with acceptable accuracy. Moreover, we conduct experiments different numbers of components with the appropriate The results in Fig. show that our model can outperform the SOTA even with less number of components. confirms that our algorithm can filter out the redundant deta-knooledde component and is more parameter-efficiennyy
achieve
CONCLUSION
This paper focuses more challenging setting in meta-learning, where tasks from non-stationary distribution are available sequentially. We propose SMM-CML, a Scalable Multi-Modal Meta- Learning algorithm where cluster similar tasks are associated with multiple components, allowing tasks to share meta- knowllnnowe while maintaining their diversity. Moreover, an IBP prior is employed to determine whether increase the number components, and an evidential sparsity method is proposed to filter out the components which have not received support information from tasks. This confirms a posterior number of -nowledgw components so that it avoids parameter inefficiency, The conducted experiment shows the effectiveness of multi-modal meta-knowledge and confirms that our algorithm can learn the needed meta- meta-know knowledge from tasks. One limitation comes from the space complexity, since model still needs to increase the number of mixture components cover more meta- meea--nowledeer The proposed evidential sparsity method can help alleviate the required space complexity.

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REFERENCES
Ron Amit and Ron Meir. Meta-learning by adjusting priors based on extended pac-bayes theory. In International Conference on Machine Learning, pp. 205-214. PMLR, 2018.
Ari Benjamin, David Rolnick, and Konrad Kording. Measuring and regularizing networks in function space. In International Conference on Learning Representations, 2018.
Luca Bertinetto, Joao F Henriques. Philip Torr, and Andrea Vedaldi. Meta-learning with differentiable closed-form solvers. International Conference on Learning Representations, 2018.
David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the American statistical Association, 1126118):899-877,2 2017.
Phil Chen, Masha Itkina, Ransalu Senanayake, and Mykel J. Kochenderfer. Evidential softmax for sparse multimodal distributions deep generative models. In Marc' Aurelio Ranzato, Alina Beygelzimer, Yann N. Dauphin, Percy Liang, and Jennifer Wortman Vaughan (eds.), Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurlPS 2021, December 6-14, 2021, virtual, pp. 11565-11576, 2021.
Matthias Delange, Rahaf Aljundi, Marc Masana, Sarah Parisot, Xu Jia, Ales Leonardis, Greg Slabaugh, and Tinne Tuytelaars. continual learning survey: Defying forgetting classification tasks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021.
Arthur Dempster. generalization of bayesian inference. In Classic Works of the Dempster-Shafer Theory of Belief Functions, pp. 73-104. Springer, 2008
Giulia Denevi, Dimitris Stamos, Carlo Ciliberto, and Massimiliano Pontil. Saline-within-ooe meta-learning. In Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d' Alché-Buc, Emily Fox, and Roman Garnett (eds.), Advances Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pp. 13089-13099, 2019.
Thierry Denæeux. Logistic regression, neural networks and dempster- shaffer theory: A new perspective. Knowledge-Baud Based Systems, 176:54-67, 2019.
Mikhail Figurnov, Shakir Mohamed, and Andriy Mnih. Implicit reparameterization gradients. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Gamett (eds.), Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, pp. 439-450, 2018
Chelsea Finn, Pieter Abbeel, and Sergey Levine Model-agnostic meta-learning for fast adaptation of deep networks. International conference on machine learning. pp. 1126-11 1135. PMLR, 2017.
Chelsea Finn, Kelvin Xu, and Sergey Levine. Probabilistic model-agnostic meta- meta-learning. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa- Bianchi, and Roman Gamett (eds.), Advances in Neural Information Processing Systems 31: Annual Conference on Neurul Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréul, Canada, pp. 9537-9548, 2018.
Chelsea Finn, Aravind Rajeswaran, Sham Kakade, and Sergey Levine. Online meta-learning. In International Conference Machine Learning, pp. 1930.1930, PMLR, 2019.
Jonathan Gordon, John Bronskill, Matthias Bauer, Richard E Turner, Jan Stühmer, and Sebastian Nowozin. Meta-lcarning probabilistic inference for prediction. In International Conference Learning Representations, 2019.
Thomas I. Griffiths and Zoubin Ghahramani. The indian buffet process: An introduction and review. Journal of Machine Learning Research, 12(4), 2011.
David Ha, Andrew Dai, and Quoe V Le. Hypernetwrrss. arXiv preprint 16Xiv:1609.099 2016.
10

---

Under review as a conference paper at ICLR 2023
Matthew Hoffman and David Blei. Stochastic structured variational inference. In Artificial Intelligence and Statistics, pp. 361-369. PMLR, 2015.
Timothy Hospedales, Antreas Antoniou, Paul Micaelli, and Amos Storkey. Meta-learning in neural networks: A survey, arXiv preprint arXiv:2004.05439, 2020.
Wenpeng Hu, Zhou Lin, Bing Liu, Chongyang Tao, Zhengwei Tao Tao, Dongyan Zhao, Jinwen Ma, and Rui Yan. Overcoming catastrophic forgetting for continual learning via model adaptation. In International Conference Learning Representations, 2019.
Ekaterina lakovleva, Jakob Verbeek, and Karteek Alahari. Meta-learning with shared amortized variational inference. In International Conference on Machine Learning, pp. 4572 4582. PMLR, 2020.
Masha Itkina, Boris Ivanovic, Ransalu Senanayake, Mykel J. Kochenderfer, and Marco Pavone. Evidential sparsification of multimodal latent spaces in conditional variational autoencoders. In Hugo Larochelle, Marc' Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (eds.), Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurlPS 2020, December 6-12, 2020, virtual, 2020.
Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterizztioo with gumbel-softmax. In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings. OpenReview.net, 2017.
