Mapping conditional distributions for domain adaptation under generalized target shift
Authors: Matthieu Kirchmeyer
Alain Rakotomamonjy
Emmanuel de Bezenac
patrick gallinari
Venue: ICLR 2022 Poster (2022)
Keywords: Unsupervised domain adaptation
generalized target shift

Abstract: We consider the problem of unsupervised domain adaptation (UDA) between a source and a target domain under conditional and label shift a.k.a Generalized Target Shift (GeTarS). Unlike simpler UDA settings, few works have addressed this challenging problem. Recent approaches learn domain-invariant representations, yet they have practical limitations and rely on strong assumptions that may not hold in practice. In this paper, we explore a novel and general approach to align pretrained representations, which circumvents existing drawbacks. Instead of constraining representation invariance, it learns an optimal transport map, implemented as a NN, which maps source representations onto target ones. Our approach is flexible and scalable, it preserves the problem's structure and it has strong theoretical guarantees under mild assumptions. In particular, our solution is unique, matches conditional distributions across domains, recovers target proportions and explicitly controls the target generalization risk. Through an exhaustive comparison on several datasets, we challenge the state-of-the-art in GeTarS.

Published as a conference paper at ICLR 2022
MAPPING CONDITIONAL DISTRIBUTIONS FOR DOMAIN ADAPTATION UNDER GENERALIZED TARGET SHIFT
Matthieu Kirchmeyer! Alain Rakotomamonjy Emmanuel de Bézenac', Patrick Gallinari!2 CNNS-ISIR, Sorbonne University; "Criteo AI Lab; 'Université de Rouen, LITIS
ABSTRACT
We consider the problem of unsupervised domain adaptation (UDA) between a source and target domain under conditional and label shift a.k.a Generalized Target Shift (GeTars). Unlike simpler UDA settings, few works have addressed this challenging problem. Recent approaches learn domain-invarian representa- tions, yet they have practical limitations and rely on strong assumptions that may not hold in practice. In this paper. we explore novel and general approach align pretrained representations, which circumvents existing drawbacks. Instead of constraining representation invariance, learns optimal transport map, im- plemented as a NN, which maps source representations onto target ones. Our approach is flexible and scalable, preserves the problem's 's structure and has strong theoretical guarantees under mild assumptions. In particular, our solution is unique, matches conditional distributions across domains, recovers target propor- tions and explicitly controls the target generalization risk. Through an exhaustive comparison on several datasets, we challenge the state-of-the-ar in GeTarS.
INTRODUCTION
Unsupervised Domain Adaptation (UDA) methods (Pan & Yang 2010) train a classifier with la- belled samples from source domain S such that its risk on unlabelled target domain T is low. This problem is ill-posed and simplifying assumptions were considered. Initial contributions focused on three settings which decompose differently the joint distribution over input and label X x Y: covariate shift (CovS) (Shimodaira) (2000) with ps (Y (X) = pr (Y (X( # (X)),, target shift (TarS) (Zhang et al.] 2013) with (Y) * PY ps (X
Pr(X and conditional shift (Zhang et al.] (013) with (X * pr(( PY = PY) These assumptions are restrictive for real-worke applications and were extended into model shift when ps(Y|X) + pT (YIX). ps( X) & (X()) (Wang & Schneider (2144] 2015) and generalized target shift (GeTarS) (Zhang et al. 2013 when pss 7 pT (Y), (Y) GeTarS where key challenge is to map the source domain onto the target one minimize both conditional and label shifts, without using target labels. The current SOTA in Gong et al. (201)); Shui al. (2021) learns domain-invariant rep-
J + PY))). We consider
Combes et al. (2020); Rakotomamonjy al. 2021 : resentations and uses estimated class-ratios between domains as importance weights in the training loss. However, this approach has several limitations. First, it updates representations through adver- sarial alignment which is prone to well-known instabilities, especially on applications where there no established Deep Learning architectures e.g. click-through-rate prediction, spam filtering etc. in contrast to vision. Second, to transfer representations, the Nomain-invarianne the original problem structure and it was shown that this may degrade the discriminativity of target representations (Liu et al., 2019). Existing approaches that consider this issue (Xiao et al. 2019 Li
constraint breaks
al.) 2020 Chen al. (009)) were not applied to GeTarS. Finally, generalization guarantees are derived under strong assumptions, detailed in Section 2.3 which may not hold in practice.
In this paper, we address these limitations with new general approach, named Optimal Sam- ple Transformation and Reweight (OSTAR), which maps pretrained representations using Optimal Transport (OT). OSTAR proposes an alternative to constraining representation invariance and per- forms jointly three operations: given pretrained encoder, (i) it learns an OT map, implemented as neural network (NN), between encoded source and target conditionals, (ii) it estimates target propor- tions for sample reweighting and (iii) it learns classifier for the target domain using source labels.

