Controlling Directions Orthogonal to a Classifier
Authors: Yilun Xu
Hao He
Tianxiao Shen
Tommi S. Jaakkola
Venue: ICLR 2022 Spotlight (2022)
Keywords: orthogonal classifier
invariance

Abstract: We propose to identify directions invariant to a given classifier so that these directions can be controlled in tasks such as style transfer. While orthogonal decomposition is directly identifiable when the given classifier is linear, we formally define a notion of orthogonality in the non-linear case. We also provide a surprisingly simple method for constructing the orthogonal classifier (a classifier utilizing directions other than those of the given classifier). Empirically, we present three use cases where controlling orthogonal variation is important: style transfer, domain adaptation, and fairness. The orthogonal classifier enables desired style transfer when domains vary in multiple aspects, improves domain adaptation with label shifts and mitigates the unfairness as a predictor. The code is available at https://github.com/Newbeeer/orthogonal_classifier

Published as a conference paper at ICLR 2022
CONTROLLING DIRECTIONS ORTHOGONAL TO A CLASSIFIER
Yilun Xu, Hao He, Tianxiao Shen, Tommi Jaakkola Computer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology {ylxu, haohe, tianxiao femittoeemmmmeem.eeeeee tommi@caail .mit. edu
ABSTRACT
We propose to identify directions invariant to given classifter that these direc- tions can be controlled in tasks such as style transfer. While orthogonal decom- position is directly identifiable when the given classifier is linear, formally de- fine a notion of orthogonality in the non-linear case. We also provide a surpris- ingly simple method for constructing the orthogonal classifier (a classifier utiliz- ing directions other than those of the given classifier). Empirically, we present three use cases where controlling orthogonal variation important: style transfer, domain adaptation, and fairness. The orthogonal classifier enables desired style transfer when domains vary in multiple aspects, improves domain adaptation with label shifts and mitigates the unfairness as a predictor. The code is available at https:// / thub. www.////eeeeeeererrtoeeeoeeeeeee Stw.ggnnnalllsssieeee.....e
INTRODUCTION
Many machine learning applications require explicit control of directions that are orthogonal to predefined one. For example, to ensure fairness, we can learn classifier that is orthogonal to sensitive attributes such as gender or race (Zemel al., 2013; Madras al., 2018). Similar, if we transfer images from one style to another, content other than style should remain untouched. Therefore images before and after transfer should align directions orthogonal style. Common these problems is the task of finding an orthogonal classifier. Given any principal classifer operating on the basis of principal variables, our goal is find a classifier, termed orthogonal classifier, that predicts the label on the basis of orthogonal variables, defined formally later.
The notion of orthogonality is clear in the linear case. Consider a joint distribution Pxy over X F and binary label Y. Suppose the label distribution is Bernoulli, i.e., Py B(Y; 0.5) and class-conditional distributions are Gaussian, P' NV My oz) where the means and variances depend on the label. the principal classifier linear, w1 Pr(Y 1|6Tx), x), any classifier W2, in the set W2 (Pr(Y- = 1|0]x) 0,0, 02 0}, is considered orthogonal to wi. Thus the two classifiers W). W2, with orthogonal decision boundaries (Fig. 1) focus on distinct but complementary attributes for predicting the same label.
Finding the orthogonal classifier is no longer straightforward in the non-linear case. To rigorously define what we mean by the orthogonal classifier, we first introduce the notion of mutually orthogonal random variables that correspond to (conditinally) independent latent variables mapped to observations through diffeomorphism (or bijection discrete). Each r.v. predictive of the label but represents complementary information. Indeed, we show that the orthogonal random variable maximizes the conditional mutual information with the label given the principal counterpart, subject to an independence constraint that ensures complementarity.
Our search for the orthogonal classifier can be framed as follows: given a principal classifier wi using some unknown principal r.v. for prediction, how do we find its orthogonal classifier w2 relying solely on orthogonal random variables? The solution to this problem, which we call classifier orthogonalization, turns out to be surprisingly simple. In addition to the principal classifier, we assume access to a full classifier wy that predicts the same label based on all the available information, implicitly relying on both principal and orthogonal latent variables. The full classifier can be trained normally, absent of constraints' We can then effectively "subtract" the contribution of wi from the full classifier to obtain the orthogonal classifier Wy which we denote as = Wz x W1. The advantage of this construction is that
'The full classifier may fail to depend on all the features, e.g., due to simplicity bias (Shah et al., 2020).

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pixY- pixt=11
X1
X0
(a) Distribution PxjY
(b) Wan
(c)
(z)
(d) W2 (1r)
Figure I: An illustrative example of orthogonal classifier in linear case. (a) is the data distributions in two classes; (b.c,d) are the probabilities of the data from class predicted by the tullpprincipal/orhhoonaa classifiers. Red and blue colors mean a probability close or The white color indicates regions with probability close to 0.5, which classifiers' decision boundaries. Clearly, wi and w2 have orthogonal decision boundaries.
do not need to explicitly identify the underlying orthogonal variables. It suffices to operate only the level classifier predictions.
We provide several use cases for the orthogonal classifier, either predictor predictor, the orthogonal classifier predictions are invariant to the principal sensitive r.v., thus ensuring fairness. As discriminator, the orthogonal classifier enforces partial alignment of distributions, allowing changes in the principal direction. We demonstrate the value such discriminators in 1) controlled style transfer where the source and target domains differ in multiple aspects, but we only wish to align domain A's style to domain B, leaving other aspects intact; 2) domain adaptation with label shift where we align feature distributions between the source and target domains, allowing shifts in label proportions. Our results show that the simple method on par with the state-of-the-ard methods in each task.
discriminator. As
2 NOTATIONS AND DEFINITION
Symbols. We use the uppercase to denote random variable (e.g., data X, label Y), the lowercase to denote the corresponding samples and the calligraphic letter to denote the sample spaces of r.v., e.g., data sample space X. We focus on the setting where label space discrete, i.e., y (1. C.), and denote the C' dimensional probability simplex A classifier X A is a mapping from sample space to the simplex. Its y-th dimension w(x),, denotes the predicted probability of label y for sample x.
Distributions. For random variables A, B, marginal/conddtional//gi distribution, i.e., (a(a) p(A= ()) PAI p(A a B = b).PAB(a, p( (AA= A= B b). Sometimes, for simplicity, we may ignore the subscript if there is no ambiguity, e.g., p(alb) an abbreviation for PA (
the notation PA. PAAB- PAB
denote the
We begin by defining the notion of an orthogonal random variable. We consider continuous X, 21, 7/2 and assume their supports are manifolds diffeomorphic the Fuclidean space. The probability density functions (PDF) are in C². Given joint distribution Pxy, we define the orthogonal random variable as follows:
Definition (Orthogonal random variables). We say Z₁ and Zz are orthogonal random variables w.r.t pxy they satisfy the following properties:
(i) There exists diffeomorphism f Z Z, X such that f(Z1, 22) = X.
(ii) 71 and Zz are statistically independent given Y, i.e., Zı Zz
The orthogonality relation is symmetric by definition. Note that the orthogonal pair perfectly reconstructs the observations via the diffeomorphism f; as random variables they are also sampled independently from class conditional distributions p( YY) and p YY For example, can regard foreground objects and background scenes in natural images as being mutually orthogonal random variables.
The definition of orthogonality can be similarly developed for discrete variables and discrete- continuous mixtures. For discrete variables, for example, we replace the requirement of diffeomor- phism with bijection.
Remark.

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Since the diffeomorphism f is invertible, we can 21 X 21 and 22 X Z2 denote the two parts of the inverse mapping so that Z1 = =1 and Z 22( Note that, for a given joint distribution Pxy, the decomposition into orthogonal random variables is not unique. There are multiple pairs of random variables that represent valid mutually orthogonal latents of the data. We can further justify our definition of orthogonality from information theoretic perspective by showing that the choice of 22 attains the maximum of the following constrained optimization problem.
Proposition 1. Suppose the orthogonal E.V. of 21 (X) w.F.f pXY exists and is denoted as 22(X). Then =Xx)= (X) is maximizer of (= Y| (X)) subject to I ==(X):=1(X)) = 0.
We defer the proof to Appendix B.1. Proposition 1 shows that the orthogonal random variable maxi- mizes the additional information about the label we can obtain from X while remaining conditionally independent of the principal random variable. This ensures complementary in predicting the label.
3 CONSTRUCTING THE ORTHOGONAL CLASSIFIER
Let 21 =x) and Zg = xx be mutually orthogonal random variables w.r.Pxy:: We call Z the principal variable and Zz the orthogonal variable. In this section, describe how we can construct the Bayes optimal classifier operating on features Z2 from the Bayes optimal classifier relying on Z1. We formally refer to the classifiers of interests as: (1) principal classifier w (1), = p(Y = 2 = 21 (z)): (2) orthogonal classifier u2(2)) y p(Y 2 = 22 (z)); (3) full classifier Wa (x)x), p(Y = y = z).