Ghassen Jerfel, Erin Grant, Tom Griffiths. and Katherine A. Heller. Reconciling meta- meta-learning and continual learning with online mixtures of tasks. In Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d' Alché-Buc, Emily B. Fox, and Roman Garnett (eds.), Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurlPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pp. 9119-9130, 2019.
Adam Kalai and Santosh Vempala. Efficient algorithms for online decision problems. Journal of Computer and System Sciences, 71(3):291-307, 2005.
Samuel Kessler, Vu Nguyen, Stefan Zohren, and Stephen J Roberts. Hierarchical indian buffet neural networks for bayesian continual learning. In Uncertainty in Artificial Intelligence, pp. 749-759, PMLR, 2021.
Diederik Kingma and Max Welling. arXiv:1312.6114 14,, 2013.
Auto-encoding variational bayes.
arXiv preprint
James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinst Barwinska, et al. Overcoming catastrophic forgetting in neural networks. Proceedings of the national academy of sciences, 114 (13):3521-3526 2017.
Alex Krizhevsky, Geoffrey Hinton, al. Learning multiple layers of features from tiny images. 2009.
Abhishek Kumar, Sunabha Chatterjee, and Piyush Rai. Bayesian structural adaptation for continual learning. International Conference on Machine Learning, pp. 5850-5860 PMLR, 2021.
Anirban Laha, Saneem Ahmed Chemmengath, Priyanka Agrawal, Mitesh M. Khapra, Karthik Sankaranarayanan. and Harish G. Ramaswamy. On controllable sparse alternatives to softmax. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Sesa-Bianchi, and Roman Garnet (eds.), Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, pp. 6423-6433, 2018.
Brenden Lake, Ruslan Salakhutdinov, Jason Gross, and Joshua Tenenbaum. One shot learning of simple visual concepts. Proceedings of the anmual meeting of the cognitive science society, volume 33, 2011.
11

---

Under review as a conference paper at ICLR 2023
Sang- Woo Lee, Jin-Hwa Kim, Jachyun Jun, Jung- Woo Ha, and Byoung- -aau Zhang. Overcoming catastrophic forgetting by incremental moment matching. In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. Wallach, Rob Fergus, S. V. N. Vishwanathan, and Roman Gamett (eds.), Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4.9, 2017, Long Beach, CA, USA, pp. 4652 4662, 2017.
Chris J. Maddison, Andriy Mnih, and Yee Whye Teh. The concrete distribution: continuous relax- ation of discrete random variables. In 5th International Conference on Learning Representations, ICIR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings. OpenReview.net, 2017.
Andre Martins and Ramon Astudillo. From softmax to sparsemax: A sparse model of attention and multi-label classification. In International conference on machine learning, pp. 1614-1623. PMLR, 2016.
Tsendsuren Munkhdalai and Hong Yu. Meta networks. In International Conference on Machine Learning, pp. 2554-255560 PMLR, 2017.
Eric T. Nalisnick and Padhraic Smyth. Stick-breaking variational autoencoders. In 5th Interna- tional Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings. OpenReview.net, 2017.
Cuong V Nguyen, Yingzhen Li, Thang D Bui, and Richard E Turner. Variational continual learning. In International Conference on Learning Representations, 2018.
Maria-Elena Nilsback and Andrew Zisserman. Automated flower classification over a large number of classes. 2008 Sixth Indian Conference on Computer Vision, Graphics & Image Processing. pp. 722-729. IEEE, 2008
Boris N. Oreshkin, Pau Rodríguez López, and Alexandre Lacoste. TADAM:
task dependent adaptive metric for improved few-shot learning. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett (eds.), Advances in Neural Information Processing Systems 31: Anmual Conference on Neural Information Processing Systems 2018, NeurlPS 2018, December 3-8, 2018, Montréal, Canada, pp. 719-729, 2018
Pingbo Pan, Siddharth Swaroop, Alexander Immer, Runa Eschenhagen, Richard E. Turner, and Mohammad Emtiyaz Khan. Continual deep learning by functional regularisation of memorable past. In Hugo Larochelle, Mare' Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan- Tien Lin (eds.), Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.
Sachin Ravi and Alex Beatson. Amortized bayesian meta-learning. In International Conference on Learning Representations, 2018.
Sachin Ravi and Hugo Larochelle. Optimization as model for few-shot learning. In 5th Interna- tional Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings. OpenReview.net, 2017.
Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In
Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. Wallach, Rob Fergus, S. V. N. Vishwanathan, and Roman Garnett (eds.), Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA, pp. 4077-4087 2017.
Michalis K. Titsias, Jonathan Schwarz, Alexander G. de G. Matthews, Razvan Pascanu, and Yee Whye Teh. Functional regularisation for continual learning using gaussian processes. CoRR, Abs/1991..11330 2019.
Joaquin Vanschoren. Meta-learning: A survey. arXiv preprint arXiv: 2018
Huaxiu Yao, Ying Wei, Junzhou Huang, and Zhenhui Li. Hierarchically structured leaanng. In Proceedings of the 36th International Conference on Machine Learning. ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97, pp. 7045-7054, PMLR, 2019.
12

---

Under review as a conference paper at ICLR 2023
Huaxiu Yao, Yingbo Zhou, Mehrdad Mahdavi, Zhenhui Li, Richard Socher, and Caiming Xiong. Online structured meta-learning. In Hugo Larochelle, Mare' Aurelio Ranzato, Raia Hadsell, Maria- Florina Balcan, and Hsuan-Tiee Lin (eds.), Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurlPS 2020, December 6-12, 2020, virtual, 2020.
Pauching Yap, Hippolyt Ritter, and David Barber. Addressing catastrophic forgetting in few-shot problems, In International Conference on Machine Learning, pp. 11909-11919, PMLR, 2021.
