Learning to Induce Causal Structure 
Authors: Nan Rosemary Ke
Silvia Chiappa
Jane X Wang
Jorg Bornschein
Anirudh Goyal
Melanie Rey
Theophane Weber
Matthew Botvinick
Michael Curtis Mozer
Danilo Jimenez Rezende
Venue: ICLR 2023 poster (2023)
Keywords: causality
deep learning

Abstract: The fundamental challenge in causal induction is to infer the underlying graph structure given observational and/or interventional data. Most existing causal induction algorithms operate by generating candidate graphs and evaluating them using either score-based methods (including continuous optimization) or independence tests. In our work, we instead treat the inference process as a black box and design a neural network architecture that learns the mapping from both observational and interventional data to graph structures via supervised training on synthetic graphs. The learned model generalizes to new synthetic graphs, is robust to train-test distribution shifts, and achieves state-of-the-art performance on naturalistic graphs for low sample complexity.

Published as a conference paper at ICLR 2023
LEARNING TO INDUCE CAUSAL STRUCTURE
Nan Rosemary Ke 1,2.3, Silvia Chiappa ', Jane Wang 1 Anirudh Goyal 1,2,4 Jorg Bornschein 1, Melanie Rey" Theophane Weber Matthew Botvinick Michael Mozer Danilo Jimenez Rezende'
ABSTRACT
One of the fundamental challenges in causal induction is to infer the underlying graph structure given observational and/or interventional data. Most existing causal induction algorithms operate by generating candidate graphs and evaluating them using either score- score-based methods (including continuous optimization) or independence tests. In our work, we instead treat the inference process as black box and design neural network architecture that learns the mapping from both observational and interventional data to graph structures via supervised training on synthetic graphs. The learned model generalizes to new synthetic graphs, is robust to train-test distribution shifts, and achieves state-of-the-art performance on naturalistic graphs for low sample complexity.
INTRODUCTION
The problem of discovering the causal relationships that govern a system through observing its behavior, either passively (observational data) or by manipulating some of its variables (interventional data), lies at the core of many important challenges spanning scientific disciplines, including medicine, biology, and economics. By using the graphical formalism of causal Bayesian networks (CBNs) (Koller Friedman| [2099 [Paarl 2009). this problem can be framed as inducing the graph structure that best represents the relationships. Most approaches to causal structure induction (or causal discovery) are based on an unsupervised learning paradigm in which the structure is directly inferred from the system observations, either by ranking different structures according to some metrics (score-based approaches) or by determining the presence of an edge between pairs of variables using conditional independence tests (constraint-based approaches) (Drton & Maathuis [017]] Glymour eet al., 2019 Heinze-Deml et al., 2018a (see Fig. [a)). The unsupervised paradigm poses however some challenges: score-based approaches are burdened with the high computational cost of having to explicitly consider all possible structures and with the difficulty of devising metrics that can balance goodness of fit with constraints for differentiating causal from purely statistical relationships (e.g. sparsity of the structure or simplicity of the generation mechanism); constraint-based methods are sensitive to failure of independence tests and require faithfulness, a property that does not hold in many real-world scenarios (Koski & Noble 2012, Mabrouk et al.) 2014).
Recently, supervised learning methods based on observational data have been introduced as an alternative to unsupervised approaches (Lopez-Paz et al. 22015 b Li al.) 2020). In this work, we extend the supervised learning paradigm to also use interventional data, enabling greater flexibility. We propose model that is first trained synthetic data generated using different CBNs to learn mapping from data to graph structures and then used to induce the structures underlying datasets of interest (see Fig. [b)) The model is a novel variant of transformer neural network that receives as input a dataset consisting of observational and interventional samples corresponding to the same CBN and outputs prediction of the CBN graph structure. The mapping from the dataset to the underlying structure is achieved through an attention mechanism which alternates between attending different variables in the graph and to different samples from a variable. The output is produced by a decoder mechanism that operates as an autoregressive generative model on the inferred structure. Our approach can be viewed as form of meta-learning, whereby the relationship between datasets and causal structures underlying them are learned rather than built-in.