3.1 CLASSIFIER ORTHOGONALLIZATION
Our key idea relies on the bijection between the density ratio and the Bayes optimal classifier (Sugiyama et al., 2012). Specifically, the ratio of densities Pxay (zli) and Pxxy (xlj), assigned to an arbitrary point
I, can be represented by the Bayes optimal classifier wxx); = Pr(Y = i|æ) as Pxir (xir
= py (x))
(xls) py (1)
Similar, the principal classifier w1 gives us associated density ratios of class-conditional distributions over Z1. For any i.j E VV, we have (= Pxi (=1())]3) py ()))
(v( These can be combined obtain
density ratios of class-conditional distribution Pza|Y and subsequently calculate the orthogonal classifier w2. We additionally rely on the fact that the diffeomorphism permits to change variables between x and z1; zz: Pxlr (xl() = Pz,y (z)li) Pzay (z2li) * (21,)(21,2)) =2), where volJy is volume of the Jacobian (Berger et al., 1987) of the diffeomorphism mapping. Taken together.
Pz,Jy (2211)
Pz, (2) PZ,yy (22)1) (21,
/
P2.1r wx(x); P2.IY (x)x)
/
wi (I) Ye (x))
(1)
Pzzlr (2 PZ (=) PZ.JY (=2li) (21,)
Note that since the diffcomorphism f is shared with all classes, the Jacobian is the same for all label- conditioned distributions on Z1, Zz. Hence the Jacobian volume terms cancel each other in the above equation. We can then finally work backwards from the density ratios of P2zsy to the orthogonal classifier:
We (x)x
/ Σ (P(Y 3
Wo (3)); wi (x);
Pr(Y = i| = 22 = Pr(Y-P(-)))) =
=
wi (x);
We call this procedure classifier orthogonalization since it adjusts the full classifier Wy to be orthogonal to the principal classifier wi- The validity of this procedure requires overlapping supports of the class- conditional distributions, which ensures the classifier outputs w. (2:;; ; wi (x)) remain non-zero for all & E Xx, V.
Empirically, usually have access dataset D {(1,, 9)) with n iid samples from the joint distribution pxy. To obtain the orthogonal classifier, we need to first train the full classifier uh based on the dataset D. We can then follow the classifier orthogonalization to get an empirical orthogonal classifier, denoted as W2 = th S W1. We use symbol x to emphasize that the orthogonal classifier uses complementary information relative to 21. Algorithm 1 summarizes the construction of the orthogonal classifier.
Generalization bound. Since w., is trained on finite dataset, consider the generalization bound of the constructed orthogonal classifier. We denote the population risk as R(w)- = -E (x.z) [log wx),y) and the empirical risk as Ri w) Yo (..D log lo:w(z;)),- For function family W whose elements map X' to the simplex C we define w inf wwe R(wx), w. inf ew R(wz). .. We further denote the Rademacher complexity of function family W with sample size |D] as RD((W)

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Algorithm 1 Classifier Orthogonalization
Input:
principal classifier w1, dataset DD
Train an empirical full classifier v on by empirical risk minimization. Construct orthogonal classifier ú, wi via classifier orthogonalization (Eq. (2)).
return the empirical orthogonal classifier ú2 = Wz w1
Theorem 1. Assume Py is uniform distribution, Yuz W takes values in (m, 1 m) with m (0. ;), and L/px/y (xlyy) (0,7) (0,+00) holds for Vx X E V. Then for any ô (0, 1) with probability at least 1- 5, we have:
1 m
2 log x [D]
|R(de s w1) R(w", w1)l s (1 (1+ +7) 29 (WV 2 log
Theorem 1 shows that the population risk of the empirical orthogonal classifier in Algorithm would be close to the optimal risk if the maximum value of the reciprocal of class-conditioned distributions 1/Px/y (alyy) and the Rademacher term are small. Typically, the Rademacher complexity term satisfies Rip (W) O(D (Bartlett & Mendelson, 2001; Gao & Zhou, 2016). We note that the empirical full classifier may fail in specific ways that are harmful for our purposes. For example, the classifier may not rely on all the key features due to simplicity bias as demonstrated by (Shah et al., 2020). There are several ways that this effect can be mitigated, including Ross et al. (2020); Teney et al. (2021).
3.2 ALTERNATIVE METHOD: IMPORTANCE SAMPLING
An alternative way to get the orthogonal classifier is via importance sampling. For each sample point P2, (#1(x))
(x, y). we construct its importance Φ(x, y) = P2,rr (=))))) Via Bayes' rule, the importance o(2, y) can
be calculated from the principal classifier by dzz,y)
pr( We define the importance sampling (IS)
w1("),
objective as Lis(w) (w) = En --xxy [0(2, y) log (1) [zyy. J. It can be shown that the orthogonal classifier wa maximizes the IS objective, i.e., W2 = arg max C However, the importance sampling method has an Achilles' heel: variance. A few bad samples with large weights can drastically deviate the estimation, which problematic for mini-batch learning in practice. We provide an analysis of the variance of the IS objective. It shows the variance increases with the divergences between Z1 's marginal and its subel-conditionet marginals, (Dy (P2, IIpz) y-1 even when the learned classifier w is approaching
the optimal classifier, i.e., w W2. Let CT. (w) == Er oaI,, v1) log (x:) be the empirical IS objective estimated with n iid samples. Clearly, Var (TT = Var (C1 (w))). While Var(C W2 is the following,
Var(Cto (w2)) = By [D) + E) log PrIza 22) l(w))
where Dy is the Pearson x2 -iveergence The expression indicates that the variance grows linearly with the expected divergence. In contrast, the divergences have little effect when training the empirical full classifier in classifier orthogonalization. We provide further experimental results in section 4.1.2 to demonstrate that classifier orthogonalization has better stability than the IS objective with increasing divergences and learning rates.
ORTHOGONAL CLASSIFIER FOR CONTROLLED STYLE TRANSFER
In style transfer, we have two domains A, B with distributions Pxy, QXy We use binary label Y to indicate the domain of data X (0 for domain A, 1 for domain B). Assume 7n and 7/2 mutually orthogonal variables w.r.t both Pxy and Qxy The goal is to conduct controlled style transfer between the two domains. By controlled style transfer, we mean transferring domain A's data domain B only via making changes on partial latent (Z1) while not touching the other latent (Zz). Mathematically. we aim transfer the latent distribution from Pz, Pz, to Qz, Pz.. This task cannot be achieved by traditional style transfer algorithms such as CycleGAN (Zhu et al., 2017), since they directly transfer data distributions from Px to Qx, or equivalently from the perspective of latent distributions, from Pz, Pz, to Qz, Qzy- Below we show that the orthogonal classifier wa. \ wi enables such controlled style transfer.
Our strategy is to plug the orthogonal classifier into the objective of CycleGAN. The original Cycle- GAN objective defines a min-max game between two generators/diicciiinatoo pairs GAB/ DA and

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0.55
0.55
PLY - B) (w)
Tost loss
0.50
Oracle
Test loss
0.50
Wal W:
Oracie
data
3000
P(Y = 1 B) (8)
PLY = G) (w)1w)
0.45
0.45
2000
PLY 1 G) (8)
0.40
0.35
-3.0
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1.0
logE,IDApzzIler (P2arr
Leamingg rate (log-scale)
Predicted probabillty
(a)
(b)
(c)
Figure 2: (a,b): Test loss versus log expected divergence and learning rate. (c): The histogram of predicted probability of different methods. The two red lines the ground-truth probabilities for P(Y 1/Brown background) and P(Y = 1/Green background).
GRA/DRA- GaB transforms the images from domain A to domain B. We use P to denote the generative distribution, i.e., GAB(X) P for X P. Similarly, Q denotes the generative distribution of generator GBA. The minimax game of CycleGAN given by
min max GAB.GBA DAB.DBA
CAB (GAB, DAB) LBA DBA) Leya GBA)
(3)
where GAN losses CAB CaN minimize the divergence between the generative distributions and the targets, and the cycle consistency loss Leye is used to regularize the generators (Zhu et al., 2017). Concretely, the GAN loss GAN is defined as follows,
LAB (GAB, DAB) Ez-qolog
DanlogDDn(Ce Pan(=)] B.- DAn
Now we tilt the GAN losses in CycleGAN objective to guide the generators to perform controlled transfer. Specially, we replace CAB in Eq. (3) with the orthogonal GAN loss CAB when updating the generator GAB:
C (GBB, DAB) = E E. -il--((DBBBBt ((DA(())
where of (())( (--Dan(e)--aanttrre and r(s) Wr)er(F) wx)w)()). Consider an extreme case where we allow the model to change all latents, i.e., (x) = I. As a result, LoeaN degenerates to the original Las since w2 W1 = and o( (DaB(r), (()) = DAB(#). The other orthogonal GAN loss CBA can be similarly derived. For given generator GAB. the discriminator's optimum is Assuming the discriminator is always trained be
Dan(z)r(s)
achieved at (x)(x) (P))) optimal, the generator GAB be viewed as optimizing the following virtual objective: LAB (GAB)
F(x) The optimal generator in the new objective satisfies the
LGGAN AAB DAB) FE g log (())=Q(s))(()"
following property.