Qiang Zhang, Jinyuan Fang, Zaiqiao Meng, Shangsong Liang, and Emine Yilmaz. Variational continual bayesian meta-learning. Mare' Aurelio Ranzato, Alina Beygelzimer, Yann Dauphin, Percy Liang, and Jennifer Wortman Vaughan (eds.), Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, 2021.
December 6-14, 2021, virtual, pp.
Zhenxun Zhuang, Yunlong Wang, Kezi Yu, and Songtao Lu. No-regret non-convex online meta- learning. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3942-3946. IEEE, 2020.
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A VARIATIONAL INFERENCE FOR META-LEARNING
Following MAML Finn et al. 2017 many Bayesian variant (Ravi & Beatson 2018 Gordon et al.
2019; lakovleva al., 2020) are proposed. To fit well with the bi-level optimization architecture, most of them consider a hierarchical bayesian inference (Amit & Meir 2018), where the Evidence Lower Bound (ELBO) of likelihood can derived as follows:
T
S p(0)
T II
log
II
D
=
log
p(D:los)p((ss))ddg d0
:==1
log I p(D:)o)))p(())))dd ]
2 (0000
Belerdd) [log (D:))))) D
(x=(((00)))))(0)) (17)
where 0 and Φ are the global parameter and task-specific parameter, respectively. Note that the low bound is derived based on the Jensen equation and the variational distributions of and e are introduced to approximate the intractable posterior. Then, the bi-level optimization is transformed as:
A* arg max E (0:+))
Σ
Blland (logp(D,lo.) D
A,A
(x0 (0)00)))pp00) (()))
(18)
So that the goal of the optimization is to seek the optimal variational distribution of 0 and Φ parameterized by vb and X, respectively.
B COMPUTING RULES IN EVIDENTIAL THEORY
There are some computing rules introduced by Dempster-Sheee Shaferr theory (Dencrux 2019p. Given two mass functions my and m2, their combination is defined according to the Dempster rule:
(m1 A m2) = 1 BnC=A
1
Σ
m1(B) m2(C),
(19)
where & is the degree of conflict between two evidences, which is defined as:
Σ (1 - mm2(C..
(20)
BOA=#
Note that Dempster's rule for the combination of mass functions is commutative and associative. Based on Dempster's 's rule for the combination between two mass functions, the combination of two corresponding contour functions pl and pla can be computed as:
(pl + pl2(2x)) pla (2x) pl2(zx)
(21)
K
And if both mass functions are simple with the same strict subset, their fusion can be defined as:
A @ A th A 4102
(22)
where nh represents the simple mass function with single strict subset and its evidential weight is W1.
C THE COMPUTATIONAL DETAILS OF FUSING MASS FUNCTION
We try to combine all the positive mass functions and all the negative mass functions, respectively. And then the two be fused produce the final result.
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Continual
Setting
c-1
Support
Doubt
Support
Doubt
Censoiity Uncertanty
Corturity Uiceriantty
Centartyy
Evidences
[Rv]
Cortainty Uncertainty
Uncertalnty
Certalitty Uncertainty
Cartantly
Uncartaincy
k 1,.., K
Dempster's rule
Merged Result
Figure 6: An intuitive explanation of our proposed evidential sparsity method. The relationship between tasks and meta-knowledge components each time provides evidences, containing support and doubt information with uncertainty. Such information can be merged by Dempster's 's rule to provide unified relationship between the occurring tasks and the components. The components not receiving support removed (i.e., the component K in the figure).
C.1 THE FUSION ACROSS TIME
Before positive fusion and negative fusion, we need to merge evidence supporting the same focal elements different times. Since the simple mass functions have the same focal set, their fusion can be calculated following Eq. 22land the weight is:
w
Σ w
W
W
(23)
where wit and W are the evidential weight of the positive and negative mass function at time i respectively. In this way, the evidence supporting the same focal element from different time be merged first:
m ([*x}) 1 my ({=+)) = 1 exp( ( my (Z) ).
m (Z) exp(- ):
(24)
(25)
C.2 THE FUSION OF +
As we define above, all the positive mass functions have the only two focal elements, {2x} and Z. Then the combination of them can be computed according to the Dempster's rule:
K
(1 x [1 - exp( Exp(-wt) wx 1
Il
expppwuj = (exp(wtt II exp( w
(26)
14
m m (Z) x -
(27)
1=1
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As the fused mass function constraint to the sum of one, the results can be computed by normalizing the terms. So that the sum of all terms is:
K
K
K
K
m (Z) m
+ II m + (fzx))
&
II
exp( ( wxt
+
(exp(wx)
11
II
expl - wkk
(28)
1m1
J6ml
km
[ml
k
K
II M==
exp( exp(-wit
Σ
(utp(w))
K +1
(29)
And the terms can be normalized as:
(erp(w [ exp( -w
m + (12+})
k=1 expl (wx
[
[E k== exp(wxt) - K6 +1
exp(wit) -
(30)
E exp(utt) K +
K erp(- -w [ exp(- -w
(²)
[Ek=re exp(wit) - K +1
(31)
1 exp(wtt) K +1
C.3 THE FUSION OF m
Different from the positive mass functions, the negative mass functions have the only two focal elements, } and Z. To compute the combination of all negative mass functions, need to compute the conflict firstly:
K
R
(1- e1p)(((-w)))
(32)
A=1
Thus, for any strict subset A of Z, its belief can be computed as:
m (A) = I (1 (1-erp(-- (e00-ww wx))] [II (cp(-wū])
(33)
[ (1 expl (wpp--wF)
And the mass belief of the complete set Z is:
I erp(-w:
K
m (2)
(34)
I ezp( 1epp(-wp w
For further fusion of the positive and negative mass functions, need compute pl- (zz), which can be defined as:
I K (1 exp(-wx))
pl ({2+))
pl ({zx))
(35)
where the plausibility of negative mass function is:
{11 (exp(-w;))
k otherwise
pl, ((=x))
(36)
Thus, the result of the fused plausibility is:
eapl - wZ
pl (fz+1) =
(37)
1 1=1 (1 e.ppp-pppp w.