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OSTAR has several benefits: (i) is flexible, scalable and preserves target discriminativity and (ii) it provides strong theoretical guarantees under mild assumptions. In summary, our contributions are:
We propose an approach, OSTAR, to align pretrained representations under GeTars. With- OSTAR jointly learns a classifier for inference
out constraining representationninvariine the target domain and an OT map, which maps representations of source conditionals to those of target ones under class-reweighting. OSTAR preserves target discriminativity and experimentally challenges the saute-of-the-ar for GeTarS.
OSTAR implements its OT map as NN shared across classes. Our approach is thus flexible and has native regularization biases for stability. Moreover it is scalable and generalizes beyond training samples unlike standard linear programming based OT approaches.
OSTAR has strong theoretical guarantees under mild assumptions: its solution is unique, recovers target proportions and correctly matches source and target conditionals at the op- timum. It also explicitly controls the target risk with new Wasserstein-base bound.
Our paper is organized as follows, In Section 2 we define our problem, approach and assumptions. In Section 3 derive theoretical results. In Section 4 describe our implementation. We report in Section experimental results and ablation studies. In Section we present related work.
2 PROPOSED APPROACH
In this section, we successively define our problem, present our method, OSTAR and its main ideas and introduce our assumptions, used to provide theoretical guarantees for our method.
2.1 PROBLEM DEFINITION
Denoting X' the input space and y = [1,... . K the label space, we consider UDA between a source S (K's pss.pssss.... ps(p,s Y)) with labeled samples $ {(x}]
(X's x V's)" and target = (Xr,yr, pr (X, X)) with unlabeled samples π = {x1 (6) H E XT.. We denote 2 c Rd a latent space and X + Z an encoder from X' Z. Zs and Z7 are the encoded source and target input domains, 33 and 27 the corresponding training sets and Z random variable in this space. The latent marginal probability induced by g D e (S, is defined A C (A) & I For convenience, p E K denotes the label marginal (Y) and pD _ (Z). In all generality, latent conditional distributions and label marginals differ across domains; this is the GeTarS assumption (Zhang et al. (2013) made in feature space rather than input space X, as in Definition] | class are different across domains with different label proportions. Operating in the latent space has several practical advantages c.g. improved discriminativity and dimension reduction.
(4) (3)
y/ }
I# (AD(A)
GeTarS is illustrated in Figure Ta and states that representations from given
Definition 1 (GeTarS), GeTarS is characterized by conditional mismatch across domains i.e. = j) + (ZY = -) and label shift i.e. P + fpy
3j
Our goal is to learn a classifier in Z with low target risk, using source labels. This is challenging as (i) target labels are unknown and (ii) there are two shifts handle. We will show that this can be achieved with pretrained representations if we recover two key properties: (i) map which matches source and target conditional distributions and (ii) target proportions to reweight samples by class- ratios and thus account for label shift. Our approach, OSTAR, achieves this objective.
2.2 MAPPING CONDITIONAL. DISTRIBUTIONS UNDER LABEL SHIFT
We now present the components in OSTAR, their objective and the various training steps. Components The main components of OSTAR, illustrated in Figure lb are detailed below. These components are learned and estimated using the algorithm detailed Section They include:
a fixed encoder g X' + Z, defined in Section 2.1
mapping Z + Z, acting on source samples encoded by g.
f P papin is the fosh-forward measure f#
(B)) for all measurable
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T ((x)) Px p.(2.))
(
Zy gX) (Ar(2,)) 12(2.1]
x
Class
00 Class 2
IN
Zy: labeled Souree
Zy Source representations mapped reweighted pk/p! 2 Lillaelleed Target
XXX XXXXXX xx
XXX
Zy Unlaheled Target representations
representations
23 - (X}))
representations
8
Optimal Transport Mapping
5 g(Xs)
&o: Target classifier
(a) UDA under GeTarS in latent space Z
(b) Mapping & reweighting source samples with OSTAR
Figure I: Illustration of our approach on a 2-class UDA problem [better viewed in color]. (a) A pretrained encoder y defines a latent space Z with labelled source samples Xo and unlabelled target samples xo under conditional and label shift (GeTars). (b) We train a target classifier fN on a new domain N, where labelled samples on conditional distributions and (ii) reweighting these samples by estimated class-ratios p / should match source and target conditionals and ply should estimate target proportions p
obtained by (i) mapping source samples with ø acting
label proportion vector ply on the simplex ÅK-
a classifier fN 2 + {l,... K} for the target domain in a hypothesis class H. over 2.
Objective g encodes source and target samples latent space such that it preserves rich informa- tion about the target task and such that the risk on the source domain is small. is fixed throughout training to preserve target discriminativity. Φ should map encoded source conditionals in 2s onto corresponding encoded target ones in Zr to account for conditional shift; NN denotes the mapped space. px should estimate the target proportions p to account for label shift. Components (ø, pl) define new labelled domain in latent space N = (Zv. PN (Z, Y)) through Sample Transfor- mation And Reweight operation of the encoded domain, as illustrated in Figure [b Indeed, the pushforward by á of encoded source conditionals defines conditionals in domain N. (x
Vk, PN (ZlY = k) - o# (ps(Z|y =
Then, P weights each conditional in N. This yields marginal distribution in N. PN (Z):
K
(2)
k))
(I)
Finally, classifier fNv is trained on labelled samples from domain N. This is possible each sample in N is projection of a labelled sample from S. fN can then be used for inference on T'. We will show that has low target risk when components & and ply achieve their objectives detailed above.
Training We train OSTAR's components in two stages. First, we train along source classifier fs from scratch by minimizing source classification loss; alternatively, can be tailored to specific problems with pretraining. Second, we jointly learn (fN, é, px)) to minimize classification loss in domain N and to match target conditionals and proportions with those domain N. As target conditionals and proportions
unknown, propose proxy problem for (ø) pk) to match instead latent marginals pr(Z) and p (Z) (T). We solve this proxy problem under least action principle measured by Monge transport cost (Santambrogio 2015), denoted C(Ø), as in problem (OT):
min C(o) e Σ e
[Ø(z) ps(z k)dz
(OT)
subject to x (Z) = Pr (7)
For any function @ e.g. parametrized by a NN, C(o) is the transport cost of encoded source condi- tionals by é. It uses cost function, y(x,y) = || yI where without loss of generality 2. The optimal C()) is the sum of Wasserstein-2 distances between source conditionals and their map-
pings. Problem (OT) seeks to minimize C(Φ) under marginal matching. We provide some further
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background on optimal transport in Appendix C and discuss there the differences between our OT problem (T) and the standard Monge OT problem between (2) and pr(Z).
Our OT formulation is key to the approach. First, it is at the basis of our theoretical analysis. Under Assumption
2 later defined, the optimal transport cost is the sum of the Wasserstein-2 distance be- tween source and matched target conditionals. We can then provide conditions these distances in Assumption to formally define when the solution to problem (OT) correctly matches conditionals. Second, it allows us learn the OT map with a NN. This NN approach: (i) generalizes beyond training samples and scales up with the number of samples unlike linear programming based OT approaches (Courty et al., 2017b) and (ii) introduces useful stability biases which make learning less prone to numerical instabilities as highlighted in de Bézenac et al. (202I); Karkar et al. (2020).
2.3 ASSUMPTIONS
In Section we will introduce the theoretical properties of our method which offers several guar- antees. As always, this requires some assumptions which are introduced below. We discuss their relations with assumptions used in related work and detail why they are less restrictive. We provide some additional discussion their motivation and validity in Appendix [
and there is a partition of the source domain 3g such that 33 = K 1 (R) and vk ps E Y k) =
Assumption 1 (Cluster Assumption S). yk, Ps Y=k >
Assumption T. inspired from Chapelle et al. 2010), states that source representations with the same label are within the same cluster. It helps guarantee that only one map is required to match source and target conditionals. Assumption [ is for instance satisfied when the classification loss on the source domain is zero which corresponds to the training criterion of our encoder g. Interestingly other approaches e.g. [Combes et al. (2020); Rakotomamonjy et al. (202T) assume that it also holds for target representations; this is harder to induce as target labels are unknown.
Assumption (Conditional matching). mapping @ solution to our matching problem in problem (OT) maps a source conditional to target one i... Ak 3j Φ# (ps ( = k)) = (2 j)-
Assumption 2 guarantees that mass of source conditional will be entirely transferred to target conditional; & solution to problem (OT) will then perform optimal assignment between conditionals. It is less restrictive than alternatives: the ACons assumption in Zhang al. (2013); Gong al. (2016)
states the existence of a map matching the right conditional pairs i.e. j = in Assumption] 2 while the GLS assumption of Combes al. (020)) imposes that ps (2 included in Assumption 2 when j = k and Φ = Id and is more restrictive.
pr pr(Z))) GLS is thus
Assumption 3 (Cyclical monotonicity between S and T). For all K elements permutation o, con- ditional probabilities in the source and target domains satisfy Σ Walps(ZlY = k), =
k)) Σ k= Wa (ZlY k). (Z = okk)) with Wz, the Wasserstein-2 distance.
Assumption 3 introduced in Raaootoammnny et al. (202I), formally detines settings where condi-
tionals are guaranteed to be correctly matched with an optimal assignment in the latent space under Assumption 2 One sufficient condition for yielding this assumption is when Yk, W2(ps
j)). This last condition is typically achieved when conditionals between source and target of the same class are "sufficiently near" to each other.
(Z k)) < (2sp(2l = k). P.r[2l
Assumption (Conditional linear independence on T). (2r k)}{ are linearly indepen-
dent implying Vk. I {Akl PT k) > jj aspr (opr(Z)--)) j)
standard and seen in Tars to guarantee correctly estimating target pro- portions (Redko et al.) 2019; Garg et al. 2020). It discards pathological cases like when jii,j,k) Ja, b) s.t. a + milder than its alternative A2Cons in Zhang et al. 2013;; Gong al. (2016) which states linear independence of linear combinations of source and target conditionals, in respectively Z.
Assumption
(Z
a x pr (Z)Y j} (1)++0) x (Z = k)). It is
3 THEORETICAL RESULTS
We present our theoretical results, with proofs in Appendix E for OSTAR under our mild assump- tions in Section 2.3 We first analyze in Section 3.1 the properties of the solution to our problem in
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OT). Then, show in Section 3.2 that given the learned components 9, and p the target gen- eralization error a classifier in the hypothesis space H. can be upper-bounded by different terms including its risk domain N and the Wasserstein-1 distance between marginals in N and T.
3.1 PROPERTIES OF THE SOLUTION TO THE OT ALIGNMENT PROBLEM
Proposition (Unicity and match). For any encoder which defines 2 satisfying Assumption 7 there is a unique solution (6, p%) (OT) and (#<ps (Ps() = Pr(Z and p/y =py-
Given encoder g satisfying assumptions, Proposition [ shows two strong results. First, the solution to problem (OT) exists and is unique. Second, we prove that this solution defines a domain N, via the sample transformation and reweight operation defined in (T), where encoded conditionals and label proportions are equal to target ones. For comparison, [Combes et al.] (2020); [Shui et al.)
(222I) recover target proportions only under GLS when conditionals are already matched, while we match both conditional and label proportions under the more general GeTarS.
3.2 CONTROLLED TARGET GENERALIZATION RISK
We now characterize the generalization properties of classifier IN trained on this domain N with new general bound. First, we introduce some notations; given encoder onto a latent space Z. we define the risk of classifier f as e (f) (x-pp(2)))](()) # yl with D E (S,
Theorem 1 (Target risk upper-bound). Given fixed encoder defining latent space Z, two domains N and T satisfying cyclical monotonicity in Z, assuming that we have yk. Ply 0, then NfN E H. where H is set of M M-Lipschitz continuous functions over Z. we have
2M
&(fN)s
(fN)
Wi (pN(2), +
min
P.y -K
Classification (C)
Alignment (A)
K Wi Σ
21
Y= = pr (2 k),
Σ
P pr =k)
(2)
min P =
R=1
Label (F.)
We first analyze upper-bound in 2 The target risk of fN is controlled by three main terms: the first (C) is the risk of JN on domain N. the second (A) is the Wasserstein-1 distance between latent marginals of domain N and T, the third term (L) measures a divergence between label dis- tributions using, as a proxy, two ad-hoc marginal distributions. There are two other terms in 2 first, Lipschitz-constant M that be made small by implementing JN as a NN with piece-wise linear activations and regularized weights; second the minimum of p which says that generaliza- tion harder when target class is less represented. We learn OSTAR's components to minimize the r.h.s of (2). Terms (C) and [A] can be explicitly minimized, respectively by training classifier IN on domain N and by learning (x.p%) to match marginals of domains N and T. Term (L)
not computable, yet its minimization is naturally handled by OSTAR. Indeed, term (L) is minimal
when p = pl under Assumption per Redko et al. (2019). With OSTAR, this sufficient condi- tion is guaranteed in Proposition by the solution to problem OT) under our mild assumptions in
Section 2.3 This solution defines a domain for which p = PT and term (A) equals zero.
We now detail the originality of this result over existing Wasserstein-bast based generalization bounds. First, our upper-bound is general and can be explicitly minimized even when target labels are un- known unlike Shui et al. (2021) or Combes et al. (2020) which require knowledge of pt or its perfect estimation. Combes et al. (2020) claims that correct estimation of pt i.e. (L) close to zero, achieved under GLS Le. (2(2) Y) Pr Pr (ZlY). However, GLS is hardly guaranteed when the latent space is learned: sufficient condition in Combes et al. 2020) requires knowing p), which is unrealistic in UDA. Second, our bound is simpler than the one in Rakotomamonjy et al. (202T): in particular, it removes several redundant terms which unmeasurable due unknown target labels.
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4 IMPLEMENTATION
Our solution, detailed below, minimizes the generalization bound in 2 It implements JN g with NNs and with residual NN. Our solution jointly solves (i) classification problem to account for term
(C) in (2) and (ii) an alignment problem on pretrained representations (OT) to account for terms Our pseudo-code and runtime / complexity analysis are presented in Appendix F
(A) and
Encoder initialization Prior to learning Φ JNN we first learn the encoder jointly with source classifier fs to yield a zero source classification loss via (3). With Leee the cross-entropy loss,
min 1 Σ Ceelfs o g(xg)) 18-9
min (fs, S) fsg
1=1
g is then fixed to preserve the original problem structure. This initialization step helps enforce Assumption T and could alternatively be replaced by directly using pretrained encoder if available.
Joint alignment and classification We then solve alternatively (i) a classification problem on do- main N w.r.t fN to account for term (Crm(C)in 2 and (ii) the problem (OT) w.r.f. (ø,p)) to account for term (A) Term (L) in ( is handled by matching pl and p through the minimization of problem OT Piv is estimated with the confusion- based approach in Lipton et al. (2018), while & minimizes the Lagrangian relaxation, with hyperparameter λOr, of the constrained optimization problem (OT)
The equality constraint in (OT) is measured through Wasserstein distance our bound in tailored to this distance, however, we are not restricted by this choice as other discrepancy measures are also possible. The objective based (2) corresponds to the following optimization problem:
min [G[NN, N) ÀOT (or(0) + L (x. 6.1N
(CAL)
subject to p = arg min 2 |1by Č
P
12
[Label propoction estimation)
p20,pEAx
Pg
where CoUN, N)
Lee
UN o 6o )
[Cussificatinn loes 2
(4)
K
and (o) A
1
Σ
l (1)
z5
(8)
[Objective function U
(5)
ktie
1=1
n
1 m
and Co pk)
sup 1
P
(3)
o(z
[Relased equality constraint in 01
(6)
n
m
PS
Ć is the confusion matrix of fN on domain N. pt E Ax is the target label proportions estimated with fN. [f N)
(4)
is the classification loss of fN Zg. derived in Appendix B which
minimizes term (C) in 2 Note that samples in domain N are obtained by mapping source samples
with Φ; they are reweighted account for label shift. Cg,r (5) defines the transport cost of à on Zz.
Implementing & with a ResNet performed better than standard MLPs, thus we minimize in practise the dynamical transport cost, better tailored to residual maps and used in de Bézenac et (2021); Karkar et al. 22020). L (6) is the empirical form of the dual Wasserstein- I distance between
ph (Z) and pr(2) and seeks enforcing the equality constraint in problem (OT) i.e. minimizing term (A) OSTAR's 's assumptions also guarantee that term (L) is small at the optimum.
Improve target discriminativity OSTAR solves transfer problem with pretrained representa- tions, but cannot change their discriminativity in the target. This may harm performance when the encoder g is not well suited for the target. Domain-invariant methods are less prone to this issue as they update target representations. In our upper-bound in (2), target discriminativity is assessed by the value of term (C) at optimum of the alignment problem (OT). This value depends on g and may be high since g has been trained with only source labels. Nevertheless, it can be reduced by updating using target outputs such that class separability for target representations is better en- forced. To achieve this goal, we propose extension of (CAL)) using Information Maximization

