Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
Authors: Jayadev Naram
Fredrik Hellström
Ziming Wang
Rebecka Jörnsten
Giuseppe Durisi
Venue: ICML 2025 poster (2025)
Keywords: Partial Domain Adaptation
Optimal Transport
Generalization Bounds
Primary area: theory->domain_adaptation_and_transfer_learning

Abstract: In many scenarios of practical interest, labeled data from a target distribution are scarce while labeled data from a related source distribution are abundant. One particular setting of interest arises when the target label space is a subset of the source label space, leading to the framework of partial domain adaptation (PDA). Typical approaches to PDA involve minimizing a domain alignment term and a weighted empirical loss on the source data, with the aim of transferring knowledge between domains. However, a theoretical basis for this procedure is lacking, and in particular, most existing weighting schemes are heuristic. In this work, we derive generalization bounds for the PDA problem based on partial optimal transport. These bounds corroborate the use of the partial Wasserstein distance as a domain alignment term, and lead to theoretically motivated explicit expressions for the empirical source loss weights. Inspired by these bounds, we devise a practical algorithm for PDA, termed WARMPOT. Through extensive numerical experiments, we show that WARMPOT is competitive with recent approaches, and that our proposed weights improve on existing schemes.

Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
Jayadev Naram
Fredrik Hellström?
Ziming Wang' Rebecka Jörnsten
Giuseppe Durisi
Abstract
domains to be deployed on smaller domains, characterized by smaller label space. This motivates the framework of unsupervised partial domain adaptation (PDA), wherein the target label space a subset of the source label space (Cao et al., 2018).
In many scenarios of practical interest, labeled data from a target distribution are scarce while labeled data from a related source distribution are abundant. One particular setting of interest arises when the target label space is subset of the source label space, leading the framework of partial domain adaptation (PDA). Typical ap- proaches to PDA involve minimizing a domain alignment term and a weighted empirical loss on the source data, with the aim of transferring knowledge between domains. However, a theo- retical basis for this procedure is lacking, and in particular, most existing weighting schemes are heuristic. In this work, we derive generalization bounds for the PDA problem based on partial op- timal transport. These bounds corroborate the use of the partial Wasserstein distance as a domain alignment term, and lead to theoretically moti- vated explicit expressions for the empirical source loss weights. Inspired by these bounds, we devise practical algorithm for PDA, termed WARM- POT. Through extensive numerical experiments, we show that WARMPOT is competitive with recent approaches, and that our proposed weights improve on existing schemes.
In PDA, the use of labeled source data from outlier classes during training typically has adverse effect test performance-a phenomenon termed negative transfer (Cao et al., 2018). To alleviate this issue, several heuristic schemes to weight the source data during training have been proposed (Zhang et al., 2018; Liang et al., 2020; Li et al., 2020; Gu et al., 2021; 2024). However, a theoretical motivation lacking for most weight selections.
In this work, we provide theoretically motivated algorithms for PDA. Specifically, we derive generalization bounds on the target population loss and devise training strategies that minimize them. The bounds that we obtain involve a partial Wasserstein distance between the empirical feature distribu- tions for the source and target data (Caffarelli & McCann, 2010). This motivates the popular strategy of learning feature map that aligns source and target features, followed by predictor trained to classify the labeled source features. We will refer to objectives that are minimized with this aim as domain alignment terms. While the standard Wasserstein distance has been widely used to analyze and design algo- rithms for domain adaptation (Courty et al., 2014; 2017b), its partial counterpart is crucial to handle the existence of outliers (Wang al., 2024). Additionally, our bounds in- clude weighted source training losses, where, in contrast to all results available in the literature, the weights arise constructively from the partial optimal transport problem as- sociated with the partial Wasserstein distance. This enables a principled weight selection for addressing negative trans-
1. Introduction
In unsupervised domain adaptation, one has access to a set of labeled source data and a set of unlabeled target data, drawn from different but related distributions. The aim is to use these data sets learn predictor that performs well on new instances from the target data distribution (Redko et al., 2019; Farahani et al., 2021). In contemporary practice, it is common for classifiers that are pre-trained on large, diverse
fer that, unlike the aforementioned heuristics, comes with a clear theoretical motivation in terms of transport plans.
Our bounds come in two flavors. First, similar to Shen et al. (2018), one bound depends on the partial Wasserstein dis- tance between the empirical distributions of the source and target features. Second, building upon the work of Courty et al. (2017a), we obtain a bound that incorporates estimates of the unknown labels of the target samples. This yields a partial transport problem involving the joint empirical distri-
'Chalmers University of Technology, Gothenburg, Sweden 2 UUniversity College London, London, England. Correspondence to: Jayadev Naram jjayadev @ chalmers.se>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
bution of features and labels. Each bound depends on two parameters, which intuitively correspond to the expected portion of outliers in the source and target data sets, respec- tively. Concretely, the parameters determine the proportion of mass from each of the data sets that is accounted for in the transport problem.
Courty et al. (2014; 2017b) proposed the use of optimal transport, and specifically the Wasserstein distance, for domain adaptation. This approach was theoretically sup- ported by Redko et al. (2017) and Shen et al. (2018), who derived bounds in terms of the Wasserstein distance be- tween the source and target input distributions. Notably, this alleviates the issues of uniform convergence associated with the H-divergence. Based on these bounds, Shen et al. (2018) proposed domain alignment term, computed using Wasserstein generative adversarial network (GAN) oper- ating on empirical feature distributions (Arjovsky et al., 2017). Courty et al. (2017a) derived a bound in terms of the joint source and target instance-labe distributions, where estimates appear in place of the unknown target labels. Damodaran et al. (2018) drew inspiration from this bound to devise an algorithm using mini-batch optimal transport the joint feature-label distribution. However, it is worth noting that the bounds reviewed so far all depend on the instance distributions, and do not incorporate the learned feature map. Hence, they do not fully motivate the typical practice of computing the Wasserstein distance between the empirical distributions of source and target features.
Inspired by the bounds, we propose a novel algorithm for PDA, termed weighted and regularized minimizer via partial optimal transport (WARMPOT), whose performance we compare against otate-of-the-art (SOTA) methods.
Contributions. In this work, we derive two new families of generalization bounds for PDA, and devise algorithms to minimize them. In particular, our bounds:
explicitly depend on the learned feature map, moti- vating, for the first time in the context of PDA, the approach of partly aligning feature distributions;
yield explicit weights for source data points with a principled motivation, giving a theoretically grounded way to tackle negative transfer;
lead to algorithms that improve upon or are comparable to recent approaches to PDA;
As mentioned, important factor in solving the PDA prob- lem is to appropriately weight the source data. Cao et al. (201 8) proposed to use heuristic class-level weights based on the predictions on unlabeled target inputs. Li et al. (2020) used these class-level weights, along with a maximum mean discrepancy loss as the domain alignment term. In addition to
give rise to weights that, when combined with the ARPM algorithm of Gu et al. (2024), lead to SOTA results for the Office- data set.
Furthermore, compared to previous bounds obtained for the more restrictive domain-adaptation problem, our proof techniques directly yield bounds that depend on the Wassser- stein distance between the empirical distributions of source and target features. In contrast, in existing bounds, such as the ones proposed by Courty et al. (2017a) and Shen al. (2018), the Wasserstein distance involves the actual source and target distributions, and an additional concentration-oft measure step is required to express such bounds in terms of numerically computable empirical distributions.
training, Zhang et al. (2018) and Cao et al. (2019) determined the weights based how well a domain discriminator can predict whether given input is from the source or target distribution. Liang et al. (2020) proposed entropy-aware weights, along with an advanced alignment strategy, while Gu et al. (2021; 2024) computed weights by minimizing the Wasserstein distance between a weighted source feature distribution and the target feature distribution. Nguyen et al. (2022) used the joint distribu- tion partial Wasserstein distance the domain alignment term, but with uniformly weighted source data. Wang et al. (2024) proposed a partial Wasserstein- an extension of Wasserstein- GAN to PDA (Wang et al., 2022). Specifically, they considered a class-level weighting scheme with the 1-partial Wasserstein distance as domain alignment term. Li & Chen (2022) derived a generalization bound for PDA in terms of model smoothness, and proposed to focus on smoothness rather than alignment to transfer knowledge between domains. Fatras et al. (2021) use mini-batch joint distribution unbalanced optimal transport (UOT) cost as domain alignment term, along with uniformly weighted source data. Chang et al. (2022) consider more general setting where the proposed algorithm uses the marginals of the UOT transport plan to compute binary weights for target samples.
2. Related Work
The problem of unsupervised domain adaptation was first formalized and analyzed by Ben-David et al. (2006). They derived generalization bound in terms of the so called H- divergence, defined in terms of a hypothesis class H. This divergence is bounded and can be efficiently estimated if the VC dimension of H is finite (Ben-David et al., 2010). Moti- vated by this bound, Ganin al. (2016) proposed domain- adversarial training, wherein an approximation of the H- divergence is used as a domain alignment term. However, the worst-case nature of the VC dimension leads to bounds that are too weak to explain generalization in deep neural networks (Nagarajan & Kolter, 2019; Zhang et al., 2021).
In order to exploit the geometry of the data distributions,
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Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
