Generative Gradual Domain Adaptation with Optimal Transport
Authors: Yifei He
Haoxiang Wang
Han Zhao
Venue: Submitted to ICLR 2023 (2023)
Keywords: Domain Adaptation
Gradual Domain Adaptation
Distribution Shift

Abstract: Unsupervised domain adaptation (UDA) adapts a model from a labeled source domain to an unlabeled target domain in a one-off way. Though widely applied, UDA faces a great challenge whenever the distribution shift between the source and the target is large. Gradual domain adaptation (GDA) mitigates this limitation by using intermediate domains to gradually adapt from the source to the target domain. However, it remains an open problem on how to leverage this paradigm when the oracle intermediate domains are missing or scarce. To approach this practical challenge, we propose Generative Gradual Domain Adaptation with Optimal Transport (GOAT), an algorithmic framework that can generate intermediate domains in a data-dependent way. More concretely, we generate intermediate domains along the Wasserstein geodesic between two given consecutive domains in a feature space, and apply gradual self-training, a standard GDA algorithm, to adapt the source-trained classifier to the target along the sequence of intermediate domains. Empirically, we demonstrate that our GOAT framework can improve the performance of standard GDA when the oracle intermediate domains are scarce, significantly broadening the real-world application scenarios of GDA.

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GENERATIVE GRADUAL DOMAIN ADAPTATION WITH OPTIMAL TRANSPORT
Anonymous authors Paper under double-blind review
ABSTRACT
Unsupervised domain adaptation (UDA) adapts a model from a labeled source domain to an unlabeled target domain in a one-off way. Though widely applied, UDA faces a great challenge whenever the distribution shift between the source and the target is large. Gradual domain adaptation (GDA) mitigates this limitation by using intermediate domains to gradually adapt from the source to target domain. However, it remains an open problem on how to leverage this paradigm when the given intermediate domains are missing or scarce. To approach this practical challenge, we propose Generative Gradual DOmain Adaptation with Op- timal Transport (GOAT), an algorithmic framework that can generate intermediate domains in data-dependent way. More concretely, we first generate intermediate domains along the Wasserstein geodesic between two given consecutive domains in a feature space, and apply gradual self-training, standard GDA algorithm, adapt the source-trained classifier to the target along the sequence of intermediate domains. Empirically, we demonstrate that our GOAT framework can improve the performance of standard GDA when the given intermediate domains are scarce, significantly broadening the real-world application scenarios of GDA.
INTRODUCTION
Modern machine learning models suffer from data distribution shifts across various settings and datasets [Gulrajani & Lopez-Paz, 2021; Sagawa et al., 2021; Koh et al., 2021; Hendrycks et al., 2021; Wiles al., 2022], i.c., trained models may face a significant performance degrade when the test data come from a distribution largely shifted from the training data distribution. Unsupervised domain adaptation (UDA) a promising approach to address the distribution shift problem by adapting models from the training distribution (source domain) with labeled data to the test distribution (target domain) with unlabeled data [Ganin et al., 2016; Long et al., 2015; Zhao al., 2018; Tzeng et al., 2017]. Typical UDA approaches include adversarial training [Ajakan et al., 2014; Ganin et al., 2016), distribution matching [Zhang et al., 2019; Tachet des Combes et al., 2020]. optimal transport [Courty al., 2016; 2017]. and self self-training (aka essuudolabeling [Liang al., 2019; 2020; Zou et al., 2018; 2019]. However, as the distribution shifts become large, these UDA algorithms have been observed to suffer from significant performance degradation [Kumar al., 2020; Sagawa al., 2021; Abnar et al., 2021; Wang et al., 2022a]. This empirical observation consistent with theoretical analyses [Ben-David al., 2010; Zhao et al., 2019], which indicate that the expected test accuracy of trained model in the target domain degrades as the distribution shift becomes larger.
To tackle a large data distribution shift, one may resort to intuitive divide-and-conquer strategy, e.g., breaking the large shift into pieces of smaller shifts and resolving each piece with a classical UDA approach. Concretely, the data distribution shift between the source and target can be divided into pieces with intermediate domains bridging the two (i.e., the source and target). In settings, e.g. gradual domain adaptation (GDA) [Kumar et al., 2020; Abnar et al., 2021; Chen & Chao, 2021; Wang et al., 2022a; Gadermayr et al., 2018; Wang et al., 2020; Bobu et al., 201 8; Wulfmeier et al., 2018]. where the intermediate domains with unlabeled data are available to the learner, Kumar et al. [2020] proposed simple yet effective algorithm. Gradual Self-Training (GST), which applies self-training consecutively along the sequence of intermediate domains towards the target. For GST, Kumar et al. [2020] proved upper bound on the target error of gradual self-training and Wang et al. [2022a] provided a significantly improved bound under relaxed assumptions, corroborating the effectiveness of gradual self-training in reducing the target domain error with unlabeled intermediate domains.

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Interneciate Domains
Targel
Doman Index
[-0
1=1
7-
[=/
15'
30'
60'
QDA
8
8
+
8
Direct UDA
Figure 1: A schematic diagram comparing Direct Unsupervised Domain Adaptation (UDA) vs. Gradual Domain Adaptation (GDA), using an example of Rotated MNIST (60 rotation).
However, although the unlabeled intermediate domains have been shown to be useful for domain adaptation both empirically and theoretically, they are scarce in many real-world applications, hinder- ing the wide deployment of GDA methods to overcome large distribution shifts. Hence, it is natural to ask, in cases of insufficient intermediate domains, is it possible to generate more intermediate domains useful for gradual domain adaptation? How should generate these intermediate domains?
In this paper, we provide an affirmative answer to the first question along with an effective solution to the second, which term Generative Gradual Domain Adaptation with Optimal Transport (GOAT). Our solution for intermediate domain generation is inspired by a theoretical insight from the generalization error of gradual self-training in Wang et al. [2022a]:
In order to minimize the generalization error on the target domain, the sequence of intermediate domains should be placed uniformly along the Wasserstein geodesic between the source and largel domains.
At a high-level GOAT contains the following steps: i) generate intermediate domains between each pair of consecutive given domains along the Wasserstein geodesic in a feature space, ii) apply gradual self-training (GST) [Kumar et al., 2020]. popular GDA algorithm, over the sequence of given and generated domains.
Empirically, we conduct experiments on Rotated MNIST and Portraits [Ginosar et al., 2015], two benchmark datasets commonly used in the literature of GDA. The experimental results show that our GOAT significantly outperforms vanilla GDA, especially when the number of given intermediate domains is small. The empirical results also confirm the theoretical insights from Wang al. [2022a]: 1). when the distribution shift between a pair of consecutive domains is small, one can generate more intermediate domains to further improve the performance of GDA; 2). there exists an optimal choice for the number of generated intermediate domains.
PRELIMINARIES
Notation We refer to the input and output space X, respectively, and use X, Y to represent random variables that take values in X. A domain corresponds to an underlying data distribution (XXYY) X' V. As we focus unlabeled samples,
((X) denote the marginal distribution of X. Classifiers from the hypothesis class H. X' + y are trained with the loss function ((..)). We use 1, to refer a re-iieesiion vector with all 1s. [[:] stands for the indicator function.