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(IM), not considered in existing for source-free adaptation without alignment (Liang et al. 2020). In this context, target predictions prone to errors and there is no principled way to mitigate this problem. In our case, have access to source samples and OSTAR minimizes an upper-bound to its target error, which avoids performance degradation. IM refines the decision boundaries of fN with terms target sam- ples, (f "") (7) is the conditional entropy of fN which favors low-density separation between classes. Denoting (.) the k-th component of the softmax function,
GeTarS methods. IM was originally introduced
K 5a g(x log(ós g(xx))))
Cliv (fN, T) (8) promotes diversity by regularizing the average output of fN og be uniform on T'. avoids predictions from collapsing to the same class thus softens the effect of conditional entropy.
K
1 Pu log fr. = DRL(P, K xx) log K; p = Exrexr [8 (JN o ((xT))
2 (NN
Σ
(8)
Our variant introduces two additional steps in the learning process. First, the latent space is fixed and we optimize fN with IM in (SS). Then, we optimize the representations in (SSg) while avoiding
modifying source representations by including 2 (fs, , S) B with a fixed source classifier fs.
min Cf( N) + Ca (fN. T) + L (fN, T)
(SS)
fx
min CfN, N) + C. (fN, + C2 (fN. T) S), fx-9
(SSg)
EXPERIMENTAL RESULTS
We now present our experimental results on several UDA problems under GeTarS and show that OSTAR outperforms recent SOTA baselines. The GeTarS assumption is particularly relevant on our datasets as encoded conditional distributions do not initially match as seen in Appendix Figure
Setting We consider: (i) an academic benchmark Digits with two adaptation problems between USPS (Hull 1994) and MNIST (LeCun et al., 1998). (ii) a synthetic to real images adaptation benchmark VisDA12 2 (Peng al. 2017) and (ii) two object categorizations problems Office3] 31 Saennoo et al., 2010), Of aeficeeome (Venkateswara et al. 2017) with respectively six and twelve adaptation problems. The original datasets have fairly balanced classes, thus source and target label distributions are similar. This is why we subsample our datasets to make label proportions dis- similar across domains as detailed in Appendix Table For Digits, we subsample the target domain and investigate three settings balanced, mild and high as Rakotomamonjy et al. (2 For other datasets, we modify the source domain by considering 30% of the samples coming from the first half of their classes as Combes et al. (2220). We report a Source model, trained using only source samples without adaptation, and various UDA methods: two Cov5 methods and three recent SOTA GeTarS models. Chosen UDA methods learn invariant representations with reweighting in GeTarS models or without reweighting in other baselines. The two CovS baselines are DANN Ganin et al. 2016) which approaches H-divergence and WD 3-0 (Shen et al. 2018) which computes Wasserstein distance. The TeTarS basclines are Wu et al. (2019) Rakotomamonjy et al. (202T); I); Combes et al. 2020). We use Wasserstein distance learn invariant representations such that only the strategy to account for label shift differs. Wu et al.] (2019), denoted wDg, performs assymet- ric alignment with parameter B, for which we test different values (B 2)). MARSc, MARSg |Raootomamoniy et al. 2021) and WD (Combes et al.) (2020) estimate target proportions respec- tively with optimal assignment or with the estimator in Lipton et al.) (2018) also used in OSTAR. We report D DI-Oracle, an oracle which learns invariant representations with Wasserstein distance and makes use of true class-ratios. Finally, we report OSTAR with and without IM. All baselines are reimplemented for a fair comparison with the same NN architectures detailed in Appendix G
Results In Table I we report, over 10 runs, mean and standard deviations for balanced accuracy, iec. average recall each class. This metric is suited for imbalanced problems (Brodersen et al.) 2010). For better visibility, results are aggregated over all imbalance settings of a dataset (line "subsampled"). In the Appendix, full results reported in Table 3 along the E target proportion
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Table 1: Balanced accuracy (1) over 10 runs, The best performing model is indicated in bold. Results are aggregated over all imbalance scenarios and adaptation problems within a same dataset.
Setting
Sourcc
DAKK
WD
W De-t
WD
HARSq
NARSO
IM-AD
O8TAR
OSTA8-IN
5I-Orocle
Digina
halencal
P1.96 3.8
9081
1.5
92.65 a 1.0
82.00
7007 T.1.
10.18:22
9181
05.80
91
97.51 0.3
90.00
sabamplod
75.00 3.1
89M0
45
N1
48
77.77
855
M1ST 4.0
98.7
10.22 4.1
88 30
96 69
0.7
96 03
ViaDAL2
origital
65.03 =: 1.0
68.72 = 0.0
11.40
67.00 2...5.0...
3621
61.58 L.5
50.07 0.0
69.24:10.6
57.61 4 D.
vhhomplal
47.57
13.00
5186:220
58.84 009
56.77
Orticw31
sabsumpled
74,00 0.5
76.13 a 0.3
76224:000
7240
20.30
8000 05
17.2
76
82.01 0.4
SL67
sabsumppedd
50,00 e 2.8
30ST a 1.05
40.8
1.
0000
3622 06 54.87 4.44
54.64 59.51 + 0.4 57.97 a nJ
estimation error for GeTarS baselines and OSTAR+IM in Figure 4 First, we note that low estima- tion error of pi- is correlated to high accuracy for all models and that DI-Oracle, upper-bounds domain-invariant approaches. This shows the importance of correctly estimating p.-. Second, we note that OSTAR improves Source (column 2) and is competitive w.r.t. IW-WD (column 9) on VisDA, Office31 and OfficeHome and MARSg (column 7) on Digits although it keeps tar- get representations fixed unlike these two methods. OSTAR+ IM (column 11) is less prone to initial target discriminativity and clearly outperforms or equals the baselines on both balanced accuracy and proportion estimation. It even improves DI-Oracle (column 12) for balanced accuracy de- spite not using true class-ratios. This (i) shows the benefits of not constraining domain-invariane which may degrade target discriminativity especially under label estimation errors; (ii) validates our theoretical results which show that OSTAR controls the target risk and recovers target proportions. We visualize how OSTAR aligns source and target representations in Appendix Figure
We perform two studies. We first measure the role of IM in our model. In Appendix Table 4 we show the contribution of (SS) (column 3) and (SSg) (columm 4) to (CAL) (column 2). We also evaluate the effect of IM on our baselines in Appendix Table even if this is not part of their original work. IM improves performance on VisDA and Office and degrades it on Digits.. s. The performance remains below the ones of OSTAR+ IM. Finally, we evaluate the impact of IM on upper-bound in (2) in Appendix Table 6 We assume that target conditionals are known to compute term (L) We observe that term [C] related to target discriminativity. and the alignment terms (A) and (L) are reduced. This explains the improvements due to IM and shows empirically
Ablation studies
that IM helps better optimize the three functions Φ, g. fN in our aoper-bound, The second study
in Table 2 measures the effect of the OT transport cost in our objective (CAL). Proposition showed that OT guarantees recovering target proportions and matching conditionals. We consider MNIST-USSP under various shifts (Line 1) and initialization gains i.c. the standard deviation of the weights of the NN (Line 2). On line 1, we note that AOT * 0 in problem (CAL) improves balanced accuracy (left) and & estimation error (middle) over XoT = 0 over all shifts, especially high ones. This improvement is correlated with lower mean and standard deviation of the normalized transport cost per sample (right). We observe the same trends when changing initialization gains in Line 2. This confirms our theoretical results and shows the advantages of OT regularization biases for performance and stability. In Appendix Table 7 we test additional values of Aor and see that OSTAR recovers the Source model under high Aor- Indeed, the latter constrains 6 Re ld.
6 RELATED WORK
UDA Existing approaches train a classifier using source labels while handling distribution shifts. We review two main approaches: the first learns invariant representations and the second learns a map between fixed samples from source and target domains. While nomain-invariant approaches are SOTA, mapping-basee based approaches avoid structure issues posed by the invariance constraint which may degrade target discriminativity (Liu et al.) 2019 Chen et al. 2019). For CovS, domain-invariant methods directly match marginal distributions in latent space (Ganin et al., 2016: Shen et al. 2018 Long et al. 2015). while mapping- based approaches learn mapping in input space (Courty et al. (0177) et al. 2018). For Tars and Tars,s, label shift requires estimating p to reweight source samples by estimated class-ratios (Zhao et al. 2019. An alternative is to use a fixed weight
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Table 2: Effect of OT on balanced accuracy 1 (left), & estimation error (middle) and normalized transport cost for/n /n (right) for MNIST-USSS.. The best model for accuracy is in bold with We consider varying shifts (Line 1) and initialization gains (stdev of the NN's weights) (Line2)
Sample
OSTAR
OSTAR Apr
MNIST-USHP initialization gain 0.02
OSTAR
Shift
Aor 0
Aon = 10
balanced
94.92 = 0.6
95. 95.12 0.6
mild high
88.28
1.5
91.77
85.21 1.6
88.55 111
balanced
mid
Nigh
balanced
mild
high
-NIST-UUSPS high imbalance
OSTAN Aur - 1ph*
Init Gain
Aor 0
Aor = 10-2
0.02
85.24 1.6
88.55 +
0.1
84.62
+ 2.3
88.41 + 1.3
0.3
83.11 2.4
89.41 + 1.G
010
0.02
Wu et al. 2019). When conditionals are unchanged i.e. Tars, pl can be recovered without needs for alignment (Lipton et al., 2018: Redko et al. 2019). Under GeTars, there is the additional difficulty of matching conditionals. The SOTA methods for GeTarS are domain-invariant with sample reweighting [Rakotomamoniy et al., 2021 Combes et al. 2020 Gong et al. 2016). An ear- lier mapping- based approach proposed in Zhang et al. (2013) to align conditionals, also under reweighting. Yet, is not flexible, operates on high dimensional input spaces and does not scale up to large label spaces it considers linear map for each pair of conditionals. Estimators used in GeTars are confusion-based in Combes et al. (2020); [Shui et al. (202T;; derived from optimal as- signment in RRakotommmonyy et al. (2021) or from the minimization of reweighted MMD between marginals in Zhang et al. (2013); Gong et al. (2016). OSTAR is a new mapping-basee based approach for GeTarS with improvements over these approaches as detailed in this paper. The improvements are both practical (flexibilityy, scalability, stability) and theoretical. CTC (Gong et al.) (2016) also oper- ates on representations, yet OSTAR simplifies learning by clearly separating alignment and encoding operations, intertwined in CTC as both encoder and maps are trained to align.
OT for UDA A standard approach is to compute transport plan between empirical distributions with linear programming (LP). LP used in CovS to align joint distributions (Courty et al. 2017a
Damodaran et al., 2018) or to compute a barycentric mapping in input space (Courty et al. 2017b);
and Tars estimate target proportions through minimization of reweighted Wasserstein dis-
tance between marginals (Redko et al. 2019). An alternative to LP is to learn invariant representa- tion by minimizing a dual Wasserstein- distance with adversarial training Shen et al. (20T8) in
CovS or, more recently. [Rakotomamonjy et al. (202T); (Shui et al. (202T) for GeTarS. Rakotoma- monjy et al. (202I) formulates two separate OT problems for class-ratio estimation and conditional alignment and Shui al. (202T) is the OT extension of Combes et al. (2020) multi-source UDA. OSTAR is the first OT CeTarS approach that does not learn invariant representations. It has several benefits: (i) representation are kept separate, thus target discriminativity is not deteriorated, (ii) single OT problem is solved unlike Rakotomamonjy et al. (2021), (iii) the OT map is implemented with a NN which generalizes beyond training samples and scales up with the number of samples unlike barycentric OT maps; (iv) it improves efficiency of matching by encoding samples, thereby improving discriminativity and reducing dimensionality, unlike Courty al. (0017b)
CONCLUSION
We introduced OSTAR, a new general approach to align pretrained representations under GeTars, which does not constrain representation invariance. OSTAR learns flexible and scalable map be- tween conditional distributions in the latent space. This map, implemented as ResNet, solves an OT matching problem with native regularization biases. Our approach provides strong generaliza- tion guarantees under mild assumptions as it explicitly minimizes a new upper-bound to the target risk. Experimentally, it challenges recent invariant GeTars methods several UDA benchmarks.