DeepPind,, Mila, Polytechnique Montreal, University of Montreal, Google Research, Brain Team, Corresponding author: nke nkelgoog e. com

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Figure I: (A). Standard unsupervised approach to causal structure induction: Algorithms use a predefined scoring metric or statistical independence tests to select the best candidate structures. (B). Our supervised approach to causal structure induction (via attention; CSIvA): A model is presented with data and structures as training pairs and learns a mapping between them.
A requirement of a supervised approach would seem to be that the distributions of the training and test data match or highly overlap. Obtaining real-world training data with a known causal structure that matches test data from multiple domains is extremely challenging. We show that meta-learning enables the model to generalize well to data from naturalistic CBNs even if trained on synthetic data with relatively few assumptions, where naturalistic CBNs are graphs that correspond to causal relationships that exist in nature, such as graphs from the bnlearn repository (www. onl eallearn.. com/ borepos i tory). We show that our model can learn mapping from datasets to structures and achieves state-of-the- datasets (Lauritzen & Spiegelhalter 1988 Sachs al. 2005 Spiegelhalter & Cowell, 1992). despite never directly being trained on such data. Our contributions can be summarized as follows:
performance on classic benchmarks such as the Sachs, Asia and Child
We tackle causal structure induction with supervised approach (CSIvA) that maps datasets composed of both observational and interventional samples to structures.
We introduce variant of a transformer architecture whose attention mechanism is structured to discover relationships among variables across samples.
We show that CSIvA generalizes to novel structures, whether or not training and test distributions match. Most importantly, training on synthetic data transfers effectively to naturalistic CBNs.
We show that CSIvA significantly outperforms sate-of-the-att causal structure induction methods such as DCDI (Brouillard et al. 2020) and ENCO (Lippe et al. 2021) both on various types of synthetic CBNs, as well as naturalistic CBNs.
2 BACKGROUND
In this section we give some background on causal Bayesian networks and transformer neural networks, which form the main ingredients of our approach (see Appendix AA. for more details).
Causal Bayesian networks (CBNs). A causal Bayesian network (Koller & Friedman, 2009 Pearl 2009) is pair M = (G, p). where G is a directed acyelic gruph (DAG) whose nodes XI,..., XN represent random variables and edges express casual dependencies among them, and where p is a joint distribution
all nodes such that p(X1,..., Xx) II,, p(X, pa.X..). where pa(X,) ) are the parents of X,, i.e. the nodes with an edge onto Xx, (direct causes of X,). An input to the transformer neural network is formed by dataset D {a+}s J==1. where is either an observational data sample or an interventional data sample obtained by performing an intervention on a randomly selected node in G. Observational data samples are samples from [.X,...,,xx) Except where otherwise noted, for all experimental settings, we considered hard interventions on a node X,, that consist in replacing the conditional probability distribution (CPD) p(X,, pa(Xx)) with a delta function 8x,,== forcing X,, to take on value x. Additional experiments were also performed using soft interventions, which con- sisted of replacing p(X,, pa()))) with different CPD p' (Xn,. pa(X.)). An interventional data sample is a sample from 6x,,=0 In P(X,) pa(X,)) in the first case, and a sample from p'(X, pa(Xx)) I Im== p( p(x,, pa(X.)) in the second case. The structure of G can be repre-
x² == (xf....,
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sented by adjacency matrix A, defined by setting the (k. 1) entry, Ag, to if there is edge from X1 to X5 and to 0 otherwise. Therefore, the n-th row of A, denoted by An.:s indicates the parents of X, while the n-th column, denoted by A. indicates the children of X,.
Transformer neural network. A transformer (Devlin et al. 2018: Vaswani et al. 2017) is a neural network equipped with layers of self-attentio that make them suited to modeling structured data. In traditional applications, attention is used to account for the sequentially ordered nature of the data, e.g. modeling sentence as a stream of words. In our case, each input of the transformer a dataset of observational or interventional samples corresponding to the same CBN. Attention is thus used to account for the structure induced by the CBN graph structure and by having different samples from the same node. Transformers are permutation invariant with respect to the positions of the imput elements, ensuring that the graph structure prediction does not depend on node and sample position.