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C.4 THE FINAL FUSION
To clarify the following derivation, we assume that:
C+
(38)
E exp(wit) K +
==
(39)
-===1 (1
Similarly, to combine the positive and negative mass function, we need to compute the conflict between them first:
K
Σ mm((a)) (
}
m
FLEA
{m² font(12a4 (12x)) [1 (1- pl [[xx((((x))]]
K
Σ
{c+ [exp(wtt -1] [1 C C-eepp( ([ppp
(40)
k=1
where A S Z. To make the following derivation clarified, let:
C 1-K 1- -- (C+ (erp(wt)))))))))))))))))))) - (expp-- -1 ))}}
(41)
Then for any k E (1,2, ..., K), the mass belief of each singleton be computed
m({=+]) = m ({2x})
Σ m (A)
+ m (Z) m ((2x))
LENE
= =C (C ({zx}) pl (12+3) }) -mm mm+(2) (Z) m - ({+x))}
(42)
where A Z. Combining Eq. 30 Eq. 31 Eq. 33 and Eq. 37 the final result of the mass singleton belief is:
ml (
c c+ c+ (erp(wx)) 1] c- (exp((-w))) ++ o- exp(--w) 1g
II (1 expl exp(-wi))
= c- expp- corp(-wx) exp(utt) -1 II -1-1rpp(-00 (rp(-=(r)
(43)
D DETAILS OF INFERENCE
In this section, we present the details of our structured variational inference for our proposed SMM- CML. The pseudo-code and the probability model are shown in Alg and Fig. respectively.
D.I VARIATIONAL DISTRIBUTION
Because of the intractability of posterior, introduce the variational distribution approximate the true posterior. To capture the dependencies among the approximate posterior distribution, we consider using the structured mean-field approximation (Hoffman & Blei 2015) instead of the traditional mean-field approximation. Specifically, the joint variational distribution can be decomposed as follows:
K
q(vi, zt, 0,, 6. D.) = 0r 21, D.) II
((w.)aa Am
(44)
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VK
Zt,k
Φ
Qt,k
K
Xi
N₂ + M
T
Figure 7: The probability model SMM-CML. The solid line denotes the generative process, and the white circle and the grey circle denote the latent variant and the observed variant, respectively.
The composed variational distributions parameterized as:
g(vL,)) = Beta( 8t,k),
(45)
me, Bern(Te,k) where Te,k Ut
(46)
(00,kk) -N o...)))
(47)
K
( 24, D,) =
Σ 1 (aLk # 0f Ac,k),
(48)
k=
where x;= = SGD, D c), and SGD,()) represents the stochastic gradient descent with J steps. That is, the required variational parameters are th {art, Br,,, (te,k; or.k, At,x) for k = I,.,, K.
Note that we replace IT Ut with 0 in the posterior, to remove the implicit order constraint in the prior. So that the optimization aims to search for the optimal variational parameter maximize the ELBO in Eq. [6
D.2 REPARAMETERIZATION
The variational posterior is obtained by optimizing the ELBO using structured variational inference. To make inference tractable, utilize three reparameterizatiooss, to infer the Gaussian distribution, beta distribution and Bernoulli distribution, respectively.
D.2.1 THE VARIATIONAL GAUSSIAN DISTRIBUTION REPARAMETERIIATTION
As we mentioned above, the variational distributions meta-knowledge from each clusters are diagonal Gaussian 02 (Nt,,, o2,)). We employ the reparameterization, which represent the meta-knowledge using a deterministic function 0., = g(E; fr,ks kt,k), where ~ (O, To apply the reparameteriizatoon, we define the standardization function and its inverse as:
S (0) 0 -e- q(e), where g(s) =N(0.,
= 0 (5)
(E) =
(49)
Note that we omit the subscripts for clarity and the remainder of this section omits them as well. Then we can represent the objective in ELBO w.r.t (0) as follows:
E (0) [[(0)) [S
(e)) (e)))
(50)
This allows to compute the gradient of the expectation in another way:
V E (o)(((0) = Egte) ( = (E (e))
(51)
D.2.2 THE VARIATIONAL BETA DISTRIBUTION REPARAMETERIIZATION
There no simple inverse of the standardization function when using the reparameterization for Beta distribution, which makes impossible apply the explicit reparameterization directly. Instead,
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Algorithm 1:
The meta-training process of SMM-CML.
Input:
Task distribution р(T), data distribution p(D[7),
the initial number of component Ko. concentration parameter a, the number of inner update step J, the inner learning rate and the outer learning rate
1: for t=l... do
2: Determine the added number: Kr,new Possion( (
3: Determine the number of component: K₁ Ki-1 + K Kinew
4: Initialize the variational beta distribution: ar, Se, Ak 1,... K₁
5:
Initialize the variational distribution of meta-knowledge: Pk, Øk, Vk 1, K2
6:
while not converge do
7:
Sample 1 ~ q(v,,x: 0r, Pr, Vk =1.... K
8:
Compute the ELBO according to Eq. 16
9:
Compute the gradient: Vpc, Vor Vk 1, .... Ke via explicit reparameterizatioo
according to Eq. 5T
10:
Compute the gradient: Vark: VBt, Vk 1, .... K₁ via implicit reparameterizzatoon
according to Eq. 55
11:
Update the variational parameters: 0ik Ot.k Vanr Vk 1, ..... Kr
12:
Update the variational parameters: BL + PE, <V VK 1, Ke
Update the variational parameters: Me, Ht, CV WK- Kr
14:
Update the variational parameters: Je. + Jt - SVOLk Vk 1, Ke
end while
16:
Update prior: p(u,,)) + q(,,,)), 1, Kr
17:
Compute the evidential weight w "',, vk
1,... K, according to Eq.
