Latent Causal Invariant Model
Authors: Xinwei Sun
Botong Wu
Chang Liu
Xiangyu Zheng
Wei Chen
Tao Qin
Tie-Yan Liu
Venue:  (2021)
Keywords: invariance
causality
spurious correlation
out-of-distribution generalization
interpretability
variational auto-encoder

Abstract: Current supervised learning can learn spurious correlation during the data-fitting process, imposing issues regarding interpretability, out-of-distribution (OOD) generalization, and robustness. To avoid spurious correlation, we propose a \textbf{La}tent \textbf{C}ausal \textbf{I}nvariance \textbf{M}odel (LaCIM) which pursues \emph{causal prediction}. Specifically, we introduce latent variables that are separated into (a) output-causative factors and (b) others that are spuriously correlated to the output via confounders, to model the underlying causal factors. We further assume the generating mechanisms from latent space to observed data to be \emph{causally invariant}. We give the identifiable claim of such invariance, particularly the disentanglement of output-causative factors from others, as a theoretical guarantee for precise inference and avoiding spurious correlation. We propose a Variational-Bayesian-based method for estimation and to optimize over the latent space for prediction. The utility of our approach is verified by improved interpretability, prediction power on various OOD scenarios (including healthcare) and robustness on security. 

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LATENT CAUSAL INVARIANT MODEL
Anonymous authors
Paper under double-blind review
ABSTRACT
Current supervised learning can learn spurious correlation during the data-fitting process, imposing issues regarding interpretability, out-of-distribution (OOD) gen- eralization, and robustness. To avoid spurious correlation, we propose Latent Causal Invariance Model (LaCIM) which pursues causal prediction. Specifically, we introduce latent variables that are separated into (a) sutput-cassatie factors and (b) others that are spuriously correlated to the output via confounders, model the underlying causal factors. We further assume the generating mechanisms from latent space to observed data to be causally invariant. We give the identifiable claim of such invariance, particularly the disentanglement of output-causative factors from others, as a theoretical guarantee for precise inference and avoiding spurious correlation. We propose to optimize over the latent space for prediction. The utility of our approach is verified by improved interpretability,, prediction power various OOD scenarios (including healthcare) and robustness on security.
-aaiionnn---asiinnn--- based method for estimation and
INTRODUCTION
Current data-driven deep learning models, revolutionary in various tasks though, heavily rely on i.i.d data to exploit all types of correlations to fit data well. Among such correlations, there can be spurious ones corresponding to biases (e.g., selection or confounding bias due coincidence of the presence of the third factor) inherited from the data provided. Such dependent spurious correlations can erode the (i) interpretability of soccision-makine i.e., extrapolation from observed to new environments, which is crucial especially in safety-critical tasks such as healthcare, and (ii) robustness to small perturbation (Goodfellow et al., 2014).
(ii) ability of didtiiiibbttion (OOD) generalization,
Recently, there is Renaissance of causality in machine learning, expected to pursue causal prediction (Schölkopf, 2019). The so-called "causality" is pioneered by Judea Pearl (Pearl, 2009), as a math- ematical formulation of this metaphysical concept grasped in the human mind. The incorporation of a priori about cause and effect endows the model with the ability to identify the causal structure which entails not only the data but also the underlying process of how they generated. For causal prediction, the old-school methods (Peters et al., 2016; Bühlmann, 2018) causally related the output label Y to the observed inpul X, which however is NOT conceptually reasonable in scenarios with sensory-level observed data (e.g. modeling pixels as causal factors of Y does not make much sense).
For such applications, we rather adopt the manner Bengio al. (2013); Biederman (1987) relate the causal factors of Y to unobserved abstractions denoted by S, i.e., Y + fy,S, Ey) via mechanism fg. We further assume existence of additional latent components denoted as Z. that together with S generates the input X via mechanism fa as X + (S, Zz Taking image classification as an example, the S and Z respectively refer to object-related abstractions (e.g., contour, texture, color) and contextual information (e.g., light, view). Such an assumption is similarly adopted in the literature of nonlinear Independent Components Analysis (ICA) (Hyvarinen and Morioka, 2016; Hyvärinen et al., 2019; Khemakhem, Kingma and Hyvärinen, 2020; Teshima et al., 2020) and latent generative models (Suter et al., 2019), which are however without separation of output (y)-saavte factors (a.k., S) and other correlating factors (a.k.a, Z) that can both be learned in kata-fitting process.
We encapsulate these assumptions into a novel causal model, namely Latent Causal Invariance Model (LaCIM) illustrated in Fig. 1, in which we assume the structural equations fa (associated with S, Z + X), fy (associated with 5 > Y) to be the Causal Invariant Mechanisms (CIMe) that hold under any circumstances with P(S, Z) allowed to be varied across domains. The incorporation of these

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priories can explain the spurious correlation embedded in the back-door path from Z to Y (contextual information to the class label in image classification). To avoid learning spurious correlations, our goal is to identify the intrinsic CIMe fa+ f.. Specifically, possibility to be precisely inferred up to an equivalence relation) of the CIMe. Notably, far beyond the scope in existing literature (Khemakhem, Kingma and Hyvärinen, 2020), our results can implicitly, and are the first to disentangle the output-causative factors (a.ka, S) from others (a.k.a 2) for prediction, to ensure the isolation of undesired spurious correlation. Guaranteed by such, we propose to estimate the CIMe by extending the Variational Auto-encoder (VAE) (Kingma and Welling, 2014) to the supervised scenario. For OOD prediction, we propose to optimize over latent space under the identified CIMe. To verify the correctness of our identifiability claim, we conduct a simulation experiment. We further demonstrate the utility of our LaCIM via high explainable learned semantic features, improved prediction power on various OOD scenarios (including tasks with confounding and selection bias, healthcare), and robustness on security.
first prove the identifiability (i.e. the
We summarize our contribution as follows: (i) Methodologically, propose in section 4.1 latent causal model in which only a subset of latent components are causally related to the output, to avoid spurious correlation and benefit OOD generalization; (ii) Theoretically, we prove the identifiability (in theorem 4.3) of CIMe fa. fy from latent variables to observed data, which disentangles output- causative factors from others; (iii) Algorithmically, guided by the identifiability, we in section 4.3 reformulate Variational Bayesian method to estimate CIMe during training and optimize over latent space during the test: (iv) Experimentally. LaCIM outperforms others in terms of prediction power on OOD tasks and interpretability in section 5.2, and robustness to tiny perturbation in section 5.3.
2 RELATED WORK
The invariance/causal learning proposes learn the assumed invariance for transferring. For the invariance learning methods in Krueger et al. (2020) and Schölkopf (2019), the "invariance" can refer to stable correlation rather than causation, which lacks the interpretability and impedes its generalization to broader set of domains. For causal learning, Peters et al. (2016); Bühlmann (2018); Kuang al. (2018); Heinze-Deml and Meinshausen (2017) assume causal factors as observed input, which is inappropriate for sensory- observational data. In contrast, our LaCIM introduces latent components as causal factors of the input; more importantly, we explicitly separate them into the output-causative features and others, to avoid spurious correlation. Further, we provide the identifiability claim of causal invariant mechanisms. In independent and concurrent works, Teshima et al. (2020) and Ilse et al. (2020) also explore latent variables in causal relation. As comparisons, Teshima et al. (2020) did not differentiate S from Z; and Ilse et al. (2020) proposed to augment intervened data, which can be intractable in real cases.
Other works which are conceptually related to us, as a non- non-exhaustive review, include (i) transfer learning which also leverages invariance in the context of domain adaptation (Schölkopf et al., 2011; 1; Zhang et al., 2013; Gong et al., 2016) or domain generalization (Li et al., 2018; Shankar et al., 2018); and (ii) causal inference (Pearl, 2009; Peters al., 2017) which proposes structural causal model to incorporate intervention via "do-calculus" for cause-effect reasoning and counterfactual learning; (iii) latent generative model which also assumes generation from latent space to observed data (Kingma and Welling, 2014; Suter et al., 2019) but aims at learning generator in the unsupervised scenario.
3 PRELIMINARIES
Problem Setup& & Notation Let X, Y respectively denote the input and output variables. The training data JeEEsran are collected from the set of multiple environments Etrain, where each domain e is associated with a distribution P (X, Y) over X x y and De = {=f, of , d" (Te(n.)) [k] : (1,.., 1-} for any k E Z The d' E {0, 1)TT denotes the one-hot encoded domain index for e, where < m : Etrain] S n : Σ- eeetrai Te. Our goal is to learn a model f X' + y that learns output (y)-causative factors for prediction and performs well on the set of all environments ED Etrain. which is aligned with existing OOD generalization works (Arjovsky ct al., 2019; Krueger et al., 2020). We use respectively upper, lower case letter and Cursive letter to denote the random variable, the instance and the space, e.g., a is an instance in the space A of random variable A. The [flA denotes the ff restricted on dimensions of A. The Sobolev space WRPP( (AA) contains all f such that La fa A=al"
i.idd ~ P with
P da < 20, Vo k.
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Structural Causal Model. The structural causal model (SCM) is defined as the causal graph assigned with structural equations. The causal graph encodes the assumptions in missing arrows in a directed acylic graph (DAG): G (V, E) with V. E respectively denoting the node set and the edge set. The Pa(k) denotes the set of parent nodes of Vi: for each Vi: E V and the X + Y E E indicates the causal effect of X on Y. The structural equations (Vi + (Pa(Pa() ex)] ev. quantify the causal effects shown in the causal graph G. By assuming independence among exogenous variables { the Causal Markov Condition states that P((Vi en )v P(, un Pa(() = pa(k)). A back-door path from Va to Vg is defined as path that ends with an arrow pointing to Va (Pearl, 2009).
4 METHODOLOGG
We build our causal model associated with Causal Invariant Mechanism (CIMe, i.e., fzs Sy) and priori about the generating process in section 4,1, followed by our identifiability result for CIMe in section 4.2. Finially, we introduce our learning method to estimate CIMe in section 4.3.
4.1 LATENT CAUSAL INVARIANCE MODEL
D
We introduce latent variables to model the abstrac- tions/concepts that play as causal factors that generate the observed variables (X, Y), which is more reasonable than assuming the X as the direct cause of Y in scenarios with sensory-level data. We explicitly separate the latent variables into two parts: the S and Z that respectively denote the y coutput)-causative and g-non-causative factors, as shown by the arrow S + Y in Fig. 1. Besides, the X and Y are re- spectively generated by S, Z and S. via structural equations (with noise) fa. fy. which are denoted as Causal Invariant Mechanisms (CIMe) that hold across all domains. The out- put Y denotes the label generated by human knowledge, egg. the semantic shape, the contour to discern the object, etc. Hence, we assume the Y as the outcome/effect of these high-level abstractions (Biederman, 1987) rather than the cause (detailed comparison with Y + S is left in supple- mentary 7.7.1). We call the model associated with the causal graph in Fig. as Latent Causal Invariance Model (LaCIM), with formal definition given in Def. 4.1.
Z
X
Figure 1: The variables marked by white (gray) color represent the unobserved (ob- served) variables. Each arrow repre- sents the causal effect from the vari- able points from on the one it points to. The C denotes the confounder of S, Z. which are the causal factors of X. The D denotes the domain in- dex, which varies across domains and characterizes the distribution of C.
The DAG for LaCIM.
As an illustration, we consider the image classification in which X, Y denote the image and the class label. Instead of X, i.e., the pixels, it is more reasonable to assume the causal factors (of X, Y) as latent concepts (S, Z) that can denote light, angle, the shape of the object to generate X following the physical mechanisms. Among these concepts, only the ones that are causally related to the object, i.e., S (e.g., shape) are causal factors of the object label, i.e.. Y. Following the physical or natural law, the mechanisms S, Z + X, S Y invariantly hold across domains. The S == 19,, Z := R denote the space of S, 7, with pa (S, Z) (that characterizes the correlation between S and Z) varying across & (e.g., the object is more associated with a specific scene than others).
We assume that the y-non-causative factor (i.e., Z) is associated with (but not causally related to) S.Y through the confounder C. which is allowed to take a specific value for each sample unit. Therefore, the back-door path Z 7-C- C-C-S--Y induces the correlation between Z and Y in each single domain. Rather than invariant causation, this correlation is data-dependent and can vary across domains, which known as "spurious correlation" ". In real applications, this spurious correlation corresponds the bias inherited from data, e.g. the contextual information in object classification. This domain-specinee s-Z correlation, can be explained by the source variable D, which takes a specific and fixed value for each domain and functions the prior of distribution of the confounder C, as illustrated in Fig. 1. This source variable D can refer to attributes/paramnter that characterize the distribution of S, Z in each domain. When such attributes are unobserved, we use the domain index the cat/dog classification task as an illustration, the animal in each image is either associated with the snow or grass. The S, Z respectively denote the concepts of animals and scenes. The D denotes the sampler, which can be described by the proportions of scenes associated with the cat and those associated with the dog. The D generates the C that denotes the (time, weather) to go outside and
substitute. Consider
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collect samples. Since each sampler may have fixed pattern (e.g. gets used to going out in the sunny morning (or in the snowy evening)), the data he/she collects, may have sample selection bias (e.g. with dogs (cats) more associated with grass (snow) in the sunny morning (or snowy evening) ). In this regard, the scene concepts Z can be correlated with the animal concepts S, and also the label Y.
Definition 4.1 (LaCIM). The Latent Causal Invariance Model (LaCIM) for e E E is defined as a SCM characterized by (i) the causal graph, i.e., the G (V, with V {C, 7, X, Y) and E F S.C C + 7, 7- - X.S + X.S S + Y); and (ü) structural equations with causal mech- anisms far fy] embodying the quantitative causal information: c fo(d" ,e,), + &
+ -,((8,z, z, Z,E2); + fa(8, in which E2; are inddpen- dent exogenous variables that induce Pfe Pf((c)d)) ), ),P] (2)e),,, (slc). (s fz. fy are assumed to be invariant across E. We call the environment -eppedddt parts: (S,S Z) and (s(S x as S, Z-prior and S, Z-inference in the following.
(a1ss, 2),Pj, (y]s). The CIMe
Remark 1. We denote LaCIM-d, and LaCIM-d as two versions of LaCIM, with the source variable de with practical meaning (e.g. attributes or parameters of P(S, Z)) observed or not. The observation of de can be possible in some applications (e.g. age. gender that characterizes population in medical diagnosis). As for the LaCIM-d with d, unobserved, we use domain index D) as a substitute.
Denote C as the space of C. We assume that the C is finite union of disjoint sets {Cr i.e. ; C,z such that for any Cr A crj E Cy, holds that p(s, 2(cr,) = p(ss, 2/cr,,) for any (s, 2). Returning to the cat/dog classification example, the C denotes the range of time to collect samples, i.e. 00: 00-24: 00. The C can be divided into several time periods C1. ..., CR. such that the proportion of concepts of (cenm)l given any e in the same period is unchanged, e.g., the dog often comes up on the grass the morning. Further, since p(æ, y/s, z) = p(ar|s, z z)p(y]s) ()) is invariant, we have for each C, that p(z, yle,,) f p(x,, vlss z)p(s, ddszz
f p(zr, yls, z)p(s, z (/rrr)dsds = p(z, ylen.) for any (z, y). That is, the {p(x, y/cre,,Ec, for each (z, y) collapse to a single point, namely p(a,, ulc.). In this regard, we have p² (x, == M(z,yldd = Σ Em p(ar, y/c, ((cr)d")) Besides, we assume the Additive Noise Model (ANM) for i.., fa(), (s, 2) + Ex (we replace fe with La without loss of generality). which has been widely adopted to identify the causal factors (Janzing et al., 2009; Peters et al., 2014; Khemakhem, Kingma and Hyvärinen, 2020). We need to identify the CIMe (i.e., fz. fy). guaranteed by the identifiability that ensures the learning method to distinguish S from Z to avoid spurious correlation, as presented in section 4.2. Traditionally speaking, the identifiability means the parameter giving rise to the observational distribution pe- (z, y|d') can
be uniquely determined, i.e., Pz(z, y dd)) = p, yld") = 0 = Ô. Instead of strict uniqueness, we rather identify an equivalent class of 8 (in Def. 4.2) that suffices to disentangle the y-causative features 5 from Z to avoid learning spurious correlation. To achieve this goal, we first narrow our interest in case when p(s, 2 (e) is exponential family Eq. (1), in which we can respectively identify the S, Z up to linear and point-wise transformations given by theorem 4.3; then we generalize to any p(s, 2 z|c) as long as it belongs to Sobolev space, as explained in theorem 4.4.
