Controllable Generative Modeling via Causal Reasoning
Authors: Joey Bose
Ricardo Pio Monti
Aditya Grover
Venue: Accepted by TMLR ()

Abstract: Deep latent variable generative models excel at generating complex, high-dimensional data, often exhibiting impressive generalization beyond the training distribution. However, many such models in use today are black-boxes trained on large unlabelled datasets with statistical objectives and lack an interpretable understanding of the latent space required for controlling the generative process. 
We propose CAGE, a framework for controllable generation in latent variable models based on causal reasoning.
Given a pair of attributes, CAGE infers the implicit cause-effect relationships between these attributes as induced by a deep generative model. This is achieved by defining and estimating a novel notion of unit-level causal effects in the latent space of the generative model.
Thereafter, we use the inferred cause-effect relationships to design a novel strategy for controllable generation based on counterfactual sampling. Through a series of large-scale synthetic and human evaluations, we demonstrate that generating counterfactual samples which respect the underlying causal relationships inferred via CAGE leads to subjectively more realistic images.

Published in Transactions on Machine Learning Research (12/2022)
Controllable Generative Modeling via Causal Reasoning
Avishck Jocy Bosc McGill University and Mila
jocy. bose @osil
Ricardo Pio Monti Meta
rpmonti@@eta.ccm
Aditya Grover UCLA
adilyag Bes, ucla edu
Reviewed on OpenReview: https: openreview. net/ forun? id= SaqISS_ L9e5
Abstract
Deep latent variable generative models excel at generating complex, high-dimensional data, often exhibiting impressive generalization beyond the training distribution. However, many such models in use today are black- trained on large unlabelled datasets with statistical objectives and lack an interpretable understanding of the latent space required for controlling the generative process. We propose CAGE, a framework for controllable generation in latent variable models based on causal reasoning. Given & pair of attributes, CAGE infers the implicit cause-effect relationships between these attributes as induced by a deep generative model. This achieved by defining and estimating a novel notion of unit-level causal effects in the latent space of the generative model. Thereafter, we use the inferred causc-effoct relationships to design a novel strategy for controllable generation based counterfactual sampling. Through series of large-scale synthetic and human evaluations, demonstrate that generating counterfactual samples which respect the underlying causal relationships inferred via CAGE leads to subjectively more realistic images.
1 Introduction
Data generation using generative models is one of the fastest growing usecases of machine learning, with suc- cess storics across the artificial intelligence spectrum, from vision (Karras et al., 2020) and language (Brown et al., 2020) to scientific discovery (Wu et al., 2021) and sequential decision making (Chen et al., 2021). How- ever, translating these successes into deployable applications high-stake domains requires an interpretable understanding of the generative process for controllable generation.
Causality presents natural framework to study and understand any generative process (Xu et al., 2020), and as result, we believe it presents an crucial ingredient for improving controllability of generative models. Specifically, we are interested in using causality for interpreting and controlling pretrained deep latent variable generative models (DLVGM), such as generative adversarial networks (GANs) Goodfellow et al. (2014) and variational autoencoders (VAFs) Kingma & Welling (2013). Prior work in controllable gencration for such models can be categorized as either augmenting the loss function with disentanglement objectives, or post-hoc control approaches based on latent space explorations. The former class of methods e.g., Chen et al. (2016); Higgins et al. (2016) often introduce undesirable trade-offs with the standard generative modeling objectives (Locatello et al., 2019) and moreover, cannot be readily applied to the fast- growing zoo of state-of-theeart pretrained models in use today by the very virtue of needing to alter the original training hodooogy
We focus on post-hoc control of DLVGMs, where control is specified w.r.t. small of interpretable meta- attributes of interest, such as gender or hair color for human faces. Here, the predominant approach for controllable generation is to identify directions of variation in the latent space for each meta-attribute and

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Figure 1: Counterfactual generation of Blonde Males using a deep generative model (Karras et al., 2020). CAGE infers the implicit causal structure as Gender -> Blonde Hair. Top: The left panel contains the original image of blond female, other panels display counterfactual samples where gender (treatment) is changed to male. Bottom: Equivalent visualizations starting from non-blond male (left panel) and counterfactually changing hair color (treatment) to blond. Generating counterfactuals that respect the implicit causal structure yiclds subjectively better samples. This corresponds to the same-dcstination sctting and is further outlined and empirically investigated in $4.1.
manipulate the latent code for an input image along these directions to achieve the desired control, e.g., Shen et al. (2020); Härkönen et al. (2020); Voynov & Babenko (2020); Khrulkov al. (2021). While this procedure may work well in practice, it sidesteps more fundamental question: what are the interactions between these attributes in the latent space of deep generative model? Certainly, real world data generation processes have rich interplay that go beyond mere correlation borrelationnbelweenn between such mela-altributes.. As a result, in practical scenarios, we do not expect that the generative model perceives these meta-attributes as independent and hence, inferring the cause-effect relationships between these meta-attributes can enhance our understanding and controllability of deep generative models.
We propose CAGE, framework for inferring implicit causc-effect relationships in doep generative models. Our framework in inspired from the potential outcomes framework of the Neyman-Rubin causal model (Neyman, 1923; Rubin, 1974) and defines new notion of generative average treatment effects (GATE). A fundamental problem of causal inference is that, by construction, we cannot observe the potential outcomes under all treatments (Holland, 1986) i.c., at any given time, any individual can be assigned only one treatment ka.k.a. the factual) but not both. However, when studying treatment effects for deep generative models, CAGE exploits the generative nature of such models to overcome this challenge and explicitly generate the counterfactual via standard latent space manipulation strategy. Thercafter, use outcome attributc classifier to quantify the difference in outcomes for the factual and counterfactual generations and thus, estimate GATE. We further leverage the generative treatment effects to define a natural measure of causal direction over pair of meta-attributes (e.g., gender and hair color) associated with a deep generative model. We refer to our overall framework for causal probing of deep generative models as CAGE.
We study the consistency of the inferred causal directions for two high-dimensional image datasets, Mor- phoMNIST (Pawlowski al., 2020) and CelebAHQ (Karras et al., 2017), with known or biologically guessed prior cause-effect relationships. Finally, we use these inferred causal directions for augmenting post-hoc strategies for controllable generation. Specifically, source manipulation, wherein we wish to select which attribute of an input image should be manipulated for maximizing image fidelity, and (b) destination manipulation, where we wish to select inpul images that can be best manipulated to satisfy a target control attribute. We show that knowledge the causal direc- tions inferred via CAGE significantly improves the generation quality over bascline controllable generation strategies on the CelebAHQ dataset. In particular, we observe average improvements of 66.7% and 83.3% source and destination manipulation scenarios respectively as measured by extensive human evaluations.
consider two scenarios for controllable generation: (a)
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1.1 Related Work and Contributions
Classical Causal Discovery
CAGE
Causal VAE/GAN
Implicit Causal
Explicit Causal Relationship
True Causal
Observational Data
Observational Data
Relationship
Relationship
Inferring
Latent Encoding
Inferring Data's 's Causal
G's Causal
Process
Generative Process
Generative
Generative Model G
Model
Observational Data
Legend
True Causal Structure
Counterfactual
Counterfactual
Inferred Causal Structure
Data
Data
Known Causal Structure
Figure 2: A visual summary of our proposed framework (middle panel) and how it relates to traditional causal discovery (left panel) and Causal VAE/GAN (right panel) respectively. Traditional causal discovery uses observational data to infer causal relationships over the true data generating mechanism. In contrast, the goal of CAGE is to infer implicit causal relationships within a deep generative model. We note that the implicit causal relationships within a generator need not match the causal relationships over the data usod to train such a model. Finally, methods such as Causal VAE/GAN train gencrative models under a prescribed causal structure, whereas the goal of our proposed framework is reliably uncover implicit causal structure over any generative model.