11
18:
Compute the mass function according to Eq.
Remove the components without support information according Eq. 15
20:
end for
there are two ways to tackle the problem: the implicit reparameterization and the Kumaraswamy reparameterization.
Implicit reparameterization. This way also utilizes the reparameterizztioo to tackle the intractable
gradient in Beta distribution:
V (on Ev, Ve
VV
(52)
without the inverse the standardization function, the term V, is difficult compute. Inspired by (Figurnov et al. (2018), we employ the implicit reparameterization to compute the gradient, the idea of which is to differentiate the standardization function S., (vx) = using the chain rule instead of
searching its inverse:
(,,,, V,S, O, 7 = -V S. (vx)) (v)) -V S,, (vx).
(53)
(54)
Note that the standardization function can be the CDF of the Beta distribution and ~ Uni 1|. Then the implicit gradient is:
V F(eii)) v(F(u)))
V Up (V F(witt)
(55)
-p(wnit)
where 7) is the PDF of the Beta distribution.
Kumaraswamy distribution. The Beta distribution of Us also be reparameterized using
Kumaraswamy distribution (Nalisnick & Smyth 2017). The Kumaraswamy distribution can be defined
p(ve; a, 9) a: og)
(56)
and then the inverse of standardization function be computed as:
S,(Uk) (1 ee1/8///a where e Unif
(57)
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The KL-Divergenee between the Kumaraswamy distribution and the Beta distribution in ELBO can be written as:
DKL axll:::x ak - (-y
(-y --y- - (((xk) B4
+ log ax
Bk
log [B(a, 3)]
Bk
+ (8 1)35
Σ m=1
B
m
5k
(58)
Ockk
Okk
where y is the Euler constant, (.()) is the digamma function, and B(,,) .) is the beta function. Following the existing work (Nalisnick & Smyth, 2017), the above the infinite term in the formula can be approximated using infinite sum of the first terms.
D.2.3 THE VARIATIONAL BERNOULLI DISTRIBUTION REPARAMETTRIIZTTON
As the Bernoulli distribution one the classic discrete distributions, the sampling requires performing an orgmar operation. But the yraa operation is not differentiable.
We employ the Concrete distribution (Maddison et al. 2017), also named Gumbel-softmax distribution (Jang et al. 2017). address the above issue. Then,
sample random variable follows:
log(T<) ) + log 1-ws
IJ
(O,, 1),
(59)
where À E (0, oo) is a temperature hyper-parameten (-) is the sigmoid function, Ty is the parameter of the Bernoulli distribution and uę sampled from a uniform distribution U. To guarantee lower bound on the ELBO, both posterior and prior Bernoulli distribution need to be replaced with concrete distribution:
DKL (K, (ll 2 DKL A)]p(( Ta. A)]
(60)
E THE ANALYSIS OF COMPLEXITY
We discuss the computational cost of our proposed SMM-CML as follows, including the time complexity and space complexity.
For time complexity, the de facto bi-level optimization mechanism in meta-lcarning requires O(n²) when updating one knollleewwedee component, where an algorithm with time complexity O(n) is linear time algorithm. If without any sparsification constraint on the number of components, it will see an unlimited increase, and thus the time complexity will be up (n²)) If with our evidential sparsification, the number of components will be limited to small constant C' with an appropriate hyperparameter y in Eq. 10 so that the time complexity will be down to O * n²) R O(1²).
Similarly, each component of meta-knowledge contains the parameter of the model, its space complexity is On)). And the total space complexity models without sparsification will be up to Onn²) for the unlimited number of meta-knowledee components when encountering many tasks. But our algorithm can alleviate this issue using the evidential sparsification to reduce down to O(n) with appropriate hyperparameter 7
F DETAILS OF EXPERIMENT
F. THE DETAILS OF BASELINES
For fair comparison, we use the widely-applied network architecture following (Yap et al. 2021 Zhang al. 202T). In what follows, describe the details of the baselines:
TOE:
Training On- Every-tttnng method (TOE) is an intuitive method, that re-initializes the meta- knowledge and trains them on all the having arrived datasets at each time. We use the same Bayesian
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meta-learning architecture algorithm. The difference between TOE and SMM-CML is that SMM-CML is only trained on the current dataset at each time instead of all the having arrived dataset in TOE and SMM-CML does not re-initialize the meta-knowledge at each time what TOE do.
TFS: Train-Fromms Crom--Scratch (TFS) is another intuitive method, which also re-initializes meta- weta-knowledde but only trains them the current dataset. Similarly, it also uses the same Bayesian meta-learning architecture as our algorithm. The difference between TFS and SMM-CML that our algorithm maintains the posterior meta-knowledge at last time as the prior at the current time instead of re-initializing them as TFS.
FTML: Follow the Meta Leader (FTML) proposed by (Finn al.] 2019) uses the Follow the Leader algorithm to fill the gap between meta-learning and online learning. However, it assumes that all the having arrived datasets are available, which is memory-consuming and conflicts with the continual meta-learning. For a fair comparison, we only train FTML on the current dataset as same as our algorithm.