A reformulated VAE is proposed to learn the CIMe practically. For generalization, note that the gap between two environments terms of prediction given x; i.e., |E,=a|Y|| = x) Eper [Y]X- = =]| = Ss p (s|x) p² (s|=) |p,, (wla)ds, is mainly due to the inconsistency of S, Z-inference, i.e., p² zl) + pf 2/x) for e' A e (for details please refer to theorem 7.1 in supplement 7.1). Therefore, one cannot directly apply the trained (p'(s, (s,(s,((x), [()))) ( to the inference model of new environment, i.e. oe (s, (s,)) p² (ylæz) for el & Etrain- To solve this problem and generalize to new environment, we note that since py. (z](s 2) and pf, (v]s) are shared among all environments, we propose inference s, z that give rise to the test sample x via maximizing the identified pyz (x|s, 2). as pseudo-likelihood of x given (s, 2). rather than using S, Z-inference model which is inconsistent among environments. Then, we feed estimated s into invariant predictor Ps, (y)s) for prediction.
4.2 IDENTIFIABILITY OF CAUSAL INVARIANT MECHANISMS
We present the identifiability claim about the CIMe fax fy. which implicitly distinguishes the y- causative factors (a.k.a, S) from others (a.ka, 2) for prediction, to provide a theoretical guarantee for avoiding spurious correlations. Notably, the S and Z play "asymmetric roles" in terms of generating process, reflected in additional generating flow from S to Y. This "information intersection" property of S. i.e., fo (ū) = [f= (s) for any (ž. 7) E fS Z) x (S( if y = fs) + Ey. is exploited disentangle S from Z. Such disentanglement analysis, is crucial to causal prediction
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but lacked in existing literature about identifiability, such as those identifying the discrete latent confounders (Janzing, Sgouritsa, Stegle, Peters and Schölkopf, 2012; Sgouritsa et al., 2013); or those relying ANM assumption (Janzing, Peters, Mooij and Schölkopf, 2012); linear ICA (Eriksson and Koivunen, 2003); (Khemakhem, Kingma and Hyvärinen, 2020; Khemakhem, Monti, Kingma and Hyvärinen, 2020; Teshima et al., 2020) (Please refer to supplement 7.6 for more broad reviews). Besides, our analysis extends the scope of Khemakhem, Kingma and Hyvärinen (2020) to categorical Y and general forms of P(S, ZIC = c) that belongs to Sobolev space, in theorem 4.4, Note that our analysis does NOT require observing the original source variable d,.
We first narrow our interest to a family class of LaCIM denoted as Pexp in which any p E Pexp satisfies that (i) the S, Z belong to the exponential family; and that (ii) the Y is generated from the ANM. We will show later that Pexp can approximate any P(S, Z)c) E W x 2) for some r 2
P exp
[LaCIM with y = (s) Eg, 2/c) PT p (s)c)
4+
Ry ETj + B((()) - AL.) for = s, and e E E, j=1
(I)
with rr (1) = II
exp
denote the sufficient statistics and natural parameters, {B.). (4 de- note the base measures and normalizing constants to ensure the integral of distribution equals to 1. Let (t)
where {T.
(t))).(f (T))(,)),,
(t1()) .... Ty (fx))) R4ex (T. := [T], ...., Ti (t:)), Vi E [4:]).
E ERR (T = : (T........ D 1. Vi E (æ:)). We define the iddieiiniiiibblity for 8 == {fa. T T as:
: Ter..... r c,0
Definition 4.2 (---identifiability). We define binary relation on the parameter space of X x Y: 0 ~y O if there exist two sets of permutation matrices and vectors, (Ma; a2) and (M ag) for s and 2 respectively, such that for any (z. E X x y,
((f Pr, (yIlff = Pj, (f.(oolff=ls((t)
T (V= (s(x)) = M.T (1/ [](())))))))))))))})))))) + ass ('( = M (VZz + a=:
We say that 0 is ~--identiffabll, for any 0, pī(z, = = (I, y) Ve E Eirain- implies 0 ~p 0.
can be shown that ~p satisfies the reflective property (0~,0) the symmetric property (if 0~p 0 then 0 ~n 0), and the transitive property (if 0, ~ep O2 and 0g ~0p 0z, then 0, ~p 0s), and hence is an equivalence relation (details in supplement 7.2). This definition states that the S, Z can be identified up to permutation and point-wise transformation, which sufficient for disentanglement of S and identifying the predicting mechanism pf, ls(=)) Specifically, the definition regarding fa implics the separation of and Z unless the extreme case when S can be represented by Z, i.e., there exists function h : S + Z such that [f= (x)) = "z [z)). This definition is inspired by but beyond the scope of unsupervised scenario considered in nonlinear ICA (Hyvärinen et al., 2019; Khemakhem, Kingma and Hyvärinen, 2020) further distinguish of S from Z. Besides, the Pfe (v (x=) = Pj, ()))))) further guarantees the identifiability of prediction: predict using (s) with 8 obtained from fz. The following theorem presents the >p-identifiability for Pexp:
Theorem 4.3 (>p--identfiaabiiity). For @ in the LaCIM (x(r) E Pesp for any E Euraius we assume that i) CIMe satisfies that fa, ftt and S are continuous and that fa, Jy are bijective; ii) the Th, are twice differentiable for any 1. = =,,2, E [yc], E [k.]; li)) the exogenous variables satisfy that the characteristic functions Eg, are almost everywhere nonzero. Under the diversity "..,).,...Ill that the A and T have full column rank for both and 2, we are ~ip identifiable.
condition on A == [PT 1 .... PT_m] E ERTxI with Pa= ==
[
Tt=
02 c, have that the = (fz. fy. T T
CR
The bijectivity of fa and fy have been widely assumed in Janzing et al. (2009); Peters et al. (2014; 2017); Khemakhem, Kingma and Hyvärinen (2020); Teshima et al. (2020) as a basic condition for identifiability. It naturally holds for fa to be bijective since the latent components S, Z, as high-level abstractions which can be viewed as embeddings in auto-encoder (Kramer, 1991). lies in lower-dimensional space compared with input X which is supposed to have more variations, i.e.,
(g. + 92 < qa). For categorical Y, the f, which generates the classification result, i.e., p(y = k(s) = (Jy)a())/ (Ex (s)), will be shown later to be identifiable.
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The diversity condition implies that i) 2 R 2 max(k; 92, k, *2, qs) + 1; and that ii) different
environments are variant enough in terms of S-Z correlation (which is also assumed in Arjovsky et al. (2019)), as necessary for the invariant one to be identified. As noted in the formulation, a larger m would be easier to satisfy the condition, which agrees with the intuition that more environments can provide more complementary information for the identification of the invariant mechanisms.
The dimensions of the ground-truth S, Z are unknown, making the check about whether m is large enough impossible. Besides, in some real applications. the training environments are passively observed and may not satisfy the condition. However, we empirically find the improvement of LaCIM in terms of both OOD prediction and intepretabilliy, if the multiple environments provided are diverse enough. Besides, training environment can be the mixture of many sub-environments, which motivates to splitting the data according to their source ID or clustering results (Teney et al., 2020) to obtain more environments, making the condition easier to satisfy.
Remark 2.
Extension to the general form of LaCIM. We generalize the identifiable result in theorem 4.3 to any LaCIM as long as its P(S, ZIC = E W x Z) (for some > 2) and categorical Y, in the following theorem. This is accomplished by showing that any such LaCIM can be approximated by sequence of distributions in Pexp, motivated by the facts in Barron and Sheu (1991) that the exponential family is dense in the set of distributions with bounded support, and in Maddison et al. (2016) that the continuous variable with multinomial logit model can be approximated by a series of distributions with i.i.d Gumbel noise as the temperature converges to infinity.
Theorem 4.4 (Asymptotic ~y-iideniifabiiity)). Consider a LaCIM satisfying that Pf. (x) s,]s,2) and pf (y]s) are smooth w.n.t s, and respectively. For each e and e E C, suppose Pe (S, [|, = c) E ( x 2) for some r 2 2, we have that P is asymptotically ~p-identifiable defined as: V 0, 3 "ddeetenieeblle Pg E P Pespp s.I. dpok(p (x. 17,6,, u)) < V, E EErains (I. E X x x
4.3 CAUSAL SUPERVISED VARIATIONAL AUTO-ENCODER
Guided by identifiability, we first provide the training method to learn fur fy by reformulating VAE in supervised scenario, followed by optimization over latent space for inference and test.
Training. To learn the CIMe and Pr. (z|s, 2), Pr, (yls) for invariant prediction, we implement the generative model to fit {pf(r, (ar, c which has been guaranteed by theorem 4.3, 4.4 to be able to identify the ground-truth predicting mechanism. Specifically, we reformulate the objective of VAE, as generative model proposed in (Kingma and Welling, 2014), in supervised scenario. For unsupervised learning, the VAE introduces the variational distribution dv parameterized by jb to approximate the intractable posterior by maximizing the following Evidence Lower Bound (ELBO):
-Co.v = Ex [[ag(=ts) log qe(s|m))) Pa(x.)) as a tractable surrogate of maximum likelihood Epta)) log (x)).
Specifically, the ELBO is less than and equal to Epc=) loggs (xr) and the equality can only be achieved
when gy (2 (z) [[]==]pp (=(2(z)) Therefore, maximizing the ELBO over Pa and qo will drive (i) gs to
learn po 2]);; (ii) P& to learn the ground- truth model (including Pp (2) to learn p(x|2)).
In our supervised scenario, we introduce the variational distribution of (s, 2|x, y) and the correspond- r pg(app..)))) ing ELBO for any e is - - [E (a,)).)))) log (a.al_a))) Similarly, minimizing L can drive Ps (x/s, 2). (ylss) to learn the CIMe (i.e. Pf, (arss, 2), Pr, (yls)), and also ( 2|1r, y) to learn p5 2 (z, v). In other words, the qo can inherit the properties of ps. As p6 2/20, y) 1 (1al (((as//))
(((a)))
for our DAG in Fig. 1. we can similarly reparameterize q8 (s, 2/2, y) as Le %. m) to Causal Markov Condition, we have that (r, v, pa palls 2)p& (s, = =)pa(u)s). (.) Substituting the above reparameterizations into the ELBO with (u:(y)s) replaced by P (o(l))), the S can be rewritten as:
According
5/ = (v.s)) [ log (x1l))) Eug (s.2
(x]ss)
Po 2 2s,
log
(2)
55 (v) x)
of (8, 2()
where of (v]=) = Ss g (s)c)po(uls)dd. The overall loss function is: La. = EecEmmi Ca
The training datasets JeEEram are applied to optimize prior model (5, inference
'The drok denotes the Pakorov distance and lim drok(Un, 1n) + 0 + ph $ p.

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Table 1: MCC of identified latent variables. Average over 20 times for each data.
Data
Data #2
Data
Data #4
Data #5
Average
Z
S
Z
S
Z
S
Z
S
Z
S
Z
S
pool-LaCIM
0.28
0.58
0.38
0.66
0.34
0.77
0.34
0.79
0.36
0.75
0.34
0.71
LaCIM-d, (Ours, m 5)
0.81
0.86
0.81
0.87
0.85
0.87
0.73
0.78
0.86
0.87
0.82
0.85
LaCIM-d (Ours, 3)
0.62
0.88
0.63
0.83
0.70
0.78
0.69
0.81
0.71
0.84
0.67
0.83
LaCIM-d (Ours, 5)
0.64
0.82
0.75
0.80
0.76
0.83
0.79
0.90
0.75
0.85
0.74 f
0.84
LaCIM-d (Ours, m = 7)
0.70
0.89
0.81
0.90
0.82
0.88
0.84
0.83
0.90
0.85
0.81 f
0.87
model (=, and generative models (o(x]s, 2), Puls)) in Eq. (2). The generative models Pa(x]s, 2). (b(( are shared among all environments, while the p z), (), are respec- tively (s, =|dg), qu (s, 2/2,, dz) and pp(s,=1d') qu (s, 2/2,, d=) for LaCIM-d, and LaCIM-d.
Inference & Test. When dy can be acquired during test for e' Eust, can predict y as arg max, P,(y) = = (s|2, di )p ().s. Otherwise, for LaCIM-d with d, unobserved, we first optimize s, 2 via (s", == arg max,,; log pr(ss, 2) and predict as arg max, (y)s'* Specifically, we adopt the strategy for optimization in Schott et al. (2018) that we first sample initial points and select the one with the maximum log Pa (x 2), then we optimize for 50 iterations using Adam. The implementation details and optimization effect are shown in supplement 7.9.
EXPERIMENTS
We evaluate LaCIM on (I) synthetic data verify the identifiability in theorem 4.3; (II) OOD challenges: object classification with sample selection bias (Non-I.I.D. Image dataset with Contexts (NICO)): Hand-Writing Recognition with confounding bias (Colored MNIST (CMNIST)); prediction of Alzheimer's ' Disease (Alzheimer's 's Disease Neuroimaging Initiative (ADNI www loni, ucla. edu /ADDI;;;; (III) Robustness on detecting images with small perturbation (FaceForensiest+))
5.1 SIMULATION
To verify the identifiability claim and effectiveness of our learning method, we implement LaCIM on synthetic data. The data generating process is provided in Supplement 7.8. The domain index 5. To verify the utility of training
D E _ is denoted as a one-hot encoded vector with multiple domains (m > 1), we also conduct LaCIM by pooling data from all m domains together, namely PooI-LaCIM for comparison. We randomly generate each. We compute the metric mean correlation coefficient (MCC) adopted in Khemakhem, Kingma and Hyvärinen (2020) measure the goodness of identifiability under permutation by introducing cost optimization to assign each learned component to the source component. This measurement is aligned with the goal of -ddiieenifiiablitty which allows us to distinguish S from Z. Table 5.1 shows the superiority of our LaCIM-d, LaCIM-d, over Pool-LaCIM in terms of the CIMe relating to S, Z under permutation, by means of multiple diverse experiments. Besides, we consider LaCIM-d on m 3, 5, 7 with the same total number of samples. It yields that more environments can perform better; and that even m still performs much better than LoOI-L To illustrate the learning effect, we visualize the learned Z in Fig. 7.8, with S left in supplement 7.8 due to space limit.
5 datasets and run 20 times for
(a) Ppool- (2ool-LaCM((2),,,, (b) PLACIM y, ds) (c) PLaCIM(2|22, y, d)
(d) Po- (2(d,)
Figure 2: Visualization of Z. From left to right are: estimated posterior by LaCI- LaCIM-d,, LaCIM-d and the ground-truth. As shown, the LaCIM-d, (Fig.(b) and LaCIM-d (F)) can identify the ground truth distribution of Z (ie., p& (zldx)) up to permutation and point-wise transformation, which validates the claim in theorem 4.3.
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5.2 REAL-WORLD OOD CHALLENGE
We present our LaCIM's 's results on three OOD tasks, with different environments associated with different values of dg. We implement both versions of LaCIM, i.e., LaCIM-d, and LaCIM-d, with lask-dependent definition of d,. In CMNIST, the da (digit color) is a fully observed confounder, and LaCIM-d, in this case is the ceiling of LaCIM-d under the same implementation. In NICO and ADNI, the LaCIM-d even outperform LaCIM-d d,, when the source variables are only partially observed.
Dataset. We describe the datasets as follows (the X denote image; the Y denote label):
NICO:
we evaluate the cat/dog classification in "Animal" dataset in NICO, a benchmark for non-i..d problem in He et al. (2019). Each animal is associated with "grass", "orrss"""""oo""" "snow" contexts with different proportions, denoted as d. E IR (cat,dog in snow). We set m 8 and m 14. The C, Z, S respectively denote the (time, whether) of sampling, the context and semantic shape of cat/dog.
CMNIST: We relabel the digits 0-4 and 5-9 as O and y = 1, based on MNIST. Then we color pe (1 -p") of images with y = (y 1) as green and color others as red. We set m
with yfi = 0.9. p = 0.8. The de is pe to describe the intensity of spursion correlation caused by color. We do not flip y with 25% like Arjovsky et al. (2019) 2, since doing so will cause the digit correlated rather than causally related to the label, which is beyond our scope. The Z, S respectively represent the color and number. The C' can also denote whether) for which the painter draws the number and color, e.g., the painter tends draw red more often than green 1 in the sunny morning.
ADNI. The data are obtained from the ADNI databaset, the y = {0, 1, 2} with 0,1,2 respectively denoting AD, Mild Cognitive Impairment (MCI) and Normal Control (NC). The X is Magnetic resonance imaging (sMRI). We set m 2. We consider two types of d,: Age and TAU (a biomarker Humpel and Hochstrasser (2011)). The S (Z) denote the disease-related (-unrelated) brain regions. The C' denotes the hormone level that can affect the brain structure development.