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Acknowledgement We acknowledge financial support from DL4CLIM ANR-99-CHA 0018-01, RAIMO ANR-20-CHIA--0021 OATMIL ANR-17-CE23-0012 and LEAUDS ANR-18-CE23- 0020 Projects of the French National Research Agency (ANR).
Reproducibility Statement We provide explanations of our assumptions in Section 2.3 and pro- vide our proofs in Appendix The datasets used in our experiments are public benchmarks and we provide a complete description of the data processing steps and NN architectures in Appendix G Our source code available at at/htpss:/// httpp:://githbbe wtwpgttt...ooommm..eeeeooe rwcheycrreeaeeee /ostar
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A ADDITIONAL VISUALISATION
We visualize how OSTAR maps source representations onto target ones under GeTarS. We note that OSTAR (i) maps source conditionals target ones (blue and green points are matched c.f. left), (ii) matches conditionals of the same class ("v" and "o" of the same colour are matched c.f. right).
75
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(a) MNNSS subsampled high imbalance
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(c) VisDA subsampled
Figure 3: t-SNE feature visualizations for OSTAR on various datasets and label imbalance. Crosses "x" denote source samples, circles "o" target samples and triangles "y" transported source samples. On the left, source samples are red, target samples blue and transported source samples green. On the right, samples from the same class have the same colour regardless domain.
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B ADDITIONAL RESULTS
We report our full results below. Aggregated results for balanced accuracy dataset and imbal- ance scenarios are reported in Table Prefix "s" refers to subsampled datasets defined in Table
Table 3: Balanced accuracy (f) over 10 runs. The best performing model is indicated in bold.
Setting
Source
W
W
MARSg
MARSC
TW
CSTAR: DT-Cracle
MUSPT
balanced
86.04 + 1.1
03.97 0.9
94.92 0.6
62.05633
7..14-6.7
94.19
96.44 0.5
96.10 0.3
96.91 +.0.3
96.95
97.12
high
86.23 28
08:00 1.0
01.14:077
85.15.199
94.80
96.07.1:00
94.00
96
96.87.10.4
Dieica USPS
-MNIST
bolanced mild high
63.01
97.64-11
63.54
17:00
+-02
62.83 2:2l::228
27.00
88,07 + 1.2
77.83 5.4
19.11:71
89.3
90.20
92.73
97.44 0.0
96.70 0.3
64.04 4.
96-37
87.04
70.17 6.0
74.47 7.4
88.24
91.54 0.9
90.81
87.
0.6
96.87
VisDA12
ViSDA
48.63
53.72
11.00.1 1.99
47.86 0.88
36.21 L18
$5.62
$6.33
51.88
39.24 1.
$7.61 03
42,46
47.57
47.32
31.83 3.0
66.00
51.86
66.77
O2ric031
IA-D
80.71 0.6
82.30 0.4
81.76+004 0.1
70.98
08:64 2.4
84.54
84.10
81.83
84.17 +
87.74 0:0.6
1.2
1W-A
58.91 0.2
58.87
0.1
50.18:10.2
60.70 1:00 0.33
00.95 0.2
63.94 0.0.1
00.80.1.0.3
60.25 0.2
89
1.1.1
60.02.1.02
<W-D
95.64 0.2
07:00 0.3
97.13+033
96.90-003
05-57
0.5
97.96 0.1
96. +.2
07.03
98.47 0.2
98.35++00
112
67.81
38.24
sAW
69.23
72:00
72,60 0.
68.94 0.0
61:64 7.2
81.00
81.05
75.84 0.7
83.91 0.0.
82.51 0.5
OFE
41.74::
012
58.96
89.96
54.67
52.18 23
66.20
62.17
0.2
61.07 1.02
90.46
09.52 0.4
63,59 0.2
LA-R
67.10
0.2
07
65.00 0.6
60.52
80.51
1.9
68:65 0.3
68-02+-05
67.76
69.85
85.54
37.34
36.81
33.10 2.3
0.2
46.10 002
46
2.1
50.56
50.53
40.96
50.82
58.67
60.14
CC-R
$4.99
54.22
0.8
54.00 t 0.5
31.31:1:25
45.00
3.5
62.69 0.1
62.41 0.1
60:74 0.2
03 0:2
62.80
38.10
96.30
48.24
48
47.78
45.08
60.84
41.10002
JPRR
66.28
3.4
59.10
0.8
70.01 0.1
68.78 1.2.
06.61 0.5
70.00
69.37 0.9
60.45
73 0.3
71.26.10.4
66.67
70:97
0.6
73.47
72.66
0.7
72.62
72.72
72.90
T5.
74.17
$0.44
06.28
$7.68
SR
39.36 2.5
45:33
06.47 1 0.4
47.11 1 0.5
45.38 1.3
45.81 1
45.30
42.66
49.07
0.9
0.3
0112
0122
8.00
0.90
8005
halenced
mild
high
Iala
mil
high
VIDAA
MNIST-USPS Setting
USPS-MNIST Setting
VisDA-12 Setting
Error
0.35
MARSg
MARSc
NH-WD
OSTAR
0.2
8.15
0.10
0.2
6.5
3W-D
Office31 Setting
officeHome Setting
Figure 4: e estimation error of p. (1). The best model for balanced accuracy is indicated with "42
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Additional ablation studies
We detail the full results for ablation studies.
Table Sepi-super learning for OSTAR and balanced accuracy (t). Best results are bold.
Sctting Objective
MNIST-USSP
balanced mild
1.0
96.91 0.3
USPS-MNIST
balanced high
2.8
97.08
VisDA12
original
60.37 +.0.
aulsimplee
63.37
offiice31
81.52-+-07
83.18
84.17
W
89.94 1.0.8
89,50
1.
94.13 0.2
<W-A
59.02 0.0.6
60.00 0.04
69.00
0.1
sW-D
96.88 0.0
97.44
0.02
98.47
0.2
<D-A
54.38 +: 1:1
0.0.6 56.58
B5
0.5
sA W
76.30 1.0
81.32
0.88
83.91 1: 0.5
Table 5: IM for MARSc, IW-WD and OSTAR on balanced accuracy (1). Best results are in bold.
Setting Model
XBRSc
MIRSc-IM
IW-XD
IN-XD+IN
OSTAR
OSTAR-IM
MNIST--USPS
0.5
90.00
97.91
95.12
mild high
95.47
+-0
98.07 0.6
05.76 0.5
94.00 0.8.
9.7
88.35 1.1
96.06 0.6
USPS-MNNIT
83.57
95:03
88 18.19:+11
98.11
mild high
83.20--224
91:50
9273
92.05
88.34
91.51
90.16
91.26
84.87
65.33
57.37
0.1
1.8.
39.24
VVIDA
61.96
07.00
50000
1.5
57.02.0077
48005 0.9
0ffico31
0..50-10..3
67.42.008
7..84
48.17
42.06
49.07
Table 6: Best value over training epochs of term (A) ()), term (C)
(L) and term (L) without
and with IM in OSTAR. Best results are in bold. Terms (A) and (L) are computed with the primal
formulation of OT using the POT package https: http://ttthotttgggooo thth io/
Alignment
Discriminativity
(A)
Temm (L.)
(C)
Setting
OSTAR
OSTAR-IE
OSTAR
OSTAR+IM
OSTAR
OSTAR-IM
MNIST -USPS
halanced
49.83
16.56
1.45
0.18
2.19 10
0.918 x 10
mild
19.13
10
10
high
38.60
21.10
12.78
0.74
1.76 10-8
0.510
USPS-MMNIST
balanced mild
235.43
86.65
7.08
0.32
4.46 x 10
0.495 10
188.07
104.66
20.99
1.05
3.98
10-3
0.399 10
high
21.02
0.82
0.616
Table 7: Detailed analysis the impact of λo on balanced accuracy (t). Best results are in bold.
MNIST -USPS initialization gain
Shin \ Aor
Apr =0
Ape 10
Apr = 10
Aor = 10-1
Aog
Apy
Source
balanced
94.92 + 0.6
96.02 0.2
95.12 0.6
89.76 + 1.2
91.06 + 1.2
87.03 1.8
86.02 1.4
mild
1.3
91.77
90.17
high
85.24 :: 1.6
85.35 4:
88.50
89.10 12
88.82
86.94 =: 1.1
86.73
16