Unsupervised Domain Adaptation (UDA) In UDA, we have a source domain and a target domain. During the training stage, the learner can access m labeled samples from the source domain and unlabeled samples from the target domain. In the test stage, the trained model will then be evaluated by its prediction accuracy on samples from the target domain.
Gradual Domain Adaptation (GDA) Most UDA algorithms adapt models from the source to target in one-step fashion, which can be challenging when the distribution shift between the two is large. Instead, in the setting of GDA, there exists a sequence of additional T unlabeled intermediate domains bridging the source and target. We denote the underlying data distributions of these intermediate domains f1 (X, (X,Y).... Y).... MT (X, Y), with (X, Y) and PT NT(X, Y) being the source and target domains, respectively. has access to S,, set of unlabeled data drawn i.i.d. from Me (X). Same UDA, the goal of GDA still to make accurate predictions on test data from the target domain, while the learner can train over labeled source data and n unlabeled data from {S.) To contrast the setting of UDA and
this case, for each domain E (1...., ......Ty T), the learner
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GDA, we provide illustration in Fig. that compares UDA with GDA, using the Rotated MNIST dataset as an example.
Gradual Self- Teaanirain (GST) When there are multiple intermediate domains available to the learner, one classic algorithm for GDA is gradual self-training first, train a classifier on labeled source data with the empirical risk minimization (ERM) principle to obtain source-trained classifier ho H. Then, fit it to the set of unlabeled target data, Si, with self-training to obtain the classifier hy:
h1 = (ho,,, S1) = arg min Σ (((((z),haz))) JEH DES,
(I)
where ST denotes the process of self-training and hz(z) represents the pseudo-label provided by ho on each unlabeled target sample x. The above process will then be repeatedly applied to each pair of consecutive domains (pt, (e++), ). until one obtains a final classifier for the target domain pr. Note that in this process, at each stage the classifier h from the last stage will be used to generate pseudo-labels for the unlabeled data St, hence the name gradual self- celf-training.
Intuitively, one can expect that when the distribution shift between each consecutive pair of intermedi- ate domains is large, the quality of the pseudo-labed obtained from the previous classifier can degrade significantly, hence hurting the final target generalization. This scenario particularly relevant when the number of given intermediate domains is relatively small.
Theoretical Guarantees Kumar et al. [2020] provided the first target domain error bound for gradual self-training under certain assumptions, theoretically justifying the effectiveness of GST. However, the error bound of Kumar et al. [2020] grows exponentially in T (the number of domains), which contradicts empirical observations that the optimal number of intermediate domains is relatively large over multiple GDA datasets [Abnar et al., 2021; Chen & Chao, 2021; Wang et al., 2022a]. To resolve this issue, Wang et al. [2022a] proved significantly improved error bound for GST under weaker assumptions than those Kumar et al. [2020], expressed as
T Vn
(rrrr) S E
TA+
(2)
VnT
where (h) stands for the population loss of classifier h E H. in domain & (i.e., (h) = Eye, E (h) & Exz,yrpu (((h(z), 7) and A is the average p- p-Wasserstein distance between consecutive domains, i.e.,
1 T
A
Σ
W, (4- (X,)) for p 2 1.
(3)
Note that in GDA, we cannot directly measure A since requires access the joint distributions of the intermediate domains, whereas only unlabeled data are available to us. In order to bridge the gap, in this paper, we make the following assumption of the intermediate domains: there exists a feature space Z such that the covariate shift assumption holds over Z, i.e., the conditional distribution of Y given the feature Z is invariant across all the intermediate domains. Note that this assumption is also consistent with line of recent works under the principle of invariant risk minimization [Arjovsky et al., 2019; Rosenfeld et al., 2020; Wang et al., 2022b] that seek to find features from the inputs such that the covariate shift assumption holds. Under this assumption, the Wasserstein distance between the joint distance W, (1 Y), (z Y)) reduces to the one between the marginal feature distribution Wp (2)-1(2), pz(Z)).
3 GENERATIVE GRADUAL DOMAIN ADAPTATION WITH OPTIMAL TRANSPORT
Inspired by the theoretical results in (2), in this section, we shall present our algorithm to automatically generate a series of intermediate domains between any pair of consecutive given domains, with the hope that when applied to the sequence of generated intermediate domains, GST could lead to better target generalization. Before presenting the proposed algorithm. we first formally introduce several notions that will be used in the design of our algorithm. First, given two distributions, one can transform one another with purh-forward operator.
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Definition 1 (Push-forward Operator). Consider continuous map T X X and arbitrary probability measures p. V on X. When 'T' pushes y. forward to v, we denote it with T# = v, which means that for any measurable set A C X.
((A) = P
A) (T
Monge [1781] formalized the optimal transport problem 1781, which is a problem to find the push-forward operator that minimizes total transport cost.
Definition 2 (Optimal Transport). Given measures over X and a cost function c: X X +> [0, 00), the optimal transport map T is the that attains the infimum of the total transport cost:
U/ c(x, ((o.T(())du())) T
inf
T=p=v/
(5)
One can create a path of measures that interpolates the given two, and the theory of optimal transport can help us find the optimal path that minimizes the path length measured with the Wasserstein metric, i.e., the sum of Wasserstein distance between pairs of consecutive measures along the path. This optimal path is termed the Wasserstein geodesic, which is formally defined below.
Definition 3 (Wasserstein Geodesic). Given two measures Vo, V1 over X optimal transport map T X + X such that vo = V1, the (constant-speed) Wasserstein geodesic between them under Euclidean metric can be defined by the path P(y,, v1) = {ux : x 1), where vx ((1 A)Id XT*)# vo, and Id is the identity mapping.
3.1 MOTIVATIONS
The target domain error bound of gradual self-training proved by Wang al. [2022a], i.., Eq. (2), has a dominant term TA, which can be interpreted as the length of the path of intermediate domains connecting the source and target. Interestingly, geodesic between the source μ4 and target pT, and we formalize our findings as follows.
find that this path is related to the Wasserstein
Proposition 1 (Path Length of Intermediate Domains). For arbitrary intermediate domains (......, PT-1, the following inequality holds.
TA Σ Wo(()-1, 2 Wy(Ho,
(6)
where the equality is obtained as the intermediate domains .1.... PT-1 sequentially fall along the Wasserstein geodesic between pa and PT.
Without explicit access to the intermediate domains, gradual domain adaptation cannot be applied. Interestingly, Proposition sheds light on the task of intermediate domain generation bridge this gap: the generated intermediate domains should fall on or close to the Wasserstein geodesic in order to minimize the path length.
3.2 COMPUTATION WITH OPTIMAL TRANSPORT
From Def. 3, we know that one has to solve an optimal transport problem to generate intermediate domains along the Wasserstein geodesic. Here, we consider the problem of optimal transport between a source domain and target domain.
Solving Optimal Transport with Linear Programming unsupervised domain adaptation (UDA), the source and target domains have finite training data. Hence, we can consider the measures of the source and target to be discrete, i.e., 140 and μır only have probability mass over the finite training data points. More formally, denoting the source dataset So {0oi am and target dataset as ST = (TT;)a the measures 140 and μTT can be expressed as
1 n
Mo
5(1o:)
pT
8 8(1r)),
where 6(z) represents the Dirac delta distribution [Dirac al., 1930]. Under the discrete the push-forward operator T that pushes μo forward to per can obtained by solving a linear program [Peyré al., 2019].