OSML: Online Structured Meta-Learning (OSML) Yao et al. 2020 maintains meta- veta-hierarchical
graph with different knowledge blocks and conducts a meta- knowledge pathway for the encountered new task. However, it employs well pre-trained convolution network to initialize the model in the original paper. As SMM-CML. and other baselines are randomly initialized, would be unfair to use the original initializing way. Therefore, we also randomly initialize the OSML model.
DPMM: Dirichlet Process Mixture Model (DPMM) (Jerfel et al. 2019) employs a Chinese Restaurant Process to conduct the mixture meta-knowledge with dynamic number of components. Note that it is not a Bayesian method and employs the point estimation to update the meta- neta-knowleedee
Bayesian Online Meta-Learning with Variational Inference (BOMVI) Yap al. 2021 is ofate-of-the-art art algorithm, which conducts a meta-knowlee knowledge distribution to address the catastrophic forgetting issue in continual meta-learning. Similarly, also employs variational inference to update the meta-knowledge.
BOMVI:
VC-BML: Variation Continual Bayesian Meta-Learning (VC-BML) (Zhang et al. 2021 is another state-of-the-ar algorithm. which also employs a truncated Chinese Restaurant Process to conduct the mixture meta-knowledge Different from DPMM, it uses the Bayesian inference conduct the mixture distribution of meta-knowledge and places an upper bound on the number of components to reduce the computational consumption.
All the baselines and our proposed SMM-CML follow the experimental setting as described in Sec.
F.2
THE DATASETS
VGG-Flowers VGG-Flowers Nilsback & Zisserman 2008) consists of 102 flower categories. Also, we randomly choose 66 categories for meta-training, 16 categories for validation and the remained 20 categories for meta-test.
& Larochelle 2017) is designed for few-shot learning, which consists of 100 different classes. Similarly, we also split the dataset into three datasets (i.e., 64 classes for meta-training, 16 classes for validation and 20 classes for meta-test) following the existing works.
htgllmaggnetmmiiimmmeeeene
et al. 2018) dataset used in our experiment is adapted from the CIFAR- 100 dataset (Krizhevsky et | 2009)) for few-shot learning, which consists of 100 classes. Following the existing works (Yap et al.) [2021] Zhang et al.) [02T), we also randomly split the datasets, where 64 classes are used for meta-training, 16 classes are used for validation and the remained 20 classes are used for meta-test, respectively.
CIFAR
et al.) 2011) is widely-used dataset. which contains 1,623 different handwritten characters from 50 different alphabets. Following the previous works (Yap et al. [022 Zhang al. 2021). we randomly split the dataset into three subsets, 1,100 characters for meta-tras meta-training, 100 characters for validation and the remaining 423 characters for meta-test.
To create the online non-stationary setting, assume the above datasets arriving and available sequentially. Moreover, we focus 5-way 5-shot task, which conducts the low-resource environment. For each dataset, form the streaming tasks via randomly sampling 5 classes with replacement
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Table 4: The conyolution neural network architecture in SMM-CML and baselines.
Layers
Output Size
Input image
28 28
The first convolution layers
14 x 14 64
The second convolution layers
x X 64
The third convolution layers
3 X 64
The forth convolution layers 1 x 64
Table 5: Some important Syper-parameen used in our experiments.
Plyper-parameter
VGG-Flowers
minilmagenet
CIFAR-FS
Omniglot
The number of outer update step The outer learning rate The number of outer update step The inner learning rate
2000
2000
2000
2000
0.001
0.001
0.001
0.001
3
3
3
1
0.05
0.1
0.1
0.1
task. And we randomly sample examples for each class in a support set and examples for each class in query set.
In our experiment, we also consider another more challenging setting, where tasks from different datasets are mixed and arrive one by one. In this setting, we conduct different tasks stream with length of 100, and then train the model on each task one by one and evaluate the performance on each dataset.
F.3 THE DETAILS OF EXPERIMENT SETTING
For each task, employ the same convolution network our base network following the previous works (Yap et al. 2 Zhang et al. 2021), which showed in Tab. 4 For our model, we use the Adam optimizer the outer optimizer and the SGD optimizer as the inner optimizer. For the Monte Carlo sampling used in our algorithm, we set the number of sampling as 5. For the initial number of set 4. All the important can be seen in Tab. 5 We ran our algorithm on NVIDIA Tesla V100 32GB GPU. It took about 54 hours train.
components in the multi-modal meta-knowledge.
F.4 ADDITIONAL EXPERIMENTAL RESULT
what follows, we present the full result on streaming datasets (i.e., VGG-Flowers, minilmagenet, CIFAR-FS and Omniglot), and change the order of datasets to verify the generality of our algorithm.
F.4.1 META-TEST ACCURACIES ON EACH DATASET AT DIFFERENT META-TRAINDE STAGE
We only show the average result at each meta-training stage and the performance on each dataset at the last meta-training stage in the main text. We additionally show the full results in Tab, 6 Although SMM-CML can not achieve the best performance on all having arriving datasets some meta-traing
meta-training stages (i.e., CIFAR- FS and minilmage), it outperforms all the baselines on the average results, which confirms the effectiveness of SMM-CML. Additionally, SMM-CML can not only maintain the performance on the old datasets, but also achieve better results on the new datasets, which illustrates that can alleviate better catastrophic forgetting than other baselines. Note that SMM-CML achieves the best performance on the current datasets at each stage (especially compared to VC-BML, where it assumes that each component is mutually exclusive), which shows that our proposed model can resolve the conflict between the learned meta-knows knowledge and the incremental meta-Knowledge, and it is expected that the multi-modas can utilize the shared neetl-knoweede to improve the performances.
Tab. also shows the detailed results of SMM-CML before and after evidential sparsification. The results show that SMM-CML still achieves comparative performance most datasets at each
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Table 6: Performance of SMM-CMI. and the baselines on each datasets each meta-training stage. The best performance on each dataset is marked with boldface and the second best is marked with underline.