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Published as a conference paper at ICLR 2022
MNIST-UUSPS high imbalance
Gain \ Aor
ÅoT
Aor = 10-3
Aor = 10-2 Aor = 10-1
Aor
Aor
0.02
85,24 1.6
85.38
88.55
89.10
1.2
88.82
86.94
0.1
84.62 1 2.3
85.84 1.841111 1.1
88,41 1 1.3
88.79
1.2
87.90
1 1.2
87.10 1 1.0
0.3
83.11
84.45
89.41
91.00
80.65
0.7
86.23
C DETAILS ON OPTIMAL TRANSPORT
Background OT was introduced to find a transportation map minimizing the cost of displacing mass from one configuration another (Villani 2008). For comprehensive introduction, refer to Peyré & Cuturi (2019). Formally, let a and 3 be absolutely continuous distributions compactly supported in _ and Rd Id + R cost function. Consider a map á IR + H that satisfies Φ#a 3, i.e. that pushes to B. We remind that for a function f. f#P is the push-forward measure fB = (B)) for all measurable set B. The total transportation cost depends on the contributions of costs for transporting each point x to (xx) and the Monge OT problem is:
min Cmonge (Ø) / c(x, ((x))da(x) JR.
(9)
s.t. 04 B
c)x,y) |x yll induces the p- Wasserstein distance, W/ 3) min a== Cmonge (0) 1/. When p 1, W1 can be expressed in the dual form Wi(a, 8) suPi|ella Ex vxx) By where ||u||z is the Lipschitz constant of function u.
OT is the extension of ( to the setting where p * pf with p unknown. This extension aims at matching conditional distri- butions regardless of how their mass differ and learns a weighting term p\ for source conditional distributions which addresses label shift settings. When PS pr, these two formulations are equivalent under Assumption if we fix ph- p.s =p.
Relationship between (OT) and the Monge OT problem 9
D DISCUSSION
Our four assumptions required for deriving our theoretical guarantees. All GeTars papers that propose theoretical guarantees also rely on set of assumptions, either explicit or implicit. Assumptions D and are common several papers. Assumption is easily met in practice since it could be forced by training a classifier on the source labels. Assumption
4 is said to be met in high dimensions (Redko al.] 2019 Garg et al. 2020). Assumption 3 says that source and target clusters from the same class are globally (there a sum in the condition) closer one another than clusters from two different classes. This assumption required to cope with the absence of target labels. Because of the sum, it could be considered as a reasonable hypothesis. It is milder than the hypotheses made in papers offering guarantees similar to ours (Zhang et al. a |2033 Gong et al. 2016)). Assumption 2 new to this paper. It states that ø maps j source conditional to unique k target conditional i.e. it guarantees that mass of a source conditional will be entirely transferred to a target conditional and will not be split across several target conditionals. This property is key to show that OSTAR matches source conditionals and their corresponding target conditionals, which is otherwise very difficult to guarantee under GeTarS both target conditionals and proportions are unknown. This assumption has some restrictions, despite being milder than existing assumptions in GeTarS. First, mass from a source conditional might in practise be split across several target conditionals. In practise, our optimization problem in (CAL) mitigates this problem by learning how to reweight source conditionals to improve alignment. We could additionally apply Laplacian regularization (Ferradans et al.) 2013 Carreira-Perpiñal & Wang 2014) into our OT map & but this was not developed in the paper. Second, it assumes a closed-set UDA problem i.e. label domains are the same Vs3 = Dr. The setting where Ys + J'r is more realistic in large scale image pretraining and is another interesting follow-up. Note that OSTAR can address empirically open-set UDA (Dr C Yss) by simply applying LI regularization
pN in our objective function (CAL). This forces sparsity and allows "loosing" mass when mapping a source conditional. avoids the unwanted negative transfer setting when source clusters, with labels absent on the target, aligned with target clusters.
17

