Continuous Transfer Learning
Authors: Jun Wu
Jingrui He
Venue:  (2021)

Abstract: Transfer learning has been successfully applied across many high-impact applications. However, most existing work focuses on the static transfer learning setting, and very little is devoted to modeling the time evolving target domain, such as the online reviews for movies. To bridge this gap, in this paper, we focus on the continuous transfer learning setting with a time evolving target domain. One major challenge associated with continuous transfer learning is the time evolving relatedness of the source domain and the current target domain as the target domain evolves over time. To address this challenge, we first derive a generic generalization error bound on the current target domain with flexible domain discrepancy measures. Furthermore, a novel label-informed C-divergence is proposed to measure the shift of joint data distributions (over input features and output labels) across domains. It could be utilized to instantiate a tighter error upper bound in the continuous transfer learning setting, thus motivating us to develop an adversarial Variational Auto-encoder algorithm named CONTE by minimizing the C-divergence based error upper bound. Extensive experiments on various data sets demonstrate the effectiveness of our CONTE algorithm.


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CONTINUOUS TRANSFER LEARNING
ABSTRACT
Transfer learning has been successfully applied across many high-impact applica- tions. However, most existing work focuses on the static transfer learning setting, and very little is devoted to modeling the time evolving target domain, such the online reviews for movies. To bridge this gap, in this paper, we focus on the con- tinuous transfer learning setting with a time evolving target domain. One major challenge associated with continuous transfer learning is the time evolving relat- edness of the source domain and the current target domain the target domain evolves over time. To address this challenge, we first derive a generic generaliza- tion error bound on the current target domain with flexible domain discrepancy measures. Furthermore, novel label-informed C-divergence is proposed to mea- sure the shift of joint data distributions (over input features and output labels) across domains. It could be utilized to instantiate a tighter error upper bound in the continuous transfer learning setting, thus motivating us to develop an ad- versarial Variational Auto-encoder algorithm named CONTE by minimizing the C-divergence based error upper bound. Extensive experiments on various data sets demonstrate the effectiveness of our CONTE algorithm.
INTRODUCTION
Transfer learning has achieved significant suc- cess across multiple high-impact application domains (Pan & Yang. 2009). Compared to conventional machine learning methods assum- ing both training and test data have the same data distribution, transfer learning allows us to learn the target domain with limited label infor- mation by leveraging a related source domain with abundant label information (Ying al. 2018). However, in many real applications, the target domain is constantly evolving over time. For example, the online movie reviews are changing over the years: some famous movies were not well received by the mainstream audience when they were first released, but became famous only years later (e.g.. Citizen Cane, Fight Club, and The Shawshank Redemption); whereas the online book reviews typically do not have this type of dynamics. It is challenging to transfer knowledge from the static source domain (c.g.. the book reviews) to the time evolving target domain (e.g.. the movie reviews). Therefore, in this paper, we study the transfer learning setting with a static source domain and continuously time evolving target domain (see Figure Dp. which has not attracted much attention from the research community and yet is commonly seen across many real applications. The unique challenge for continuous transfer learning lies in the time evolving nature of the task relat- edness between the static source domain and the time evolving target domain. Although the change in the target data distribution in consecutive time stamps might be small, over time, the cumulative change in the target domain might even lead to negative transfer (Rosenstein et al. 2005).
Negative traenfer
Target doenala
Figure 1: Illustration of continuous transfer learn- ing. It learns predictive function in Dr, us- ing knowledge from both source domain Ds and historical target domain Dr, (i 1,... , 1). Directly transferring from the source domain Ds to the target domain Dr, might lead to negative transfer with undesirable predictive performance.
Existing theoretical analysis on transfer learning (Ben-David showed that the target error is typically bounded by the source error, the domain discrepancy of marginal data distributions and the difference of labeling functions. However, it has been ob- served Zhao et al., 2019, Wu et al., 2019) that marginal feature distribution alignment might not guarantee the minimization of the target error real world scenarios. This indicates that in the context of continuous transfer learning, marginal feature distribution alignment would lead to the sub-optimal solution (or even negative transfer) with undesirable predictive performance when di- rectly transferring from Ds to the target domain Dz, at the t time stamp. This paper aims to bridge the gap in terms of both the theoretical analysis and the empirical solutions for the target domain with a time evolving distribution, which lead to novel continuous transfer learning algorithm as
al.) 2010 Mansour et al., 2009

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well as the characterization of negative transfer. The main contributions of this paper are summa- rized as follows: (1) We derive a generic error bound for continuous transfer learning setting with flexible domain divergence measures; (2) We propose a label-informed domain discrepancy measure (C-divergence) with its empirical estimate, which instantiates a tighter error bound for continuous transfer learning setting: (3) Based on the proposed C-divergence, we design novel adversarial Variational Auto-encoder algorithm (CONTE) for continuous transfer learning; (4) Extensive exper- imental results on various data sets verify the effectiveness of the proposed CONTE algorithm.