CAUSAL STRUCTURE INDUCTION VIA ATTENTION (CSIVA)
Our approach is to treat causal structure induc- tion as a supervised learning problem, by train- ing neural network to learn to map obser- vational and interventional data to the graph structure of the underlying CBN. Obtaining diverse, real-world data with known causal relationships in amounts sufficient for super- vised training is not feasible. The key contri- bution of this work is to introduce method that uses synthetic data generated from CBNs with different graph structures and associated CPDs that is robust to shifts between the train- ing and test data distributions.
AN.N
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3.1 SUPERVISED APPROACH
Attention across sample
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We learn distribution of graphs conditioned on observational and interventional data as fol- lows. We generate training data from a joint distribution K(G. D), between a graph G and a dataset D comprising of S observational and interventional samples from a CBN associated G as follows. We first sample set of graphs { with nodes Xi..... X from a com- mon distribution t(g) as described in Section (to simplify notation, in the remainder of the paper we omit the graph index when refer- ring to nodes), and then associate random CPDs to the graphs as described in Section 4 This results in a set of CBNs {MH For each CBN M', we then create a dataset Di = [ where each element x² == (z] (zi...., .., "N T either an observational data sample or an interventional data sample obtained by performing an intervention on randomly selected node in g'.
Attention over nodes
X2²
X5
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Figure 2: Our model architecture and the structure of the input and output at training time. The input is a dataset D = {xr == (af....,*" ) of S samples from a CBN and its adjacency matrix A.
The output is prediction À of A.
Our model defines a distribution i(g) (D; Q) over graphs conditioned on observational and interven- tional data and parametrized by @. Specifically, {(A D; +) has the following auto-regressive form: ((A);;) = II N² (Ar; A, = fe(AA,.... 1): D)), where a<:: is the Bernoulli distribution with parameter p, which is function fe built from an encoder-decoder architecture explained in Section 3.2 taking as input previous elements of the adjacency matrix A (represented here as an array of 2 elements) and D. is trained via maximum likelihood estimation (MLE), i. O* argmin (OO), where (())- = E y [In { D: O)]. which corresponds to the usual cross- cross-entropy (CE) loss for the Bernoulli distribution. Training is achieved using a stochastic gradient descent (SGD) approach in which each gradient update is performed using a pair (Di, A The data-sampling distribution t(G D) and the MLE objective uniquely determine the target distribution learned by the model. In the infinite capacity case, i(- DD; 0+) ((-1 ()) To see this, it suffices to note that the MLE objective L(O) can be written as (())- ED-, [KL DD; O); [))] + G, where KL is the Kullhack-Leebler
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divergence and is constant. In the finite- capitiiy case, the distribution defined by the model E(-1D;0", DD; 0*) is only an approximation of t(- (-))
3.2 MODEL ARCHITECTURE
The function fe defining the model's probabilities is built using two transformer networks. It is formed by an encoder transformer and by decoder transformer (which we refer to as "encoder" and "decoder" for short). At training time, the encoder receives as input dataset DD and outputs a representation that summarizes the relationship between nodes in G The decoder then recursively outputs predictions of the elements of the adjacency matrix A' using as input the elements previously predicted and the encoder output. This is shown in Fig. 2 (where we omitted index i, in the remainder of the section). At test time we obtain deterministic predictions of the adjacency matrix elements by taking the argmax of the Bernoulli distribution for use imputs the decoder.
3.2.1 ENCODER
Our encoder is structured as an (N + 1) x (S + 1) lattice. The N x S part of the lattice formed by the first rows and first S columns receives a dataset D = [(ff. ...,, xxv) T }==11 This is unlike standard transformers which typically receive as input a single data sample (e.g.. sequence of words in neural machine translation applications) rather than a set of data samples. Row N + 1 of the lattice is used specify whether each data sample is observational, through value 1, interventional, through integer value in (1...., N} to indicate the intervened node.
T
The goal of the encoder is to infer causal relationships between nodes by examining the set of samples. The transformer performs this inference in multiple stages, each represented by one transformer layer, such that each layer yields a (N + 1) (S + 1) lattice of representations. The transformer designed to deposit its summary representation of the causal structure in column S + 1.