Proposition 2. The global minimum of CAB L is achieved if and only if P2 = Qz, (z))2,)),
We defer the proof to Appendix B.4. The proposition states that the new objective CAB converts the original global minimum Qz, Qz in LAB the desired one Qz, Pzg- The symmetric statement holds for OGAM (GBA).
4.1
EXPERIMENTS
4.1.1 SETUP
Datasets. (a) CMNIST: We construct C(olOTsJMNIST dataset based on MNIST digits (LeCun & Cortes,
2005). CMNIST has two domains and 60000 images with size (3, 28, 28). The majority of the images in each domain will have digit color and a background color correspond to the domain: domain A "red digit/green background" and domain B "blue digit/brown background". We define two bias degrees Aas A to control the probability of the images having domain-associated colors, egg. for an image in domain A, with Xa probability, its digit set be red; otherwise, its digit color is randomly red or blue. The background colors are determined similarly with parameter As. In our experiments,

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Table 1: CMNIST
Table 2: CelebA-GH
Z1 acc.
Zz acc.
Z1 acc.
Zz acc.
FID
Vanilla
90.2
14.3
Vanilla
88.4
16.8
38.5
W1 (MLDG)
94.9
91.1
We (MLDG)
90.9
53.0
46.2
Wx wi (TRM)
89.2
96.2
Wa wn (TRM)
92.2
56.5
39.8
Ww Wi (Fish)
93.9
98.0
W2 wh (Fish)
88.8
42.9
43.4
IS (Oracle)
90.0
100
IS (Oracle)
89.1
40.6
44.3
Wx wi (Oracle)
94.9
100
W/ wh (Oracle)
93.7
57.4
39.7
we set As = 0.9 and A 0.8. Our goal is to transfer the digit color (21) while leaving the background color (Z2) invariant. (b) CelebA-GHI We construct the (alebA-G(enderrH((in)) dataset based on the gender and hair color attributes in CelebA (Liu et al., 2015). CelebA-GH consists of I 110919 images resized to (3, 128, 128). In CelebA-GH, domain A non-blond-hains males, and the domain B is blond-haired females. Our goal is to transfer the facial characteristic of gender (21) while keeping the hair color (22) intact.
Models. We compare variants of orthogonal classifiers to the vanilla CycleGAN (Vanilla) and the importance sampling objective (IS). We consider two ways of obtaining the wi classifier: (a) Oracle: The principal classifier W1 is trained on an oracle dataset where Z1 is the only discriminative direction, e.g., setting bias degrees Xa= 0.9 and A O in CMNIST such that only digit colors vary across domains. (b) Domain generalization: Domain generalization algorithms aim to learn the prediction mechanism that invariant across environments and thus generalizable to unseen environments (Blanchard al., 2011 1; Arjovsky et al., 2019). The variability of environments is considered as nuisance variation. We construct environments such that only YIZı is invariant. We defer details of the environments Appendix E. We use three domain generalization algorithms, Fish (Shi et al., 2021), TRM (Xu & Jaakkola, 2021) I) and MLDG (Li al., 2018), to obtain w1. We indicate different approaches by the parentheses after Wa wi and IS, e.g., W2 wi (Oracle) is the orthogonal classifier with wi trained on Oracle datasets.
Metric. We three metrics for quantitative evaluation: 1) Z1 accuracy: the success rate of transferring an image's s latent 21 from domain A to domain B; 2) Z- accuracy: percentage of transferred images whose latents 22 unchanged; 3) FID scores: standard metric of image quality (Heusel et al., 2017). We only report FID scores on the CelebA-GH dataset since it is not common to compute FID on MNIST images. Zi, Z2 accuracies are measured by two oracle classifiers that output an image's latents 2 and z2 (Appendix 1.13)
For more details of datasets, models and training procedure, please refer to Appendix E.
4.1.2 RESULTS
Comparison to IS. In section 3.2, we demonstrate that IS suffers from high variance when the di- vergences of the marginal and the label-conditioned latent distributions are large. We provide further empirical evidence that classifier orthogonalization method is more robust than IS. Fig. 2(a) shows that the test loss of IS increases dramatically with the divergences. Fig. 2(b) displays that IS's test loss grows rapidly when enlarging learning rate, which corroborates the observation that gradient variances are more detrimental to the model's generalization with larger learning rates (Wang et al., 2013). In contrast, the test loss of w, wi remains stable with varying divergences and learning rates. In Fig. 2(c), we visualize the histograms of predicted probabilities of Wa wi (light color) and IS (dark color). We observe that the predicted probabilities of wa wi better concentrates around the ground-truth probabilities (red lines).
Main result. As shown in Table and 2, adding an orthogonal classifier to the vanilla CycleGAN significantly improves its 22 accuracy (from 14 to 90+ in CMNIST, from 16 to 40+ in CelebA-GH) while incurring slight loss of the image quality. We observe that the oracle version of orthogonal classi- fier (w, N wi (Oracle)) achieves best 22 accuracy on both datasets. In addition, class orthogonalization is compatible with domain generalization algorithms when the prediction mechanism / is invariant across collected environments.
In Fig. 3, we visualize the transferred samples from domain A. We observe that vanilla CycleGAN models change the background colors/hair colors along with digit colors/facial characteristics in CMNIST and CelebA. In contrast, orthogonal classifiers better preserve the orthogonal aspects Zz in the input images. We also provide visualizations for domain B in Appendix C.

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Input
Ingert
Onelle MLDO Vanilla
6
9
2
6
7
7
Oncic MLIXI Vanilts
6
9
2
6
1
7 3
7
0
6
2
6
7
7
(a) CMNIST: Domain A
(b) CelebA-GH: Domain A
Figure 3: Style transfer samples on the domain A of CMNIST and CelebA-GH. We visualize the inputs (top row) and the corresponding transferred samples of different methods.
5 ORTHOGONAL CLASSIFIER FOR DOMAIN ADAPTATION
In unsupervised domain adaptation, we have two domains, source and target with data distribution Ps Pt. Each sample comes with three variables, data X, task label Y and domain label U. We assume U is uniformly distributed in {0 (source), 1 (target)). The task label is only accessible in the source domain. Consider learning model h = go f that is the composite of the encoder f : X + Z and the predictor 9: + V. A common practice of domain adaptation is to match the two domain's 's feature distributions (x()
via domain adversarial training (Ganin & Lempitsky, 2015; Long et al., 2018; Shu al., 2018). The encoder f is optimized to extract useful feature for the predictor while simultaneously bridging the domain gap via the following domain ad adversarial objective:
min max Lff D) == Eo-p.llog D(f(=)(()))))))))))) Ex-p, [log D((((x))) 1 1>
(4)
where discriminator D: 2 [0, distinguishes features from two domains. The equilibrium of the objective is achieved when the encoder perfectly aligns the two domain, iee, p.(f(=)) p. (f()))). We highlight that such equilibrium not desirable when the target domain has shifted label distribution (Az- izzadenesheli et al., 2019; des Combes et al., 2020). Instead, when the label shift appears, it is more preferred align the conditioned feature distribution,iee, ps (v)))) = pe(s(x)lv), Vy y.
Now we show that our classifier orthogonalivation technique can be applied to the discriminator for making it "orthogonal to" the task label. Specifically, consider the original discriminator full classifier
Wz, i.e., for given f. W. (x)) arg max [(S, D) p.(((=))pp((a))) We then construct the principle classifier wi that discriminates the domain purely via the task label Y, i.e., w/ (x)) Pr(U OJY = (x)). Note that here we assume the task label Y can be determined by the data i..., Y = y(X). We focus on the case that Y is discrete so wi could be directly computed via frequency count. We propose to train the encoder I with the orthogonalized discriminator w> Wi,
p.(f(=))
min f( (w, wi (-)) Ex.pp [lo((,, w. > wi (+))]+ + Ex-p, [log( - (w, w/ (x))) (5)
Proposition 3. Suppose there exists random variable Zz = xx) orthogonal to y(X) Y w.rl p(((),U) Then I achieves globul optimum if and only if it aligns all label-conditioned feature distributions, i.e., p.([())lv)) pc (<((()))))) Vy )
Note that in practice, we have no access to the target domain label prior (Y) and the label y(x) for target domain data x ~ Pt. Thus we use the pseudo label ý as a surrogate to construct the principle classifier, where j(z) arg.ax, h(z), is generated by our model h.
5.1 EXPERIMENTS
Models. We take well-known domain adaptation algorithm VADA (Shu al., 2018) our baseline. VADA is based on domain adversarial training and combined with virtual adversarial training and entropy regularization. We show that utilizing orthogonal classifier can improve its robustness to label shift. We compare it with two improvements: (1) VADA+W, \ Wi which is our method that plugs in the orthogonal classifier discriminator in VADA; (2) VADA+IW: We tailor the SOTA method for label shifted domain adaptation, importance- inportance-weighted domain adaptation (des Combes et al., 2020), and apply it to VADA. We also incorporate two recent domain adaptation algorithms for images CyCADA (Hoffman et al., 2018) and A ADDA (Tzeng et al., 2017)- into comparisons.

