Domain-Free Adversarial Splitting for Domain Generalization
Authors: Xiang Gu
Jiasun Feng
Jian Sun
Zongben Xu
Venue:  (2021)
Keywords: domain generalization
adversarial splitting
meta-learning
image recognition

Abstract: Domain generalization is an approach that utilizes several source domains to train the learner to be generalizable to unseen target domain to tackle domain shift issue. It has drawn much attention in machine learning community. This paper aims to learn to generalize well to unseen target domain without relying on the knowledge of the number of source domains and domain labels. To achieve that goal, we unify adversarial training and meta-learning in a novel proposed Domain-Free Adversarial Splitting (DFAS) framework. In this framework, we model the domain generalization as a learning problem that enforces the learner to be able to generalize well for any train/val subsets splitting of the training dataset. This model can be further transformed to be a min-max optimization problem which can be solved by an iterative adversarial training process. In each iteration, it adversarially splits the training dataset into train/val subsets to maximize domain shift between them using current learner, and then updates the learner on this splitting to be able to generalize well from train-subset to val-subset using meta-learning approach. Extensive experiments on three benchmark datasets under three different settings on the source and target domains show that our method achieves state-of-the-art results and confirm the effectiveness of our method by ablation study. We also derive a generalization error bound for theoretical understanding of our method.

Under review as a conference paper at ICLR 2021
DOMAIN-FREE ADVERSARIAL SPLITTING FOR DOMAIN GENERALIZATION
Anonymous authors Paper under double-blind review
ABSTRACT
Domain generalization is an approach that utilizes several source domains to train the learner be generalizable to unseen target domain to tackle domain shift issue. It has drawn much attention in machine learning community. This paper aims to learn to generalize well unseen target domain without relying on the knowl- edge of the number of source domains and domain labels. We unify adversarial training and meta-learning in novel proposed Domain- Free Adversarial Splitting (DFAS) framework. In this framework, we model the domain generalization as a learning problem that enforces the learner to be able to generalize well for any train/val subsets splitting of the training dataset. To achieve this goal, we propose a min-max optimization problem which can be solved by an iterative adversarial training process. In each iteration, it adversarially splits the training dataset into train/val subsets to maximize domain shift between them using current learner, and then updates the learner on this splitting to be able to generalize well from train-subset to val-subset using meta-learning approach. Extensive experiments on three benchmark datasets under three different settings on the source and target domains show that our method achieves state-of-the-art results and confirm the effectiveness method by ablation study. We also derive generalization error bound for thcoretical understanding of our method.
INTRODUCTION
Deep learning approach has achieved great success in image recognition (He al., 2016 Krizhevsky et al. 2012 SSionyyan & Zisserman 2014). However, deep learning methods mostly succeed in the case that the training and test data are sampled from the same distribution (i.e., the i..dd. assumption). However, this assumption is often violated in real-world applications since the equipments/envirrnn
ments that generate data are often different in training and test datasets. When there exists distribution difference (domain shift (Torralba & Efros 2017) between training and test datasets, the performance of trained model, i.e., learner, will significantly degrade.
To tackle the domain shift issue, domain adaptation approach (Pan & Yang 2010: Daume III & Marcu 2006; Huang et al.) 2007) learns a transferable learner from source domain to target domain. Domain adaptation methods align distributions of different domains either in feature space (Long ct al. 2015 Ganin et al. 2016)) or in raw pixel space (Hoffman et al. 20188), which relies on unlabeled data from target domain training time. However, many applications, it is unrealistic access the unlabeled target data, therefore this prevents to use domain adaptation approach in this setting, and motivates the research on the learning problem of domain generalization.
Domain generalization (DG) approach (Blanchard et al. 2011: Muandet et al. 2013 commonly uses several source domains train a learner that can generalize to an unseen target domain. The underlying assumption is that there exists a latent domain invariant feature space across source domains and unseen target domain. To learn the domain invariant features, (Muandet et al., 2013 Ghifary et al. 2015 Li et al. 2018b) explicitly align distributions of different source domains in feature space. (Balaj et al. 2018 Li et al. 2019b 2018: Dou et al. (019) split source domains into meta-train train and meta-test simulate domain shift and train learner meta- ineaallearnnng approach.
(Shankar et al.) 2018 Carlucci et al., 2019 Zhou et al. 2020; Ryu et al. 2020) augment images or teatures to train learner enhance generalization capability.