Generalization bounds for domain adaptation containing weighted source loss term reported in the works of Ta- chet des Combes et al. (2020) and Luo & Ren (2024). These bounds do not use a generalized Wasserstein metric, such as the partial Wasserstein metric, as domain alignment term. Furthermore, the bounds rely on class-level weights defined in terms of unknown data distributions. These weights are then estimated using method developed by Lipton et al. (2018). However, these estimates are only guaranteed to be accurate if the so called generalized label shift assumption (Tachet des Combes et al., 2020) holds exactly, i.e., if the feature representation Z = f(X) of the instance X with la- bel Y is such that P(Z = y) = Q( = y) for source distribution and target distribution Q.
interested in determining a hypothesis w within suitably chosen hypothesis class W (to be introduced later) that minimizes the population target loss
2 (w) (xx,x) (X]
(1)
To do so, in the PDA setup considered in this paper, we have at our disposal a vector Z = (z),-.., Zn, E _",, with 2, = (zz. Ve), of labeled source instances drawn in- dependently from Pz. Additionally, we have a vector = (#1,.... ...) of unlabeled target instances drawn in- dependently from Qx which is the marginal distribution on X induced by Q2- Let Σ = (Z1,... En.) 2ne, with 2 = (ā.z ) be the corresponding vector of labeled target instances. Since this vector is not available to the learner, the learner cannot evaluate the empirical target loss
In this work, similar to Nguyen et al. (2022); Wang et al. (2024), we use the partial Wasserstein distance as the do- main alignment term. However, unlike prior work, we derive generalization bounds that both corroborate the choice of domain alignment term and lead to a theoretically motivated weighting scheme for the empirical source loss. Further- more, unlike some of the empirical weighting methods used in the literature (e.g., class-level weights), our approach readily extends beyond classification settings.
1 _ Thg
Ls(w)
Σ ((w(Zz).
(2)
To overcome this issue, we will obtain upper bound on this quantity in Section 3.2. This bound contains partial Wasserstein distance term and a weighted version of the empirical source loss, in which the weights are a function of the optimal coupling measure IT obtained when solving the optimization problem in the definition of the partial Wasserstein distance. This definition is provided below.
3. Theoretical Results
Definition 3.1 (Figalli, 2010, Eq. (2.1), Caffarelli & Mc- Cann, 2010, Eq. (1.8).. The partial Wasserstein distance with parameter between two measures² Px and Qx on (X, Σx) is defined as
We now present our main theoretical results. First, in Sec- tion 3.1, we formalize the problem setup and introduce the notation. In Section 3.2, we obtain bounds on the empir- ical target loss for fixed sample, which leverage to obtain generalization bounds in Section 3.3. Some useful definitions and results are recalled in Appendix A.
inf PW (Px, (PTx (x) / c(x,,) 3) (l(,, (3)
where e X' x X + R+ is the so-called cost function (typically a metric) and [= Qx) is the set of all non- negative measures II on X x X' for which I( x 2) a and, for which, for all measurable sets A, B C X, we have that (I x X) I (A) and II(X' x B) S Q (B).
3.1. Problem Setup and Proposed Approach
We next introduce the notation that we will use throughout this section. We let Z X' x y be the source domain, where X' C I is the input space, equipped with the sigma- algebra Σx, and y C TR is the source label space, equipped with the sigma-algebra Ny. We consider a joint probability distribution Pz on (z, Σx x X Ey), called the source distri- bution. Similarly, we let Z X' x y' be the target domain, where Y equipped with the sigma-algebra Ey, is un- known subset of y/ Furthermore, we introduce second joint probability distribution Q2 on Ex Ex @ ) termed
For the case in which the measures of interest are discrete and supported on m and n mass points, respectively, it is convenient to express Px and Qx x as vectors of dimension m and n. and the (coupling) measure as an m X nonneg- ative (coupling) matrix IT with entries IT; Then, one can rewrite (3) as
the target distribution.' !
min IPW, (Px, (Px (Px =1
cla,, Uy (4)
hypothesis is a measurable function w : + y, In order to discuss feature alignment, we express each hypothesis as g o f, where is a feature extractor and g is a classifier. Throughout the paper, will for simplicity consider bounded loss functions é: y x y + [0, 1j. We are
where [a ((Px,Qx)= is the set of nonnegative matrices that
satisfy² 111, Px. 11 S <Qx and 1 11, a.
We do not require that the two measures are probability mea- sures. In particular, in our setup we will have Px 2 1. 'Here and throughout the paper, vector inequalities should be interpreted entry-wise.
In the remainder of the paper, we will not specify sigma- algebras if they are clear from the context.
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Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
3.2. Bounds on the Empirical Target Loss
with II being the optimal coupling matrix in the definition of a 3 Pd Q$). Finally,
Next, we present our main theoretical results: two bounds on the empirical target loss (2). Generally speaking, the bounds consist of four terms: (i) weighted average of the loss computed on the labeled source instances, (ii) partial Wasserstein term, (iii) a total variation term that allows us to make the bound explicit in the empirical target loss, and, similar to most theoretical bounds for domain adaptation available in the literature (Ben-Davite et al.. 2006; Courty et al., 2017a; Shen al., 2018), (iv) non-computable term that dictates the difficulty of the PDA problem under consideration.
b = =mm max (()((( 9'E9
(8)
=E
(2.9)
where 9 denotes the set of classifiers g' associated hy- potheses in W. and where, with an abuse of notation, z U ž denotes the set of all labeled source and target instances.
The proof of Theorem 3.2 provided in Appendix B. The weights p; and qj arise constructively from the partial trans- port problem corresponding to IW ( p Q.). The under- lying intuition is that, when partially transporting the source data to the target data, the source samples that play the dom- inant role in the transportation plan should be the ones that are most similar to the target samples. Conversely, outliers are expected to be nearly ignored, leading to small values of the corresponding pa and 9j. The connection between this transportation view and the actual prediction problem of interest is formalized in the proof of the bound, detailed in Appendix B.
Drawing inspiration from Shen et al. (2018), we first present a bound in Theorem 3.2 in which the partial Wasserstein distance is between the empirical distributions of the source and target features. Then, drawing inspiration from Courty et al. (2017a), we extend it in Theorem 3.3 to the case in which the partial Wasserstein distance is between the joint empirical distribution of source features and labels and the joint empirical distribution of target features and predicted labels. While the natture-based approach can capture covariate shift, where only the marginal distributions on the input differ, a joint distribution- based approach is beneficial in the case of labeling distribution shift, i.e., when the conditional distribution on labels given inputs also differ. Both bounds are in terms of the IPW,, distance. Partial domain adaptation is achieved by inflating the empirical source distribution by a parameter 1/3, where 0 < B I Intuitively, the parameter 3 relates to the fraction of source instances we want to associate to target instances, whereas the parameter a corresponds to the fraction of target instances we want to consider. Hence, / allows for partial domain adaptation, while can be used to avoid outliers in the target set.
Note that when = 1, we have that 9j = 1/n, and, hence, the third term on the right-hand side of (5) disappears. Set- ting a 1 is reasonable if we do not expect any outliers in the target set, and, hence, we want to consider all target instances equally. In contrast, a < and B = 1 may be suitable for the so called open set adaptation problem, where the target label space includes additional classes beyond the source label space (Panareda Busto & Gall, 2017).
Next, present an analogous bound, in which joint feature- label distributions are used in place of feature-only distribu- tions in the partial Wasserstein term.
Theorem 3.3 (Joint distribution based bound). Assume that the loss function E is a metric on y and Ç-Lipschitz in each argument. Consider the set W of hypotheses w 9 f for which g is "-Lipschitz with respect to the Euclidean distance. Let P = 1 -i= 8 and Qr = 1 Σドn )1(2)),0(E,)) be the empirical joint source the and estimated joint target distributions, respectively, for the hypothesis w = g J. Then, for all w E W and all a E (0.
Theorem 3.2 (Feature-based bound). Assume that the loss function E is a metric on y and consider the set W of hy- potheses w- = g o f for which g is -ipsschit with respect to f. Let pf- nh be the empirical source and target feature distributions, re- spectively, with feature extractor f. Then, for all w E W
1 Ei. f(z)) and Q = -
n. EiE 87(2,)
and all a, 3 E (0, 1).
Lg(w)= E
R2
Pc (w(()))))) 91) + 2 Pwa (1P/.0.) Q!)
Lg(w)=
S
Di Elv(())) yi)
-
1 IP,, (HP! QF
a
a
1=1
The
n1.
+ i. (9)
+ +2L,, (5)
2
j=1
n
a
2
j=1
n
where the cost function in IPWa is ē(x,,) = 711((x)
where the underlying cost cost, function for IW, is
fi)]], and the weights {pi} and {ag} } are given by
c((x, y), (ž. 7) = (x1lf(x) - f())l + ely, W). (10)
p:- 1ml' i= 1, n2
(6)
the weights { and } are given by
Φ = (cr) j == I,...,n.. n2g
P. = 11'',,)) i= 1,..., n
(11)
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Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
i = (((r))))))) ;=1,...,
(12)
(Z, 2)
with To being the optimal coupling matrix in the definition of F 3 P1. 28)
A
L (W)] S E (R(W, 2, 2) + 8mz
and
+ DRI(Pwiz.r Qw) + log x
(15)
i (( }
f = min D'EG
+
(13)
a
where DKL denotes the Sellback-Leibler (KL) divergence.
with E given in (63) (see Appendix C).
We provide the proof of Lemma 3.4 in Appendix D. Note that, in Section 3.2, we derived functions that can be combined with Lemma 3.4 to yield generalization bounds for PDA. We present these bounds below, beginning with a feature- based bound, obtained by combining Theorem 3.2 and Lemma 3.4.
The proof of Theorem 3.3 is provided in Appendix C. Note that the cost in (10) coincides with the one proposed in Courty al. (2017b). However, is important to note that the values of the weights can differ between the two bounds. Indeed, while the weights are given by similar expressions, the underlying optimal coupling matrix differs in general. The same considerations on the role of the weights detailed after Theorem 3.2 also apply to Theorem 3.3. Furthermore, while both Ly in (13) and Ly in (8) relate to the difficulty of the PDA problem under consideration, they are incom- parable in general. Finally, some terms in (5) have extra factor of 2 compared to the corresponding terms in (9), and the underlying cost function in the partial Wasserstein dis- tance in (9) has an extra term. This, in addition to the slight difference in assumptions on the loss, means that the two bounds given in (5) and (9) cannot be compared general beyond the discussion above.
Corollary 3.5. Suppose that the assumptions of Theo- rem 3.2 and Lemma 3.4 hold, and consider the same nota- fion as used therein. Furthermore, denote the decomposition of the hypothesis W as W = G F. Then, for every choice of X > 0 and d E (0, 1), with probability at least 1 - over Z p Z 2