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Algorithm I Generative Gradual Domain Adaptation with Optimal Transport (GOAT)
Require: S0 ( mh S {aTs) Encoder &; Source-trained classifier ho
ENCODE DATA: S/ {zor (z )eem]] SV {2r, E(rry))se
OPTIMAL TRANSPORT (OT): Solve for the OT plan y E mx between S and S7. CUTOFF: Use a cutoff threshold to keep ((++m) elements of 7* above the threshold and zero out the rest / / /Only applies the entropy entropy-regularize version of OT INTERMEDIATE DOMAIN GENERATION:
for 1, ,,, ,T do
Initialize an empty set S?? for each non zero element My of 7" do
T-t zoi + 2zy;; Add (z, 7hy) to St
GRADUAL DOMAIN ADAPTATION:
for t = 1, ,.... do
h = ST(h.-1, //Can also apply sample weights to losses based on Tiy
Output: Target-adapted classifier hT
Proposition 2. Consider pay source data {Dei ) and pe over target data ser,li=r- Given a transport cost function c X' x X + [0, oc). there exists an optimal transport map T", which satisfies T Ugo pr.. Furthermore, for i [m]. T maps zni as follows,
Tg8(20))
Σ (1r)())))
(8)
where 7 E R2200 is the optimal transport plan, a non- non-negative matrix of dimension m X n. The plan 7" can be obtained by solving the following linear program,
7 arg min
Σ (a,c(soi WTy)
71n
1 1m and 7 1m 1, m
1 n
YER
20
Generating Intermediate Domains with Optimal Transport Proposition 2 demonstrates that one can use linear programming (LP) to solve the optimal transport problem between source dataset and a target dataset. With the optimal transport plan 7", one can leverage Definition to generate intermediate domains along the Wasserstein geodesic. Specifically, fort 1,..., T 1, the measure of the intermediate domain can be obtained by the following push-forward
T T-'-d T
-1a ld + - T
Σ b
T T
Lpi +
t 7TT
(10)
Intuitively, in can be interpreted as discrete measure over n,y data points with data weights assigned by Ti. where Tly- = Ex, 1[ny > 0] is the number non-zero entries in the matrix 7"
Space Complexity Clearly, one needs to the optimal transport plan matrix 7" ER <Rmx. in order to generate intermediate domains with (10). Thus, the space complexity appears be ((m)). However, by leveraging the theory of linear programming, one can show that the maximum number of non-zero elements of the solution 7 the I.P (9) is at most m + [Peyré et al., 2019]. Thus, be reduced to O(m n1) when using sparse matrix format to store y
the space complexity
Time Complexity For simplicity, let us consider m n. Then, the current time complexity of solving the LP (9) is known to be O(n log(n)) [Cuturi, 2013; Pele & Werman, 2009].
3.3 PROPOSED ALGORITHM
We present proposed algorithm in Algorithm Notice that Algorithm 1 directly generates intermediate domains between the source and target domains. However, in practice, there might a few given intermediate domains that can be used by GDA. In this case, simply treat each

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Source
Generated Intermediate
Target
Saurce
Intermediate
Target
3
3
3
3
3
3
3
3
3
9
9
9
9
9
9
q
4
4
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6
6
6
6
6
/
X
X
X
X
/
/
1
!
(a) Generation in the input space.
(b) Generation in the feature space.
Figure 2: Random samples from the generated intermediate domains.
Path of Intermodiate Domains X
given Intermediate Domain in X
X
X
Generated Intermediate Domains in
g
E(pr)
8(Ir)
Z
E(io))
E(P)
Generated Intermediate Domains in
Illustration of the intermediate domain generation in GOAT (a) without any given intermediate domain, (b) with given intermediate domain.
Figure 3:
pair of consecutive domains source-target domain pair in sub-level, and apply Algorithm 1 iteratively to the pairs of consecutive given domains from the source to target.
In the following, we explain some keys to our algorithm design.
Fast Computation of Optimal Transport (OT) The super-cubic time complexity of solving the LP in (9) essentially prevents this optimal transport approach from being scaled up to large datasets. To remedy this issue, we propose to solve an approximate objective the OT problem (9) when takes too long solve the original OT exactly. Specifically,
add entropy regularization term to the objective (9), turning it to strictly convex, and the time complexity of solving this regularized objective is reduced to nearly (nn)) from the original O(n² log n) [Cuturi, 2013]. However, the solution to this regularized objective, i.e., the OT plan > not guaranteed to have at most n +m elements anymore. Thus, the space complexity increases ((m)) from O(m n.). We design cutoff trick to zero out entries of tiny magnitude in 7 (see details in Algo. 1). reducing the space complexity back to (m n). More details regarding this part are provided in Appendix B.
Intermediate Domain Generation in a Feature Space The intermediate domain generation approach described above directly generates data the input space X. However, the generation does not have to be restricted to the input space. One can show that with Lipschitz continuous encoder E: Z mapping inputs to the feature space Z (i.e., <- E(x) for any input æ)), the order of generation bound (2) stays the same! (the proof is provided in Appendix A).
Feature Space vs. Input Space. We use an encoder by default in Algorithm since we empirically observe that directly generating intermediate domains in the input space is usually sub-optimal (see 4a for a detailed analysis). To give the readers an intuitive understanding, we provide a demo of Rotated MNIST in Fig. 2: if we apply the intermediate domain generation of Algorithm in the input space, the generated data do not approximate the digit rotation well; when applying the algorithm in the latent space of VAE (fitted to the source and target data), the generated data (obtained by
Some in the bound get multiplied by factor of the Lipschitz constant of

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Table I: Accuracy (%) on Rotated MNIST.
Table Accuracy (%) on Portraits.
Given
# Generated Domains of GOAT
A Given Domains
# Generated Domains of GOAT
Domains
0 (GST)
(GST)
50,3=0.7
48.5422
47.2=1.7
48.2=2.7
47.542288
73.3=1.3
74.0+1.3
73.542.2
73.6#2.5
74.2=2.5
$6.311.9
55.212.6
54.611.6
57.1:2.2
56.2.119
74.51:1.
76.4.13
75,5126
76.8
74.711.77
61.642.11
68.04114
67.0+2.2
68.1422.
70.3122
77.0+133
77442.11
79,4424
7,99:
77.240.9
3
66.3422.0
74.0+11
74,4=18
732242.0
74.042.3
80.7+2.3
809916
818+13
82.31.3
81.3+1.55
75.5+2.0
83.842.0
84.0=16
86.4220
82.7+188
82.0=14
82881.55
83.61:
824214
818816
the decoder of the VAE) captures the digit rotation accurately. Fig. 3a explains this superiority of the feature space the input space with schematic diagram: the input-space Wasserstein geodesic may not approximate the ground-truth distribution shift (e.g., rotation) due to the linearity of optimal transport; with proper encoder &, the feature-space Wasserstein geodesic can capture the distribution shift more accurately.
Leveraging Given Intermediate Domain(s). With given intermediate domain, we generate interme- diate domains with GOAT between the two pairs of consecutive domains, respectively, Fig. 3b shows that this approach can make the generated domains closer to the ground-truth path of distribution shift, explaining why GOAT benefits off given intermediate domains.
