Learning Neural Causal Models from Unknown Interventions
Authors: Nan Rosemary Ke
Olexa Bilaniuk
Anirudh Goyal
Stephan Bauer
Hugol Larochelle
Chris Pal
Yoshua Bengio
Venue:  (2020)
Keywords: deep learning
graphical models
meta learning

Abstract: Meta-learning over a set of distributions can be interpreted as learning different types of parameters corresponding to short-term vs long-term aspects of the mechanisms underlying the generation of data. These are respectively captured by quickly-changing \textit{parameters} and slowly-changing \textit{meta-parameters}. We present a new framework for meta-learning causal models where the relationship between each variable and its parents is modeled by a neural network, modulated by structural meta-parameters which capture the overall topology of a directed graphical model. Our approach avoids a discrete search over models in favour of a continuous optimization procedure. We study a setting where interventional distributions are induced as a result of a random intervention on a single unknown variable of an unknown ground truth causal model, and the observations arising after such an intervention constitute one meta-example. To disentangle the slow-changing aspects of each conditional from the fast-changing adaptations to each intervention, we parametrize the neural network into fast parameters and slow meta-parameters. We introduce a meta-learning objective that favours solutions \textit{robust} to frequent but sparse interventional distribution change, and which generalize well to previously unseen interventions. Optimizing this objective is shown experimentally to recover the structure of the causal graph. Finally, we find that when the learner is unaware of the intervention variable, it is able to infer that information, improving results further and focusing the parameter and meta-parameter updates where needed.

Under review as a conference paper at ICLR 2020
LEARNING NEURAL CAUSAL MODELS FROM UNKNOWN INTERVENTIONS
Anonymous authors Paper under double-bling review
ABSTRACT
Meta-learning over a set of distributions can be interpreted as learning different types of parameters corresponding to short-term vs long-term aspects of the mech- anisms underlying the generation of data. These are respectively captured by quickly-changing parameters and slowly-changie new framework for meta-learning causal models where the relationship between each variable and its parents is modeled by a neural network, modulated by struc- tural meta- parameteasssssss which capture the overall topology of a directed graphical model. Our approach avoids discrete search over models in favour of a continu- ous optimization procedure. We study a setting where interventional distributions are induced result of a random intervention an unknown ground truth causal model, and the observations arising after such intervention constitute one meta-example. To disentangle the slow-changing aspects of each conditional from the fast-changing adaptations to each intervention, we parametrize the neural network into fast parameters and slow meta- veeta-prameters. We introduce a meta- meta-learning objective that favours solutions robust to frequent but sparse interventional distribution change, and which generalize well to previously unseen interventions. Optimizing this objective is shown experimentally to recover the structure of the causal graph. Finally, we find that when the learner is unaware of the intervention variable, it is able to infer that information, improving results further and focusing the parameter and meta meta-parameter updates where needed.
peeaa-parameters We present a
single unknown variable of
INTRODUCTION
major challenge of contemporary deep learning to generalize well outside the assumptions of independent and identically distributed data, when we care about generalization or fast adaptation to distributions other than the main training distribution. For this purpose, we propose an approach that starts by distinguishing between: (a) underlying causal model, (b) observational distributions derived from it, an (c) interventional distributions arising from interventions upon its variables, whether they be known or unknown. Fortunately, these distinctions can be addressed by the paradigm of Structural Causal Models (SCMs) (Pearl 1995 Peters et al., 2017) and a wide body of associated literature. Unlike other frameworks using SCMs also account for interventions performed by agents other than an experimenter. By treating the interventions of other agents as unknown interventions that lead to changes in the underlying data distribution, the present work is contribution towards the use of meta-learning for causal model induction.
Estimating the underlying causal structure from data is an open and challenging problem (Pearl 2009; [mbens & Rubin 2015). A lot of prior work has examined learning causal structure based on observational data (Chickering 2002 Tsamardinos et 2006; Goudet et al.) [2017] Hauser
Bühlmann| 2012; Spirtes et al. 2000; Sun et al. [007]] Zhang et al. 2012: [Shimizu et al. 2006
Hoyer et al.) | Daniusis
2012] Budhathoki Vreeken [017]] Kalainathan al. 2018).
However, many real-world datasets have an inherent distributional heterogeneity due to different
interventions to the variables composing the model. these situations interventional approaches are
needed (e.g. [eckerman et al. 1995 Cooper & Yoo, 1999: Hauser & Bühlmann, 2012 Peters et al.) 2016 Rothenhäusler et al.] 2015 Ghassami et al. 2017); 7}. Established approaches for causal inference conditional independence testing, which is hard (Shah & Peters] 2018). Furthermore, most of these approaches either assume full knowledge of
often either relying restrictive assumptions
the intervention or make strong assumptions about its form (Heinze-Demi al. 2018

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However, in the real world, interventions are also not always performed by experimenter. They can be performed by other agents, or by environmental changes in ways that are unknown, or by naive learner (like robot) which does not know precisely yet how its low-level actions change high-level causal variables. In this paper, we look at the setting where interventions are unknown, and our goal is to discover causal graphs given nnknowo-intervention samples. The challenging aspect of this setting is to not only learn the causal graph structure, but also predict the intervention accurately, In this setting, we need make sure to: (I) avoid an exponential search over all possible DAGs, (2) handle unknown interventions, (3) model the effect of interventions, and (4) model the underlying causal structure.
One possibility for learning a causal structure (through SCM modelling) is to perform many ex- periments which one executes interventions. Thus, such interventions modify the effect of the intervened upon variable from its parents in the corresponding DAG, which the model has to quickly adapt to. We can make parallel connections to meta-learning. where the inner loop can be consid- ered as fast adaptation to the distribution change, and outer loop can be considered as learning the stationary petaaparaseters of the model. For causal induction, one can consider each distribution which arises result of an intervention as a meta-example, and use meta-learning objective for fast adaptation in response to an intervention. One can think of model parameters as being composed of slow- and fast-changing parameters. The slow parameters are analogous the meta-parameters in meta-learning and are used for (1) intervention prediction in order to handle the unknown intervention and for (2) modeling the underlying causal structure. On the other hand, the fast parameters are used soanenttilly-ggowiine space of all
to model the effect of interventions. An explicit search over the possible DAGs is avoided by modeling the conditionals of the structural causal model using function approximators, with one neural network per variable. The belief over whether one node is direct causal parent of another node j corresponds a dropout probability for the i-th input network j (which predicts variable j given its parents). This cheaply represents all 2M² possible model graphs, with the graph search implicitly achieved by learning these dropout prohabilities. We thus propose new method for fast adaptation and learning of neural causal models by framing the problem in a meta-learning setting, similar (Dasgupta al. 2019 Bengio al. 2019).
Our contributions Our key contributions can be summarized as follows:
Handle causal induction to the case where the variable on which soft intervention took place is not known by the learner, and show that better results can be obtained when the learner attempts to infer that information and uses it to appropriately change parameters and meta-pane peta-paarmmttes.
We bypass the issue of having to optimize over and represent exponentially large set of discrete causal graphs by learning an efficiently parametrized ensemble of SCMs,
Show that our algorithm correctly identifies the causal graph and use the learned graph for generalization to an unseen environment.
2 PRELIMINARIES
A Structural Causal Model (SCM) (Peters et al. 2017) over a finite number M of random variables
X; is set of structural assignments
X fi((patt.)) N.), Vi e MO....,-1)
where N; is jointly-independerr noise and pa(i, C)) is the set of parents (direct causes) of variable i under configuration C of the SCM directed acyclic graph, i.e., C' MXMM with Gj- = 1 if node i has node j as parent (equivalently, X, E XXp((t.C); i.e. X, is a direct cause of X,). The n-th power of an adjacency matrix, C counts the number of length-n walks from node to node of the graph in element cjj- The trace of the n-th power of an adjacency matrix, Tr(C"), ). counts the number of length-n cycles in the graph. Causal structure learning is the recovery of the ground-truth C' from observational and interventional studies.
Functional and structural meea-persseeters Let consider the simplest SCMs, those with M = 2 random variables (lets say random variables A and B). Only three DAGS exist relating them; they are - B, B A or A B. These can be represented as the following. If A causes
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B, then the SCM between A and B can be represented as B föp (CBA A,) where свΛ 1, eø ~ NB is a sample of an independent noise factor. If B causes A, then the SCM can be presented as A (CAB B,),), where CaB 1, ελ NA is sample of an independent noise factor. If A and B are independent, then we have the same equations but we set CAB = 0 and CBΛ = 0.
We may now think of learning the structural causal model as learning the two probabilities of CAB or CBA being (versus 0), representing our belief in the causal relationship between and B. We can parametrize these probabilities differentiably using P(CAB = 1) = (TAB) and P(CRA = 1) (TBA), with real numbers and (2) T-e- We can also parametrize the structural equations So, and fog in a differentiable manner, using conditional probability tables (CPTs) or neural networks.
Hence, the problem becomes simultaneously learning the structural meta-parameters and the functional meta-pan vecta-parmmtter 0. Functional meta-parameters may be easily learned by maximum likelihood or proxy, and using backpropagation. However, the are more difficult to infer. [Bengio al.) (219)) propose learn them by using observations of how the distribution changes sparsely, with the transfer generalization (the adaptation rate after the sparse change) being the training objective for y. Simultaneously inferring both and 7 still more difficult. An M --variable SCM over random variables x;, i E (O...., M - }} can induce a super-exponential number of adjacency matrices C. The super-exponential explosion in the number of potential graph connectivity patterns and the super-exponentially growing storage requirements of their detining conditional probability tables make CPT-based parametrizations of the structural assignments fi increasingly unwieldy as M scales. As shown below, neural networks with Cty masked inputs can provide more manageable parametrization. For more background about different kinds intervention we ask the reader to refer Appendix A.3
PROPOSED FRAMEWORK: META LEARNING FOR CAUSAL INDUCTION
Our framework disentangles the slow-changing meta- pertaeaaramttess which reflect the stationary prop- erties discovered by the learner, and the fast-changing parameters, which adapt in response to interventional changes in distribution. We consider two kinds of meta-parameters: the causal graph structure y and the model's s slow weights Sstow, along with the meta-learning objective for both of them. We also consider one kind of parameter: the model's fast weights. 0 We will call 0 = fllow + mas the sum of the slow, stationary meta-paranee veta-parameters and the fast, adaptational parameters.
3.1 TASK DESCRIPTION
Our task setup deviates from most common deep learning modeling setups. For the purposes of this work, we restrict ourselves inference of SCMs of M categorical random variables causally related via a DAG. The model is permitted to see (I) data from the original ground-truth model, and (2) data from modified ground-trui -ruth model with a random intervention applied. In our experiments, at most one intervention is concurrently performed. When an intervention is performed, a single node is randomly and uniformly chosen among all M nodes, and its ground-truth distribution soft- soft-intervened upon. The learner model is aware of the samples having come from an intervention distribution, but is not aware of the identity of the intervention node, and so must predict it. Each run of sampling steps under given intervention is referred to as an episode. The learner, over a large number of episodes, will experience all nodes being intervened upon, and should be able to infer the SCM from these interventions.
or manually-provided ground-truth
3.2 CAUSAL INDUCTION AS AN OPTIMIZATION PROBLEM
We first explain how we mitigate the problem of searching in the super-exponential set of graph structures. If there are M such variables, the strategy of considering all the possible structural graphs as separate hypotheses is not feasible because it would require maintaining O(2M²) models of the data. We note that we can cheaply choose any of the 2 possible DAGs through suitable independent Bernoulli choices Ciy associated with each edge i + j of the causal graph, i.e., sampling all the ciy" 's independently. Then we only need to learn the 2 coefficients "ij and we implicitly
maintain a distribution over the MM² models corresponding to all the possible draws of cıj- Note that slight dependency between the Gj is induced if we require the causal graph to be acyelic (which
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allows one to sample the X using ancestral sampling). To enforce that constraint it is not sufficient to require Cyyy; = (both cannot be 1). We deal with this problem with a regularizer acting on the y's in order to favour acyclic solutions (Zheng et al. 2018
In our approach, each random vari- able's structural assignment is mod- eled via X; := (Cso x Xo, Ca1, x X1...., Cim x Xm. Gs). where fo. () is a neural network (MLP) with parame- ters 8t. and Cúj (in(sigmoodd(x.)), Through this construction we can frame the causal induction problem as an optimization problem, with 0 optimized maximize the likelihood of data under the model but opti- mized with respect meta-learning objective arising from changes in distribution because of interventions. There are few benefits for learning a parametrized ensemble of SCMs. Such an ensemble is analogous to an ensemble of neural nets differing by their binary input dropout masks, which select what variables are used as predictors of another variable.
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Mooking tepest valuns with proveze cortgrraccss
XL.
Figure MLP Model Architecture computes conditional probabilities Á, O given their parents using stack of three indopendent MLPs. The MLP input layer uses an adjacency matrix sampled from (er( as an input mask to force the model to make use only of parent nodes predict their child mode.
(fork3) SCM. model
3.3 FAST ADAPTATION BY META-LEARNING
Fast and slow weights To disentangle an environment's stable, unchanging properties (the causal structure) from unstable, changing properties (the effects of an intervention), we proposed in to to distinguish between the model's functional meta-parameters Oslow and parameters Orast. The sum of these weights, 0 = Sslow + ftsst parametrizes the MLPs computing the conditionals P, pa))) p 0,). The fast weights and the slow weights terminology is drawn from Hinton & Plaut] (1987). )). The construction of 8 sum of initial, slow weights plus zeroed, fast weight that then allowed to quickly adapt during a transfer episode is due to MAML (Finn et al. (017). The ability to generalize out-of-distribution by adapting
transfer distribution can then be measured by the likelihood after adapting the fast weights on transfer data.
Since intervention is generally not persistent from transfer distribution another, the model's functional parameters (0tust are reset after each episode of transfer distribution adaptation. The meta-parameters (0slow, ?) are preserved, then updated after each episode. Inspired by Bengio et al.) (2019), the meta-objective for each meta-example over some intervention distribution Diut is the tollowing "meta-tree mneta-transfer loss:
R - -Ex--Dum [log BC- II (X; Oslow)) )]]
(2)
where X is example sampled from the intervention distribution Dines C' is adjacency matrix drawn from our belief distribution (parametrized by 7) about graph structure configurations and
(x,(X) P(X, (Xpttcc)) 0
is the likelihood of the i-th variable X; of the sample X, when predicting it under the configuration C' from the set of its putative parents, Xpa(i.C)-
Structural Parameter Gradient Estimator Because a discrete Bernoulli random sampling pro- cess is used to produce the configurations under which the log-likelihood of data samples is obtained, we require a gradient estimator to propagate gradient through to the structural meta- meta-parameters.
These equations differ from Bengio al. 2019 in that the t indices and products were dropped, because the computation is online.
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We adopt for this purpose the gradient estimate of Bengio al. (2019) (but see foomooe 73
9ij Ee(o(Tag) Cij ) (X)
(4)
Cof((
where the (R) superscript indicates the values obtained for the k-th draw of C. This gradient is estimated solely with fow because estimates employing have much greater variance.
Acyelic Constraint We add regularization term to the loss term that discourages the model from having length-2 eycles in the learned adjacency matrix. The regularizer term is
JDAG
vi,j {0, ................................ M-13
and is derived from Zheng al. 2018). The details of the derivation are in the Appendix.
3.4 MODEL DESCRIPTION
Learner Model We model the M structural assignments X; == fi(X C, No) (Eq. D of the SCM as set of M multi-layer perceptrons (MI.ss), in [nnneio et al., in shape but do not share any parameters, since they are modeling independent causal mechanisms. Each possesses an input layer of M N neurons (for M one-hot vectors of length N each), single hidden layer chosen arbitrarily to have max(4M, 4N) neurons with LeakyReLU activation of slope 0.1, and an output layer N neurons representing the unnormalized log- pag--obbabilites of each category. To force Si to rely exclusively on the direct ancestor set pu(i, C') under adjacency matrix C (See Eqn. 2 the one-hot input vector X, for variable X;'s MLP masked by the Boolean element Cij- The functional parameters of the MLP are the set 0 [WOshin B0 W1 into Blin
al. (2019). The MLPs are identical
An example of the multi-MLP architecture with M = 3 categorical variables of N = 2 categories is shown in Figure I
Ground- Truth Model
our experiments, ground-truth SCM models are parametrized as a set of MLPs of the same size as the learner models, thus avoid having manually define the Conditional Probability Tables (CPTs). Ground-truth models exist in two variants that differ mainly in initial- ization: synthetic, where the 0 are randomly initialized and the either aandomly-iniiialle pre-specified; and real-world, where the 0 and are both initialized so as to closely replicate the CPTs of given Bayesian networks.
or
3.5
INTERVENTIONS
To execute an intervention variable X,, we reinitialize X, ground-truth MLP parameters randomly while leaving other variables' MLPs untouched. copy the old parameters is saved, allowing the intervention to be undone by resetting the parameters back to their original values.
Soft interventions
Predicting interventions After intervention on X,, the gradients into the learned model's '' 7/ and the slow weights for the i-th conditional are false, because they do not bear the blame for X,'s outcome (which lies with the intervener). We find that ignoring this issue considerably hurts or slows down meta-learning, suggesting that we should try to infer on which variable the intervention took place. For this purpose, we take advantage of the fact that the conditional likelihood of the intervened variable tends to have a poorer relative likelihood under Dint. so we pick the variable with the greatest deterioration in likelihood as our guess.
3.6 TRAINING ALGORITHM
The structural meta-parameters (1ai) represent the belief in the hypothesis that node i has node j as a direct causal parent. We may sample from this belief, obtaining different configurations (causal structures) of the causal graph. We hypothesize that the correct configuration enables better adaptation to slight change in distribution, e.g. resulting from a soft intervention. Hence, we evaluate different configurations under the transfer distribution; those giving higher transfer likelihood under Oaw get
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higher reward and their probability is increased. The functional (meta- pparmmetaasamsessss are trained as usual by gradient ascent on the log- likeliiiiiood. The details of the training algorithm is in Section [A.2 in the Appendix.
Synthetic datasets use either a specified edge structure for y, or randomly initialize "Ky such that is Boolean, strictly lower-triangul and each row has expected sum of 1-5 (and therefore each node expects 1-5 direct ancestors). Real-world datasets, specified as CPTs, must first be converted or
approximated by a near-ground-truth MLP. We use the graph 's proper edge structure to initialize 7, and learn 0 by training individually each MLP in the set to replicate the correct pre-softmax logits. In practice, excellent reproductions of the CPT can be achieved.
Stability of training Our model requires simultancous training of both the structural and the functional pettaparameterss but these are not independent and do influence each other, which leads to instability in training. For example, if (74y) R 0 incorrectly, the i-th MLP does not learn to use input x, and vice-versa, if the i-th MLP has not learned to properly use input Xj. this will favour pushing (?%y towards O. To overcome this instability, we pretrain the model under observational data (from the distribution of the data before interventions) using dropout on the inputs. This ensures that the functional meta- meta-parameters Osw are not too biased towards certain configurations of the meta meta-parameters y.
RELATED WORK
The recovery of the underlying structural causal graph from observational and interventional data is a fundamental problem (Pearl 1995, 2009). Different approaches have been studied, score-based, constraint-based and asymmetry-base based methods. Score-based methods search through the space of all possible directed acyclic graphs (DAGs) representing the causal structure based on some form of scoring function for network structures (Chickering, (2022] samarddnos et al. (2066) Goudet et al.)
2017 Hauser & Bühlmann 2012) Heckerman et al. Cooper & Yool 1999 Hauser & Bühlmann
2012). Constraint-basee methods (Spirtes et al. 2000; Sun et 2 /hang et al. | (2012) infer the
DAG by analyzing the conditional independence of data. |aaton & Murphy (2007b) use dynamic
programming techniques to accelerate Markov Chain Monte Carlo (MCMC) sampling in a Bayesian approach to structure learning for discrete variable DAGs, Asymmetry -symeetry-based methods (Shimizu et al.
2006 Hoyer al. 2009 Daniusis al. 2012, Budhathoki Vreeken 2017 Mitrovic al.) 2018) assume asymmetry between cause and effect in the data and try to use this information to estimate the causal structure. Recently Peteers et al. (2016); Ghassami et al. (2017) proposed to exploit invariance across different environments to infer causal structure, but are difficult to scale to large graphs due to the necessary iteration over the super-exponential set of possible graphs.
For interventional data, it is often assumed that the models have access to full intervention information, which is rare in the real world. Rothenhäusler et al. (2015) have investigated the case of additive shift interventions, while [aato& & Murphy (2007a) have examined the situation where the targets of experimental interventions are imperfect or uncertain. This is different from our setting where the intervention is unknown to start with and is assumed to arise from other agents and the environment.
Learning based methods have been proposed (Guyon a Lopez-Paz et al.
2015) and there also exist learn causal signals from
recent approaches using the generalization ability of neural networks
purely observational data (Kalainathan et al.] 2018 Goudet et al.) (018).
Neural network methods
equipped with learned masks, such as (Ivanov al., 2018 et al. 2019 Yoon et al. 2018 Douglas et al. 2017), exist in the literature, but only a few (Kalainathan et al.) 2018) have been adapted to causal inference. This last work is, however, tailored for causal inference on continuous variables and from observations only. Adapting it to fiiscrete-varibble setting is made difficult by its use of a Generative Adversarial Network (GAN) Goodfellow et al. (2014) framework.
Turning now to meta-learning, Dasgupta et al. (2019) have used it to learn to make predictions under interventions. However, their approach does not induce a causal graph, neither explicitly nor via decoding. Thus, it cannot be used for general causal discovery, but only to make predictions of variable values. Most similar to our work, Bengio et al. (2019) proposes a meta-learning framework for learning causal models from interventional data. However, the proposed method (Bengio et al. 2019) explicitly models every possible set of parents for every child variable and attempts to distinguish the best among them. Because there beyond trivial graphs. In our work, we bypass this restriction by modeling the edge between any variables as dropout probability and hence our model only scales quadratically with the graph size.
Nombinatorially-maan
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5 EXPERIMENTAL SETUP AND RESULTS
Our experiments aim to evaluate the proposed method to recover the correct causal structure and which elements of the method matter. We first evaluate our model synthetic dataset where have control over the number of variables and causal edges in the ground-truit SCM. This allows us verify after learning with what aceuracy understand the performance of our algorithm under various conditions. We then evaluate our method on real world datasets collected from the BnLearn dataset repository, and show that the proposed approach recovers the true causal structure. We then show that the trained models can correctly predict the consequences of previously unseen interventions on the rest of the graph. We also perform ablations showing how important is each component of the model.
recover the individual decisions about Cij and
5.1 SYNTHETIC DATASETS
Figure 2: Leamod odges at three different stages of training. Left: Chain graph with variables. Right: Fally-connected DAG graph with variables.
We first evaluate the model's performance several randomly-initialized SCMs with specific, representative graph structures. For M 3-varialle DAGs, we consider every possible connected graph: chain3, fork3, collidee Liderr and confounder33 (See Fig. 7 in appendix). They exhibit every graph sub-structure that can exist in larger graphs, and must mastered before tackling larger graphs. Since the number of possible DAGs grows super-exponentially with the number of variables, for MM > 3 up to a selection of representative and edge-case graphs are chosen. The cha nM and fullM graphs (M 3-8) the minimally and maximally connected M-variable graphs, while the remaining graphs are randomly generated with varying sparsity level (1-4 expected number of parents per node). The details of the setup can be found in Appendix A.5
Results The model can successfully recover the correct edge structure for all synthetic graphs con- sidered. The learning curves plotting the average cross-entropy (CE) loss for the learned edges against the ground-truth model for M = 3 are shown in Figure 7 The fully-connected con confoure fonnded graph is particularly easy to learn, but all 3-variable graphs are learned perfectly. Plots of the same process on 4- through 8-variable graphs were similarly encouraging, with all models converging to a negligible loss. The results are, however, sensitive to some hyperparameters, notably the DAG penalty and the sparsity penalty.
5.2 REAL-WORLD DATASETS: BNLEARN
The Bayesian Network Repository is collection of commonly- used causal Bayesian networks from the literature, suitable for Bayesian and causal learning benchmarks. We evaluate our model on the Earthquake (Korb & Nicholson, 2010), Cancer (Korb & Nicholson 2010) and Asia (Lauritzen & Spiegelhalter
1988) datasets (M =5, 5 and 8-variables respectively,
Figure 3: Farthquake: Leamed edges three dif- forent stages training.
imum 2 parents per node) in the BnLearn dataset repository.
The ground-truth SCM given for each dataset, and the functional parameters are represented as conditional probability tables (CPTs). We learn a wear-ground-trutte MLP from the dataset's s CPT and the ground-truth data generator. We also insert (greater than 1) temperature factor in order to increase the likelihood of sampling some very rare events in the CPTs. Details of the setup can be found in Appendix A.S.I
The model can successfully recover the correct edge structure for all BnLearn graphs considered up to and including 8-variable Asia. Figures and illustrate what the model has learned at several stages of learning. In these figures, the a (Tiy) and Cıy adjacency matrix elements plotted as colored squares and dots. Off-diagonal terms are unknown, and appear yellow.
Results
Figure 4: Learned edges at different stages of training.
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COLLERN
Statnn
Tarka crncoradact
Figure Left: Cross entropy (CE) for edge probabiity between learned ground-truils graphs 3-variable SCMs. Error #1 over PRNG seeds 1.5. Middle: Edge CE loss for the chain graph with 4.7 variables. Right: Edge CE loss for 3 3-variable graphs with no dropout during pretmining, showing the importance dropout.
dattal, LEAG
Foeth LDAG
Foekk LEAG
1D40
1EA0
LIAA
Figure study
possible variable graphs. Boih graphs show cross-entzupy loss learned ground edges training time. Left: Models that infer the interveation (prediction. bold) vs models that have knowledge of the trae interveation (ground truth, long dash) vs models that use no knowledge the intervention at all (no prediction, short dash), Result suggests inferring the intervention works almest well knowing the true intervention. Right: Comparisons of model trained with and without DAG regulatizer (Tnwо), showing that DAG regularizer helps convergence.
The color of the squares indicates belief in the presence or absence of an edge (1)7)) while the color of the dot indicates the ground truth Cij. Red indicates (belief in) an edge; Blue indicates (belief in) the absence of an edge. Yellow indicates maximum uncertainty At the beginning of training, the main diagonal is priori known to be clear of edges, and therefore solid blue.
As training progresses, the beliefs approach the ground truth, which visually appears as the squares converging towards the color of the dot within them. When they coincide, the dot vanishes. An erroncous belief stands out as red dot blue square or, vice-versa, blue dot red square. Because we pre-sort the nodes in Bnlearn datasets so that they are topological order, the model must learn lower triangle. This corresponds to completely blue uppper triangle. Anti-causal violations are easily recognizable as non-blue squares in the upper triangle.
Baseline comparisons We compared our method to ICP (Peters et al. 2016) and Eaton & Murphy (2077)). Eaton & Murphy (2007a) handles uncertain interventions and Peters et (2016) handles unknown interventions. However, neither attempt to predict the intervention.
Table 1: Baseline comparisons: Cruss entropy (lower is better) for edge probability on learned and ground truth edges on Asia graph.
compared Peters al 2016). (Faton Murby 2007a Zheng et 2018)
Our method
(Ettnn & Murphy 2007a)
(Peters et al. 2016
(Zheng et al. 2018)
0.0
0.0
10.7
3.1
Importance of Dropout To perform initial pretraining for an observational distribution, sampling adjacency matrices is required. One may be tempted make these "fully-connected" (all-ones except for a zero diagonal), to give the MLP maximum freedom to learn any potential causal relations itself. We demonstrate that pretraining cannot be carried out this way, and that it is necessary to "drop out" each edge (with probability 0.5 in our experiments) during pre-training of the conditional distributions of the SCM. We attempt to recover the reriiously-rcoovrrbbbl graphs chain3, fork3 and confounder3 without dropout, but fail to do so, shown in Figure S
Generalization to Previously Unseen Interventions It often argued that learning approaches based on prediction do not necessarily yield models that generalize to unseen experiments. since they do not explicitly model changes through interventions in contrast causal models use the concept
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Table 2: Evaluating the consequences of previously unseen intervention: (lest lig-iikelitood under intervention)
fork3
chain3
confounder3
collider3
Our Model
-0.4502
-0.3801
-0.2819 -0.3628
-0.4677
Baseline
-0.5036
-0.4562
-0.5082
of interventions to explicitly model changing environments and hold thus the promise to work even under distributional shifts (Pearl 2009, /Schölkopf et al. 2012 Peters al. 2017).
To test the robustness causal modelling previously unseen interventions (new values for an intervened variable), we evaluate well-trained causal model against non-causal variant model where all Cty 1, * j. both cases, an intervention performed, and the models, with knowledge of the intervention, are asked predict the rest. For this purpose, batch of samples X are drawn from Di and their average log- lig-likelihoods are computed and contrasted. The intervention variable's contribution to the log-likelihood is ignored.
For all 3-variable graphs (chain3, fork3, col collider 10lidde3,, confounder the causal model attributes higher liglliiel to the intervention distribution'ss 's samples than the non-causal variant, thereby demonstrating causal models' superior generalization ability in transfer tasks. collecso these results.
Importance of Inference After the inter- vention has been performed, the learner draws data samples from the intervention distribution and computes the per-variable average log- prgpobbabylltyy under sampled ad- jacency matrices. The variable consistently producing the least-likely outputs is pre-
Table Intervention Prediction Accuracy: (identify which variable the intervention took place)
3 variables
variables
5 variables
8 variables
95 %
90 %
81 %
63
dicted to be the intervention node. Experiments over all 3-variable DAGs show that this prediction mechanism functions well in practice, yielding far above-random accuracy in correctly predicting the intervention node (Table 3). the model performance dropped significantly without the predication (Figure Left) and is comparable to having the ground-truth intervention (Figure 6 Right).
Effect of DAG Regularizer: To promote the acyclicity of C, we include a DAG regularizer This significantly improves cross-entropy of the solution (wrt. ground truth DAG) on all 3-variable graphs, as illustrated in Figure 6 and was therefore included in all experiments with M >> 3 variables.
CONCLUSION
In this work, we introduced framework for fast adaptation and slow learning of neural causal models. We demonstrate through experiments that the principle of optimizing an outto--distribution meta-learning objective enables the learner to recover the causal graph structure for graphs with more than two variables. To achieve this we introduce an efficient parametrization of the belief regarding the underlying graph structure, implemented as an adaptive form of dropout on the inputs of MI.Ps computing the conditionals of the model. This relies on pre-training the conditionals using agnostic beliefs and by approximately inferring which variable the intervention took place. We believe that our approach of treating seemingly observational data as being derived from an environment with agents executing interventions could represent important change in modelling perspective with deeper implications.

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Under review as a conference paper at ICLR 2020
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A APPENDIX
A.1 DAG PENALTY DERIVATION
Recall, from Zheng et al. 2018):
Theorem 1. A matrix W E Rdxd is DAG if and only if
h(W) = Tr(eWoW) - d =
For the two-variable (d 2) graph with adjacency matrix
0
a(w1))
W =
((w2))
have
A
Tr(exp(A)) = Tr Σ
n!
7=0
0² (wi2)
Tr(exp( ( W)) = Ir
n!
a² (w=2)
n=0
There only be odd-length paths in graph. Because, in two-variable graph with self-edges, all even-length paths are cycles and none of the odd-length paths are, we have
(
Tr
(2k)
o (w21)
k=0
Even n, Tr of
21TT
0
Tr
(2k))
(221)
k=
Odd m.
80
02(112)0²(waa)
0
(2k)!
0
(w
km0
22h (w12)o2h (w21)
(2k)
cosh(a (w12 (w21))
A pairwise generalization to multinode graphs over all i + j is:
JDAG
Σ (osh(a(mr))aa(
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A.2 TRAINING ALGORITHM
In this section, we describe the training algorithm in detail.
Algorithm 1 Training Algorithm
1: procedure TRAINiNG(Categee Distribution D, with M nodes and N categories)
2: Let integer from 0 to M
4:
for Kprettain steps do I D Ber(o((1) log P(z|c) Blw + Adam Oslows Vb
b. Pretraining Loop D. Sample data from D Sample config from structure distribution Compute log log- probability of data given config
5:
6:
8:
> Update 8soo
9:
for kimterention Steps do
D Interventions Loop
10:
IN < randint (0, M - 1)
11:
D=s D with intervention node I if predicting intervention then L, 0 Vi
12:
13:
14:
for hipeddict steps do Dim
> Prediction Loop b Draw batch of data from D Draw config from structure distribution b Accumulate NLL for every node separately
15:
16:
Ber (
I, log P. (arlos;; (alles::stu.v
N arg
19:
ganmagrads, logregrets = [], I] for Repuade steps do
20:
b Transfer Episode Adaptation Loop
21:
D
22:
gammagrad, logregret 0,
kerz steps do c ~ Ber (7))) L = log Pz(2lcs (sww) Vi
Configurations Loop
24:
25:
26:
gammagrad += (77)
Accumulate yy gradient Accumulate regret
27:
logregret
Σ Li
28:
gammagrads append (ganmagrad) logregrets... wwtpgretts..ppndddeeeeee (1 ogregret)
29:
30:
J f AMavEa Lam Aspare (sp + ADAG LoaG(T)
D Regularizers "Y gradient estimator
31:
V, V.JJ + gammagradswy wtw.odddwwwyyoooooeeeereeee sottmaax
32:
7 +- Adam(y, V.)
b Update
A.3 PRELIMINARIES
Interventions In a purely-observationa setting, is known that causal graphs can be distinguished only to a Markov equivalence class. In order to identify the true causal graph intervention data is needed (Eberhardt et al.] (012). Several types of common interventions may be available (Eaton & Murphy 2007a). These are: No intervention: only observational data is obtained from the ground truth causal model. Hard/perfect: the value single or several variables fixed and then ancestral sampling is performed on the other variables. Soft//mpperfect: the conditional distribution of the variable on which the intervention is performed is changed. Uncertain: the learner is not sure of which variable exactly the intervention affected directly. Here make use of soft interventions for several reasons: First, they include hard interventions as a limiting case and hence are more general. Second, many real-world scenarios, a soft one. We also deal with special case of uncertain interventions, where the variable selected for intervention is random and unknown. We call these unidentified or unknown interventions.
more difficult to perform hard intervention compared to
Causal sufficiency The inability to distinguish which causal graph, within Markov equivalence class, the correct one in the surely-observationn setting is called the identifiability problem. In setting, all variables observed (there are no latent confounders) and all interventions
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random and independent. Hence, within setting the true causal graph is always identifiable in principle (Eberhardt et al. [012 Heinze-Deane Deml et al. 2018). We consider here situations where single variable is randomly selected and intervened upon with soft or imprecise intervention, its identity is unknown and must be inferred.
A.4 EXPERIMENTAL SETUP
For all datasets, the weight parameters for the learned model is initialized randomly. In order to not bias the structural parameters, y initialized 0.5 the beginning training.
A.5 SYNTHETIC DATA
SCM with n variables modeled by n feedforward neural networks (MLPs) as described in section
3.1 For simplicity, assume acyclic causal graph such that could easily sample from it. Hence, given any pair of random variables and B, either - B, B + A or A and B are independent.
The MLP representing the ground-truth SCM has its weights 0 initialized use orthogonal initialization with gain 2.5 and the biases are initialized using uniform initialization between -1.1 and 1.1, which was empirically found to yield "interesting" yet learnable random SCMs.
Ulird
collider
fork3 corfouncer
fork3
cellider3
3
Rumber
Figure 7: Left: Every possible 3-variable connected DAG, Right: Cross entropy for edge probability between learned and ground-truth SCM for all 3-variable SCMs.
A.5.1 BNLEARN DATA REPOSITORY
The repo contains many datasets with various sizes and structures modeling different variables. We evaluate model 3 of the datasets in the repo, namely the Earthquake (Korb & Nicholson 2010), Cancer (Korb & Nicholson 2010) and Asia (Lauritzen & Spiegelhalter 1988) datasets. The ground-truth model structure for the Cancer (Korb & Nicholson, 2010) and Harthquake (Korb & Nicholson 2010) datasets are shown in Figure 8 Note that even though the structure for the 2 datasets seems be the same, the conditional probability tables (CPTs) for these 2 datasets very different and hence results in different structured causal models (SCMs) for the 2 datasets.
Figure S: Lell: Ground Truth SCM for Cancer. Middle: Groundrruth SCM for Earthqunke, Right: Groundiruth SCM for Asiu.
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A.5.2 TRAINING GROUND-TRUTE DATA GENERATOR
Because a CPT capable of representing any distribution, and MLPs are strictly less powerful in this respect, it may not be possible to learn perfectly the distribution with our MLP learner model. We therefore train a near-ge near-ground -round- MLP to replicate as closely as possible the CPT's probability table, and then use this trained MLP are the ground-truie SCM data generator.
Training is by 1000 iterations of full-batch gradient descent with learning rate 0.001 and momentum 0.9, with all possible parent values masked with the ground-truth vector 7/j. The objective is to minimize mean squared error between the ML.'ss logits and the log-probability as drawn from the CPT. If the CPT contains zero, it is approximated by a logit of 100.
Given that some of the CPTs contain very unlikely events, we have found it necessary to add a lemperature parameter in order to make them more frequent. The near sor-grouun-truth MLP model's s logit outputs are divided by the temperature before being used for sampling. Temperatures above 1 result in more uniform distributions for all causal variables; Temperatures below I result in less uniform, sharper distributions that peak around the most likely value. We find empirically that a temperature of about 2 is required for our BnLearn benchmarks.
A.6 EFFECT OF SPARSITY
We use a L1 regularizer on the structure parameters " to encourage a sparse representation of edges in the causal graph. In order to better understand the effect of the L1 regularizer, we conducted ablation studies on the I.1 regularizer. It seems that the regularizer has an small effect on rate of converges and that the model converges faster with the regularizer. This is shown in Figure
C.23
0.03
quarilyy 6.1
spen D.:
apandily
Eobodo
Figure 9: Effect of Sparsity: On 5 variable, 6 variable and varioble Nodes
A.7 EFFECT OF TEMPERATURE
As noted in section A.5.1 we have introduced temperature hyperparameter in order to encourage the groundtruth model to generate some very rare events in the conditional probability tables (CPT's) more frequently. We run ablation studies to understand the importance of the temperature term. A temperature of 1 corresponds to no changes to the underlying CPTs. As shown in Figure [2, for the Cancer (Korb & Nicholson 2010) dataset, a temperature of 2 improves the accuracy of causal graph recovery.
cencer_temp2
cancer templ
Crest 11tro0 (C21
0.6
0.4
022
Number Eploodos
Figure Cruss entrupy for edge probability between and ground truth SCM Cancer varying temperatures.
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arrqquake
1.2
Datrry
0.
5030
10000
15:00
20004
20203
20533
32
200
10302
12303
2000
2400
3000
Number Ephodm
Figure entropy edge prohability leamed ground-trath The Asin datasct with 8 variables
Farthquake variables. Right:
12+
0
DANas, preditios
1.0
DlSss, grouad
Gamma atrpp (ong
0K
5K
10K
15K
Steps
Figure 12: Left: 5CM for crosst graph. Right: Cross entropy for edge probability between learned and ground-truth SCM for chaimb and comparing the leaming precess with prodiction edges ground-truth intervention
17


Reviews:
[{"review_id": "rylbZRDpYS", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a MAML objective to learn causal graphs from data. The data in question is randomized but the algorithm does not have access to the identity of the intervention variable. So there is an added layer of complexity of deciphering which variable was intervened on. The MAML objective, in this case, links the causal structure to the slow-moving parameter theta_slow.\n\nThe novelty of the paper seems to be in the application of the MAML framework to causal discovery which is interesting to me. I think a little theory about the sensitivity of the claim of ' theta slow changes relate to the causal structure ' is important. Even showing empirically which sort of graphs and functions become issues for the model would be useful.\n\nHere are my issues with the paper:\n - No error bars for cross-entropy are reported in the experiments.\n - The acyclic regularizer does not reject large length cycles than 3.\n - The ability to predict interventions seems to drop off sharply as the number of nodes increases. This suggests an inability to scale to more than 20 variables.\n - The experimental setup of uniformly sampling an intervening variable seems artificial to me.\n - MLP-specification of the SCM also seemed a bit artificial to me. \n\nOverall, the experiments look reasonable and the method itself seems interesting although further work is needed to show it is useful.\n\n(writing comments) The paper could use a more structured re-write. I had trouble tracking terms around the paper. For example, there seems to be a difference between P_i and P because the former uses theta_i and the latter only uses theta_slow only.\n\n---------------------------------\n\nUpdated score to 6 after rebuttal.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "Skx29AZdFH", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes an SCM-model based on masked neural networks to capture arbitrary conditional relationships combined with meta-learning-style adaptation to reflect the effects of various unknown interventions. Overall the paper is well written and easy to follow, but some conceptual issues remain.\n\n\n- How come there is hardly any discussion of the identifiability issue beyond the few sentences in A.3. This is one of the key issues in learning SCMs and it is strange that the concept of \"faithfulness\" is not even mentioned in the paper.\n\nIn general, there is hardly any discussion of what conditions are required for the proposed estimates to even be valid. The authors seem to be optimistically assuming that their neural network + metalearning model will somehow pick up on the correct structure, without any actual conceptual investigation of this issue.\n\n- The massive downside of neural nets is all the various hyperparameters one has to set (eg. architecture, optimizer, activations, etc). In this setting, how do the authors propose selecting hyperparameter values? How does the reader know the authors did not simply tune their hyperparameters to best match the underlying ground truth (I assume the proposed methodology has many more hyperparameters and thus more degrees of freedom here).\nI would like to see the empirical performance of different variants of your model with different hyperparameter values to assess its sensitivity to these choices. \n\n- Why does one even care about the graph being acylic in this setting?\nThe mere fact that the authors require a regularizer to ensure acylicity suggests this approach is prone to mis-identifying the ground truth structure (which is always acyclic in the experiments).\n\n- One main reason for SCM modeling in science and policy-making is for analysts to better understand the data generating phenomena.  However your use of neural networks here seems to hamper interpretability, so how do you reconcile this issue? Also is your sparsity regularizer satisfactory to confidently diagnose presence/absence of an edge (in constrast to statistical hypothesis tests, say based on conditional independence). Isn't this heavily influenced by the particular sparsity-regularizer value that happened to be selected?\n\n\n- Related papers that utilize the same idea of predicting a variable conditioned on subset of other variables via neural network + masking strategy:\n\nIvanov et al (2019). VARIATIONAL AUTOENCODER WITH ARBITRARY CONDITIONING. \nhttps://openreview.net/pdf?id=SyxtJh0qYm\n\nLi et al (2019). Flow Models for Arbitrary Conditional Likelihoods.\nhttps://arxiv.org/abs/1909.06319\n\nYoon et al. GAIN: Missing data imputation using generative adversarial nets. Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, 2018. http://proceedings.mlr.press/v80/yoon18a.html\n\nDouglas et al. A universal marginalizer for amortized inference in generative\nmodels. arXiv preprint arXiv:1711.00695, 2017\n\nFor clarity, the authors should highlight the differences of their approach from these works (beyond the causal setting).\n\n- Given the lack of theoretical / conceptual guarantees that the methodology will work, our faith in the proposed methodology rests entirely on the empirical experiments.  However, I find these a bit too basic to be very convincing, and would at least like to see more methods being compared (in particular for the simulated graphs as well).\n\n- The authors should describe what are the underlying interventions in each dataset a bit more.\n\n- The Figures should be better explained (took me a while to figure out what dots/colors represent).\n\n- Why do the authors report cross entropy loss in Table 1? To my knowledge this is not a standard metric for measuring the quality of structure-estimates.\n \n- Instead of ICP (which is constrained to be linear which is unrealistically simple), why don't the authors compare against nonlinearICP (which is more flexible like their neural networks):  \n\nHeinze-Deml et al (2018). Invariant Causal Prediction for Nonlinear Models. https://arxiv.org/pdf/1706.08576.pdf", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}, {"review_id": "HJxO6r5HYr", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper develops a learning-based causal inference method that performs multiple tasks jointly: \n\ni) scalable discovery of the underlying Structured Causal Model (SCM) by modeling both structural assignments and the SCM as a continuously parameterized chain of distributions, \n\nii) identification of the intervened variables, which are not known to the model a-priori unlike the mainstream causal inference setups,\n\niii) achieving the two aforementioned goals using meta-learning driven heuristics, i.e. interventions cause distributional shifts. \n\nWhile the paper adopts the core design choices from recent prior art (Bengio et al., 2019), the proposed methodology (especially ii)) is sufficiently novel to be published as a main-track conference paper. The paper is very well-written, follows a concrete and easy-to-follow story line. It solves multiple ambitious problems end-to-end and justifies the methodological novelty claims by a properly conducted set of experiments. The paper also successfully employs simple and useful but forgotten old techniques such as fast/slow parameter decomposition in the proposed model pipeline.\n\nThe intervention prediction heuristic is splendid. It is simple, sensible, and has been proven by experiments to be very effective. I would rate this as the primary novelty presented in this paper.\n\nThe paper can be improved if the below relatively minor concerns are addressed:\n\n i) It would be informative if the paper had a paragraph discussing also the fundamental limitations of the approach more openly. For instance, the choice of the neural net architecture used for the structural assignment might have a huge impact on the outcome, especially because the same architecture is repetitively used for all variables of the SCM. Furthermore, treatment of each variable with a fully independent neural net could cause overparameterization as the SCM grows in number of variables.\n\n ii) The paper makes a strong scalability claim across the variable size thanks to independent Bernoullis assigned on the adjacency matrix entries. However, it reports results only for very small SCMs. It is understandable that given the premature stage of the causal inference research might not grant standardized data sets at a larger scale, but at least lack of this quantitative scalability test could be acknowledged and the related claims could be a little bit softened.\n\n iii) I do not buy the argument in the first paragraph of Sec 3.5 about why the structural assignment functions need to be independent. As the model does not pose a distribution on neural net weights, sharing some weights (i.e. conditioning on them) would only bring conditional independence across the variables. I do not see a solid reason to try to avoid this. What is wrong for multiple variables to share some functional characteristics in their structural assignment? After all, some sort of conditional independence will be inevitable in modeling. If the variables share the same architecture, this is also conditional independence, not full independence. Relaxing the independence assumption and allowing some weight sharing could be beneficial at least for scalability of the model, could even bring about improved model fit due to cross-variable knowledge transfer.\n\nOverall, none of the aforementioned three weaknesses is fundamental. In the status-quo, this is a spectactular research paper and my initial vote is an accept.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a metalearning objective to infer causal graphs from data based on masked neural networks to capture arbitrary conditional relationships. While the authors agree that the paper contains various interesting ideas, the theoretical and conceptual underpinnings of the proposed methodology are still lacking and the experiments cannot sufficiently make up for this. The method is definitely worth exploring more and a revision is likely to be accepted at another venue.
Decision: Reject