EPwiz Lo (W)) B+
[W)]=B+EPwr) B+EPWIZIE
Pi Σ
e(W(X.), Y.)
2 + p. (bP5 QF) +
11
+2 (16)
a
nte
Q
3.3. PAC-Bayes Generalization Bounds
where (X,, Y.) = Zi are the entries of Z. we let S denote the projection on X of Z, and we use the shorthand
In Section 3.2, we derived bounds on the empirical target loss for a fixed hypothesis in the PDA setting. However, these results cannot be applied directly to learned hypothe- ses. To proceed, we will use the PAC-Bayesian approach (McAllester, 1999; Catoni, 2007), which will allow us to obtain loss bounds for a learned hypothesis. These bounds hold with high probability over the choice of the training source and target samples.
A (Pu(P Qw) + log 1 à 8n4
B
(17)
X
Next, we present a joint distribution bound, which follows by Theorem 3.3 and Lemma 3.4.
Corollary 3.6. Suppose that the assumptions of Theo- rem 3.3 and Lemma 3.4 hold, and consider the same nota- tion as used therein and in Corollary 3.5. Then, for every choice of X > 0 and 8 E (0,1), with probability at least 1- over
While wide array of PAC-Bayes generalization bounds are available (Alquier, 2024; Hellström et al., 2025), we restrict ourselves to the following one for simplicity.
over Z ~ p Z ~ 27
Lemma 3.4. Suppose that there exists a function R: W x 2n x n such that, for all (w, 2, x) W x n nn,
The (( (X,)),), Y2)
E Erwiz.r [qq
(W)( B E
Lg(w) S R(w, 2, 2).
(14)
+ 1 PW.
P
F Q# 2:
1
1 14
4
+LF
(18)
Let Qw be a prior distribution on W and Pwiz,T a poste- rior distribution on W given the labeled source samples* Z - and the unlabeled target samples T. Here, T is the projection on x of 2 ~ Q7 Then, for every fixed >>0 0 and d E (0,1), with probability at least 1 - over
2
The proof techniques that we use to derive these generaliza- tion bounds yield several key advantages compared to prior results obtained in the more restrictive context of domain adaptation by Shen et al. (2018) and Courty et al. (2017a). First, when relating the loss in the target domain to the loss
WWe denote by P2. the fold product of Pz. Similarly, Q7 stands for the foldd product of Qz.
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Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
in the source domain in Section 3.2, we work directly with the empirical measures. Consequently, the partial Wasser- stein distances in our results are fully empirical. In contrast, in earlier derivations for the case of domain adaptation, the Wasserstein distance is between population measures, which need to be related to their empirical counterparts via step. This step adds an addi- tional term the bound, the need for which is obviated by our approach. Furthermore, while most existing domain- adaptation algorithms compute domain alignment terms the basis of learned feature distributions, the underlying generalization bounds available in the literature depend on the fixed input distribution. In contrast, our bounds explic- itly depend on the learned features, directly motivating the feature-based approaches most commonly used in practice.
of WARMPOT. WARMPOT with = 1. Here, the source distribu- tion is scaled so that its total mass is 1/3 > 1. Of this mass, unit is aligned with the entire target distri- bution, whose total mass is 1. The PWAN algorithm (Wang et al., 2024) uses this alignment approach along with the heuristic class-level weights of the BA algorithm (Liang et al., 2020), detailed in Section 5.3.
a amncentraiionof-meaasss
The proposed WARMPOT algorithm can then be inter- preted as aligning fraction of the f-scaled source dis- tribution with an fraction of the target distribution. The use of two parameters a and / allows for asymmetry in this domain alignment process, which is necessary the proportion of outliers differs between the source and target data sets.
4. Algorithms for PDA: WARMPOT
5. Experiments
Now, motivated by the bounds on the empirical target loss in Section 3.2, we propose family of algorithms for the PDA problem. Specifically, focusing on the first two computable terms of (9) in Theorem 3.3, we consider the following optimization problem:
We now experimentally evaluate our proposed algorithm, WARMPOT, for PDA tasks. Specifically, in Section 5.1, we detail our experimental setup. Section 5.2, we discuss the implementation details of WARMPOT. In Section 5.3, we investigate the impact of our proposed weight choice. Then, in Section 5.4, we compare WARMPOT against existing PDA methods. Finally, in Section 5.5, we visualize the weights used in WARMPOT and provide insight into their role. Additional details on the experiments are provided in Appendix E.
min Σ (all((as)))) + PW (3e5 )
(19)
Here, the cost function in the definition of the partial Wasser- stein distance is c((z, y), (ž, W)) = Ta (x)) - f())l + (p. 8)) and the weights PJ are given by (11). By set- ting 1/2 = 0, we observe that P P/. Qt:) reduces to PWW ( Pl. Qf), which is the domain alignment term appearing in (5) in Theorem 3.2. Thus, the optimization problem in (19) allows to draw inspiration from both Theorem 3.2 and Theorem 3.3.
5.1. Setup
In the experiments, we focus on the Office-Home data set (Venkateswara et al., 2017), which consists of images of 65 objects belonging to different domains: Art, Clipart, Product, and RorldWorld. To construct a PDA task, we con- sider source data set consisting of all labeled samples from one domain and a target data set consisting of unlabeled samples from the first 25 classes of another domain. We consider all 12 possible combinations of source and target domains. This PDA setup has been widely studied and was considered by, among others, Nguyen et al. (2022); Wang et al. (2022); Gu et al. (2024).
The parameters 7/ and 172 determine the impact of the inter- feature and inter-label distances respectively, which act as regularizers, while a and 3 control the alignment between the source and target distributions.
We refer to an algorithm that minimizes (19) as weighted and regularized minimizer via partial optimal transport (WARMPOT). To clarify the role of the parameters and the relation to prior work, we discuss two extreme cases:
5.2. Implementation of WA MARMPT
WARMPOT with 3 = 1. In this case, the partial Wasserstein term aligns an a fraction of the source distribution with an a fraction of the target distribution. This is reminiscent of the MPOT algorithm, which uses a batih approximation of PW solve the PDA problem (Nguyen et al., 2022). The key differ- ences are that (i): MPOT uses a different cost function, where the inter-feature cost is given by the squared dis- tance [z) - f and (ii): MPOT uses uniform weights for the source sample losses, rather than the Pi
We consider hypotheses w = go ʃ consisting of a ResNet50 (He al., 2016) feature extractor f pretrained on ImageNet (Russakovsky et al., 2015) and a fully connected network with hidden layer of dimension 256 as the classifier g. We solve (19) using stochastic gradient descent, and compute PW. P QT) for each mini-batch following the method proposed by Nguyen et al. (2022). Throughout, we set to be the cross-entropy loss and the parameters of the domain alignment term to (a, B) = (0.8. 0.35). The values of all
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Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
Table 1. Test accuracy on the Office-Home dataset using the weight choices described in Section 5.3.
Woighting schemee
A-C
A-PP
A-R
C-AA
C-+
C-R
P-A
P-C
P-R
R-A
R-C
R-+P
Avg
MPOT weights
(2.2(114)
(1.0)
(16)
(1.0
77003.1)
(1.3
61.7 61.7 (1.3)
(0.9)
(1.5
80.0 (0.7)
65.9
8388 (0.5)
(0.4)
BA-US weights
(1.5
82.6 (2.0)
(9.4
75.5 (0.8)
(64
(0.2
(6.(1.))
81.3 (L.1
67.4
86.7 (0.9)
(0.4)
ARPM weights
(1.9(1.0)
75.8 (0.6)
(0.4)
(9.7
737(1.4)
(..2
(2.4
(2.6)
81.2
80.5 (0.6)
(0.8)
83.9 (0.7)
72.9 (0.3)
WARMPOT (ours)
62.5(1.2)
(1.1)
(0:3)
(1.1 (22
82.3(1.3)
(1.6
(3.1 88.(1.0)
81.1
(1.0
86.6 (0.6)
77.6 (0.7)
other hyperparameters are provided in Appendix E. The results of a sensitivity analysis on and / are discussed in Appendix G.
similar experiment using the ImageNet + Caltech data set (see Appendix F) confirm these findings. Interestingly, the ARPM weighting strategy can be seen as a variation of our weighting strategy. We discuss this connection in further detail in Appendix H.
5.3. Comparison with Existing Weighting Schemes
In this section, we evaluate our weighting strategy. Specifi- cally, we compare our choice for the weights in (11) with the following weighting schemes:
5.4. Comparison with State of the Art
Next, we compare the performance of WARMPOT with the performance of alternative algorithms proposed in the literature for the Office-Homs data set. The results of our analysis are detailed in Table 2.6 First, we focus on compar- isons to algorithms that, similar to WARMPOT, rely cost function of the form given in (19). Specifically, we con- sider MPOT (Nguyen et al., 2022) and PWAN (Wang et al., 2024). As shown in the table, WARMPOT achieves higher average test accuracy compared to PWAN and similar to that of MPOT.
MPOT weighting strategy (Nguyen et al., 2022), with uniform weights p.: = 1/n,-
US weighting strategy (Liang al., 2020), where
11 [ = wil/nz.
P
(20)
Here, Jty is a target instance and yi is the label of the ith source instance 24. ARPM weighting strategy (Gu et al., 2024), where
In the same table, broaden the comparison to wider range of algorithms. As shown in the table, among the available algorithms, the best performance on Office-Home is achieved by ARPM (Gu et al., 2024), which relies on several heuristic loss terms beyond the weighted source loss. Motivated by our results in Section 5.3, which indicate that the WARMPOT weighting strategy yields better perfor- mance than that of ARPM, we also consider an algorithm that is identical to ARPM except that it uses the WARM- POT weights. We refer to this approach as ARPM+our- weights. Interestingly, ARPM+our-weighh improves upon SOTA performance. This indicates that our theoretically motivated weighting strategy can lead to gains for PDA al- gorithms that involve weighted source loss. We detail the choice of hyperparameters used to obtain the test results of ARPM+our- weights in Appendix E.
min W1 PEA 0.20
nhe Σ
po))(
1
Σ
(21)
ntt 1=1
Here, p- = ) and A is given by
1
p
p. 1,
pi
<
(22)
n2
n2
1=1
with being hyperparameter.
The results are presented in Table 1. Note that, in our weight- ing strategy, the weights are computed for each mini-batch at every iteration. On the contrary, in BA US and ARPM, the weights are computed on the entire dataset but only every n iterations. We set this weight update interval to be n 500 for BAAUS and ARPM as suggested in Liang et al. (2020) and Gu et al. (2024) respectively. The experiments are re- peated for 6 random seeds, and we report the average and the standard deviation. As seen in the table, WARMPOT results in better performance than MPOT and ARPM, and yield performance comparable to BAA UUS This illustrates that the weighting strategy suggested by the theoretical re- sults reported in Section 3 is indeed effective. Results from
5.5. The WARMPOT Weights
In order to illustrate the behavior of the WARMPOT weights, we focus on single task from the Office- data set, namely, the P-+A task. In Fig. 1, we present the weights P. in (11) obtained by evaluating the transport ma- trix fi achieving PW. ( PL. QT) over the entire data set,
TThe test accuracy scores accompanied by standard deviation in Table 2 are obtained by reproducing the results reported in the corresponding papers.
'Open source Python implementation of WARMPOT: htttps://githune httpp:///ggttub...///aaaa22221..
7