Gradual Domain Adaptation (GDA) on Generated Intermediate Domains With generated data of intermediate domains, one can run a GDA algorithm consecutively over the source- innee-intttmmiatttatieeeee domains in the feature space. As for the choice of GDA algorithm, we adopt Gradual Self- Training (GST) [Kumar et al., 2020], since it is simple, popular, and powerful. Nevertheless, one can freely apply any other GDA algorithm on top of the generated domains.
4 EXPERIMENTS
Our goal of the experiment is to demonstrate the performance gain of training on generated intermedi- ate domains in addition to given domains. We compare method with gradual self-training [Kumar et al., 2020], which only self-trains a model along the sequence of given domains iteratively. In Sec. 4.4, we further analyze the choices of encoder & and transport plan > used by Algorithm 1. More details of our experiments are provided in Appendix C.
4.1 DATASETS
Rotated MNIST A semi-synthetic dataset built on the MNIST dataset [LeCun & Cortes, 1998], with 50,000 images as the source domain and the same 50,000 images rotated by 45 degrees the target domain. Intermediate domains are evenly distributed between the source and target.
Portraits [Ginosar et al., 2015] A real-world gender classification dataset consisting of portraits of high school seniors from 1905 to 2013. Following Kumar et al. [2020], the dataset is sorted chronologically and split into a source domain (first 2000 images), intermediate domains (next 14000 images), and a target domain (last 2000 images).
4.2 IMPLEMENTATION
Our code developed in PyTorch [Paszke al., 2019], and our experiments are run on NVIDIA RTX A6000 GPUs We use a convolutional neural network (CNN) of 4 convolutional layers of 32 channels followed by 3 fully silll--coneected layers of 1024 hidden neurons, with ReLU activation. We also adopt common practices of Adam optimizer [Kingma & Ba, 2015], Dropout [Srivastava et al., 2014], and BatchNorm [Toffe& & Szegedy, 2015). To calculate the optimal transport plan between the source and target, we use the Earth Mover Distance solver from [Flamary et al., 2021]. Notice that the number of generated intermediate domains is hyperparameter, and we show the performance for 1,2,3 or 4 generated domains between each pair of consecutive given domains. Following practices [Kumar et al., 2020], in self-training, we filter out the 10% data where the model's prediction is least confident at.
As for the implementation of GOAT (Algorithm 1), we take the first two convolutional layers as the encoder ε, and treat the last layers the classifier h. Sec. 4.4 explains why use this encoder.
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GOAT with Different Encoders (E)
GOAT with Different Transport Plans (r")
72
70
60
68
AccuRDy
66
64
Tost
62
Transport Pian
GOAT
Random
Traiing (GST)
Uniform
58-
Gradial Swif-Triiiig (GST)
OT (Ours)
Oracie
Irput
Cone-1
Comm22
Cone3
Conv44
1
2
3
Neural Layer (as Encoder)
Number Generated Domains
(a) Choices of the encoder (E) in Algorithm (b) Choices of the transport plan (y") in Algorithm
Figure 4: Ablation studies on Rotated MNIST with 2 given intermediate domains. (a) Differeni neural net layers ay encoders for the intermediate domain generation of GOAT. One see that input space is not suitable for intermediate domain generation, and second convolutional layer (CONV-2) is optimal. (b) Different transport plans the intermediate domain generation of GOAT. Obviously, our optimal transport (OT) plan significantly outperforms the baseline transport plans (Random & Uniform), and its performance is even close to the oracle.
4.3 EMPIRICAL RESULTS
We empirically compare our proposed GOAT with Gradual Self-Training (GST) [Kumar et al., 2020]. The results on Rotated MNIST and Portraits are shown in Table and Table 2. Each experiment is run 5 times with 95% confidence interval reported. The leftmost column corresponds to the performance of GST only on given intermediate domains, which is equivalent to GOAT without any generated intermediate domain.
In Table and 2, the columns ("# Generated Domains of GOAT") represent the number of generated intermediate domains between each pair of consecutive given domains, while the rows ("# Given Domains") indicate the number of given intermediate domains.
Observations i) From the columns of both Table and 2, we can observe that the performance of GOAT monotonically increases with more given intermediate domains, indicating that GOAT indeed benefits from given intermediate domains. ii) From the rows of both Table and 2, we can see that with fixed number of given domains, our GOAT can consistently outperform Gradual Self-Training (GST). The only exception is the case of Rotated MNIST without any given intermediate domain, which might be due to the challenge illustrated in Fig. 3(a).
Overall, the empirical results shown in the two tables demonstrate that our GOAT can consistently improve gradual self-training (GST) with generated intermediate domains when only a few given intermediate domains are available.
4.4 ABLATION STUDIES
Choice of Encoder (E) Here, we study how the choice of the encoder (i.e., feature space) affects the performance of GOAT. Since
CNN, we can take each network layer as the feature space. Specifically, we consider the four convolutional layers and input space as candidate choices for the encoder. Once choosing layer, we take all layers before it (including itself) as the encoder. In this ablation study, we use Rotated MNIST dataset with 2 given intermediate domains, and let GOAT generate 4 intermediate domains between consecutive given domains. From Fig. 4a, we can observe that directly applying GOAT in the input space performs significantly worse than the optimal choice, CONV-2 (i.e., the second convolutional layer). This result justifies our use of an encoder for intermediate domain generation (instead of directly generating in the input space). Notably, Fig. 4a shows that deeper layers are not always better, showing clear sereease-tee--ecrreee accuracy curve. Hence, we keep using CONV-2 as the encoder for GOAT in all experiments.
Choice of Transport Plan (y*) our Algorithm the data generated along the Wasserstein geodesic are essentially linear combinations of data from the pair of given domains, with weights (for each combination) assigned by the optimal transport (OT) plan y* To validate that the performance gain of GOAT indeed comes from the Wasserstein geodesic estimation instead of just linear combinations, we conduct an ablation study on GOAT in Rotated MNIST with 2 given intermediate domains. Specifically, consider four approaches to provide the transport plan y*: i) a random transport plan (weights are sampled from a uniform distribution). ii) a uniform transport plan (weights are the same for all combinations), iii) the optimal transport (OT) plan provided by Algorithm 1, iv) the oracle
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transport plan', which is the ground-truth transport plan in this study. For a fair comparison, When constructing the random and uniform plans, we ensure that the number of non conzeero elements is the same as that of the oracle plan (i.e., keeping the number of generated data the same). See more details in Appendix C.
From Fig 4b, observe that, in general, the random and uniform plans do not obtain non-trivial performance gain compared with the baseline, the vanilla Gradual Trainingg (GST) without any generated domain. In contrast, our OT plan is significantly better and achieves similar performance as the oracle, demonstrating the high quality of the OT plan and justifying our algorithm design with the Wasserstein geodesic.