The rest of the paper is organized as follows. Section introduces the notation and our problem definition. We derive generic error bound for continuous transfer learning setting in Section B Then we propose a novel C-divergence in Section ! followed by instantiated eπror bound and a novel continuous transfer learning algorithm in Section The experimental results are provided in Section T We summarize the related work in Section and conclude the paper in Section g
2 PRELIMINARIES
In this section, we introduce the notation and problem definition of continuous transfer learning.
2.1 NOTATION
We use X' and y to denote the input space and label space. Let Ds and Dr denote the source and target domains with data distribution ps(x, v) and pr(x, y) over X' x ) respectively. Let H be hypothesis class on X, where a hypothesis is function X - y. The notation summarized in Table 3 in the appendices.
2.2 PROBLEM DEFINITION
Transfer learning (Pan & Yang 2009) refers to the knowledge transfer from source domain to target domain such that the prediction performance on the target domain could be significantly improved as compared to learning from the target domain alone. However, some applications, the target domain is changing over time, hence the time evolving relatedness between the source and target domains. This motivates us to consider the transfer learning setting with the time evolving target domain, which is much less studied compared to the static transfer learning setting. We formally define the continuous transfer learning problem as follows.
Definition 2.1. (Continuous Transfer Learning) Given a source domain Ds (available time stamp j = 1) and a time evolving target domain (Dr, J==1 with time stamp j. continuous transfer learning aims to improve the prediction function for target domain Dr... using the knowledge from source domain Dş and the historical target domain Dr, (j 1,.........
Notice that the source domain Ds can be considered a special initial domain for the time-evolving target domain. Therefore, for notation simplicity, we will use Dí, represent the source domain in this paper. It assumes that there are my, labeled source examples drawn independently from a source domain Dr, and my, labeled target examples drawn independently from target domain Dr, at time stamp j.
3 A GENERIC ERROR BOUND
Given static source domain and time evolving target domain, continuous transfer learning aims to improve the target predictive function over Dr.., using the source domain and historical target domain. We begin by considering the hinary classification setting, i.e., )/ {0, 1). The source error of hypothesis h can be defined as follows: eT, (h) ((x.y)<<r,(( (x.y) [c(h(x),)) y)] where L(:) is the loss function. Its empirical estimate using source labeled examples is denoted êτ, (h). Similarly, we define the target error éT, (h) and the empirical estimate of the target error ér, (h) over the target distribution Pr, (x, y) time stamp j. A natural domain discrepancy measure over joint distributions on X x y between features and class labels can be defined as follows:
dj (Dr,, Dr) sup PrDr, [Q] Prp,[Q]]
QEQ
where Q is the set of measurable subsets under PT, (x, v) and (x. v Then, the error bound of continuous transfer learning given by the following theorem.
Theorem 3.1. Assume the loss function L is bounded with 0 < L < M. Given source domain Dro and historical target domain (Dr, st Yi for h E H. the target domain error ETe on D:41 is
Note that it is slightly different from T. variation divergence in (Ben-David et al. 2010 with only marginal distribution of features involved.
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bounded as follows.
CT (h) = Eer-eer (h) + M Ln Drin))
3=0
3=0
where 2 is the domain decay rate indicating the importance of source or historical target domain over Dr.,. and µ = E
Remark. In particular, we have the following arguments. (1) It is not tractable to accurately es- timate d1 from finite examples in real scenarios (Ben-Davie et al. 2010 (2) This error bound could be much tighter when considering other advanced domain discrepancy measures, e.g., A- distance (Ben- Davei et al. 2007), discrepancy distance (Mansour et al. 2009) etc. (3) There are two special cases: when μ = O, the error bound of Dr.,. would be simply determined by the latest historical target data Dr. and if goes to infinity, DIc+ is just determined by the source data Dro because inluitively the coefficient μ 1²-3/ j of historical larget domain data Dr, converges to zero.
Corollary 3.2. With the assumption in Theorem 3.1 and assume that the loss function C is symmetric Y) and obeys the triangle inequality, Then
(i.e., C(y1.92) 92) C(y2. 91 for y1, y/2
(1) if A-distancc (Ben-Davike et al.
2007 is adopted to measure the distribution shift, i.e. duaH SupH,EEH Prero [xx)) 7 h Prd,, [h(x) 7 h Hx(x)l] we have:
Cry (h) + M Σ 3 (duau(Dr, Dr + X3) Nt) 3=0
(h) -
where X; minncn ET, (h) (2) if discrepancy distance (Mansour et a. 2009) is adopted to measure the distribution shift, i.e., datise Dr) = WAE IEDr, (h(()),
(h) ET
h' (x())) ED, (C(M(x), (x))], we have:
er (h) S
1 (h) + Σ n² (daisc (Dr,, Dr...) ) + 02y)
5=0
1=0
where Q2 EDr, (xx E ED. (z(h)(x), h (x)) ED [2(h)+2(x)) 8)}, and h = arg min CT, (h) for = 0, ... 1.
The aforementioned domain discrepancy measures mainly focus on the marginal distribution over input features and have inspired line of practical transfer learning algorithms (Ganin et al., 2016 Chen et al. 2019). However, recent work (Wu et al.) 2019 Zhao et al.) 2019) observed that the minimization of marginal distributions cannot guarantee the success of transfer learning in real sce- narios. We propose address this problem by incorporating the label information in the domain discrepancy measure (see next section).