---

Thcoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
Table 2. Test accuracy on the Office-Home dataset.
Algocillom
A-4C
A-1P
C-AA
C-4P
C-RR
P-1A
P--C
P-4R
R-AA
R-C
R-sP
Avg
ResNe:-0 et 201,) 2016)
46.3
67.5
75.9
99.1
59.9
62.7
58.2
41.8
74.9
67.
48.2
74.2
61.4
ADDA (Trong et 2017)
45.2
68.8
79.2
64.6
600
68.3
57.6
38.9
77.5
70.3
45.2
78.3
62.8
CDAN+ (Long et 2015)
47.5
65.9
75.7
57.1
63.4
39,6
44.3
72.4
49.9
72.8
60.7
IWAN (Zhang 2018)
53.9
54.5
78.1
61.3
48.0
63.3
54.2
52.0
$1.3
76.5
56.8
$2.9
63.6
PADA (Cavd al. 2018)
$2.0
67.0
78.7
52.2
53.8
$9.0
$2.6
43.2
78.8
73.7
$6.6
77.1
62.1
ETN (Clo 2019)
59.2
77.0
79.5
62.9
75.0
55.4
54.4
75-7
57.7
84.5
70.5
DRCN (Li 2020)
56.0
76,4
$3.0
62.1
71,0
70.8
49,8
80,5
77.5
59.1
79.9
69.0
BA'US (Liang 2020)
60.6
83.2
58.4
71,8
728
83.4
75.5
61.6
86.5
79.3
02.8
36.1
76.0
ISRA-BAUUS (Xin 2021)
64.7
83.0
$9.1
75.7
75.5
85.4
78.5
64.2
63.1
81.3
65.3
86.7
78.2
SHOT++ (Linng 2021)
65.0
85.8
78.8
87.3
66.0
79.9
SPDA (Giu et 2022)
64.2
87.8
S3.0
74.3
75L1
79.1
79.4
58.9
KS.1
M1A4
67.4
34.1
77.1
APDA (I 2022)
61.7
86.9
90.5
77.2
76.9
83.8
79.5
63.8
63.5
85.0
65.8
$6.2
78.8
(Yang 2013)
85.6
75.6
86.8
67.4
79.5
RAN (Wu ot al., 2023)
63.3
83.1
99.0
75.0
74.5
82.9
78.0
61.2
86.7
79.9
63.5
35.0
76.8
JUMBOT (Fatnas al., al21) 2021)
62.7
77.5
34.4
76.0
73.3
80.5
74.7
60.8
85.1
80.2
66.5
83.9
75.5
STCPDA
2003)
87.8
77.2
85.6
79.5
SLM (Sahoo et all, 2023)
61.1
$1.0
91.4
76.5
75.0
81.8
74.6
55.6
87.8
82.3
57.8
83.5
76.0
SAN++ (Cro al., 2023)
61.3
81.6
88.6
72.8
76.4
81.9
74,5
57.7
K7.2
79.7
63.8
86.1
76.0
IDSP [L. Chen, 2022)
60.8
80.8
57.3
69.3
M0.3
74.7
59.2
83.3
77.8
61.3
85.7
74.9
MOT (Lao Ren, 2023)
63.1
86.1
92.3
78.7
85.4
89.6
79,8
62.3
89.7
83.8
67.0
89.5
80.6
AR (Gl et al., 2021)
67.4
85.3
90.0
77,3
70.6
85.3
79.0
64.8
89.5
80.4
66.2
86.4
78.3
ARPM et ol., 20144
(2.5)
(1.4
(2.7
79.9 (2.1)
(2.6)
(7.0
(3.1))
(9.6
89.6 (0.3)
(1.0)
69.5 (2.2)
89.7 (1.1)
81.7 (0.7)
MPOT (Nguyen et all, 2022)
64.6
80.6
87.1
76.4
77.6
83.5
77.0
63.7
87.6
81.4
68.5
87.3
73.9
PWAN (Kling it all, 2034)
63.3 (1.8)
84.1 (1.8)
(0.6)
16.7 (1.1
(2.0)
83.8 (1.89
76.6
(2.7
86.7 (0.8)
80.1 (0.6)
64.4 (0.5)
86.6 (0.7)
77.3 (0.5)
WARMPOT (ouns)
62.5 (1.2)
(1.0)
89.5 (0.3)
(1.2
82.3 (1.3) (2.2)
(1.5()..))
88.0 (3.4
81.1 (0.7)
66.5 (1.0)
86.6 (0.6)
77.6 (0.7)
ARPM weights
69.0 (1.9)
87.2
(0.7)
(1.0(1.)
(3.4
(6.0
(9.9(2.2)
69.1 (0.5)
90.2
86.6 (0.8)
69.7 (2.3)
88.7 (1.0)
82.0 (0.4)
at the end of the training process. For illustrative purposes, we normalize the P. by 1/(8n, ) to obtain values in the in- terval [0, 1]. The left plot in Fig. shows the distribution of all weights on source instances. The middle and the right plots show the distribution of the normalized weights on the shared and the outlier class instances, respectively. The weight proportion assigned to outlier source samples in WA WARMPOTT is just 6.04%, even though 58.41 1% of the source samples belong to outlier classes. A total of 64.4% outlier class instances have been assigned to the smallest bin, which helps in minimizing the effect of negative transfer.
This, along with an additional ablation study, corroborates the utility of our proposed weights.
It should be noted that an exact minimization of our bounds is prohibitively expensive from a computational standpoint, and hence, WARMPOT relies on some approximations. Additional performance gains may be obtained by optimiz- ing this implementation. Furthermore, the SOTA algorithm ARPM includes additional loss terms that aim to reduce pre- diction uncertainty and improve robustness. Such quantities are not explicitly present in our generalization bounds, but are studied by Gu al. (2024, Thm. 1). A promising direc- tion for future research is explore these aspects within theoretical framework, potentially enabling more powerful algorithms.
6. Conclusion
In this work, we obtain generalization bounds for PDA tasks. In particular, our bounds depend on a partial Wasserstein distance, and hence provide a theoretical motivation for us- ing as domain alignment term. While several existing algorithms in the literature take such an approach, a theo- retical justification was previously missing. Furthermore, our bounds constructively give rise to explicit source data weights, which can help alleviate negative transfer. In con- trast, prior work used heuristic weight choices, which were not directly motivated by theoretical considerations.
A Acknowledgements
This work was partially supported by the Wallenberg AI, Au- tonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. The compu- tations were enabled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS), partially funded by the Swedish Research Council through grant agreement no. 2022-06725.
Inspired by our bounds, we propose the algorithm WARM- POT to minimize them. Through numerical experiments, we demonstrate that WARMPOT is competitive with recent approaches to PDA. Furthermore, we show that the perfor- mance of the SOTA algorithm ARPM is improved when its weighting scheme is replaced with that of WARMPOT.
Impact Statement
This paper presents work whose goal is to advance the field of machine learning. There are many potential societal consequences of our work, none which we feel must be
8

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Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
Shared classes
Outlier classes
All weights
Percentage of total weights = 93.96%
Percentage of total weights = 6.04%
1,0000
1,000
1,000
800
6000
600
600
400
400
400
200
200
200
0
0.2
0.8
Figure 1. The distribution of WARMPOT weights for the task AA Most of the weights of the outlier classes are close to zero, suggesting that most of the outliers successfully omitted when training the classifier.
specifically highlighted here.
IEEE Trans. Pattern Anal. Mach. Intell. (PAMI), 45(2): 1766-1780, Mar. 2022.
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Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
A. Preliminaries
recall some definitions that are used in the main text as well as the appendices. will also establish useful lemma. Definition A.I. A function p= X' x X' + IR,, is called a metric on 2' if it is real-valued, finite, and nonnegative, and if for all a, b, 6. E X:
(i p(a, b) = 0 if and only if a = b, (ii p(a, b) p(b, a) (symmetry).
(iii) p(a, c) - p(a, b) + p(b, (triangle inequality).
We shall also the so-called reverse triangle inequality,
lp(a, b) pla, c(b,c)
(23)
which can be readily obtained from the properties Definition A.I.
Definition A.2. A function IR + +''is is --Lisschitz with respect to a metric on y for all al gd
p(g(t), g())) I >/lt - +'Il-
(24)
Definition A.3. The total variation distance between two discrete probability distributions P and Q on Z is defined as
TV(P,Q) = Σ PP()) Q(=))
(25)
=EZ
Lemma A.4. Assume that the loss function 8: 3 y + R4 is metric on Y. Let W be the set of all hypotheses g such that g is q-Lipschitz with respect to l. Let f and U E be defined as in Theorem 3.2. Then, for all pairs (x. y) and _ (I, 35) in U 2, we have
[[(w(x).)) ((w(E), 0) $ 2)1lf(=) f())l 2Ly.
(26)
Proof. Consider two arbitrary pairs (x, y) and (#, in U Furthermore, fix f and let g" be a classifier achieving L, in (8). Then,
=(gg)))))((
(27)
(28)
(29)
=(g +L,
(30)
(31)
<>]l/(x) [())1)-)(00//()()(((()(6))-0011-
+
(32)
$ 2>)]f((x) [())))+)+(((((())))())))))))))))
(33)
$ 2x]]f()) [()) ((o(((())) + +(. (f())) + L
(34)
$ 2x) /(*) + ((w(())))) 2L.
(35)
Here, (27) follows because w =0 f: in (28) we used the triangle inequality; in (29) we used the max-min inequality (Boyd Vandenberghe, 2004, Eq. (5.46) as well as the definition of Ly: in (30) we again used the triangle inequality; (31) we used that g is Li-iicchit (32) we used the triangle inequality: in (33) we used that g is >-iischhit, (34) we used the triangle inequality; and finally, (35) follows again from the max-min inequality and the definition of Ly. Similarly, starting with (w()), 17) and proceeding analogously, we conclude that
((w (5) ). < fl f()))l f(w(x),)) v) +2L,.
(36)
Combining (35) and (36), we obtain the desired result.
12

---

Thcoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
B. Proof of Theorem 3.2
Note that there may be duplicate features in {f(z)) and {= Hence, strictly speaking, P/ and Qf are probability vectors whose dimensions are given by the number of distinct features, and multiplicities need be accounted for. However, in our proof, this yields the same result as we treat the duplicate values as separate features with identical cost values. Hence, for simplicity but without loss of generality, we assume that the features in both {f(x)) and {f(i,)) distinct. This allows us to view Pf/ and Q as probability vectors of dimensions n, and ny respectively, where all entries of each vector are equal, i.e., P = _ n,
and Q = - IT
Now, define the nh X n4 cost matrix with entries Ctp = fx,) - f(,))l, We consider the partial Wasserstein distance between 1 3 P and Qf, which is given by (see the definition in (4))
FW (BP).0.) NEr. (f
min
Ciyllyy
(37)
am jm
where To ( = {IT
E Rnax. I1, S spp 11 I, s 0 1h, 11 = =a..
Let II* ETa jpf. Q!)
attain the minimum in (37). Then
2PW
3
1
ps. of
20 11
(38)
n.
2711f(x;)- - f(
(39)
(le(w(())) - ((wo(z)) - -2L,)
(40)
((w(G))))) ((w(xx)) u.) -2L
(41)
mi T
The
{1o(T,))
((w(z)) 21:) Σ I1; 22,
Σ>
I
(42)
i=1 j=1
(w)(,,) 0y ((I+)) 1In.) (1) 22, 11
(43)
((((ū)) (lo(x:) M.p: 2L
(44)
Here, in (38) we used (37); in (39) we used the definition of the cost function; (40) follows from Lemma A.4; and finally, in (44) we used that 1", II',,, = a by definition, and we set p. = (II*,,, )is i = 1...., n2 and qj = ((II))
j = 1,..., ny. The inequality just obtained can be rewritten as
Σ ((w(Z)))ūua)
(woas) + 2PW a
( + 2abs.
1
(45)
Now, let q = =[g1... .g....., In.]" and define the following empirical distributions on
The
Q:
82
Q4
Li 82,-
(46)
Using that the loss function is supported on [0, 1). we perform the following change of measure (Ohnishi & Honorio, 2021, Lemma 4):
E ExX,Y) (( V)) s & (xxx))-g((((XX),.. TTV(+,, Q9)
(47)
13