RELATED WORK
Unsupervised Domain Adaptation (UDA) One of the most popular approaches is invariant repre- sentation learning, which minimizes the distribution distance between the source and target domains under certain metric in some feature space with either adversarial training [Ajakan et al., 2014; Ganin et al., 2016] or divergence minimization [Sun & Saenko, 2016; Zhang et al., 2019]. The goal of this approach is to learn representations invariant over the source and target domains, and enjoys theoretical guarantees [Ben- -ani- et al., 2010; Zhao et al., 2019]. Self-training (i.e., sreudo-labeling another popular approach for UDA [Zou et al., 2018; 2019; Liang et al., 2019; 2020], which utilizes a -ource-trained model to offer pseudo-labels labels for unlabeled target samples and further finetunes the model with these pseudo-labels. is notable that even having achieved remarkable successes over wide range of datasets, modern UDA approaches may still become ineffective as the distribution shifts are large enough [Kumar et al., 2020; Sagawa et al., 2021; Abnar et al., 2021].
Gradual Domain Adaptation (GDA) Most UDA approaches (c.g.. the ones introduced above) adapt models directly from the source to the target in a one-step style, so they are also addressed as Direct UDA [Kumar et al., 2020; Chen & Chao, 2021]. Gopalan et al. [2011] introduces the notion of intermediate domains to the area of domain adaptation. In fact, Gopalan et al. [201 I] studies the standard UDA task with labeled source data and unlabeled target data, and fits PCA the source and target data to obtain two linear representations of the two domains. They constructed the intermediate domains via linear interpolation between the two PCA subspaces, and then concatenated the linear representations of all domains to a classifier upon them. A series of follow-up works of Gopalan al. [2011] 1] were proposed [Zheng et al., 2012: Gong al., 2012; Hoffman al., 2014; Cui et al., 2014] in the following years. In the deep learning era, some empirical GDA algorithms are proposed [Gadermayr et al., 2018; Wulfmeier al., 2018: Gong et al., 2019] in computer vision, leveraging modern deep learning techniques, but they lack theoretical guarantees. Recently, Kumar et al. [2020] proposes gradual self-training (GST), a simple yet effective GDA algorithm that enjoys theoretical guarantees, building the self-training technique [Yarowsky, 1995; Lee al., 2013]. Abnar et al. [2021]; Chen & Chao [2021] develop more powerful empirical GDA algorithms based on GST, and Wang et al. [2022a] significantly improves the theoretical guarantees of GST. In parallel to GST, another GDA approach is proposed [Wang et al., 2020], which also enjoys certain theoretical guarantees.
6 CONCLUSION
Gradual domain adaptation (GDA) attempts to address large distribution shifts between the source and target domains by adapting models along a sequence of intermediate domains. However, GDA becomes unreliable as the number of given intermediate domains is insufficient. To address this limitation of GDA, we propose a novel algorithmic framework, Generative Gradual Domain Adaptation with Optimal Transport (GOAT), which automatically generates intermediate domains along the Wasserstein geodesic (between consecutive given domains) and applies GDA on the generated domains. Our algorithm is motivated by the recent theory of Wang et al. [2022a] on gradual domain adaptation, and we develop practical computation techniques to implement the algorithm. Empirically, we show that GOAT can significantly outperform vanilla GDA when the given intermediate domains are scarce. Essentially, our GOAT is promising framework that augments GDA with generated intermediate domains, leading GDA to be applicable to more real- world scenarios.
'The target data of the Rotated MNIST dataset are obtained by rotating training data. Thus there is a one-to-one mapping between source and target data. The oracle plan is built from the one-to-one mapping, i.e., an element Y/y is non-zero if and only if xoz is rotated to ærz--

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REFERENCES
Samira Abnar, Rianne van den Berg, Golnaz Ghiasi, Mostafa Dehghani, Nal Kalchbrenner, and Hanic Sodghi. Gradual domain adaptation in the wild: When intermediate distributions are absent. arXiv preprint arXiv:2 106.060600 2021.
Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, and Mario Marchand. Domain- adversarial neural networks. arXiv preprint urXiv:1412.44444 2014.
Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Pazz Invariant risk minimization. 2019.
arXiv preprint
Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Vaughan. A theory of learning from different domains. Machine learning, 79(1)=151-175, 2010.
Andreea Bobu, Eric Tzeng, Judy Hoffman, and Trevor Darrell. Adapting to continuously shifting domains, 2018, URL https: / opwnreview..v.em net. Lor.mm
Hong-You You Chen and Wei-Lun Chao. Gradual domain adaptation without indexed intermediate domains. Advances in Neural Information Processing Systems, 34, 2021.
Nicolas Courty, Rémi Flamary, Devis Tuia, and Alain Rakotomamoniy. Optimal transport for domain adaptation. IEEE transactions on pattern analysis and machine intelligence, 39()):1833-1855 2016.
Nicolas Courty, Rémi Flamary, Amaury Habrard, and Alain Rakotomamoniy. Joint distribution optimal transportation for domain adaptation. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
Zhen Cui, Wen Li, Dong Xu, Shiguang Shan, Xilin Chen, and Xuelong Li. Flowing riemannian manifold: Domain adaptation by shifting covariance. IEEE transactions on cybernetics, 44(12): 2264 2273, 2014.
Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. Advances in neural information processing systems, 26, 2013.
Paul Adrien Maurice Dirac et al. The principles of quantum mechanics. Oxford university press, 1930.
Rémi Flamary, Nicolas Courty, Alexandre Gramfort, Mokhtar Z. Alaya, Aurélie Boisbunon, Stanislas Chambon, Laetitia Chapel, Adrien Corenflos, Kilian Fatras, Nemo Fournier, Léo Gautheron, Nathalie T.H. Gayraud, Hicham Janati, Alain Rakotomamonjy. levgen Redko, Antoine Rolet, Antony Schutz, Vivien Seguy, Danica Sutherland, Romain Tavenard, Alexander Tong, and Titouan Vayer. Pot: Python optimal transport. Journal of Machine Learning Research, 22(78):1-8, 2021. URL http: jmlr. org/ hwtpg//apers////e / 20/22/2204555 20-451 html.
Michael Gadermayr, Dennis Eschweiler, Barbara Mara Klinkhammer, Peter Boor, and Dorit Merhof. Gradual domain adaptation for segmenting whole slide images showing pathological variability. In International Conference on Image and Signal Processing, pp. 461-469. Springer, 2018
Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. Domain-adversarial training of neural networks. The journal of machine learning research, 17()):099-20300 2016.
Shiry Ginosar, Kate Rakelly, Sarah Sachs, Brian Yin, Crystal Lee, Philipp Krahenbuhl, and Alexei A. Efros. A century of portraits: A visual historical record of american high school yearbooks, 2015. URL https: /tap::///axiiie o abs/ /1111.1111 02575.
Boqing Gong, Yuan Shi, Fei Sha, and Kristen Grauman. Geodesic flow kernel for unsupervised domain adaptation. In CVPR, pp. 2066-2073. IEEE, 2012.
Rui Gong, Wen Li, Yuhua Chen, and Lue Van Gool. Dlow: Domain flow for adaptation and generalization. In CVPR, pp. 2477-2486, 2019.
10

---

Under review as a conference paper at ICLR 2023
Raghuraman Gopalan, Ruonan Li, and Rama Chellappa. Domain adaptation for object recognition: An unsupervised approach. In ICCV, pp. 999-1006. IEEE, 2011.
Ishaan Gulrajani and David Lopez-Pazz In search of lost domain generalization. In International Conference Learning Representations, 2021.