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Published as a conference paper at ICLR 2022
E PROOFS
4 unique solution (o. pk) (OT) and @# (ps (Z))) = Pr (Z and ph Pr-
Proposition (.. For any encoder g which defines Z satisfying Assumption 1 2
3
there is an
Proof. Fixing satisfying Assumption 1 3ad
3 and 4 we first show that there exists solution (6.P))
to (TT) Following Brenier (1991) as Z C IR" we can find K unique Monge maps { (k) 1
1k=1
((h) Vk 4# (2 k)) pr Y
k) with respective transport costs Wi (ps( (
k). (Z k)). Let's define o as Ak 0 T(k) where - is the partition of Zs in
Assumption (.. p satisties the equality constraint to (OT) thus we easily deduce existence.
Now, let (db p be solution to (T)), let's show unicity. We first show that 3. Under As-
sumption 2
is the Monge formulation of the optimal assignment problem between (ps (Z = k)) with C the cost matrix defined by Cjj W2(ps ( (T = -)) At the optimum, the transport cost is related to the Wasserstein distance between
OT
k)) and (P (P (ZlY
source conditionals and their corresponding target conditionals i.c. C(o) = Σ W2 W2(ps Φ (2s(Z = k))). Suppose J(i.j). * s.t. o# (ps(Z() = i)) = pr(Z = j) and j)) ( i) and Vk. # i (ps (Z = k))
## (ps (
pr
k))
Assumption 3 implies E Wa = (2 = k)) S Exgis W2(ps (
pr k), Y k)) + (Z = P). ( j)) + W2(ps(ZlY = j), Pr ( = 1))
Thus C(o. Z) S C(bb, 2) whereas Φ solution to (OT). has minimal transport cost. Thus @ 3.
Now let's show p pu under Assumption 4
We inject @4 (Zl)) (2 into OT
K
K
K
PN
k pr (2k))
Σ PT Y
pT PT
(Z])) +
Y-kk k - pt p)=²) pT = + pk- = (pX
Given a fixed encoder g defining a latent space 2, two domains N and T satisfying cyclical monotonicity in Z, assuming that we have Yk, pN of M Lipschhtt continuous fanctions over Z, we have
Theorem 1
Y= O. then VfN E H. where H is a set
(fx)) s
(xNN
2M
wi ((xx(2).Pr2))
mink
K
PNv
Claesification C
Alignment
K
K Σ
+ 2M (1 (1+
1 min 1 pk=n
)
Wi
Y =k pr (Z k), p
pT
(z k)
k=
Label
(
Proof. We first recall that (fx) B(nn))eno22,,(((((())))))) where L is the 0/1 loss. For 2 z/Y conciseness, Vz E 2, Pr .PN LLzk will refer to the vector of (2](2) [px (z), k)]k= respectively. In the following, @ denotes the element-wis. product operator be- tween two vectors of the same size.
VfN. ((x.)(fN)+ (fv( + -c
K
S {f()))+ +
Σ
(pr(z,k) (pr(2,))) 1o-NNN(,, (2) x C ((fx(2), k6) dzdy
s <(((N) + /
K
Y=k
pr(zlk) pxx Y=k - pN(e/k) x c ((Frle)) k))
Pr
k=1
<CN( +
pr
YT
Pr ② c2.k)
x
-xx
PPN
x|1
@ Ca Lam)
18

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Published as a conference paper at ICLR 2022
((NN + PT PT oC -D"V YT (PT)
YT ZIY
2|1 - +DN (PT o - -DN (PN 3 o de
VT
2/y
s {((( + / (1)) - -P%
x/y
@ + XT (1D7" -PN x 1 (zah)) )
PT
{((( + / ( Dx
Y Czk
dz + / PN (PP
PPN 2|Y
2 dz
We now introduce a preliminary result from Shen et al. (2018). VfN E H. M-Lipschitz continuous, (Sx) - ((x) S 2M Wi (rn
(z)
Assuming that h is M M-Lipschitz continuous we apply this result in the following
K
1 (0i - -Dx") (pT) y @ Czk) dz = /
Σ
pp -
pk - pT(() x (ff((z), k
2 k=1
K > k=1
=f((N) (G(( where (z)
PN
prr
Y=k
S 2M . (1(
1 PNv T (PT7'
-PN ) 3 Ca. dz =
/ Σ
K
pN
=k
(pr(z)k) -N(((1)))) x < (z)
2 k=1
$ Σ
(pN(2k)) x C (IN(2)), k)
wk xxpx"** S
k=1
K Σ
< 2M
Wi Wi (Pr((2))).xx(211))
k=1
Thus, VNN M-Lipschitz continuous
K
{f S (fN) + 2M x Σ (m((((ZI). + 2M x wi (r))
We rewrite the second term to involve directly latent marginals, Proposition 2 in Rakotomamonjy et al. (202)) shows that under cyclical monotonicity, if Vk: p.y Y= >0. O.
K
Wi Σ pk * (z2))
Σ
y "wi (PN(2ZZ))).pr(211)
pN
This allows to write
K
K
K
Wi (px(2))),77((113 <
Y== Wi (pN(2)3),PT
pinPN
pN k=1
k=
k=1
K
= wi
(Z W/ -W((p((2),NN2)
We then use the triangle inequality for the Wasserstein distance Wi
(f S (&
2M
(p;(2))px(2) + 2M Wi (pr(Z).p(()) Y
Wi
mink k-
K
PX
S (fx) +
2M
Wi (px(2))P((2) + 2M+
Mi (px(2).7((2)
min K
PNv
=
min P/
k x=
19

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Published as a conference paper at ICLR 2022
(fN) Let Lie be the cross-entropy loss. Given classifier h, feature extractor 9 and domain N, the mapping of domain S by (Ø, pk).
Derivation of the reweighted classification loss (C)
(fx)=
Pi é
(z v) Coo (fM(Z), dzdy J2Y p
Y=
PN (Nx(z))) Cce (fx(z),
dzdy
2.y
PN Y
Y=y
" (2lU)) (
Y=
{
PN
Y Y 2 (ps(2)))) (fx(z), u) dddy
Y=y Ps
Y=y Ps
2.y Ps
2.y Ps
F PSEUDO-CODE AND RUNTIME / COMPLEXITY ANALYSIS
We detail in Algorithm T our pseudo-code and in Algorithm how we minimize CAL with respect
to (N) using the dual form of Wasserstein- distance 66. Our method is based on a standard backpropagation strategy with gradient descent and uses gradient penalty (Gulrajani al. 2017).
Algorithm Training and inference procedure for OSTAR
Training:
5= {xg M/
(1) 12 1. T - (x
pm Jーm1 z = (6 g( 1, 1 2q (x(x()) )mm
fs, fN E H classifiers; feature extractor; & latent domain- mapping. critic. Nº: number of epochs, N,; epoch to update pk= Ngi epoch to update g
1: Train [s,9 g on S to minimize source classification loss
D
2: Initialize p K
1k 3: for Nepoch SN, do
4: if Nepoch mod N, = then
5: Compute p with estimator in Lipton al 2018 (Z Zz))
CCA w.r.t. P
6: Average ply with cumulative moving average
7: if Repoch I Ng then Train Φ v. fN with (S, ?)
CAL
SS
w.r.t. Φ fN w.r.t. fN
8: else Train v, fN. g with (S,T)
(CAL
+ SSg
Inference: Score xT with fN g (
Algorithm Minimize (CAL) w.r.t. (Ø,fN)
(i) (n)n (x (x5' : Us Ji=1
(1) Im J'= y feature extractor, á domain- v critic, JN classifier.
Parameters of o, U, fN: ts Ge, fy and learning rates agg, a. asxx
Nater: batches per epoch, Nb; batch size, N,: critic iterations
1: for Niter Niter do
(8) 2: Sample minibatches x u {xs Us JEl, xB (Zu J Ji=1 from S,
3: Compute 2 g( (x5) N do g and z (x
4: Compute class ratios: Wy PN/P3s Ps
5:
for nhe < N₂ do
6:
Sample random points B' from the lines between B
2B
pairs
7:
Compute gradient penalty forad with =N: zB
BB
(Gulrajani al. 2017
8:
Compute L = -ii'' w W
N Li=1 ZT
No.
9:
0, + 0, ag Vo.
C9 ld
farad
10:
Compute Cor ΣK k==
Σgg (E(I.N) [l((z 2s
# k)iE[L.Nl
O, 10- a Vo,
L9 +
12:
Compute Lg(JN, N) = w
w w
No
01x t 8fx afx Volx (f(,N,N
20