---

Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
where the total variation distance ( introduced in Definition A.3. This implies that
2 or, equivalently,
R2
(w)(3,) -
((w(2,). 0,)g TV(Q2. +TO: Q
(48)
1-1
LE - -
Σ
n2
(w((z)).0) + TVQQ2, Q2)
(49)
01 j=l
Finally, we note that
1
TV(Q..09) =
(50)
74
We substitute (50) in (49) and (49) in (45) get the desired result.
C. Proof of Theorem 3.3
As in Appendix B, we assume for simplicity that we can view Pf and Q" probability vectors of dimensions and na respectively, where all entries of each vector are equal, i.e., P [ 1 and Q = - 1
now consider an n, X ny cost matrix C' with entries Ct5- = (x)) - f(,,)) + E(v,, w((z))). We consider the partial Wasserstein distance between 1/2 P and QF, which is given by (see the definition in (4))
n1
PW, (4P.00)
min neT. ( (L
Ciyllyn
(51)
j=1
where r. (3P/ Q+") = (1I E R
111,, S P S QF, 1 111,, = a). We define Q2 and Q3 in the
same way as (46), where a = i?..... "a.....,,,,, 2n Then, given the feature map f, for every fixed hypothesis w' E W that can be decomposed as w' g' f. we have
aL=(w) S <TV((Q, Q +
g ((((w())).
(52)
j=1
<aTV(QQ, Q
i (l(w(T)), w (z,))) E(w (1)))
(53)
The
2 Σ
=aTV Q9)
Σ p (xx), vi) > (w (5z). + (
(54)
Here, (52) follows from (49); in (53) we used that the weights { are nonnegative as well as triangle inequality; to
obtain (54) we just summed and subtracted the term E pul(ws (xx), y:). We now focus on the last two terms of (54). Let
14

---

Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
I1" be the coupling matrix achieving 1 (5P2. (
Q+)
We have
Dee
D2
(fl(w(ōz), w/ (5,)) Σ
pil(w' (
Σ
(w((T,), w (53))
(w'((x)) (24)) yi) Σ Mi
(55)
M (Aw(())) (a,)) E(w (2x)); vi))
(56)
ity (w((ū,) w (2,)) ((w (zi).
(57)
m [(w(T)), w
(58)
+ +1((w(a)). w (z:) (w (2.). 2t)l
(59)
my (T)l)((;)))))) - s(;))1 -(=)) +(=)))+
(60)
= FW.
P 21 Q"
(61)
Here, (55) follows from the definitions of f; in (11) and in (12); (59) follows because the loss is Ç-Lipschitz and because of the reverse triangle inequality (23); and (60) follows since g' is 7- --Lipschitz with respect to the Euclidean distance.
By substituting (61) into (54) and decomposing w' as w' = g' o f. we obtain
1 LE(w)= S -P..
1 P/. 0r + TV(Q2, Q + Σ
pe eld ((x))))))) v)) +
Σ L (()
(62)
1=1
Next, we define
nhe
L
Σ (( 1=1
= min 6'6g
ll (f(e)))) )) (63)
nhe
P ((g(((()))) vi)
min
+ min g'EgT
g'Eg
a
a
We now minimize over g' in the two summations of (62), and note that
The Ps Elg' Elg
min y'Eg
(f(==)))
Σ (
} +E (64)
i
((((s(1)))
min g'Eg
We obtain the desired result by recalling that (xf()))) = w(xx). by using the definition of Ly. and by writing out the explicit form of (VQ Q2), as per (50).
15

---

Thcoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
D. Proof of Lemma 3.4
The proof follows that of Alquier (2024, Thm. 2.1), with adjustments to account for the non-standard posterior and the use of R(W, Z, Z) in place of the empirical loss.
For fixed w E W, we let w = Lag (w) E( (X)) : Note that the ((w((x)),) are supported on an interval Yj). of width 1, are independent and identically distributed, and satisfy E[e(w(X,)) Y2)] L (w). Hence, we can apply Hoeffding's ' inequality (Wainwright, 2019, Prop. 2.5) to find that, for every > 0,
E Ez ( 1 Ses
(65)
Now, note that Σラ= U1(0] = -n,(Lez (w) - L (w)). Hence, setting t = A/n, for some >> 0, we have
R 2 6 (Tazz (1a2(())(12(10)) S
x²
8mz
(66)
Now, it follows from the upper bound in (14) that
E Ez- (
(1 (w) (()
< Eg Zz e (1)
(67)
By combining (66) and (67) and averaging over Z ~ P2 and W ~ Qw we find that, collecting factors on the left-hand side,
(1)
E Zz 2n 2- WwQQw
Bny
< 1.
(68)
In the remainder of the proof, we will suppress the explicit distributions in the expectation notation when they are clear from context. We now apply the Donsker- Donsker-Varadhan Varadhan variational formula (Alquier, 2024, Lemma 2.2) to conclude that, for a given
posterior Pwiz,r.
2, AB PwI.-
-Pa 11Q
22 Bng
< 1.
(69)
Next, by the Chernoff bound (Wainwright, 2019, Eq. (2.5)), we have that for every
22
P 2.2 AEW AEW-PuizrlLog (W) - R(W, z, 2) D (PwIz,r Qw)
8m
[ (BW w -k...., Q[ -
< E 2,2
(70)
S -
(71)
Setting log(1/0), we thus conclude that with probability at most over Z Pm, 2 277
X²
AEW-Pwij.. (W) R(W, Z, 2)] (KL((wuz.pppp [QQ)
log 8*
(72)
8nt
We obtain the desired result by considering the complementary event and re-arranging terms.
E. Additional Details on the Experiments
We compute the domain alignment term (W..PP in (19) using the entropic partial Wasserstein solver from the POT library (Flamary al., 2021), with regularization constant 7.0 in all the experiments, which is selected avoid numerical instabilities. We set the maximum number of iterations to 5000. Following Nguyen al. (2022), we linearly increase from 0.01 to Omax for the first 2500 iterations, and keep it constant for the last 2500 iterations. Through parameter search, we obtained the following values for the hyperparameters: max = 0.8, 771 = 0.125, 72 = 1.75, 3 = 0.35. We use a batch size of 65, and set the learning rate of stochastic gradient descent to 0.001. We used the same values for these hyperparameters in all our experiments. For PTMM+our-weight. we set 1/B = 3, while all other hyperparameters are the same as those in ARPM (Gu et al., 2024).
16

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Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
Table 3. Test accuracy on the ImageNet - Caltech dataset using the weight choices described in Section 5.3.
Weighting scheme
Test accuracy
MPOT weights
78.6 (1.2)
BA*US weights
84.7 (0.7)
ARPM weights
79.2 (1.4)
WARMPOT (ours)
84.8 (0.1)
Table 4. Test accuracy on the ImageNet - Caltech datasct.
Algorithm
Test accuracy
ResNet-50 (He et al., 2016)
69.7
DAN (Long al., 2015)
71.3
DANN (Ganin et al., 2016)
70.8
IWAN (Zhang et al., 2018)
78.1
PADA (Cao et al., 2018)
75.0
ETN (Cao et al., 2019)
83.2
DRCN (Li et al., 2020)
75.3
BA BA$US (Liang al., 2020)
84.0
ISRA+BA*US (Xiao et al., 2021)
85.3
SLM (Sahoo et al., 2023)
82.3
SAN++ (Cau al., 2022)
83.3
AR (Gu et al., 2021)
85.4 (0.2)
ARPM (Gu et al., 2024)
84.1 (1.4)
PWAN (Wang al., 2024)
86.0 (0.5)
WARMPOT (ours)
84.8 (0.1)
ARPM--orrwcightt
85.1 (0.9)
F. Additional Numerical Results
In this section, discuss the results obtained by repeating the experiments described in Section 5.3 the ImageNet Caltech dataset, where ImageNet (Russakovsky al., 2015) consists of 1000 classes and Caltech-256 (Griffin al., 2022) consists of 256 classes.
We first compare different weighting schemes. Following Gu et al. (2024), we set the weight update intervals of ARPM and BA US to 2000. The experiment is repeated for 3 random seeds, and we report the average and the standard deviation. The results are presented in Table WARMPOT results in better performance than MPOT and ARPM, and yields performance comparable BAA
Then we compare WARMPOT against alternative algorithms. We set ammax 0.08, m1 0.92, 7/2 5.47, = 0.72, 5.59 for WARMPOT. The results are shown in Table 4. Once more, we observe that ARPM+our-weights achieves better performance than ARPM, highlighting the effectiveness of the WARMPOT weights.
G. Sensitivity Analysis for Alignment Parameters
In order assess the impact on performance the hyperparameters in the domain alignment term, we conduct a sensitivity analysis Ouaxx B, the ImageNet - Caltech dataset. In this analysis, set the following values for the other hyperparameters: 7/1 0.92, 5.47, 5.59. In the experiment on Ommax we set B 0.72, while in the experiment on 3, we set O'ma 0.08.
As seen in Fig. 2 (right), varying over the entire range (0, 1] has a limited impact on performance, with variations not exceeding 2%. We observe a similar trend whenever Omma varied within the range (0, 0.1). When Omax > 0.1, however, we see a significant drop in performance, caused most likely by the large number of outliers the source sample. These
17

---

Thcoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
90
90
max 0.08
80
80
B = 0.72
Test Accuracy
Test Accuracy
70
70
60
60
0.1
0.3
0.2
0.4
0.6
0.8
O'max
B
Figure 2. The effect of changing alignment parameters Omex and accuracy of ImageNet - Caltech.
results indicate that the specific choice of these parameters has minor impact over a range of reasonable values.
H. Relation between WARMPOT and ARPM Weights
The weights p (fi...., "i....,,n,) Pn,) used in the ARPM algorithm of Gu et al. (2024) are defined as the solution of the following Wasserstein- type problem between the source and target distributions:
min W PEA
p.d f(Z)) Σ 1=1
Σ E5
(73)
j=1
Here, the constraint set A defined as
pi 20
(74a)
Re
p==1
(74b)
A
1
<
(74c)
71.
T1
where is hyperparameter. Interestingly, the weights P = ...,) of WARMPOT, given in (11), can be equivalently
expressed, for the case a 1, as the solution of the same optimization problem as in (73), but with the constraint set A replaced by I defined as:
Ú 2 0
(75a)
R,
Σ
P =1
(75b)
r
1=1
(75c)
Bn,
Here, 3 is hyperparameter. Note that the only difference between the two sets of constraints is that they use different ways to control the magnitude of 6. in (74c) and (75c). Compared to the ARPM constraint A, our constraint I' is not only simpler and more theoretically grounded, but also more intuitive: (75c) controls the maximum target sample mass that can be matched to a single source sample, while the corresponding ARPM constraint (74c) is harder to interpret. As demonstrated our numerical results in Table 2, our weights also lead better performance.
18

---

Theoretical Performance Guarantees for Partial Domain Adaptation via Partial Optimal Transport
Note finally that ARPM solves (73) only approximately: it first solves Wasserstein-1 problem with fixed ps 1/n,, using Wasserstein-GAN, and then fixes this learned Wasserstein-GAN and updates the weights pt-
19