Compared Baselines. We compare with (i) Cross-Entropy (CE) from X + Y (CE x Y), (i) domain-adversarial neural network (DANN) for domain adaptation Ganin et al. (2016), (iii) Maximum Mean Discrepancy with Adversarial Auto-Encoder (MMD-AAF -AD- for domain generalization Li et al. (2018), (iv) Domain Invariant Variational Autoencoders (DIVA) Ilse et al. (2019) (v) Selecting Data Augmentation (SDA) llse et al. (2020), (vi) Invariant Risk Mnimization (IRM) Arjovsky et al. (2019), (vii) CE (X, d,) - Y, (viii) VAE with causal graph C -> V - {X, Y} with V mixing S, Z and we call it sVAE for simplicity. We only implement SDA on CMNIST, since the intervened-datt generation of SDA requires explicitly extracting the S, Z. which is intractable in ADNI and NICO. For fair comparison, we keep the model capacity (numer of parameters) in the same level.
Implementation Details. For each domain e, we implement the reparameterization with P3 p s', = (s), (2). to transform the (s, 2) into isotropic Gaussian; then the generative models correspondingly modified as (=)())(()))1(
(s))) according to rule of change of variables. The optimized parameters are (x) Ps (z]s,, 2), Po(y))), (pi=.. Je), with the encoder o (s, a|xx being sequentially composed of: i) the sequential of Conv-BN-ReLU- MaxPool blocks that shared among Errann followed by ii) the sequential of ReLU-FC for the mean and log- variance of S, 7 that specific to e. The structure of pi=s, is FC-ReLU--FC. The decoder Po(x)), 2) is the sequential of upsampling. several -CCon--BN-ReLD blocks and Sigmoid. The predictor po (B)l)) is sequential of FC-BBN--RRel classification. The network structure and the output channel size for CMNIST, NICO and ADNI are introduced in supplement 7.11, 7.12, 7.13, Tab. 13, 14. We implement SGD as optimizer: with learning rate (Ir) 0.5 and weight decay (wd) le-5 for CMNIST; Ir 0.01 with decaying 0.2x every 60 epochs. wd 5e-5 for NICO and ADNI (wd is 2e-4). The batch-size are set to 256, 30 and 4 for CMNIST, NICO, ADNI. The "FC". "BN" stand for Fully-Connected, Batch- Natch-Normalization.
(x))))((()(( -s))} (p=)-
blocks, followed by Softmax (or Sigmoid) for
Results. We report accuracy over three runs for each method. As shown in Tab. 2 our LaCIM-d performs comparable and better than others on all applications, except the 99.3 achieved by SDA on CMNIST, which is comparable to the result on the original MNIST. This is because during training,
We also conduct this experiment with flipping y in supplementary 7.11. *On NICO, we implement ConvNet with Batch Balancing as specifically benchmark in He et al. (2019). The results 60 + m 8 and 62.33 + 3.06 on 14.
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the SDA implemented data augmentation with random colors, which decorrelate the color-label When S cannot be explicitly extracted in general case, the SDA is not tractable.
Discussions. The advan- tage over invariant learn- ing method (IRM) and CE (X, d.) + Y which also takes de into prediction can be contributed to the identifi- cation of true causal mecha- nisms. Further, the improve- ment over sVAE is benefited from our separation of y-- causative factors (a.ka, S) from others to avoid spu- rious correlation. Besides, shown from results NICO, a larger m (with the total number of samples n fixed) can bring further benefit, which may due to the easier satisfaction of the diversity condition in theorem 4.3. One thing worth particular mention that on NICO and ADNI (when d, denotes TAU), our LaCIM-d performs comparable and even better than LaCIM-d,, due the existence of unobserved partial variables. For example, each de only contains one attribute each time in ADNI. For completeness, we conduct experiments with fully observed confounders in supplement 7.13. Besides, we apply our method intervened data, the result of which can validate more robustness of LaCIM, as shown in supplement 7.12.
Table 2: Accuracy (%) of OOD prediction. Average three runs.
Dataset
NICO
CMNIST
ADNI
(m
Age
T:
CB Y
59.00
1.70
97.87 4 0.19
62.000 2: 2.20
64.58 2: 0.90
DANN
50.33 4.3
60.00
2.65
07.
0.13
60
1.83
64.58
0.90
MMID-AR
61.33-288
06.33
62.48
66.62
0.00000000
DIVA
60.67 2.08
1.58 58.67
07.07 0.10
61.57
3.30
1:.00:0:0000
SDA
99.37
IRM
01:07 4.16
60.00
3,00
0.222
69.49 1.00
05.10
CR X.d,
67.38660
1.00 64.000
09.00 0:22
02.43=0.02
65,02
0:00
60.33
1.87
4.091
LaCIM-l, (Ours)
62.00
1.73
68.00
2.64
08.81
1.0.4
65.08
1.50
66.14
0.91
LaCIM (Ours)
62.67 1: 0.58
08.67
2.64
98.78
0.20
66.44
0.90 68.23 0.90
Interpretability. We visualize learned S as side proof interpretability. Specifically, we select the s* that has the highest correlation with y among all dimension of S, and visualize the derivatives of s* with respect the image. For CE x - y and CE (x. d',) + y, we visualize the derivatives of predicted class scores with respect to the image. As shown in Fig. 5.2, LaCIM (the 4th column) can identify more explainable semantic features, which verifies the identifiability and effectiveness the learning method. Supplement 7.12 provides more results.
(a) Cat on grass
(b) Cat snow
(c) Dog grass
(d) Dog on snow
Figure 3: Visualization via gradient Simonyan et al. (2013), From the left to right: original image,
CE X +, CE dX, Y and LaCIM-d.-
5.3 ROBUSTNESS SECURITY
We consider the DeepFake-related security problem, which targets on detecting small perturbed fake im- ages that spread fake news. The Rossler et al.
Table 3: Accuracy (%) of robustness Face- Forensics++. Average over three runs.
(2019) provides Forensieess dataset from 1000 Youtube videos for training and 1,000 benchmark images from other sources (OOD) for testing. We split the train data into m environments accord- ing to video ID. The considcrable result in Tab. 5.3 verifies potential value on security.
XE 82.8 # 0.99
LaCIMed (Ours)
88.4 + U.59
84. 47 0.90
6 CONCLUSIONS & DISCUSSIONS
We incorporate the causal structure as prior knowledge in proposed LaCIM, by introducing: (i) latent variables and explicitly separate them into y-causative factors (a.ka, S) and others (a.k.a, Z) which are spuriously correlated with the output; (ii) the source variable d, that explains the distributional inconsistency among domains. When the environments are diverse and much enough, we can successfully identify the causal invariant mechanisms, and also y-causative factors for prediction without a mix others. Our LaCIM shows potential value regarding robustness to OOD tasks with confounding bias, selection bias and others such healthcare and security. A possible drawback of our model lies our requirement of the number of environments (which may be not satisfied in some scenarios) for identifiability, and the relaxation of which is left in the future work.

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7 SUPPLEMENTARY MATERIALS
7.1 O.O.D GENERALIZATION ERROR BOUND
Denote Eply/=] := Jn yp(yla)dy for any æ; y E X'y. We have
upp syy according to that p(])s) is invariant across E, we can omit pº in Ep [y/a s] and denote g(S) := E| Then, the OOD bound pp (u/x) - Eyr> (u)==),, V(x, y) is bounded as follows:
Theorem 7.1 (OOD genearlization error), Consider two LaCIM per and pe,, suppose that their densities i.., pe1 (:) and pf² (s)ar) are absolutely continuous having support (-00, oo). For any (x. v) X x V, assume that
(S) is Lipschitz-contnnune function; (s) == pel (2/z)
is differentiable and Eyes [T)(()]((S) M1
0o with μl
Ep (o(S) = ((sf(s)
then have [Eps (2)1z) - Epe
|| 1| Varge (S| = z).
When E Etrain and Etest, the theorem 7.1 describes the error during generalization on es for the strategy that trained on e₁. The bound is mainly affected by: (i) the Lipschitz constant of g, i.e., |!g]booi (ii) I/ which measures the difference between yf1 (s, and pf* z); and (iii) the Vargeı (S]z) that measures the intensity of x - 2). These terms can be roughly categorized into two classes: (iii which are related to the property of CIMe and gave few space for improvement; and the (ii) that describes the distributional change between two environments. Specifically for the first class, the (i) measures the smoothness of [s(y))) with respect to The smaller value of || || implies that the flatter regions give rise to the same prediction result, hence easier transfer from ej e2 and vice versa. For the term (iii,, consider the deterministic setting that Ey (leads to Varge1 (5(z)= = 0), then can be determined from æ for generalization if the f is bijective function.
The term (ii) measures the distributional change between posterior distributions pt (s|xr and (s(z). which contributes to the difference during prediction: |Ege (y/=) Ey (y]=r) = -Is (1 (s|xr) pd (s|x)))r,, (y s)dd. Such change is due to the inconsistency between priors pr1 (s, 2) and paa (s, 2). which is caused by different value of the confounder dg.
Proof. In the following, we will derive the upper bound
E Epe1 [Y]X= - Epe2 [X|X =x]
< |ل
|| Varpei (S|X = x).
where (s) =
p (slar) and g(s) is assumed to be Lipschite-continous.. pr (elp)
To begin with, note that
E| = E S| B[o(S) = /f (())p(s(x)ds
Let P1 p (s)x), (2(2)x) - (s]a:). For ease of notations, we use P1 and P2 denote the distributions with densities Pi (s|z) and P2(s|a:) and suppose S, ~ Pi and S2 Pz. where is omitted the following analysis is conditional on fixed X =x.
Then we may rewrite the difference conditional expectations as
Eyr Y x] Epe= Y z] E(g(S2)) E(g(S))),
where ([g((;)) = / (s) ds denotes the expectation over Pj-
Let 11 Ep [g( x] [[) [ [s(s) Then
Eyez y|X xr] REpr | = E(g(S))) [g((S)) -m1]-
Further, we have the following transformation
E [g((S)) pi] = [(((s) ( [0l0)--w-hnn)pppp0)>=>>- -1)),
(3)
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In the following, we will use the results of the Stein kernel function. Please refer to Definition 7.2 for a general definition. Particularly, for the distribution P1 ~ Pi (s/2x) the Stein kernel Ti (s) is
1
(E(S)) t)p ())
(4)
Pi (1(
where (E()) = f's pi (s/x)dss Further, we define (TI o (s) as
(T) = 1
1 pi (5) (E(g(S))) g(())) (1(x))) / (M1 - ( ((t))p((t())dd (5)
Under the second condition listed in Theorem 7.1, we may apply the result Lemma 7.3. Specifically, by the equation (8), we have
E ((((S)) (1) [1))T(((i)) E ((T1 (S))
Then under the first condition in Theorem 7.1, we can obtain the following incquality by Lemma 7.4,
[ ( E
(Si)
SE
(
TI (Si) [
TI
(6)
S Bl (T' m (S1) $ E
In the following, show that the Stein kernel non-negative, which enables (S))]
E [n()) According the definition, (s)s) (i(l)) l" (B(S)) (t)((t))), where B(S))
I t. (tt Let (i(s) f (t(t) dtt be the distribution function for Pj. Note that
L (()))pi(t))ddt ((()) E(S))
[ tp (t)r)d
Fi()) (1((]]]
Fi(s) di ((S) 2 (())
The last inequality is based on E(SiSS) 5) E(Si) S that can be proved as the following 1"
[ (t((rr di / lp (()x)))
1
-1 1 pi (1(())) S. (p)(1x)dt
(1)
(+)))
p1(t)))dt - $ $f.
L./0 pi(t(x) 0.
Fi(s)
Therefore, Ti (x) 2 and hence E [/m (Si)1) = E [TI (Si)) (6).
Besides, by equation (9), the special case of Lemma 7.3, we have
E (S:)] = Var(S1) = Var,-. (S) = 2).
To sum up.
E ( (S)))
|1g
T
1. (S.) = |1g 11 Var (S|X x).
Definition 7.2 (the Stein Kernel Tp of distribution P). Suppose X~ with density p. The Stein kernel of P is the function (x) defined by
1
Tp(x)
(E(X) v)y(y)dy,
p(x))
where Id is the identity function for Id(x) More generally, for function h satisfying [[]](X]] < oc, define (Tp h)(x) as
-/1/// J (B(M(X)) - hy)))(y)dd
(TP (tpoh)(x) plx))
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Lemma 7.3. For differentiable functioo ip such that Bil(TP (1))))) (X)Il cc, we have
E [[TP - ()) (X)) [[(h(X))))))) (x(()))))
(8)
Proof. Let pm =: E(N(X)) As ( pm)=
(Tp (TDo))(x)
1
-1 (pn h(u))p(u)dy - [ (mn-h(y)) h(y))((ddd
p(x)
Then
[[ o (x)x /
(1))))))((())(())))))) [-- (Tp o ())))) (x)p
x (uh h(y) p(y)) (x)dddx 1.
[ (mn - h(y))p()) ( J.
% (un h(y))p()) of (x)dzdy
(un h(y) (()))(()) (x)dzdy
I'" (h(y) pm va)ld (c) +
(h(v) pn)plu) (xc)drd
(Myy) ph)p()) ( ((xx)dx) dy=
(M(v mn)p(U)(((()) (o()))
(hlv) mn)p( E((()))X)))))))))) E(((X));(())
Particularly, taking h(X) = X and (X) = X - E(X), we immediately have E(rp(X)) = Var(X)
Lemma 7.4. Assume that [E([X) < oo and the density p is locally absolutely continuous on oo), and is a Lipschitz continuous function. Then we have [fx| S llo' for
fx(x) (Tp h)))) 1 (B(( h(y))p(y)dy
Tp(x)
1 (I((X) y))(y)dy
Proof. This a special case of Corollary 3.15 in Döbler et (2015), taking the constant =
7.2 PROOF OF THE EQUIVALENCE OF DEFINITION 4.2
Proposition 7.5. The binary relation ~op defined in Def. 4.2 is an equivalence relation.
Proof. The equivalence relation should satisfy three properties as follows:
Reflexive property: The 0 ~p 0 with Mg. M, being identity matrix and a,, ag being 0.
Symmtric property: If ~p 0, then there exists block permutation matrices M2 and M, such that
(z) M,T (I (z)) (z)) (+) (1 a., Mf az
Pr. (vIlf=)5 P; - (1 (=)).
The we have M,- and M,- are also block permutation matrices and such that:
T ([in)s (x)) = - ([/m) (z)) + (-a.), T ((in) (x)) M ([/m) + (-az),
P7, (wllf Ps. (yllffl5 (z))
Therefore, we have ~ 0.
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Transitive property: if 01 ~p 02 and 02 ~ 0g with 8, := then we have
{fr fg. T
T r P
(V2,))) ((2,)) +
(V2,,)))) MēT ((fa,) +
((2,)))))) - M.T (U?)) + +
for block permutation matrices M..M. M2, M² and vectors a3 ag 03 a². Then we have
T ( - M ((12,))~(())) (M, +
((2,)) = M ((f) (Mfaa)
+
Besides, it is apparent that
Prz Pr3(o1((2) = pr (x)) P3g (v)) (x))- Therefore, we have 01 ~ip Og since M, and M are also permutation matrices.
(10)
With above three properties satisfied, we have that ~pp is equivalence relation.
7.3 PROOF OF THEOREM 4.3
In the following, we write (z. as p(z, yld") and also I (d',
(,dd), Ze.i Ze(d²). To prove the theorem 4.3, we first prove the theorem 7.6 for the sim- plest case when cld = dd, then we generalize to the case when C == U,C,. The overall roadmap follows: first prove the --iieentifiabiiity in theorem 7.9, and the combination of which with lemma 7.12, 7.11 give theorem 7.6 the simplest case when cld = d". Then we generalize the case considered in theorem 7.6 to the more general case when C == U,C,,
Theorem 7.6 --nniieetiffabbllit)).. For in the LaCIM (f. y) E for any E &urain we assume that (1) the CIMe satisfies that fa. I'z and f" are continuous and that fax fy are bijective; (2) that the T are twice differentiable for any t == 3, 3, E [0r,,j E [ks]; (3) the exogenous variables satisfy that the characteristic functions Ez. Ey are almost everywhere nonzero: (4) the number of
environments, i.e,, m 2 max(qs * kig 9s * *kg) + 1 and [ - p ..... dem
Itm., -I = have
full column rank for both and z, we have that the parameters = (far Sor T T are "pp identifiable.
To prove theorem 7.6, We first prove the -iiiettiiiabblly that defined as follows:
Definition 7.7 ----iieniiiiaiiitt)). The definition is the same with the one defined in 4.2, with Ms. M2 being invertible matrices which are not necessarily to be the permutation matrices in Def. 4.2.
Proposition 7.8. The binary relation ~A defined in Def. 7.7 is an equivalence relation.
Proof. The proof is similar to that of proposition 7.5.
The following theorem states that any LaCIM that belongs to Pexp is A-iienttifiale.