4 LABEL-INFORMED DOMAIN DISCREPANCY
In this section, we introduce novel label-informed domain discrepancy measure between the source domain Dr, and target domain Dr, its empirical estimate, and transfer signature based on this measure to identify potential negative transfer. The use of this discrepancy measure in continuous transfer learning will be discussed in the next section.
4.1 C-DIVERGENCE
For hypothesis H. we denote 1 (h) to be the subset of X such that x E 1(h) + h(x) = In order to estimate the label-informed domain discrepancy from finite samples in practice, instead of Eq. L.. we propose the following C-divergence between Dr, and Dr, taking into consideration the joint distribution over features and class labels:
de (Dr, Dr) = sup Prp., [(I(h), = 1) = 0 Prp, [(T()), = 1)U(T(h), = 0}] REW
(2)
where I(h) is the complement of I(h).
In this case, we assume 1 for any μ
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We show that some existing domain discrepancy methods (e.g.,Ben- Ben- David et al. (207)) can be seen as special cases of this definition by using the following relaxed covariate shift assumption.
Definition 4.1. (Relaxed Covariate Shift Assumption) The source and target domains satisfy the relaxed covariate shift assumption if for any h E H. PrDr, [y [hh]] PTp, [y [(h)]= Prly I(h)]
Notice that it would be equivalent to covariance shift assumption (Shimodaira) 2000 Johansson et al. 2019 when I(h) consists of only one example for all h E H (see Lemma A.6 for details).
Lemma 4.2. With the relaxed covariate shift assumption, for any h H. we have:
dD Dr) = sup
PrDm. [I(h]] - PrD. (hh)) -Sh n + -rrrr by = 1] PrDm, by 1]
hEH
where Sp = Prly = [h) Priy = (1]]
Remark. From Lemma 7.2, we can see that in the special case where S₁, is a constant for all h E H and PrD, ly 1] PrDzy yy 11, the proposed C C-divergence is reduced to the A-distance (Ben- David et al. 2007) defined on the marginal distribution of features. More generally speaking, C- divergence can be considered as weighted version of the A-distance where the hypothesis whose characteristic function has a larger class- cuas--seprrability (i.e., (Sy)) receives a higher weight. Intu- itively, compared A-distance, C-divergence would pay less attention class-inseparabee regions in the imput feature space, which provide irrelevant information for learning the prediction function in the target domain.
Moreover, the following theorem states that in conventional transfer learning scenario with a static source domain and a static target domain, the target error is bounded terms of C-divergence across domains and the expected source error.
Theorem 4.3. Assume that loss function L is bounded, i.e., there exists a constant M > 0 such that 0 < L S M. For hypothesis h E H. we have the following bound: (T) S eTo (h) + M de(Dro, Dr)
4.2 EMPIRICAL ESTIMATE OF C-DIVERGENCE
In practice, it is difficult to calculate the proposed C-divergence based on Eq. as it uses the true underlying distributions. Therefore, we propose the following empirical estimate of the C- divergence between Dr, and DDr follows. Assuming that the hypothesis class H is symmetric (i.e., 1 h E Hf h E H), the empirical C-divergence is:
de (Do DT) 1
minn bER
Σ I[(x,V) E Dr.
1
Σ
I[(x, v) E Dr!
(4)
mTa
mt (xx)):R(x)
(xx)):h(xx)/
where Dr, and Dr denote the source and target domains with finite samples, respectively. [[a] is the binary indicator function which is if a is true, and 0 otherwise.
The following lemma provides the upper bound of the true C-divergence using its empirical estimate.
Lemma 4.4. For any 8 E (0, 1), with probability at least - 8 over mr, labeled source examples Br, and mr labeled target examples BT, we have:
log 2
log
dc(Dr,,Dy) S de Dr) + (RET. (LH) + RB, (LB) +
V
+
2nnT
where (Ls(L (B& E (Bro, Br}) ) denotes the Rademacher complexity Mansour et al. and L. = ((x,)) [[((x) = v) h E H} is class of functions mapping 2 = X x
2009 over
y to 10.
4.3 NEGATIVE TRANSFER CHARACTERIZATIOO
Informally, negative transfer considered as the situation where transferring knowledge from the source domain has a negative impact on the target learner (Wang et al.) 2019): ET (A(TT, Dr))= > (A(A((, Dr)) where A is the learning algorithm. cr is the target error induced by algorithm A. implies that it only considers the target data set for target learner. In this paper, we define transfer signature to measure the transferability from source domain to target domain as follows.
TS(Dr
inf (er (A(DT,, Dr)) er (A(0, Dr)))
AEg where G is the set of all learning algorithms. We state that source domain knowledge is not trans- ferable over target domain when (PDrr((DTa) > O. Specially, since A(DT, Dr) learns an optimal classifier using both source and target data, we can define er er (A((r, Dr)) er(h))
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where ha arg H(A)) ( + (1 a)en (h) and H(A) is the hypothesis space induced
by A. When we only consider the target domain with = 1, <r (A(( Dr)) = <r(hT) where h arg min)E (T(h). Then we have the following theorem regarding the transfer signature.