Embedding of the input. Each data-sample element =, is embedded into a vector of dimensionality H. Half of this vector is allocated to embed the value x;, itself, while the other half is allocated to embed the unique identity for the node X,. The value embedding is obtained by passing x whether discrete or continuous, through an MLPUsing an MIP for discrete variable is a slightly inefficient implementation of node value embedding, but ensures that the architecture is general. encoder specific to node X,, We use a node-specift embedding because the values of each node may have very different interpretations and meanings. The node identity embedding is obtained using standard ID transformer positional embedding over node indices. For column S + 1 of the input, the value embedding is a vector of zeros.
Alternating attention. Traditional transformers discover relationships among the elements of data sample arranged in a one- mne-dimensional sequence. With our two-dimensional lattice, the transformer could operate the entire lattice once discover relationships among both nodes and samples. Given an encoding that indicates position n, in the lattice, the model can in principle discover stable relationships among nodes over samples. However, the inductive bias to encourage the model to leverage the lattice structure is weak. Additionally, the model is invariant to sample ordering, which desirable because the samples are iid. Therefore, we arrange our transformer in alternating layers. In the first layer of the pair, attention operates across all nodes of a single sample (af,..... ..i.....,')) .rx)T T to encode the relationships among two or more nodes. In the second layer of the pair, attention operates across all samples for a given node (r-) ................................ = ) to encode information about the distribution of node values. Alternating attention in transformers was also done in Kossen et al. (021).
Encoder summary. The encoder produces summary vector esum with H elements for each node XR, which captures essential information about the node's behavior and its interactions with other nodes. The summary representation is formed independently for each node and involves combining information across the 5 samples (the columns of the lattice). This is achieved with a method often used with transformers that involves weighted average based on how informative each sample is. The weighting is obtained using the embeddings in column S + 1 to form queries, and embeddings in columns 1, 1,...,S S provide keys and values, and then using standard key-value attention.
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3.2.2 DECODER
The decoder uses the summary information from the encoder to generate a prediction of the adjacency matrix A of the underlying G. It operates sequentially, at each step producing binary output indicating the prediction Ak, of Ak,i proceeding row by row. The decoder is an autoregressive transformer, meaning that each prediction As is obtained based on all elements of A previously predicted, as well as the summary produced by the encoder. Our method does not enforce acyclicity. Although this could in principle yield cycles in the graph, practice we observe strong performance regardless (Section 6.4), likely due to the fact that training and evaluation graphs typically studied (e.g., ER-1 1 and ER-2) are very sparse. Nevertheless, one could likely improve the results c.g. by using post-processing (Lippe et al. 202T) or by extending the method with an accept-reject algorithm Castelletti & Mascaro 2022 Li et al.) 2022).
Auxiliary loss. Autoregressive decoding of the flattened N x N adjacency matrix can be difficult for the decoder to learn alone. To provide additional inductive bias to facilitate learning, we added auxiliary task of predicting the parents An,: and children A.,, of node X,, from the encoder summary, csm This is achieved using an MLP to learn a mapping fn. such that (ex")) = (An,: A.,.). While this prediction is redundant with the operation of the decoder, it short-circuits the autoregressive decoder and provides strong training signal to support proper training.
SYNTHETIC DATA
In this section, we discuss identifiability and describe how synthetic data were generated.
Identifiability. Dataset D' associated CBN M is given by D {x*} ) where x= (zf....,.7%) is either an observational or interventional data sample obtained by performing a hard intervention on a randomly selected node in G As discussed in Eberhardt et al. (2006), in the limit of infinite amount of such single-node interventional data samples, G is identifiable. Note, that identifiability here means the ability to recover the exact graph given the data. As our model defines a distribution over graphs, its predictions are meaningful even when the amount of data is insufficient for identifiability: in this case, the model would sample graphs that are compatible with the given data. Empirically, we found that our model can make reasonable predictions even with a small amount of samples per intervention and improves as more samples are observed (Section 6.4)
Graph distribution. We specified a distribution over G in terms of the number of nodes N (graph size) and number of edges (graph density) present in G. As shown in Zheng et al. (2018 8y: Yu et al. (2019): Ke et al. (2020a), larger and denser graphs are more challenging to learn. We varied N from 5 to 80. We used the Erdōs-Red Rényi (ER) and the scale-free free (SF) metrics to vary density and evaluated our model on ER-1 and ER-2 graphs, as in Yu et al. 2019); Brouillard et al. (2020); Scherrer et al. (2021). We generated an adjacency matrix A by first sampling lower-triangulat lar matrix to ensure that it represents a DAG, and by then permuting the order of the nodes to ensure random ordering.