Theorem 7.9 (~~ d-a-identifiabiiity). For 0 in the LaCIM (1 y) E Pexp for any Exrain- we assume (1) the CIMe satisfies that fas fy are bijective; (2) the T5 are twice differentiable for any 8, 2, i [gel.j E [Be]; (3) the exogenous variables satisfy that the characteristic functions of Ez: Ey are almost everywhere nonzero; (4) the number of environments. i.e m 2 max (gx * k. 9: kz) +
for
and [[T - 1 I_, ..., T - -I'r 1] have full column rank for t 8, z, we have that the parameters {fz. fy. T T2} are ~p identifiable.
Ty, T Te share the same observational Proof. Suppose that = (sa, fy, T and Ō = (Jas distribution for each environment Evaain i.e.,
Pl..G.TT.r..T... yld") = Pi.... (.
(11)
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Then we have
Pf P (xld") - T.t.-...T.r....aa..... (x)ld"
(12)
1 Psa
=|, =dd =1d==) ydsd= 1 SSxz Pi. ".,./r.,,,,,,.. , ios (s, 2 (d) ddddz
sxz
(13)
Pea (
(14)
= 1 pc. (x - 2 T r T (f (o)ld")vold (0))
(15)
xx -.r...rr.. Pe, (I z)de / PT i T F Ia (3) )e_ - z)dr
(16)
C.....T.rs...e.... ( = ( p.,.D,TT.TT .. fi
Pe, (xldd)
(17)
Flor:...r.r...ll)9,,... (w) FluT. r. rx
(w)
(18)
F. T F F. I(w) Flir- (w) Fl F..
(19)
Pr..,TT.T.., (x/d²) where (s] = Pr r- -. r f. (xldd)
(20)
=
for any square matrix X and function f with "J" standing for the The
Jacobian. The pr. (z) in Eq. (16) is denoted as PT. (
'*' in Eq. (17) denotes the convolution operator. The FI.] in Eq. (18) denotes the Fourier transform, where D. (w) F(p:_](w), Since we assume that the Pe, (w) is non-zero almost everywhere, we can drop it to get Eq. (20). Similarly, we have that:
Ph.TTr r (yld') = Pf..i.. (vlld))
(21)
/ Pso (uls)pr r- (sld (
pf. (-
(22)
J pc, (u BTT. Ug (T) (1) volJ (ū)dg
(23)
= -f, Pey (y - DP r (fy (v) df volJJ 1 (g)dg
(24)
J fr. r Sy (0) pe, (y dy - PT. i (p, (u - dy
(25)
r' Pe, (yldd d (fi. f. Pee (ylat))
(26)
(w) = F FluT f, (w)
(27)
FlēT r fo (w) Fl "i,,,. k,ww) Pr.rr (1) = p.. (v),
(28)
(29)
and that
P... = Pf.. T Fo T= r- wdd
(30)
{[ PJ. (1)
ddd
Sx2
= =1 sx2 pf(0)s, (v) ( F r I- (8, )dszz
Jdsdz
(31)
/ polu - (( (0)))) (oll (F)do
(32)
= /
- ( (f
(33)
Jszz PT. ir ú)do Jsxz - . h (o
( (old)))(((
(u
f)dō (34)
(FT.P.TTTT.A = T. (v)
(35)
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Pr r (.):))()))(
(36)
F Fl -...--.-ff--,
(37)
(r.P..TTr:,)))) = (.......r...)))
(38)
where := [x" y T.....E== |=] h(v) llfelz (z) (T) According to Eq. (29), have
4, Σ
log vol.J, F, (v) +
log (f log A,(d*) + Σ
T
(f (1))
1=1
j=1
b
= log vol. f (v) +
log B. log (d' +
Σ
(f
j=1
(39)
Suppose that the assumption (4) holds, then we have
(T)) (d Σ log Ar(d+t) (f) F' (d*)) + Σ log A,(d
(40)
Ad(d
Ac(ee)
for all & E [m]. where [(d) = [(d) - I. Denote b,(k) = E, (1(11) (
for & E [m], then
have
F (T)' = T TH +
(41)
Similarly, from Eq. (20) and Eq. (38), there exists dz by such that
TTT ((als= (z)) + (UE))) = (()) (T(())
+ b2 + bar
(42)
where (k))== = Σ; (((d-))2((d")
for k E [m]; and that, 7,(d" (2(((1)
T T ( (v)) + T (I (z())) =i +F)+++++++ (Tf
+ b. + da.
(43)
Substituting Eq. (41) Eq. (42) and Eq. (43), have that
F T (f =
+ b., T" (T = =F" (' + b.-
(44)
According to assumption (4), the F
and F
have full column rank. Therefore, we have that
((f = [T F
T ( 12()) + [T
b
(45)
T(((7>)s(x)) (T'T".
-1 =
T ( (s()))
(TT -1
b.-
(46)
T', (y)) TT' F
T J, (v)) ( -1 b..
(47)
Denote M2 = (T (TT M, = (T' T and a, = (T'T" -1 (r'r" (TT
r
b, 0g
-1 b.. The left is prove that M, and M, are invertible matrices. Denote F f (x). Applying the (Khemakhem, Kingma and Hyvärinen, 2020, Lemma 3) we have that there exists k. points al ...,,,k, _1 ...., ah such that ((T>) (x!)). ([f "ls (a²+)) for each i e [gs] and ((T*) (+t)), (T (T>(((==))s [s,""
(z. for each i E [gx] are linearly independent.
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By differentiating Eq. (45) and Eq. (46) for each ø¹ with i [a,] and St with i E [q:] respectively, we have that
(JT+(E"), ((a))..,) JT (x = Mo (Jroofh "oreof (t'), (o')("))) ...,, No to (2tt1)
(48)
(Jra (Ur)(""))))))) ( Jr ((he)) = M2 (Jrioft' ($)),.., J of o8. (ah..
(49)
The linearly
independence
of (ss (T Is, (a.)))
and
((T)) 12, (z))), (T lIs (2) imply that the (JT- (z (.., (3 and (JT= (T. (5%x)) are invertible, which implies the invertibility of matrix M, and M,. The rest is to prove Pr, (yl lf (z)) = Pj, (ullf=lss (z)). This can be shown by applying Eq. (31) again. Specifically, according to Eq. (31), we have that
[ Pe. (I 1)p(ollflss (f)rrr.r.rrrr(((((())))))))
{ Pea z) -ollo--))(ll(() () 11 ( (z)dz.
(50)
Denote Tr.. Pí, (ullfel
volJJ 15 (T), we have
J -
T.. Pr. F Sun (d)dz
(51)
VV.. Pe, (x]dd' (1. p. ... T Syseo Pe, (x)dd)
(52)
= Fll D T
T- F f (w) Pp_ (w) - (w)
(53)
Flr p T r fydf
F. r. T., F.
(54)
lT P (z) F. . f f.. (z)
(55)
Pr
(z) prr (z)
vol. volJJ
Pj. (v) |f (z) Piu f i f (f (z)
(z)l" voll (z). (z).
(56)
Taking the log transformation on both sides of Eq. (56), we have that
logp,, (vl wllffll' (x)) + log T. (f (z) + log vol. 5x)
=, (11 (x)) + log Pi i. (f (x)(dt) + log vold (z).
(57)
Subtracting Eq. (57) with 322 from Eq. (57) with vı, have
Ps, (uellfll1 (x)) Pj (12 (v
(58)
Pí, (yrlfflls (x)) Pj, (uilfals'' (z))
Pr. (va [[ (x)) (x))
dy2 / Pj, (22 (x)) dyz
Jyp (w) 1a (x))
(59)
y Pf (vn 15 (x))
Pz. (n [/a] (x)) = Pf, (vi (v))
(60)
for any 31 E. This completes the proof.
Understanding the assumption (4) in Theorem 7.9 and 7.6. Recall that we assume the con- founder de in LaCIM is the source variable for generating data in corresponding domain. Here we also use the C to denote the space of d, (since d, : c), then we have the following theoretical conclusion that the as long as the image set of C is not included in any sets with Lebesgue measure the assumption (4) holds. This conclusion means that the assumption (4) holds generically.
Theorem 7.10. Denote hd= = (Ti, - -i (dd I' Kt.ke (d) - (i.)))))) MCC) =
hS) @ (Z) C. th, + Ra.ek, then assumption (4) holds if h(C) is not included in any zero-measure set of R4 @ Ra.ak. Denote r. == Os K2 and r= 9z kz-
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Proof. With loss of generality, assume that T3 fzz Denote Q the set of integers q such that there exists d".."..,""" that the rank (d")). h (dd"u11)) = min(g, rz) and rank([h" (Ih" "ank((""(d"").... (d")).... hi (dddvt))) min(g,rs). Denote max(Q). We discuss two possible cases for u, respectively:
Case 1. and (d ) 1e (deell) are linearly independent.
Ts I Tz. Then there exists d²
d ...,, d s.t. (d..))...,,((
Then Vc, we have (d) E (d=²), b... 1e (f(e++)) Therefore, so have (d) @ (d) E [L( (d"), )....... (d @ R' 1 [R's x @ (d"),, (d ))}, which has measure O Oin R $ R
(dh=)) b,.. 1e (d)ww)) )) or (d)
Case 2. To S < Tz. Then there exists d=a ...., deuil s.t. htt (d h² (d are linearly independent and rank([h" (d")), ...., h1 (d )]} rss Then Ve, we have h²t (d) E L(h"(d")) ). ..., (dd'=+::) )), which means that h2 (d) + ht (d E L(h"(d" ) h (d + IR',, which has measure 0 in R R
The above two cases are contradict to the assumption that h(C) not included any zero-measure set of R + R
Lemma 7.11. Consider the cases when k; 2 2. Then suppose the assumptions in theorem 7.9 are satisfied. Further assumed that
The sufficient statistics Tú, are twice differentiable for each i [ga] and E [ka].
Jy is twice differentiable.
Then have Mg in theorem 7.9 is block permulation matrix.
Proof. Directly applying (Khemakhem, Kingma and Hyvärinen, 2020, Theorem 2) with fz, A,b, T, replaced by fg, Mg. Ug, T Ty
Lemma 7.12. Consider the cases when k; Then suppose the assumptions theorem 7.9 are satisfied. Further assumed that
The sufficient statistics TV are not monotonic for E [g+].
is smooth.
Then we have Mg in theorem 7.9 is block permutation matrix.
Proof. Directly applying (Khemakhem, Kingma and Hyvärinen, 2020, Theorem 3) with Sz, A, b, T, replaced by Sq, Ma 0s, T
Proof of Theorem 7.6. According to theorem 7.9, there exist invertible matrices M, and M2 such that
AT<
('(("='ss()) = (f= + a.. (f (v)) [0.0-M,TT-- M. (v))
(v)) + as,
where T= T T and
M, 0 0 M. )
(61)
By further assuming that the sufficient statistics Tú, are twice differentiable for each E [g=] and j E [kt]] for k, 2 2 and not monotonic for k; = Then we have that M, is block permutation matrix. By further assuming that Tú, are twice differentiable for each i [n,] and [k_]] for
kg 2 and not monotonic for k and applying the lemma 7.11 and 7.12 respectively, we have that A is block permutation matrix. Therefore, M, is also block permutation matrix.
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Proof of Theorem 4.3. We consider the general case when C := Cr, in which each C, can be simplified as a representative point c,.. For environment d", let Pa= = (P(C=g]d"), P(C== crgld")] )] be the vector of probability mass of C' in the environment d". And Errain has environments with indexes d= ,.mm The latent factors (S, 2) belongs to the exponential family distribution p(s, z/c)- - Pr- (4) (")Prr (s). Suppose that 0 = fy: , } and = {f2 T Th share the same observational distribution for each environment, i.e,, po y u|dd) = (1, y (d), then
_d
we have that
n
po (z, vlc,,) P(C= Σ
P(ccool = E po (c.c..lc.)P(CCec(())))
(62)
(lcm)) y [m)] then Eq. (62) can be written as AAe,v O. Denote A == Pa. E Rmxn According the diversity condition, we have that A
Let An,y - (10 w/c1) - -p ycl).
have full column rank, therefore we have that An. = 0, i.e. po (x, mle,) = P2 ylc, for each E [R]. The left proof is the same with the one in theorem 7.6.
and the (o2) [c)))]] .....
az
7.4 PROOF OF THEOREM 4.4
Proof of Theorem 4.4. Due to Eq. (62), it is suffices to prove the conclusion for every C, E {ce f Motivated by Barron and Sheu (1991, Theorem 2) that the distribution p" 2) defined on bounded set be approximated by sequence of exponential family with sufficient statisties denoted as polynomial terms, therefore the T are twice differentiable hence satisfies the assumption (2) in theorem 4.3 and assumption (1) in lemma 7.11. Besides, the lemma 4 in Barron and Sheu (1991) informs us that the KL divergence between Pa, 2|c,) (0o := (fz. fy. T", T", r. D and po_ (s, z|c,) (0 == (fe, fy, T T ri, r (the peo (5, 2|e)), pe, (s, 2|c] belong exponential family with polynomial sufficient statistics terms) can be bounded by the & norm of ((T'(c,) (e))) ((T(())) - T Therefore, Ve 0, there exists open set of [(c)) such that the (p(, po,, =]e))) < e. Such an open set is with non-zero Lebesgue measurement therefore can satisfy the assumption (4) in theorem 4.3, according to result theorem 7.10. The left is
to prove that for any p defined by LaCIM following Def. 4.1, there is sequence of [pm In E Pexp
such that the (P.P.) -> 0 that is equivalent to Pr. 4 p. For any A, B, we consider to prove that
In - p(x E A, Blc,) Pa E A, E Blc,) + O,
(63)
where (n E A, yn E Blc,) = Js p(x E Als,))p(Un E (ls) 2/c, (dsdz with
+ Eg,))/ i= 1,,,..,k
yn (1) (x (s) + TT,))TTT)'
(64)
for E RR denoting the k-dimensional one-hot vector for categorical variable and Ey.1..... are Gumbel i..dd According to (Maddison et al., 2016, Proposition 1) that the you (i) S v() with
exp(fn,(s))
plu(1) =) 1) = E, exp(((xx=())) as Tn +.
(65)
As long as fy is smooth, we have that the p(y, (s) is continuous. We have that
In ppl A Bler) Jove p(x E Als, (1s,z)((um E (s, A Blc) p(x E A, yn E Blc)]
+ +1plzE A, E Blc,) 1
p(x E
E 2l,,)
J82Z
-16
plxE E Al8, (ply E Bls) - -pln E Bls)) p(s,)))))))))d)d )dazz
SX2
+ +11 Sx2 p(o E Als,2)p((u E Bls) - (=
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p( E Als, e Bls) plyn Bls)) p(s, 2,c, dsda Un MMMMM,
pe E Al8, 2) (ply Bis) plun E Bls)) ddadz U
In
p(e 2)(un e Bls) - (n (1,2))))) M,X.,
- p( E Als, 2) Bls) (0ss, -lass)))(pp 2l,,) JMM.XM)
(66)
T
d For In if y is itself additive model with y = (s) + Eys then we just set yn y, then we have that n,,1 O. Therefore, only consider the case when y denotes the categorical variable with softmax distribution, i.e., Eq, (65). Ve,- C [c..... en) and Ve > O, there exists M and "
such that p(s, 2 E M x le,) I c; Denote M, - u M and M, - U M we have that p(s, M, x M2 le) _ 2c for all E.C. Since Vsa E M,. EN,, such that vn 2 No,, we have that p(y E Blss) ply E B1)) S from that yn 4 Besides, there exists open set Os, such that Vs Os, and
ply E B|si) - ply e B|st)l S |p(un E B(ss) - p(yn Blsi)] Se.
Again, according to Heine-Borel theorem, there exists finite s, namely 81, 81 such that M, C U/=10(s)), Then there exists N e (Nx Nor) such that Vn 2 N, we have that
Ip(u B|s) plyn E }) S 3e, Vs M₁.
(67)
Therefore, In,1 I SM.XM, 3cp(x E 2)s, c)dsdz S 3e. Hence, I,,1 + 0 as - co. Besides, we have that In,2 I JM, 2cp(s, 2/c, dsdd $ Therefore, we have that Ssxz p(a A|s, (p(y E Bls) - plun E Bls)) p(s, 2/c, dsdz - 0 as n cc. For In,3, we have that
SM p( Als, E Bis) Ms Me) (p(s, Pn(s, 2(cr)) dadz M,XM.
1n,
S p(u E a)v, E Bls)p(s, 2/c))
(M Ms x M2(c7) -1)
dadz
M.XMM
In,3,1
|
1 (p(8, E M, Melc)) -1) dadz
plæ E 2ls,z)p(y, E Bls)p(s,zle) M, X.M
1,,3,2
(68)
The T,,3,1 S T Denote Mss 2/c,)) 1991, Theorem 2), there exists sequence of (s,s, 2/c) defined on compact support M, x M, such that Vc, C, we have that
X,M according to (Barron and Sheu, M(a,EEM, MM,
(s, 4 p(s, 2(cr).