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Setup. We focus visual domain adaptation and directly borrow the datasets, architectures and domain setups from Shu et al. (2018). To add label shift between domains, we control the label distribution in the two domains. For source domain, sub-sample 70% data points from the first half of the classes and 30% from the second half. We reverse the sub-sampling ratios on the target domain. The label distribution remains the same across the target domain train and test set. Please see Appendix E.2 for more details.
Results. Table reports the test accuracy seven domain adaptation tasks. We observe that VADA+W, Wi improves over VADA across all tasks and outperforms VADA+IW on five out of seven tasks. We find VADA+IW performs worse than VADA in two tasks, MNIST-SVHN and MNIST- hypothesis is that the domain gap large between these datasets, hindering the estimation of importance weight. Hence, VADA-IW is unable adapt the label shift appropriately. In addition, the results show that VADA+Wa > w1 outperforms ADDA on six out of seven tasks.
Our
Table 3: Test accuracy on visual domain adaptation benchmarks
Source Target
MNIST MNIST-M
SVHN MNIST
MNIST SVHN
DIGITS SVHN
SIGNS GTSRB
CIFAR STL
STL CIFAR
Source- Only
51.8
75.7
34.5
85.0
74.7
68.7
47.7
ADDA
89.7
78.2
38.4
86.0
90.6
66.8
50.4
CyCADA
82.8
39.6
VADA
77.8
79.0
35.7
90.3
93.6
72.4
53.1
VADA + IW
71.21
87.11
34.5
90.7 1
95.41
74.07
53.81
VADA + W1
79.11
88.01
40.51
90.61
95.21
74.5 1
54.11
ORTHOGONAL CLASSIFIER FOR FAIRNESS
We are given a dataset D = {[(2,, Di. 4z) that is sampled iid from the distribution pxYu, which contains the observations æ X' the sensitive attributes e l and the labels y V. Our goal learn classifier that is accurate w.r.t and fair w.r.t u. We frame the faimees problem as finding the orthogonal classifier of an "totally unfair" classifier wi that only uses the sensitive attributes for prediction. We can directly get the unfair classifier from the dataset statistics, i.e., wi(z), p(Y y/U = u(x)). Below we show that the orthogonal classifier of unfair classifier meets equalized odds, one metric for fairness evaluation.
Proposition 4. If the orthogonal random variable Uf = u(X) w.r. pxy exists, then the orthogonal classifier Wx Wh satisfies the criterion of equalized odds.
We emphasize that, unlike existing algorithms for learning fairness classifier, our method does not require additional training. We obtain a fair classifier via orthogonalizing an existing model wz which simply the vanilla model trained by ERM on the dataset D.
6.1 EXPERIMENTS
Setup. We experiment on the UCI Adult dataset, which has gender as the sensitive attribute and the UCI German credit dataset, which has age as the sensitive attribute (Zemel al., 2013; Madras al., 2018: Song et al., 2019). We compare the orthogonal classifier Wa N Wi to three baselines: LAFTR (Madras al., 2018), which proposes adversarial objective functions that upper bounds the unfairness metrics; L-MIFR (Song et al., 2019), which uses mutual information objectives to control the expressiveness- fairness trade-off;; ReBias (Bahng al., 2020), which minimizes the SSilber--Schiidt Independence Criterion between the model representations and biased representations; as well as the Vanilla (w,) trained by ERM. We employ
faimeess metrics demographic parity distance (Ape) and equalized - defined in Madras et al. (2018). We denote the penalty coefficient of the adversarial odds distance - or diasing objective as 7, whose values govern a trade-off between prediction accuracy and fairness, in all baselines. We borrow experiment configurations, such as CNN architecture, from Song et al. (2019). Please refer Appendix E.3 for more details.
Results. Tables 4, 5 report the test accuracy, App and AçO on Adults and German datasets. We observe that the orthogonal classifier decreases the unfairness degree of the Vanilla model and has competitive performances to existing baselines. Especially in the German dataset, compared to LAFTR, our method

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Published as a conference paper at ICLR 2022
has the same Ago but better Ape and test accuracy. It is surprising that method outperforms LAFTR, even though it is not specially designed for fairness. Further, our method has benefit of being training free, which allows it to be applied to improve any off-the-shelff classifier's 's fairesss without additional training.
Table 4: Accuracy v.s. Fairness (Adults)
Table 5: Accuracy v.s. Fairness (German)
Acc.t
Ape 4
Aso L
Acc.1
APp
ABo
Vanilla
84.5
0.19
0.20
Vanilla
76.0
0.19
0.33
LAFTR (7 0.1)
84.2
0.14
0.09
LAFTR (7 = 0.1)
73.0
0.11
0.17
LAFTR (7 0.5)
84.0
0.12
0.07
LAFTR (7 = 0.5)
72.7
0.11
0.19
L-MIIF 00.05)
81.6
0.04
0.15
L-MIFR ( = =0.1)
75.8
0.10
0.21
I.MMIF (y=0.))
82.0
0.06
0.16
L-MIFR ( = =0.05)
75.6
0.08
0.18
ReBias (7 100)
84.3
0.15
0.11
ReBias (7 100)
73.0
0.07
0.17
ReBias (7= = 50)
84.4
0.17
0.16
ReBias (7 = 50)
75.0
0.10
0.20
W. WI
81.6
0.12
0.12
WP Wi
75.4
0.09
0.18
RELATED WORKS
Disentangled representation learning Similar to orthogonal random variables, disentangled represen- tations are also based on specific notions of feature independence. For example, Higgins et al. (2018) defines disentangled representations via equivalence and independence of group transformations, Kim & Mnih (2018) relates disentanglement to distribution factorization, and Shu et al. (2020) characterizes disentanglement through generator consistency and notion of restrictiveness. Our work differs in two key respects. First, most definitions of disentanglement rely primarily on the bijection between latents and inputs (Shu et al., 2020) absent labels. In contrast, orthogonal features must be conditionally independent given labels. Further, our work orthogonal features remain implicit and they are used discriminatively in predictors. Several approaches aim to learn disentangled representations in un- supervised manner (Chen et al., 2016; Higgins et al., 2017; Chen et al., 2018). However, Locatello al. (2019) argues that unsupervised disentangled representation learning is impossible without proper inductive bias.
Model de-biasing A line of works focuses preventing model replying on the dataset biases. Bahng et al. (2020) learns the de-biased representation by imposing HSIC penalty with biased representation, and Nam al. (2020) trains an unbiased model by up- up-weighting the high loss samples in the biased model. Li et al. (2021) de-bias training data through data augmentation. However, these works lack theoretical definition for the de-biased model in general cases and often require explicit dataset biases.
Density ratio estimation using a classifier Using a binary classifier for estimating the density ra- tio (Sugiyama et al., 2012) enjoys widespread attention in machine learning. The density ratio estimated by classifier has been applied to Monte Carlo inference (Grover et al., 2019; Azadi et al., 2019), class imbalance (Byrd & Lipton, 2019) and domain adaptation (Azizzadeneshell et al., 2019).
Learning a set of diverse classifiers Another line of work related to ours is learning a collection of diverse classifiers through imposing penalties that relate to input gradients. Diversity here means that the classifiers rely on different sets of features. To encourage such diversity, Ross et al. (2018; 2020) propose a notion of local independence, which uses cosine similarity between input gradients of classifiers as the regularizer, while in Teney et al. (2021) the regularizer pertains to dot products. Ross al. (2017) sequentially trains multiple classifiers to obtain qualitatively different decision boundaries. We defer more discussions of the advantages of classifier orthogonalization over existing methods to Appendix F.
8 CONCLUSION
We consider finding a discriminative direction that is orthogonal given principal classifier. The solution in the linear case is straightforward but does not generalize to the non-linear case. We define and investigate orthogonal random variables, and propose a simple but effective algorithm (classifier orthogonalization) to construct the orthogonal classifier with both theoretical and empirical support. Empirically, we demonstrate that the orthogonal classifier enables controlled style transfer, improves existing alignment methods for domain adaptation, and has a lower degree of unfairness.

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ACKNOWLEDGEMENTS
The work was partially supported by an MIT-DSTA Singapore project and by an MIT-IBM Grand Challenge grant. YX was partially supported by the HDTV Grand Alliance Fellowship. We would like to thank the anonymous reviewers for their valuable feedback.
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A MORE PROPERTIES OF ORTHOGONAL RANDOM VARIABLE
In this section demonstrate the properties of orthogonal random variables. We also discuss the existence and uniqueness of orthogonal classifier when given the principal classifier. Below we list some useful properties of orthogonal random variables.
Proposition 5. Let Zi and Z2 be any mutually orthogonal w.r.t Pxy, then we have
(i) Invariance: For any diffeomorphism 9. (Zz)) and ZZ)) are also mutually orthogonal.
(ii) Distribution-fre:: Z1 L2 are mutually orthogonal w.r. Pxay if and only if there exists a diffeo- morphism mapping between X and X'.