Conditional probability distributions. We performed ancestral sampling the underlying CBN. We considered both continuous and discrete nodes. For continuous nodes, we generated contimuous data using three methods following similar setups in previous works: (i) linear models (linear data) (Zheng et al. 2018 Yu et al. 2019). (ii) nonlinear with additive noise models (ANM data) (Brouillrr al., 2020 Lippe et al. (022)), and (iii) nonlinear with non-additive noise models using neural networks (MN data) (Brouillard et al. 2020 Lippe et al. 2021). For discrete nodes, we generated discrete data using two different methods: MLP (MLP data) and Dirichlet (Dirichlet data) conditional- probability table generators. Following past work (Ke et al. 2020a; Scherrer et al. 2021). we used a randomly initialized network. The Dirichlet generator filled the rows of conditional probability table by sampling categorical distribution from Dirichlet prior with symmetric parameters a. (We remind the reader that this generative procedure is performed prior to node ordering being randomized for presentation to the learning model.) Refer to Appendix A.2 for details.
5 RELATED WORK
Methods for inferring causal graphs can broadly be categorized into score- based, constraint-based and asymmetry- based methods. Score-based methods search through the space of possible candidate graphs, and ranks them based on some scoring function (Chickering 2002 Cooper & Yoo! 1999
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Goudet et al. 2017: Hauser & Bühlmann 2012: Heckerman et al.
1995 T samardinos al. 2006
Huang et al. 2018 Zhu et al. 2019). Recently, Zheng et al. (2018); Yu et al.
2019); Lachapelle et al.
(2019) framed the structure search continuous optimization problem. There exist score- based
methods that use a mix of continuous and discrete optimization (Bengio et al. 2019 Zhu et al.
2019 Ke al. 2020a Lippe al., 2022 [Scherrer al. 2 Constraint-based methods (Monti
ct al. 2019 Spirtes et al., 2000] Sun et al., [007]] Zhhng et al. [012] Zhu et al. 2019) infer the
DAG by analyzing conditional independencies in the data. [Eaton & Murphy (2007) use dynamic
programming techniques. Asymmetry based methods (Shimizu et al., 2006 Hoyer et al. 2009
Peters al., 2011 Damiusis et al.) 2012 Budhatthki & &rreeenn 2017 Mitrovic al. 2018) uses
asymmetry between cause and effect to estimate the causal structure. Peters et al.
(2016); Ghassami
(et al. (2017); Rooas-Carulla et al.
2018); Heinzee-eml et al. (201 8a) exploit invariance across
environments. Mooij et al. (2016) propose modeling framework that leverages existing methods.
Learning-basee methods have been proposed (Bengio et al. 2019 Goudet
Data Type
RCC
DAG-EQ
CSIvA
et al. 2018: Guyon 2013 [014
Observational Interventional
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Kalainathan et al. 2018; Ke et al.
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2020a by Lachapelle et al. [2022
Linear dependencies
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2015b) Wang et al. Lopez-Paz et al.
Non-linear dependencies
X
V
2019p. In particular. Wang et al. (202Ib) use transformers. These works are concerned with learning only part of the causal induction pipeline, such as the scoring function, and hence are significantly different from our work, which uses an end-to-end
[021b]/Zhu et al.
Zhu et al. (2019);
Table I: Data-type comparison between CSIvA and other supervised approaches causal structure induction (RCC
(Lopez-Paz et al. 2015a b) and DAG-EQ (Li et al. 2020
supervised learning approach to learn to map from datasets to graphs. Neural network methods
equipped with Iearned masks cxist (Douglas al.) [017]] Goyal et al.)