Applying again the Heine-Borel theorem, we have that Ve, IN such that Vn 2 N, we have
p(s,2(cr) (n(s sc.
(69)
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which implies that 1,,3,2 - - oo combining with the fact that p(z, y/s, is continuous with respect to S, 2. For 1, we have that
nnA V
M.M. p(æ E 2)s,2)p(Un B|s)p(s, 21c.) - | I/ p(s, s,
(70)
M,XM,
where the first equality is from that the (n(ss 2,), is defined on M2 x M, Then have that
IS p(ar E Als, 2)p(yn Bls) (p(s z/c,) pm pn 2|c))
0, as n + co.
SXZ
The proof is completed.
7.5 REPARAMETERIZATION FOR LACIM-d
We provide an alternative training method to avoid parameterization of prior =(s, to increase the diversity of generative models different environments. Specifically, motivated by Hyvärinen and Pajunen (1999) that any distribution can be transformed to isotropic Gaussian with the density denoted by PGau: we have that for any e E Etrain, we have
{
pf (x.1)
Pf. (zls, 2), (vls)p(s,, )dsdz dsddz
sxX
| p(=1(p3) (s)))))((s)))))))))))))))) (z))())))) (pp))aa 2''s,'' d'''''''', dz',
SxZ
with 8 z = 13(s)) I).
We can then rewrite ELBO for LaCIM-d for environment
as:
Ca (x)()))) I- log g (vlx))
Lo (v)(())))ss)) log
(s(( (pE) (8)
+ (x.()
- -Bg (s.l)))))))
Lu (1lxx)
2 (8, 2/xr
(72)
7.6 IDENTIFIABILITY
Earlier works that identify the latent confounders rely strong assumptions regarding the causal structure, such as the linear model from latent to observed variable or ICA in which the latent component are independent Silva al. (2006), noise-free model Shimizu al. (2009); Davies (2004). The Hoyer et al. (2008); Janzing, Peters, Mooij and Schölkopf (2012) extend to the additive noise model (ANM) and other causal discovery assumptions. Although the Lee et al. (2019) relaxed the constraints put on the causal structure, it required the latent noise is with small strength, which does not match with many realistic scenarios, such as the structural MRI of Alzheimer's Disease considered in our experiment. The works which also based on the independent component analysis (ICA), i.e., the latent variables are (conditionally) independent, include Davies (2004); Eriksson and Koivunen (2003); recently, series of works extend the above results to deep nonlinear ICA (Hyvarinen and Morioka, 2016; Hyvärinen et al., 2019; Khemakhem, Kingma and Hyvärinen, 2020; Khemakhem, Monti, Kingma and Hyvärinen, 2020; Teshima et al., 2020). However, these works require that the value of confounder of these latent variables fixed, which cannot explain the spurious correlation in single dataset. In contrast, our result incorporate these scenarios by assuming that each sample has a specific value of the confounder. Other works assume discrete distribution for latent variables. such as Janzing, Sgouritsa, Stegle, Peters and Schölkopf (2012); Kocaoglu et al. (2018); Sgouritsa et al. (2013). However, in the literature, no existing works can disentangle the features from others, in the scenario of avoiding spurious correlation in order for OOD generalization.
7.7 COMPARISON WITH EXISTING WORKS
7.7.1
- OR + Y?
Many existing works Rojas-Carulla et al. (2018); Khemakhem, Monti, Kingma and Hyvärinen (2020); Isse et al. (2020; 2019) assumed y - S(X) the causal direction. Such an difference from
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can mainly be contributed to the generating process of Y. Different understanding leads to different causal graph. The example of digital hand-writing in Peters et al. (2017) provides a good explanation. Consider the case that the writer is provided with a label first (such as "2") before writing the digit (denoted as X), then it should be Y - X. Consider another case, when the writing is based on the incentive (denoted as S) of which digit write, then the writer record the label Y and the digit X concurrently, in which case it should be X + S + Y. For Y + S, the Y is thought to be the source variable that generates the latent components and is observed before X. In contrast, we define as ground-truth labels given by humans. Taking image classification as an example, it is the human that give the classification of all things such as animals. In this case, it can be assumed that the label given by humans are ground-truth labels. This assumption can be based by the work Biederman (1987) in the field of psychology that humans can factorize the image X by many components due to the powerful perception learning ability of human beings. These components which denoted as S, can be accurately detected by humans, therefore we can approximately assume that it is the S generating the label Y. Consider the task of early prediction in Alzheimer's Disease, the disease label is given based on the pathological analysis and observed after the MRI X. Such a labelling outcome can be regarded as the ground-truth which itself is defined by medical science. The corresponding pathology features, as the evidences for labelling, can also thought as the generators of X. In these cases, it is more appropriate to assume the Y as the outcome than the cause. For example, the Peters et al. (2016); Kuang et al. (2018) assumed Xs + Y. As an adaptation to sensory-level data such as image, we assume S + Y with S are latent variables to model high-level explanatory factors, which coincides with existing literature Teshima et al. (2020). Another difference lies in the definition of Y. The Invariant Risk Minimization (we will give a detailed comparison later) Arjovsky et al. (2019) assumes that X - S - Y by defining the Y as the label with noise. The S denoted as the extracted hidden components by observer.
7.7.2 COMPARISONS WITH DATA AUGMENTATION & ARCHITECTURE DESIGN
The goal of data augmentation Shorten and Khoshgoftaar (2019) is increase the variety of the data distribution, such as geometrical transformation Kang et al. (2017); Taylor and Nitschke (2017), flipping, style transfer Gatys al. (2015), adversarial robustness Madry et al. (2017). On the other way round, an alternative kind of approaches is to integrate into the model corresponding modules that improve the robustness some types of variations, such as Worrall et al. (2017); Marcos et al. (2016).
However, these techniques can only make effect because they are included in the training data for neural network to memorize Zhang et al. (2016); besides, the improvement is only limited to some specific types of variation considered, As analyzed in Xie et al. (2020); Krueger et al. (2020), the data augmentation trained with empirical risk minimization or robust optimization Ben-Tal et al. (2009) such as adversarial training Madry et al. (2017); Sagawa et al. (2019) can only achieve robustness on interpolation (convex hull) rather than extrapolation of training environments.
7.7.3 COMPARISONS WITH EXISTING WORKS IN DOMAIN ADAPTATION
Apparently, the main difference lies in the problem setting that (i) the domain adaptation (DA) can access the input data of the target domain while ours cannot; and (ii) our methods need multiple training data while the DA only needs one source domain. For methodology, our LaCIM shares insights but different with DA. Specifically, both methods assume some types of invariance that relates the training domains to the target domain. For DA, one stream is assume the same conditional distribution shared between the source and the target domain, such as covariate shift Huang et al. (2007); Ben-David et al. (2007); Johansson et al. (2019); Sugiyama et al. (2008) in which PYY|X) are assumed to be the same across domains, concept shift Zhang et al. (2013) in which the P(X assumed to be invariant. Such an invariance related to representation, such as & ((X) in Zhao et al. (2019) and P(Y|$(X)) in Pan al. (2010); Ganin et al. (2016); Magliacane et al. (2018).
However, these assumptions are only distribution-leve rather than the underlying causation which takes the data-generating process into account. Taking the image classification again as an example, our method first propose a causal graph in which the latent factors are introduced as the explana- tory/causal factors of the observed variables. These are supported by the framework of generative model Khemakhem, Kingma and Hyvärinen (2020); Khemakhem, Monti, Kingma and Hyvärinen (2020); Kingma and Welling (2014); Suter et al. (2019) which has natural connection with the causal
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graph Schölkopf (2019) that the edge in the causal graph reflects both the causal effect and also the generating process. Until now, perhaps the most similar work to us are Romeijn and Williamson (2018) and Teshima et al. (2020) which also need multiple training domains and get access to few samples in the target domain. Both work assumes the similar causal graph with us but unlike our LaCIM, they do not separate the latent factors which can not explain the spurious correlation learned by supervised learning Ilse et al. (2020). Besides, the multiple training datasets in Romeijn and Williamson (2018) refer to intervened data which may hard obtain in some applications. We have verified in our experiments that explicitly disentangle the latent variables into two parts can result in better OOD prediction power than mixing them together.
7.7.4 COMPARISONS WITH DOMAIN GENERALIZATION
For domain generalization (DG), similar to the invariance assumption in DA, a series of work proposed to align the representation ((X) that assumed to be invariant across domains Li et al. (2017; 1);; Muandet et al. (2013). As discussed above, these methods lack the deep delving of the underlying causal structure and precludes the variations of unseen domains.
Recently, a series of works leverage causal invariance to enable OOD generalization on unseen domains, such as llse et al. (2019) which learns the representation that is innainninvariaat. Notably, the Invariant Causal Prediction Peters et al. (2016) formulates the assumption in the definition of Structural Causal Model and assumes that Y X + Ey where Ey satisfies Gaussian distribution and S denotes the subset of covariates of X. The Rojas-Carulla et al. 2); Bühlmann (2018) relaxes such an assumption by assuming the invariance of fy and noise distribution Ey in Y < fy(Xx, Ey)) which induces P(Y|Xs). The similar assumption is also adopted in Kuang et al. (2018). However, these works causally related the output to the observed input, which may not hold in many real applications in which the observed data is sensory-level, such as audio waves and pixels. It has been discussed in Bengio et al. (2013); Bengio (2017) that the causal factors should be high-level abstractions/concepts. The Heinze- and Meinshausen (2017) considers the style transfer setting in which each image is linear combination of shape-related variable and contextual-relater variable, which respectively correspond to S and Z in our LaCIM in which the nonlinear mechanism (rather than linear combination in Heinze-Demn and Meinshausen (2017)) is allowed. Besides, during testing, our method can generalize to the OOD sample with intervention such as adversarial noise and contextual intervention.
Recently, the most notable work is Invariant Risk Minimization Arjovsky al. (2019), which will be discussed in detail in the subsequent section.
7.7.5 COMPARISONS WITH INVARIANT RISK MINIMIZATION ARJOVSKY ET AL. (2019) AND REFERENCES THERE IN
The Invariant Risk Minimization (IRM) Arjovsky et al. (2019) assumes the existence of invariant representation (X) that induces the optimal classifier for all domains, i.e., the | Pa(Y)) is domain-independent in the formulation of SCM. Similar to our LaCIM, the Pa(Y) can refer to latent variables. Besides, to identify the invariance and the optimal classifier, the training environments also need to be diverse enough. As aforementioned. this assumption is almost necessary to differentiate the invariance mechanism from the variant ones. To learn such an invariance, a regularization function is proposed.
The difference of our LaCIM with IRM lies in two aspects: the direction of causal relation and the methodology. For the direction, as aforementioned in section 7.7.1. the IRM assumes X + S rather than the S, - X in our LaCIM. This is because the IRM defines Y as label with noise while ours definie the Y as the ground-truth label hence should be generated by the ground-truth hidden components that generating S. Such an inconsistency can be reflected by experiment regarding to the CMNIST in which the number is the causal factors of the label Y. rather than only invariant correlation. Besides, in terms of methodology, the theoretical claim of IRM only holds in linear case; in contrast, the CIMe fa, fy are allowed to be nonlinear.
Some other works share the similar spirit with or based on IRM. The Risk fisk--xtrapolation (REx) Krueger et al. (2020) proposed to enforce the similar behavior of m classifiers with variance of which proposed as the regularization function. The work in Xie et al. (2020) proposed a Quasi- distribution framework that can incorporate empirical risk minimization, robust optimization and
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REx. It can be concluded that the robust optimization only generalizes the convex hull of training environments (defined as interpolation) and the REx can generalize extrapolated combinations of training environments. This work lacks model of underlying causal structure, although it performs similarly to IRM experimentally. Besides, the Teney et al. (2020) proposed to unpool the training data into several domains with different environment and leverages Arjovsky et al. (2019) to learn invariant information for classifier. Recently, the Bellot and van der Schaar (2020) also assumes the invariance be generating mechanisms and can generalize the capability of IRM when unobserved confounder exist. However, this work also lacks the analysis of identifiability result.
We finish this section with the following summary of methods in section 7.7.4 and the IRM, in terms of causal factor, invariance type, direction of causal relation, theoretical judgement and the ability to generalize to intervened data.
Table 4: Our LaCIM with related works.
Causal Facior
Direction
Imaatincee Type
Theoresical Judgement
Intervenion
Poters (2010)
Subset covaritees
g
Linear medel with Gaussian noise
Identitability
Rous Carulla al (2018)
Subset of covuritees X
X +
Linear Model
Kuang (2018)
X
Nonlincar
Contounder Balancing
Uthimann (2015)
Subset of covurates X
Ag
Nonlinear
liemiiabilitt
Arjovaly al. (2019)
Latent vanables
8 wa
Lincer
Identitabiiiity
LaCIM (Ours)
Luteat vastables S.
Nonlinaar
ldeniitability
7.8 IMPLEMENTATION DETAILS AND MORE RESULTS FOR SIMULATION
Data Generation We set m ng = 1000 for each e. The generating process of d, E R94,
Z E IR= R2,, X E R and Y E R& is introduced in the supplement 7.8. We set qd, = qe
q: = 4y = 2 and Or = 4. For each environment E [m]] with m = 5, we generate 1000 samples
D = (.,, ind SPr (x/1 s)pr,, (ul 8 ()) (s, dld2)dsdz. The d = (N, Jax, xaa, ) + 5 t) 2:
the 2|d; ~ N (Hot.. (s, = (lg)) ob (s, (())) with Moz.. A dī and log oor.. = 4 de
(AM., Ag. are random matrices); the x = ~ N (poo (2r (o 2), oz (r]ss with poz. = h(Ap M(A22, h(A4²2 [ ))] and log oet h(A h(Ag2 11 [s ,2 (ID) (h is
....iii......iiiii....... A are random matrices); the y|s is similarly to z/s, with A , -. respectively replaced by AW. A,. 1.a.,.2,3...1__ -.g..11.22
LeakyReLU activation function with slope = 0.5 and Ag.
Implementation Details We parameterize po 2 2/d), (s 2/2 y, p=(x|s, 2) and po as 3- layer MLP with the LeakyReLU activation function. The Adam with learning rate 5 x 10 4 is implemented for optimization. We set the batch size as 512 and run for 2,000 iterations in each trial.
Visualization. As shown from the visualization of S is shown in Fig. 7.8, our LaCIM can identify the causal factor S.
40-
(a) Ppool Laccoa (b) PLaCIa y, de)
(c) (L y. v.d.) de)
(d) Pa- (s|d.)
Figure 4: Visualization of S. From left to right are: estimated posterior by pool-LaCIM, (LCM y). by LaCIM with as input: (S)CIM( y, d,d,), by LaCIM with D as input: (saCTM( x, y, d); the ground-truth pa- (s]da).
The sefting when C' can take a value in a sample-leve. We consider the generation process of D as Dr {æ,, kddd f f (r)s,))),, 2)p,, (vls)p(s, 2 c)p(cldd )dsdzde, with ge := 2. The generation is the same except that the after obtaining d,, we additionally generate e with c == AVd,,, The results summarized in Tab. 7.8.
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Table MCC of identified latent variables for p (x. y) Average over 20 times for each data.
p(a)s, 2 ()p(yl))pppp)))))))))) )dddddddddddd...
)dedsdz.
Data
Data #2
Data
Data #4
Average
Z
S
Z
S
Z
S
Z
S
Z
S
S
pool-LaCIM
0.26
0.61
0.26
0.67
0.44
0.70
0.51
0.78
0.58
0.77
0,41
0.71
LaCIM-d, d, (Ours, m = 3)
0.70
0.79
0.72
0.79
0.69
0.74
0.74
0.85
0,64
0.88
0.70
0.81
LaCIM-d, (Ours, = 5)
0.73
0.85
0.70
0.89
0.85
0.91
0.81
0.84
0.83
0.93
0.78 1
0.89
LaCIM-d, (Ours, 0.92
0.90
0.83
0.90
0.84
0.93
0.85
0.94
0.83
0.90
0.86
0.91
7.9 IMPLEMENTATION DETAILS FOR OPTIMIZATION OVER 8, Z
Recall that we first optimize s" according to
, arg max log Pa (r((ss,))
8,2
We first sample some initial points from each posterior distribution q (s)ar) and then optimize for 50 iterations. We using Adam optimizer, with learning rate 0.002 and weight decay 0.0002. The Fig. 7.9 shows the optimization effect of one run in CMNIST. As shown, the test accuracy keeps growing iterates. For time saving, we chose optimize for 50 iterations.
Optimization Effect
0.991
0.988
0.955
50
100
150
200
250
300
Iteration
Figure 5: The optimization effect in CMNIST, starting from the point with initial sampling from inference model g of each branch. As shown, the test accuracy increases iterates.