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Published as a conference paper at ICLR 2022
Complexity / Runtime analysis In practise on USPS - MNIST, the runtimes in seconds
NVIDIA Tesla V100 GPU machine are the following: DANN: 22.75s, WDg=0: 59.25s, MARSc: 72.87s, MARSg: 2769.06s, IW-WD: 74.17s, OSTAR+ IM: 89.72s. We observe that (i) computing Wasserstein distance (WD is slower than computing H-divergence (DANN), (ii) runtimes for domain-invariant GeTarS baselines are slightly higher than for WD as proportions are addition- ally estimated, (iii) domain-invariann which uses GMM, (iv) our model, OSTAR+ IM, has slightly higher runtime than GeTarS baselines other than MARSg. We now provide some further analysis on computational cost and memory in OSTAR. We denote the number of classes, d the dimension of latent representations, n and m the number of source and target samples.
GeTarS baselines have similar runtimes apart from MARSg
Memory: Proportion estimation is based on the method in Lipton et al. (018)) and requires storing the confusion matrix C and predictions p with memory cost of O(K²). Encod- ing is performed by a deep NN into IR and classification by a shallow classifier from R& R Alignment is performed with a ResNet & : Rd Rd in the latent space which has an order magnitude less parameters than g. Globally, memory consumption is mostly defined by the number of parameters in the encoder g.
Computational cost comes from: (i) proportion estimation, which depends on K, n and is solved once every epochs with small runtime; (ii) alignment between source and target representations, which depends on d, n and N₂ (the number of critic iterations). This step updates less parameters than domain-invariant methods which align with g instead of o; this may lead to speedups for large encoders. The transport cost 2oT depends on d and adds small additional runtime; (iii) classification with fN using labelled source samples, which depends on d, K, n. In a second stage, we improve target discriminativity by updating the encoder y with semi-supervised learning; this depends on d, K, n. +
G EXPERIMENTAL SETUP
Datasets We consider the following UDA problems:
Digits is a synthetic binary adaptation problem. We consider adaptation between MNIST and USPS datasets. We consider subsampled version of the original datasets with the following number of samples per domain: 10000-2609 for MNIST-UUSPE 5700-20000 for USPS- The feature extractor is learned from scratch.
VisDA12 is 12-class adaptation problem between simulated and real images. We con- sider a subsampled version of the original problem using 9600 samples per domain and use pre-trained ImageNet ResNet-50 features http://cststtteeee bu. edu/ ftp/ adaasdaaaaec sda17/clf/
Office31 is object categorization problem with 31 classes. We do not sample the original dataset. There are domains: Amazon (A), DSLR (D) and WebCam (W) and we consider all pairwise target domains. We use pre-ted ImageNet ResNet-50 http//llob//mmmaeeeeeeee
features https: cttt:/thh.cooooecooooooooooeeeee http:///.w....nonnaaaa//aneeeeee
data/dataset.nm mc]]
is another object categorization problem with 65 classes. We do not sample the original dataset. There are 4 domains: Art (A), Product (P), Cli- part (C), Realworld (R) and consider all pairwise souree-target domains. We use pre- pre-trained ImageNet ResNet-50 features [attps:// com/ haamm//iddongwagg// Eransfer leaaffrreattee lea rarnnn http:/llob/maattr/daaaaaaa
md
Imbalance settings We consider different class-ratios between domains to simulate label-shift and denote with "s" prefix, the subsampled datasets. For Diqits., we explicitly provide the class- ratios as |Raootommonnj et al. (2021) (e.g. for high imbalance, class 2 accounts for the 7% of target samples while class 4 accounts for 22% of target samples). For vi Visd 12, Offfice3 and of ceHmee, subsampled datasets only consider a small percentage of source samples for the first half classes as [ombes et al. (2020) (c.g. Office3l considers 30% of source samples in classes below 15 and uses all source samples from other classes and all target samples).
21

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Published as a conference paper at ICLR 2022
Table 8: Label imbalance settings
Dataset
Configuration
ps
PF
Digits
halanced
[10
subsampled mild subsampled high
(0. 1,2,3,6)
0.06. (4.5)
0.2, [7. 8.9) 0.1
(,,
0.07, (1.5) 0.22
VisDA12
original subsampled
10 11) 100%
{0-11): 100%
[0 5) 30% (5
100%
113 100%
Office31
subsampled
15)
30%
{15
31 100%
{0 313 100%
Officetone
subsampled
[0 32) 30% {33 61) 100%
{0- 64) 100%
Hyperparameters Nomain-invariin methods weight alignment over classification; we tuned the corresponding hyperparameter for 3- in the range [10 4, 1000001 and used the one that achieves the best performance on other models. We also tuned Aor in problem (CAL) and fixed to 10-2 on Digits and 10-5 VisDA12 2, Officel and OfficeHome, Batch size is N,, 200 and all models are trained using Adam with learning rate tuned in the range [10-4, 10-31- We initialize NN for classifiers and feature extractors with normal prior with zero mean and gain 0.02 and with orthogonal initialization with gain 0.02.
Training procedure We fix Ne the number of epochs to 50 on Digits, 150 on VisDA12 and 100 on Office31, OfficeHome; OSTAP requires smaller N. to converge. Critic iterations are fixed to No = which worked best for all baseline models; for CSTAR higher values performed better. For all models, we initialize fs.g for 10 epochs with (3). Then, we perform alignment either through domain-invariance or with a domain-mapping until we reach the total number of epochs. GeTars models TW-WD, MARSc, MARSg, OSTAR perform reweighting with estimates refreshed every N,, 2 epochs the first 10 alignment epochs, every N₂, = 5 epochs after. OSTAR minimizes (CAL) + (SS) for 10 epochs on Digitss, Officee1, OfficeHome and epochs on VisDA12, then minimizes CCL + SSg for remaining epochs.
Architectures For Digits, our feature extractor y composed of three convolutional layers with respectively 64, 64, 128 filters of size 5 x 5 interleaved with batch norm, max max-pooling and ReLUU. Our classifiers (fs. fN) are three-layered fully-connected networks with 100 units interleaved with batch norm, ReLU. Our discriminators are three-layered NN with 100 units and ReLU activation. For VisDA12 and Offi office31. 31, CfficeHome, we consider pre-trained 2048 features obtained from ResNet-50 followed by 2 fully- connectdd networks with ReLU and 100 units for VisDA11 256 units for Office 31, Of ceHome. Discriminators are 2-layer fully-connected networks with respectively 100/1 units sisDA12, 256/1 units on Of c131, offi ceHome interleaved with ReLU, Classifiers are 2-layer fully- cully-connected networks with 100/K units on VisDA12, single layer fully-connected network with K units on Office3l, Office é a ResNet with 10 blocks of two fully-comnen colyecconeeete layers with ReLU and batch-norm.
Implementation of target proportion estimators OSTAR and IW-WD use the confusion based estimator in Lipton et al. (2018) and solve a convex optimization problem ((4) in Combes et al. (2020) and CAL w.r.t p for OSTAR) which has an unique solution the soft confusion matrix C of full rank. We implement the same optimization problem using the parallel proximal method from Pustelnik et al. (201)) instead of cvxopt used in Combes et al. (2020). MARSc and MARSg (Rakotomamoijy et al. 2021) use linear programming with PCT to estimate proportions with opti- mal assignment between conditional distributions. Target conditionals are obtained with hierarchical clustering or with a Gaussian Mixture Model (GMM) using sklearrf MARSg has some compu- tational overhead due to the GMM.
hattp://evxptt.ozg/
https://bothonot.aithhn https://scikittlerr...o
22