2021 Ivanov et al. 2018
Li et al., 2019; Yoon et al.] 2018). but only a few have been adapted to causal inference. Several transformer models (Goyal et al.) 2022 Kossen et al. 2021 Müller et al. 2021) have been proposed for learning to map from datasets to targets. However, none have been applied to causal discovery, although Löwe et al. (2022) proposes neural-network based approach for causal discovery on time-series data. A few supervised learning approaches have been proposed either framing the task as kernel mean embedding classification problem (Lopez-Paz al. 2015a b) or operating directly on covariance matrices (Li et al.) 2020) or binary classification problem of identifying v-structures (Dai et al. 202T). These models accept observational data only (see Table []. and because causal identifiaillity requires both observational and interventional data, our model is in principle more powerful.
EXPERIMENTS
We report a series of experiments of increasing challenge to our supervised approach to causal structure induction. First, we examined whether CSIvA generalizes well on synthetic data for which the training and test distributions are identical (Section 6.1). This experiment tests whether the model can learn to map from a dataset to a structure. Second, we examined generalization to an out-of-distribution (OOD) test distribution, and we determined hyperparameters of the synthetic data generating process that are most robust to OOD testing (Section 6.2). Third, we trained CSIvA using the hyperparameters from our second experiment and evaluated on a different type of OOD test distribution from several naturalistic CBNs (Section 6.3). This experiment is the most important test of our hypothesis that causal structure of synthetic datasets can be useful proxy for discovering causal structure in realistic settings. Lastly, we performed set of ablation studies to analyze the performance of CSIvA under different settings (Section 6.4). All models are trained for 500k: iterations, please refer to Appendix A.3 for details the hyperparameters.
Comparisons to baselines. We compared CSIvA to a range of methods considered to be state-of-the- art in the literature, ranging from classic to neural-network based causal discovery baselines. For both in-distribution and OOD experiments, we compare to 4 very strong baselines: DAG-GNI (Yu et al. 2019), non-linear ICP (Heinze-Demi et al. 20188), DCDI (Brouillard et al. 2020), and ENCO (Lippe) et al. 202T). For OOD experiments to naturalistic graphs, we compare to additional baselines (Chickering 2002 Hauser & Bühlmann) 2012 Zheng al. 2018 Gamella & Heinze-Deml 2020

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(a) Results on N 30 and ER-1 graphs.
(b) Results on N 30 and ER-2 graphs.
Figure 4:
Hamming distance H. between predicted and ground-truth adjacency matrices on 5 different data types: Linear, ANM, NN, MLP and Dirichlet data, compared to DCDI (Brouillard et al. 2020) and ENCO (Lippe et al. 2021) on N = 30 graphs, averaged over 128 sampled graphs. The dotted line indicates the value of the all-absent baseline. CSIvA significantly outperforms all other baselines for all data types.
Li al. 2020). Because some methods only observational data and others do scale well to large N, we could not compare to all methods in all experiments. See Appendix AA.4for for further discussion of alternative methods and conditions under which they can be used.
6.1 IN-DISTRIBUTION EXPERIMENTS
We begin by investigating whether CSIvA can learn to map from data to structures in the case in which the training and test distributions identical. In this setting, supervised ap- proach has an advantage over unsupervised ones, as can learn about the training distribu- tion and leverage this knowledge during testing. We evaluate CSIvA for N 80 graphs.
We examined the performance on data with increasing order of diffi culty, starting with linear (continu- data). before moving non-linear cases (ANM. NN, MLP and Dirichlet data). See Fig. 4 for comparisons be- tween CSIvA and strong baselines on all data types (N 30), showing that
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CSIvA significantly outperforms base- lines for a wide range of data types.
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Results on continuous linear data are presented in Tables 4 and results on continuous ANM data are presented in Table results continuous NN data are reported in Table 7 in Appendix. Results on MLP data are shown in Fig. b) (Appendix). For all data types and all graphs, CSIvA signif- icantly outperforms non-linear ICP, DAG-GNN, DCDI and ENCO. Differences become more apparent with larger graph sizes (N 2 10) and denser graphs (ER-2 ER-1).