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Conventional domain generalization methods assume that the domain labels are available. But in a more realistic scenario, the domain labels may be unknown (Wang et al.] 2019). To handle this domain-five setting, Carlucci et al. (2019) combines supervised learning and supf-supe learning to solve jigsaw puzzles of the training images. Matsuura & Harada (2020) divides samples into several latent domains via clustering and trains a domain invariant feature extractor via adversarial training. Huang et al. (2020) discards the dominant activated features, forcing the learner to activate remaining features that correlate with labels. Another line of works (Volpi al., 20118] Qiao et al. 2020) tackle the single source setting that the training set comprises single domain, and the train and test data are from different domains.
In this work, we focus on general learning scenario of domain generalization as follows. First, we do not know the domain label of each data and do not assume that there are several domains in the training dataset. Second, we do not assume that the training and test data are from different domains (e.g., styles). However, the previous domain-free DG methods (Matsuura & Harada, 2020) commonly evaluate on the datasets (e.g., PACS) composed of several domains though they do not use domain labels in training.
In our domain-free setting, we do not assume and know the domains in the training dataset, we therefore model domain generalization as a learning problem that the learner should be able to generalize well for any splitting of train/val subsets, i.e., synthetic souree/target domains, over the training dataset. This explicitly enforces that the trained learner should be generalizable for any possible domain shifts within the training dataset.
To achieve this goal, we propose an adversarial splitting model that is a min-max optimization problem, due to the difficulty of enumerating all splittings. In this min-max problem, we adversarially split training dataset to train/val subsets by maximizing the domain shift between them based on the given learner, and then update learner by minimizing the prediction error on val-subset using meta-learning approach given the splitting. By optimizing this min-max problem, we enforce the learner generalize well even in the worst-case splitting. We also investigate nomormllliottion of features in our domain generalization method. It is surprisingly found that I-znormalization can improve performance of learner and mitigate gradient explosion in the meta-learning process of DG. We further theorectically analyze the underlying reasons for this finding. This proposed domain generalization approach is dubbed Domain- Free Adversarial Splitting, i.e., DFAS.
To verify the effectiveness of our method, we conduct extensive experiments on benchmark datasets of PACS, Office-Homs, and CIFAR- 10 under different settings with multiple/single source domains. In experiments that the training data are from several source domains, our method achieves state-of- the-art results on both PACS and Office-Home datasets. We also find that our method significantly outperforms baselines in experiments that the training data are from single source domain PACS and CIFAR-10. We also confirm the effectiveness of our method by ablation study.
Based on domain adaptation theory, also derive an upper bound of the generalization error unseen target domain. We analyze that the terms in this upper bound are implicitly minimized by our method. This theoretical analysis partially explains the success of our method.
2 RELATED WORKS
We summarize and compare with related domain generalization (DG) methods in two perspectives, i.e., DG with domain labels and DG without domain labels.
DG with domain labels. When the domain labels are available, there are three categories of methods
for DG. First, (Manndet et al. 2013 Ghifary et al., 2015) Li et al.) 2018b) Piratla al.) 2020) learn domain invariant features by aligning feature distributions or by common/specific feature decomposition. Second, (Li et al.) 2019a; BBlaji et al. 2018 Li et al. 2019b 2018a| Dou et al.. 2019; Du et al.) 2020 b) are based on meta-learning approach that splits given source domains into meta-train and meta-test domains and trains learner in an episodic training paradigm. Third, (Shankar ct al. 2018 Carlucci et al. 2019 Zhou et al. 2020 Wang al. 2020) augment fake domain data to train learner for enhancing generalization capability of learner. Our method may mostly relate to the above second category of methods. But differently, consider the DG problem in fomain--free setting and adversarially split training dataset to synthesize domain shift in a principled min-max optimization method, instead of using -ave-one-dommin-out splitting in these methods.
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DG without domain labels. When the domain label is unavailable, enhance generalization ability of learner, Wang et al. (2019) extracts robust feature representation by projecting out superficial patterns like color and texture. Carlucci et al. (2019) proposes to solve jigsaw puzzles of the training images. Matsuura & Harada 2020) divides samples into several latent domains via clustering and learns domain invariant features via adversarial training of feature extractor and domain discrimi- nator. Huang et al. (2020) discards the dominant activated features, forcing the learner to activate remaining features that correlate with labels. Volpi et al. (2018) and Qiao al. (2020) propose adversarial data augmentation to tackle the setting that the training set comprises a single domain. In methodology, these methods either explicitly force the learner to extract robust features (Wang et al., | Matsuura & Harada 2020 Huang et al. (020) or augment new data to increase training data (Carlucci al., 2019; Qiao et al., 2020; Volpi et al. 2018). While our method novel meta-learning approach for DG by introducing adversarial splitting of training dataset during training. without relying on data/domain augmentation.
METHOD
In our setting, since we do not assume and know the domains in the training dataset, the training data could be independently sampled from several underlying source domains or just from single source domain. We denote that can generalize well on unseen target domain.
{( yi )i) as the training dataset. Our goal is to train the learner with S
In the following sections, we introduce details of our proposed model in Sect. 3.1 followed by its optimization method in Sect. 3.2] We also investigate Ez-normalization for domain generalization in Sect. 3.3 Theoretical analysis for our method is presented in Sect. Experimental results are reported in Sect. 5 Sect. conclude this paper.
3.1 DOMAIN-FRER ADVERSARIAL SPLITTING MODEL
As mentioned in Sect. [ we model DG as learning problem that enforces the learner to be able to generalize well for any train/val subsets splitting of the training dataset. The learner is trained using meta-learning approach (Finn et al. 2017). To formulate our idea mathematically, we first introduce some notations. We denote as funetion/learner (f could be deep neural network, egg. ResNet (He et al. 2016)) that outputs classification score of the input image, 1 as the loss such as cross-entropy. S, and S. as the train-subset and val-subset respectively such that S, U S, and S, n.S,- = 0. The formulated optimization problem for domain generalization is
min OF
Σ ((((w);; S.) + R(w) S.ET
[&l
s.t. ((w) arg min C(0; St, w), S, = S S..
is the set of all possible val-subsets of S with length of 1 g. S. = S - S, is train-subset paired with each S.. [(((w); S.) = [s. E(ew)es, ((fm(m) (x), v) is the loss on S., where (ww is the parameters of f. L(O; St, w) is C(0; S,) with 0 initialized by w and R(w) is regularization term. In the optimization model of Eq. (]. the parameter 0(w) of learner trained on S₁ is treated as a function of the initial parameter w. To force the learner trained on S, to generalize well on S,. we directly minimize the loss L( dubbed generalization loss, on val-subset S., w.r.f. the parameter (ww) trained on S. Solving Eq. ① will force the learner to be able to generalize well from any train-subset to corresponding val-subset.
In Eq. (1), Dg (S. S. c S, -
Since [T| may be extremely large, infeasible to enumerate all possible train/val splittings. Thus, we propose the following adversarial splitting model instead,
min max L(0(w); S.) R(w) w S.eT:
s.l. ((w) arg min L(); Sr, w); S, = S
In the min-max problem of Eq. 2 the train/val (S splitting is optimized to maximize the generalization loss to increase the domain shift between train and val subsets by finding the hardest splitting to the leamnr. While is optimized by minimizing the generalization loss of learner over
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the splitting. Solving this adversarial splitting optimization model in Eq. enforces the learner to be generalizable even for the worst-case splitting. We therefore expect that the trained learner is robust to the domain shifts within the training dataset. For the regularization term R(w), set to be the training loss on St (i.e., R(w) = [(w; St)). which additionally constrains that the learner with parameter should be effective S. (Li et al. 2018a). The effect of the hyper-parameter will be discussed in Appendix A.2