Theorem 4.5. Assuming the loss function 2 is bounded with 0 < S M. we have er er(h), S eT (h7) + 2(1 add Dr)
Furthermore,
((())(PT))) S 2(1 - (81--) M) Dr)
Remark. We have the following observations: (1) Larger C-divergence between domains is often associated with higher transfer signature, which indicates that negative transfer can be charac- terized using the proposed C-divergence; (2) Empirically, the larger amount of labeled target data could increase the value of resulting in the learned classifier relying more the target data, which is consistent with the observation (Wang et al. 2219). One extreme case is where a = 1, implying have adequate labeled target examples for standard supervised learning the target domain without transferring knowledge from the source domain.
5 PROPOSED ALGORITHM
In this section, we derive the continuous error bound based our proposed C-divergence, followed by a novel continuous transfer learning algorithm (CONTE) by minimizing the error upper bound. Notice that in the context of continuous transfer learning, we also the proposed C-divergence between the target domain at adjacent time stamps to measure the change in distribution over time.
5.1 CONTINUOUS ERROR BOUND WITH EMPIRICAL C-DIVERGENCE
The following theorem states that for a bounded loss function L, the target eтror in continuous trans- fer learning can be bounded in terms of the empirical classification error within source and histori- cal target domains, the empirical C-divergence across domains as well as the empirical Rademacher complexity of the class of functions LH = ((x, y) + [x) = y] : h E H}.
Theorem 5.1. (Continuous Error Bound) Assume the loss function L is bounded with 0 < & S M. Given a source domain Dr, and historical target domain {DDr, 1: h E H and 8 E (0, 1), with probability at least 6, the target domain error CT+ Dre4. is bounded as follows.
6Te41 (1) s P
Σ n Er, (h)
M En
1 de Dr.m. ) MA
J=0
where A Eÿ=u RB., (LH) + REn... (Ln) 3/
log $
3,
log $
M log 2mr;;
2
3.2 using existing domain diver- gence measures (Ben-David et al. (2007): Mansour et al. (2009 our bound consists of only data- dependent terms (e.g., empirical source error and C-divergence), whereas previous error bounds are determined by the error terms involving the intractable labeling function or optimal target hypothe- sis (see Corollary 3.2
Remark. Compared continuous error bounds in Corollary
5.2 CONTE ALGORITIM
For continuous transfer learning, we leverage both the source domain and historical target domain data to learn the predictive function for the current time stamp. To this end, we propose to minimize the error bound Theorem 5. for learning the predictive function on Dr, Furthermore, we aim to learn a comain-invarian and time-invariant latent feature space such that the C-divergence across domains and across time stamps could be minimized. Therefore, we present an adversarial Variational Auto-encoder (VAE) algorithm with the following overall objective function:
(1b, T2, Te+1) Σ u (Eode (T), Ti+1) de Dr..)) + XLELBO (1,, #=0
(6)
where (T, Ti+1)) represents the classification error over the labeled examples from Dr, and Dr. dD Dr...) is the empirical estimate of C-divergence across domain. Thus the first two terms of Eq. 6 are associated with é7, (A)+dc(Dr,,) Dr...) in the error bound of Theorem 5. The third term (ELBO Te+) is the variational bound in the VAE framework (see Figure when learning the latent feature space and À > 0 is a hyper-parameterr In this case, have μ [0. 1] because we assume that the data distribution of a time-evolving target domain shifts smoothly over time. Then instantiate the terms of Eq. G follows.
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Inspired by semi-supervised VAE Kingma et al. 2014). we propose learn the feature space by maximizing the following likelihood across domains.
(og,e(X.)) y) (L pe y)) + (xix,)) [logpe pe (x, y, z) log (o y)] where & and 0 are the learnable parameters in the encoder and decoder respectively, and z is the latent feature representation of the input example (x, y). (T.(-Il-) is Kullback-Leibler divergence.
The evidence lower bound (ELBO), a lower bound on this log- ogglikellhoood can be written as follows.
(x, y) (xlxa) [log (x y|2)] + KL (go y) [p()))
(7)
where (x, y) I log po (x. y). Similarly, we have the following ELBO to maximize the log- likelihood of po (x(x) when the label is not available:
(x) = Σ, (go(U)xx) Ex, - (vlx) [log go (o(y)))]]]
(8)
y where po(x, y, z) pa (xly, (y)p(()l))(() with prior Gaussian distribution p(z) /N I). There- fore, the variational bound LELBO JT Te+1) 1) is given below.
mTy +
"I++1 Uo.o Us)
LELBO (T T++1) =
(x,o(Xi,
(9)
i= where WTe41 is the number of unlabeled training examples from Pr.... Besides, the classification error Lete Te+1) can be expressed as follows.