(ii) Existence: Suppose the label-conditioned distributions Py C²,, supp(p,)) X, X, V. For C² and zı (X) diffeomorphism to Euclidean space, then Vi V, there exists difseomorphism map- ping (z1(X), (=2xX)=22(()) (x,( such that =1(X) (X){Y = i. Further if there exists difseomorphism mappings between 22,5, then the orthogonal r.v. of 21 (X) exists.
In the following theorem, we justify the validity of the classifier orthogonalization when z (x) satisfies some regularity conditions.
Theorem 2. Suppose wi (I); = pY = (1)) If the label-condition distributions Pys, and 21 satisfy the conditions in Proposition 5.(ii),, then wzz is the unique orthogonal classifier of w1.
Note that construction of wa w1 does not require the exact expression of 21 (X), its orthogonal or data generation process. The theorem states that the orthogonal classifier constructed by classifier orthogonalization is the Bayesian classifier for any orthogonal random variables of 2 (X).
B
PROOFS
B. PROOF FOR PROPOSITION
Proposition 1. Suppose the orthogonal r.v. (X) w.r.t pxy exists and is denoted as 22(X), Then x(X)= z2 (X) is maximizer of (((()))\|a(())) subject to (xx());=(X))))==
Proof. For any function 2, by chain rule we have ((, (x);; Y (X)) (X)) and
(x2; +
= [aa +(
Further, the defini-
tion of the orthogonal r.v. ensures that there exists differmorphism f between X and 22
which implies [
((a((X),2(X);))
Hence by data processing inequality we have:
- (=)(x))) ((=2(X): 2 I(= 7
((2x(X): [(=(X):\\==xx))
The above inequality shows that 2 22 maximize the mutual information [(=(X)) Furher, by definition of the orthogonal r.v., = 2g is independent of 21 conditioned on Y, i.e. 2(X) 1 Y which is equivalent (=2(X:: (X)lY) (er)))))=0) O.
B.2 PROOF FOR THEOREM
Theorem 1. Assume Py is uniform distribution, Vu,, W takes values in (m, m) with m (0. })) and l/pxly (xly) (0,7) +0,+ holds for Yx X e V. Then for any 8 (0, 1) with probability
at least 1- 5, we have:
1
2 log 1
R(ū, > w1) R(w,, wn s (1 + +))
29 (W) 2 log
m
[D]
Proof. Denote the empirical risk of w as
1
R(w)) = f((w)
Σ
log ((x:)))))))
DD]
(ED
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and the population risk
R(w)= BB)( [log (x))()
Step 1: bounding excess risk (R()))) R( (.w)))
By the classical result in theorem 8 in Bartlett & Mendelson (2001), we know that for any we E W,
EE (0, 1), since - llog w()), log ), with probability at least
[f(w)) - R(w)) = -
x2
log( (wx:)(())))))))))))())))))) (o(log((())))))) 29,, + 2log 2 log B
Σ s=1
Since 1x infw,ew eW R(wx), /² inf R( (ww)) and [R(().)) R((xs))
2n (W)
+
2 2log 1 V-- 21og 2 [((((w;))))))))))){))))) R(w?) (WV + 22ogg 1 m
2 llo
we have
log 3
IR(0,) R( ([(w))= s 2x + log
(6)
Step 2: bounding [Rus w1) R(w w1)) by [R( (.)) R(w
w<xx)
w(x))/
R(w w1)= = -Ep(nw)
log
1en (xr),
=R(w)= Ep(a.))
log (z) ( y wi (1))
En wx))
Then we have:
[n(x= > w1) - -R(w" > wi)l
wx (2))
log (xg 10 (z) of 21 (x)
{((((0)) R(w()l + Ep(x,)) lo (x)
Ep(x,w)
y wi (2)()))))
Eg @b (x) wn (
=[R(wz)~ R(w + Ep(a..))
log
(2)
Ey (x),
(x)x)
wx(x)y
(x),
log min (x)x
<IR(w)) - R(w + max
Bx(z,o)
log max D¹
Ep(a.s)
(x(x)
wi(a)y y'
(1)
(x)
Ev @e (t) We define the ratio v) (x)y We define y(x) == arg min, r(x, v). 7(x) == arg max, r(, y)- (f
We have r(x, (I,z(z))= S Ey (x(()),///1))))) $ r(x, (()). Let assume アニェ) 7 for all x, For any function y/ : XX' + V/ we have the following bound, where we have importance weight 6(x.z) == p(az) if y= y (x) otherwise 0,
Ex Logr((, y As result, have
= so..
R(w)))
Ey th (z) Ey (I) (z)
log
max (HunX(//D)orraaggoo//////orrtttd
R((w))
Thus
| w1) R(w > (ww) s [R(w.) R(w!)l
1e (x)
(x),,
w/ (x),,)
1 (*)
max Bp(2)0)
log max 8f
Bp(a..)
log min (x)
(z)
v'
(w))
(x)
s (1+7)|R((i.)) R(w))l

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Combining Eq. (6) and Eq. (7), know that with probability 1 6, we have:
2 log 3
|[ > wi) R(w3 wr)l s (1 +)) 29 + 2og
B.3 PROOF FOR PROPOSITION AND THEOREM 2
Proposition 5. Let Z1 and Zy be any mutually orthogonal w.r.t Pxy. then we have
(i) Invariance: For any difseomorphism 9. g(1)) and g (Z)) are also mutually orthogonal.
(ii Distribution-fre:: Zı, L2 are mutually orthogonal w.r. Pxty if and only if there exists a diffeo- morphism mapping between X and X'.
(ii) Existence: Suppose the label-conditioned distributions Py EC¹, , suppp(p) = V.VyyEy For C² and 21 (X) diffeomorphism to Euclidean space, then Vi V, there exists diffeonorphism map- ping (z1 (X), xx)( such that 21(X) X = i. Further if there exists diffeomorphism mappings between 22,58, then the orthogonal r.v. of 21 (X) exists.
Proof. (i) Since Z1 and Z2 are independent, the independence also holds for g(Z1) and 9(Z)). In addition, we construct the diffeomorphism between (g( (1), g (Z2)) and X f(g( ZZ)..( ((2))
I( (g(2))), (o(Z))) X. Apparently ( (Z2)) is a diffeomorphism.
(ii) We denote the diffeomorphism between X' and X as f. Then the diffeomorphism mapping between Z2) and X is f f. Thus Z1, Z2 are mutually orthogonal w.r.t X'.
(ii) We will use constructive proof.
Step 1: Denote dim(X') = d, dim(z1(X)) k. then we can prove that the manifold 2) (X), Rd- is diffeomorphism of X. We denote the diffcomorphism as f. and denote f(x) [=)) where 21 X' 21 (X), t: Rd-k
Step 2: For X|Y = y. let (t((2), [())),,,,(())))))......", : [0.1]. f1...., d k} be the CDF corresponding to the distribution of t(x), -()) 21 2x),,
(1)))-122(X))
(()));;;(+); F,())) ((0))))))))))) 1), (
The conditions p C², p(x) 0, Va e X' ensure that the PDF of s(X) is also in C2 and takes values in (0, oo). Thus the above CDFs are all diffeomorphism mapping. By the inverse CDF theorem, for any 21 (x), h(((();=1 (X) = 21 (x)) is uniform distribution in (0, 1) Hence the random variable defined by the mapping h, ie., 22,y X) h(t(X); (X)), is independent of 21 (X) given ==1.
In addition, by the construction of 22,y (x:) know that there exists a diffeomorphism f between (Z))))(())) and (21 (X), 22, (x)) such that (X), = (x), = X is also a diffcomorphsim mapping.
(X)). Then f
(X),
Step 3: From the condition we know that for every y y. there exists diffeomorphism function my such that (z) 22, (x). Apparently Vy, 022 (X) ) 1 (X) = y by (2) 1 z (x)) Y y. Further, (31, 22,1) a diffeomorphism. Hence 22,1 (X) the orthogonal r.v. of 21 (X) w.r PXY.
Theorem 2. Suppose wi ( il21(x)) the label-condition distributions Pys, and 2 satisfy the conditions in Proposition 5.(ii,, then wx Wi is the unique orthogonal classifier of W1.
Proof. By the existence property in Proposition 5, we know that for X, there exists a orthogonal r.v. of 21 (X) and denote it as 22 (X1). By the proposition we know that there exists common diffeomorphism f mapping (=i(Xly), 22( (2 (ly)) to Xly, Vy V.
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Further, by change of variables and the definition of orthogonal r.V., have (o (P.a(ar) pa(x)) (Pa(a)) -x(>); mi(s),
P5.a(*2)
(5 ( (f)( py(a) P5)(2)) wa(a), (z)y) Thus via the bijection of clas-
sifier and density ratios we know that we w1(2); = -Y
B.4 PROOF FOR PROPOSITION
Proposition 2. The global minimum of LAB (G is achieved if and only if Pz,, (2))2,22222)) (22).