Dan Hendrycks, Kevin Zhao, Steven Basart, Jacob Steinhardt, and Dawn Song. Natural adversarial examples. In CVPR, pp. 15262-15271, 2021.
Judy Hoffman, Trevor Darrell, and Kate Saenko. Continuous manifold based adaptation for evolving visual domains. In CVPR, pp. 867-874, 2014.
Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. International Conference on Machine Learning, pp. 448-45, 2015.
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR, 2015.
Diederik P Kingma and Max Welling. Auto-encoding variational bayes. ICLR, 2014.
Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Bal- subramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, et Wilds: benchmark of in-the-wild distribution shifts. In International Conference on Machine Learning. pp. 5637-5664. PMLR, 2021.
Ananya Kumar, Tengyu Ma. and Percy Liang. Understanding self-training for gradual domain adaptation. In International Conference Machine Learning, pp. 1468-5479 PMLR, 2020.
Yann LeCun and Corinna Cortes. MNIST handwritten digit database. 1998. URL attp://yann. lecun. com/ exdb/mnist..
Dong-Hyun Lee al. Pseudo-label: The simple and efficient semi-supervised learning method for deep neural networks. In Workshop on challenges in representation learning. JCML, 2013.
Jian Liang, Ran He, Zhenan Sun, and Tieniu Tan. Distant supervised centroid shift: A simple and efficient approach to visual domain adaptation. CVPR, pp. 2975-2984, 2019.
Jian Liang, Dapeng Hu, and Jiashi Feng. Do really need access the source data? source hypothesis transfer for unsupervised domain adaptation. In ICML, pp. 6028-6039, 2020.
Mingsheng Long, Yue Cao, Jianmin Wang, and Michael Jordan. Learning transferable features with deep adaptation networks. In International conference on machine learning, pp. 97-105. PMLR, 2015.
Gaspard Monge. Mémoire sur la théorie des déblais et des remblais. Histoire de l'Académie Royale des Sciences de Paris, 1781.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. Advances in Neural Information Processing Systems, 32: 8026-8037, 2019.
Ofir Pele and Michael Werman. Fast and robust earth mover's 's distances. In 2009 IEEE 12th international conference on computer vision, pp. 460-467. IEEE, 2009.
Gabriel Peyré, Marco Cuturi, et al. Computational optimal transport: With applications data science. Foundations and Trendy@ in Machine Learning, 11(--6):355607,,22 2019.
Elan Rosenfeld, Pradeep Ravikumar, and Andrej Risteski. The risks of invariant risk minimization. arXiv preprint arXiv:2010.05761, 2020.
11

---

Under review as a conference paper at ICLR 2023
Shiori Sagawa, Pang Wei Koh, Tony Lee, Irena Gao, Sang Michael Xie, Kendrick Shen, Ananya Kumar, Weihua Hu, Michihiro Yasunaga, Henrik Marklund, Sara Beery, Etienne David, lan Stavness, Wei Guo, Jure Leskovec, Kate Saenko, Tatsunori Hashimoto. Sergey Levine, Chelsea Finn, and Percy Liang. Extending the WILDS benchmark for unsupervised adaptation. In NeurlPS 202/ Workshop on Distribution Shifts: Connecting Methods and Applications, 2021. URL https: http:://ppenreviie
net net/forum?idd wtwpruumiid-2EEHtxXXXo0
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. simple way to prevent neural networks from overfitting. The journal of machine learning research, 15()):1229-1958 2014.
Dropout:
Baochen Sun and Kate Saenko. Deep coral: Correlation alignment for deep domain adaptation, 2016.
Remi Tachet des Combes, Han Zhao, Yu-Xiang Wang, and Geoffrey Gordon. Domain adaptation with conditional distribution matching and generalized label shift. Advances in Neural Information Processing Systems, 13:1967-11288) 2020.
Eric Tzeng, Judy Hoffman, Kate Saenko, and Trevor Darrell. Adversarial discriminative domain adaptation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 7167-7176, 2017.
Hao Wang, Hao He, and Dina Katabi. Continuously indexed domain adaptation. In ICML, volume 119 Proceedings of Machine Learning Research, pp. 9898-9907. PMLR, 13-18 Jul 2020.
Haoxiang Wang, Bo Li, and Han Zhao. Understanding gradual domain adaptation: Improved analysis, optimal path and beyond. In International Conference on Machine Learning, pp. 22784-22801 PMLR, 17-23 Jul 2022a. URL https: pwweeooccddiiooooooooooooooo ngs. ..s.s.n ml www.pressseee 62//662226 222//ang22. html.
Haoxiang Wang, Haozhe Si, Bo Li, and Han Zhao. Provable domain generalization via invariant- feature subspace recovery. In International Conference on Machine Learning, volume 162, pp. 23018-23033. PMLR, 2022b. URL https::/pprceeetee wang22x, html.
mwr. mmr.ppress/v162
Olivia Wiles, Sven Gowal, Florian Stimberg, Sylvestre-Alvise Rebuffi, Ira Ktena, Krishnamurthy Dj Dvijotham, and Ali Taylan Cemgil. A fine-grained analysis on distribution shift. In International Conference on Learning Representations, 2022.
Markus Wulfmeier, Alex Bewley, and Ingmar Posner. Incremental adversarial domain adaptation for continually changing environments. In 2018 IEEE International conference on robotics and automation (ICRA), pp. 4489 4495. IEEE, 2018.
David Yarowsky. Unsupervised word sense disambiguation rivaling supervised methods. In 33rd annual meeting of the association for computational linguistics, pp. 189-196, 1995.
Yuchen Zhang, Tianle Liu, Mingsheng Long, and Michael Jordan. Bridging theory and algorithm for domain adaptation. In International Conference on Machine Learning, pp. 7404-7413, PMLR, 2019.
Han Zhao, Shanghang Zhang, Guanhang Wu, José MF Moura, Joao P Costeira, and Geoffrey J Gordon. Adversarial multiple source domain adaptation. Advances in neural information processing systems, 3 118559985700 2018
Han Zhao, Remi Tachet Des Combes, Kun Zhang, and Geoffrey Gordon. On learning invariant representations for domain adaptation. In International Conference on Machine Learning, pp. 7523 -532.. PMLR, 2019.
Jingjing Zheng. Ming- Yu Liu, Rama Chellappa, and P Jonathon Phillips. A grassmann manifold-based domain adaptation approach. ICPR, pp. 2095-2099, IEEE, 2012.
Yang Zou, Zhiding Yu, BVK Vijaya Kumar, and Jinsong Wang. Unsupervised domain adaptation for semantic segmentation via balanced self-training. In ECCV, pp. 289-305, 2018.
Yang Zou, Zhiding Yu, Xiaofeng Liu, B. V.K. Vijaya Kumar, and Jinsong Wang. Confidence regularized self-training. In ICCV, October 2019.
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A THEORETICAL ARGUMENTS
On Proposition 1 The inequality in (6) holds true since the Wasserstein distance metric W, is known to enjoy the property triangle inequality. In (6), the equality is obtained as the intermediate domains Ma....., A sequentially fall along the Wasserstein geodesic between (Mu and pa,, since the geodesic is defined as the shortest path of distributions connecting di and for under the W, metric.