Reviews:
[{"review_id": "Ca2Dv7Runk", "reviewer": "Reviewer_QCqK", "summary": "The paper presents (PAC) bounds and on the (expected) empirical loss using partial Wasserstein distance in either the marginal (features only) or joint (features and labels). The first two terms of the loss are the loss weighted provided by the marginal of the partial transport plan and the partial Wasserstein distance itself. Minimizing these leads to an algorithm that is performant on a benchmark dataset and when the weights are used in a more complicated heuristic achieves state of the art.\n\n## update after rebuttal\nFurther experiments suggested by reviewer confirm that the approach achieves performance. Limitation due to bias caused by batch sampling, which has been noted in the continuous conditional flow matching case, could be noted.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their helpful and constructive comments.\n\n> Q1. **For the ground distance in the label space the loss should be a metric, but this is not specified. Is a metric actually required for $\\ell$? For instance, cross-entropy/log loss/KL divergence doesn't satisfy the requirements for a metric.**\n\nThe theoretical derivations require the loss function to be symmetric in the model\nprediction and true label, and that the triangle inequality holds.\nNote, though, that in our numerical experiments we consider loss functions that do not necessarily\nconform to the theoretical assumptions. The fact that we still observe good\nperformance indicates that these assumptions are not critical for the general insights\nobtained from our theoretical results to hold.\n\n> Q2. **Line 820 \"Through a parameter search,\" on what dataset and what performance metric? Especially in a dataset that is claimed to be unsupervised, this could be 'information leakage' from the test set performance. Generally, the percent of outliers is unknown.**\n\nIn order to assess the impact of this hyperparameter selection, we have conducted a\nsensitivity analysis for the $\\alpha_{max}$ and $\\beta$ parameters on ImageNet $\\rightarrow$\nCaltech dataset.\nWe set the following values for hyperparameters: $\\eta_1 = 0.92$, $\\eta_2 = 5.47$, $\\varepsilon = 5.59$.\nIn the experiment on $\\alpha_{max}$, we set $\\beta = 0.72$ and in the experiment on $\\beta$, we set $\\alpha_{max} = 0.08$ (see Appendix E in our paper for the notation used).\n\nAs seen in [Figure 1](https://gofile.io/d/PvEK7c), the impact of varying $\\beta$ is not\nlarge (about 2%) over the entire range of $0<\\beta\\le 1$.\nSimilarly, in [Figure 2](https://gofile.io/d/ZsHkUK), when $0<\\alpha_{max}\\le 0.1$, the performance difference is not large (about 3%).\nHowever, for $\\alpha_{max}>0.1$ there is a significant drop which is due to the\nlarge number of outliers in source sample.\nUsing a larger value of $\\alpha_{max}$ results in a positive weight over outlier instances\nthus degrading the performance.\nThe results indicate that the specific choice of these parameters has a minor impact\nover a range of reasonable values.\n\nFurthermore, note that for the _ARPM+our-weights_ algorithm, we use the same parameters\nas the original ARPM algorithm.\n\n> Q3. **Although empirical measures are used, the results would appear to be for the whole sample not the mini-batch. The implications of the reduced sample size from the mini-batch isn't clear in the bounds. Is there an effect?**\n\nAs the reviewer notes, we solve the mini-batch partial optimal transport problem during\ntraining rather than the full-data transport problem due to computational considerations.\nTo study the effect that this has on the resulting weights, we compared:\n\n  (i): the average weights applied to each sample during training that arise from the\n  mini-batch partial optimal transport problem.\n  \n  (ii): the weights arising from the full-data transport problem at the end of training;\n\nThe results indicate that the two approaches lead to very similar weight distributions,\nand in particular, the weight proportion assigned to shared samples is 86.37% with\nmini-batch weights and 90.54% with full-sample weights. This aligns with discussions in\n_Improving mini-batch optimal transport via partial transportation_ by Nguyen et al. (2022)\nregarding the possibility of using mini-batches to approximate full-data transport problems.", "claims_and_evidence": "The theory is developed rigorously and the the experimental results (one dataset but many source/target pairs) shows consistent performance.", "methods_and_evaluation_criteria": "For the ground distance in the label space the loss should be a metric, but this is not specified.", "theoretical_claims": "I did not check them rigorously but they seem logical.", "experimental_designs_or_analyses": "The second regards how hyperparameter search can be performed on a dataset that its unsupervised...", "relation_to_broader_scientific_literature": "Domain transfer and partial domain alignment is a very practical problem that is well established. The contributions are meaningful and will have impact if reproduced.", "essential_references_not_discussed": "Not that I noticed.", "other_strengths_and_weaknesses": "The paper is clear and well written. It should have significance because it is an important problem due to its widespread nature.", "other_comments_or_suggestions": "-", "questions_for_authors": "Is a metric actually required for $\\ell$? For instance, cross-entropy/log loss/KL divergence doesn't satisfy the requirements for a metric. \n\n\nLine 820 \"Through a parameter search,\" on what dataset and what performance metric? Especially in a dataset that is claimed to be unsupervised, this could be 'information leakage' from the test set performance.  Generally, the percent of outliers is unknown.\n\n\nAlthough empirical measures are used, the results would appear to be for the whole sample not the mini-batch. The implications of the reduced sample size from the mini-batch isn't clear in the bounds. Is there an effect?", "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": "", "extra_scores": {}}, {"review_id": "Go4S6kYzJA", "reviewer": "Reviewer_FSCj", "summary": "This submission studies the generalization bound and empirical model for the partial domain adaptation problem, where the label spaces across domains are different. The key idea of this paper is to use the weight deduced from the partial transportation mass, which is claimed to be able to filter the outliers (that are redundant for the target domain). Theoretical bound is provided to show the target error can be upper-bounded by weighting source risk, partial Wasserstein discrepancy, estimation bias on target distribution, and (intractable) worse error. The proposed method is compared with other SOTA PDA and OT methods on a PDA dataset.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their careful reading of the paper, relevant references, and constructive comments.\n\n>_C1 and Essential References_:\n  \nWe thank the reviewer for pointing out these relevant references. While all of them are\nrelevant, consider similar settings and techniques, and deserve to be reviewed in the\nintroduction, there are a few key differences between our work and the mentioned papers.\nFirst, to the extent that the mentioned papers present bounds in terms of a weighted\nsource loss, they all rely on classwise weights defined in terms of unknown data\ndistributions. These are then estimated using a method based on _Detecting and correcting\nfor label shift with black box predictors_ by Lipton et al. (2018). However, these\nestimates are only guaranteed to be accurate if GLS holds exactly, i.e., if the feature\nrepresentation $Z=g(X)$ of the input $X$ is such that $P(Z|Y=y)=Q(Z|Y=y)$ for source $P$\nand target $Q$. Our results require no such assumptions, and yield explicitly computable\nweights from only the observed data. Additional differences are detailed below.\n\n 1. The results in [r1] that contain the weighted source loss are bounds on distribution\ndiscrepancies, and not target loss. Furthermore, the weights therein are class-level\nweights that depend on the unknown underlying distributions, and the domain discrepancy\nis measured in terms of Jensen-Shannon divergence. In [r2], the same kind of weights are\nused, and domain discrepancy is measured in terms of metrics like the total variation.\nFinally, the risk bound in [r3] does not include source loss weights, and depends on the\n1-Wasserstein distance rather than its partial counterpart considered in our paper.\n 2. While [r4] considers a more general setting, aiming for, e.g., private class discovery,\nthe proposed algorithm is not accompanied by any theoretical analysis. Furthermore,\nas pointed out by the reviewer, they use unbalanced rather than partial optimal\ntransport, and consider binary rather than soft weights. \n\nIn summary: the listed references are very relevant, and we have updated the discussion\nof related work to include them. However, our theoretical results provide test risk bounds\nin terms of empirically computable weights and apply without any assumptions on the\nspecific form of distribution shift. We have updated our stated contributions to clarify\nthese points and avoid overclaiming the merits of our work relative to prior art.\n  \n>_C2, C3, and Q2_:\n  \nFirst, note that we assume $\\beta\\in(0,1]$. Hence, the mass of the scaled source\ndistribution is always at least $1$. When $\\alpha=1$ and all entries of $Q_{\\tilde X}$\nequal $1/n_t$, the condition below Eq. (4)\n\n$\\mathbf{1}^T_{n_s}\\Pi\\mathbf{1}_{n_t}=1$\n\nrequires the $n_t$ column sums of $\\Pi$ to add up to $1$,  whereas the condition\n\n$\\Pi^T \\mathbf 1_{n_s} \\leq Q_{\\tilde X}$\n\nlimits each row sum to be $1/n_t$ at most. Combined, they imply that $q_j=1/n_t$.\n\nRegarding guaranteeing that outliers are ignored: The parameter $\\beta$ needs to be chosen\nto be small enough to enable the algorithm to ignore outliers. In particular, if\n$\\alpha=1$, $\\beta$ can at most equal the outlier proportion. However, this still does not\n_guarantee_ that outliers are ignored. In the same way, while the approach of [r4] is\ndesigned to avoid outliers, there is also no guarantee that this works. Empirically, as\nshown in Fig. 1, WARMPOT does significantly downweight outlier samples in practice.\n\nThe essential advantages of WARMPOT for PDA compared to the approach of [r4] are: _(a):_\nWARMPOT is endowed with theoretical guarantees, _(b):_ the transport plan for POT is more\ninterpretable than the one for unbalanced OT, and _(c):_ our soft weights enable different samples\nof the same class to have different influence on the final hypothesis.\n\n>_Q1_:\n\nAs the reviewer notes, Thm. 3.2 includes only a feature cost whereas Thm. 3.3 additionally\nincorporates a label cost. Intuitively, one would expect the feature-only approach to\nsuffice when the distribution shift is restricted to covariate shift and a setting where\nthe supports of the input distributions mostly overlap. However, in cases of label shift,\nincorporating the label cost may be helpful. The factor 2 in Thm. 3.2 essentially\ncompensates for the absence of the label loss, and stems from the use of Lemma A.4 (where\nwe need to use triangle inequality twice). A similar point applies to the\nuncomputable terms that capture the difficulty of the specific task ($L_f$ and $\\tilde\nL_f$). Whether or not one of the bounds leads to better performance is a largely empirical\nquestion, and our experiments indicate that, for the tasks under consideration, it is\ngenerally beneficial to include a label cost.\n\n>_Q3_:\n  \nWe have added results for ImageNet $\\rightarrow$ Caltech (see response to 9cQ1 for details).\n\n>_Other:_\n\nWe now use $\\tilde p_i$ and $\\tilde q_j$ for the weights in Thm. 3.3 to avoid confusion, and clarify the difference compared to $p_i$ and $q_j$.", "claims_and_evidence": "There are several concerns regarding the claims: \n\nC1. The merits seem to be overclaimed, e.g., the generalization bound for PDA, partial alignment. Note that several works have also derived bounds for PDA with OT or even more general framework, where the idea of partial alignment and weighted risk estimation is also presented. More details are discussed in the *Essential References Not Discussed* part.