Proof. By definition, r(x) following:
Thus we can reformulate the criterion CAB as the (=) Q(=x(=))
wa(xx
P(z(()))
P(x)
LoeAn (GAB) = E. E.. log P(x) (())(())
P(21, 22)
E 22 log
P(21, + Q22)(((((()))))) 2/2 (()))
P(x))
P 22) (22))( (22) -log
2 - log E
#1
22)
where we get the lower bound by Jensen's inequality. The equality holds when 22) ((z))
B.5 PROOF FOR PROPOSITION
Proposition 3. Suppose there exists random variable Zz 2x(X) orthogonal to v(X) Y w.t p(((),UU) Then f achieves global optimum if and only if it aligns all babbl-conditioned feature distributions, i.e., (f((f))l)) p)((())))) Vye
Proof. By definition, optimal classifier wo (1)o = P. ()))))))) principle classifier w1(x)o = (,(x)))
P.((()))
(x(x))) By the definition of orthogonal r.v., (Y,U) and s(X) have bijection between them. It suggests, conditioned on Y, the supports of f(X) y y non-overlapped for different y in both domains p, and It means if ps (f(x))v) 0 then Vy (=(())))))))) = 0.
Thus the orthogonal classifier we w1 satisfies:
p=(f(x) 1( (f(x)) + pi(f(z)),
(wy ww) P=(U(T)) (r(U())) Pc(U(x))
Ey Pa((())))))
Eg (f(=)))))) + EXPI((()))P ( P.(U(x))
/
(=()(z))
Pa(U(x))
(((F())((=)))AA(())) /
Po((())
Po()(x))
pc(((z)
(=((()))((z)
(=(())))(=)) ((((((((()))
Thus the objective in Eq. (5) can be reformulated as,
C(S, (w,, \ wi )-))
(((())))()))
(x((()))(z))
-Er-p.
P. (f(f ()))))))))))) P:( (.((()))(())) Eu--pe flog
P. (.(())))((z) + (A((()))(())
Ps ()()ll)) (.((()))))) Eyy-p Ex-pa log
ps (c)
+
-Ey-p. E (xlo) log
(=((l))
(c(f())lv
Ps ()(())) + +p(((())l)) (f(
Ps (=((())))) + p.(s(T)lv)
2 --Eyrp
log (xyps
Ey-pe log (egeepp()tl))
(x(()))
= 2log
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The lower-bound holds due to Jensen's inequality. The equality is achieved if and only if Pa( invariant to 2, for all y, Le., Vy E } (f((()))))) - (=(f(=)ly)
is
p=([(())))
B.6 PROOF FOR PROPOSITION
Proposition 4. If the orthogonal random variable Uf u(X) w.r. Pxv exists, then the orthogonal classifier Wn > wi satisfies the criterion of equalized odds.
Proof. We denote the orthogonal random variables as (x),, and w2 > (I) (X) U|Y, we know that wx W1 (X) 1 UJY. Hence the prediction of the classifier wi wi is conditional independent sensitive attribute U on the ground-trune label Y, which meets the equalized odds metric.
Pr[v]>2(z)]+ ( Since
C EXTRA SAMPLES
Inpur
Input
3
4
9
4
Waillle
6
8
2
6
7
MIDI Vanilla
Oncice MEDG
6
2
6
7 7
0
6
2
6
7
7
(a) CMNIST: Domain A
(b) CMNIST: Domain B
Inguut
Input
MEDG Vanilts
MI.DG
(c) CelebA-GH: Domain A
(d) CelebA -HH: Domain
Figure 4: CMNIST and CelebA-GH
D EXTRA EXPERIMENTS
D.1 STYLE TRANSFER
Table 6: CelebA
Z1 acc.
Zy acc.
FID
Vanilla
75.3
87.0
36.2
W7s
WT WMLMG
81.8
93.5
35.2
Wyy W1-TRM
76.3
92.5
33.5
W2 -Fishish
74.2
93.3
33.1
IS ---arclee
73.3
88.1
36.5
W' wh -Oracle
84.0
95.2
38.5
We show transferred samples for both domains in Fig. on CMNIST and CelebA-GH.
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D.2 STYLE TRANSFER ON CELEBA
Unlike the split in CelebA-GH dataset, domain A is males, and domain B is females. The two domains together form the full CelebA (Liu et al., 2015) with 202599 images. Note that there exists large imbalance of hair colors within the male group, with only around 2% males having blond hair. Table 6 reports the /22 accuracies and FID scores on the full CelebA datasets. We find that our method improves the style transfer on the original CelebA dataset, especially the Z2 accuracy. It shows that our method can help style transfer in real-world datasets with multiple varying factors.
E EXTRA EXPERIMENT DETAILS
E. STYLE TRANSFER
E.1.1 DATASET DETAILS
We randomly sampled two digits colors and background colors for CMNIST. For CelebA-GH, we extract two mabsets-male with non-blond hair and female with blond hair-by the attributes provided in CelebA dataset. For all datasets, we 0.8/0.2 proportions to split the train/test
For CelebA dataset, following the implementation of Zhu et al. (2017) htttps://gitee junyanz/ Cyyean//CyyleGAN) we use 128-layer UNet the generator and the discriminator in PatchGAN. For CMNIST dataset, we use a 6-layer CNN as the generator and a 5-layer CNN as the discriminator. We adopt Adam with learning rate 2e-4 the optimizer and batch size 128/32 for CMNIST/CelebA.
com/
E. 1.2 DETAILS OF MODELS
We craft datasets for training the oracle wi (x) (r)Y) For CMNIST, oracle dataset has bias degrees 0.9/0. For CelebA-GH, oracle dataset has {male, female] classes and the hair colors under each class are balanced.
For each dataset, we construct environments train the domain generalization models. For CMNIST, the two environments are datasets with bias degrees 0.9/0.4 and 0.9/0.8. For CelebA-GH, the two environments consist of different groups of equal size: is (non-blond-haired males, blond-haired males, blond-haired females, non-blond-haired females,
females) and the other [non-blond-hal blond-haired males). Note that the facial characteristic of gender (71) is the invariant prediction mechanism across environments, instead of the hair color.
E. EVALUATION FOR 21/22 ACCURACY
To evaluate the accuracies on Z1, Zg latents, we train two oracle classifiers wi Pr(Y|<a(<:) w (x) = (r(Y)z²(x)) on two designed datasets. Specifically, w1/wy are trained on datasets Dj/D, with Zz as the only discriminative direction. For CMNIST, D2 is the dataset with bias degrees 0.9/0 and D. is the dataset with bias degrees 0/0.8. For CelebA-GH, D1 is the dataset with {male, female) as classes and the hair colors under each class are balanced. D2 is the dataset with (blond hair, non-blond hair) as classes and the males/females under each class are balanced.
The Z1 accuracy on dataset D for transfer model G/ is:
Zi accuracy =
Σ I
arg max wi (x) f arg max wi (G(x))
D
ZED
y
And Za accuracy is:
Z1 accuracy
Σ
arg max (x)) arg max w (G(x))
D
ZED
where I is the indicator function. Z accuracy measures the success rate transferring the Z₁ and Zz accuracy measures the success rate of keeping Zr aspects unchanged.
E.2 DOMAIN ADAPTATION
The full names for the seven datasets are MNIST, MNIST-M, SVHN, GTSRB, SYN-DIGITS (DIGITS), SYN-SIGNS (SIGNS), CIFAR-10 (CIFAR) and STI-10 (STL.. We sub-sample each dataset by the designated imbalance ratio to construct the pairs.
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We use the 18- la-layer neural network in Shu et al. (2018) with pre-process via instance-normalizatie Following Shu et al. (2018), smaller CNN is used for digits (MNIST, MNIST-M, SVHN, DIGITS) and traffic signs (SIGNS, GTSRB) and larger CNN is used for CIFAR-10 and STL- 10. We use the Adam optimizer and haper-parameters in Appendix B in Shu et al. (2018) except for MNIST + MNIST-M, where find that setting X 1, Aa 1e- 1 largely improves the VADA performance over label shifts. Furthermore, we set Ad = 1e 2, ie., the default value suggested by Shu et al. (2018), to enable adversarial feature aligmmel alignment.
E.3 FAIRNESS
The two datasets are: the UCI Adult dataset which has gender as the sensitive attribute; the UCI German credit dataset which has age as the sensitive attribute. We borrow the encoder, decoder and the final fo layer from Song et al. (2019). For fair comparison, we use the same encoder+fc layer as the function family Wo of our classifier since our method do need decoder for reconstruction. We use Adam with learning rate le-3 as the optimizer, and a batch size of 64.
The original ReBias method trains set of biased models to learn the de-biased representations (Bahng et al., 2020), such as designing CNNs with small receptive fields to capture the texture bias. However, in the fairness experiments, the bias is explicitly given, i.e., the sensitive attribute. For fair comparison, we set the HSIC between the sensitive attribute and the representations as the de-bias regularizer.
F LEARNING A DIVERSE SET OF CLASSIFIERS
In this section discuss the difference and advantages of the classifier orthogonalization, compared to methods that learn a diverse set of classifiers. We also provide a counter-cxample to show that the orthogonal input gradient does not lead to a pair of orthogonal classifiers. We show that orthogonal classifiers might not in optimal solution set of minimizing input gradient penalty.