Reviews:
[{"review_id": "_AakxrFlLC5", "reviewer": "Reviewer_9ZXq", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strength:\n1. The paper is well-motivated from the Generalized Target Shift setup and uses optimal transport to map the source distribution to the target.\n2. The proposed approach is supported by theoretical analysis that minimizes the upper bound of the target risk.\n3. The proposed approach demonstrates sota results over the baselines on three datasets.\n\nWeaknesses:\n1. The theory and implementation are disconnected with the use of information maximization. The main argument is the use of optimal transport for joint alignment and classification, and it is not clear why it requires information maximization if the joint alignment and classification are successful. Because the information maximization (IM) approach is proposed in the context of source-free transfer, it is not clear whether the use of IM hampers the learning of optimal transport.\n2. In the experiments, the IM component in SS and SSg substantially improves (+9%) the results of joint alignment and classification, while the main focus of this paper is joint alignment and classification. But in the main empirical results in Tab. 1, only results of OSTAR are presented. It will be more informative and convincing to also include the results of CAL alone to help fully understand the impact of optimal transport.", "summary_of_the_paper": "The paper proposes an approach for Generalized Target Shift (GeTarS) where both conditional and label shift are present in the target domain. The proposed approach, Optimal Sample Transformation and Reweight (OSTAR), uses optimal transport to transform the latent space and uses information maximization to refine the classifier's decision boundaries. Theoretical analysis shows the proposed approach minimizes the the components in the target risk's upper bound. Empirical studies are carried out on three datasets and ablation studies were carried out to analyze the impact of MI.", "summary_of_the_review": "The idea of using optimal transport to address the problem of generalized target shift is interesting, but the main argument of this paper, optimal transport and its theoretical analysis, is not sufficiently evaluated and is confounded by the additional component information maximization. Based on the empirical studies, it appears information maximization has substantially more contribution than optimal transport, and the outcome of optimal-transport (CAL) is missing in the main results (Tab. 1).", "strength_and_weaknesses": "", "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": "Vgvnbslrd-o", "reviewer": "Reviewer_FHUc", "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": "Strengths:\n* The core of the approach is well motivated and theoretically appealing\n* Unlike many other deep learning DA papers, this one clearly puts forward assumptions and bases theoretical results on them\n* The domain generalization bound provides interesting insights regarding class proportion and imbalance\n* The experimental section is quite thorough, provides meaningful error estimates, various ablations, and achieves impressive SOTA results\n* The paper is overall very well written, and is quite fun to read\n\nWeaknesses:\n* Although the risk bound motivates Wasserstein distance minimization between latent marginals and class-weighted conditional distributions, the actual optimization objective used combines a plethora of terms, so it's not clear how relevant the theoertical results are for the actual method proposed\n* There is not (theoretical or empirical) runtime/complexity analysis \n* The intuition/motivation for some of the assumptions could use a bit more detail. Some of them are taken ipso facto from prior work, without much justification. It would be useful to have it here (even if in the appendix). This is especially true for Assumption 2, as per question below. \n* One aspect that is not discussed enough is the effect of the latent encoder g in the overall performance of the method, e.g., what happens if g is not rich enough (e.g., initial layers of a NNEt $f\\circ g(x)$, for which $g$ is shallow and $f$ is deep). Moreover, given two possible encoders g1, g2, how should one choose between them? is there a way to easily verify which one leads to the 4 Assumptions being satisfied? Should one try to estiamte the bound for both of these and choose the one with the tighter one?\n\nQuestions/Comments/Suggestions:\n* Objective OT is *almost* the Monge OT problem between $p_T(Z)$ and $p_S(Z)$ (by moving sum inside and using law of total prob), except for the fact that the pushfoward constraint is not exactly $\\phi_{\\sharp} \np_S(Z) = p_T(Z)$, but rather one that uses the class proportions $p_N$ rather than $p_S$. I was hoping to get more clarity on the reason for this discrepancy in the paper or Appendix, but it seems it isn't really discussed anywhere. \n* Assumption 2 feels potentially very restrictive: it says that the $\\phi$ must exactly preserve mass across clusters. How realistic is this? Furthermore, does the j have to uniquely assigned to a k? Or can two source clusters push their mas to a target cluster? If not, this seems to immediately rule out DA between datasets where the target domain has fewer labels that source domain (which, fine, is not that common in classic DA, but is very common in large-scale image pretraining/fine-tuning). But beyond that, \n* Is the bound computable? Given g, $f_N$, $p_N$, (C) and (A) seem straightforward to estimate from samples. What about (L)? The second distribution relies on $p_T$ and $p_T(Z|Y=k)$ which are not directly available in UDA. \n* In related work, OT approaches for UDA: it is stated that Rakotomamonjy et al. (2020) solves GeTarS with OT, buth then two lines later the paper claims OSTAR is the first one to to this. Is the *mapping* the keyword here? If so, would be nice to empahsize that this is the main difference between this paper and Rakotomamonjy et al. (2020). \n\nTypos:\n* It seems that Asssumption 4 should read $p_T(Z| Y=i)$ (not k).\n* Pg 8 conforts->supports/confirms?", "summary_of_the_paper": "This paper proposes a method for unsupervised domain adaptation under conditional and label shift, one of the most challening versions of DA. At the core of this method is an optimal transport problem that seeks a mapping between source and target domains *and* a class proportion vector that minimize a total transportation cost between the domains. Based on this formulation, and a set of 4 sensible assumptions, the paper provides theoretical results in the form of unicity of the solution and classic generalization bound on the target domain risk. The implemented algorithm is inspired (though not identical) to this objective, which is extended with a few other terms: a constraint on the the label proportions based on a confusion matrix, a classification loss for the mapped source samples, and a relaxed version of the OT's pushforward constraint, and finally, two objectives that seek to improve the discriminativity of target representations. The paper then proceeds to validate the propose method in a wide array of benchmark DA datasets, with various levels of class unbalance, whereby it shows that the method repeatedly and significantly outperforms SOTA UDA alternatives. ", "summary_of_the_review": "This paper makes a solid contribution towards the challening problem of unsupervised domain adaptation under label and conditional shift. Based on a sound and (to the best of my knowledge) novel variant of the parametric OT formulation, the method proposed achieves remarkable results across a wide range of benchmark tasks. All things considered, I think this paper would be a good contribution to ICLR and would be relevant to the OT and DA communities. ", "strength_and_weaknesses": "", "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": "7Y2B3L2GOts", "reviewer": "Reviewer_cdrt", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Pros:\n(1)\tThe proposed OSTAR well handles the GeTarS case and improves the target discriminativity. \n(2)\tThe paper gives both the theoretical and empirical analyses for the proposed OSTAR.\n(3)\tThe experiments show improvements over existing baselines. \n\nHere are some comments/questions for improvements:\n(1)\tIn introduction, the paper gives two limitations of existing works, discriminativity and strong assumptions. Please clarify (1) what’s the advantage of OSTAR compared with those UDA methods that explicitly preserve the data discriminativity in domain-invariant feature learning, e.g., [ref1] and [ref2]; and (2) please clearly state the strong assumptions or indicates the related works.\n\n[ref1] Structure preservation and distribution alignment in discriminative transfer subspace learning\n\n[ref2] Discriminative Transfer Feature and Label Consistency for Cross-Domain Image Classification\n\n(2)\t As $g$ is learned by eq (3) and fixed afterwards. It is unclear how to guarantee the target discriminativity is preserved when learning $g$. \n\n(3)\tRegarding eq (OT), please also give reference for “least action principle measured by monge transport cost”.\n\n(4)\tAccording to eq (9), the constraints in eq (OT) should be $p_N^\\phi(Z) = p_S(Z)$? If so, It is unclear how to link $p_T(Z)$ with $p_N^\\phi(Z)$. \n\n(5)\tI understand that each of assumption 1-4 is less restrictive than the corresponding one in the related work. However, proposition 1 requires $Z$ satisfying all the four assumptions. In this sense, it is unclear how significant the proposition 1 is in terms of the flexibility.\n\n(6)\tSome important references are missing, e.g., [ref3] and [ref4]. \n\n[ref3] Flexible transfer learning under support and model shift\n\n[ref4] Generalization Bounds for Transfer Learning under Model Shift\n\n(7)\tIn the overall objective, why only $\\mathcal{L}^g_{OT}$ is weighted by an importance factor? How about the term $\\mathcal{L}^g_{wd}$ is also weighted? Similarly, for each term in eqs. (SS) and (SSg), how about they are weighted?\n\n(8)\tRegarding $\\lambda_{OT}$, the authors only conduct 0 and 0.01 cases. How about the other values? More general, It is interesting to see the sensitivity analyses on such an importance factor. \n\n(9)\tFor table 1, it is better to show the average results for each dataset. \n", "summary_of_the_paper": "In this paper, the authors unsupervised domain adaptation (UDA) where both the label conditional and marginal distributions are different between the source and target domains (GeTarS). The authors propose an approach, OSTAR, to align pretrained representations under GeTarS. A new theoretical bound under some assumptions is derived. Experimental studies also show the effectiveness of OSTAR. ", "summary_of_the_review": "Overall, the paper is well-written and makes some contributions. For more detailed comments, see above. ", "strength_and_weaknesses": "", "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": {}}]

Author rebuttal:

Meta-review: This paper considers the generalized target shift setting for domain adaptation and proposes an optimal transport map-based approach to it. The considered setting for domain adaptation is rather general and of practical use. The proposed method seems sensible, as supported by the theoretical identifiability and empirical results. 

It is worth noting that the way to cite previous work seems to be improved. For instance, in the first paragraph of Introduction, the authors reviewed various settings for domain adaptation. For model shift, the authors cited previous work. However, when discussing covariate shift, target shift, and generalized target shift, the authors did not cite the original work that provides the categorization. For completeness, the authors may want to consider including the setting of conditional shift as well, which has received a number of applications in domain adaptation in computer vision. I believe the categorization of target shift, conditional shift, and generalized target shift was provided by Zhang et al. (2013). This work should also be cited when the authors give the problem definition in Section 2.1. The quality of the paper will be even better if the authors cite previous work in all the right places--this may also make the authors' contribution clearer.
Decision: Accept (Poster)