In conventional adversarial machine learning, adversarial training is imposed on adversarial samples and learner to increase robustness of the learner to adversarial corruption (Goodtellow et al. (015)). While in our optimization model of Eq. (2). adversarial training conducted on data splitting and learner to force the learner to be robust to domain shift between train/val subsets. Our model bridges adversarial training and meta-learning. It is general learning framework for domain generalization and is complement to adversarial machine learning.
3.2 OPTIMIZATION
This section focuses on the optimization of Eq. (2) Since Eq. 2 is a min-max optimization problem, alternately update S, and by fixing the other known. We should also consider the inner loop for optimization of ((w) in the bi-layer optimization problem of Eq. (2) We next discuss these updating steps in details. The convergence and computational cost of this algorithm will be also
discussed in Appendix A.3 A.3 and A.4 respectivvy.
Inner loop for optimization of O(w). We adopt finite steps of gradient descent approximate the minimizer 0( (ww) of the inner objective [(0; St) with initial value w. This approximation technique has been introduced in machine learning community several years ago (Sun & Tappen 201 Finn
et al. 2017 Fan et al., (2018). For convenience of computation, following (L.i et al. 2018a Dou et al. 2019 we only conduct gradient descent by one step as
(ww) w a vo St)le=m
where a is the step size of inner optimization and its effect will be discussed in Appendix A.2
Optimizing w with fixed S.. For convenience, we denote g'u = VoL(0; (:),:::))a=w Fixing S, (S, is then fixed), wo can also be updated by gradient descent, i.e.,
w = vyV 0 (E(w - ag'ui Sy) + +R(w).
(4)
where 17 is the step size of outer optimization.
Finding the hardest splitting S, with fixed Fixing w, to find S, E Hg to maximize L(w ag'u; Sy), we do first order Taylor expansion for L(w aghi S.) by L(w ogly S,) x L(w; S.) (gu Iw), where dh VoL(0; S. lo= and (:--) denotes the inner product. From the definition of L, ohc and Iw, the optimization problem of maxs,er, (L(w; S.) - a (gin gü) can be written (V..(f((( s.)) This problem is equivalent to
as maxs,er, |s.T Eonyess I (f)( the following splitting formulation:
Σ
(ff((r)))) - (V (V s.t. A= S. E r'e,
max
(5)
(R.))SS
where we introduced an auxiliary variable A. Eq. 5 can be solved by alternatively updating S, and A. Given A, we compute and rank the values of (z) y) a ( (f A) for all (z, y) E 5 and select the largest & samples to constitute the S.. Given S. (S, is then given), we update A by A = d [T Eiayees, V (=((fu(() v). We also discuss details and convergence of this alternative iteration Appendix C Since computing gradient w.r.f. all parameters is time and memory consuming, we only compute gradient w.r.f. parameters of the final layer of learner f.
3.3 L2 12-NORMALIZATION FOR EXTRACTED FEATURE
Ly Eo-oommalization has been used in face recognition Liu et al. 2017 Wang et al. 2018 and domain adaptation (Saito et al., 2019 Gu et al. 2020). but rarely investigated in domain generalization. We investigate Lę nooommalliotooon in domain generalization this paper. It is found surprisingly in experiments that L no-oomaaliztoonn not only improves the performance of learner (see Sect. 5.4 but
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also mitigates gradient explosion (see Sect. 5.5 that occurs frequently during the training of meta- learning for DG (Finn et al. 2017; Dou et al. (019). We next discuss details of Lg- Lg-oommalization in method and analyze why L2 normalizaiiio mitigates gradient explosion.
Feature -ogrorrmaiiition. The Lo-moaaalizttion is used as a component of our learner f. Specif- ically, decompose f into feature extractor f (e.g., the convolutional layers of ResNet), the transform f" representing L--normalization and classifier fo, i..,, f = f f o f The feature of input image extracted by fe is fed to f to output an unit vector 2, i.e.,
(²)(((x))
(a)
The classifier f consists of unit weight vectors W [W1, W2: ... _wx] where K the number of classes and ||well = 1, vk. f takes 2 as an input and outputs the classification score vector am,s(W7 z). om,a() (.) is the marginal softmax function defined by
(2))]] = exp(s(wī - mll (=I((==))))
k = 1,2, K,
(6)
E exp(s(wT<= mlI mll(w=y)))
where y is the label of x, [:]x indicates the k-th element, L[a) is the indicator function that returns 1 is true, 0 otherwise, and s are
indicating margin and radius respectively.
Analysis of mitigating gradient explosion. We find that Lg-normalization mitigates gradient explo- sion in the training of meta-leammn for domain generalization. For the sake of simplicity, we analyze gradient norm of loss w.r.f. parameters of f in the meta-learning process of domain generalization, with f fixed function. Without loss of generality, we consider the case that K = 2 (i.e., binary classification), 8 = and m = O. In this case, we have the following proposition.
Proposition Under the above setting, if the inpul feature of I is Ln=normalized, the gradient norm of loss w.r.t parameters of Je in the meta-learning process of DG is bounded.
Sketch of proof. Given feature 2, the loss of binary classification is [(w; 2) = -y -ylog(o(w"
y) log(1
aww 2)), where is the sigmoid function. Let w w a aVw.(wu 2).
then V V (w); = (T aH) m² [(w';=), where H is the Hessian matrix. The gradient
norm || w Cww' :2)l S ||4 - all | w ((w'22)) I (1 + Jal [[]]) ||V wt [(w";2)]. Since
V C(w'; 2) = (p y)z and H = p(1 p)zz" where p = ow 2), || suPai]] ||/ S
122 ull s ||=|| and |/V w/ ((w';) 2) I | If || 1, we have [(''(w)) z)]l S
supa:||u||=
1 1+ |a.
According to Proposition] D L2 normalizzaion can mitigate gradient explosion under the above setting. The analysis of gradient norm of loss w.E.t parameters of both f and fo in the meta-lcarning process much more complex, left for our future work.
4 THEORETICAL ANALYSIS
This section presents theoretical understanding of our method. We first derive generalization error bound on target domain in theorem T for the general setting of meta-learning for DG. Then, based on theorem we theoretically explain the reason the success of our method.
Without loss of generality, we consider binary classification problem. We denote H. as the set of all possible i..,, H {hr w E RM,, where M is the number of parameters. For any S. E Te and S. = S S.. we let H.s learning approach for DG is to find a function in Hs, to minimize classification loss on S.. Note that, although the training samples in S may be sampled from several distributions, they can still be seen as being i.i.d. sampled from mixture of these distributions. We next respectively denote P = Ed PaPa as the mixture distribution with Ba representing the sampling ratio of the d-th source domain, (f) B(a.))= [I (v) as the generalization error on distribution Q of unseen target domain, E (f) [S.] E (Z.))ss [() as the empirical error on S.., VC(H) as the VC-dimension of H. and (-(-) as the prediction rule such as the Bayes Optimal Predictor. Based on the domain adaptation theory (Ben-David et al. 2007: 2010) and inspired by the analysis in (Saito et al. 2019), we have the following theorem.
[fo(ww (ww) arg ming L(0; St, w), w R² J. The meta-
Theorem 1. Let y be constant, assume Eq ([(L(())))))))))) 2 S, E Tg and S. S - S,, for any (0, 1), with probability at least - 25, we have Vf Hs.,
Ep (u,(())))))))) then given any
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Table 1: Results of MSDS experiment PACS based ResNet18 and ResNet50.
Base-
DFAS
Backbone
Target
D-SAM
JiGen
MASF
MMLD
MetaReg
RSC
line
(ours)
A
77.3
79.4
80.3
81.3
83.7
83.4
80.2
84.2#
C
72.4
75.3
77.2
77.2
77.2
80.3
75.5
79.5
ResNet18
P
95.3
96.0
95.0
96.1
95.5
96.0
95,9110.
95.8 L.0.1
S
77.8
71.4
71.7
72.3
70.3
80.9
70.1 1
82,1 =0.4
Avg
80.7
80.5
81.0
81.8
81.7
85.1
80.4
85.4
A
82.9
87.2
87.9
86. 1
89.1 =0.1
80.5
79.2
82.2
79. 79.2
84.6 +022
ResNet50
P
95.0
97.6
97.9
97. 97.6
96.8=002
S
72.3
70.3
83.5
70.3
85.6=0.3
Avg
82.7
83.6
87.8
83.3
89.0
88 ()) + +2/
8(2++
C2 (c2+ +Cs,
2)
where
B(S.) C1 f'EHs,
inf
Σ Lu(r()))))))
(8)
[yp
(w.v)ES.
C supsper sup supreHs_s: EoLLur(e)wi> C2 sups,er's VCCH Vt st) log VC(H st
2nf
C Z sue inf M' 1sp (f' + + (f')), H fV Hs-s,) V1 is loss-related indicator defined by
vi(f(=)) {
fl(f(z),)) >
(9)
otherwise
Proof is given in Appendix D The assumption of Eo (I(()) > Ep (()) in theo- rem] iss realistic because the data of not accessed at training time, and the learner trained on data of P should have smaller classification loss P than Q. In theorem C1. C2, C3s are constants U1 to f. In Eq. ( the generalization error @ (f) on Q can be bounded by the empirical error We (f)
on S., the term (S,)) that measures the discrepancy between P and Q, and the last two constant terms Eq.
To obtain lower 60 (f), we need to minimize equivalent to
S (f) and B(S)). Minimizing B(S,) w.r.t S, is
max inf S.el SEHS,
Σ
(10)
S.
(a.u)eS.
Intuitively, Eq. (10) means to find a S. E r such that the infimum ratio examples in S. having loss greater than y is maximized. This min-max problem of Eq. (TO) for computing the error bound bears the similar idea as our min-max problem. Our adversarial splitting model in Eq. (2) can implicitly realize the goal of Eq. [10 and meanwhile ensure lower e (f) for any S...
The maximization in Eq. 10) corresponds to our adversarial splitting that finds the hardest val-subset S, for the learner in Eq.
The infimum in Eq. (10) corresponds to the minimization of the loss in Eq. (2) on S, w.r.f. the learner parameterized by ((w) Instead of using indicator function in Eq. (10). in our model of Eq. (2), we choose differentiable classification loss for easier optimization.
5
EXPERIMENTS
We verify the effectiveness of our method in three types of experimental settings: Multi Source with Domain Shift (MSDS) that the training data are from several source domains and there exists domain

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Table 2: Results of MSDS experiment Office- Home based R and ResNet50.
Base-
DFAS
Base-
DFAS
Backbone
Target
D-SAM
JiGen
RSC
Backbone
Target
line
(ours)
line
(ours)
Ar
58.0
53.0
58.4
56,7=03
62.0=022
Ar
64.9
70.2
Cl
44.4
47.5
47.9
47.6=02
48.6
CI
51.8=002
53.5
ResNet18
Pr
69.2
71.5
71.6
71,4=0.1
71.4=0.1
ResNet50
Pr
76.5
77.4=00.1
Rw
71.5
72.8
74.5
72.9
75.2.0.1
Rw
79.4
80.2
Avg
60.8
61.2
63.1
62.2
64.3
Avg
68.2
70.3
Table 3: Results of SSDS experiment on PACS based on ResNet18.
Method
A-CC
A-PP
A-+
C-A
C-P
C-5
P-A
P-C
P-5
S-+A
S-CC
S-P
Avg
JiGen
57.0
96.1
50.0
65.3
85.5
65.9
62.4
27.2
35.5
26.6
41.1
42.8
54.6
SagNet
67.1
95.7
56.8
72.1
85.7
69.2
69.8
35.1
40.7
41.1
62.9
46.2
61.9
BaseLine
63.7 95.66* 63.5**
72.0
86.5="
73.3*
19..2*411.6*11
60.3=2
49.3*1
62.4
DFAS (ours) 67.55-294.5 67.1
69.0
86.5
73.8 70.2
36.11*55.11
26.97
3
57.4
66.6
shift between training and test data, Single Source with Domain Shift (SSDS) that the training data from a single source domain and there exists domain shift between training and test data, and Same Source and Target Domain (SSTD) that the training and test data from same single domain. The source codes will be released online.
We conduct experiments on three benchmark datasets. PACS (Li et al. 2017) contains four do- mains, including art painting (A), cartoon (C), photo (P), sketch (S), sharing seven classes. Office- Home (Volpi et al., 2018). dataset widely used in domain adaptation and recently utilized in domain generalization, consists of four domains: Art (Ar), Clipart (CI), Product (Pr), Real World (Rw). sharing 65 classes. Both of these two datasets are utilized conduct experiments settings of MSDS and SSDS. CIFAR-10 (Krizhevsky et al. 2009) is taken for the experimental setting of SSTD.
5.1 TYPE I: MULTI SOURCE WITH DOMAIN SHIFT (MSDS)
In the setting of MSDS, following (Carlucci et al. 2019). we use leave-one-domann validation, i.e., training on three domains and testing on the remaining unseen domain, on PACS and Office-Home Note that the domain labels are not used during training. We adopt ResNet18 and ResNet50 (He et al.) 2016) pre-trained on ImageNet (Russakovsky et a 2013). For each of them, the last fully-connected layer replaced by bottleneck layer, then the corresponding network is taken feature extractor f Full implementation details are reported in Appendix B
cross-
We compare our method with several state-of -ate-offthhe-rtt methods, including D-SAM (D' Innoceeete Caputo 2018), JiGen (Carlucci et al., 2019), MASF (Dou et al.[ (209)), MMLD (Matsuura & Harada 2020). MetaReg (Balaji et al. (2018). RSC (Huang et al. (020). The results on PACS and Office-Home reported in Table T and Table 2 respectively. Our DFAS achieves state-of-the-an results based on both ResNet18 and ResNet50 on both PACS (85.4%, 89.0%) and Office-Home (64.3%, 70.3%), outperforming RSC by 0.3% and 1.2% on PACS based on ResNet18 and ResNet50 respectively, and by 1.2% on Office-Home based on ResNer18 (these methods do not conduct experiment on Office-Home using ResNet50). Compared with Baseline that directly aggregates three source domains to train learner with standard fully-connected layer as classifier fe, our method of DFAS improves its performance by 5.0% and 5.7% on PACS based on ResNet18 and ResNet50 respectively, and by 2. 1% and 2.1% on Office-Homs based on ResNet18 and ResNet50 respectively. In Table [ on PACS, DFAS significantly outperforms Baseline in almost all tasks except when P taken as target domain. Note that, in the task that domain S, of which the style is extremely different from rest three domains, is target domain, our DFAS boosts the accuracy of Baseline by 12.0% and 15.3% based on ResNet18 18 and ResNet50 respectively. This indicates that our method can generalize well when domain shift is large. Office- Home is challenging for DG since the number of classes is larger than other datasets. As office-Home These
shown in Table 2 our DFAS outperforms Baseline stably in almost all tasks performance improvements demonstrate the effectiveness our method in the case that training data from multi-source domains and the unseen target domain is different from source domains.
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5.2 TYPE II: SINGLE SOURCE WITH DOMAIN SHIFT (SSDS)
We conduct this type of experiment on PACS based on ResNet18. In this experiment, we train learner on one domain and test on each of the rest three domains, resulting total 12 tasks. Implementation details are shown in Appendix B Our method is compared with related methods, including Baseline that directly trains learner with standard fully-connected layer as classifier fe on the source domain, Jien (Carlucci et al., 2019) and SagNet (Nam et al.) 2019). Results are reported in Table [ Our method of DFAS outperforms Baseline and SagNet by 4.2% and 4.7% respectively. We observe that our method outperforms Baseline in 10 tasks among all 12 tasks. The performance boosts are large in tasks when domain S is set to be source domain. These performance improvements demonstrate the effectiveness of our method in the case that training data are from single souree domain and the unseen target domain is different from source domain.
Table 4: Results of SSTD experiment on CIFAR-10 10 based on ResNet18.
Training Size
100
600
1000
3000
5000
10000
Total
Avg
JiGen
38.5
62.9
66.1
76.5
79.2
83.9
91.2
71.2
MMLD
40.5
59.8
65.1
75.8
78.8
82.5
90.7
70.5
Baseline
39.3
62.3
66.4 2
76.3
82.0
85.8
94.9
72.4
DFAS (ours)
43,9
64.3**
69.4
80.0
83.9
87.2
95.1 4.0
74.8
5.3
TYPE III: SAME SOURCE AND TARGET DOMAIN (SSTD)
We also apply our DG method to the common recognition task that the training and test data are from same domain, Le., SS7D, CIFAR-I0 dataset. To investigate the effect of training size, we sample different sizes of training data from the provided training set (i.e., source domain). Implementation details are in Appendix B As shown in Table 4
our DFAS outperforms Baseline, JiGen and MMLD by 2.4%. 3.6% and 4.3% respectively on average. The results of JiGen and MMLD are obtained by running their codes on CIFAR-10. 10. We observe that DFAS outperforms Baseline and compared methods in all different numbers of training data. In general, the performance boost is larger when the number of training data is smaller. This may be because the learner is more possible to be overfitting when the training size is smaller and our DFAS is designed to extract better generalizable features.
5.4 ABLATION STUDY
To further verify the effectiveness of each component of our method, we conduct additional ablation experiments on PACS dataset based on R8sNet18 in both MSDS and SSDS setting. The results are reported in Table and Table
Table Additional ablation results on PACS in MSDS setting.
L2 norm Rand-split
Adv-split
A
C
P
S
Avg
X
X
80.2
75.5
95.9
70.1
80.4
82.5
77.5
95.2
75.6
82.7
X
80.6
76.8
95.6
78.1
82.8
V
83.1
77.0
94.8
79.3
83.6
V
x
83.2
78.6
95.8
79.5
84.3
V
84.2
79.5
95.8
82.1
85.4
In Table and 6 I--norm denotes feature Izz-normalization defined Sect.3.3. Rand-split means the random splitting strategy that randomly split the train/val subsets each step of updating the parameters of learner. Adv-split denotes the adversarial splitting model that we update the worst-case splitting by solving the maximization problem in Eq. (5) per epoch in training process.
Effectiveness of Lg-normalization. Table 5 Lg-norm (82.7%) outperforms Baseline (80.4%) by 2.3% and Lg-norm Adv-split (i.e., DFAS) (85.4%) outperforms Ady-split (83.6%) by 1.8% in MSDS setting. In Table 6. L2-norm 1%) outperforms Baseline (62.4%) by 0.7% and L2- norm Adv-split (i.e., DFAS) (66.6%) outperforms Adv-split (65.1%) by 1.5% in SSDS setting.
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Table 6: Additional ablation results PACS in SSDS setting.
I--norrm
Rand-split
Ady-split
A-xC
A-PP
A-S
C-^
C-P
C-8
PーA
X
63.7
95.6
63.5
72.0
86.5
73.3
68.4
64.5
95.4
67.2
69.8
87.2
73.3
67.2
X
65.8
94.8
64.4
71.2
85.2
74.2
68.7
V
67.6
95.2
66.2
76.9
88.4
73.5
70.4
X
67.2
95.3
63.7
70.8
85.3
75.5
70.0
67.5
94.5
67.1
69.0
86.5
73.8
70.2
noom
Rand-split
Adv-split
P->C
P-S
S->
S->
S-+P
Avg
X
32.7
42.2
41.6
60.3
49.3
62.4
31.7
38.5
44.3
67.9
49.8
63.1
X
32.2
39.5
52.4
64.6
56.6
64.1
V
30.5
35.3
52.7
67.7
57.3
65.1
X
37.2
45.9
53.5
64.5
55.8
65.4
V
36.1
52.1
56.7
67.9
57.4
66.6
These performance improvements demonstrate that the feature Lzznormaliiztiin is useful in domain generalization.
Effectiveness adversarial splitting model. Table Ady-split (83.6%) outperforms Baseline (80,4%) by 3.2% and L2-norm + Adv-split (i.e., DFAS) (85 .4% outperforms L2-norm (82.7%) by 2.7% in MSDS setting. In Table Adv-split (65. %) outperforms Baseline (62.4%) by 2.7% and L2- Lo-norm Adv-split (i.e., DFAS) (66.6%) outperforms L2-norm (63.1%) by 3.5% in SSDS setting. These results indicating that our proposed adversarial splitting model is effective.
Effectiveness of adversarial splitting over random splitting. In Table [ Lg-norm + Adv-split (85.4%) outperforms L2-norm + Rand-split (84.3%) by 1.1% and Adv-split (83.6%) outperforms Rand-split (82.8%) by 0.8% in MSDS setting. In Table Lz-norm Adv-split (66.6%) outperforms L2-norm + Rand-split (65.4%) by 1.2% and Adv-split (65.1%) outperforms Rand-split (64.1%) by 1.0% in SSDS setting. These results demonstrate that the adversarial splitting model outperforms the random splitting strategy in different experimental settings.
Due to space limit, add more ablation experiments Appendix A. to further compare different splittings, including adversarial splitting, Nomain-labee based splitting and random splitting.
5.5 MITIGATING GRADIENT EXPLOSION BY Lz-NORMALLIZATION.
To show that Lo-normalization can mitigate gradient explosion, we conduct the same experiments independently for 50 times respectively with Iz-normalization and without Læ-normalization. Then we count the numbers of occurrences of gradient explosion that are reported in Table From Table 7
we can observe that Lę nz-ommallzzaiinn can mitigate gradient explosion.
Table 7: The number of occurrences gradient explosion. w/o Lz-normalization w/ Inznormalizzatinn 12/50
Times
0/50
CONCLUSION
In this paper, we unify adversarial training and meta-learning in a novel proposed Domain-Fres Ad- versarial Splitting (DFAS) framework to tackle the general domain generalization problem. Extensive experiments show the effectiveness of the proposed method. We are interested in deeper theoretical understanding and more applications of our method in the future work.

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A ANALYSIS
A.1 COMPARISON OF ADVERSARIAL SPLITTING AND DOMAIN-LABEL-BASDD SPLITTING
In the section, we compare our adversarial splitting with domain-label based splitting that is com- monly used in meta-learning based DG methods. Due to variations of style, poses, sub-classes, etc., the internal inconsistency within dataset is complicated. Domain-label partially capture the inconsistency, while cannot cover all possible internal inconsistency. Our adversarial splitting method does not rely on the domain label. It iteratively finds the hardest train/val splitting to the learner to maximize the inconsistency and train the learner generalize well for the hardest splitting, in an adversarial training way. This strategy more flexibly investigates the possible inconsistency within training dataset, adaptively the learner, and can potentially enhance the generalization ability of leamer.
Table Values of objective function in Eq. (5) of Adv-split and Label-split. Learner
W1
W2
W/3
W4
Ady-split
4.15
4.28
2.61
1.28
Label-split
3.72
2.82
1.29
0.21
We first empirically show that the domain-label-baae splitting (denoted as Label-split) as our adversarial splitting (Adv-split) to the learner in Table 8 In Table 8 we report the values of objective function in Eq. (5) of Adv-split and Label-split by fixing the learner with different network parameters w; at different epoch (Ith, 2th, 5th and 10th) in the training process. Larger value in the table indicates that the splitting harder to the learner (i.e., network). can observed that the domain-label-based splitting (Label-split) is not as hard as Adv-split to learner.
hard
We also conduct experiments on PACS in MSDS setting to fairly compare different splittings, including adversarial splitting (Adv-split), lomain-labellbate splitting (Label-split) and random splitting (Rand-split). The results are reported in Table 9 Table shoows that adversarial splitting outperforms random splitting and homain-label--e splitting when training data is from multiple domains.
When the training data are from only single domain, adversarial splitting also performs well (as in Table 3) However, domain-labe basd- splitting cannot be used in this setting, since there is no domain label available.
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Table 9: Results of different splittings.
Target
A
C
P
S
Avg
Rand-split
80.6
76.8
95.6
78.1
82.8
Label-split
81.2
75.9
94.7
80.1
83.0
Adv-split
83.1
77.0
94.8
79.3
83.6
84
84
84-
ADD
83
83
81
81
0.3
0.5
0.7
0.9
16-3
0.10
0.15
0.20
0.25
0.30
(a) Effect of
(b) Effect
(c) Effect of m
Figure 1:
Effect of hyper-parameters of ç, and m. We use ResNet18, and domain is taken as target domain PACS dataset in the setting of MSDS.
A.2 EFFECT OF SYYPERPPARAENT
Effect of hapee-par ç. In Fig. (aa) we show the performance of our method when varying the hyper-parameter &, i.e., length of the val-subset S,, in adversarial splitting of training dataset. The best result obtained when ç 2
|8| and the results are similar when & ranges from 0.3 0.7.
[S]
Effect of hyper-parameter a. We evaluate the effect a in MSDS setting on PACS dataset in Fig- [(b) From Fig. [(b] the ACC is stable the values of in large range of 1e-6 to le-4. Small a results in small step-size for parameter updating in meta-learning framework, and limits the benefits from meta-learning and adversarial splitting. Larger a results in larger step-size for gradient descent based network updating, which may fail to decrease the training loss from the optimization perspective.
Effect of hyper-parameter m. The effect of m is evaluated in MSDS setting on PACS dataset in Fig. I(c)
Fig. I(c) shows that the result sensitive the value of m.
A.3 CONVERGENCE
We testify the convergence of DFAS with errors and losses in different tasks in Fig. 2 In Fig. 2(a) and Fig. 2(b),, we show the classification error curves on target domains (A and Ar respectively) in the setting of MSDS. In Fig. [(c) we show the training loss of DFAS in task A in MSDS setting. These training curves indicates that DFAS converges the training process. We also observe that
DFAS has better stability than Baseline, in Fig. ((a) and Fig. 2(b)
A.4 COMPUTATIONAL COST OF ADVERSARIAL SPLITTING AND RANDOM SPLITTING
We compare the computational cost adversarial splitting and random splitting in this section. Since only update the worst-case splitting per epoch, instead of at each step of updating parameters, the computational cost is only slightly higher than that of random splitting. To show this, we compare the total training times of the adversarial splitting and random spitting in the number of steps (20000), as in Table 10
From Table TO the training time of Adv-split only 5.6% (0.33/5.90) higher than Rand-split.
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Baseline DFAS
Baseline DFAS
3.5
ERROR
ERADR
0.4
2.5
0.3
1.5
0.5
0.0
35
0.2
5
15
25
35
45
1.00
0.0
STEP
STEP
STEP
Target error of task A
(b) Target error of task Ar
(c) Loss of task A
Figure 2: Curves of target emrors and losses of task A (PACS) and Ar (Office-Homel during training based on ResNet50 in the setting of MSDS.
Table 10: Total training time (hour) of the adversarial splitting (Adv-split) and random spitting (Rand-split).
Adv-split 6.23h
Rand-split
Time
5.90h
A.5 VISUALIZATION OF FEATURE SPACE
We visualize the feature space learned by our method of DFAS and Baseline (shown in Fig. [3, by I-SNE (Maaten & Hinton| 2008). It appears that DFAS yields better separation of classes and better alignment of distributions of source and unseen target domains, which possibly explains the accuracy improvements achieved by our DFAS.
B IMPLEMENTATION DETAILS
For the setting of MSDS, we use ResNet18 and ResNe150 (He et al. 2016) pre-trained on Ima- geNet (Russakovsky al. 2015). For each of them, the last fully-connected layer is replaced by bottleneck layer, then the corresponding network is taken as feature extractor f". The dimension of the bottleneck layer is set to be 512 when the backbone ResNet18 as in (Saito al. 2019). and 256 when the backbone is ResNet50 as in (Gu et al. 2020). Following (Gu et al. 2020), is set to 7.5 for PACS and 10.0 for Office-Horn Homee is
0.2 for PACS and 0.I for Office-Home.
181 set to 2. SGD with momentum 0.9 is utilized to update parameters of learner. The learning rate of classifier and bottleneck layer is 10 times of convolutional layers, which is widely adopted in domain adaptation (Long et al. 2015 Ganin et al.) 2016). Following (Ganin et al.) 2016), the
learning rate of convolutional layer adjusted by 1) [1+ 10775.77 where p the optimizing progress
0:001
linearly changing from to 1. The learning rate a of inner loop optimization is set to 10 10-5, The parameters are updated for 20000 steps and the hardest val-subset is updated per 200 steps. The batchsize is set to 64. The running mean and running variance of Batch Normalization (BN) layers are fixed as the pre-trained values on ImageNet during training, which is discussed in (Du al.) 2020a). Due to memory limit, when implementing experiments based on ResNet50, we adopt the first order approximation (Finn et al. 2017) that stops the gradient of g' in Eq. for reducing memory and computational cost.
For the setting of SSDS, we conduct experiment based on ResNet18 on PACS. The implementation details are same MSDS. For the setting of SSTD, we conduct experiment based on ResNet18 18 CIFAR-10. The parrempprreeters of s and m are set to 8.0 and 0.2 respectively. Other implementation details are as MSDS except that, in BN layers, the running mean and running variance updated.
We implement experiments using Pytorch Paasze et al. 2019 single NVIDIA Tesla P100 GPU.
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A (target) (source) (source)
dog
elephant
giraffe
(source)
guitar
horse
house
person
(a) Baseline (domain)
(b) Baseline (class)
dog
elephant
giraffe
(target)
guitar
(source)
horse
(source)
house
(source)
person
(c) DFAS (domain)
(d) DFAS (class)
Figure 3: The t-SNE visualization of extracted features, using our proposed DFAS (a-b) and Baseline (c-d) on PACS dataset in MSISS setting. In (a) and (c), the different colors indicate different domains. In (b) and (d), the different colors indicate different classes
C OPTIMIZATION ALGORRIHM FOR FINDING THE HARDEST S,
C.1 OPTIMIZATION ALGORITHM
To solve the problem of
Σ ((( A) s.t. A= gins S. ETg,
max S (E.M)ES.
(11)
we design an alternative iteration algorithm in Sect. 3.2 Specifically, we initialize A with gradient of sample randomly selected from S. Then we alternatively update S, and A.
Given A, S., is updated by solving
Σ
- (V..
max S.
(2,0)ES.
(12)
s.l. S. C,
where the constraints are derived from the definition of Tg. Equation (2) indicates that the optimal
S. consists of & samples that have the largest values of l (z). v) (V))(f((())) A). Thus we compute and rank the values of (f.. - a (V (x), y). A) for all (x, y) E S and select the largest samples constitute the S..
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Given S, (S, is then given), update A to satisfy the constraint A Ahe in Eq. 11). then A is
1
A Dw
Vul(fu(z).g))
(13)
(a.v)ES,
C.2 CONVERGENCE EXPERIMENTS
We show empirically the convergence of this alternative iteration algorithm in Fig. with the values of objective function in Eq. 11 Fig Figure 4 shows that the values of objective function converges after only a few iterations.
3.8
Objective Function
Function
3.7-
3.7
3.6
3.6
3.5
3.5
Iteration
Iteration
(a) Run
(b) Run 2
Figure 4: Convergence of the alternative iterations for finding the hardest S,, (a) and (b) respectively show the values of objective function in Eq. (11 in different runs with different initializations.
We also check the splitting changes when the value of objective function converges. To do this, count the ratio of changed sample indexes in S, at each iteration, as in Table I1 Table [1 shows that splitting not changed when the value objective function converges.
Table 11: Ratio of changed sample indexes in S, at each iteration.
Iteration
2
3
4
5
6
Ratio (%)
44.0
4.5
0.0
0.0
0.0
0.0
C.3 TOY EXAMPLE.
We present a toy example in this section check if this algorithm can find the optimal solution. The toy example is an 2-dimensional classification problem, shown in Fig. Different colors indicate different classes. A fully-connected layer without bias is used as the network (learner). We split the data of the first class (blue points) with our algorithm. The candidate solutions with corresponding objective function values are given in Table 12
Table 12: Candidate solutions (sample indexes) with objective function values of the toy example.
Candidate S,
(0.1,2)
(0.1.3)
(0,1,4)
(0,1,5)
(0,2,3)
(0,2,4)
(0,2,5)
Objective function
6.71
7.65
7.88
6.41
5.60
6.83
5.36
Candidate S.
(0,3,4)
(0,3,5)
(0,4,5)
(1,2,3)
(1,2,4)
(1,2,5)
(1,3,4)
Objective function
7.77
6.30
6.52
7.27
7.50
6.03
8.45
Candidate S. (1,3,5)
(1,4,5)
(2,3,4)
(2,3.5)
(2,4,5)
(3,4,5)
Objective function
6.98
7.20
7.39
5.93
6.15
7.10
The solutions in the iteration process of our algorithm are reported in Table to (1, (1,3,4), which is the optimal solution in Table 12
The solutions converge This indicates that algorithm find the
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Figure 5: Toy data.
Table 13: Solutions of the iteration process.
Iterations
1
2
3
4
Solution
(0,2,5)
(1.3,4)
(1,3,4)
(1,3,4)
(1,3,4)
Objective function
6.89
8.43
8.45
8.45
8.45
optimal splitting for the toy example. We also give the code of this toy example as below, and the reader may rerun it to verify the results.
Code the toy example:
import torch
import numpy np
import matpletlib. pyplot plt
from sklearn. datasets import make_blobs
#####
np ro..rancm.eede (eed
data make make_bloke blobs (n_ samples=12, centers=2)
x1 data[0] [data [1]ca[]]=] [1]==0]
x2 data[0] [data [1] --1]
plt plot (xl]: 0] 11,
plt.plot x2)
t= 0
for x in xi:
pet.text [0], st),
t+=1 samples.pdf" sampless.
plt.savefig
###
model torch.nn. Linear 2, euda
torch. int.iiii..tiiiiiii...... uniform. (model.weight)
log = open Loogtttoottxtt'"'1
log. w'weight: str (model. weidel..eeigh..pp. cpu datadaaaannn numpy ()) An")
feat torch. Tensor (data((])) cuda .uda) ()
label torch. LongTenoo (data cuda
loss_ = corchhnn... roycssst
Grads []
Loss []
for i in range (Len (feat))
out =
torch. nn. funct onal. near (feat [1] view (1, -1), weight=model. www.mmooeel.eeeooo....m bias- NoneeNe)e)
1o88_x loss_ fun (out, 1abel [i] view (-1,)
17

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Under review as a conference paper at ICLR 2021
grad corchhautoore grad (lcss_ moddel..eighht [0]
Loss. append (loss_x, cpu () adtaaaaaaa. dump. ())
Grads fodds.pper append (grad. () data. numpy ())
##we split the data of the first class Loss np.array (Loss) [data
Grads np.array (Grads) [data [1] 0]
save Grwds.aarryy.. Grads)
savesvve' Loss Lpyp.
Loss np. ng.load(tto2/oos.nnnn) [data[1] [data 0]
Grads np. bor.ooddooo__raad..ppppppp [data[1] -- 0]
alpha = 001
def adv adv_loss loss (val_ (0,
train_index np. np.del (np. arange (len (Loss)) ) loss = menn (Loss [val_indxx]]
grad_val np. mean (Grads [val_ axis=0)
grad_train np.mean (Grads [train_index] axis=0)
return loss - supharmp.sum
#fffbrutt force searching
Solutions
Values []
Igenerate solutions
for i in range (len [Loss--]-
for j in range len (Loss) -1):
for k range i+1, len (Loss))
Solutions. aatprinnn.pppdd (Ii, i,k]
for val_idex in Solutions:
Values append (adv_loss (val_idex))
optimal_ide np.array
inn = Solutions (optimalll optimal_value max (Values)
print (*all possible solutions:"
Selutions) "natiionn",SLLutionnss
print ("objective function values of possible solutions:" Values) print ("optimal
"Inobjective function value
of optimal pttuuiiion..ootmmaa___..
log. write (stal possible
"objective function values of possible solutions" :alluess "optimal sclution" :oppinn":ootmmmmsslltt "objective function value of optimal solution" :oputioon::ptimmll.. iomtimal_allllle
##ffff our aigorithm
Grads [np. random. randint (1en (Loss)
values_tmp
solutions_tmp []
mtr_indes
sixesss) rrp.random.chae choice imp.arange (len (Loss) size=len (Loss) //2, raaeaceaaae)
mp..ean (Loss- alpha=np. sum (Grads*A. reshape (1,2,2) 2)) axis= (1,2))
(Loss-
salles__tppapppedd
solutions_tmp.qae append (np. arange (len (Loss) ) mtt_ .mtr_index)
for i in rango
D up. ( (Arass
Loss Losa-alpha+D
idx_sort np.argsort (Loss_)
mtr index idx idx_sorti (Less) // 2]
mte index idx sort [Le (Less) // 2
18

---

Under review as a conference paper at ICLR 2021
values twp.pappppp..pe..... append (Loss [mte_ index] mean ())
solutions_tmp append append (mte_index)
A = np. mean (Grads [mtr_index], axis=0)
print (*our optimal "olutioon:
"(nojjetiivee function
value of our optimal solution: seaatiionn.vllus__ttgg.. vaalossttmpp11] log.write (str ((o optimal solution":soluttiaa [tsssmmmp]]]]]___
"objective function value of our optimal solution "aauuiion=valleeo :values_tmp [-1])))
print (solutions__i tmp,, values_tmp
D PROOF OF THEOREM
We first introduce VC-dimension based generalization bound and domain adaptation theory in Appendix D.I then present two lemmas in Appendix D.2 and finally give the proof of Theorem Appendix D.3
D.1 PRELIMINARY
VC-dimension based generalization bound.
Theorem A-1. (Abu-Mostafa et al.) 2012) Let S be the set of training data i.i.d. sampled for distribution P. For any E (0, 1), with probability at least 1- 8, we have vh (h X - {0, 13) in hypothesis space H.
(ep(h) - {s(h)]
8 |s]
(VC(H) log VC(H) +5)
2e S1|
+ 4
(14)
where (p(h) = B(r.))=(([(((>)))) and ēs(h) = [S E(uwess ((
Domain adaptation theory.
Theorem A-2. (Ben-David et al. 2007 2010 For any h in hypothesis space H. have
ca(h) s cp(h) + du (P,Q)+X)
(15)
2
where infmee((<<(M) + cc())) and
dn(P, Q) = 2 sup Ep 1] - Ee(h 1] kEH
(16)
is H-divergence.
D.2 LEMMAS
Lemma A-1. For any S, E Tg and S. S Su. 45 (0, 1), with probability least have ff E Hsp
8 S.al
2e [. VC(H)) + 8)
1c)(5) -
VO(Hg) ) log
(17)
where (f) = B( is generalization enror on distribution P, E (f) = TSTT E(SneS Lui((n) is empirical error, H.. {v f Hs,), VC(H is the VC-dimension of Hy and v())) is the prediction rule such as the Bayes Optimal Predictor, i.e.
I
= [[())))=[()))2)))
Proof:
19

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Under review as a conference paper at ICLR 2021
From the definition of Hg. for any f E Ns. there exists hy E Hss such that hy v Applying Theorem A-I with probability at least 8, we have vf Hs..
())()) - E (f) (1y - -E. (1y)
8
2e [e +
4
(18)
VC(H)) log VC(H)) 8
Lemma A-2. For any S. E [g and St = S Su, let cp(g) inf JENs, cl and &g (h) = inf JeHs, 6s (f), then Vō E (0, 1), with probability at least 1 6, we have
(g) 2 E (h)
8
(V((g) log vC(H5,)
2e S.
+
(19)
\ |5o1
Proof:
From the definition of g and h, we have (3 2 E (h). v8 E (0, 1), with probability at least 1 we have
xg(0) 55. (h) (g)
5 (g) + + (0) 5 (h)
(0)
E5 (g)
(20)
8
2c Se| 4 (VCHH)) log VC(H).
|Sp1
Thus, Eq. holds.
D.3 PROOF OF THEOREM I
Proof:
We denote by nt the hypothesis space such that vh H
h(x) v,((()) { if if((f()) 9) n
(21)
otherwise.
for f E H. Then
dy (P,2) =2 sup h
|E [ 1| Eg/h=
2 sup feH
= Ep [f]
1I
2 sup SEH
-
(22)
sup feH
[Ellum
< 2 sup Eg B -2 inf Ep Er SEM
fek
In the fourth equation, we utilize the assumption that Eo 2 Epl Given any S, Tg and S, S.,, replace H by Hs, then
d Ha.
(P <2 sup Bellureh.>>>1 -2 2 inf JEHS.
fEHs,
(23)
<20 - inf Ep(4n(())))))))) JEHs,
20

---

Under review as a conference paper at ICLR 2021
where C1 supstrr, supsENs-e, Be (((()))))))) Applying Theorem A-2 for any Ns.
we have
(f) 0 (f)+
inf Ep Selluureee))nn))+xx366... MEHa
(24)
where XT(),) 2 inf (f) + (f)) Let C3 sups,er, (*()))
2
sups,Er inf pnHs 16p (f' + ( (f'), we have
Ny 2 ( - 9 (f) C1
inf M'Hss
(pHla(((r + C3-
(25)
Applying Lemma A- to the first term of right side in Eq. 25 vō (0, (), with probability least 1 6, we have v E Hs
M cp
(f) S
8
VC(H log
2e Se VC(H)))
+
4 8
(26)
|
Applying Lemma A-2 to the third term of right side in Eq. 1 6, we have
25
4g E (0, 1), with probability at least
inf Ep([
2
[(((()))))))))
S'EH..
f ee,
1S
(x.y)
(27)
8 S
(VC(H) log
2e Se| VC(H..
4
+
and thanks to union bound, for any E (0, 1), with probability at least 1 20, we have V H.s.
Combining Eq. 25
26
2e S.|
(f) 55 (f) B(S,) ((SS)+22 +
VC(H log
+ C3.
(28)
Wi VC(H
8
where B(S.) Cj inf Ess.
E(anyes. [((r())))))) Using the fact that [Sol] & and let 20g
[8+]
C2 supseer' VC(H s/) log
have
vC(H
8
4 8
( - ()) + B(S - -2
C₂
+ C3.
(29)
S.
21


Reviews:
[{"review_id": "kAQhEcf1NQ5", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "##########################################################################\nSummary:\n\nThe paper proposes a new approach for domain generalization which minimizes the generalization error across train-validation split with the largest domain gap. The paper further gives theoretical bound on the generalization error of the proposed method.\n\n##########################################################################\nPros:\n\nThe paper proposes a new approach for domain generalization. The idea of perform meta-learning from the train to validation split is already employed by previous works (Balaji et al., 2018; Li et al., 2019b; 2018a; Dou et al., 2019). The novelty of the paper falls mostly on performing meta-learning on the train-validation split with the largest domain gap. \n\nThe paper gives a theoretical bound on the generalization error to unseen target domain.\n\n##########################################################################\nCons:\n\nThe paper needs to update the model parameter, the initialized parameters and the train-validation split, which may not converge or converges very slowly. Though empirical results show that the convergence of the method is fast, some theoretical demonstrations are needed for the convergence speed of different updates.\n\nThe contribution of the paper is a little incremental. Meta-learning based domain generalization methods are not novel. The only novelty of the paper is performing meta-learning on the train-validation split with the largest domain gap, which is an incremental contribution over meta-learning based domain generalization (demonstrated by the ablation study).\n\nAs shown in Table 1, Baseline w/L2 outperforms Baseline by 2.3% but DFAS outperforms Baseline w/L2 only by 2.7%. This means that the main performance gain is from L2 while the other contributions has very small performance gain (less than 1%). However, L2-normalization (Finn et al., 2017; Dou et al., 2019) is a widely-used techniques for meta-learning and domain generalization, which is not counted as a novel contribution for the paper. The authors need to further demonstrate that the main contribution: meta-learning on the train-validation split with the largest domain gap, has huge performance gain.\n\n##########################################################################\nI lean to rejecting the paper since the performance gain is mostly falls on the L2-normalization but the main contribution of the paper: performing meta-learning across the train-val split with the largest domain gap, does not have much performance gain.", "extra_scores": {}, "extra_text": {}}, {"review_id": "iaQUmPu090w", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "$Paper$ $summary$\n\nThis paper focuses on domain generalization, targeting the challenging scenario where the training set might not include different sources; even under the presence of different sources, the problem formulation does not takes into account domain labels. The proposed solution is based on meta-learning, following the path drawn by Li et al. AAAI 2018; the Authors propose to adversarially split the training set in meta-train and meta-validation sets, and then update the current model in a direction that fosters good generalization performance on the meta-test. Results on standard benchmarks are encouraging.\n\n$Pros$\n\n- The proposed idea is rather interesting, enabling to apply meta-learning solutions also in absence of domain labels. In particular, I like the idea of finding meta-train and meta-test splits in an adversarial fashion. This is crucial, since randomly splitting the training set in meta-train and meta-validation would not be helpful, since it would lead to episodes where meta-train and meta-test are iid.\n\n- The Authors provide a theoretical interpretation of their approach (due to my background, I was not able to properly review it though)\n\n$Cons$\n\n- My main concern is related to the way the maximization problem is tackled, in the objective in Eq. (5). I have reviewed Appendix C, but I cannot understand how convergence would require so few iterations. Even restricting the size of $S_v$ as mentioned in Section 3., the number of possible sample combinations that generate couples ($S_v$, $S_t$) is huge -- if I understood the process correctly, then with $|S_v|=K$ and $|S|=N$, the number of combinations is $\\binom{N}{K}$ -- and for each of them the gradients that lead to the meta-update are different. Could the authors comment on this? Am I missing something? Related to this point, I am also concerned by the ablation study in Table 5 - where it is shown that, while helpful, the adversarial strategy does not help very significantly in the whole picture.\n\n- The overall writing could be improved. Sentences like \"this model can be further transfored to a minimax problem\" in the Abstract are not properly exposed, and there are several examples throughout the manuscript. There is a misconception related to prior work: Carlucci et al. 2019 (as well as Volpi et al. 2018) also tackles the case where the training data only comprises a single source domain. This should be clarified in the Introduction/Related work. \n\n$Review$ $summary$\n\nI like the idea this paper starts from, and I like the proposed solution. I still do not properly understand how the maximization problem at the core of the method is approached, and I believe that the paper needs some exhaustive proof checking to improve the overall writing. I look forward reading the Author response and iterating the discussion.\n\n---- Post-rebuttal comments----\n\nI thank the Authors for their explanations. Yet, I still believe that this work is not ready for publication. Random splitting and adversarial splitting perform very comparably (1% is not a lot), in my opinion casting some doubts on how meaningful the solution found to the proposed optimization problem is. The Authors included a toy example in the Appendix, but this did not mitigate my concerns on the original manuscript's experiments. I still believe that the core idea is very interesting, and hope that it will be further investigated by the Authors for a subsequent submission.  ", "extra_scores": {}, "extra_text": {}}, {"review_id": "c-BE-OIml_", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes to unify adversarial training and meta-learning in domain-free generalization where labels of source domains are unavailable. To maximize the domain shift between the subsets of meta-train and meta-val, adversarial training is leveraged to find the worst-case train/val splits. Extensive experiments on benchmark datasets under different settings demonstrate the effectiveness of the proposed method. \n\nPros:\n+ The idea of adversarial train/val splitting in meta-learning is interesting.\n\n+ The paper provides extensive experiments on benchmark datasets under different settings with multiple/single source domains and achieves state-of-the-art results on both PACS and Office-Home datasets.\n\n+ The paper provides an upper bound of the generalization error on unseen target domains, which is implicitly minimized by the proposed method.\n\nMajor Cons:\n- The effectiveness of adversarial train/val splitting is not well justified. Typically, meta-learning based methods (MASF and MetaReg) have no overlap between domains used for meta-train and meta-val. This is how the domain discrepancy between meta-train and meta-val can be modeled. However, DFAS has the issue that the same domain may be used in both meta-train and meta-val, which significantly limits the domain transportation and may yield limited domain generalization capacity in comparison with MASF and MetaReg. So why DFAS outperforms MASF and MetaReg in terms of domain generalization? More experiments and discussion are suggested to justify this point. \n\n- The difference between adversarial splitting and domain-label-based splitting is unclear.  Additional experiments are suggested to empirically show the difference.\n\n- The proposed adversarial train/val splitting is computationally expensive since it needs to rank all samples in each iteration. According to the ablation results on PACS (Tab.5), improvements over randomly selected train/val splitting seem quite limited. \nMore details about the convergence of alternative iteration should be provided. \n\n- Fig.4 shows the objective function remains unchanged after the 3rd iteration. Will the splitting results be changed after this stage?\n\nMinor Cons:\n- It would be better to see an ablation study on the effect of margin m in Eq.6.\n\n- Standard deviation of the reported results should be provided.\n\nAfter rebuttal:\nIt is highly appreciated that the authors provide additional evidence in response to my reviews. My previous concerns about effectiveness and efficiency are addressed to some extent, however, this also means the original submission needs significant modification to address the concerns. I like the idea in general and the problem is well-motivated but it needs more work for a complete version. I would encourage the authors to further improve the quality of the paper.\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "Ccecr0x1VeZ", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper provides a novel way to combine meta-learning and adversarial training for domain generalisation. Different from existing methods, the authors propose to split the training dataset into train/val subsets in an iteratively adversarial way, regardless of domain labels, by which the model can be trained to learn to generalise well from training subset to val subset via meta-learning in each iteration.\n\nReason for score:\n\nOverall, I vote for accepting. It’s ingenious that the paper proposes to ignore domain labels to enhance the learned model’s generalization ability towards domain shift problems. The major concern of mine is that the paper provides limited explanation of how the meta-learning algorithm, Model-Agnostic Meta-Learning (MAML), is utilized in the training process(see detailed in cons).\n\n1. The paper proposes to discard domain labels when training a model to generalise well, by which the original training set containing several domains can be seen as a large labelless domain and then splitted into training/val subsets to simulate domain shift, hence MAML can be used here for the training process.\n\n2. The authors incorporate the min-max optimization method following adversarial training to let the model be more effective in learning domain shifts between training/val subsets.\n\n3. One of the paper’s contributions is that it surprisingly find that L2 -normalization can help mitigate gradient explosion in the MAML algorithm.\n\n4. The paper provides comprehensive experiments of different domain shift settings as well as theoretical proofs to evaluate the proposed method, which are quite solid and convincing.\n\nCons:\n\n1. It would be better if the paper provides more details in ablation study, for example, it mentions that DFAS-3 finds the hardest val-subset only based on loss by setting \\alpha = 0 in Eqn. (5), there can be more analysis about the hyper-parameter of α .", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper is proposing a domain generalization method based on the intuition that an invariant model would work for any split of train/val. Hence, the method uses adversarial train/val splits during training. The paper is reviewed by three expert reviews and none of them championed the paper to be accepted. I carefully checked the reviews and the authors' response and agree with the reviewers. Specifically:

- R#1: Argues that the paper is not ready for publication. Also argues the optimization problem is only a motivation as it is not directly solved. This is an important issue and it needs to be addressed in a conclusive manner.
- R#2: Argues empirical studies do not show the value of train/val splitting. I partially disagree with this issue but it is clear that more qualitative and quantitative study is needed to properly justify the proposed method.
- R#3: Argues the contribution is not enough for publication. The paper is clearly novel but the contribution and novelty is not presented in a clear manner. Moreover, the empirical study does not complement the novelty. Hence, I disagree with the comment.

Overall, I believe the paper proposes an interesting idea. However, the presentation and empirical studies need to be improved significantly. I recommend authors to address these issues and submit to the next conference.


Decision: Reject