The goals of learning diverse classifiers include interpretability (Ross al., 2017), overcoming simplicity bias (Teney et al., 2021), improving ensemble performance (Ross et al., 2020), and recovering confound- ing decision rules (Ross et al., 2018). There is direct trade-off between diversity and accuracy and this is controlled by the regularization parameter. In contrast, in our work, given the principal classifier, our method directly constructs classifier that uses only the orthogonal variables for prediction. There is no trade-off to set in this sense. We also focus on novel applications of controlling the orthogonal directions, such style transfer, domain adaptation and faimesss Further, notion of orthogonality is defined via latent orthogonal variables that relate to inputs via a diffcomorphism (Definition I) as opposed to directly in the input space. Although learning diverse classifiers through input gradient penalties efficient, it does not guarantee orthogonality in the sense we define it. We provide an illustrative counter-example in Appendix F, where the input space is not disentangled nor has orthogonal input gradients but corresponds to a pair of orthogonal classifiers in our sense. We also prove that, given the principal classifier, minimizing the loss by introducing input gradient penalty would not necessarily lead to an orthogonal classifier.
We note that one clear limitation of our classifier orthogonalization procedure is that we need access to the full classifier. Learning this full classifier can be impacted by simplicity bias (Shah et al., 2020) that could prevent it from relying all the relevant features. We can address this limitation by leveraging previous efforts (Ross et al., 2020; Teney et al., 2021) to mitigate simplicity bias when training the full classifier.
F.1 RELATIONSHIP TO ORTHOGONAL INPUT GRADIENTS
The orthogonality constraints on input gradients demonstrate good performance in learning a set of diverse classifiers (Ross et al., 2017; 2018; 2020; Teney et al., 2021). They typically use the dot product of input gradients between pairs of classifiers as the regularizer. We highlight that when facing the latent orthogonal random variables, the orthogonal gradient constraints on input space no longer guarantees to leamn the orthogonal classifier. We demonstrate it on simple non-linear example below.
Consider a binary classification problem with the following data generating process. Label Y is uniformly distributions in 1), Conditioned on the label, we have two independent k-dimensional latents
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Z1, Z2 such that Z (yjuu Ik), Z2 y (yyu,, Ig). Na R Mg R are the means of the latent variables, and & > m. The data X is generated from latents via the following Zz
diffeomorphism X = [
( (k<m)) where is an arbitrary non-linear diffeomorphism on RR
We can see that 2, 22 are mutually orthogonal w.r.t the distribution Pxy- Now consider the Bayes optimal classifiers w1, w2 using variable Z1, Z2. We have w/ (2) Py 21 (1|z1) (2 21) and (2( = PYiZ, (1/22) a (2 22) where the sigmoid function. Apparently, they are pair of orthogonal classifiers. Besides, we denote the classifier based on the first m dimensions in for prediction as W3, ie., w3(xr) = Py)((Z))m (1)(=))mm) (2((1) ).m (=1)am).
Given the principal classifier w1, we consider using the loss function in Ross et al. (2018; 2020) to learn the orthogonal classifier of W1, by adding input gradient penalty term to the standard classification loss:
L ( (w) = (w) + XIE[cos AB[cos"(Vz>u (V (z)), [x)]
Le is the expected cross- cros--entrppy loss, cos the cosine-sine bosiie-similarity and is the hyper-paraman parameter for the gradient penalty term (Ross et al., 2018; 2020). Below show that (1) the orthogonal classifier wy does not necessarily satisfy E[cos² ( u1 (z), (2w2(x)) (2) the orthogonal classifier wy is not necessarily the minimizer of Lu,, (w).
Proposition 6. For the above (Z1, Z2), their corresponding Bayes classifiers W1, W2 satisfy B[V,w] Vz)"V==z2=x)) 10 and only if (11)mml2
Proof. X, xxx====ZZ11 - (0 (10Z)-(0z)) D.
Zz
Zz are the two components of X. Note that 21 '
(0 22
(0(--m)) ) By chain rule, the input gradient of w1
dž1 v1, Pr(Y 1 = (1))
Wr (x)
dx
-Vagg Tmnk (k+m)xk (1 (1+6-210 2uf 2p11
e-2
-2n 22(0)
where mock is the submatrix of Lxck. Similarly we get V2 = (mxx (ktm)om (1+e 2ug =2(z))z -2 Together, the dot product between input gradients is
2e 2nF (1) uT
2e 20g 22(11)
W.wi v))))=))))))))
24 2x
Imxk
(k+m))x (mxxm) (k+m))
(1+ e 2ng 22(2))2
4e 4e 2p 21 (2) (x) =x(x2) (#1 ( Y
(1+e 2n (1(()) + 241 (1))22
1e-2 (1)))) s2(=)) (1+e 2o (10)))((1+ 2n sg(0)yz
Since
0, B[V wi (x) T (x)(()) if and only if (M1 ) = O.
Proposition first shows that the expected imput gradient product of the two Bayes classifiers of underlying latents is non-zero, unless the linear decision boundaries on 21, 22 are orthogonal. Hence, given the principal classifier W1, the expected dot product of input gradients between wi and its orthogonal classifier Wa does not have to be zero.
Proposition 7. The orthogonal classifer w2 does not minimize Lw, (w) if (μ1).m = n22 and uTVz, ( (H1)) < O. Particularly, we show that Lw, (ws) < Lw, (w2) in this case.
Proof. Let w3(x) = PY](( (1((2)) The input gradient of w3(x) is
dz1
V=ws(x)= (V_w)(r) mm
((r)).m Pr(Y= = 11( (21 (2)()))
dx
(VZzgg (II k Inxm
2(1)) T
2(1r) )mm.
(k+m))m (1 e 2(11)
21) 241
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Published as a conference paper at ICLR 2022
When (f1).m U2, the dot-produet between Vews(2), Vr (x) is
Vawi (z)) (n))(x)= = 2e 24- ( uf
(Vag (x)))
(1 e 219 (10)2
mick
(k+m)) x
V (vn1 (vm1 (x1))
2 ):th 21 (1).
2((1):
mmx
(1 1 211))
(kmm)xm
4e 2n 21 ( (ar)...
(1) u2 112 +HI Va, g (x1)" Vv, ' (xx1)xm (Fr )mm))
2( 2())) 21 (
When (u1)m M2, we know that 2 22 thus we()))))))))))))))))) = (w(2)) Note that if pi Va, g (21)² (Z1)"v,,g"*
equally predictive of the label and (11)m 0, we have
E[cos
Cros"(Va'lr (VaW1 (I), (xw2(())) (V (V [zu)(=)]]:
E[cos 2 (V. (I), (x)2(x))]
#2 11
[a. (21))
A1 11 (7)
#2 lle
(ktm))kk
(k+
(1l u2 112 +H7 V g ( (V2n (x1))
E
(Ver 9 (x11) g
(vagg (
Mz 112ll|
(k+m))uk 12 11
I
Iink
(k+m)uk
[[[os 1 (V (z), [2))(())]
The strict inequality holds by I4 V 22 9 (x)) T "Vzg -1 (x) )oom (u1)cm <0
(x11)((x1))
#2 12 =1l Va.g (x1g-(<11)
#2 + (mink/ (k+m)ak
112 ll2
Imac
(k+m)xc
(k+m))x
Vzi (zig)((z))
112 112 + (Lmcc)) (k+m))x
( 112 112
0 (k+m)xk
2|l
[mnk] (k+m)ak
#2 lla
To verify that uf -
"1 - (x1 (xm (11 om O does exist, pick h: = 3, = 2, g
2 -3
3
1 -1 5
x. and fnn 1
We have of Va (x1) (x1) "v, (x1) Dom -2 in this case.
1
-1
1
Together, we have Lun (ws) Luv (wz).
Proposition 7 shows that the orthogonal classifier wy is not the minimizer of the loss with input gradient penalty. The results above also hold for un-normalized version of input gradient penalty Tency et al. (2021) by similar analysis.