\n\nC2. In line 171, the is a claim that “when $\\alpha=1$, we have that $q_j = 1/n_t$”. This claim seems to be problematic: if $\\beta$ is not small enough (i.e., $1/ (\\beta n_s) $ is not big enough) or $n_t$ is small (i.e., $1/n_t$ is large), then this claim indeed cannot be satisfied due to the inequality constraint of POT. \n\nC3. In line 174, assume that $q_j = 1/n_t$ (even it seems to cannot be guaranteed), then why this equality can ensure that the outliers are ignored? Considering the same case in C2 while the mass of source outliers is large (i.e., the mass sum of shared samples in $P_s / \\beta$ is still smaller than the total mass requirement $\\alpha$), there always seems to be transport masses from outliers.", "methods_and_evaluation_criteria": "The methodology and evaluation criteria are generally appropriate.", "theoretical_claims": "The theoretical results and proofs look correct.", "experimental_designs_or_analyses": "The comparison experiment is only conducted on a single dataset, while a consistent improvement over different datasets is necessary for demonstrating the empirical performance of the proposed method.", "relation_to_broader_scientific_literature": "The key idea is related to the recent progresses on sample-level weight estimation for (label) shift, the generalization bound for PDA (with optimal transport), where the main difference is that this work considering the partial optimal transport framework.", "essential_references_not_discussed": "The missing references (which are closely related to this submission) can be summarized from the following aspects:\n\n1. Generalization bounds. In fact, from the views of distribution shift (particularly the label shift), both PDA, Open-Set DA (OSDA), and universal DA (UniDA) can be considered as the label shift (LS) or generalized label shift (GLS), where these extreme shift scenarios imply that the supports of label distributions are different. Therefore, there are actually works that provide the same innovation, i.e., 1) GLS bound: upper-bound with weighted source risk, shift on (conditional) representation distribution, and shift on label distribution [r1,r2] 2) GLS bound under optimal transport as discrepancy measures [r3]. \n\n2. The idea of employing the marginals of transport plan (of POT) as tool for automatically address the extreme shift have been studied in Unified OT [r4], which solving a more general setting (i.e., PDA, OSDA and UniDA) and considering the hard threshold for weight (where this work considers soft weight $p_i$).\n\n**References**\n\n[r1] Tachet des Combes, Remi, et al. \"Domain adaptation with conditional distribution matching and generalized label shift.\" Advances in Neural Information Processing Systems 33 (2020): 19276-19289.\n\n[r2] Luo, You-Wei, and Chuan-Xian Ren. \"When Invariant Representation Learning Meets Label Shift: Insufficiency and Theoretical Insights.\" IEEE Transactions on Pattern Analysis and Machine Intelligence (2024).\n\n[r3] Kirchmeyer, Matthieu, et al. \"Mapping conditional distributions for domain adaptation under generalized target shift.\" International Conference on Learning Representations. 2022.\n\n[r4] Chang, Wanxing, et al. \"Unified optimal transport framework for universal domain adaptation.\" Advances in Neural Information Processing Systems 35 (2022): 29512-29524.", "other_strengths_and_weaknesses": "**Pros:**\n\n1. The organization is clear and easy to follow.\n\n2. Theoretical analysis is provided for the proposed OT framework.\n\n**Cons:**\n\n1. The interpretations of the main technique are unrigorous and insufficient.\n\n2. The related works are not properly discussed, which makes it hard to assess the essential merits.\n\n3. The empirical improvement is limited and the experiments.", "other_comments_or_suggestions": "1. The definition of $p_i, q_j$ could be confusing due to the incomplete definition of $\\Pi^*$. It is necessary to justify that what problem does $\\Pi^*$ correspond to (i.e., $(\\mathbb{PW}_\\alpha (\\cdot,\\cdot))$)? Though they are defined in the proof part of the appendix, it should be clarified in the main body.", "questions_for_authors": "I would like to authors to address my concerns on the claims and missing references, which are justified in detail in *Claims And Evidence* part and *Essential References Not Discussed* part. Besides, here are additional questions:\n\nQ1. It seems that the joint distribution-based bound in Thm. 3.3 induces larger error than Thm. 3.2. Though Eq. (5) has a factor of 2 for $\\mathbb{PW}$, note that the discrepancy in Eq. (9) is induced by the joint cost function (which naturally combines the two costs, i.e., feature cost and label cose). Thus, it is generally the same for the partial Wasserstein term while Eq. (9) induces more complex term that are intractable, i.e., term in Eq. (11). Some justifications are highly appreciated.\n\nQ2. The main difference between this submission and the existing GLS correction method [r4] is that this work adopts POT as a discrepancy metric. However, as discussed in the *Claims And Evidence* part, the $\\mathbb{PW}$ is not definitely guaranteed to exclude the outliers. Thus, how to understand the essential advantages of the proposed method?\n\nQ3. The improvement over ARPM is only 0.2\\% on Office-Home dataset, while no other empirical results are provided. This result seems to imply that the essential function of the proposed method is generally admitted by the existing ARPM. Justifications or additional experiments are highly appreciated.", "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": "", "extra_scores": {}}, {"review_id": "gjHRTD04Vs", "reviewer": "Reviewer_9cQ1", "summary": "This paper deals with Partial Domain Adaptation (PDA), a setting where source and target domain distributions differ, and where the target domain label space is a subspace of the source domain label space. The authors propose to tackle this important problem through Optimal Transport (OT), an established field of mathematics that has previously contributed to domain adaptation in general. The authors use this framework not only to propose new algorithms, but also to provide theoretical generalization bounds.\n\n---\n\n__Post-rebuttal.__ The authors did a good job on their rebuttal, especially, they provided new experiments on a large scale adaptation task. Overall, this is a good paper with strong theoretical contributions and convincing experiments. Hence, my final score is __4. Accept__", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their careful reading and helpful comments.\n\n> _Methods_ and _Weaknesses_:\n**While the authors provide a comprehensive comparison with the state-of-the-art, they do so in a single benchmark, i.e., the Office-Home benchmark. In my view the authors should complete their experiments with at least other benchmark (preferably Imagenet $\\rightarrow$ Caltech or VisDA2017, which are more complex and large scale than Office 31).**\n\nWe have now benchmarked our algorithm on ImageNet $\\rightarrow$ Caltech. The test\naccuracy of WARMPOT for this data set is 84.8%, while ARPM+our weights achieves 85.1%.\nFor reference, the test accuracy of ARPM is 84.1%. Hence, this additional experiment\nprovides further indication that WARMPOT, and specifically its weights, provides a\nrobust approach for PDA tasks.\n\n|Algorithm|ImageNet $\\rightarrow$ Caltech|\n|-|-|\n|ResNet-50|69.7|\n|DAN|71.3|\n|DANN|70.8|\n|IWAN|78.1|\n|PADA|75.0|\n|ETN|83.2|\n|DRCN|75.3|\n|BA3US|84.0|\n|ISRA+BA3US|85.3|\n|SLM|82.3|\n|SAN++|83.3|\n|AR|85.4 (0.2)|\n|ARPM|84.1 (1.4)|\n|PWAN|86.0 (0.5)|\n|WARMPOT (ours)|84.8 (0.1)|\n|ARPM+our-weights|85.1 (0.9)|\n\n> _Essential References_:\n**The authors do a good job summarizing partial Optimal Transport. However, this _is not_ the only way of tackling partial DA. Especially, the authors do not discuss, nor compare the use of unbalanced Optimal Transport (Fatras et al., 2021) for partial domain adaptation**\n\nIn Table 4 of Nguyen et al. (2022), a comparison is made between their MPOT approach and\na mini-batch unbalanced OT (UOT) approach. The results indicate that the POT-based\napproach works better. We have revised the paper to extend the discussion of UOT\napproaches and include numerical comparisons.\n\n> _Comment 1_:\n**The term $L_f$ looks like $\\lambda$ in (Redko et al., 2017) and other theoretical DA works. Could the authors provide additional discussion on the potential similarities?**\n\n$L_f$ indeed plays a similar role as, e.g., $\\lambda$ in Redko et al. (2017) and admits\nthe same interpretation, in the sense that it is related to the difficulty of the domain\nadaptation problem. However, while $\\lambda$ is the smallest achievable sum of population\nlosses, $L_f$ is the smallest achievable loss maximized over the empirical source and\ntarget set. Meanwhile, $\\tilde L_f$ relates to the minimal achievable target loss, and\n$\\Xi$ measures the performance gap between considering source and target tasks jointly\nor separately.\n\n> _Comment 2_:\n**The cost in (10) looks a lot with the cost of the joint cost proposed in (Courty et al., 2017; reference in the main paper). I think the authors could add some discussion about the similarities as well. This discussion could also make links between the feature importance factor $\\zeta\\gamma$ weighting the features in the ground-cost.**\n\nThe cost in (10) is indeed the same as the one proposed in Courty et al., 2017. We will\nmake this point clearer in the revised version of the manuscript. From a\ntheoretical perspective, the factor $\\zeta\\gamma$ (which appears both in our cost and the\none of Courty et al., 2017) corresponds to Lipschitz parameters that are generally not\navailable (see below). Hence, from a practical point of view, we treat it as a\nhyperparameter in our algorithm.\n\n> _Comment 3_:\n**Given the practical applications of their work, I think the authors could comment on the restrictiveness of their hypothesis. For instance, they assume 1) the encoder $g$ is $\\gamma$-Lipschitz, 2) the label loss function is a distance, and $\\zeta$-Lipschitz. In common DA practice, neither of these 2 hypothesis are met. For instance, since WGANs, we know that enforcing $\\gamma$-Lipschitzness on neural nets is tricky. Furthermore, neural nets are often trained with the cross-entropy loss, which is not a metric on $\\mathcal Y$.**\n\nIndeed, the Lipschitz assumptions required for our theoretical results are often not\nsatisfied for loss functions used to train neural nets in practice. Such Lipschitz\nassumptions are commonly invoked in the literature to obtain generalization bounds that\ndepend on Wasserstein distances. Mathematically, the Lipschitz assumption allows one to relate the\nloss function to the cost appearing in the Wasserstein metric (see Lemma A.4).\nNote that, in our numerical experiments, we consider loss functions that do not necessarily\nconform to the theoretical assumptions. The fact that we still observe good\nperformance indicates that these assumptions are not critical for the general insights\nobtained from our theoretical results to hold.", "claims_and_evidence": "Here is a list of highlevel claims made in the introduiction,\n\n1. \"we provide theoretically motivated algorithms for PDA\"\n2. \"we derive generalization bounds on the target population loss and devise training strategies that minimize them\"\n3. \"our bounds give rise to weights that, when combined with the ARPM algorithm of Gu et al. (2024) lead to SOTA\nresults for the Office-Home data set.\"\n\nClaims 1 and 2 are well supported by the theoretical parts of the paper. While claim 3. is also true it is fairly limited in scope. See my comments on the next section about.", "methods_and_evaluation_criteria": "In this point, the paper falls short on the acceptance criteria. While the authors provide a comprehensive comparison with the state-of-the-art, they do so in a single benchmark, i.e., the Office-Home benchmark. For instance, other papers such as (Gu et al., 2024) and (Cao et al., 2018) have considered the following (on top of Office-Home),\n\n- Office 31\n- Imagenet -> Caltech\n- VisDA2017 (Real -> Synthetic and Synthetic -> Real)\n\nWhich provide more thorough comparisons. In my view the authors should complete their experiments with at least other benchmark (preferably Imagenet -> Caltech of VisDA2017, which are more complex and large scale than Office 31).", "theoretical_claims": "The authors provide a series of 2 theorems, alongside 1 lemma and 2 corolaries. Overall, they provide generalization bounds in terms of the partial Wasserstein distance between the distributin of extracted features. These generalization bounds are novel, and in line with previous research on domain adaptation theory.