7.10 IMPLEMENTATIONS FOR BASELINE
For the CE X + Y and the CE.X, de + Y, they both composed of two parts: (i) feature extractor, followed by (ii) classifier. The network structure of the feature extractor for CE X - Y the same with that of our encoder; while the extracted features for CE X, d - Y is the concatenation of the features encoded from X - S, Z via the network with the same network structure of our encoder; and the network with the same structure of our prior network for LaCIM-d. The network structures of the classifier for both methods are the same to that of our p& (y) s). The IRM and SDA adopt the same structure as CE X + Y. DANN adopt the same structure of CE X - Y and a additional domain classifier which is the same as that P (B(s)) sVAE adopt the same structure as LaCIM-d, with the exception that the (y(l)) () is replaced by ()]=,) s). MMD-AAE adopt the same structure of encoder, decoder and classifier as LaCIM-d and a additional 2-layer MLP with channel 256-256-dim, is used to extract latent 2. The detailed number of parameters and channel size on each dataset for each method are summarized in Tab. 13, 14.
7.11 SUPPLEMENTARY FOR COLORED MNIST
Implementation details The network structure for inference model is composed of two parts, with the first part shared among all environments and multiple branches corresponding to each environment
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for the second part. The network structure of the first-part encoder is composed of four blocks, each block is the sequential of Convolutional Layer (Conv), Batch Normalization (BN), ReLU and max- pooling with stride 2. The output number of feature map is accordingly 32, 64, 128, 256. The second part network structure that output the mean and log-variance of S, Z is Conv-bn-Rel.U(2555 - Adaptive (1) FC(256, 256) ReLU + FC(256, gt=s,a) with FC stands for fully-connected layer. The structure of Pt=s.s in Eq. (72) is FC(gr. 256) + ReLU + FC(256, qt). The network structure for generative model pa (rrss,, 2) is the sequential of three modules: (i) Upsampling with stride (ii) four blocks of Transpose- Traanpose-Convoluttii Convolution (TConv), BN and ReLU with respective output dimension being 128, 64, 32, 16; (iii) Conv-BN-ReLUC Reigmoigo with number of channels the output as 3, followed by cropping step in order to make the image with the same size as input dimension, ie., 3 x 28 x 28. The network structure for generative model Po(U)s) is commposed of FC (512) - BN + ReLU - FC (256) - BN + ReLU - FC (D1). The I==., is set to 32. We implement SGD as optimizer with learning rate 0.5, weight decay 1e- - 5 and we set batch size as 256. The total training epoch is 80.
We first explain why we do not flip with 25% in the manuscript, and then provide further exploration of our method for the setting with flipping y.
Invariant Causation v.s. Invariant Correlation by Flipping y in Arjovsky et al. (2019) The is further flipped with 25% to obtain the final label in IRM setting and this step is omitted in ours. The difference lies in the definition of invariance. Our LaCIM defines invariance the causal relation between S and the label Y, while the one in IRM can be correlation. As illustrated in Handwritting Sample Form in Fig. 7.11 in Grother (1995), the generting direction should be Y - X. If denote the variable by flipping Y as Y (a.ka, the final label in IRM), then the causal graph should be X T Y + Y. In this case, the Y is correlated rather than causally related to the digit X. For LaCIM, we define the label as interpretable human label (which can approximate to y for any image r) and represented by Y in our experiments. The reason why we do not define the Y as ground-truit label is that (i) the prediction is only based on the extracted components of image which may be determined not only by the ground-truth label; (ii) the learning of ground-truth is interpretable that relevant to human. For example, if writer is provided with digit "2" but he wrote it mistakenly as "4", then it is more interpretable that we can predict the digit as "4" rather than "2". For the digit with ambiguous label from the perspective of image, even if we predict it mistakenly, it is also interpretable in terms of prediction given the information of only digit. Returning back to the IRM setting, the label is flipping without reference to the semantic shape of digit. Therefore, the flipping may happen to noiseless digits rather than noisy and unsure ones, making the shape of number less semantically related to the label.
Experiment with IRM setting We further conduct the experiment on IRM setting, with the final label y defined by flipping original label with 25%, and further color pf proportions of digits with corresponding color-label mapping. If we assume the original ground- truth label to be the effect of the digit number of S, then the anti-causal relation with Z and Y can make the identifiability of S difficult in this flipping scenario. Note that the causal effect between S and Y is invariant across domains, therefore we adopt to regularize the branch of inferring S to be shared among inference models for multiple environments. Besides, we regularize the causal effect between S and 2 to be shared among different environments via pairwise regularization. The combined loss is formulated as:
ENp L + 2m2
(B (paa))pp-((e (2a)) E (x. |
with (s, in Eq. (72) factorized as qeg (z)qu,(6 (s) and Ps shared among m environments. The appended loss coincide with recent study Eiss--xxtopootttinn (REx) in Krueger et al. (2020), with the difference of separating y-causative factors S from others. We name such training method as LaCIM-REx. For implementation details, in addition to shared encoder regarding S, we set learning rate as 0.1, weight decay 0.0002, batch size as 256. we have that p(y)ær) = Ss qw. (s (r)p (v)pp(s) for any x. We consider two settings: setting#1 with m2 and p = 0.9, nd = -0.8; and setting#2 with m 4 with nd1 0.9, nd2 0.8, p.a = 0.7, p" 0.6. We only report the number of IRM since the cross entropy performs poorly in both settings. As shown, our model performs comparably with LaCIM-d, and better than IRM Arjovsky al. (2019) due separation of S znd Z.
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Figure 6: Hand-writting Sample Form. The writer print the digit/character (i.e., X) with the label (i.e., Y) provided first.
Table 6: Accuracy (%) of Colored MNIST on IRM setting in Arjovsky et al. (2019). Average over three runs.
IRM
LaCIM-d, (Ours)
LaCIM-REx (Ours)
m 2
67.15 = 3.79
68.16 1 2.13
67.57 + 1.37
69.37 1.14
69.55 1.60
69.50 + 0.57
7.12 SUPPLEMENTARY FOR NICO
Implementation Details Due to size difference among images, we resize each image into 256x256.
The network structure of pa(2, s|d,), (0(2,, s/2, d.), (x(2z 8), Pa())s) s) for cat/dog classification is the same with the one implemented early prediction of Alzheimer's Disease with exception of replaced by 2D ones. For each model, train for 200 epochs using sgd, with learning rate (Ir) set to 0.01, and after every 60 epochs the learning rate multiplied by decay parameter that set to 0.2. The weight decay coefficients parameter is set to 5 10-1 The
3D comvolution/Deccnoolliii
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Table 7: Training and test environments (characterized by dz)
cat% on grass
dog% on grass
cat% on snow
cat% on snow
Training Environment
Env#1 (d)*)
0.6
0.4
0.1
0.9
Env#2 (d?*)
0.8
0.2
0.1
0.9
Env#3 (d.*)
0.5
0.5
0.2
0.8
Env#4 (d5a)
0.8
0.2
0.2
0.8
Env#5 (dx*)
0.7
0.3
0.2
0.8
Env#6 (d22)
0.8
0.2
0.3
0.7
Env#7 (d'77
0.7
0.3
0.3
0.7
Env#88 (dg*)
0.9
0.1
0.3
0.7
Env#9 (d.²)
0.4
0.6
0.3
0.7
Env#10 (d.20)
0.6
0.4
0.3
0.7
Env#11 (d
0.4
0.6
Env#12 (d
0.4
0.6
0.4
0.6
Env#13 (d)
0.7
0.3
0.4
0,6
Env#14 (d214)
0.8
0.2
0.4
0.6
Testing Environment
Env Test d
0.2
0.8
0.8
0.2
Table 8: Comparison constructed interventional dataset in terms of ACC.
Method ACC
CE X +
IRM
DANN
NCBB
MMD-AAE
DIVA
LaCIM-d
52.50
50.00
49.17
49.17
49.17
50.00
55.00
batch size set to 30. The training environments which characterized by can be referenced in Table 7.12. For visualization, we implemented the gradient-based method Simonyan et al. (2013) to visualize the neuron (in fully connected layer for both CE x + and CE (x, d,) + y/ in s layer for LaCIM-d,) that is most correlated to label y.
The d, for m environments We summarize the d. rn m As shown, the value of de in the test domain is the extrapolation of the training environments, iee. the d is not included in the convex hull of [d% 114 1:=1-
and m 14 environments in Table 7.12.
More Visualization Results Fig. 7 shows more visualization results.
Results on Intervened Data. We test our model and the baseline on intervened data, which each image is generated by intervention Z, i..., taking specific value of Z. This intervention breaks the correlation between and Z, thus the distribution which can be regarded as a specific type of OOD. Specifically, we replace the scene of image with the scene from the another image, shown in Fig. 8. We generate 120 images, including 30 images of types: cat on grass, dog on grass, cat on snow, and dog grass. We evaluate LaCIM-d, CE X Y, IRM, DANN, NCBB, MMD-AAF, and DIVA methods on this intervened dataset. As shown in Tab 9, our LaCIM-d can performs the best among all methods, which validate the robustness of our LaCIM.
7.13 DISEASE PREDICTION OF ALZHEIMER'S 'S DISEASE.
Dataset Description. The dataset contains in total 317 samples with 48 AD, 75 NC, and 194 MCI.
Denotation of Attributes d,. The E R includes personal attributes (e.g., age Guerreiro and Bras (2015), gender Vina and Lloret (2010) and education years Mortimer (1997) that play potential
Table 9: Comparison on constructed interventional dataset in terms of ACC.
Method
CE X +
IRM
DANN
NCBB
MMD-AAE
DIVA
LaCIM-d
ACC
52.50
50.00
49.17
49.17
49.17
50.00
55.00
31

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Under review as a conference paper at ICLR 2021
(a) Cat on grass
(b) Cat on snow
(c) Dog grass
(d) Dog snow
Figure 7: Visualization on the NICO via gradient- based method Simonyan et al. (2013) for CE X Y, CE (X, d,) + Y and LaCIM. The selected images (a) cat grass, (b) cat snow, (c) dog on grass and (d) dog on snow.
risks of AD), gene (54 allele), and biomarkers (e.g., changes of CSF, TAU, PTAU, amyloid,, cortical amyloid deposition (AV45) Humpel and Hochstrasser (2011)).
Implementation Details For LaCIM-d,, we parameterize inference model (s, da), Po (s, 2 2.),), Ps (z(222 s) and ps(y]s) and S, Z E R64, For (8,2 dg), we concatenate outputs of feature extractors of X and d,: the feature extractor for is composed of four Battoooluooo-Battee
32

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Under review as a conference paper at ICLR 2021
Dog snow
Cat on snow
Dog on grass
Cat on grass
Figure 8: The constructed interventional dataset which includes of dog snow, dog grass, cat snow, and dog on grass.
Normalization-Rel (CBNR) blocks and four Convolution-Bace
(CBNR-MP) blocks with structure 64 BNR + 128 CBNR-MP + 128 CBNR - 256 CBNR-MP + 256 CBNR 512 CBNR-MP 512 CBNR - 1024 CBNR-MP: the feature extractor of is Normalization ReLU (FC-BNR) blocks with structure 128 256 + 512. The concatenated features are further transformed by four 64 FC-BNR to generate (x, d.) and log (.2 d.). For the prior model (s(s, (d,), it shares the same structure with- (v.
composed of three Fully Connection-
out feature extractor of z. For P& (2r 8, :), the network composed of three DeConvolution Batch Normalization-RelL. (DCBNR) blocks and three Convolution- Cenvouutoon-Bacch Normalization-Rel
(CBNR) blocks, followed by a convolutional layer, with structure 256 DCBNR + 256 CBNR - 128 DCBNR (s) the network is composed of 256 FC-BNR - 512 FC-BNR + FC-BNR. For prior model (s(s, (Ms, diag(o", (da))) the (r, d.) and (x, d,) are parameterized by Multi Perceptron Neural Network (MLP). The decoders (=)s,))) are (y(s)) parameterized by Deconvolutional neural network. For all methods, train for 200 epochs using SGD with weight decay 2 x 10 and learning rate 0.01 and is multiplied by 0.2 after every 60 epochs. The batch size is set to 4. For each variable in biomarker vector E R each person may have multiple records, and we take its median as representative to avoid extreme values due to device abnormality.
+ 128 CBNR - 64 DCBNR - 64 CBNR 48 Conv. For p (2)
As for LaCIM-d, we adopt the same decoder (=(x22, and classifier Po(y)s) For qu (s, 2|r,, d). we adopt the same network for the shared part; for the part specific to each domain, M4, (x. d) and log a. (I, d) are generated by the sub-network which is composed of 1024 FC-BNR + 1024 FC-BNR 4s.s FC-BNR. The z, can be reparameterized by (I, d) and log a., d) are fed into a sub-network which is composed rid of the constraint of Gaussian distribution. Then the reconstructed images and predicted label computed by Po (2)2,, s) and p (y' s) which have the same network structure of LaCIM-C with the
9=.s FC-BNR - 1024 FC-BNR - q... FC-BNR to get
2,
33

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Under review as a conference paper at ICLR 2021
Table 10: Training and test environments (characterized by c) in early prediction of AD
Training Env#1
Training Env#1
Test
Age
Number of AD
17
17
14
Number MC1 Numher of NC Average value of dg (years):
68.75
72.78
81.74
TAU
Number of AD
11
22
15
Number of MCI Number of NC Average value dg:
75
78
41
215.34
286.69
471.72
The d, variable in training and test. The selected attributes include Education Years, Age, Gender (0 denotes male and denotes female), AV45, amyloidg and TAU. We split the data into training environments and test according to different value of d,, The Tab. 7.13 describes the data distribution terms of number of samples, the value of d, (Age and TAU).
7.13.1 EXPERIMENTS WITH COMPLETE OBSERVABLE SOURCE VARIABLE
In image-based diagnosis, the personal attributes, genes and biomarkers are often available. Therefore. we consider the setting when d, can be fully observed. In this case, the value of d, is person-by- person. Therefore, the number of environments m equal to the number of samples. In this case, the dataset turns to { Vi, di J The expected risk turns to:
Polyls)) log n (y)ar, dz) qu 2),)))
Pa(ar|s (8,(6,22da)
Le = (ptsal)))
log go (vlx, Ego
(73)
And the corresponding empirical risk is:
Ev.o 1 [ log Qe (31 (2 do,s) (r
go (yils)
po(edss (s, 2/da,))
log Io (3i xy do (v 2 2/2,, daa))
n
(74)
The d, here is re-defined as the 9-dimensional vector that includes all attributes, genes and biomarkers mentioned above. We re-split the data into 80% train and 20% test, according to different average value of specific variable in the whole vector dg.
The d, variable in training and test We implemented OOD tasks in which the value d, is different between training and test. Specifically, repeatedly split the dataset into the training and the test according to a selected attribute d, for three times. The average value of these attributes in train and test are recorded in Table 7.13.1.
Experimental Results We conduct OOD experiments with source variables Age, Gender, amyloids and TAU different between training data and the test. The results shown in Table 12.
7.14 SUPPLEMENTAR FOR DEEPFAKE
Implementation Details. We implement data augmentations, specifically images with 30 angle rotation, with flipping horizontally with 50% probability. We additionally apply random compressing techniques, such as JpegCompression. For inference model, we adopt Efficient-B5 Tan and Le (2019). with the detailed network structure as: FC(2048, 2048) - BN + ReLU - FC(2048, 2048) - BN + ReLU - FC(2048, ))= The structure of reparameterization. i.e., Pt= is FC(gt=a.s 2048)
+ BN - ReLU + FC(2048, 2048) + BN - ReLU FC(2048, qt=,,;). The network structure for generative model, i.e., (z]s,) is TC 256) + TConv-BN-Rell LeL
128) + TConv-Bl -CConv-BN-Rell BN-ReLU(125 64)+ TConv-BN- -ee--NNRRRLLL,,, +
32)
TCCon-BN-Ree..U32. 16) TConv-BN-Rel.U(16, 16) Con--BN-Rel.U(16 3) Sigmoid,
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Under review as a conference paper at ICLR 2021
Table 11: Training and test environments (characterized by c) in early prediction of AD
Education Years
Age
Gender (0/1)
AV45
amyloidæ
TAU
Setting #1
Training Test
15.34 19.44
70.56
1.29
1.21
745.61
249.38
81.74
1.83
1.56
1322.47
471.72
Setting #2
Training Test
15.34 19.43
70.62
1.29
1.21
743.01
250.21
81.19
1.83
1.57
1332.94
446.67
Setting #3
Training Test
15.34 19.44
70.62 81.19
1.29 1.83
1.21 1.56
743.39 1331.4
254.7
446.9
Table 12: Accuracy (%) of OOD prediction on ADNI. Average over three runs.