Graphs
Figure 3: Results on Dirichlet data for 30 N < 80. Hamming distance between predicted and ground adjacency matrices, averaged over 128 sampled graphs. CSIvA significantly outperforms DCDI and ENCO, both of which are very strong baselines. The difference in perfor- mance increases with N.
The Dirichlet data requires setting the values the parameter a. Hence, we run two sets of experiments this data. In the first set of experiments, we investigated how different values of impact learning in CSIvA. As shown in Table 10 the appendix, CSIvA performs well on data with a < 0.5, achieving H 2.5 in all cases. CSIvA still performs well when 1.0, achieving
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H. < 5 size 10 graphs. Learning with & >> 1 is more challenging. This is not surprising, >> 1 tends to generate more uniform distributions. which are not informative of the causal relationship between nodes. In the second set of experiments, we compared CSIvA to DCDI and ENCO, as they are the strongest performing baselines. We run the comparisons on graphs of size up to < 80. To limit the number of experiments run, set 0.1, as this allows the conditional probabilities to concentrate on non non-uniform distributions, which is more likely to be true in realistic settings, As shown in Fig. our model significantly outperforms DCDI and ENCO on all graphs (up N I 80). As with other data types, the difference becomes more apparent as the size of the graphs grows (N 50), as larger graphs are more challenging to learn.
Soft interventions and unknown interventions. We additionally evaluated CSIvA on soft interven- tions, and on hard interventions for which the model did not know the intervention node (unknown interventions). In both cases, we focus on Dirichlet data with = 0.1 and a 0.25 for ER-1 1 and ER-2 graphs of size N S 20. Results for the performance of CSIvA on soft interventions are shown in Table 13 in the Appendix. CSIvA performed well on all graphs, achieving Hamming distance H < on all graphs. For details about the data generation process, refer to Appendix Section A.5.7
Results for the performance of CSIvA on unknown interventions are shown in Appendix Table T6 CSIvA achieves strong performance on all graphs, with Hamming distance H < 7 on all graphs. Note that CSIvA loses almost no performance despite not knowing the intervention; the biggest difference in Hamming distance between known and unknown interventions on all graphs is H < 1.0. For more details, refer to Section A.5.8 in the Appendix.
6.2 OUT-OF-DISTRIBUTION EXPERIMENTS
In this set of experiments, we evaluated CSIvA's ability to generalize with respect to aspects of the data generating distribution that are often unknown, namely graph density and parameters of the CPDs, such as the values of the Dirichlet distribution.
Varying graph density. We evaluated how well our model performs when trained and tested on CBNs with varying graph density on MIP and = 1 Dirichlet data and the number of nodes to N = with variables able take on discrete values in (1, 21,2,3, The graphs in training and test datasets can take ER degree E {1, 2, 3). Results are shown in Table 17 (Appendix) for the MLP data and Table 18 (Appendix) for the Dirichlet data. For the MLP data, models trained on ER-2 graph generalizes the best. For Dirichlet data, there isn't 't one value of graph density that consistently generalizes best across graphs with different densities. Nevertheless, ER-2 graphs give a balanced trade-off and generalizes well across graphs with different sparsity.
Varying a. We next trained and evaluated on data generated from Dirichlet distributions with E {0.1, 0.25, 0.5). Results for ER-1 graphs with N = are found in Table 19 There isn't value of a that performs consistently well across different values of a for the test data. Nevertheless, 0.25 is a balanced trade-off and generalizes well across test data with 0.1 < a < 0.5.
6.3 SIM-TO-REAT -EAL EXPERIMENTS
In this final set of experiments, we evaluated CSIvA's ability to generalize from being trained on MLP and Dirichlet data to being evaluated on the widely used Sachs (Sachs et al., 2005), Asia (Lauritzen & Spicgelhalter 1988) and Child (Spicgelhalter & Cowell 1992) CBNs from the bnlearn repository, which have N 11, protocol from Ke al. (2020a): Lippe al. (202)) Scherrer et al. (2021) of sampling observational and interventional data from the CBNs provided by the repository. These experiments are the most important test of our hypothesis that causal structure of synthetic datasets can be a useful proxy for discovering causal structure in realistic settings.