\n\nI checked the appendix provided by the authors, and, as far as my knowledge goes, the proofs of theorems 3.2 and 3.3 are correct. I did not check the proof of lemma 3.4.", "experimental_designs_or_analyses": "The experimental design an analysis is in lign with current domain adaptation practice, and are, in this regard, correct. The authors could have explored more benchmarks to validate their method, as I highlighted in `Methods and Evaluation Criteria`", "relation_to_broader_scientific_literature": "The current paper goes in a similar direction to previous papers (Fatras et al., 2021; Khai et al., 2022) that propose Optimal Transport techniques for partial domain adaptation. An important feature of this paper is that the authors provide generalization bounds in terms of the partial Wasserstein distance.\n\n(Fatras et al., 2021) Fatras, Kilian, et al. \"Unbalanced minibatch optimal transport; applications to domain adaptation.\" International Conference on Machine Learning. PMLR, 2021.\n\n(Khai et al., 2022) Nguyen, Khai, et al. \"Improving mini-batch optimal transport via partial transportation.\" International conference on machine learning. PMLR, 2022.", "essential_references_not_discussed": "The authors do a good job summarizing partial Optimal Transport. However, this **is not** the only way of tackling partial DA. Especially, the authors do not discuss, nor compare the use of umbalanced Optimal Transport (Fatras et al., 2021) for partial domain adaptation\n\n(Fatras et al., 2021) Fatras, Kilian, et al. \"Unbalanced minibatch optimal transport; applications to domain adaptation.\" International Conference on Machine Learning. PMLR, 2021.", "other_strengths_and_weaknesses": "Here I give a summary of strenghts and weaknesses. Please use this list when writting your rebuttal. If the authors provide a rebuttal that answers the following weaknesses, I will raise my score accordingly.\n\n__Strenghts__\n\n1. Sound theoretical analysis with novel results for Partial DA\n2. New algorithm with promising results on Office-Home benchmark\n\n__Weaknesses__\n\n1. The most important, the empirical evaluation of this paper is very limited. The authors should complete their empirical validation with other benchmarks in Partial DA", "other_comments_or_suggestions": "__Comment 1.__ The term $L_{f}$ looks like $\\lambda$ in (Redko et al., 2017) and other theoretical DA works. Could the authors provide additional discussion on the potential similarities?\n\n__Comment 2.__ The cost in (10) looks a lot with the cost of the joint cost proposed in (Courty et al., 2017; reference in the main paper). I think the authors could add some discussion about the similarities as well. This discussion could also make links between the feature importance factor $\\xi \\gamma$ weighting the features in the ground-cost\n\n__Comment 3.__ Given the practical applications of their work, I think the authors could comment on the restrictiveness of their hypothesis. For instance, they assume\n\n1. The encoder $g$ is $\\gamma-$Lipschitz\n2. The label loss function is a distance, and $\\xi-$Lipschtiz\n\nIn common DA practice, neither of these 2 hypothesis are met. For instance, since WGANs, we know that enforcing $\\gamma-$Lipschitzness on neural nets is tricky. Furthermore, neural nets are often trained with the cross-entropy loss, which is not a metric on $\\mathcal{Y}$.\n\n(Redko et al., 2017) Redko, Ievgen, Amaury Habrard, and Marc Sebban. \"Theoretical analysis of domain adaptation with optimal transport.\" Machine Learning and Knowledge Discovery in Databases: European Conference, ECML PKDD 2017, Skopje, Macedonia, September 18–22, 2017, Proceedings, Part II 10. Springer International Publishing, 2017.", "questions_for_authors": "N/A See previous section", "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": "", "extra_scores": {}}, {"review_id": "FgCy33rxxU", "reviewer": "Reviewer_3wvs", "summary": "The paper studies the problem of Partial Domain Adaptation (PDA), where the target label space is a subset of the source label space. The authors propose a theoretically grounded approach based on Partial Optimal Transport (POT) to tackle PDA, deriving generalization bounds that justify the use of the Partial Wasserstein Distance (PWD) as a domain alignment term. These bounds also provide explicit weight formulations for the empirical source loss, distinguishing their method from prior heuristic weighting strategies. The authors introduce WARMPOT, an algorithm that leverages the derived bounds to optimize domain adaptation performance. They validate WARMPOT through extensive numerical experiments, demonstrating competitive results compared to state-of-the-art (SOTA) methods. The paper claims that their weighting strategy improves upon existing approaches and provides better theoretical justification for PDA method.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comprehensive evaluation and helpful suggestions.\n\n> _Questions for authors_:\n  **Given that solving the partial optimal transport problem is computationally expensive, how does WARMPOT compare in training time to previous approaches?**\n  \nIndeed, solving the optimal transport problem is generally computationally expensive.\nOne common way to circumvent this problem is via mini-batch approaches.\nWe follow this approach.\nSpecifically, in the numerical experiments reported in the paper, we use the mini-batch partial optimal transport framework proposed in _Improving mini-batch\noptimal transport via partial transportation_ by Nguyen et al. (2022).\nThe most well-performing algorithms in the literature, including PWAN and ARPM, also\nrequire one to solve optimal transport problems as part of their algorithms.\n\nWe would also highlight that different weighting strategies come with different\ncomputational costs. In WARMPOT, the weights, which need to be computed per mini-batch,\nare obtained directly from the solution of the partial optimal transport problem without\nany additional overhead.\nOn the contrary, a weight update in both the BA3US and the ARPM weighting strategy involves the entire dataset.\nAdditionally, we see from the tables inserted below (see final comment for further details) that there is a trade-off between performance and computational\ncost for these weighting strategies. Such a trade-off is not present in WARMPOT.\n\n> _Claims and Evidence_:\n  **Additional sensitivity analyses on weight selection would further strengthen the claims.**\n\nWe have added sensitivity analyses for the $\\alpha_{max}$ and $\\beta$ parameters on ImageNet\n$\\rightarrow$ Caltech (see response to QCqK).\n\n> _Weakness 1_:\n  **Limited exploration of additional datasets beyond Office-Home.**\n\nWe have added results for ImageNet $\\rightarrow$ Caltech (see response to 9cQ1 for details).\n  \n> _Weakness 2_:\n  **Lack of error bars to support experimental results.**\n\nIn order to address this point, we have re-run our experiments using additional random\nseeds for OfficeHome (6 seeds) and computed average and standard deviation. We have also conducted a weighting-scheme comparison on\nImageNet $\\rightarrow$ Caltech using 3 random seeds. In the tables shown below, we\nconsidered for BA$^3$US\nand ARPM the weight update interval (in epochs) indicated in the parentheses.\nWe see from the table that the performance of BA$^3$US depends on the update interval (the\nsmaller the interval, the better the performance)\nGu et al., 2024\nrecommend using 500 and 2000 as update interval for ARPM on OfficeHome and\nImageNet $\\rightarrow$ Caltech, respectively. Adopting the same update interval for\nBA$^3$US, to maintain a fair comparison,\nwe see that, as claimed in the paper, WARMPOT results in better performance than MPOT and\nARPM(500), and yields performance comparable to BA$^3$US(500) (overlapping confidence\nintervals) on both datasets.\n\nAlso, the confidence intervals of the best performing algorithms in Table 2 reported in the paper (ARPM and ARPM+our weights) overlap.\n\n  |Weighting scheme|BA$^3$US(100) weights|BA$^3$US(500) weights|BA$^3$US(750) weights|WARMPOT (ours)|MPOT weights|ARPM(500) weights|\n  |----|-----|-------|-|--|--|--|\n  |Avg. test acc. on OfficeHome|78.1 (0.4)|77.6 (0.4)|77.4 (0.4)|77.6 (0.7)|76.0 (0.4)|72.9 (0.3)|\n\n  |Weighting scheme|BA$^3$US(750) weights|BA$^3$US(1500) weights|BA$^3$US(2000) weights|BA$^3$US(5000) weights|WARMPOT (ours)|MPOT weights|ARPM(2000) weights|\n  |----|-----|-------|-|--|--|--|-|\n  |Test acc. on ImageNet $\\rightarrow$ Caltech|86.1 (0.6)|85.0 (0.2)|84.7 (0.7)|83.1 (1.4)|84.8 (0.1)|78.6 (1.2)|79.2 (1.4)|", "claims_and_evidence": "* WARMPOT is theoretically justified – the paper derives generalization bounds that incorporate PWD as a domain alignment term and explicitly define source loss weights. Evidence: Theoretical analysis and derivations in Section 3​\n.\n* WARMPOT outperforms existing PDA methods – The proposed algorithm achieves better accuracy compared to methods like MPOT, PWAN, and ARPM. Evidence: Empirical results in Section 5.4, showing improved performance on the Office-Home dataset​\n.\n* The weighting strategy is more effective than existing ones – WARMPOT's weights successfully reduce the influence of outlier classes, improving adaptation. Evidence: Table 1 shows that the weighting strategy leads to better classification accuracy and mitigates negative transfer​\n\n* Most claims are well-supported by theoretical arguments and experimental validation. However, additional sensitivity analyses on weight selection would further strengthen the claims.", "methods_and_evaluation_criteria": "The authors adopt the Office-Home dataset as the main benchmark, comparing WARMPOT against existing PDA algorithms. Evaluation focuses on classification accuracy across different domain shifts. The Partial Wasserstein Distance is used to measure alignment between source and target distributions. The chosen dataset and metrics are well-aligned with the problem, and comparisons with recent PDA approaches (e.g., MPOT, ARPM) provide a meaningful assessment​.", "theoretical_claims": "The paper presents theoretical results regarding generalization bounds for PDA, particularly: \n\n* Feature-based bound (Theorem 3.2) – Establishes a bound on the target loss using PWD between empirical feature distributions. \n\n* Joint distribution-based bound (Theorem 3.3) – Extends the previous bound to account for target labels.", "experimental_designs_or_analyses": "The experimental design includes comparisons on a standard PDA dataset, tuning of key hyperparameters $\\alpha,\\beta$, and ablation studies on weighting strategies. A notable strength is the comparison against both heuristic and theoretically motivated weighting methods, providing insight into the benefits of WARMPOT's approach​. However, additional statistical significance tests or error bars would improve confidence in the reported results.", "relation_to_broader_scientific_literature": "The paper builds upon prior work in domain adaptation, optimal transport, and partial optimal transport (POT). The paper propose a theoretically grounded approach based on Partial Optimal Transport (POT) to tackle PDA, deriving generalization bounds that justify the use of the Partial Wasserstein Distance (PWD) as a domain alignment term.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Some weaknesses:\n\n* Limited exploration of additional datasets beyond Office-Home.\n\n* Lack of error bars to support experimental results.", "other_comments_or_suggestions": "No.", "questions_for_authors": "Given that solving the partial optimal transport problem is computationally expensive, how does WARMPOT compare in training time to previous approaches?", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper addresses the problem of Partial Domain Adaptation and proposes to solve the problem using a partial optimal transport 
framework. The main contributions of the paper are a new algorithm for matching source/target distributions and
theoretical generalization bounds providing support to the approach.

All reviewers found that the paper deserves acceptance as it proposes novel and solid contributions.
Decision: Accept (poster)