(Ti 1++1) Σ i=
"Ty + L (Vi, qo(-1x;)
(10)
where (-) is the discriminative classifier formed by the distribution go (v(x)) in Eq. @ and L(:-) the cross-entropy loss function in our experiments. To estimate the C-divergence, we first define
h to be two-dimensional characteristic function with h(x,, y) 1 < h(x) y <> (1(x) = 1,
} V {h(x) = 0, = 0} for h E H. Then the empirical C-divergence in Eq. 44 can be rewritten as
follows.
de de PT..))
min h
Σ
T[(x, v) Dr,5+
1
Σ
T((x,V) E Draal
mT,
MT++1 (x.y)::(xx.))-1
(x.y)):(xx.))=0
Note that the latent feature representation z learned by (z) x, v) could capture the label-informed information of example (x. Thus, the hypothesis h can be considered as the composition of
feature extraction 9s and a domain classifier F,, i.e, h(x,y) y) = F, (z)())))))). Formally, the
empirical estimate of C-divergence is given below.
de(Dr;; D =1 min F3
Σ Σ T Dr,1+ M/T+1 xxJ5(x)=1
T/a E Dra..)
(11)
mT;
(x))
The benefits of CONTE are twofold: first, it learns the latent feature space using both input x and output y/, second, it minimizes tighter error upper bound based on C-divergence in Theorem P. This framework can also be interpreted as a minimax game: the VAE learns domain-invariant and time-invariant latent feature space, whereas the domain classifier Fj aims to distinguish the exam- ples from different domains and different time stamps. In this paper, we adopt the gradient reversal layer (Ganin et al.) 2016) when updating the parameters of domain classifier F, and thus CONTE can be optumized by back proppaaapagttion in an end-to-end manner (see Algorithm M in appendices).
However, we observe that (1) it is difficult estimate the C-divergence with only limited labeled target examples from D (2) when learning the latent features z, combining the data x (e.g., one image) and class-label y directly might lead to over-emphasizing the data itself due its high dimensionality compared to y. To address these problems, we propose the following Pseudo-label Inference, i.e., we infer the pseudo labels of unlabeled examples using the classifier (a(y)x) (x) for each training epoch. Using labeled source and target examples well as unlabeled target examples with inferred pscudo labels, the C-divergence could be estimated in a balanced setting. Furthermore, to enforce the compatibility between features and label y. adopt pre-encoder step to learn dense representation for the input x, and then learn the label-informed latent features z.
6 EXPERIMENTAL RESULTS
Synthetic Data: We generate a synthetic data set in which each domain has 1000 posi- tive examples and 1000 negative examples randomly generated from Gaussian distributions ([ cos 0, 1.5 sin 0JT_ 0.5 12x2) and ([1,5 cos (-0), 1.5 sin (-0)]²' (.00)""OO. 00. I2x2), respectively. We l = 0 for the source domain (denoted as S1), and 0 = 1 (1 = 1, 1,..,,,, ,t) for the time evolving target domain with t = 8 time stamps (denoted as T1, ..., T8).
Image Data: We consider the following two tasks: digital classification (MNIST, SVHN) and image classification (Office-31 with three domains: Amazon, DSLR and Webcam; and Office-Home with

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four domains: Art, Product, Clipart and Real World). Since standard domains are static in these data sets, we will simulate the time-evolving distribution shift on the target domain by adding noise (e.g., random salt&pepper noise, adversarial noise, rotation). Take SVHN-MNIST will use SVHN as the static source domain, and MNIST as the target domain at the first time stamp. By adding adversarial noise the MNIST images, we obtain a time-evolving target domain (denoted as TI, .... TII in Table D. For Office-31 and Home,, we add the random salt&pepper noise and rotation to generate the evolving target domain. More details can be found in the appendices.
example,
Baselines: The baseline methods are as follows. (1) SourceOnly: training with only source data: (2) TargetERM: empirical risk minimization (ERM) on only target domain: (3) DAN (Long et al.
2013). CORAL (Sun & Saenko 2016). DANN (Ganin et al., (2066). ADDA (Tzeng et al. 2017), WDGRL (Shen et al. 2018). DIFA (Volpi al. 2018) and MDD Zhang et al. 2019): training with
feature distribution alignment. (4) CONTE: training with label-informed distribution alignment on
the evolving target domain while μ E (., 0.2, 0.4, 0.6, 0.8, 1). (5) CONTE one-time transfer
learning variant of CONTE that directly transfers from source domain to current target domain. We
fix = 0.1, and all the methods use the same neural network architecture for feature extraction.
6.1 EVALUATION OF C-DIVERGENCE We compare the proposed C-divergence with conventional domain discrepancy measure A- distance (Ben-David et al. 2007) on a synthetic data set with an evolving target domain. We as- sume that the hypothesis space H consists of linear classifiers in the feature space. Figure P shows the domain discrepancy and target clas- sification accuracy for each pair of source and target domains. We have the following obser- vations. (1) The classification accuracy on the target domain significantly decreases from target domain TI to T8. One explanation is that the joint distribution p(z, 8) on the time evolving target domain gradually shifted. (2) The A-distance increases from S1-TI to S1-T4, and then decreases from S1-+T4 to S1-T8. That is because it only estimates the difference of the marginal feature dis- tribution p(x) between the source and target domains. (3) The C-divergence keeps increasing from S1-TI to S1-T8, which indicates the decreasing task relatedness between the source and the tar- get domains. Therefore, compared with A-distance the proposed C-divergence better characterizes the transfcrability from the source to the target domains.