21


Reviews:
[{"review_id": "iqQMDHv-bDS", "reviewer": "Reviewer_fQv4", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strong (+) and weak (-) points\n- (+) The authors propose and define a new concept: orthogonal classifiers.\n- (+) The concept is general and justified/motivated by needs of seemingly disparate areas of research (style transfer, domain adaptation, fairness). This non-obvious connection between multiple areas is interesting in itself.\n- (+) Demonstration of empirical benefits in multiple applications.\n- (-) There are quite a few missing references to methods that also learn multiple complementary classifiers. So it's difficult to evaluate the novelty/advantage of the proposed method.\n- (-) Imprecisions in the writing (see advice/comments below). I believe these could be fixed by the authors for the final version.\n\nDetailed comments and advice for improvements (in the order of their occurrence in the paper):\n\n- In the second paragraph of the introduction:\n  - \"joint distribution\": maybe name it (P(X, Y))\n  - the standard notation P(...) = (...) might be easier to read than (...) \\sim (...)\n  - \"label-dependent\" need a hyphen\n  - \"the orthogonal classifer (...) must satisfy\": it sounds like an established concept, whereas this is something that you define for this paper. So, rather say: \"we define the classifier w2 to be orthogonal if (...)\"\n  - Given a classifier w1, there are multiple orthogonal classifiers w2, but the current writing implies there is a single w2.\n\n- The concepts of \"principal classifier\" and \"principal variables\" are used in the abstract and intro without being defined or given a reference about. Only Sec. 3 makes it clearer what they refer to. In the intro, maybe use a non-technical word like \"a GIVEN classifier\" ?\n- Fig. 1: \"principle\", typo for principal ?\n- Proposition 1: the first notation I(A,B) uses a comma, the second a semicolumn.\n- Sec. 3: \"Similar, ...\" -> \"similarLY\"\n- Sec. 3: you should define volJf right after its first use (before \"Taken together\"). The equation before \"taken together\" would also be easier to read with some spacing between the three terms.\n- The equation after \"taken together\" is not numbered. Also add spacing between terms for readability.\n- The following sentence has a grammatical issue. I don't understand its second part. What is the subject of \"ensures\" ?\n\"The validity of this procedure is subject to overlapping support of class-conditional distributions, ensures that wx;w1 remain non-zero.\"\n- How can we make sure that this condition is satisfied ? Training a \"full classifier\" does seem trivial to me. Existing work has shown that neural networks can focus on only a few predictive features (see e.g. Pitfalls of Simplicity Bias in Neural Networks, Shah et al. 2020).\n- \"would be closeD\" -> close\n- Proposition 4: \"exits\" -> \"exists\" ?\n- Missing references: Ross et al. and more recently Teney et al. also proposed methods to learn multiple classifiers that use the input data in different ways.\n  - [Ensembles of Locally Independent Prediction Models](https://ojs.aaai.org//index.php/AAAI/article/view/6004)\n  - [Learning Qualitatively Diverse and Interpretable Rules for Classification](https://arxiv.org/abs/1806.08716)\n  - In this one, the authors train multiple classifiers sequentially to focus each on different input features.\n[Right for the right reasons: Training differentiable models by constraining their explanations](https://arxiv.org/abs/1703.03717)\n  - [Evading the Simplicity Bias](https://arxiv.org/abs/2105.05612)\n  - The multiverse loss may also be related, although admittedly more distantly. It serves to increase the number of distinct discriminative directions of a learned representation by duplicating a cross-entropy loss over multiple linear classifiers with an orthogonality constraint on their weights.\n  - [The multiverse loss for robust transfer learning](https://arxiv.org/abs/1511.09033)\n  - [Maximal multiverse learning for promoting cross-task generalization of fine-tuned language models](https://aclanthology.org/2021.eacl-main.14/)\n- The first key difference with disentanglement: \"most disentanglement definitions do not take the diffeomorphism (...) into account\". I'm not sure this is correct. At least colloquially, disentanglement is about \"breaking down\" observations into independent factors that drive the generative process, hence collectively give rise the observations. In other words, the space of the union of these disentangled factors do map, via a diffeomorphism, to the observation space. What do the authors think ? Is there really a \"key difference\" here ?\n- \"Several works aim to learn the disentangled representations\": no need for \"the\"\n- Missing word: \"Density ratio estimation using A classifier\"\n- Missing word: \"Using A binary classifier (...)\"\n- Conclusion: \"the orthogonal classifier makes accurate predictions on the orthogonal aspects of the principal classifier.\" What does \"aspect\" mean ? I feel the sentence is not very useful as a conclusion because of using such a vague term.", "summary_of_the_paper": "- The paper introduces the notion of \"orthogonal classifiers\": classifiers that rely on orthogonal variables. It starts with the simple linear case, and adds a definition that also applies to the non-linear case.\n- The paper proposes two methods to identify a classifier orthogonal to a given one.\n- It then describes 3 use cases: style transfer, domain adaptation, and fairness.\n", "summary_of_the_review": "Preliminary recommendation: unsure, because of the missing references to existing methods that achieve something comparable.\n\nQuestion to the authors:\n- Can you comment on differences/advantages of their approach compared to existing ones ?\n- Can you discuss the necessity/feasibility of learning a \"full classifier\" ?\n\n**Edit after authors' response: I am satisfied with the precisions added by the authors to the manuscript and I strongly recommend it for acceptance.**", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "EGWrzJc5km_", "reviewer": "Reviewer_dNRJ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "- Strength\n  - The paper presents rather a simple, but principled and novel way of constructing orthogonal classifier for any non-linear classifiers.\n  - The paper is written with a good flow.\n  - The effectiveness of the proposed technique has been confirmed on various applications, including style transfer, domain adaptation, and fairness. \n\n- Weakness\n  - For domain adaptation experiments, it seems authors assume access to the marginal distribution of label variables for the test set. This seems additional information the proposed method is using on top of VADA. Would you please confirm? Also, what if these marginal distributions between domains are exactly the same? Are there going to be any improvement?\n  - Is label distribution of the test set for domain adaptation experiments uniform or does it follow the distribution of the target domain used for training?\n  - For domain adaptation experiments, one baseline could be post-processing the source-only model, similarly to the procedure for constructing fair classifier in Section 6.", "summary_of_the_paper": "The paper presents classifier orthogonalization technique that works for non-linear classifiers. The idea is to find a method that orthogonalizes the full classifier w.r.t. the principal classifier so that the resulting classifier is statistically orthogonal to principal classifier. The algorithm turns out to be very simple, only requiring access to the full classifier $P(Y|x)$ and the principal classifier $P(Y|z(x))$, where $z$ is a control variable you want to orthogonalize against. The paper also discusses an alternative method of classifier orthogonalization using importance sampling. \n\nThe effectiveness of the proposed classifier orthogonalization technique has been demonstrated through 3 applications: controlled style transfer, domain adaptation, and classifier fairness. For controlled style transfer, authors modified the CycleGAN's generator update step by orthogonalizing discriminator w.r.t. the controlled style variable. For domain adaptation, they modified VADA, an adversarial domain adaptation approach, by orthogonalizing discriminator based on the label-based principal classifier, so that discriminator wouldn't discriminate the domain based on the frequency count of labels from each domain. For fairness, full classifier is orthogonalized by the sensitive attribute classifier.", "summary_of_the_review": "Overall I find that the proposed method novel and principled. As paper has demonstrated the proposed method could have great impact multiple application domains. As mentioned in the review, some information could be verified through discussion, but I am leaning towards acceptance for my initial rating.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "CiNgKo92u7h", "reviewer": "Reviewer_SmEk", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "Strengths:\n\n1. The paper does an excellent job of defining orthogonal classifiers and provides a clean solution to learning these models.\n\n2. The mapping of multiple different problems is extremely interesting and provides a strong example of how to apply the approach to different problem settings.\n\n3. The proposed approach appears to yield improvements for both the controlled style transfer and domain adaptation problems.\n\nWeaknesses:\n\n1. The controlled style transfer problem seems somewhat contrived. Providing a reasonable example of where this would be applicable would significantly increase the value demonstrated in this section.\n\n2. Given the controlled style transfer problem is novel, it's difficult to determine whether the empirical results are significant as the baseline compared against is designed for a non-controlled style transfer problem.\n\n3. Comparisons to a wider range of domain adaptation approaches (e.g. Cycada, HIDC, etc.) would make the results more convincing.\n\n4. Empirical comparisons to de-biasing techniques would significantly strengthen the experimental section.\n\n5. The results for fairness are somewhat uninteresting as the proposed approach simply reduces to weighting of labels for discrete sensitive attributes.", "summary_of_the_paper": "Given a defined notion of orthogonality for random variables, this paper proposes a straight-forward approach to constructing classifiers orthogonal to a given classifier. Examples of mapping problems to the orthogonal classifier setting are provided for domain adaptation and fairness as well as the newly proposed problem of controlled style transfer.", "summary_of_the_review": "I am overall in favor of accepting this paper. The proposed formulation and solution are well presented and the application to different problems is interesting.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper introduces the concept of classifier orthogonalization. This is a generalization of orthogonality of linear classifiers (linear classifiers with orthogonal weights) to the non-linear setting. It introduces the notion of a full and principal classifier, where the full classifier is one that minimizes the empirical risk, and the principal classifier is one that uses only partial information. The orthogonalization procedure assumes that the input domain, X can be divided into two sets of latent random variables Z1 and Z2 via a bijective mapping. The random variables Z1 are the principal random variables, and Z2 contains all other information. Z1 and Z2 are assumed to be conditionally independent given the target label. The paper outlines two approaches to construct orthogonal classifiers that operate only on Z2. The approach is highlighted in three applications: controlled style transfer, domain adaptation, and fair classification.

The reviewers all found the proposed method to be principled and compelling. Beyond clarification questions and some discussion on related work, the reviewers raised a few issues that were subsequently addressed: 1) Additional baselines for domain adaptation and fairness. 2) Controlled style transfer being a new task with no established baselines, and 3) The feasibility of training a proper “full classifier” that minimizes the empirical risk, and its necessity in the approach. The authors addressed these concerns and updated the paper, to the satisfaction of the reviewers. All of them unanimously recommend acceptance.
Decision: Accept (Spotlight)