Our proposed framework lies at the intersection of causal inference and deep generative modeling. While CAGE borrows important concepts and ideas from each field, we also make a series of distinctive contributions to each as outlined in Figure
Traditional causal discovery methods are depicted in the left panel of Figure 2. These methods operate over observational data and seek to infer the underlying causal relationship which generated the data. The intrinsic appeal of causal discovery methods is that they are able to infer the latent causal structure of complex datasets, thus providing an explicit description of the underlying generative mechanism. However, in order to achieve this, a series of (potentially restrictive) assumptions are typically required, such as assuming causal relationships are linear (Shimizu et al., 2006; Hyvärinen & Smith, 2013; Poters & Bühlmann, 2014; Peters et al., 2016) or follow lightly parameterized non-linear relationships such as the additive noise model of Hoyer et al. (2008). CAGE, depicted in the middle panel, borrows ideas from causal discovery in order infer latent causal structure within a deep generative model, G. This is fundamentally different to learning the causal structure of the data itself, which is the goal of causal discovery. Specifically, this is because the implicit causal structure within deep generative model will only respect the true causal structure of the data under strict conditions (Klindt et al., 2020; Khemakhem et al., 2020; Gresele et al., 2021; Monti et al., 2020), which do not apply for a wide range of popular deep generative models, for example many state-of-the-art GANs or Normalizing Flows. In the absence of such guarantees, the implicit causal structure within G can be arbitrarily different from the cansal relationships which generated the observed data.
Finally, the right panel of Figure 2 highlights related avenue of research which aims to train deep generative models under prescribed causal structure. Examples include CausalGAN, CausalVAE, and counterfactual gencrative networks (Kocaoglu et al., 2017; Yang et al., 2021; Saucr & Geiger, 2021). Despite sharing some similarities, these methods are based on instilling a given causal structure within a generative model, G, by changing the objective function. In stark contrast, the objective of CAGE is to infer such causal structure over any generator, G without introducing any additional constraints or tradc-offs whilst training since we only look to understand the implicit causal associations after training. As such, CAGE is applicable any deep generative model, and provide examples with VAFs, Normalizing Flows, and GANs in this work.
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2 Background and Preliminaries
2.1 The potential outcomes framework
We briefly overview the potential outcomes framework of Neyman (1923) and Rubin (1974), upon which we base our proposed method the next section. In its simplest form, this framework considers the causal effect of assigning a binary treatment, T' e {0, 1}. Such a framework posits the existence of potential outcomes for the ith individual both whilst receiving treatment, (i), and when treatment is withheld, Yr=o(i). The causal effect for given individual, i, is defined as the difference in these potential outcomes.
The "fundamental problem of causal inference" is that whilst we may posit the existence of both potential outcomes, (1) and (i)) we only ever observe the outcome under one treatment (Holland, 1986). For this reason, causal inference is often performed at the population level, by considering quantitics such as the average treatment effect (ATE):
E, (1)=1(1) (r))
(1)
Equation (1) is the expected difference in potential outcomes of individuals receiving treat T and = 0. Under standard assumptions of unconfoundedness, positivity, and stable unit treatment. value assumption, the ATE is identifiable and can be estimated by a statistical estimate of the associational difference in expectations (Rubin, 1978). Please see Appendix $A for a light introduction causal inference it pertains to our framework CAGE.
2.2 Deep latent variable generative models
The goal of generative modeling is to learn an underlying data distribution given training dataset. For high-dimensional data, this can be done effectively via deep neural networks by learning mapping G' from unobserved factors of variation Z (i.c., latent variables) to observed variables X' (e.g., raw pixels). The mapping G is parameterized using deep neural networks, and can be learned via a variely of training objectives, such as adversarial training (e.g., generative adversarial networks (GAN; Goodfellow et al. (2014)), exact maximum likelihood estimation (e.g., Normalizing Flows (Dinh et al., 2017)), and variational Bayes (e.g.. Variational Autoencoders (VAE; Kingma & Welling (2013); Rezende & Mohamed (2015)). For our current work, can any such latent variable generative model. Our experiments will exhibit this generality by considering powerful families of flows (Song et al., 2019) and GANs (Karras et al., 2020).
In addition to G (i.e., the decoder), we will assume the existence of an encoder ENC X + Z that can map any observed datapoint to its latent vector representation. For instance, for flows, the encoder is given by the inverse of the generator mapping, ENC = G-1. For VAEs, the encoder is the variational postcrior distribution trained alongside G. For GANs, either the encoder can be trained during training (e.g.. Donahue et al. (2016); Dumoulin et al. (2016)), after training (e.g., Richardson al. (2021)), or implicitly learned by backpropogating through G to find latent vector E Z that best represents target input x, measured via reconstruction loss (e.g., Lipton & Tripathi (2017); Bora et al. (2017)) or pretrained domain classifier (e.g., (Karras et al., 2020)).
3 Causally aware controllable generation
In this work we approach the challenge of controllable generation from the perspective of causality. Tn particular, we posit the existence of implicit causal structure within deep generative model, G. We study this structure over metadata altributes, for example gender and hair color described in Figure 1, with the underlying motivation that an effect attribute is by construction leads to more fine-graines control of G whilst the converse is true for causal attributes. For example, deep generative model may internalize that changing the meta-attribule gender may causally materialize the meta-attribute pertaining to the presence or absence of a mustache, but not vice-versa. Our goal is therefore to understand the implicit causal structure if any encoded within a deep generative model by asking the following question:
In the context of a generator, G, does considering the counterfactual over a given "causol" attribute (e-g-, gender) lend to signifioont measurable changes in "effect" altribute (e.g., hair color)?
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Formally, we assume full (whitebox) access to a pretrained latent variable deep generative model G + X. We further assume finite dataset of n annotated observations D = - mi (i), m2(1)) Here, the examples (x.( (NN} are drawn from the model's 's training distribution and we annotate each example xD(i) with binary metadata for two variables m1(1), m2 E 10. These two annotated variables can be drawn from a larger unobserved (w.r.t. G) causal process. For example, xD(i) could correspond to a high- dimensional image and m1 mi) and m²(i) could be any two attributes of the image, such as gender expression and hair color in Figure 1. Apriori, we do not know which of the variables (if any) is cause or effect but seek to identify their relationship within G.
We note that any causal conclusions obtained under our proposed framework only reflect implicit properties of a generator G, as opposed to properties of the data on which the generator was trained. That is to say, it is perfectly permitted, for G to have learned a causal relationship that is inconsistent with the observational data used for training. As we shall see in $4, shedding light on the implicit causal relationship learned by G is abundantly useful for the purposes of controllable generation. As result, this work-by design-is in sharp cont rast, with the majorily of causal inference literature that focuses on identifying the generative mechanisms of fixed dataset opposed to our focus on the generative model itself.
3.1 Overview of the CAGE framework
We propose to extend the potential outcomes framework described in $2.1 to accommodate DIVGMs. Our approach is premised on computing generative average treatment effect for a candidate causal attribute, me. on a given effect attribute, me, for a given generator, G. Unlike traditional approaches to treatment effect estimation, the key observation in CAGE is that we can simulate the counterfactuals under a given treatment using the generative model G. In practice, this amounts to traversing the latent space of G using pre-specified strategy. For instance, we may move orthogonally to learned mela-attribute hyperplanes or even consider a more exotic non-lineat latent manipulation strategy. It is important re-emphasize here the distinction between between simulating counterfactuals with respect to the true causal relationships in the data versus CAGE which uses the generative models own latent space to perform counterfactual queries. In order to investigate the cause-effect relationship between two meta-attributes eeg. gender and hair color, let us initially arbitrarily assign gender to be the treatment mi = me (cause) and hair color to be the outcome my = mm (effect). We note this choice is arbitrary and the overall algorithm will also consider the reverse assignment. (i.e., my = m. and m2 = me) and if needed, also reveal independence between the variables via a statistical test. Then we may simulate the effects of treatment assignments by manipulating the gender attribute in the latent space and measuring the corresponding change in outcomes. We next discuss our strategies for counterfactual manipulation. The overall pseudocode for CAGE is summarized in Algorithm 1.
3.2 Simulating Treatments via Latent Counterfactual Manipulations
In order to simulate the assignment of counterfactuals, we follow a three-step procedure.
Step 1: Our first step is to train a latent space classifier for the attributes using D. Here, we use the encoder
to first project every imput xi) in D to its latent encoding (D) = (i)() Given the latent encodings (i) and their annotations for m. and m. attributes, we train a (probabilistic) classifiers, Φe : 2 - [0, 1], to discriminate between binary causal and effect attribute values given latent representations.