0.3
Target accuracy A Aciaaanss C-Iheegeeee
siom
StoT2
Stot3
Stot4 domain
Shore
51-577
010
Target
Figure 2: Comparison of domain discrepancy and target accuracy
6.2 EVALUATION OF ERROR BOUND
When there is only one time stamp involved in the target domain, Theorem i. reduced to the standard error bound in the conventional static transfer learning setting. We empiri- cally compare this reduced error bound with the existing Rademacher complexity based error bound in (Mansour et al., 2009) (see Theorem A.4 in appendices for being self-contained). We use the 0-1 loss function L and assume that the hypothesis space H consists of linear classi- fiers in the feature space. Figure 3 shows the estimated error bounds and target error with the time evolving target domain (i.e.. S1-T1I, evolving target domain to ensure that the baseline bound is meaningful most of the time) where we choose h -hj-- It demonstrates that our C-divergence based error bound is much tighter than the baseline. Notice that when transferring source domain S1 to target domain T8, our error bound is largely determined by the C-divergence, whereas the baseline is determined by the difference be- tween the optimal source and target hypothesizes. Furthermore, given any hypothesis h E H. we may not be able to estimate the baseline bound when the optimal hypothesis is not available.
Target anor -assolin bound
Out bourd
Stot3
StooT Target domain
Stuuts
Stutt
S1-777
Storm
Figure 3: Comparison of error bounds
S1-ST8 in new synthetic data set with slower time
6.3 EVALUATION OF CONTINUOUS TRANSFER LEARNING
Tables D and 2 provide the continuous transfer learning results on digital and office-31 data sets where the classification accuracy on target domain is reported (the best results are highlighted in bold). It is observed that (1) the classification accuracy using SourceOnly algorithm significantly
"The results for other existing discrepancy measures follow similar pattern and thus omitted for brevity
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decreases the evolving target domain due the shift of joint data distribution p(x, y) target domain: (2) the performance of static baseline algorithmms is largely affected by the distribution shift in the evolving target domain, and even worse than TargetERM in some cases (e.g.. on T6- T11 from SVHN to evolving MNIST); (3) CONTE significantly outperforms CONTE,, as well as target domain by large margin (i.e., up to 30% improvement on the last time stamp of target domain) because it effectively leverages the historical target domain information to smoothly re-align the target distribution when the change of target domain distribution in consecutive time stamps is small.
other competitors
Table 1: Transfer learning accuracy from SVHN (source) to time evolving MNIST (target)
Target Domain
TI
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
SourceOnly
0.6998
0.6738 0.6336
0.5692
0.4747
0.4110 0.3087
0.2220
0.1481
0.0828 0.0764
TaggetERM
0,7451
0.6997 0.6618 0.6314 0.6368
0.6359 0.6695 0.7133
0.7214
0.7450 0.7512
CORAL
0,8349 0,8410 0.7633 0.7063 0.6496 0.5900 0.5031 0,5101
0,4337 0,4156 0,4502
DANN
0.8666 0.8356 0.8018 0.7529 0.7309 0.6641 0.6614 0.5618 0.5204 0.5082 0.4594
ADDA
0.8667 0.8487 0.7982 0.7187 0.6804 0.5397 0.4366 0.3473 0.2636 0.1659 0.1259
WDGRL
0.8990 0.8602 0.8247 0,8222 0.7452 0.6877 0.6481 0.5896 0.5145 0.4952 0.5196
DIFA
0.9164 0.8993 0,8713 0.8273 0.7935 0.6661 0,5956 0.4381 0,3479 0.2448 0.1332
CONTE
0.9747
0.9552 0.9514 0.9279 0.8801 0.8833 0.8691 0.6979
0.7030
0.7415 0.7316
CONTE
0.9747 0.9740 0.9803 0.9864 0.9908 0.9940 0.9950 0.9965 0.9970 0.9967 0.9975
Table 2: Transfer learning accuracy on Office-31
Amazon + Webcam
Webcam + DSLR
T1
T2
T3
T4
T5
T1
T2
T3
T4
T5
SourceOnly
0.7490
0.2255
0.2282
0.1275
0.1503
0.9651
0.4309
0.3329
0.1611
0.2027
TargetERM
0.5584
0.3933
0.4215
0.3396
0.3732
0.4966
0.4201
0.4188
0.3248
0.4067
DAN
0,8537
0.5007
0.4993
0.3638
0,4470
0,9772
0.7302
0.6161
0.4765
0,5302
CORAL
0,8711
0.5235
0.4819
0.3195
0.4054
0.9812
0.7289
0.6671
0.4846
0.5221
DANN
0,8389
0.4993
0.4121
0.3973
0.3382
0.9651
0.7356
0.6416
0.4510
0.5490
MDD
0.8940
0.6738
0.5490
0,5141
0.4295
0.9724
0.8738
0.7315
0.5047
0,5289
CONTE
0.9154
0.6376
0.5758
0.4591
0,4846
0.9785
0.8591
0.7289
0,4926
0,5557
CONTE
0.9154
0.8134
0.8081
0.7611
0.7826
0.9785
0.9235
0.9208
0,8886
0.9154
7 RELATED WORK
Transfer Learning: Transfer learning (Ying et al. 2018: Jang et al. 2019) improves the perfor-
mance of learning algorithm on the target domain by using the knowledge from the source domain. It is theoretically proven that the target error is well bounded (Ben-David et al. 20100) Mansour et al.)