Meta-Training Stage
Algorithms
VGG-Flowers
minilmagenet
CIFAR-FS
Omniglot
Average
FIML
76.84 + 1.75
76.84 = 1.75
OSML
79.61 + 1.50
79.61 + 1.50
DPMM
78.97 + 1.52
78.97 1.52
VGG-Flowers
BOMVI
77.05 + 1.80
77.05 1.80
YC-BML
83.71 = 1.58
83.71 1.58
SMM-CML
85.11 + 1.46
85.11 1.46
FTML.
76.51 + 1.92
44.97
1.77
60.74 + 1.85
OSML
76.19 1.68
56.11 1.77
66.15 = 1.73
DPMM
76.65 + 1.79
56.45 :: 1.74
66.55 = 1.77
minilmagenet
BOMVI
75.75 = 1.97
45.12 1.74
60.44 = 1.86
VC-BML
76.47 + 1.41
59.71 1.75
68.09 1.58
SMM-CML
81.71 + 1.42
57.19 1.66
69.45 1.54
FTML.
75.11 + 1.84
54.89 + 1.66
70.13 + 2.07
66.71 + 1.86
OSML
78.29 + 1.63
57.36 + 1.54
69.07 2.01
68.24 - 1.73
DPMM
75.60 + 1.76
55.79
70.15 2.07
67.18 1.86
CIFAR-FS
BOMVI
74.08 + 1.60
47.55 + 1.84
37.07 1.86
59.57 1.77
VC-BML
79.04 1.54
1.74 59.17
71.00 1.93
69.87 1.74
SMM-CML
79.25 + 1.48
58.96
1.69 73.89
70.72
FTML
63.04 1.0
37.27 + 1.69
47.95 + 1.99
99.31 + 0.28
61.89 = 1.49
OSML
70.68 + 1.83
40.67 1.50
51.89 2.04
99.35 + 0.24
65.63 1.40
DPMM
65.20 + 1.67
48.53 1.63
60.15 + 2.30
99.16 + 0.29
68.26 1.47
Omniglot
BOMVI
73.19 + 1.86
16.28 1.62
58.99 + 2.14
97.71 + 0.53
69.04 1.54
VC BML
1.76
48.53
39.14
99.21 0.47
1.52
SMM-CML
71.92 + 1.86
50.07 1.66
64.50 1 1.83
99.36 0.22
71.46
meta-training stage, compared to before evidential sparsification. further confirms the effectiveness of our proposed evidential sparsification.
F.4.2 ADDITIONAL EXPERIMENTAL IN DIFFERENT ORDER
To further confirm the generality of our model, we change the order of datasets in the streaming tasks. We conduct the experiments
order, where the model is trained chronologically CIFAR-FS minilmagenet and VGG-Flowers. The results are shown in Tab. / The result on the streaming tasks with different order shows that SMM-CML still outperforms other baselines, which further confirms the generality of SMM-CML.
Omniglot,
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Table 7: Performance of our SMM-CML and the baselines on each datasets at each meta-training stage. The best performance (without 'original") on each dataset is marked with boldface and the second best (without 'orgiiaal)) is marked with underline.
Meta-Training
Algorithms
Omniglot
CIEAR-FS
miniImagenet
VGG-Flowers
Average
Stage
FTML
99.25 +: 0.24
99.25 = 0.24
OSML
98.20 +: 0.39
98 28.20= = 0.39
DPMM
97.15 + 0.48
97.15 + 0.48
Omniglot
BOMVI
97.35 + 0.73
97.35 + 0.73
VC-BML
99.28 +: 0.48
99.28 = 0.48
SMM-CML
99.43 +: 0.21
99 43 + 0.21
orginall
00.21
99.31 = 0.25
FTML
1.122+0770
67.08 1.87
81.60 1.32
OSMI.
96.09 00.99+0.53
66.20 2.02
81.15 1.28
DPMM
93.31 0.880
60.88 2.03
77.10 1.42
CIFAR-FS
BOMVI
97.68
56.29 = 2.00
76.99 1.22
VC-BML
97.72 0.38
72.8 + 1.74
85.26 = 1.06
SMM-CML
17.66+00.3
75.39 + 1.71
$6.53 + 1.05
original
98.15 0.339
73.82 = 1.75
85.99 = 1.07
FTML
96.63
68.60 = 1.79
73.30
OSML
9.79
69.20 1.72
55.13 1.81
73.12 1.44
DPMM
95.01
2.14
55.49 1.74
71.81 1.52
minilmagenct
BOMVI
97.01 0.70
59.25 1.76
46.21
1.66
67.19 1.3T
VC-BMI.