On Proposition 2 This linear program (LP) formulation of optimal transport is also called Kan- torovich LP the literature. One can find details and proof of Kantorovich LP [Peyré al., 2019].
On the Encoder With Pe-Lipschitz continuous encoder & : X' 2 mapping inputs to the feature space E (i.e., 2 +- E()) for any input x), the order of the generation bound (2) stays the same. The reason is follows: The bound (2) linear in terms of Ph. 3, which is the Liphschitz constant of the classifier h; With the encoder &, one can effectively view the whole encoder-classifier model as f X + y' such that f(z) = h(E())); Then, the Liphschitz constant f obviously PEPl since f is a composite function of h o E; Finally, replacing h with f in the analysis of Wang et al. [2022a], one can see that the order of the bound (2) stays the same, with some terms getting multiplied by factor of pe (i.e., equivalent to replacing the term Ps. with PePs in the bound).
B MORE DETAILS ON THE PROPOSED ALGORITHM
To reduce the On log n) complexity of the exact OT calculation O( we can solve the entropy-regularized OT problem Cuturi [2013] instead. Consider source data fæod 1 and target
data { the regropy-rrgularizeed OT plan 7 under the transport cost function is obtained by solving
X arg min
Σ
M.so((Di A
Σ Ti. log ni,r in
yER m
20
(II)
s.t. 7In
1m and n 11m n 1n
m
where À is regularization coefficient. The low computational complexity comes at the cost of a dense optimal transport plan, i.e., 7 is generally dense matrix rather than a sparse one Thus, ((m)) non-zero entries will be generated in 7 and this quadratic space complexity becomes intractable for large datasets. To remedy this issue, we design two methods to zero out insignificant entries in Y to reduce the space complexity:
1. Small-value cutoff. Although the transport plan 7 resulted from entropy-regularizee OT is dense, most entries still have values close to 0. Those entries of tiny magnitude can be zeroed without having noticeable impact on the final results.
2. Confidence cutoff. Consider the one-hot encoded matrix of source labels Yo e {0, m and the reguaay-re OT plan 7 The logits of target prediction by optimal label transport is
Pr 75 Yo-
(12)
Then, we can calculate a confidence score for each target prediction by the logits. Using certain confidence threshold, the target samples that the transport plan unconfident with can be filtered out, making the transport plan more sparse.
With proper choices of cutoff values, those methods can reduce the space complexity from O (mm) to Om + n) without noticeable compromise on final performance.
"The dependence on Ph. is hidden with the big-O notation in (2)
As we discussed in Sec. 3.2, > has most + non-zero entries, thus it is sparse matrix.
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C MORE DETAILS ON EXPERIMENTS
Network Implementation. For the 4-layer CNN encoder used in experiments on Rotated MNIST and Portraits, we use convolutional layers with kernel size and SAME padding. During self-training. we train on each domain for 10 epochs. Empirically, we verify that regularization techniques are important for the success of gradual self-training, including using dropout layers and early stopping.
For the VAE used to produce Fig. 2, we use 4 convolutional layers with kernel size 3 and max- max-poolingg followed by culne-con layer with 128 neurons the encoder. For the decoder, we use four deconvolutional layers with kernel size 3 [Kingma & Welling, 2014]. We use ReLU activation for the layers. The encoder and decoder are jointly trained on data from source and target in an unsupervised manner with the Adam optimizer [Kingma & Ba, 2015] (learning rate as 10-4 and batch size as 512).
Encoder Pretraining. We pretrain an encoder on the given domains. During pretraining, we use 3-layyr MLP on top of the encoder and perform self-training the given domains. Specifically, we first fit the model on the source domain, then iteratively use the model to pseudo-label the next domain and self-train it. After pretraining, the MLP is discarded and the encoder is fixed to provide features for the downstream tasks.
OT ablation. When designing different plans, we make sure that the number of non-zero entries equal so that in the domains generated by those plans, the amount of data is the same. For the random plan, we first initialize a zero matrix, then sample the same amount of entries as the ground-truth plan in the matrix, and fill in weight value between 0 to uniformly at random. For the uniform plan, use the same procedure except that we fill in the same weight for each sampled entry. the end, we normalize the matrix.
14


Reviews:
[{"review_id": "yEWNJIhhK3", "reviewer": "Reviewer_d4Li", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "This paper tackles the setting of unsupervised domain adaptation when the shift between the source and target is large and thus gradual domain adaptation is needed. They propose a gradual domain adaptation algorithm that creates intermediate domains for the gradual adaptation and then applies (standard) gradual self training on these created domains. Authors propose to create domains in extracted feature space (e.g. on the embedding level). They show that it does better than creating domains using the original input space. The idea of creating intermediate domains spans from theoretical bounds: synthetic domains minimize the average Wasserstein distance between the pairs of intermediate domains.  \n", "summary_of_the_review": "I read authors response and am increasing my score to accept\nOverall very well written paper with somewhat weak experimental results. I am willing to be convinced otherwise if authors are able to provide additional results (please see below)\n\nMethod:\n1) Is the number of domains generated a hyperparameter? Why do you go only up to 4 (or rather 4*number of intermediate domains provided)? Can you instead find the optimal value of it using the bounds in (2)? (e.g. min over T (1/T + T/sqrt(n) + 1/sqrt(nT))?\n2) How many data points do you geenerate for each intermediate domain - is it the number of non zero elements in optimal transport matrix (so m+n approx, where m and n are source and target number of points?)\n3) This method sounds extremely expensive. Apart from OT which you can solve approximately, you train additional T models to convergence. Can you add wall time comparison to Tables 1 and 2, I am trying to understand the bulk part of it (is it 4 times slower for 4 intermediate domains? Or more?)\n\nExperiments:\n1)  A bit surprised that only two datasets are used, but seems that a lot of papers on GDA use those 2 datasets. Though this popular paper https://arxiv.org/pdf/2002.11361.pdf also has experiments on Gaussian synthetic data. Do you have experiment results on such data by any chance?\n2) Why not to add UDA like DANN or MMD to Table 1 and 2. You seem to say that they don't make sense since the shift is large. Do they underperform the 0-self training entry?\n3) is in Table 1 and 2 \"0 domains\" entry correspond to self training? \nLooking at table 1 it seems that when there are 0 given intermediate domains, you don't imporve over the self training at all apart from one case (portrait on 4). All drastic improvements are only when there are some intermediary domains already provided. Why is this?\n4) Also i do find that you report # of generated domains this way confusing - for 3 given domains and 3 generated, you actually generated 9 domains right?\n\nMinor:\n- (6) treats T delta as the length, but it was the AVERAGE (so 1/T) in (3). \n- intro: performance degrade => performance degradation\n\n", "strength_and_weaknesses": "Strength:\n- clarity: well written and easy to follow\n- good idea inspired by the theoretical bounds - find the optimal intermediate domains\nWeaknesses\n- Experiments don't compare with UDA baselines\n- Experiments are not convincing when there are no intermediate domains provided. May be more intermediate domains needed? \n- The method is expensive", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and 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": {}}, {"review_id": "eqe7LjI-0M", "reviewer": "Reviewer_SoJb", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "This paper tackles the domain adaptation problem, i.e., transferring a model trained on a source domain to a new target domain for which unlabeled data is available. It does this using self-training (pseudo-labeling, noisy student) where the source-trained model is used to generate labels for the target domain. The paper builds on theoretical insights which suggest self-training works best when the input data shifts gradually from the source to target domain (gradual self-training). This theory suggests that the ideal case would involve walking the shortest path between all the joint data distributions of the intermediate domains.\n\nThe authors transform this theoretical insight into a practical algorithm by:\n\n* making a covariate shift assumption which allows them to operate in a latent space, rather than the joint distribution of the data\n* generating intermediate domains that lie close to the Wasserstein geodesic between the source and target domain\n* adding a regularization term to the optimal transport problem to reduce computational complexity, in combination with thresholding the values in the transport plan to minimize space complexity", "summary_of_the_review": "I think this is a good paper. It's main shortcoming is the experimental evaluation, which is limited to very small networks on small and artificial datasets. If such experiments are added, I would be able to support acceptance of this paper more strongly.", "strength_and_weaknesses": "I like this paper.\n\nThe idea seems well-grounded in theory. It seems to me that the authors are minimizing a lower bound though (the right-hand side of equation 6). I am also curious to know what cost function ($c$ in proposition 2). Is it L1 or (squared) L2-distance? Did the authors consider evaluating these different cost functions?\n\nI appreciate the simplicity of the resulting algorithm: Pick a feature space to interpolate in, map both the source and target domain to this space, solve the optimal transport problem, and now just self-train your network as you slowly walk this path. This makes a lot of intuitive sense from the perspective of neural networks learning smooth \"manifolds\" of data, and GOAT exploits this well.\n\nExperimentally, one concern I have is that GOAT does not seem very effective when there are no intermediate domains available (which seems to me the most common setting in practice). In particular, in table 1 the best approach for going from 0 to 45 degrees rotated MNIST seems to be to not generate any intermediate domains, and the gain for portraits is within margin of error. On the other hand, these tasks seem small and artificial enough (especially rotated MNIST) that I don't want to draw too strong conclusions from it.\n\nSince GOAT does not require any intermediate domains, I would love to see GOAT evaluated using a few more domain adaptation datasets (e.g., the WILDS 2.0 dataset). This would give a much clearer signal as to how this algorithm performs in practice on real-world datasets. Is there a reason the authors did not do this? Does the EMD solver not scale to this size?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and 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": {}}, {"review_id": "aTf9yOJ_U9G", "reviewer": "Reviewer_3XRG", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "When the distribution gap between the source domain and the target domain is very large, some intermediate domains are usually used to make the model gradually adapt to the target domain. The author proposes the GOTA method to generate intermediate domains based on Wasserstein geodesic. Empirically, this GOAT framework can improve the performance of standard GDA(Gradually Domain Adaptation) when the given intermediate domains are scarce, significantly broadening the real-world application scenarios of GDA", "summary_of_the_review": "The generation theory of the intermediate domain is interesting, but there are some issues with the derivation. In addition, the author's experiment is relatively simple and lacks justification for the existing DA datasets", "strength_and_weaknesses": "Strengths:\n(1)\tThis paper gives a clearly theoretical analysis which explanation of the definition and generation method of the intermediate domains\n\nWeaknesses:\n(1)\tIs there some problem with Equation 10. How does it become the mixup of samples? please give a detailed proof of this problem.\n(2)\tThe experiment is insufficient. The authors mentioned [1] in related work, which also relies on intermediate domain generation, however, is compared in this work. \n(3)\tWhile the generated intermediate domains by the proposed GOAT method is demonstrate in Figure 2 for the rotated MINIST data, other more complicated scenarios are not shown. \n(4)\tIn Table, given domains=0, GOAT does not bring improvement, why? \n(5)\tSome papers have also achieved very good results in constructing intermediate domain at the input level [2,3]. Discussion and comparisons are needed.\n[1] Gong R, Li W, Chen Y, et al. Dlow: Domain flow for adaptation and generalization[C]//Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2019: 2477-2486.\n[2] Na J, Jung H, Chang H J, et al. Fixbi: Bridging domain spaces for unsupervised domain adaptation[C]//Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2021: 1094-1103.\n[3] Na J, Han D, Chang H J, et al. Contrastive Vicinal Space for Unsupervised Domain Adaptation[J]. arXiv preprint arXiv:2111.13353, 2021.\n", "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": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "F-_Ne0s7A7", "reviewer": "Reviewer_tuhk", "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": "This paper aims to address the task of gradual domain adaptation (GDA) which aims to address large distribution shifts via adapting models along a sequence of intermediate domains. To overcome domain shifts (especially when the given intermediate domains are missing or scarce), the authors propose a generative gradual domain adaptation with optimal transport (GOAT). Specifically, the authors start by generating intermediate domains along the Wasserstein geodesic, then adapt the source-trained classifier to target by considering the intermediate domains. Experimental results demonstrate the effectiveness of the proposed GOAT. However, I still have concerns about this paper. My detailed comments are as follows. ", "summary_of_the_review": "The idea of generating intermediate domains for unsupervised domain adaptation is interesting to me. Moreover, the proposed GOAT is backed by a theory guarantee. However, the datasets and baselines for verifying the GOAT are insufficient for me. Thus, I recommend the weak rejection of this paper. If the authors convince me of the effectiveness of the GOAT, I would be glad to change the reviews. ", "strength_and_weaknesses": "*Positive points\n\nThe task, gradual domain adaptation, which the authors focus on, is truly important and practical in reality since it mitigates the limitation of UDA whenever the distribution shift between the source and the target is large.\n\nThe proposed GOAT adapts a progressive manner for shifting between different distributions along Wasserstein geodesic, which is guaranteed by a theory from wang et al[1].\n\nThe visualization results in figure.2 are intuitive for the semantic meaning of the generated intermediate domains, which helps readers to understand this paper better.\n\n\n*Negative points\n\nHow to decide the number of generated domains of the proposed method? More discussions are needed.\n\n1. In table 1, it is observed that the performance of GOAT suffers performance degradation with a large # Generated domains (i.e., # Given Domains=4, # Generated domains=4). Besides, when given 3 domains GOAT achieves the best performance with 2 generated domains, why is it the case? Why more intermediate domains would not boost the performance of GOAT? \n\n2. But in Figure 4 (b), with the increase of generated domains, the test accuracy tends to increase, including Random, Uniform and OT settings. It seems that the contradictions in the experimental results.\n\nIt is necessary for the authors to provide more comprehensive and fair results.\n\n1. The experiments are only conducted on two datasets (4 datasets are used in [1], 3 datasets are used in [2]), I would like to see the performance of the proposed GOAT in more datasets for further verification of the effectiveness of GOAT.\n\n2. This paper only considers one baseline (GST, published in 2 years ago) for the comparison experiments. More recent state-of-the-art such as [1, 2] should be considered. \n\n**Reference\n\n[1] Understanding gradual domain adaptation: Improved analysis, optimal path and beyond. In ICML 2022.\n\n[2] Kumar, A., Ma, T., & Liang, P. (2020, November). Understanding self-training for gradual domain adaptation. In ICML 2020.\n", "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": {}}]

Author rebuttal:

Decision: Reject