2009 followed by a line of practical algorithms (Shen et al. 2018 Long et al. 2017 [018 Saito
al. 2018 Chen et al. 2019) with covariate shift assumption. However, is observed that this
sumption does not always hold in real-world scenarios (Rosenstein et al. 2005 Wang et al. 2019).
Multi-source Domain Adaptation: Multi-source domain adaptation improves the target prediction function from multiple source domains (Zhao et al. 2018 Hoffman et al. 2018 Wen et al. 2020). It is similar to our problem setting as source and historical target domains can be considered as multiple "source" domains when modeling the target domain at the current time stamp. However, only limited labeled target examples domain adaptation requires that all source domains have adequate labeled examples.
provided in our problem setting, whereas multi-source
Continual Learning: Continual lifelong learning Parisi et al. 2019 Rusu et al. 2016 Hoffman
ett al. 2014 Bobu et al., 2018) involves the sequential learning tasks with the goal of learning a predictive function on the new task using knowledge from historical tasks. Most of them focused on mitigating catastrophic forgetting when learning new tasks from only one evolving domain, whereas our work studied the transferability between a source domain and a time evolving target domain.
CONCLUSION
In this paper, we study continuous transfer learning with time evolving target domain, which has not been widely studied and yet is commonly seen in many real applications. We start by deriving a generic error bound of continuous transfer learning with flexible domain discrepancy measures. Then we propose novel label-informed C-divergence measure the domain discrepancy incorpo- rating the label information, and study its application in continuous transfer learning, which leads to an improved error bound. Based this bound, we further propose generic adversarial Variational Auto-enccder algorithm named CONTE for continuous transfer learning. Extensive experiments on both synthetic and real data sets demonstrate the effectiveness of our CONTE algorithm.
8

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Under review as a conference paper at ICLR 2021
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Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference computer vision and pattern recognition, pp. 770-778, 2016.
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Judy Hoffman, Mehryar Mohri, and Ningshan Zhang, Algorithms and theory for multiple-source adaptation. In Advances in Neural Information Processing Systems, 2018.
Yunhun Jang, Hankook Lee, Sung Ju Hwang, and Jinwoo Shin. Learning what and where to transfer. In International Conference on Machine Learning. 2019.
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10


Reviews:
[{"review_id": "bQJ9AwlGFzm", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "\nThe paper proposed a transfer learning setting where the target domain varies/evolves over time and the source domain is considered static. The paper uses C-divergence to measure label-dependent domain discrepancy between source/previous target domain and the current target domain and provided a theoretical bound. The paper also used supervised VAE for CONTE algorithm and included C-divergence as a part of the objective function.\n\nThe paper address an interesting and an important practical problem on non-stationary target distribution. The proposed (semi-supervised) divergence measure and it’s relation to the previously proposed (unsupervised) divergence measures such as (A-distance and disc distance) is well-motivated. Please consider the following questions to help me understand the problem. \n\n### Major Questions:\n\n* The reason for the label dependency is motivated as explained but I believe that the usage of adversarial semi-supervised VAE was not properly motivated and related it the C-divergence.  \n\n* Sec 5.2 needs a little clarification for us readers to understand it clearly.  For instance, How is minimizing Eq 6 minimizes Continuous Error bound in Theorem 5.1. The first two terms in Eq 6 is related to the bound but Why is the ELBO term needed and how it is related to the bound. Eq 7-9 follows directly from the Kingma et al.’s Semi-supervised VAE so no explanation is needed.\n\n* Please help me to understand how to get Eq 4 from Lemme 4.2. I couldn’t find the related steps in the Suppl. On a related question, it seems that, in Eq 11 or the one above … if the F-classifier doesn’t distinguish between the domains (across the time or between source and target domains), ie., if F-classifier always say  that the example is from one of the two domains, then one of the two terms in the bracket will be 0 and the other will be 1 and it will make the distance measure to 0. \n\n* I believe the problem setting is very closely related to the Continual learning setting,  addition. To the Catastrophic forgetting, most recent work in Continual learning focuses on Negative transfer (such as Gradient Episodic Memory). The paper will be stronger if the key comparison to baselines in Continual learning is added.\n\n### Minor Questions:\n\n* Notation issue in Eq.9 for the unlabeled U(x) to estimate the log-likelihood of the marginal distribution of the features. ‘y’ shouldn’t be in the U function.\n\n* Overloaded use of notation F in Eq. 11 and in Suppl. \n\n* In many practical applications, for instance product reviews, if a new version of the product released periodically with new sets of features. Even though, the review data is  not only from the same target distribution but sample the different regions of this same target distribution. Can the current distance measure and  the CONTE algorithm be adapted to this problem setting.\n\n* Related question to the one above: Can we automatically learn \\mu from the data? If there is transfer from Ti to Tt+1 but not from Tt to Tt+1, will the proposed method can be adapted to this problem setting. Does the information from Ti is retained in Tt from previous learning cycle, such that, it will be used by Tt+1 task?