ACC (%)
Setting#1
Setting#2
Settin#33
Setting#1
Setting#2
Setting#3
Method
source
Education Years
AV45
CE X + +
61.9 0.0
66.7 1 1.6
63.0
0.9
67.7 0.9
66.1
3.3
66.1 1: 1.8
DANN
62.4 0.9
62.4
0.9
63.0
1.8
0.9
67.2 + 0.9
66.1
CE (X, de) Y
67.2 + 1.8
66.7
3.2
63.0
1.8
66.1
3.3
1.8
64.0
0.9
SVAE
67.2 = 0.9
67.2
0.9
67.2
0.9
65.6
1.8
66.7
65.1
1.6
LaCIM-ds (Ours)
69.8 + 1.6
68.8
+ 0.9
69.8
1.6
69.3
67.7 +
67.7
0.0
OOD source
Age
Gender
CE +Y
63.6 2.6
65.6 6.0
648
6.9
60.5
1.8 1.8
60.5
DANN
60.8 = 1.8
0.0
5.0
58.5
19.511. 1.5
61.5
60
CE [X, d.) + Y
60.4 2.9
2..
6..4=38
63.2
- 0.9
64.1 6.
SVAE
58.2 + 0.9
58.7
64.1
+ 0.9
65.6
1..8
64.1 0.9
LaCIM-d, (Ours)
64.0
+ 2.4
70.4 + 2.4
66. 3.7
65.6
0.9
67.2 = 1.8
68.2 = 0.9
ACC (%)
Setting#1
Setting#2
Setting#3
Setting#1
Setting#2
Setting#3
Method
OOD source
amyloid
TAU
CE X + Y DANN
59.2 0.9
63.5 4.2
63.1 5.1
6..6
64.1
0.0
66.0
1.1
60.8 + 0.9
60.8 + 0.9
60.8 0.9
64.6 0.9
65.1 0.9
64.6
+ 0.9
CE (X, de) + Y
64.6 + 1.8
64.6 3.7
64.2 2.4
64.6 0.9
6..
67.0
1.3
sVAE
+ 0.9
63.5
68.2
0.9
68.8
2.7
67.2
LaCIM-d, (Ours)
68.3
+ 1.6
66.1
+ 1.8
2.4 65.6
69.8
0.9
71.4
+ 1.8
68.8
- 0.0
followed by cropping the image to the same size x 224 x 224. We set gt=s,= as 1024. We implement SGD as optimizer, with learning rate 0.02, weight decay 0.00005, and run for 9 epochs.
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Under review as a conference paper at ICLR 2021
Table 13: General framework table for our method and baselines on Data (CMNIST, NICO, ADNI, DeepFake Dataset. We denote the dimension 2 28 dim We list the output dimension (e.g. the channel number) of each module, if different from the one in Tab. 14.
LaCTMat
Data Eng fist x m Des Dela ped
0.923
wil. 0.0000
Dati Eng x a Dale x m
Dats..DDe Dood
18,35M 19:70M
wit 0.0001
Ele Data
Doo
wd: 0.0002
Das Deba
Ir,
Doy De..
LaCiMdd
Enne -in Pata
FC-BN RecaN-R MeL.ur5122250
(C)256..im 22todiimm dim Dod Dis Batt
pricr Eng Dains
000001
Eine DatDat Line
FC 1556. 1550.dmm:.
Dod Doe Data
prior DagDa
16,31M
Ir 0000
wd: 0,0005
Enc Enneott
PC11500 aa)
Doo Dos
price: Eng
0000
wd: 0,0002
wd:
P9 (at Fg)
Data (al ag 4an mg)
Pata a dex Ey)
DIVA
1itt (itg (s)
( to
Pa (at
wet 0.00001
Dhan (naa (xald))
(ayl lb)
Data (=y
Ba (2y (x)
14,07
In G.o
wat 0.000
po (agld) lel)))
Pla (2 (w)
paaarza lx)
Data (sx
35.22M 4
IF C100S
we: 0.0001
Pay
Pag
9g
"gg
DANN CLS.Ctsaaa... DANN CLS Data
DAN. DAN.-AANNN.... Data
DANN-CE DANN-CI S
DatData
L.MM
wid: 0.0022
Enc
19. ESM
In 1.0S ud:: CLOKKSS
EacData
311,31M
0.01
we:: 0.002
DANN-CSP
Eng Io (224)
FC-BN-Re LRR (024)
Enc Dati w...n.add.egg
PC7200 2500 =2 2
Do Deu
FCUOA 1026) 2
De:Data, Dec
101122 1020 - Dec Data, Deli
MMD-AEE
1233
wel: 0.0001
19,200
In 0.2
wel: 0.00111
Eng Datas
Ir: 0.03
we: 0.0002
EatDat Data D Data
Eic UncDDiin Dali Dww.n.ninn kin Dake FCv312,88m Dec Des-Cr CI
Paka
FO:512,8m ) Dec Cipaaoo
Ir: wdt 0.000s
wel n.00K
EacDDi Data pac #Cr15studiing Doe-CBB Dho-Cg.DDatt Data
wil 0,0000
TM
1011MM In 0.01
25,27M
In 0000
CE
Data PO256 (m)
Dec Do
Dute Date KCpitzddddm Dax- CC-CCDDDo Deca
wi 0,00022
Date P000E3,aim aimm))
Do-CCBDess Dets
wil 0,00002
LT-M
In 0.01
36.27M
0001
in:
CB
Meihod
Dani CANSSTT
Farums
hopper Panms
DuieNICO
of Paras (m =
hyper Purams
Dutt:DDN
PypessPummsss
36

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Under review as a conference paper at ICLR 2021
Table 14: Network Structure of Modules used in method and baselines.
Method
CMNIST
NTCO
ADNI
- 128,3,1.1)
ReLUKdiminngr.22....3 128.3.1.1)
Conv. 2d..1122,26226222220
ReL.
Conv-BN-Re BN-ReeLUddiiippp (4....... 6413,I.1)
MaxPool(2)
MoxPooli2)
MaxPool(2)
Conv. BN ReL.U(256,226631111
Conv3d Rel.Ui(256.256.3....
Conv-RN-Rel.a (664,
MaxPool(2)
MaxPool(2)
MaxPool(2)
Enc Enc
CCN-BBN-ReLLUS122:122...1
MaxPool(2)
Cony BoN-BBN.a. RedUUS122_51223...))
Conv3d BN ReLU,512.511233111.
Con--B..Relti256...
MaxPool(2)
MaxPoolr2)
AdaptivePool(!)
Con--NN-cLLU:/12.5112...3
Flatten()
Conv BN Rel.7
Conv3d (N............S11220033.....
AdaptivePool())
AdaptivePool(C)
Flatten))
Flatten()
UnPlarten()
UnFlattenO)
Upsample(16)
Upsample(6)
UnFlatien()
Upsample(2)
Toonx-BN-ReLLUde 1
Com-BN--RdDDU3562266333
Cow.bd-BN--Rel1U0266666666
Cco--BN-ReLLU2256122...222
Toom--BN-ReL BN---RRRLUU1122622222000
Dec
Teonv BN RNN-RLLUV643322222.00
Toony Bow--BN-R. RN12
Con-BBNReel
Ww..BBNRR...((64444......
Conv3d-BN.Ral
Comv(16,333111)
SigmoidO
Tcom-BN--edLUIG455222222
Toom:4LBN--eeL1U6666642222
Comr34-BN--LLLUU1666.......
2ropping(28)
Conv(32,33311.)
Com33d664.13.1.))
Sigmoid()
SigmoidO)
RCBN-RRLLUd 256)
FC-NN-ReLUid, 256j
Ency
Data
FC-BN-ReLUCd, HC-BN-R::LU(122,550 256)
FC-BN-ReLUV256.111
FC-BN-ReLU1256, 512)
FC-BN-ReLUV5122512
FC-BN-ReLL
FC-BN-R:LUddmmm,... 3112)
HC-BN-KLUUddmm.... a:12
EC-BN-RdLUddmma.... .122
Doão
FC. 256) FC(256.2) Rel U'dim 512)
FC.BN.Re1.11112. 256)
PC-
FCC256,2
FC(256,2)
FC- Lidim 1024)
FC.NN Reddieelldddmm... 512)
Dec-CEData Data
FC-BN-ReL 256) FC(256.2)
FC-BN-ReLUKr102,, 2048)
FC-BN-ReLU
FC(2048.2)
FCY256.2)
DANN-CLS
BeL 32)
FC-BN-ReLU(122, 2048)
FC-BN-ReLU( 1024)
FC BN ReL.U(32,
PC BN Rel.U
FC BN ReL.U
+ Data
FC-ReLUrdim 256)
FC-RcLUidim,. 1024)
FC-RoLU(dim, 1024)
RCLUU222,56, dimi,..
FC-ReLU11024 dim.
FC-ReLU dim,
Enc
Data
FC-ReLU(256, 256) FC-ReL, dim....
FC-RoLU(IU24, 1024)
FC-ReLU(1024, 1024)
FC FC-RRLL dim...)
FC-ReLU dim,,
Rel.Ui(1024)
UnFlattenO)
FC-BN-ReLU((024)
FC-BN-ReLU(1024)
Upsample(8)
UnPlattca()
UnFlatten()
Upsample(16)
Upsample(8)
Conv3d BN ReL.1664,1283.....)
Po
(ata (a)
TComv "Comy-BN-Rell (144
TComy Bonv-BN.Bae (64,
Upsample(64)
Upsample(24)
Upsample(24) www.-RN.Re...i/28....o Conv(256, 2,."),1.1.0
TCony3d BN Rct.u((12111......11
TCony BN Rcl.U(128..5655..000
Upsample(266)
Coav(266, 3.1..))
Upsamplo(48)
Cow:3ULBNN-RLLU11122.222.3.
Comv3dr32, 1,1.100
Pa (zald) (f, lv) PH,
PC-BN- Rew--RLLUddiaaa. 64) FC(64,64);; FC(64,64)
FC WC-BBN-eLLLddmm..... 64) FC(64464)
HC-BN-ReLLUrdmma, 64)
FC(64.64); FC(64.64)
ConviceBN-RcLLUL.........
Com-BN--ee1UUG.32222..1
MaxPool(3)
Con--BN-R:LU/3332511.00 MaxPool/2)
MaxPool(2)
Conv3d-BN-Raad BON03UU-BN-RRLLUU222252222222
102 (xal q 1a
Conv-BN Nan--BN--RLLUU32664..2222
Conv3d BN Re1.U/256,,_6631111
(ata(2,
Conv: Bonv-BN.Be ReL.U(32.6445.1.0 MaxPool(2) Flartent)
MaxPool(2)
MaxPooli2)
(2, Rdy
Bata
MaxPool(2)
Conv3d-BN-ReLLUd Bittca-BBN-ReLU/256.222525
Cow.3U-BN-ReLL//26..2.....
FCU124, 64); FC(1024, 64) Data
Hattent)
MaxPool(2)
PCU002 64); FC(1024, 64) Date
Flatten()
64); 64) Data
37


Reviews:
[{"review_id": "j-2ZzdvD-Hk", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "I would like to thank the authors for the interesting work they proposed. I tried to explain my concerns below and I am open to changing my score with their feedback. \n\n1) My main concern is whether the considered causal graph indeed captures the phenomenon that the authors are attempting to address. What I mean is that, yes, indeed C induces correlation between Z and Y. But only in the combined dataset where we do not fix C. For each realization of C=c, Z and Y become independent. This is in contrast with what we experience in practice, e.g., we see cows on pasture sceneries more often than not in a single dataset. So the correlation should exist even when we only look at a single dataset. Then this correlation may change in a different dataset where we see cows on the beach. In both datasets, there is spurious correlation which needs to be resolved. The only reason there is spurious correlation in data obtained from this graph is because we assume C is not conditioned on, i.e., we are given mixed data from multiple environments. One of the proposals of IRM was to make use of this environment information, rather than work with the combined data and still there are many challenges there. Theirs is one attempt to address this. In short, I am not certain if this causal graph is suitable to model the phenomenon we are facing in ML today, which the authors talk about in the introduction. \n\n2) Another concern is about the invertibility of f_x to obtain s,z. If s can be uniquely obtained from x, then how does C affect p(y|x) across different environments? Writing down Bayes rule, normally posterior is affected by p^e(s,z) but when p(x|s,z) is deterministic, this is not the case: For each x, one can uniquely go to s only to then go to y, without C playing any role in the process. Then I think this invertibility assumption might make the effect of C void. Please comment on this. \n\n3) When the new (test) environment is one of the environments that we trained on (c^e is known) then we can just use the training data p(y|x) in that environment. Why is this analysis needed then for that case? \n\n4) When the new environment is not known (paragraph before Section 5), the authors propose optimizing s and z and to maximize the data likelihood for the trained model and finding the most likely y given this s. This is connected to the second condition of Definition 4.2, correct? In other words, under the assumptions, this inference will give the correct distribution p(y|x) in ANY new environment. Please verify if I am understanding the result correctly here. But then again, based on 2) above, I am having a hard time seeing the value of the identifiability result under these assumptions.\n\n5) Assumptions are revealed sequentially one by one and too late in the paper. It would also be great if the authors could make an itemized list of assumptions (ANM, exponential family, invertibility of functions etc.).\n\nThe following are my detailed feedback and further questions:\n\nThe authors assume a simple causal graph where features (image) and labels do not cause each other but are caused by latent factors that are affected by the environment (different interventions). To get to p(y|x) in a way that does not depend on the environment to be able to do inference on new data, the authors attempt to learn the mechanism to generate S from X and Y, where S is not observed, for each environment. They show that under a number of assumptions, an identifiability result arises, i.e., there is a set of models which all give the same \\sum p(s|x)p(y|s), implying that we can predict Y from X for any new environment and this can be recovered from p^e(x,y) given enough number of diverse environments. \n\nThe narrative is a little overwhelming and unnecessarily confusing in that, very simply, authors propose using a specific causal graph, with specific modeling assumptions within the SCM framework. Their main contribution is the identifiability result which shows that under the given assumptions, we can identify an invariant mechanism p(y|x) that does not depend on the environment (if i am understanding the theorem correctly).\n\nExperimental sections are too briefly described and should be expanded on. Also please comment of Figure 2.\n\nPlease explain the variables corresponding to those in the considered causal graph in each real experiment. This is very important to assess the validity of the analyzed causal graph and the assumed missing edges between the variables. For example, what are the variables in CMNIST? I believe Color should cause Image but not the Label and Digit should cause both the Image and the Label. But the narrative is different than this. \n\nAdditive noise model, which is a huge assumption is only revealed in page 4, whereas claims about identifiability of the SCM are made starting abstract. This should be given much before. ANM assumption is made for the SCM of both X and Y.\n\n\"Information intersection property\" in first par. of Sec. 4.2\nf_y^{-1}(y')=[f^{-1}]_S(x') for all x',y' will not be true unless the exogenous variables are zero since y=f(x)+e. What am I missing here? Maybe the authors mean invertibility from f(x) back to x rather than from y to x.\n\nPlease explain the notation in Definition 4.2 in detail. What is [f_x^{-1}]_S(x)? Without these, it is very hard to parse the statement. I was able to decode these eventually but they need to be introduced for ease of reading. \n\nOther notation is also very confusing. Boldface capital T is a matrix but also a function that takes inputs. Again, I can decode what authors meant but a more rigorous and precise notation is needed. \n\n\"It is shown in supplement 7.2 that ∼p is an equivalence relation.\"\nCan you please elaborate in the main text? \n\nThere is some mismatch with the experiments and the theory. Additive noise models and the exponential families enable the identifiability results whereas experiments use a VAE formulation. Why not assume the additive noise models and the exponential families in the experiments as well? Then I am sure other inference methods could be used to discover the latent space from the exponential family and still leverage the proposed theory for identification. Did you try this and does it not perform as well? Please comment if possible.\n\nCan you give intuition on why the identifiability result requires a condition on the rank of \\Gamma matrix but not on the other parameters? For example, what if all T's are zeros? Then no condition on Gamma should be sufficient since it doesn't appear in the equations anymore. How is this and similar corner cases are covered by the theorem?\n\n\"For causal prediction, the old-school causal learning frameworks (Peters et al., 2016; Buhlmann, 2018) causally related the output label Y to the observed input X, which however is NOT conceptually reasonable in scenarios with sensory-level observed data (e.g. pixels in image classification).\"\nI understand what this implies but it is very implicit. Please consider elaborating: Modeling pixels as variables of a causal graph does not make much sense. \n\n\"For such applications, we rather adopt the manner in\"\nThis divergence is also confusing. The hinted difference is not a fundamental difference but a difference in how one models the causal structure. \n\n\"into a novel causal model\"\nI would consider changing this statement. The authors assume a specific causal graph, calling it a novel causal model does not seem accurate. \n\nFigure 1 is interesting because it makes the strong assumption that the features X do not cause the label Y. It also makes the strong assumption that S and Z cannot cause each other. Please comment on these non-causality restrictions imposed by the graph.\n\n\"Notably, far beyond the scope in existing literature (Khemakhem, Kingma and Hyvarinen, 2020), our results can implicitly, and are the first to disentangle the output-causative factors (a.k.a, S) from others (a.k.a, Z) for prediction, to ensure the isolation of undesired spurious correlation.\"\nI believe the related work for identifiability of causal factors should be much richer than this. Please consider including identifiability of SCMs in linear and related settings from the causal inference literature. \n\nI do not think there is a need for a new name \"Causal Invariant Mechanisms (CIMe)\" where what the authors mean is to learn the structural equations in the SCM alongside the distributions.\n\n\"From the perspective of causality, the confounder C blocks the back-door path from Z to Y , making the Z spuriously correlated with Y .\"\nMaybe splitting into two sentences will help with clarity here. The path induces the correlation, not blocking the path by C. \n\n\"brutal-force\"->\"brute-force\"\n\nIn (1), it looks like the joint distribution on S,Z is assumed to factorize as p(z)p(s), i.e., they are independent. But this is per environment, indicated by the index e. I believe it is better if the authors could write this as p(s,z|c) instead of p^{e}(s,z).\n\nThe arxiv identifier of Tan et al. is in boldface.\n\nI think the authors should discuss identifiability of SCMs especially in the ANM setting more explicitly. It would also help to discuss that once the causal graph is known, under the ANM assumption one can trivially obtain the SCMs. The challenge arises when there are latent factors, which is the setting considered here.\n\nWhat is assumption (4) mentioned in the proof? Is it the fullrank condition written in caps? Please cross-reference. \n\nCausal Markov condition is invoked before (3) to factorize interventional distribution, but this connection between different environments and them essentially representing different interventions is not made before. I think this will be very helpful if the authors enable the connection. \n\n%%% After the Author Responses and Paper Updates %%%\n\nI would like to thank the authors for very seriously considering my recommendations and genuinely attempting to implement many of them, even in their proofs. I do think their updates made the paper stronger in general. A minor general note is that many of the newly edited sections have typos and could benefit from proof-reading. Following are my final remarks:\n\nAfter going through the paper again, I realized a step in the proofs which indicates an extra assumption that is not mentioned in the main paper. The proof in lines (14,15) of Section 7 (supplementary material) seems to assume that the exogenous variable has a fixed distribution, i.e., despite two different models inducing the same observed distribution having two different functions f_x and \\tilde{f_x}, their exogenous noise \\epsilon_x must have identical distributions for the proof to go through. Similar steps are used for the exogenous noise of Y as well. These steps can only be explained by the very strong assumption that the exogenous noise term of every variable has a fixed distribution across different causal models. Such an assumption is not mentioned anywhere in the paper as far as I can see - this has to be definitely addressed. Moreover, I do not think the identifiability result is very useful when obtained under such a strong assumption. Unfortunately, I cannot recommend acceptance due to this. But I encourage the authors to pursue this direction and seek out ways to relax this condition.