+
69.05 1.68
59.25
1.86
74.86 1.37
SMM-CML
97.10
70.67 4 1.97
59.97
1.70 60.13
75.91 1.38
onginall
96.74+00.69
71.35 = 1.72
76.07 1.40
FTML
93.69 0.83
58.27 1.81
45.75 1.52
80.32 + 1.77
69.51
1.48
OSML
91.79
59.05 1.80
1.64 46.51
81.71 + 1.69
69.77
1.57
DPMM
93.21 1.03
61.55 1 1.82
45.01 1.56
1.72
70.12
VGG-Flowers
BOMVI
97.13
=
47.24 1.81
2.04
69.68
1.57
VC-BML
92.80
58.36 1.87
47.09 1.78
82.92
70.29 1.48
SMM-CML
04.0740.84
58.76
83.31 1.50
71.22 1.36
onginal
08.25110.811
19.79
84.1611.56
24


Reviews:
[{"review_id": "N_ztYvt0EGw", "reviewer": "Reviewer_yxSH", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "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": "This paper proposes a continual meta-learning framework, Scalable Multi-Modal Continual Meta-Learning. Specifically, the authors employ the Indian Buffet Process for sharing meta-knowledge across different tasks and encourage evidential sparsity for parameter efficiency. The experiments validate the proposed method under the online non-stationary setting.", "summary_of_the_review": "Given the strength and weaknesses, I tend to rate this paper as marginally below the acceptance.", "strength_and_weaknesses": "Strength\n\n1. The paper aims at a very interesting and important topic.\n2. The use of IBP prior to handle the update of the number of mixture components looks novel.\n\nWeaknesses\n\n1. The writing can be improved. For example, it would help if the authors can elaborate on why the proposed method can allow the sharing of meta-knowledge and why previous methods cannot (as this is one major contribution summarized by the authors). The term \"multi-modal\" throughout the paper also seems obscure to me. Plus, by \"multi-modal knowledge\" I guess the authors do not want to mean something like knowledge from image and text but want to mean something relating to informativeness as in [1]. It would help if the authors can make this point more clear.\n2. I am not sure about the novelty of section 4.3. In section 4.3, many equations (Eqs 14-17) are very similar to section 2.2 of [1] without explicitly citing [1]. However, I am not an expert in this field. I would like to bring it to other reviewers' attention and would defer to their comments.\n\n\n[1] Masha Itkina, Boris Ivanovic, Ransalu Senanayake, Mykel J. Kochenderfer, and Marco Pavone. Evidential sparsification of multimodal latent spaces in conditional variational autoencoders. In NeurIPS 2020.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "MUVKYTqtte", "reviewer": "Reviewer_XxCX", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper presents a scalable multi-model continual meta-learning algorithm. This method associates a cluster of similar tasks with a set of meta-knowledge components instead of one single component in previous approaches. If I understand correctly, “multi-modal” in the title is due to a set of components being used. The authors proposed to use Indian buffet process to determine whether to add new components into the mixture when learning for a new task and leverage evidential theory to remove redundant components from the mixture. They evaluated their methods using four benchmark datasets. ", "summary_of_the_review": "The proposed idea may be interesting. However, poor presentation really affects a thorough understanding of the paper.     ", "strength_and_weaknesses": "Strength\n+ The idea is innovative and may be of value\n+ Based on the reported empirical results, the proposed method seems working. \n\nWeakness\n- The paper is poorly written, very difficult to follow. Especially in the mathematic derivation, the overloading and inconsistency choice of notations cause a lot of confusion. For example, in 3.2, what exact A is? Is it an element in Z or a set? The definition of the plausibility of A in Eq. (5) is confusing. In the paragraph following Eq. (7), what does exact k represent, a cluster or component? Awkward sentences and poor choices of worlding show up in so many places. \n- If I understand correctly, the posterior calculated to determine whether to remove components is only conditioned on the data at current time point as one has no access to previous data in a continual learning setting. If this is the case, the removal of components could easily lead to catastrophic forgetting.  \n- There is lack of clarity in the description of the setting of their experiment. Specially, at what point, the performance on a task was evaluated? Was it right after the training for that task was completed or after the training for all tasks were done. If the former, catastrophic forgetting was not properly evaluated. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "8Uw-tq8926F", "reviewer": "Reviewer_MeHS", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper studies the continual meta learning, widely used in low-resource setting.  Compared with the existing works, this work is motivated by two points. One is existing works assume the number of components of meta knowledge is mutually exclusive. Two, existing works usually only use a prior determined by Chinese Restaurant Processs, but do not make a posterior decision on number of components. To address the two issues, this method proposes an IBP prior to determine whether to increase the number of components, then leverages evidential theory to filter out the uninformative components.  \n\n\n--- post-rebuttal ---\\\nI have read the authors' response and other reviewers' comments. I think the current form of this paper is not ready for ICLR publication.", "summary_of_the_review": "Overall I think the idea of applying evidential theory to address the inappropriate mutual exclusiveness of mixture meta knowledge components is interesting. While unfortunately there are several concerns on the presentation and experiments, that needs to be addressed and clarified.", "strength_and_weaknesses": "Strength \n\n-Few shot continual learning is a hot research topic, the authors successfully identify two weaknesses of the existing works, and proposed a principal solution to solve them. \n\n-Evidential theory is a well suited equipment to solve the mutual exclusiveness among mixture components.  \n\n-Detailed theoretical derivation is provided.  \n\n \n\nWeakness \n\n-I have a little bit of hard time following this paper. For instance, the title highlights “scalable” and “multi-modal”. However, I do see much description on these two terms, except in title and abstract. Not sure the meaning of “multimodal”. “Scalable” I guess it refers to the ability of filtering out the components. However, the current experiments do not well support this claim. \n\nAnother part is the technical description in evidential theory. I am not clear how to map the concepts in evidential theory to continual meta learning problem without explanation. This prevents me from understanding the technical details intuitively. Also, some terms in evidential theory lacks definition, e.g. focal set in page 5. \n\n \n\n-In page 1, the authors use user profiling as an example to explain the unreasonable assumption on mutual exclusiveness of meta knowledge components. However, the chosen experimental datasets don’t reflect this to my understanding. Please explain (or better qualitatively illustrate) how the proposed method addresses the mutual exclusiveness on these test data. \n\n \n\n-The experimental results of VC-BML in Table 2 seems lower than the reported numbers in the original paper. Please clarify if there is any difference in experimental setting.  \n\n \n\n-Regarding the scalable claim, I would recommend conducting longer sequence of tasks/datasets to show the value of the proposed m method. The current reported results show a marginal improvement over existing methods. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject