Stablizing Adversarial Invariance Induction by Discriminator Matching
Authors: Yusuke Iwasawa
Kei Akuzawa
Yutaka Matsuo
Venue:  (2020)
Keywords: invariance induction
adversarial training
domain generalization

Abstract: Incorporating the desired invariance into representation learning is a key challenge in many situations, e.g., for domain generalization and privacy/fairness constraints. An adversarial invariance induction (AII) shows its power on this purpose, which maximizes the proxy of the conditional entropy between representations and attributes by adversarial training between an attribute discriminator and feature extractor. However, the practical behavior of AII is still unclear as the previous analysis assumes the optimality of the attribute classifier, which is rarely held in practice. This paper first analyzes the practical behavior of AII both theoretically and empirically, indicating that AII has theoretical difficulty as it maximizes variational {\em upper} bound of the actual conditional entropy, and AII catastrophically fails to induce invariance even in simple cases as suggested by the above theoretical findings. We then argue that a simple modification to AII can significantly stabilize the adversarial induction framework and achieve better invariant representations. Our modification is based on the property of conditional entropy; it is maximized if and only if the divergence between all pairs of marginal distributions over $z$ between different attributes is minimized. The proposed method, {\em invariance induction by discriminator matching}, modify AII objective to explicitly consider the divergence minimization requirements by defining a proxy of the divergence by using the attribute discriminator. Empirical validations on both the toy dataset and four real-world datasets (related to applications of user anonymization and domain generalization) reveal that the proposed method provides superior performance when inducing invariance for nuisance factors. 

Under review as a conference paper at ICLR 2020
STABILIZING ADVERSARIAL INVARIANCE INDUCTION BY DISCRIMINATOR MATCHING
Anonymous authors Paper under double-blind review
ABSTRACT
Incorporating the desired invariance into representation learning is key chal- lenge in many situations, e.g., for domain generalization and privacy/fairness con- straints. An adversarial invariance induction (All) shows its power on this pur- pose, which maximizes the proxy of the conditional entropy between representa- tions and attributes by adversarial training between an attribute discriminator and feature extractor (Xie et al. 2017). However, the practical behavior of All is still unclear as the previous analysit sis assumes the optimality of the attribute classifier. which is rarely held in practice. This paper first analyzes the practical behavior of All, indicating that All has theoretical difficulty bound of the actual conditional entropy, and All catastrophically fails to induce invariance even in simple cases as suggested by the above theoretical findings. We then argue that simple modification to All can significantly stabilize the adversarial induction framework and achieve better invariant representations. Our modification is based on the property of conditional entropy; is maximized if and only if the divergence between all pairs of marginal distributions over representa- tions between different attributes is minimized. The proposed method, invariance induction by discriminator matching, modify All objective to explicitly consider the divergence minimization requirements. Empirical validations on both the toy dataset and four real-world datasets (related to applications of user anonymization and domain generalization) reveal that the proposed method provides superior per- formance when inducing invariance for nuisance factors.
maximizes variational upper
INTRODUCTION
Extensive studies have demonstrated that deep neural networks (DNNs) can uncover complicated variations in data to provide powerful representations that are useful for classification tasks (Hinton al. 2006 Krizhevsky et al. 2012). However, in some scenarios, the learned representation should be invariant to some attribute of the input data. motivating example is a task called domain gen- eralization (Blanchard al.] (01IT) which requires learning a domain-invariant representation that applies to unseen domains (e.g.. the data of an unseen user or different image sources). When prac- titioners apply DNNs data that include large amount of user information (such images with usernames (Edwards & Storkey 2016) or data from wearables (Iwasawa et al. (017)). the desired representations should not include user identifying information. For legal and ethical reasons, ma- chine learning algorithms must make fair decisions that are independent of sensitive variables such as gender, age, or race (Louizos et al., 2016). Therefore, this study aims to answer the following question: how can we systematically incorporate the desired invariance into representation learning?
Invariance induction is systematic solution this problem. which often introduces an additional regularization term that measures the level of invariance (Edwwrdd & Storkey 2016 [wasawa et al.) 2017 Xie et al.] 2017). One theoretically sound metric is the conditional entropy attributes given representations of data H (a]z) = Ep(s.a) |- log p(a)z)) where and denote the random variables of attributes and representations, as it is maximized if and only if the representations are invariant to attributes. However, exact calculation is intractable p(a[2) is unknown in general.
Recently, Xie et al. (007)) introduced a sensible way to approximate the conditional entropy (Fig- ure [ (a)), which uses probabilistic attribute classifier 4s. where & represents the parameter of the classifier, to approximate the conditional probability distribution p()/z). One can estimate the de- sired conditional entropy by simply substituting p(a) 2) to the approximated conditional distribution

---

Under review as a conference paper at ICLR 2020
label dasaiter//
encoder
whiir label loss
arbiral
cassifirre,
attribute classter
atrilutte disstication
dasoifcttion
g
perception
dooimbunar motching
(a) Adversarial invariance induction
(b) Proposal
Figure 1: Comparison between the previous method and the proposed method. As with All, the proposed method ensure the invariance to the nuisance attribute by deceiving the external attribute classifier, but the proposed method deceives it deferentially. The detail is explained in Section 4.
(al=). The estimated conditional entropy is then used to update the weights of the encoder fo so that the updated representations increase the approximated conditional entropy. Since fo and qs is an adversarial relationship, this framework is called adversarial invariance induction (All). A simi- lar approach was extensively used domain generalization, fair-prediction, and
contexts (Edwards & Storkey 2016 Motiian et al. 2017 Xie al. 2017 Twasawa al. 2017).
Build upon the above achievements, this paper examines a way to improve the stability of the ad- versarial invariance induction framework. Note that, Xie et al. (2017) already demonstrated that the above procedure (alternatively optimizing attribute classifier and feature extractor) possesses an equilibrium where the encoder maximizes the true conditional entropy. However, their analysis im- practically assumes (s(a]2) p)a]z), so little is known about the practical behavior of All and how to improve it. Indeed, several prior works report unsuccessful results of All (Xie et al., 2017 Moyer et al. 2018). To further examine the practical behavior of All, we first analyze the behavior of All from the lens of the variational approximation. Our analysis shows that All maximizes the variational upper bound of the conditional entropy H (a]z). and All objective itself is not upper- bounded. This result suggests that the optimization of AII, without the assumption of the optimality of (a(a)), does not need to maximize the conditional entropy, does not give any guarantees. Empirical validations show that All catastrophically fails to induce invariance even in satisfactory simple situations, as suggested by the above finding.
the maximizing upper bound
We then argue that the simple modification to All attains better property from the optimization perspective while achieving the same goal asymptotically. Our modification is based on the prop- erty of the conditional entropy, i.e., it is maximized if and only if the divergence between all pairs of marginal distributions over between different attributes are minimized (under the uniform assump- tion of p(a)). This property suggests that the invariance induction algorithm should also minimize the divergence, which is not considered on All and possibly induce unstable behavior. In contrast, our proposed method, invariance induction by discriminator matching (IIDM), explicitly considers divergence minimization requirements by minimizing the proxy of the divergence between the marginals, which push the representations with different attributes are recognized similarly by the discriminator (as shown in Figure (b)). We discuss the relationship among our proxy, the diver- gence between the marginals, and non-saturating heuristics used in different but related community (Goodfellow et al. 2014).
The main contributions of this paper can be summarized follows. (1) We highlight the prac- tical issues of AlI, which is a state-of-the-aar framework for invariance induction. (2) We pro- pose modification to All by explicitly consider the property of the maximum conditional en- tropy. Empirical validations on the toy dataset and four real-world datasets demonstrate that the proposed method provides superior performance and faster convergence the level of in- variance induction. For example, in the experiments on toy dataset, the proposed method con- verges to mostly the maximum conditional entropy, while the optimization of All is catastroph- ically unstable. Gif visualizations available at https: / ive. ve. gooqle le com http//pen?ll ?id= 1N2kuTCfW (eWp::/wwwEECee.xTxxSSSSSeSo
(see Section 4.3 for more detail).
2 RELATED WORKS The goal of the invariance induction is to learn attribute-invariant spaces given training dataset made of tuples of { (xm Vn: [n)] (supervised setting) or pairs of {#n, Dn) } (referred as
N
2

---

Under review as a conference paper at ICLR 2020
unsupervised setting (Jaiswal et al. 2018)), where in is observation and yn is a target of Fn, an A is a realization of a categorical random variable a. This setting arises in many application areas. For example. [wasawa et al. (2017) considers the wwwacy-preservingg activity recogmition problem using the data of wearables, where one wants to learn user invariant representations while keeping the information about activity into the representations. In this case, the attribute corresponds to some sensitive user information, and y is the activity label. In the context of fairness-aware decision making, one needs to ensure the algorithm away the sensitive attributes a (such as gender, age, or race) from the representations (Edwards) & Storkey 2016). To build a classifier robust the domain shifts (called domain generalization Blanchard et al. (2 learning representations invariant to domain shifts is a popular approach (the domain label is the attribute in this case) (Muandet et al. 2013 Ghifary et al. 2015).
make fair decisions, which often done by statistical
Assume fo is function (encoder) that parameterized by neural network, which maps observations x to representations = E 2. Typically, the invariance induction algorithm use differentiable dependency measurements and use it as a regularization term to learn fo: E [L((fo(T)) Vn.) + (fo()))))) an)], where a À is the weighting parameter, V somehow measures invariance of the representations regarding the attribute, and L is loss function that represents how much information about y is present in the representations.
Adversarial invariance induction (AII) is recently proposed approach for measuring the V by an external neural network. That is, if the external network can accurately predict a from 2 = (x)). All considers have considerable information about The external neural network is often called as a discriminator or adversary in this context. Information from the discriminator is used to update the weights of the encoder fo that the updated representations have less information about a. Mathematically, the method optimizes following min-max game:
min max Ep(x,o) I- log qo (vnlzn= 0,0 Φ
+ Alog (=(anl=n=f((x.))
(1)
where (vl-) and (al-) is a conditional probability distribution approximated by categorical classifier parameterized by vi (for y) and & (for a) respectively. By alternatingly (or jointly with the gradient reversal layer (Gan et al., 2016)) optimizing 0 and o, this framework ensures that there is little or no information about the nuisance attributes in the representations.
As Xie et al. (017)) demonstrated, the above min-max game can be regarded as a way to estimate the conditional entropy H(a and use the estimates to induce the invariance. The key advantage of All that this framework does not depend on the pre-defined metries such as L2 distance and maximum mean discrepancy (Zemel et al. 2013, Li et al. 2014 Louizos et al.) 2016)), which is often easy to compute yet less powerful. fly, Xie et al. (2017) demonstrated that the above adversarial game has an equilibrium where the encoder maximizes the true conditional entropy, under the assumption that qo (a) 2) correctly estimates the true p(alz). However, their analysis assumes the optimality of qs (a]z), which is rarely held in practice. This paper analyzes the behavior of All without such an assumption to examine the practical behavior of AlI.
It is noteworthy that the above formulation resembles the original formulation of GAN (Goodfellow)
et al. 2014) and domain adversarial networks (DAN) (Ajakan et al. 2014 Gan et al. 2016), which used practically. Instead, they often incorporate several heuristics, e.g., non-saturating heuristics in GAN, and asymmetric mapping in domain adaptation (Tzeng et al.] 2017). These works motivate us to replace the min-max game of the adversarial invariance induction problem.
Although no prior works in the invariance induction community have been explicitly considered yet, one can transfer the non-saturating heuristic used in the GAN via label flipping:
min E 0
log (a = (xww))
ayzan
which also enhances the discriminator to misclassify the attribute of the data, In the remainder of this paper, refer this version the non-saturating version and denote it by NS. As discussed in the GAN community, this objective provides meaningful gradients because it not saturated even if the discriminator is a supremum. Later, we discuss the connection between the proposal and this objective, and how our proposal attains better properties while inheriting the non-saturating nature of the NS objective.
3

---

Under review as a conference paper at ICLR 2020
3 PRACTICAL BEHAVIOR OF ADVERSARIAL INVARIANCE INDUCTION
In this section, we analyze the practical behavior of All framework. As mentioned above, the prior analysis of AlI focus on the ideal situation where one can access the perfect attribute classifier, i.c. (s(a]z) = p(alz). What happen if the optimality does not holds? As the optimally rarely holds in practice, it is important to analyze the behavior under the absence of the optimality.
3.1 VARIATIONAL. BOUNDS
To examine the practical behavior All, we first derive the general relationships between the con- ditional entropy and All framework, without assuming (s(a|z) p(alz). In the remainder of this paper, we assume the use of alternating optimization to solve the adversarial game. At each itera- tion, All first updates the attribute classifier x times by maximizing eq. while fixing fo. All then updates the encoder by minimizing the eq while fixing 96-
The conditional entropy and All objective has the following relationships:
H
[- pann((n))))
Epo(zn.en) I- log lon -Dxr ( I (po( [- (a
Hy(al2),
(4)
where DKL represents the Mullback-eeible (KL) divergence, which is greater than zero by def- inition, and pe (2) is an empirical distribution representations. Maximizing Hg yields the encoder update in All framework exactly. The update of the discriminator is equivalent to min- imizing this KL divergence. Therefore, the update of the discriminator tightens the bound. The bound tight when D (p(aa=)]ll O. In this case, maximizing (alz)) 22) is equivalent to maximizing the true conditional entropy, recovering the previous analysis (Xie et al. 2017).
Unfortunately, it is not practical assume the optimality of (al=) due the limitation of the capacity of the ái and computational costs to increase k. In such a practical situation, All maximizes the upper bounds of the conditional entropy, as shown in the eq. This challenge All as maximiz- ing the upper bound of the function of interest f does not guarantee the maximizing the f in general (in our case, the conditional entropy). For example, in the most exaggerated case, moving the entire distribution in specific direction increases the upper bound while the actual conditional en- tropy remains constant. The above observations do not immediately mean that maximizing Hy (alz) (which is done in All) is useless. However, it implies that there are regions where maximizing (ala)) does not increase H (alz).
3.2 DIVERGENCE MINIMIZATION REQUIREMENT
is noteworthy that Ha(a)z) itself is not upper-boundee as assigning zero probability for correct pair of 2, makes H, (a)=)) infinity. As the true conditional entropy has maximum values by defini- tion, it suggests that All's optimization has an incentive to update the parameter of the encoder even when the conditional entropy is successfully maximized. Why All has such incentive? Here, we derive divergence mininization requirements for maximizing the conditional entropy and how it challenges All optimization.
Specifically, the maximum conditional entropy holds the following property:
Proposition We assume that is a uniform categorical random variable. The maximum condi- tional entropy H (alz)) is log K. and H(azz) maximized if and only if p(=(aa)) p(zla;) for all a, f aj E A and E Z.
The proof is shown in the Appendix This proposition means that maximizing conditional entropy is asymptotically equal to minimizing pairwise divergence. Assume D is divergence measurement over a space of a possible probability distribution (i.e., (plllg) 2 0 for all p, 4 and D(pllg) = if and only if p q). For simplicity, we denote the marginal distribution over random variable associated with an attribute a = a; as (z). Then, the following corollary holds.
Corollary 1. If fo gives an attribute invariant representation (i.e., conditional entropy H(a is maximized), then D (1p (2)(2))pp = 0 for all a,, 0j and vice versa.
The above analysis suggests that the invariance induction algorithm should also minimize the diver- gence between marginal distributions of different attributes a. However, the original All does not
4

---

Under review as a conference paper at ICLR 2020
(4,00) A
go(alz)
(b) All
(c) Proposal
Figure 2: Visualizing how All and the proposed method update the encoder. The arrow in (b, c) represents the gradient vectors. (a) Both methods utilize (a(a]z), which represented counterplot in the figure. (b) All update fa by keeping a distribution away from the decision boundary without considering the information ( p (z). (c) Our proposal considers both the decision boundary and the information from the marginal distribution of different attribute Po 1 (z).
consider such constraints as it only keeps the distribution (²(2) away from the non-desired point where a discriminator correctly predicts the attribute. Figure (a, b) visualize how All updates the feature extractor, assuming there three gaussian distribution where each distribution corresponds to different attributes. The arrow in the figure represents the direction of the gradient when updating the fe (updating the distribution po(z)). The gradient vector suggests that All has an incentive to move the distribution regardless of whether it aligns marginal distributions of different attributes.
INVARIANCE INDUCTION BY DISCRIMINATOR MATCHING
4.1 DISCRIMINATOR MATCHING
Based the analysis, we propose simple modifications All considering the divergence minimiza- tion requirements. One possible approach to induce this divergence minimization requirement is to cast the representations on some distribution whose divergence is easy to compute. For example, KL divergence is easy to compute if both marginal distributions are Gaussian. However, casting the distribution harms the representative power of the deep encoder as selecting the proper prior is difficult in general. Moreover, if the representations have much information about the nuisance attribute and, therefore, the marginal distributions between different attributes do not have supports in the representation spaces, D(pllq) generally does not exists.
We propose a simple alternative by defining the proxy of the divergence using the learned critic function qd. Specifically, we consider divergence over of (a), where (a) = (a) 2)d2. In the remainder of the paper, we assume use Kl.-divergence as divergence measurements, though it could be a design choice in practice. IIDM minimizes the following discriminator matching loss for all pairs of a, and a a a;:
Vam (p' (z) (1)(()))) = BB (5 (
(5)
Intuitively speaking, IIDM restricts the update of the encoder to consider the location the marginal distributions of different attributes, in addition to the decision boundary (which is all the information source of the All). In other words, IIDM alleviates the possibility that the encoder moves the distri- bution regardless of whether it aligns marginal distributions of different attributes. Note that, such a divergence minimization perspective missing in the original All, as mentioned in Section 3.2. For reference, Figure compares how All and the proposed method update the feature extractor.
Formally, the discriminator matching objective Vam is related to the divergence D DKL (ps()))))) through the divergence between (g)
Firstly, we can derive
D (kl(((a))l((())) S E 2, [DxL (())
by using the Jensen's 's inequality. Also, based on the data processing inequality of the f-divergence (Gerchinovitz et al. 2017; Barber et al. 2018), the following inequality holds:
DeL (=))))))))) 2 DKI (xa)((())))
See the Appendix B for more detail.
5

---

Under review as a conference paper at ICLR 2020
Algorithm Pseudo code of IIDM
Require:
Initialize weights of neural networks {8, o, 16). and the centroid Q(0) with all data S.
while training() do
while repeat times do
update the attribute classifier by eq. with the bath of data B ~ p(x,y.a).
end while
sample the batch of data B ~ p(x, y, a).
split the bath of data B into B BK- where B, the batch of data with the attribute ag-
Qi s --1 + (1 7) 1 B, L (o) E A).
update the feature extractor and label classifier by eq. 10with the batch of the data B. end while
As Vam is an upper bound of the divergence (klla(aa))la minimizing Vam surely min- imizes the KL-divergence. Besides, in special case where the attribute classifier is invertible, Vam (0, ②) 0 ensures that (b(2) (2), as minimizing Dx (g's (a)lla((a)) is equivalent to min- imizing z ph (=)]lp( (Barber et al. 2018). In other words, minimizing Vam has the same fixed points with conditional entropy maximization if the discriminator invertible. Restricting the invertibility of the discriminator is an interesting direction. However, we did not add such reg- ularization restricting neural networks is difficult general and open research areas (Jacobsen) et al., 2018 |Behrmann et al., 2018) Ardizzone et al. (218)). Instead, we empirically validate that the proposed method reliably learns invariant representations even without such a regularization.
It is worth mentioning that the discriminator matching loss is closely related to the non-saturating version of AlI. Specifically, the proposed method minimizes
E (e)
(a) 2,)] =B (pgr((()
Σ LEEA
(a(a) (o(a))og))(();))) + +C,
and the NS objective (Bq. 2 can be rewritten as
(Ns(0;d)
Σ Σ E xx(())
Σ --((0)oo000(003))
agEAnsta
LaEA
where p² (a) is equal to 1.0 for 0; and zero otherwise. Ignoring the constant term, the only difference is whether to push (a)zz) to 9' (a). or true marginal (a)). Because the discriminator matching loss considers qo(a * 12)) similar to the NS objective, its gradient does not vanish even if 96 is the supremum. This property encourages faster convergence with the NS loss. It differs from NS as it considers the divergence minimization requirements, while NS only considers the current decision boundary of the discriminator but does not consider information from (=) directly (similar to All). Therefore, NS push representations as far as possible as long as it successfully changes the discriminator's prediction regardless of whether it aligns marginal distributions.
4.2 ALGORITHM
One implementation choice is how to calculate o' (a) in Eq. through Monte Carlo approximation: (a) = Pb (a)))(())))) an unbiased estimation, the variance is large the number of samples is small. The average can be calculated from all samples (or sufficiently large number of samples from each K attributes) at ev- cry iteration. However, requires additional computation other than standard mini-batch estimation. Moreover, the computation becomes intractable as the number of attribute values grows.
4.1 The straightforward approach is (= Although
We address these issues by using the moving centroid mechanism. Instead of estimating % every time with sufficiently large samples, the proposed method stores the moving average of 9 (a):
(i(a) = (Qi=1(a) + (a).
(8)
where Q! previous centroid, (a) is the new estimation of the centroid based on a single batch, and 7 is the decay parameter for controlling the speed which the centroids change. We initialized Of by computing the centroids of all training data points.

---

Under review as a conference paper at ICLR 2020
(a) All (1, 3, 35, 40, 200 steps each)
(b) IIDM (1. 3, 200 steps each)
Figure 3: Visualizing behaviors of All and IIDM on toy datasets hhttps: /tttis://driv... http:///vvvveee.llle....
com/ http://pwon.aaa..aetacooococoeeo
contains gif version).
NAA
14
+0 0.5
3.
(b) All
(C) NS
(d) IIDM
Figure 4: Quantitative comparison of AII, NS and IIDM (proposed method) on the toy datasets.
Then we can use the standard mini-batch method to calculate the discriminator matching loss. As with the All, IIDM incorporates alternating optimization. Specifically, as with All, IIDM firstly update the attribute classifier de by
max E al) yo
(9)
and update the encoder and the classifier by
min LE ()(zza) 0,0
log au (vwn=/== +
Σ [
(10)
a.tan
where & the parameter of the classifier, the 1st term is classification loss (which is same with AII), and X is weighting parameter.
4.3 SIMULATION
To clarify the practical issue of All and the benefit of the proposal, we design the toy dataset, which comprises samples from several Gaussian distributions with different means ([sin R m) cos( m)], and E 2, ... K) and the same variance, assuming that each distribution corresponds different attributes. We test All on this synthesized dataset while varying the number of distributions and (number of updates of á per iteration). Specifically, a batch size 128 and update 4e and fo iteratively using stochastic gradient descent with a learning rate=0.1. Figure visualizes how distributions move during the optimization of (a) All and (b) IDM on synthesized data (K = 3 and x=1). The Figure/rrpressee configuration, where solid lines represent the approximated conditional entropy by using post-hoc classifier Geval. which is parameterized by the neural network having the same architecture as that of qg. For reference, the theoretical maximum value of the conditional entropy (gray line) and the negative log-likes ligllikeldoood of do (dashed lines) are also depicted. Figure compares the performances of (b) All (c) NS, and (d) IIDM on different configurations of ( K, x). where color denotes different K, marker denotes different and the dashed line denotes optimal values.
first train the discriminator 100 times with
the quantitative results on this
We can make the following observations. (1) As shown the visualization of All and the vibration of the red lines in Figure 4a. All is catastrophically unstable, even such simple case. Figure Da also indicates that, at some points (c.g., around 50 iterations), the approximated conditional entropy becomes significantly large, whereas the true conditional entropy remains constant decreases. The issuc may be alleviated if the discriminator has a sufficiently large capacity and suggested by faster convergence of All when k 8.
trained many times at each iteration, However, the toy dataset is much simpler than real datasets, and it is fair to say that identifying the supremum is more challenging. (2) NS is superior to AII, but still unstable near the optimal point, as indicated by the vibration of the plots. (3) IIDM consistently reaches the theoretical maximum values all configurations, as depicted in Figure d.

---

Under review as a conference paper at ICLR 2020
4.4 REMARKS
Continuous nuisance attribute: One may question the applicability of our method to the contin- uous attribute case, e.g., the goal is to learn agg-iinariannt representations. Firstly, the treatment of continuous V.V. is an open question. The most straightforward yet practically used answer is dis- cretize the continuous R.V. For example, (Xie et al. 2017) divides an age variable into two groups. We can apply the proposed method with similar discretization.
Semantic aliguments: One well-known problem of invariant feature learning is determining how to incorporate semantic alignment, i.e. how to align only the pair of samples that have the same semantic (the target label). For this purpose, (Li et al. 2018b) proposes adversarial training based semantic alignment method, which prepares multiple domain classification networks where each classifier specialize for each class y. Another merit of the proposed method is that we can enforce semantic alignment with a simple modification, without any additional computational costs. Individ- ually, semantic alignment can be carried out by merely computing the centroids for each (attribute, label) tuple and aligning the (o of {a, y, a between only centroids of the same label y' = but different attributes a' j a. Since most applications of invariant feature learning require that Ly is also minimized, we also test this modification for all the later-described experiments.
EXPERIMENTS
5.1 EXPERIMENTAL SETTINGS
In addition to the simulation results, we provide experimental results on two tasks (four datasets) relevant to invariant feature learning: (1) user anonymization (Opportunity and USC datascts), and (2) domain generalization (MNISTR and PACS datasets). All experiments were implemented in PyTorch and were run on GPUs (either GTX 1080 or Tesla V100).
For user anonymization tasks, Opportunity and USC datasets were used. This task is to learn anonymized representations (z that does not contain user-identifiable information) while maintain- ing classification performance. The Opportunity dataset (Sagha et al., (011)) consists of sensory data regarding human activity in a breakfast scenario. Each record consists of 113 real-value sensory readings, excluding time information. We considered the task of recognizing 18 classes" Following previous studies (Yang et al., 2015; [wasawa et al. 2017). we use a sliding window procedure with 30 frames and 50% overlap. The number of samples coder using convolutional neural networks (CNN) with three SetoolutiooReLL..... -pollin repetitions followed by one fully connected layer and classification by logistic regression, following previous studies (Yang et al. 2015; Iwasawa et al. 2017). The discriminator is a simple feedforward network with 800-400 hidden units. The USC-HAD dataset is another activity recognition dataset that con- sists of 14 subjects (Zhng & Sawchuk 2012). The data include 12 activity classes that correspond to people's most essential and daily activities. MotionNode, which is DOF inertial measurement unit, is used to record the output from accelerometers that record six real sensory values. The sliding window procedure, with 30 frames and a 50% overlap, produced 172,169 samples.
57,790 in total. We parameterize the
The MNISTR and PACS are two typical datasets of domain generalization tasks. The MNISTR dataset, derived from MNIST, was introduced by (Ghifary et al., 2015). Its labels comprise the ten digits; domains are created by rotating the images in multiples of 15 degrees: 0, 15, 30, 45, 60, and 75. The domains are labeled with the angle by which they are rotated, c.g., M15 and M30. Each image is cropped to 16 x 16 pixel in accordance with a previous study (Ghifary et al. 2015
Similar to (Ghiiarry et al. 2015). we used two convolution layers with 32 and 48 filters of 5 x kernels, followed by max-poo mox-pooling layer and two fully connected layers with 100 hidden units. A discriminator with 100 hidden units is connected to the output of the first fully connected layer. The PACS dataset is relatively new benchmark dataset designed for cross-domain recognition (Li et al. 2017). The dataset has 9991 images in total across seven categories (dog, elephant, giraffe, guitar house, horse, and person) and four domains of different stylistic depictions (photo, painting,
'open door 1, open door 2, close door 1, close door 2, open fridge, close fridge, open dishwasher, close dishwasher, open drawer 1, close drawer 1, open drawer 2, close drawer 2, open drawer 3, close drawer 3, clean table, drink from cup, toggle switch, and null
"walking forward, walking left, walking right, walking upstairs, walking downstairs, running forward, jump- ing, sitting, standing, sleeping, clevator up, and elevator down
SSpecifically, we used the dataset distributed at withttss: tht.b/gtttttuug//// com//hiff f/mtae
8

---

Under review as a conference paper at ICLR 2020
Table I: Performance comparison of user anonymization tasks.
The value is the lowest user-
classification accuracy with specific performance degradation (0.01, 0.03 points) from CNN.
dafaset
Spp-S1
Opp-S2
Opp-S3
Opp-S4
threshold
0.01
0.03
0.01
0.03
0.01
0.03
0.01
0.03
0.01
0.03
0.973
0.973
0,984
0.983
0.983
0.683
ATI
0.631
0.517
0.590
0.590
0.694
0.659
0.589
0.586
0.512
0.179
Al[+GP
0.521
0.521
0.673
0.510
0.580
N8
0.635
0.452
0.614
0,523
0.484
0,484
0.499
0.482
None
None
IDM
0.462
0.417
0.415
0.415
0.409
0,409
0.456
0.486
0.499
0.499
IDM+
0.502
0.433
0.474
0.474
0.495
0.495
0.631
0.461
0.478
0.478
cartoon, and sketch). The diverse depiction styles provide a significant domain gap. We use the ImageNet pre-trained AlexNet CNN (Krizhevsky al. (012) as base network, following previous studies(LL et al., 2018a). A discriminator with 1024 hidden units is connected to the output of the last fully connected layer.
Baselines: To demonstrate the efficacy of the proposed method, we compared it with the following methods. (1) A CNN trained on the aggregation of data from all source domains. Although there are special treatments for domain generalization, (Li et al. (017) reports that CNN outperforms many domain generalization methods on the PACS dataset. (2) All (Xie et al. 2017), is a main baseline. (3) AII+GP uses variant of All with additional gradient penalty regularization used in GAN Mescheder et al. 2018). (4) ReyGrad is a slightly modified version of All, which uses the gradient reversal layer (Ganin al. 2016) to train all the networks (encoder, classifier, and discriminator) at the same time. (5) NS is a non-saturating version of All introduced in section 3 of this paper. (6) state-of-the-art method in domain generalization tasks. Note that it does not intend to learn invariant representation, so we use CrossGrad only for comparing domain generalization performance. (7) IIDM is our proposal. We used the gradient penalty as well. We also tested semantic alignment version and denoted it as IIDM+
CrossGrad (Shankar et al.
2018) is regarded
Optimization: For all datasets and methods, we used RMSprop for optimization. For all datasets except PACS, we set the learning rate to 0.001 and the batch size to 128. For PACS, we set the learning rate to 5e - 5 and the batch size to 64. The number of iterations was 10k, 5k, 20k, 30k; and 50k for MNISTR, PACS, Opp, and USC, respectively. For a fair comparison, hyper- parameters were tuned on a validation set for each baseline. For the method, we optimized weighting parameter À from {0.001, 0.01, 0.1, 1.0), except for MNISTR, for which it was optimized from {0.01, 0.1, 1.0, 10.0}. The value of a for CrossGrad was selected from {0.1, 0.25, 0.5, 0.75, 0.9). Unless mentioned otherwise, we set the decay rate 7 to 0.7.
pecersarill-ttannnnn -baed
Evaluation: In all the experiments, we selected the data of one or several domains for the test set and used the data of disjoint domain as the training/validattin of the disjoint domain into groupings of 80% and 20%. We denote the test domain by a suffix (e.g., MNISTR-MO denotes that the model is trained with the data from M15, M30, M45, M60, and M75 and evaluated on MO). We conducted 20 validations during training at equal intervals. In each validation, we measured the label classification accuracy (Y-acc) and the level of invariance. We empirically measured the level of invariance by training post-hoc classifier D Deval that tries to predict a over learned representations, following previous studies (Xie et al. 2017; Iwasawa et al, 2017]. Specifically, we trained the classifier with 800 hidden units Ih iterations (by RMSprop optimizer, with a learning rate of 0.001 and batch size of 128) with the data that are used to train the encoder and evaluate attribute classification accuracy on the validation dataset.
data. Accurately, we split the data
5.2 RESULTS
User anonymization: comparess the user-anonymizztion performance. The value represents the lowest cuss-caaasiiiiaatio accuracy (the lower the better) with specific performance degradation compared to CNN on classification accuracy. For example, the columns with 0.01 represent the lowest user-llassification accuracy with less than 0.01 point performance degradation. The best per- formance is underlined and highlighted in bold, and the second-best performance is only highlighted in bold. IIDM+ represents the variants that use the semantic alignment extension introduced in sec- tion 4. The results show the clear benefit of our proposal to inducing user invariance. Specifically, IIDM performs best on seven out of ten configurations. Note that the value with 'None' represents the method always reduce the label classification performance significantly.
summarizes the classification performance on two different Domain generalization: Table 2
datasets: MNISTR, and PACS. The top row of each table represents the test domain. We report

---

Under review as a conference paper at ICLR 2020
Table 2: Classification accuracies on unseen domains.
MNISTR
(b) PACS
MB
M15
M30
M44
M6O
M25
photo
art
cartoon
skeech
Avg
CNN
$4.0.1 1.7
99.1 0.5
97.6.109 0.9
91.9111
1.8
97.5 0.5
$7.7.1 1.7
92.97
80.81
58.11
62.71 2.6
60.61 4.5
65.57
RevGrad
1.6
97.92
92.14 0.8
2.2
85.94
82.92
4.6
64.06
A11
0744+09
91.0+
All+GP
97.91
CreeGrad
85.31
98.9
0.5
07.61.08
9091
1.0
082
0.4
87,51
2.0
93.09
81.4 18
S8 4.7
60.5 3.1
60.1
13
65.15
I1M
S5.01 1.6
9x
08.1
94.37
0.8
95.0
077
88.91
13
82.0 12
61.7 15
654 07
103T
0.5
66,9
IDM+
88.31 0.9
98.62
0.5
98.11 0.6
93..
18
98.1
0.9
36.91
255
93.85
84.8
0.6
6231
1.6
6488
1.5
60.21
2.5
68.04
0.5
0.5
0.5
A-acc
A-acc
0.4
ARICPIO
AI-CPIOLS
(L.)
0.4
0.3
IZM T5
HEM STER
#iter
#lter
Aiter
(a) A
(b)
(c) architecture of
Figure 5: Comparison of All and HIDM with different configurations on MNISTR dataset (MO as test domain). The number in parenthesis represents the corresponding configuration.
the mean accuracy well as the standard error of three different seeds (five seeds for MNISTR
and three seeds for PACS). The best performance underlined and highlighted bold, and the second-best performance is only highlighted in bold. We make the following observations. (1) IDM and IIDM+ demonstrate the best or comparable performance on all conditions except sketch a simpler
domain. Although the semantic alignment extension does not help the performance task (MNISTR), it improves the performance on PACS dataset, giving approximately 1.0 point per- formance gain. (2) RevGrad and All often fail improve performance even when compared with a standard CNN. The score of All+GP suggests that gradient penalty helps to improve the perfor- mance, but the improvements are lower than our proposal. (3) The Wilcoxon rank-sum test shows that IIDM is statistically better than CNN, RevGrad, AII, AII+GP, and CrossGrad with < 0.01.
Figure 5 compares All and IIM on different (a) weighing parameter 7, (b) the number of the discriminator update and (c) the network architecture of the discriminator. The dataset used is MNISTR with MO as a test domain. each figure, color represents a different method (red: All, blue: IIDM) and marker denotes different configurations. The value represents the attribute classi- fication accuracy (the lower the better invariant) by post-hoc classifier qeval (a]z). For À we used 1.0 by default. For and the architecture, sults show that our proposal is consistently to learn better invariant representations regardless of the choice of the hyperparameters. These results suggest that our proposal is better than searching such hyperaarameters. Note that, λ = 10.0 for All seems to attain better invariance, it was degenerated the random representations and gives a random performance on the classification of y.
used default setting described in Section 5.1. The
6 CONCLUSION
This paper presents method for invariance induction, called IIDM, by analyzing and extending current state-of-the-an art adversarial invariance induction framework. This paper first examines the in- stability issue of All both theoretically and empirically, indicating that All has theoretical difficulty as it maximizes variational upper bound of the actual conditional entropy, and this fact leads All to catastrophically fails even in simple cases. We then argue that simple modification to All can sig- nificantly stabilize the adversarial induction framework and achieve better invariant representations. The fundamental principle of our proposal is that desirable invariance induction algorithm should also minimize the divergence between marginal distribution p(z) between different attributes, as it requirement of maximum conditional entropy (Corollary 1) and missing All optimization. IIDM minimizes discriminator matching loss (Eq. 5). which is proxy of the divergence between the marginals. On toy dataset, we compare our proposal with the adversarial invariance induction framework, and show that proposal significantly stabilizes the optimization (Figure annd Figure 4). Two world tasks muser-anonymizationn in Table and domain generalization in Table also supports that our proposal achieve better invariance induction.
10

---

Under review as a conference paper at ICLR 2020
REFERENCES
Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, and Mario Marchand. aovaiinadverssaial neural networks. In Annual Conference on Neural Information Processing Systems (NIPS) Workshop Transfer and Multi-Task Learning: Theory meets Practice, 2014.
Lynton Ardizzone, Jakob Kruse, Sebastian Wirkert, Daniel Rahner, Eric W Pellegrini, Ralf Klessen, Lena Waier-Hein, Carsten Rother, and Ullrich Köthe. Analyzing inverse problems with invertible neural networks. arXiv preprint 1808.044 04730, 2018.
David Barber, Mingtian Zhang, Raza Habib, and Thomas Bird. Spread divergences. arXiv preprint arXiv:1811.0896 2018.
Jens Behrmann, David Duvenaud, and förn-Henrie Jacobsen. Invertible residual networks. arXiv preprint arXiv:1811.0099) 2018.
Gilles Blanchard, Gyemin Lee, and Clayton Scott. Generalizing from several related classification tasks to a new unlabeled sample. In NIPS, pp. 2178-2186, 2011.
Harrison Edwards and Amos Storkey. Censoring representations with an adversary. In ICLR, 2016.
Chuang Gan, Tianbao Yang, and Boqing Gong. Learning attributes equals multi-source domain generalization. In CVPR, 2016.
Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. Domain-adversarial training of neural net- works. The Journal of Machine Learning Research (JMLR), 17(1):2096-2033 2016.
Sebastien Gerchinovitz, Pierre Ménard, and Gilles Stoltz. Fano's inequality for random variables. arXiv preprint arXiv: 1702170029999 2017.
Muhammad Ghifary, W Bastiaan Kleijn, Mengjie Zhang, and David Balduzzi. Domain generaliza- tion for object recognition with multi-task autoencoders. Proceedings of the IEEE International Conference Computer Vision (ICCV), pp. 2551-2550 2015.
Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Proceedings of the Annual Conference on Neural Information Processing Systems (NIPS), 2672-2680, 2014.
Geoffrey E Hinton, Simon Osindero, and Yee-Whye Teh. A fast learning algorithm for deep belief' nets. Neural computation, 18(7):1527-13) 2006.
Yusuke Iwasawa, Kotaro Nakayama, Ikuko Eguchi Yairi, and Yutaka Matsuo. Privacy issues regard- ing the application of dnns accovity-rooonnitionn using wearables and its countermeasures by use of adversarial training. In Proceedings of the IJCAI, pp. 1930-1936, 2017.
Jörn Henn-Herik Jacobsen, Amold Smeulders, and Edouard Oyallon. i-revnet: Deep invertible networks. arXiv preprint Xrviv: 1802.07088 2018
Ayush Jaiswal, Rex Yue Wu, Wael Abd-Almageed, and Prem Natarajan. Unsupervised adversarial invariance. In Advances in Neural Information Processing Systems, pp. 5092-5102, 2018.
Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep con- volutional neural networks. In Proceedings of the Anmual Conference on Neural Information Processing Systems (NIPS), pp. 1097-1105, 2012.
Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M Hospedales. Deeper, broader and artier domain generalization. In IEEE International Conference on Computer Vision (ICCV), pp. 5543-5551. IEEE, 2017.
Da Li, Yongxin Yang. Yi-Zhe Song, and Timothy M Hospedales. Learning to generalize: Meta- learning for domain generalization. In AAAI, 2018a.

---

Under review as a conference paper at ICLR 2020
Ya Li, Xinmei Tian, Mingming Gong, Yajing Liu, Tongliang Liu, Kun Zhang, and Dacheng Tao. Deep domain generalization via conditional invariant adversarial networks. In Proceedings of the European Conference Computer Vision (ECCV), pp. 624 009,,,9,000 2018b.
Yujia Li, Kevin Swersky, and Richard Zemel. arXiv:1412.5244 2014.
Learning unbiased features. arXiv preprint
Christos Louizos, Kevin Swersky, Yujia Li, Max Welling, and Richard Zemel. The variational fair auto encoder. Proceedings of the International Conference on Learning Representations (ICLR), 2016.
Lars Mescheder, Andreas Geiger, and Sebastian Nowozin. Which training methods for gans do actually converge? In International Conference on Machine Learning, pp. 3478-3487, 2018.
Sacid Motiian, Quinn Jones, Seyed Iranmanesh, and Gianfranco Doretto. Few-Shot Adversarial Domain Adaptation. Advances Neural Information Processing Systems(NIPS), pp. 6673- 6683. Curran Associates, Inc., 2017.
Daniel Moyer, Shuyang Gao, Rob Brekelmans, Greg Ver Steeg, and Aram Galstyan. Invariant representations without adversarial training. 2018.
Krikamol Muandet, David Balduzzi, and Bernhard Schölkopf. Domain generalization via invariant feature representation. In International Conference Machine Learning (ICML), pp. 10-18, 2013.
Hesam Sagha, Sundara Tejaswi Digumarti, José del R Millán, Ricardo Chavarriaga, Alberto Ca- latroni, Daniel Roggen, and Gerhard Tröster. Benchmarking classification techniques using the opportunity human activity dataset. In Proceedings of the INSS, 2011.
Shiv Shankar, Vihari Piratla, Soumen Chakrabarti, Siddhartha Chaudhuri, Preethi Jyothi, and Sunita Sarawagi. Generalizing across domains via cross-gradient training. International Conference Learning Representations (ICLR), 2018.
Eric Tzeng, Judy Hoffman, Kate Saenko, and Trevor Darrell. Adversarial discriminative domain adaptation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 7167-7176, 2017.
Qizhe Xie, Zihang Dai, Yulun Du, Eduard Hovy, and Graham Neubig. Controllable invariance through adversarial feature learning. In Proceedings of the Anmual Conference Neural Infor- mation Processing Systems (NIPS), 2017.
Jian Bo Yang, Minh Nhut Nguyen, Phyo Phyo San, Xiao Li Li, and Shonali Krishnaswamy. Deep convolutional neural networks multichannel time series for human activity recognition. In International Joint Conference on Artficial Intelligncce pp. 0995-4001, 2015.
Rich Zemel, Yu Wu, Kevin Swersky, Toni Pitassi, and Cynthia Dwork. Learning fair representations. In Proceedings of the 30th International Conference on Machine Learning (ICML), pp. 325-333, 2013.
Mi Zhang and Alexander A Sawchuk. USC-HAD: daily activity dataset for ubiquitous activity recognition using wearable sensors. International Joins Conference on Pervasive and Ubiqui- tous Computing (UbiComp), pp. 1036-1043. ACM, 2012.
12

---

Under review as a conference paper at ICLR 2020
A
PROOF OF THE PROPOSITION 1
Proof. Using the Lagrange multiplier method, the derivative
= Σ p(a, 2ogp(al2) X(1
p(alz))
(11)
aEA
equal zero for the maximum entropy Ha)e)). Solving the simultaneous equations, we can say p(a1|z) p(02 2) _... p(axl=) p for all 2 when the conditional entropy is maximized,
and based on the definition, the conditional entropy becomes log k
From Bayes' law and the uniform assumption of p(a), p(z(a;) = p(z)a;) holds Va, # aj E and E Z.
B RELATIONSHIPS BETWEEN DISCRIMINATOR MATCHING AND THE DIVERGENCE MINIMIZATION REQUIREMENTS
B.1
PROOF OF THE EQ.
Proof.
DRzle(())lo((a))
Edtla) E (a(a) log (=(c))alo(()
Esb(a) E Ls (() llog (al2)
= E sobit)
The inequality shows that the discriminator matching objective is variational upper bound of the KL--divergence between (a) and (a). Therefore, minimizing the proposed objective surely minimize the Kl.-divergence between (a) and (a(a). Note that, this inequality holds even when ((a|2) is not optimal.
B.2 DATA PROCESSING INEQUALITY OF THE F-DIVERGENCE
Theorem 1. Consider a channel that produces given x based on the p(y)x)). For any f-divergence Dy(pl14) Byg [s()]
Dy(p(v)ll(()) < Dy(p(=))la(x))
(12)
Proof. Using p(y(z), we consider the following joint distributions.
p(z, y) p(u)=))(x))
<(x, v) = p(yl=))(x),
The divergence between the tow joint distributions is
p()))))(()) p (())))
Dy (p(a, (2 v)) = (..()
D)(p(=))l(())),
(13)
as p(z, v) and q(z, y) has common channel p(y]=), and p(y(z) (z) is canceled. For other direction,
p(x, y)
Dy(((,, w)lla(z,)))
Eg(x)
By(=lu)
((<ly)<()))
p(),y)
Beg(n)
((x))))(())
p(v)
B
= Dy(p(v)lla()))
q(v)
The inequality uses Jensen's inequality. Combining these two equations, Dy(p(v)ll(())) <
D)(p()))la())))
(14)
n
13

---

Under review as a conference paper at ICLR 2020
As the KL divergence is also the family of the f-divergence, by replacing the p(y|æ) to (o(2)))), p(x) pz(2), ((x) vi (2), p(y) aa(a), and ply) to (o))).
DKL (RI 2 (ra(d)(a)l)(())
(15)
which is equivalent to eq. 7
It is noteworthily that if one can assume the common inverse mapping p(æly) for both joint distri- butions p(z, y) and g(,, y), then
= 1
(16)
The proof is straightforward results with the above proof. In our case, p(y[2) is parameterized by neural networks. In this case, if the (a|a) is invertible, then the inverse mapping exists. Intuitively speaking, the data process do does lose information, though the data transformed to better explaining the attribute information (as go is trained to discriminate the attribute).
14


Reviews:
[{"review_id": "rkg8J4Vg9r", "reviewer": "AnonReviewer3", "summary": "", "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "** Summary\nThe paper studies the problem of representation learning under invariance constraints (i.e., the representation should be invariant wrt some attributes). The authors first review the adversarial invariance induction (AII) approach and they point out its limitations and then they propose a novel variant, which introduces an explicit regularization to minimize pairwise divergence (i.e., different attributes should lead to the same conditional distribution over the learned representation). The authors support the modified objective function both from a formal point of view and with an extensive empirical validation\n\n** Evaluation\nThe paper lies a bit outside my area of expertise. Although the paper tries to capture intrinsic limitations of the AII approach and build a more stable algorithm, my impression is that too many elements in the discussion and derivation remain too vague at the current stage and they would require better and clearer explanation.\n\nDetailed comments:\n1- In many parts of the paper the notation is not very rigorous and sometimes it may create confusion. In general, the writing needs to be improved in many parts:\n- In eq. 1, it is not explained what the expectation is wrt. It should be x drawn from p(x). But it would be better to make it explicit.\n- The setting defined in sect.2 should be more rigorous: it is not clear what y is and what are the attributes we would like to be invariant for. This notation is not fully consistent with eq.1.\n- In the first paragraph of Sect.3, it would be very helpful to have a more complete sketch of the algorithm. In general, you mention in the Sect. 2 that you focus on the supervised case, but then it is not clear whether this is the case across the paper.\n2- While the intuition behind using the divergence is sound, as it is mentioned, it seems to suffer from the same issue as the original AII: minimizing a lower bound does not guarantee that we are minimizing the actual objective. Having the support of 0, does not seem to make it much more sensible.\n3- As the description of the algorithm advances in Sect.4.2, it is clear that many additional choices need to be made in order to have a full workable algorithm (e.g., how to estimate q_\\phi^i). In the paper, the actual algorithm is never reported in detail and this makes the experiments very hard to reproduce in my opinion.\n4- At the best of my understanding, the algorithm may become more and more intractable as the number of attribute values grows. In fact, you need to check the divergence for each pair of values and estimated distributions.\n5- The empirical analysis seems well executed and a good level of detail is reported on how the datasets are managed and the experiments are set up.\n\nMinor comments:\n- The caption of Fig.1 is not very clear. Many elements at this stage of the paper are not defined yet (e.g., \"our proposal minimize the proxy of divergence ...\").\n- p2: \"as the assumption of ... is rarely holds\", remove \"is\"\n- p3: \"(encoder) that parameterized\", remove \"that\"\n- last paragraph of Sect.2 is very confusing.\n- p4: \"updation\" is not a word\n- p4: \"In general, minimizing the upper bound...\" does not seem correct.\n- p4: \"even such a simple case\" -> \"even in such a simple case\"\n- Sect 4.1 \"the above theoretical\", I would not really say there is much theory behind the analysis in the previous section.\n- p5 \"an ttribute\" -> \"an attribute\"\n- p6 \"the average discriminator's perception\" what is the perception?", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "HJg0D4d0YH", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 1, "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": "", "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 points out that the practical behavior of AII assumes the optimality of the attribute classifier, which is rarely held in practice. And claims that the paper analyzes the practical behavior of AII both theoretically and empirically, indicating that AII has theoretical difficulty as it maximizes variational upper bound of the actual conditional entropy. Then it argues an ugly modification based on a wrong property of conditional entropy. \n\n- The paper says that it analyzes AII theoretically and empirically. But it only shows the practical drawback of AII intuitively without any theoretical proof.\n- In Section 3, the paper says : 'In general, maximizing the upper bound of the function of interest $f$ does not guarantee the minimizing the $f$ '. \n- Also in the section 3, the paper says : 'Figure 2-(b) visualizes how distribution move during the optimization of AII on synthesized data'. And I want to ask why the caption of Figure 2-(b) is IIDM ?\n- The proposition 1 on which the modification proposed in the paper is based will be not true when the distribution of attributes is not uniform by Bayses' s law which is used in the so called proof of the proposition 1. Which means that $p(z|a_i)=p(z|a_j) \\Leftarrow p(a_i|z)=p(a_j|z)$ if and only if $p(a_i)=p(a_j)$.\n- In the proof of equation 4 which is the main theorem of the total work. We can find $-\\sum q_{\\phi}^i(a)\\log\\mathbb{E}_{p_{\\theta}^j(z)}[q_{\\phi}(a|z)]\\ge -\\sum q_{\\phi}^i(a)\\mathbb{E}_{p_{\\theta}^j(z)}[\\log q_{\\phi}(a|z)]$.  The inequality direction is reversed. \n", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "BJxoDJtnKS", "reviewer": "AnonReviewer2", "summary": "", "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "*Summary*\n\nThe paper proposes a new method to learn data-driven representations, being invariant to some specific nuisance factors which are detrimental for the selected (supervised) classification task.\nAuthors build upon the existing probabilistic framework termed Adversarial Invariant Induction (AII) from (Xie et al., 2017). \n\nThey claim to explore it under a both theoretical and practical point of view, demonstrating the limitations of maximizing a variational upper bound on conditional entropy as a proxy to achieve invariance. \n\nLeveraging these observation, authors propose a novel method, called “invariance induction by discriminator matching” (IIDM) that is based on a regularized classification loss, penalized by a Kullbach-Leibler divergence between conditional distributions of the nuisance factor.\n\nExtremely convincing experiments are carried out on a synthetic and a real benchmark in multi-source domain generalization (PACS).\n\n\n\n*Pros*\n1. The genesis of the proposed IIDM is extremely paced since smoothly derived from the AII framework.\n2. Experimental results on a synthetic benchmark (a version of rotated MNIST) and on a popular benchmark for domain generalization (PACS) proved the effectiveness of IIDM\n\n\n\n *Cons*\n1. The paper is hard to get, if the reader is not familiar with related literature\n2. It is not fully clear from the paper which parts are original and which are inherited from prior work.\n3. The structure of the paper needs to be improved (check my comments in the section beneath)\nSome of the proposed methodologies are not clear (IIDM+)\n\n\n\n\n*Detailed Comments*\n\nThe problem considered by authors is surely interesting and addressing a popular topic in computer vision and deep learning. \n\n1. Unfortunately, the paper, as it is is hard to get for scholars which are not expert of the AII formalism, which, in my opinion is not enough detailed. Therefore, in my opinion clarity is something that authors should try to work hard on: for instance, during the rebuttal time, authors can write from scratch an entire new Section in which they explain in plain terms the main outcomes of their paper, without entering too much into technical details.\n\n\n2. Additionally, the structure of the paper needs, in my opinion a major re-styling, still for the sake of better readability:\n2.a. A visualization of the proposed method (for instance, using flow-diagrams) in comparison with the existing AII should help in rapidly getting the factors of novelty of the proposed IIDM method. I would also encourage authors to add a pseudo-code\n2.b. Since authors claim two major contributions (understanding AII + IIDM), I would like to see those two contributions thoroughly presented and dissected in two separated sections of the paper. I am not fully convinced with the actual writing style in which the two contributions seem to be intertwined together.\n\n\n3. Although already convincing, the experimental part can be improved:\n3.a. Instead of a version of rotated-MNIST, authors can test on the “digits-five” setting (MNIST, MNIST-M, SVHN, UPS, SYN) as done in several works like http://openaccess.thecvf.com/content_cvpr_2018/papers/Xu_Deep_Cocktail_Network_CVPR_2018_paper.pdf.\n3.b. In addition to multi-domain generalization, authors could also have tried more classical unsupervised domain adaptation settings or, even, single-source domain generalization as in https://papers.nips.cc/paper/7779-generalizing-to-unseen-domains-via-adversarial-data-augmentation.\n\n\n\n*Final Evaluation*\n\nI think that the main aspect that authors should face during the rebuttal is to make the paper more easy to read and better separate the two contributions (understanding AII and IIDM). What I am not convinced at all about the writing style of the authors since when reading the paper I am not always capable of understanding what is novel (since proposed by authors) and what is inherited from prior work. But, maybe, the reason for this is that I am not an expert of the specific related field - but, even so, I think that the paper needs to be understood from the broadest audience possible.\n\nInstead, I am familiar with multi-source/single-source domain generalization (and adaptation) and, after my careful analysis of the experiments, I see a lot of potential in the approach. I would me more than interested in checking the performance of the proposed method over some of the novel benchmarks that I have recommended. It would be nice if authors add more experiments, but I know that this is always a complicated request during a rebuttal period.\nGlobally, if I were asked to only rate the experimental part, I would have promoted for full acceptance. Unfortunately, the theoretical part of the paper is not fully clear to me and, therefore, I am not confident in calling for a full acceptance only based on the experiments. \n\nIn brief, I would go for a weak reject, looking forward to the authors’ rebuttal and the opinion of the other Fellow Reviewers.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a modification to improve adversarial invariance induction for learning representations under invariance constraints. The authors provide both a formal analysis and experimental evaluation of the method. The reviewers generally agree that the experimental evaluation is rigorous and above average, but the paper lacks clarity making it difficult to judge the significance of it. Therefore, I recommend rejection, but encourage the authors to improve the presentation and resubmit.
Decision: Reject