Step 2: Since we are interested in casual discovery for the generative model, in the second step, we use G to create dataset of annotated examples by sampling k latent, vectors 7 = - (z()) from the prior of the generative model. Let X = - [x(1) = G(z(i)) denote the corresponding generations. Since sampling latents from the prior generative model are typically inexpensive, the size of our generated dataset can be fairly large. For cach generated example x(i) E X, we obtain its factual treatment by simply using the latent space classifier as mi) (x)(()())) > 0j.
Step 3: For a generated sample, x(i) X, assume without loss of generality, that m. = 0 is the factual treatment obtained via Step 2. We can derive analogous expressions for factual treatments m. 1, but we skip those for brevity. Finally, we define the counterfactual latent with respect to setting the treatment
For stochastic encoders such as in VAEs, we consider the mean of the encoding distribution.
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mc 1
mc
z(i)
h Outcome Hyperplane
m
z(i; do 1))
he Treatment Hyperplane
Figure 3: Linear Counterfactual Manipulation in the Latent Space of Generative Models in CAGE.
attribute m. as using the do operator as:
do(mm = 1)) = Tr(2(i), ax),
(2)
where Te latent space interpolation strategy E R4 is hyperparameter which is positive scalar that controls the extent to which manipulate. Note, that ne itself is very general and it could be simple linear interpolation if the latent space is linearly separable or more exotic approach that exploits the known geometric structure of the latent space. A counterfactual sample can subsequently be obtained by simply pushing forward the counterfactual latent through G:
x(i; do = 1)) ((z(ii =))).
3.3
Case Study: Linear Latent Counterfactuals
As concrete example, we consider instantiating our CAGE with counterfactuals crafted by lincarly in- terpolating across a linear decision boundary. We outline below how such latent linear counterfactuals are materialized in the following steps:
Step We first train a probabilistic linear classifiers de, d. d., 2 [0, 1] on to minimize the cross-entropy loss and thus, discriminate between binary causal and effect attribute values given latent encodings zD(i) and their annotations for m. and m.- Note, that will use &e in $3.4 quantify treatment effects. Concretely, by restricting ourselves to linear classifiers obtain a unit hyperplane that encodes the classification boundary. We use h, to denote the normal vector to the hyperplane for the meta-attribute m.-
Step 2: After gencrating our dataset X = (x(1) = G(z(1)) using G we can freely and cheaply apply our trained latent linear classifier to obtain factual treatments for each generated sample x(i) E X as mi( [d)(2(i)) > 0).
Step 3: As outlined in the general CAGE framework for generated sample, x(i) X denote m, 0 the factual treatment obtained via Step 2. and define the counterfactual latent-i. me 1 1--using the do operator as:
z(i: do(m. = 1)) = z(i) + aah.
(4)
where n. z(i) + oh., Equation corresponds moving linearly along the hyperplane normal, he which encodes attribute m. for simulating the counterfactual. Such approach is premised the assumption that

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Z is linearly separable with respect to semantically meaningful latent attribute, which was first observed and empirically validated for GANs (Denton et. al., 2019). A counterfactual sample can subsequently be obtained by simply pushing forward the counterfactual latent through G:
x(i; do = 1)) - G(z(i; do(m. 1))).
(5)
Figure 3 illustrates such a lincar counterfactual manipulation strategy.
3.4 Quantifying Treatment Effects via Proxy Classifiers
We now prescribe an estimate to quantify treatment effects using our trained latent classifiers At a high level we seek to measure the difference in treatment effects when manipulating one meta-attribute, mes on the other meta-attribute me via simulating counterfactuals. By considering both directions m. + me and me + me and leveraging the fact that causal treatments leave measurable imprints on effect variables and not vers-versa we can assign approxinale measure of the causal relationship by considering the difference between the two directions at population level.
In order to estimate treatment effects- c.g. the effect attribute, me, for the counterfactuals, To this end, we propose the use of binary classifiers trained to detect the presence of an effect variable. Intuitively, using Equation 5 and the corresponding geometric manipulations required to obtain interventional samples x(i; do = 1)) from a generative model, G we can simulate the impact of prescribing a treatment to an individual. We dub this the generative individual treatment effect (GITE) estimate TGITE which is defined
m. + m. we require an estimate of the presence or absence of
Terre(m, + m.) me (t; do(m. = 1)) - me do(m, 0)).
(6)
Similar to the ATE estimate eqn. 1 can define population level estimate by taking an expectation over the generated examples. We term this generative average treatment effect (GATE) of m. on m. defined as:
TCATE(M. + m2) = E. do(m. = 1)) me do(m = =0]]
(7)
To materially compute treatments effects required in equations 6 and 7 e.g. me do(m. 1)), we propose two candidate options that directly act on the counterfactual latent:
(1) Latent space classifiers are trained the latent space to detect the presence or absence of the effect variable in the counterfactual latent as:
m do 1)) Φ (z(i; do(m, = =))).
(8)
(2) Observed space classifiers are trained in the observation space and perform classifications after generating the samples corresponding to the counterfactual latent as:
me mi do = 1)) ve do = 1)))
(9)
As we can see from Equation 7, our GATE estimate differs from the conventional ATE estimates in Equa- tion 1 in the use of generative models and classifiers to explicitly generate the counterfactual. For reliable estimation, it is reasonable to expect some standard assumptions from causality to apply to even our setting, such as unconfoundedness, positivity, and SUTVA. In addition to these standard assumptions, the quality of the generative model and classifiers play an important role. In $3.3, for the generative model, we de- signed our counterfactual manipulation scheme assuming that the latent space is linearly separable in the attributes of interest. Further, implicit in framework is the assumption that the generative model can generalize outside the training set to include the support of the counterfactual distribution which is neoded for Equation& to provide a reliable effect estimate of the counterfactual. On the use of classifiers, we need accurate and calibrated classifiers (ideally, Bayes optimal) for both manipulating latent vectors in Equation and quantifying generative average treatment effects Equation 7. While impossible to test in in-full veracity- the above assumptions on real-world distributions, there is significant empirical evidence in the last. few years that suggest that modern deep generative models and classifiers indeed satisfy the above
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requirements for many practical usecases (Denton et al., 2019; Brown al., 2020). The theoretical properties and these necessary assumptions of CAGE are further discussed in $B.
ÂT as a measure of causal direction. Finally, we note that the preceding sections have focused on the challenge of quantifying if a variable, me: has a causal effect on second variable, me. This assumes knowledge of the causal ordering over variables As such, we can extend our approach to define measure of causal direction over any pair of variables by considering the difference in absolute GATE scores when either variable is considered the treatment:
AT = =G + me) | TGATE(me + +m))
(10)
The Ar score determines the magnitude of the difference outcomes when either attribute is prescribed as the trealment. Intuitively, when using the true causal attribule, according to G, the treatment we expect signature of the interventions to be large in magnitude. Conversely, the magnitude of disturbance measured when manipulating the true effect variable should be markedly smaller. Thus Ar >> 0 corroborates that the chosen causal ordering is indeed the one supported by G, while a Ar < 0 implies that the chosen causal ordering may in fact be reversed.
A null distribution for treatment effects. A further important consideration relates to how we might determine whether estimated GATE is statistically significant. To address this, we can obtain empirical sample of GATE scores under the null hypothesis where the altribule me has no causal association with me- A simple manner through which this can be obtained is via randomization and permutation testing. We can randomly shuffle the values of a causal attribute, Me, thereby removing any potential causal association. This corresponds to performing interventions over the latent space of which are effectively random projections. This process is repeated many times to obtaín an empirical distribution for a GATE under the null hypothesis. Given an empirical distribution of GATEs under the null, we can obtain p-value for our observed GATE.
Algorithm Obtaining CAGE scores ÂT
Inputs: Generative model G, Dataset D, Attributes m., m2, Counterfactual data size k me + [mm], m + [ma] (// Randomly Assign meta-attributes as cause me and effect me}
2: for (x))))mm.)), mf) in D do
3:
ZD(1) = ENc(xx(i)
D. = Dz (i). mi))
5: end for
de Bz.mmii)DD, [[(2, m. d.)] /// Train probabilistic meta-attribute classifier de)
7: de + B..m, [(2, me (i): 6.) /// Train probabilistic meta-attribute classifier de)
for E [k] do
9:
Z = z() ~ N(0, /// Sampling from G's prior}
xu G(x(1))
11: end for
12: for =(1) E Z do
13: domm. 1)) (z(z( a) 1// Counterfactual latent by applying the treatment
14: d(i:do(m, = 0)) = ne. (z(i), (((()))))) -a) (// Counterfactual latent by applying the treatment m. 0)
15: d(ime 1)) (e(2(i), a) (// Counterfactual latent by applying the treatment me 1}
16: do(m. = 0)) (2(),, -a) /// Counterfactual latent by applying the treatment m. = 0}
17:
(do(m. 1)) m. m. 1)) Um. (i; do m. 1)) d. (z(f; do 1)))}
18: me me(do (d = 0)) = me (do m. = 1)) (m fmaa( do m. = 0)) Φe (z(i; do(m: = 0)))
me
me (d 1)) ma (do( m 1)) U (m do mm 1)) de (z(i: = 1)))}
20: ma (do(m. = 0)) = m. (do (do(me me 0)) U do = 0))o. (z(1; do(m. = 0))}
end for
22: (TA - Me) = E, |mte do(me 1)) - me me do = 0))
23: (GATE(M, - m.) = E, (i; do(m. 1)) m. ; do dm = 0))
24: AT = [TCATTR(m m.)l (OOATE(m. ++ m.)l
25: return: AT

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4 Experiments
We evaluate the ability of CAGE towards inferring causal relationships in pretrained deep generative models on both synthetic and high- high-dimensional datasets such as high- reshutenonution images of faces. As outlined in $3.4 the estimate for the causal association uncovered by CAGE is predicated on a few key assumptions. Our experiments section is organized as follows: we first test the empirical caliber of the causal estimate returned by CAGE by designing a series of unit tests that explicitly probe each testable assumption. Concretely, through our experiments we seek to answer the following questions:
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Figure 4: Left-1: A scatterplot of MoG with points colored by the value of causal attribute, me. Left- 2: Histogram of GATE values under mull hypothesis, Ho. and vertical line denoting estimated GATE. Left-3 Generating counterfactuals of MorphoMNIST digits by reducing the thickness of each digit without changing the average intensity. Lcft-4,5 Progressively decrcasing digit thickness leads to background pixels being illuminated as the digits get thinner.
(Q1) Generator Causal Discovery: Can CAGE recover the true causal structure on synthetic datasets? (Q2) Linear Separability: To what degree the latent space linearly separable in meta-attributes of interest?
(Q3) Statistical Significance: How statistically significant is GATE under the mull distribution?
Our empirical result summarizing these core experimental questions is presented in table 3. Next, we turn to our main experimental findings in $4.1 on controllable generation leveraging causal insights supplied by CAGE in human studies under two settings that we term as same source or same destination manipulation.
Q1. Generator Causal Discovery
Mixture of Gaussians. We consider two synthetic datasets with known causal relationships within the data. In our first experiment we consider the following toy setup: data is generated according to two distinct mixture distributions as shown in the left panel of Figure 4 where each color denotes a mixture. We define the causal altribule, ma; to be which mixture each sample is drawn from (i.e., from the mixture of 8 Gaussians, in blue, or the mixture of Gaussians, in orange). We further define an effect variable, me, be defined as one, if the mean of the (z. y)-coordinates is less than 2.5 and zero otherwise. In this manner, we see that me has causal influence over me by determining which mixture each sample is drawn from. We employ Masked Autoregressive Flow (Papamakarios et al., 2017) as the deep generative model. The middle panel of Figure 4 visualises the distribution of TGATE under the null as well as the estimated GATE, TGATE (m. + me), which is significantly larger in magnitude while the right plots AT as a function of a, positive values corrobrate that m. -> me.
MorphoMNNIS.. In our second synthetic experiment we evaluate the causal relationships learned by power- ful deep normalizing flow models a synthetic dataset based MNIST dubbed MorphoMNIST (Pawlowski et al., 2020). Here, the original MNIST digits are modified to respect causal structure whereby the stroke thickness of the digits is a cause to the brightness (i.c. T + 1). Specifically, thicker digits are brighter while thinner digits are dimmer under the prescribed causal graph. We train a powerful normalizing flow in Mintnet (Song et al., 2019) and interrogate the direction of causality any between thickness and intensity by first projecting all test set samples to their corresponding latent vectors. Figure shows qualita- tive examples of counterfactuals, where we observe that the generated counterfactuals reduce the thickness while maintaining the average intensity. Specifically, the model generates digits that contain holes but to
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compensate for the drop in intensity the model instead increases the intensity of the background pixels. Such generations are in line with expectations as Ar > 0 (see $C.2) indicates that the model has learned that T+ I and as such manipulating the causal variable propagates influence to the effect variable manifesting itself as an increase in background intensity.
Q2. Linear Separability
We test the linear separability of latent classifiers on a powerful generative model StyleGAN2 (Karras ct al., 2020) which is pretrained on the FlickrFacesHQ (Karras et al., 2019) but finetuned using 10% of CelebaHQ which contains annotated meta-attributes such as Gender and Haircolor for photo-rcalistic facial images. For latent linear classifiers we use an SVM based classifier as found in the widely used Sci-kit learn library (Pedregosa et al., 2011). In Table 1 we report the test accuracy of our latent SVM based classifier and observe that many attributes of interest exhibit high degrees of linear separability. However, certain attributes such as Pointy Nose and Brown Hair are less lincarly separable and as a result the causal relationships inferred by CAGE are less reliable. In particular, meta-al tributes such as Gender and Goatee are quile linearly separable while meta-attributes such Brown Hair and Pointy Nose are less linearly separable and any causal estimate returned by CAGE is less reliable.
Q3. Statistical Significance
We now perform causal discovery over a pair of mota-attributes within the latent space of a pretrained StyleGAN2 G. In Table 3, we report the direction of causality found using Ar on pairs that pass the linear separability unit test. above. We observe that AT is consistently capable of diagnosing the causal relationship implicit to G. For example, for the causal relationship: Gender - Mustache, we observe counterfactull samples where gender (top row) is manipulated-icc. from male to female-the mustache altribute also changes. In contrast, when the effect variable (botiom row) is manipulated there is a reduced impact on the perceived gender. For each pair, we also generate counterfactual samples by manipulating one attribute and qualitatively observing downstream effects on the other. To ascertain that our perceived scores are significant we plot a histogram AT scores under 100 random latent projections which allows us reject the mull hypothesis if the true Ar (black line) is significant (e.g. p-val 0.05).
Correlation vs. Causation. To determine the extent to which correlated attributes affect CAGE, we com-- pute the Maximum Mean Discrepancy (MMD) metric (Gretton et al., 2012) between the latent conditional and counterfactual distributions as a function of the interpolant strength in the final column in Table 3. Concretely, we filter the generated samples by their attribute values (as determined via a pretrained latent space classifier) as an estimate for the conditional p(z/m, = 1) and compare it with p(z]do(m, = 1)) obtained by generating counterfactuals. We find that in all cases the MMD is large, signifying that the conditional and counterfactual distributions are quite different. For example, the attributes Gender and Smiling are highly correlated in the CelebAHQ dataset (14.2% males vs. 32.8% females are smiling) and in the gener- ated examples (10.0% males vs. 36.0% females are smiling, with labels determined by pretrained classifier) but CAGE infers the two attributes to be independent as substantiated with the generated counterfactuals
Latent Causal Discovery Baselines. As noted in Section 1.1, the primary objective of CAGE is to infer causal structuve over metadata attributes, m,, encoded a generator, G, as opposed to over observations, X. As such, CAGE looks solve fundamentally different problem from traditional causal discovery methods. Despite these differences (Shimizu et al., 2006) 2. Additive Noise Model (Hoyer et al,, 2008), and 3. DAGs with No Tears (Zhang et al., 2018). We adapt such basclines so as to be comparable with CAGE by having them operate on the class probabilities learned by the proxy classifiers described in $3.4. We note that this is required both for a meaningful comparison with CAGE, but also as a result of the poor scaling properties of many causal discovery algorithms which cannot be directly applied over image data. In general, we find little agrocment among the basclines which is expected as they each make different assumptions and as result this inhibits their easy application to downstream causally informed controllable generation.
nonetheless include series of latent causal discovery baselines: 1. LINGAM
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4.1 Sample Quality of Counterfactual Generation
We now investigate the visual fidelity of counterfactual samples crafted by manipulating the causal variable or the effect variable as uncovered through CAGE in CelebaHQ.
Experiment Design. For cach pair of meta-attributes, we consider two experimental setups: (1) same source and (2) same destination manipulation. In the same source setting, the starting point for generation is a source image (e.g., male with no mustache) which is then manipulated twice by the causal variable and the effect variable respectively to produce two different images (e.g., female without a mustache and male with a mustache). In the same destination setting, reverse the previous process by attempting to arrive at the same set of meta-attributes for two different real starting images after manipulation. For example, let's say we wish to generate images of men without mustaches. Here, we may start from a female with no mustache and change its gender, or start from a male with a mustache and remove the mustache. An illustration of these settings can be found in Fig. 8, For each attribute pair and both of the above settings, we generate 100 pairs of images which are evaluated using human annotators solicited through Amazon Mechanical Turk to rale images on the basis of their realism. See $C for details and $E for example samples.
Results. We report findings in Table 2 where we find in the same source setting generating counterfac- tuals by modifying the causal variable leads to subjectively better visual samples as determined by human annotators in all but two seltings (e.g. Gender + Bald). Conversely, in the same destination setting mod- ifying the effect variable is preferable to the causal one when evaluating for sample quality. These results highlight that depending on how G used downstream e.g. a same source setting like data augmentation for rare classes a higher degree of visual fidelity can through controllable generation using a causal attribute. On the other hand, when doing conditional generation in tasks such as in-painting it is more desirable to modify the effect variable to produce higher quality visual samples.
Attributes
SVM Classifier Accuracy
CAGE direction
Same Source t
Same Dest.
Gender
90.0
Gender + Mustache
231/300"
51/300*
Mustache
79.7
Gender - Bald
156/300
138/300
Bald
70.8
Goatce
83.2
Gender + Goatee
254/300*
91/300"
Rosy Cheeks
72.8
Age < Bald
281/300*
56/300*
Age
75.1
Gender + Rosy Cheeks
183/300*
97/300*
Pointy Nose
64.6
Brown Hair
65.4
Gender + Blond Hair
157/300
126/300*
Smiling
79.6
Blond Hair
84.8
Table 2: Counterfactual Sample Quality Results. Each cell rep- resents the percentage of annotators that preferred the counter- factual generated by modifying the cansal variable in each pair of attributes. We add a to denote statistical significance at level a = 0.05, after Bonferroni correction.
Table 1: Latent Space SVM Classifier Test Accuracies on StyleGAN2 finetuned on CelebaHIQ.
5 Conclusion
We proposed CAGE, a framework for inferring causo-effect relationships within the latent space of deep generative model via geometric manipulations for causally guided controllable generation. CAGE is well suited to wide family of modern deep generative models trained complex high-dimensional data and does not require any altering of the original training objective nor hard to obtain counterfactual data. Empirically, we find CAGE reliably extracts correct causc-offect relationships in controlled scttings like MorphoMNIST. On high-resolution image data such as CelebAHQ, CAGE reveals cause-effect relationships that are best supported by generated counterfactual samples. Finally, using causal insights supplied by CAGE we also perform a large scale human evaluation of controllable generation revealing that higher quality samples are generated by controlling the causal variable in same souree settings while manipulating the effect variable leads to better fine-grained generation in same destination settings.
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Acknowledgements
The authors would like to thank all anonymous reviewers for their feedback. In addition, the authors thank the workers of Amazon Mechanical Turk for assisting with the qualitative evaluation of generated samples. AJB performed this work during an internship at Meta Al. This work was supported by a Sony Faculty Innovation Award for AG. Finally, the authors would also like to thank Mcta AI for the computing resources that made this work possible.
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Table 3: Causal Discovery over G for various pairs attributes, (M,,m2). each pair, top row corresponds to taking my as the treatment whilst the bottom utilizes as the treatment. Please see appendix $F contains additional generated samples and a qualitative description of visual fidelity for each pair attributes. Note, the results from this experiment are distinct from the same source or same destination experiments in 4.1 and summarizes the findings from answering Q1-Q3 posed earlier.
Attributes
Counterfactual samples
GATE
MMD
ÅT 3/1 + m2
my: Gender ma! Mustache
25
550
315
Ar: Y71 -> 7712
mi1 Gender
m2:t Bald
AT: + 121
Hale CoatCoatee
my: Gender Goatee
305
155
AT: m1 1712
Malu Kuy Dhocks
mi. Gender
mg: Rosy Chee
0.0
t
084
022
300
15%
Ar: mi 6- m2
L 2
20
mi: Age
ma: Bald
Ar: mi - m2
my: Gender
mg: Blonde
Ar: m7 1. m22
012
my: Gender
mai Smiling
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A Review: Casual inference and connections to generative models and CAGE
In this section we provide brief overview of the causal inference methodology we rely upon for CAGE well provide intuitive explanations for our work. CAGE relies upon the framework of structural equation models (SEMs) which can be used to formalize causal knowledge as well as answer interventional and counterfactual queries. We provide a brief description below, for further details please see (Pearl, 2009) as well as (Pawlowski et al., 2020) and (Khemakhem et al., 2021) for applications the context of deep generative models.
Structural Equation Models: For d-dimensional random variable, x (d))...,,a), SEM is defined collection of structural equations:
Sj: Bj = (p), j = i=,....,d
which dictate the generative mechanism for observed data, We write pa, to denote the causal parents of variable 2j and introduce mutually independent latent noise variables (also referred to as latent disturbances) ny for cach 2j to accommodate non-deterministic causal relationships.
The set of structural equations, S ( ".S,,...,Su. Sa), serves two purposes: first it encodes causal structure over observed random variables x following a given distribution Px. As a concrete example, T& E paj then z is a direct cause of variable zy. Second, it also defines a generative mochanism to sample new obscrvations from the distribution Px by simply ilerating through the set of structural equations, S.
Interventions and the do Operator:
One of the fundamental benefits of encoding causal structure via SEMs is that it we can directly paramcterize interventions over the associated causal graph (Pearl, 2009). Formally, this involves re-defining the structural equation over the variable which we will intervene over. For example, we considered the intervention of setting the Ij a, this would redefine S, as follows:
S₁ I3 a.
Pearl (2009) denotes such a deterministic intervention as do(x; = a).
Counterfactuals: Counterfactual queries seek lo understand statements of the form: what would the value for variable x; have been if variable zj had taken value a, given that we have observed x Xobo? It therefore follows that counterfactual and interventional queries closely related, with the fundamental difference being that an interventional query seeks to marginalize over latent variables, whereas counterfactual query is evaluated conditional on latent variables inferred from observed x = Xabs As such, counterfactual query first infers latent disturbance variables, ni, and then proceeds in the same manner as an intervention.
Parameterization of Interventions and Counterfactuals in CAGE: In the context of CAGE, we consider counterfactual queries over binary metadata variables, ma- As described Section 3, we first infer the latent associated latent disturbance variables by projecting a given image, x, to latent encoding Exc(x). Thereafter, it is possible to define intervention over given binary metadata attribute, do(m.. a) in a variety of ways. In Section 3.3 we consider interventions parameterized by linear projection over the latent variable, z such that z(do (m. = 1)) z + oh, where the vector h is defined via training a linear classifier (see Figure 3).
B Extending Average Treatment Effects to Generative Models
For average treatment effects, we are interested in computing
Eg m_ (i)
There are two important distinctions between the use GATE in CAGE and a traditional ATE computation:
First, we are interrogating a generative model, we can observe both Ym.=1(1) (1) and Yi). This is significant difference to standard ATE calculations, which can only observe one outcome.
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Second, we are interested in understanding implicit causal structure in the generator, G, the expec- lation over samples from the prior of G as opposed to over samples from an observational dataset. This is an important distinction as it disambiguates the causal structure present in an observation dataset, which is the focus of traditional causal discovery mothodology, with any causal structure implicit within the generalor, which our primary concern.
For unbiased estimation of ATE (a causal quantity) via statistical quantities, it is standard to assume mnconfuundedness, positivity, and SUTVA. For the use of GATE in CAGE, we implicitly make further assumptions such as:
1. Counterfactual parameterization: The latent space of a generator, G, must be linearly separable with respect to the attributes of interest. This assumplion is required as CAGE paramelerizes the assignment of counterfactual treatments as linear manipulations over the latent space of a generator. We note that such an assumption has been shown hold in practice for deep generative models (Denton al., 2019; Ramaswany al., 2021).
2. Complete support of the generator: The generator, G, can generate high quality samples over the entire support of its latent space, Z. This assumption is required to guarantee that linear manipulations over the latent space of a generator continue to produce high quality images, such that our resulting conclusions are not the result of e.g., image artefacts.
Sufficiently expressive and accurate classifiers: CAGE is premised on the use probabilistic classifiers in order to quantify the presence aaneence/bsencoe absence of an effect variable on counterfactual exain ples and thus compute treatment effects as specified in equations (6-7). We require accurate and calibrated classifiers (ideally, Bayes optimal) for computing GATE.
C Model Details
For our experiments, whenever possible, we used the default settings found in the original papers of all chosen models. In particular, used the default settings for both Mintnet (Song al., 2019) and Style- GAN2 (Karras et al., 2020) which were pretrained on MNIST and FlickrFacesHQ respectively. For Mintnet we finetuned on a MorphoMnist dataset for 250 epochs using the Adam optimizer with default settings. Similarly, we also finetuned StyleGAN2 on CelebAHQ for 2000 iterations using 10% of CelebAHQ, Our synthetic experiments on the other required us to train a Masked AutoRegressive Flow (Papamakarios et al., 2017) that consisted of 10 layers with blocks per layer. The Masked AutoRegressive Flow was trained for 5000 iterations using 5000 data samples.
C.1 Human Evaluation of Counterfactuals on Amazon Mechanical Turk
To study the visual quality of generated ualt by manipulating the cause or effect variable as determined by CAGE we solicit human evaluators on the Amazon mechanical turk platform. In particular, collect 100 generated pairs for each DAG in both the same source and same destination settings. Each total of 300
pair of generated samples is then evaluated by different human annotators giving annotations per DAG. A capture of the UI provided to the human annotators is depicted below 5.
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Instructions:
ISNDRING BACKGROUND
liney to you baing metl person's picture briefl.. right pleases select most restogic imagge using yourr jatgement.
Bottom Image
Figure 5: UI provided to AWS Turkers.
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C.2 AT Mixture of Gaussians and MorphoMNIST
To illustrate the fact that a deep generative model may learn a different causal relationship than the ground truth data generation process we consider the same mixture of gaussian experiment as the main paper. We consider both causal ordering and anti-causal ordering over imput variables that fed into a Masked Autoregressive Flow model which is then trained using maximum likelihood. In each case CAGE reports AT 0 indicating for the same ordering of variables which indicates that it is possible for the MAF model to learn different internal causal ordering than the ground truth.
Samples colored by causal attribute, m
Estimated scores (a
Samples with causal attribute m²(0) =1
ATE score
160
Samples with causal attribute =
120
100
60
20
-0.0125
-0.0100 -0.0075
-0.0050 -0,0025
0.0000
0,0025
0.0050
Estimated
Samples colored by causal attribute, mc
Estimated ATE scores (a
Samples causal attribute m,(i)
ATE score
160
Samples with caus al attribute m,(i) =
140
p-ralue: 0.97
120
80
40
-4
-0.0125 -0.0100 -0.0075
-0.0050 -0 00,022
0.0000
0.0255
0.0500
Estimated ATE
Figure AT plots on Mixture of Gaussians for both causal and anti-causal input variable orderings for Mixed Autoregressive Flow generative model.
We plot the Ar a function of interpolant strength below for the MorphoMNIST dataset. As observed Ar > 0 which indicates that under generator G thickness is the cause of intensity which matches the true synthetic data generation process.
A note on baseline methods for CelebHQ
We compared CAGE to well-ostablished causal discovery algorithms when looking to infer structure over a deep generative model, G. As noted in $4, the causal discovery baselines considered operate over the class probabilities output by probabilistic classifiers trained as $3.4. We note that while it would be possible to learn causal structure using bascline methods, such as LINGAM, over the metadata (c.g., hair color and gender), this would not necesarly provide any insights into the causal structure implicit within the generator, G. For this reason, instead focus applying causal discovery methods the output of proxy classifiers.
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3.0
2.5
Delta-ATe Telta-ATE-score
2.0
1.5
1.0
0.5
0.0
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
alpha
Figure 7: AT plot on the MorphoMNIIT dataset as function of a.
D Additional Baseline Experiments
In this section we perform causal discovery in the latent space using various popular approaches in the literature. As noted in the main paper these approaches cannot be applied to the observation space which are high dimensional images and are not baselines.
Method
ANM
DAGs with No Tears LinGAM
Gender, Mustache Gender, Bald
Gender, Goatee
Gender, Rosy Cheeks
Age, Bald
Pointy Nose, Brown Hair Gender, Smiling
Hard vs. Soft. As we have access to ground truth labels it is tempting to consider whether using hard labels opposed to the classifiers probability is belter suited to computing our AT metric. In the table below we repeat our causal discovery experiment over CelebAHQ. As observed, all baselines provide unreliable estimates to the causal relationship between variables when compared to the main table which is computed using soft labels. Finally, we found it useful to assign soft labels provided by the classifiers used the ATE computations for all augmented images during group DRO training.
Baselines without using CountorFactuals We now turn to the use of Counterfactuals when computing our baseline scores. Specifically, allempt causal discovery purely using observation data and labels with both hard and soft labels, which is in contrast to the main paper which considered baselines which had access to classifier probabilities on counterfactual data, The table below shows this result of this ablation.
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Method
ANM
DAGs with No Tears LinGAM
Gender, Mustache
Gender, Bald
Gender, Goatee
Gender, Rosy Checks
Age,
Pointy Nose, Brown Hair Gender, Smiling
Method
ANM
DAGs with No Tears LinGAM
Gender, Mustache
Gender, Bald
Gender, Goatee
Gender, Rosy Chocks
Age, Bald
Pointy Nose, Brown Hair
Gender, Smiling
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Same Source
Starting image
do(mustache 1)
do(gender 0)
Same Destination
do(gender 1)
do(mustache = 0)
Starting image
Starting image 2
Figure 8: A visual description of the same source and same destionation settings. In the same source setting we start with the same image which is then intervened using cach mota-attribute separately. In tch same destination setting we start out with two images which are then intervened in such a way that the output of both interventions arrives the same combination of meta-attributes. For example, we may wish to generate samples of men without mustaches by intervening on starting image of a female and then changing gender starting with an image of male with mustache and intervening the mustache meta-attribute.
E
Qualitative Samples for Same Source and Same Destination
In this section we provide selected qualitative examples that were used for our human evaluation study on Amazon Mechanical Turk. In each visualization we show a panel three images; the leftmost image is the source image while the middle and rightmost image correspond to manipulating the first and second meta-attribute respectively. We also give a visual description of both the same source and same destination setting below:
E.1 Same Source
Gender- Mustache.
G Goatee.
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Figure 9: Left: The target image used for source manipulation. Mid: Manipulating the causal Gender variable. Right: Manipulating the effect Mustache variable.
Figure 10: Left: The target image used for source manipulation. Mid: Manipulating the causal Gender variable. Right: Manipulating the effect Goatee variable.
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Gender-> Bald.
Figure 11: Left: The target image used for source manipulation. Mid: Manipulating the causal Gender variable. Right: Manipulating the effect Bald variable.
Gender-+ Blonde Hair.
Figure 12: Left: The target image used for source manipulation. Mid: Manipulating the causal Gender variable. Right: Manipulating the effect Blonde Hair variable.
-ender-+ Rosy Cheeks.
Figure 13: Left; The target image used for source manipulation. Mid: Manipulating the causal Gender variable. Right: Manipulating the effect Rosy Cheeks variable.
Ager- Bald.
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Figure 14: Left: The target image used for source manipulation. Mid: Manipulating the causal Bald variable. Right: Manipulating the effect Age variable.
E.2 Same Destination
Gender-> Mustache.
Figure 15: Left and Mid Left: The target images used for destination manipulation. Mid Right: Manipulating the causal Gender variable. Right: Manipulating the effect Mustache variable.
Gender~+ Goatee.
D
Figure 16: Left and Mid Left: The target images used for same destination manipulation. Mid Right:
Manipulating the causal Gender variable. Right: Manipulating the effect Goatee variable.
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-ender~> Bald.
Figure 17: Left and Mid Left: The target images used for same destination manipulation. Mid Right:
Manipulating the causal Gender variable. Right: Manipulating the effect Bald variable.
-ender-> Blonde Hair.
Figure 18: Left and Mid Left: The target images used for same destination manipulation. Mid Right:
Manipulating the causal Gender variable. Right: Manipulating the effect Blonde Hair variable.
Gender-, Rosy Cheeks.
Figure 19: Left and Mid Left: The target images used for same destination manipulation. Mid Right:
Manipulating the causal Gender variable. Right: Manipulating the effect Rosy Cheeks variable.
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Age < Bald.
2
2
Figure 20: Left and Mid Left: The larget images used for same destination manipulation. Mid Right:
Manipulating the causal Bald variable. Right: Manipulating the effect Age variable.
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F
Additional Generated samples
We now provide additional qualitiative examples of the generated counterfactuals on the same set of DAG's as considered in the main text. In the first row of all settings we generate counterfactuals by manipulating the first attribute variable (e.g. Gender) while the second row corresponds to counterfactuals where the second attribute is manipulated. The third row can either be the first or second attribute depending on context. We note that the generated samples may expose unintended biases that may be present in CelobAHQ and be learned by the generator. While CAGE does not introduce any sources of biases it should be noted that generating counterfactuals in this manner further exacerbate these initially hidden biases. As result, CAGE can serve as important aid in empirically illuminating important ethical concerns that should be taken into account for any purposeful uses of generative models.
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Gender, Mustache.
We consider generating samples by either manipulating the mustache attribute (top two rows) and then manipulating gender. As we can see, when the Gender male we are able to semantically preserve the gender increase the presence of mustache (top row). Similarly,
find that when the gender is female
are still able to semantically preserve the (subjective) gender as manipulate the mustache attribute (middle row). Finally, in the third row we change the gender of from female to male and notice that although the mustache attribute was not explicitly manipulated it appears as we manipulate gender more aggressively to male (bottom row). These results suggests that the generative models learned causal relationship is that gender causes mustache.
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Gender, Goatee.
We consider generating samples by either manipulating the goatee attribute (top two rows) and manipulating gender. In the first row we that manipulating the goatee attribute leads visual and subjective change the samples perceived gender (top row). Similarly, we find that when the gender is already male increasing the presence of goatee does not reverse the gender to female expected (middle row). Finally, in the third row we change the gender of from female to male and notice that there is no increase in goatee in the generated samples. These results suggests that the generative models learned causal relationship is that gender the cause goatee.
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Gender, Smiling.
We consider generating samples by hir manipulating the smiling attribute (top two rows) and manipulating gender. In the first rows we that increasing the smiling attribute does not cause (subjective) change the samples perceived gender. Conversely, in the last row manipulating samples gender from female to male preserves the smiling attribute (or lack thereof) in the generated samples. These results suggests that the generative models learned causal relationship is that gender is independent of smiling.
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Age, Bald.
We consider generating samples by either manipulating the baldness attribute (top two rows) and then ma- nipulating age (last row). In the first row we that increasing the baldness attribute does considerably change the age of the generated sample. On the other hand, in the last row increase the age of the can- didate and observe that while the baldness attribute increases it is not the extent as samples found in the first two rows. These results suggests that the generative models learned causal relationship. in contrast to our biological prior, is that baldness is the cause of age.
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Pointy Nose, Brown Hair.
We consider generating samples by either manipulating the pointy nose attribute (top two rows) and ma- nipulating brown hair. In the first two rows we that increasing the pointy nose attribute does cause change in the samples hair color from dark brown to more blonde. Conversely, in the last row manipulating samples hair color to be more brown does not appear to significantly change the pointy nose attribute in the generated samples. These results suggests that the generative models learned causal relationship is that poinly nose causes brown hair.
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Gender, Rosy Cheeks.
We consider generating samples by either manipulating the rosy check attribute (top two rows) and ma- nipulating gender. In the first two rows change in the samples (subjective) gender-ie. both females and males retain their gender their rosy cheeks. On the other hand, in the last row manipulating samples gender from female to male appears to slighltly decrease the rosy check attribute in the generated samples. These results suggests that the generative models learned causal relationship is that gender couses rosy cheeks.
that increasing the rosy cheek attribute does not cause
increase
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Gender, Blonde Hair.
We consider generating samples by either manipulating the blonde hair attribute (top two rows) and ma- nipulating gender. As observed in the first two rows increasing the blondeness of a samples hair does change the perceived gender from female to male. In sharp contrast, the last row the gender from male to female leads to the samples hair color changing towards being more blonde. These results suggests that the generative models learned causal relationship is that gender causes blonde hair.
that manipulating
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Ethical Implications of CAGE
As deep generative models grow in their practicality and deployment in high-stake applications, it is impor- lant to lake a critical view of the inherent biases implicitly encoded in these models (Sheng et al., 2019; Choi et al., 2020; Bender et al., 2021). Our framework allows for a principled study of such biases through a causal analysis, potentially paving the way to mitigate the undesirable harms of deep generative models.
For example, our approach highlights important failings of the generative model itself-e.g. when cause- effect relationships for scientific applications. In such cases, the use of these models in any high-stakes downstream application can be very costly without any intervention. Examples range from the current use of generative models for accelerating scientific discovery as well as societal applications in training fair classifiers. Ironically, in all such applications, the use of generative models can worsen the problem at hand by introducing new blindspots through the use generated data that does not conform to the desired cause-effect relationshpp. Explicitly inferring these failings provides researchers and practitioners with an opportunity to develop algorithms and evaluation metries that go beyond the current focus visual fidelity or likelihoods as well derive better data acquisition strategies for finctuning generative models.
inconsistent with biological priors for human subjects or with physical relationships
On the other hand, we acknowledge that like many other efforts in generative modeling, our endeavors for better understanding these models can potentially be exploited by malicious actors for generating fake content, commonly known as Deepfakes (Korshunov & Marcel, 2018). Finally, causality has proven to be a highly powerful tool for formalizing important concerns of machine learning models with regards to fairness, robustness, and interpretability. However, as highlighted in our limitations, any causal framework including ours itself implicitly encodes some assumptions that may not hold in practice. Hence, researchers and practitioners should exercise due caution and appropriate judgment in interpreting the findings.
We would also like to highlight our proposed framework and its application to natural images has been constructed for illustrating this research and not for any business application.
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Reviews:
[{"review_id": "vj8nt1MgVC", "reviewer": "Reviewer_9whC", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The idea to detect the reasoning process of deep generative models is very interesting. It provides a new possibility that how causal inference (mainly used to help human make decision) can help AI.\n2. The methodology is clean and reasonable, for the defined task. \n3. The comparison of three approaches in Figure 2 is insightful.\n\nWeaknesses:\n\nMy main concern is on the \"controlable generation\" part, which shows how GATE can help us on particular AI tasks. Intuitively, if we know the NN is reasoning along certain directions, then we'd better control the generation process by manipulating the cause. However, the current sec 4.1 fails to demonstrate this. Some suggestions: 1) explicitly define the generation tasks for both \"same source\" and \"same destination manipulation\" setting by explaining what are the two features and the possible control pipelines; 2) show the generative images if we do not manipulate along the reasoning directions and make quantitative comparisons; 3) discuss how the accuracy of classifiers influence the generation and the insights; 4) conduct experiments for different kinds of deep generative models and discuss the insights.\n", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this paper, authors propose to discover how deep latent models \"causally reason\"  two conceptual features. They measure the causal reasoning direction by the cause effect direction between the two features. Since the target is the reasoning of NN but not the real data generation process,  counterfactual data can be created by manipulating the latent variables with the help of classifiers of the two features over the latent space. Then, we overpass the challenge in standard causal inference tasks. Afterwards, the authors design the detailed implementation and call this algorithm GATE. The statistical significance can be tested by the p-value of GATE under the null. Finally, authors leverage the causal direction to controlly generate images. The results are subjectively more realistic.", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "Please see my suggestions above. I expect the authors can respond to at least the first two items. ", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "rnhIb8ZZFC", "reviewer": "Reviewer_1ieq", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n\nThe paper tackles a problem that has not been studied extensively. Currently researchers lack a deep understanding of what samples can deep generative model generate and what concepts are learned, and this method is a good probe to these questions. \n\nThe idea is simple but effective. By moving latent variables across the plane defined by the classifier, counter factual samples can be effectively generated. Experiments on generating counter factual examples are impressive.\n\nWeakness:\n\n1. Insufficient background knowledge. For readers with little or no experience in causal inference, this paper is a bit hard to follow. It would be better to have some explanation of terminologies, such as counter factual, do operator, etc. Also, the method should be explained in a more intuitive way that audience from general ML background can understand.\n\n2. Some recent related work in controllable generation and counterfactual generation is not discussed. In particular, with the help of a latent classifier, [1] propose to formulate an EBM on the latent space for controllable generation. In addition, [2] propose method for creating counterfactual examples.\n\n[1] Controllable and Compositional Generation with Latent-Space Energy-Based Models, https://arxiv.org/abs/2110.10873\n[2]  Counterfactual Generative Networks, https://arxiv.org/abs/2101.06046", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper propose a method called CAGE, which infers the cause-effect relationships within the latent space of deep generative models. It is done by geometric manipulation of latent vector with the help of a latent classifier. The authors evaluate the method with both synthetic toy data and high dimensional human face data. It is shown that CAGE can reliably extract correct cause-effect relationships in the setting where such a relationship is known, and in more complex setting such as CelebAHQ, the method can generate counterfactual examples to support the discovery of cause-effect relationships.", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "As I said above, the paper can be better if background knowledge of causal inference and more intuitive introduction of the method are added. In addition, related papers mentioned above should be discussed.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "odO2K1ZsVR", "reviewer": "Reviewer_FwcQ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength \n1. I agree with the author's opinion that traditional causal discovery techniques can damage generalization ability.\n2. There are rich experimentation and clear visualization. \n3. Appendix E is well explained.\n\nWeakness \n1. The motivation is confusing. The reasonability behind the assumption “we do not expect that the generative model perceives these meta-attributes as independent” needs to be clarified. Intuitively, meta-attributes are independent, (e.g., gender and hair color are unrelated), so meta-attributes have no cause-effect relationships.\n2. The intuition of the work should be clearly presented. The authors claim that using causal insights can obtain fine-grained generation. However, by reading the Introduction part, it is totally unclear why and how can implicit causal relationships achieve the goal.\n3. The merits of the proposed one are unclear. In the authors’ words, implicit causal structure within a deep generative model cannot well respect the true causal structure, so why do the authors still try to infer latent causal structure within a deep generative model?\n4. It is necessary to explain these datasets and experiments at the beginning of section 4 so that the reader has a clearer sense of the functions of these experiments. \n5. In Section 4.1, Experiment Design, there are two experimental setups: same source and same destination manipulation. In Table 3, it is unclear which one refers to the same source or destination. \n6. The caption 'The target image used of source manipulation' in the figures shown in Appendix D is not mentioned and explained earlier, causing some confusion. \n7. Given that the other places are written as 'Figure x.' it is suggested to change the last sentence of subsection 3.3 from 'Fig. 3' to 'Figure 3'. \n8. Two candidate options that act directly on the counterfactual latent are proposed in subsection 3.4. I am not quite sure which option is adopted in the calculation. \n", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper propose CAGE, a framework for CAusal probing of deep GEnerative models. Compared with traditional causal discovery and causality, CAGE inferred implicit causal relationships in deep generative models and was able to discover the implicit causal structure of any generative model. The authors mainly consider two scenarios of control generation, source manipulation, and destination manipulation. The knowledge of causal directions inferred by CAGE can significantly improve the generation quality on the CelebAHQ dataset. Furthermore, the improvement of the source and destination manipulation scenarios was measured by extensive human evaluation. In addition, the authors extended the potential outcome framework to fit deep latent variable generative models. By computing the difference in treatment effects between the generative average treatment effect (GATE) of two variables to measure the candidate causal attributes. \n\nAlthough the topic is interesting, the motivation and the intuition of the work are rather confusing. I pose a negative attitude towards the manuscript unless the authors can address my concerns.\n", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "1. Highlight the contribution and clearly present intuition.\n2. Add a short overview of the experiments at the beginning of section 4. \n3. Add a clearer description of Table 3. \n4. Add clear explanations to things that appear for the first time. \n5. Pay attention to some format uniformity. \n6. Need to clarify some options in the paper. \n", "broader_impact_concerns": "I see no broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "S886MVGZuG", "reviewer": "Reviewer_inuw", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper addresses an interesting question, namely how generative models can be made more interpretable and controllable using methods from causal inference. I did not find any significant technical errors, and the paper is mostly well written. Given the interest in causality and generative modelling I'm confident some readers would be interested in this paper.\n\nThe notion of intervention and causality employed here is precisely defined, in fact more precisely than in the wider causality literature. This is because in this setting the intervention is actually defined as a transformation on latent space (adding a vector h_c), whereas typically it is only operationally defined (e.g. give a patient a treatment). However it is not so clear that the result is in line with our intuition. For example, what does it mean to say that gender causes hair color (section 3)? Isn't every combination equally well possible? And couldn't we imagine intervening on one without changing the value of the other? So as the authors mention, the notion of causality as employed in this paper really only says something about the model, and how the operation of adding h_c changes the causal variables, rather than something about causality in the real world. As such one could argue about how important or interesting the notion of causality used in this paper is, but that would lead to a rather subjective discussion.\n\nThe main experiment (4.1) shows that according to human raters, sample quality is generally better when applying an intervention on a value deemed cause rather than on one deemed effect (in the same source setting). This seems interesting but I wonder what the explanation for this is? Could it be related to something simple like different frequency of occurence in the dataset (e.g. few to no women with mustaches)?", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The goal of this paper is to use ideas from the causality literature to make pretrained latent variable models like GANs and VAEs more interpretable and to make their generation process controllable. The paper focuses on causal relations between a predetermined set of binary attributes such as gender or hair color that are provided alongside the main data (e.g. images of faces). Causal effects among such attributes can be defined by giving an intervention operation in the latent space. In practice this means taking the latent vector z representing an image, adding a vector h_c which is the normal of a classifier hyperplane (e.g. moving in the \"blonde\" direction of latent space), and then measuring the effect on the attributes using a latent-space classifier. This is a unique setting, because normally we can only execute one intervention per unit (image, person, ...) whereas here we can apply any number. Using this notion of intervention, a new controllable generation method is presented. Using a study with human raters, it is shown that intervening on the cause variable generally leads to better samples.", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "Minor issues:\n- Top of page 5, the union operation creates a big set with lots of vectors and metadata but no pairing. Better to use a set of tuples or something.\n- Page 6, typo: \"factual treats\"", "broader_impact_concerns": "No concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper addresses an interesting question on how generative models can be made more interpretable and controllable with causal inference. All reviewers agree the problem to be interesting and worth investigating. At the same time, the reviewers have raised many questions, most of which focus on the clarity and motivation. The authors provide detailed explanations on all the questions, which I believe have sufficiently addressed the issues. In the revision, I would expect the authors to take the comments from all the reviewers and incorporate them to revise and improve the current version. This will make the paper more clear and stronger.
Decision: Accept (poster)