\n\nI had brought up the point that C is not used to induce correlation within a dataset, which takes away from the essence of the practical problem they are trying to address. In order to address this, i.e., to be able to handle the confounding within each dataset, the authors added an extra condition on the effect of confounding: They assume that once p_1(x,y)=p_2(x,y), this implies p_1(x,y|c)=p_2(x,y|c) for all c. This extra assumption allows the authors to use the machinery they developed as is with this additional argument.\n\nUnfortunately, this assumption, much like the others, is also presented in between the lines. I think it will be really helpful if the authors could explicitly write down their assumptions in a theorem environment (\n\\begin{assumption} ... \\end{assumption}\n) and make them very explicit rather than only within the theorems or in-text: Assumption 1: ANM Assumption 2: Exponential family Assumption 3: ... etc.\n\nThis relates to the implicitness of another key assumption made in the paper: \\Gamma matrix is assumed to be full rank. This intuitively suggests that the experimental conditions are sufficiently different. But this is an algebraic statement and is hard to interpret. I would recommend the authors to think about how to interpret this condition, i.e., assess how it impacts the conditional distributions - exponential assumption allows them to make an algebraic assumption here, rather than probabilistic; but a probabilistic interpretation would be more intuitive.\n\nSome of the typos that I can see: \"and functions the prior of distribution of C\"->\"and functions as the prior of distribution of C\"\n\ntypo in (62): \"R\"->\"r\"\n\n\"We generalize the identifiable result in theorem 4.3 \"->\"We generalize the identifiability result in theorem 4.3 \"\n\nMany of the arXiv citations are actually published in various venues, please go through the bibliography and update.\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "ehMiJJCn6W2", "reviewer": "AnonReviewer4", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "# Summary of the review\n- The authors make a strong case regarding their approach to causally robust supervised learning. Their contributions are thorough and carefully situated in the relevant subfields. I believe the authors' research will be of interest to the relevant research communities.\n\n# Summary of the paper\n- The authors suggest a causal latent variable model for high dimensional supervised data, such as images (input, $X$) with labels (output, $Y$), which can be used to obtain robust predictions as well as learn disentangled representations for high dimensional input given heterogenuous observational data, collected in different environments $e \\in \\mathcal{E}$. In their generative model, the authors distinguish two types of latent variables. Continuing the image classification example, a group of latent variables (called $S$ in the paper) correspond to the object level abstractions such as shape and contour, that causally determine the label $Y$ as well as the image $X$. The other group of latent variables (called $Z$ in the paper) correspond to contextual information such as pose or lighting, that causally only effect $X$ but also can be spuriously correlated with $Y$ (see below). Accordingly, the foundation for the authors' model is the robust / invariant / transferable functional relationships between the latent variables and observed variables, $f_x: (S, Z) \\to X$ and $f_y: S \\to Y$ that are constant across $e$, whereas $P^e(S, Z)$ is subject to change given the environment $e \\in \\mathcal{E}$. The authors name $f_x$ and $f_y$ Causal Invariant Mechanisms (CIMe). The model is completed by an observed confounder $C$ that causally affects both $S$ and $Z$, and corresponds to information determining the distribution of $P^e(S,Z)$. $C$ enables the backdoor path that spuriously relates $Z$ and $Y$. When $C$ is inaccessible, the authors assume that a binary index $D$ is accessible instead, which serves as a label for the environment which observations were made. This completes the Latent Causal Invariance Model (LaCIM).\n- The overall aim of the authors approach is to formulate the learning problem such that when a function mapping input to output $f: X \\to Y$ is to be learnt, this learned function is not biased by the spurious correlations between $X$ and $Y$ that likely would not generalize to new environments or tasks. \n- Based on their model LaCIM the authors:\n    - demonstrate the $\\sim_p$ identifiability of $f_x$ and $f_y$ \n    - propose variational inference methods to approximate these functions given $C$ or $D$\n    - conduct experiments demonstrating how their method performs on out-of-distribution (OOD) generalization tasks, interpretability tasks, and adversarial robustness tasks.\n\n# Strenghts of the paper\n- The authors model accommodates different types of high-level latent variables that might affect the observations causally or induce spurious relationships between them. The model also makes clear what relationships are expected to be invariant due to being causal mechanisms, an how spurious relationships can be blocked by conditioning on environment properties (if accessible), and. The authors also provide a variant of their model given only the label/index of the environments are accessible.\n- Building on the work by Khemakhem et al. (2020), the authors prove the identifiability of the inveriant causal relationships between latent and observed variables, given certain conditions on the number and diversity of environments observed. \n- They provide a variational-autoencoder algorithm for training for the case when confounders are observed and unobserved. \n- They provide extensive experiments to 1- verify their theoretical claims, 2- show that their model shows the desired robust prediction properties under different settings 3- demonstrate their method estimates meaningful latent variables.\n- The authors make extensive comparisons with the literature both methodologically and experimentally. Actually they provide a more thorough review in the supplementary material. They do not mention this additional review in the main text, which I think they should.\n- Though being dense, the paper is easy to follow (save for occasional typos, see below).\n \n# Questions and potential improvements\n- As the authors note, their identifiability conditions rely on the existence of a virtually unknowable number of different domains. How should we expect the performance of their algorithms to degrade when the number of domains are lower than this unknown number? An initial idea for this could be obtained through a simulation study, where the dimensions of the latent variables could be controlled.\n- In the experiments, the LaCIM-D variant of the algorithm seems to perform better in the cases where the back-door paths are less likely to be exhaustively blocked. Given finding such a set of variables should be even more difficult in realistic scenarios, do the authors see a realistic use for LaCIM-C? If so, how would they compare the two algorithms in terms of their applicability in different tasks? Do they see a case when the LaCIM-D would be disadvantageous?\n- The authors mention that given insufficient number of environments, the environments/datasets at hand could be e.g. clustered and treated as heterogenuous (as does e.g. Buhlmann 2018). How successful would the authors expect such an approach to be? Though such an approach could be helpful to some extent, it is unlikely that any homogenous data with $\\mid \\mathcal{E} \\mid = 1$ can be successfully treated as heterogenous after some processing.\n- The authors mention the diversity condition at Pg. 5 without having introduced it before. The authors might want to clarify this aspect of their exposition.\n\n## Minor comments, typos\n- Section 3: The authors' back-door path explanation is hard to understand, they might want to replace it with something to the effect of: \"an unblocked, undirected path between $V_a$ and $V_b$ that has edges going into $V_a$ and $V_b$\".\n- Section 3: $d^e \\in \\mathbb{R}^m$ could be replaced by $d^e \\in {0, 1}^m$.\n- Section 3 and later: The abbreviation for structural causal models could be capitalized: SCM\n- Section 4: \"and priori\", \"less and equal\"\n- Section 4.1: \"the confounder $C$ blocks the back-door path from $Z$ to $Y$, making $Z$ and $Y$ spuriously correlated\": the authors might want to restate this part, when the path is \"blocked\" ($C$ is observed/conditioned on) $Z$ and $Y$ are no longer spuriously correlated.\n- Section 4.1: By \"brute force data fitting\" the authors must mean an algorithm that tries to directly learn a functional relationship between $X$ and $Y$. They could express this part a little more clearly.\n- Section 5: title \"Experiment\"s\n- The word \"condition\" in Theorem 4.3 is capitalized, I am assuming by mistake.", "extra_scores": {}, "extra_text": {}}, {"review_id": "be-W-K8dA4-", "reviewer": "AnonReviewer3", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "### Summary\n\nIn this manuscript, the authors introduce a method for supervised learning in an out-of-distribution (o.o.d.) setting.\n\nIn particular, the authors consider the case where labelled data pairs $(x, y) ~ p^{e}(x, y)$ are available for training sampled from multiple environments / interventions $e \\in \\epsilon_{\\mathrm{train}}$ and the goal is to learn a predictive model $\\hat{y} = f(x)$ that will generalize to data sampled from environments $e \\in \\epsilon_{\\mathrm{test}}$ not present in the training set.\n\nIn a nutshell, the authors propose to tackle this problem by fitting a deep generative model, in their case, a variational autoencoder, that models both features $x$ and targets $y$ as being jointly generated by a set of latent factors $(z, s)$. Crucially, their model removes the connection between the latent factors $z$ and the target $y$, encouraging disentanglement of causative (ideally, captured by $s$) and non-causative (ideally, captured by $z$) factors at the latent variable level. In their model, variation across environments is limited to the prior over the latent factors $(z, s)$, which is modelled conditional on the confounded $c$ (when known) or environment/domain indicator $d$ (when the true confounder is unknown) as $p(z, s | c) = p(z | c)p(s | c)$ (resp. for $d$).\n\nFrom a theoretical standpoint, the model builds heavily on recent advances on nonlinear independent component analysis (ICA), such as Hyvärinen et al. 2019 and Khemakhem et al. 2020. To frame the problem in the language of that line of work, the authors introduce additional assumptions, some of which include:\n(1) The (causal) mechanisms generating $x$ from $(z, s)$ and $y$ from $s$ must be invariant across training environments.\n(2) Noise in those mechanisms must be additive.\n(3) Additionally, assumptions analogous to those in e.g. Theorem 1 of Khemakhem et al. 2020.\n\nProvided all those assumptions apply, the authors are able to adapt results in that line of work to their proposed model and prove similar identifiability results for their new setting.\n\nFinally, the authors test their approach on a simulated toy dataset and three real-world datasets (a variation of the colored MNIST task introduced by Arjovsky et al. 2019, the NICO dataset from He et al. 2019 and a medical imaging classification task from the Alzheimer’s Disease Neuroimaging Initiative, ADNI). Results suggest that the proposed approach slightly outperforms basilines such as IRM (Arjovsky et al. 2019) or DANN (Ganin et al. 2016), among others.\n\n\n### High-level assessment\n\nOut-of-distribution generalization is, in my opinion, a topic of utmost relevance for the machine learning community.\n\nFrom a methodological perspective, the authors here propose a novel application of existing ideas, models and identifiability results (e.g. Hyvärinen et al. 2019 and Khemakhem et al. 2020, among others) which they adapted to disentangle latent factors that jointly affect $x$ and $y$ from those which only affect $x$ while being able to borrow most of the theoretical machinery from that line of work to provide similar identifiability results. Because of this, I believe the contribution to be a sound, sufficiently innovative step worthy of publication, albeit arguably leaning on the incremental side.\n\nAll in all, I lean slightly towards recommending acceptance of the manuscript. Nonetheless, I also believe the paper has substantial room for improvement in key areas such as (i) soundness and depth of the experimental results and (ii) clarity of exposition, description and writing style. Those shortcomings, coupled to the incremental nature of the contribution, prevent me, for the moment, from giving a more enthusiastic endorsement of the manuscript for publication.\n\n### Major points / suggestions for improvement\n\n1. To the best of my knowledge, the manuscript is currently lacking key information to allow a reader to reproduce the results. In particular, I could not find any clear description of the architectures and training procedures used for all baselines and how these differ from (1) their original publications and/or (2) the architecture and training procedures of the proposed approach.\n\n2. Related to the previous point, I believe it would be essential that the authors disentangle the effect of model architecture / model capacity from the actual methodological contribution. Because of the aforementioned issue, it is at this point not possible for me to assess whether the experiments were carried in such a way that all baselines have a similar capacity or whether a part of the performance improvements achieved by the proposed approach could be explained away by the model having more parameters and/or expliciting using for prediction additional variables (e.g. the domain indicators) than some of the baselines.\n\n3. I believe the methods in Isle et al. 2020 and Teshima et al. 2020 are sufficiently relevant and should be included as additional baselines to better assess the practical impact of the differences between the proposed approach and those methods.\n\n4. In my opinion, the manuscript is at the present moment not particularly clear. For example, in sections such as 4.3, I believe that the writing does not make it sufficiently clear which statements are supposed to be modelling choices / approximations / compromises for computational tractability and which statements follow mathematically without loss of generality. All in all, I would also recommend having the manuscript proof-read for English style and grammar issues.", "extra_scores": {}, "extra_text": {}}, {"review_id": "aMdBiXX42P7", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a VAE based model for learning latent causal factors given data from multiple domains. Similar to [Kingma and Hyv¨arinen, 2020], it utilizes additional labels as supervision signals and learns the model using a Bayesian optimization approach given a fixed hypothetical causal structure. The identifiability is obtained by assuming the casual mechanism to be domain invariant, which is partially supported by some empirical experiments.\nI have three concerns for the current version of the paper.\n1.\tDirectly mitigating the identifiability result of [Kingma and Hyv¨arinen, 2020] to this model seems to be inappropriate. The result of [Kingma and Hyv¨arinen, 2020] shows that the sufficient statistics of the latent code are recoverable. However, it does not mean the causal structure is unique: additional transformations may be allowed to be applied to the adjacency matrix. As a result, it is in question whether your causal mode learns the correct factors under given structure.\n2.\tThe symbols are a bit confusing. Some important concepts stay intuitive and lack rigorous mathematical definitions. For example, “output-causative” “cross-domain causal effect” stability measure etc should be defined. A clear table listing the symbols and its meaning would be helpful.\n3.\tThe empirical results of table 1 do not fully support your conclusion “true causal factors are learnable”. Simply computing a MCC to the original factors is not enough to me. Some experiments, like examining the vulnerability and performance of the system under the condition that the latent factors are controlled or intervened, for the claimed “invariant causal model” are better to be included to convince readers.\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This work introduces a method for supervised learning that takes a data-generating process into account. While the paper proposes an interesting approach to learning a causally invariant model, the reviewers had several concerns about the proposed method. I thank the authors for having the paper revised, addressing the reviewers' comments. However, there are still legitimate issues unresolved about the specific theoretical results and assumptions made throughout the work.  I share similar concerns, and, therefore, recommend rejection. Still, I would like to encourage the authors to address the reviewers' problems in the paper's next iteration. 

Decision: Reject