\n\ni have read the author(s)' comments and I have updated the ratings based on their replies. Thanks for very extensive clarification. Adding these comments in the final revision would significantly improve the quality of the paper.", "extra_scores": {}, "extra_text": {}}, {"review_id": "XdJtgahGN29", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "\n(1) This article is more detailed in its theoretical analysis, and the C-divergence has advantages over other divergence from the theoretical analysis. \n\n(2) This article proposes a new continuous transfer learning setting. It theoretically analyzes the upper bound of the target domain error, and utilizes the C-divergence to measure the joint distribution discrepancy in the two domains.\n\n(3) Some parts of the presentation needs to be improved, such as the presentation of negative transfer in section 4.3. How to use other methods such as DAN to this new setting? It should be described in more details in section 6. How to set the target domain label in section 6?\n\n(4) Although the results outperforms some other methods, the experiments are not sufficient. In experiments, the authors only do experiments in some easy task in DA, and no experiments are shown for the effect of the hyper-parameter mu.\nDetailed Evaluation:\n \n(5) About formula (10), why does the cross-entropy of T_{t+1} calculate t+1 times?\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "IHE-bkafZDp", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "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:\n\nThis paper studies how to transfer the information in the static source domain to the time-evolving target domain. This paper proposes a domain discrepancy measure and an algorithm for continuous transfer learning. The results seem to be interesting and the problem this paper studies is important. However, the domain rate in the main results and algorithm could be easily generalized which can make the results more broadly applicable. Moreover, it needs more clarification about the motivation of using the C-divergence measure in the time-evolving target domain.\n\nMajor comments:\n\n1. Domain decay rate: There is a domain decay rate in all the main results and the algorithm. From the proof of all the main results, the rate $\\mu^{t-j}/\\bar{\\mu}$ can be easily generalized to any arbitrary weight $\\omega_j$ with the constraint $\\sum_{j = 0}^t \\omega_j = 1$ and $\\omega_j \\geq 0$. Hence, there is no assumption about $\\mu$ in the loss function. Allowing general $\\omega_j$ is more appealing for many reasons. \n\nFirst, this paper indexes the source domain by $T_0$. The CONTE algorithm assigns similar weights to the source domain ($\\mu^t$) and target domain at time 1 ($\\mu^{t-1}$), but a different weight to the target domain at time $t$ ($\\mu^0$) (when $\\mu \\neq 1$). This algorithm seems to implicitly assume the target domain at time 1 is more similar to the source domain than the target domain at time $t$, which seems unnatural and requires further justification. However, using  $\\omega_j$ does not have this issue. \n\nSecond, it is possible that the time-evolving domain has a cyclic or seasonality pattern. Even in the experiment setup in Section 6, $\\sin(\\theta) = \\sin(\\frac{i\\cdot \\pi}{t})$ is cyclic. It may make more sense to assign a larger weight to the time period whose domain is more similar to the current domain. \n\n2. Potential improvement of the algorithm: Following the previous point, the CONTE algorithm could potentially be improved. Since the domain discrepancy measure $d_c(\\cdot, \\cdot)$ can be estimated, we could assign the weight $\\omega_j$ inverse proportional to $d_c(\\cdot, \\cdot)$ and the error could be potentially reduced. \n\n3. C-divergence measure: It seems the C-divergence measure does not involve the time concept and is generally applicable to transfer learning tasks beyond continuous transfer learning? Then why is the C-divergence measure particularly of interest in the context of continuous transfer learning and why is it more preferred to use the C-divergence measure rather than the conventional measures, such as A-divergence in this setting? Some discussions on this point and numerical comparisons will be helpful. ", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes transfer learning where the target domain data is evolving along time.  They use both labeled and unlabeled data to learn domain and time-invariant features based on a discrepancy measure they introduce.  Their proposed algorithm uses VAE to learn such features.  Reviewers have mixed response, although the author feedback did help.  
The main limitation with the paper is that it does not seem to be aware of the very extensive literature on continuous domain adaptation.  The related work only discusses papers on transfer learning, multi-source domain adaptation, and continuous learning.  But ignores papers on continuous domain adaptation which are much more related to this paper.  The most recent of these that appeared in ICML 2020 also attempts to learn time invariant features using adversarial methods.  Unfortunately, the reviewers seem to be also unaware of this literature:

1.  Continuously Indexed Domain Adaptation,  Hao Wang, Hao He, Dina Katabi, ICML 2020
2.  Active Adversarial Domain Adaptation 
3.  Continuous Domain Adaptation using Optimal Transport
4. Learning to Adapt to Evolving Domains - NeurIPS 2020
5.  Judy Hoffman, Trevor Darrell, and Kate Saenko. Continuous manifold based adaptation for evolving visual
domains. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 867–874,
2014.
6 Massimiliano Mancini, Samuel Rota Bulo, Barbara Caputo, and Elisa Ricci. Adagraph: Unifying predictive and
continuous domain adaptation through graphs. In Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition, pages 6568–6577, 2019.

7 Atsutoshi Kumagai and Tomoharu Iwata. Learning future classifiers without additional data. In Thirtieth AAAI
Conference on Artificial Intelligence, 2016.


Decision: Reject