8 and N 20 nodes respectively. We followed the established
We emphasize that all hyperparameters for the MLP and Dirichlet data and for the learning procedure were chosen a priori; only after the architecture and parameters were finalized was the model tested on these benchmarks. Furthermore, to keep the setup simple, we trained on data sampled from single set of hyperparameters instead of a broader mixture. Findings in Section 6.2 suggest that ER-2 graphs with a 0.25 work well overall and hence were chosen. Results are reported in Table 2 comparing to range of baselines from Squires et al. (2020): Heinze- Deml et al. (2018b); Yuu et al.) (2019): Gamella & MleineeDemml (2020): Brouillard et (2020): Lippe et al. (202 h and others.
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CSIvA trained on both MLP and Dirichlet data signifi- cantly outperforms all other methods on both the Asia and Sachs datasets. This serves as strong evidence that our model can learn to induce causal structures in the more realistic CBNs, while only trained on synthetic data.
Asia
Sachs
Child
Number of nodes All-absent Baseline
8
20
8
25
GES (Chickering 2002
33
DAG-notedd (Zieng ef al.
2018
14
22
23
DAG-GNN Yu et al. 2019
8
13
20
GES (Hauser & Bühlmann 2012)
31
ICP
(Peters et al.
2016)
8
17
27"
Non-Imear ICP
Henzze-Dee Deml al.
2018b
8
16
23*
DAG-EQ (Li ei al.
2020
16
DCDI-DSF [Brou lad el al.
2020
7
33
18
6.4 DETAILED ANALYSES
ENCO (Lippe al.
2021)
9
14
In order better under- stand model performance, and which model components were most critical, we per- formed further analyses. We describe here subset of the studies performed; please re- fer Appendix [A.7 for the full results, including ablations.
CSIvA (MLP data)
3
6
11
CSIvA (Dirichlet data)
3
5
10
Table 2: Results on Asia, Sachs and Child data: Hamming distance H. between predicted and ground-trute trrth adjacency matrices. *To maintain computational tractability, the number of parents considered was limited to 3.
Acyclicity. We analyzed the generated (sampled) graphs for acyclicity under several settings. We found that none of the generated graphs contains any cycles. For details, refer to Appendix A.7.3
Visualization of generated graphs. We visualized some of the generated graphs in Figure and Figure 8 in the Appendix. Note that samples are randomly drawn, and all are acyclic.
Identifiability upon seeing intervention data. Here, investigate how varying the proportion of interventional data impacts the performance of the proposed model. As shown in Figure (Appendix), the model's performance improves (Hamming distance decreases) almost monotonically as the amount of interventional data increases, from 0 100%. This is clear indication that our model is able extract information from interventional data for identifying the graph structure, For more details on the results and the how the experiments are conducted, please refer to Section AA..4
Amount of training cases. We conducted experiments to better understand how much synthetic data is needed for different graph sizes. For smaller graphs (N < 25), there is a small improvement for using more than 10k training cases. For larger graphs (N > 25) performance consistently improves the model uses more training data (more details in Appendix A.7.6).
Ablation studies. We investigate the role of various components in CSIvA. We found that all the different components play important role in model. For details, see Appendix A.7
Computation time. Our models are trained for 500k iterations. Training can take up to 8 hours, and then inference takes minutes. Training time of CSIvA can be amortized for multiple test cases; training time for baseline models are proportional to the number of test cases. For more details, please refer to Appendix A.7.9)
DISCUSSION
In this paper. we have presented a novel approach towards causal graph structure inference. Our method is based on learning from synthetic data in order to obtain a strong learning signal (in the form of explicit supervision), using a novel transformer- based architecture which directly analyzes the data and computes distribution of candidate graphs. Through a comprehensive and detailed set experiments, we demonstrated that even though only trained on synthetic data, our model generalizes to out-of-distribution datasets, and robustly outperforms comparison methods under wide range of conditions. A direction of future work would be to use the proposed framework for
learning causal structure from visual data. This could be useful, e.g. in RL setting in which an RL agent interacts with the environment via observing low level pixel data (Ahmed et al. 2020 Ke
al. 2021 Wang et al. 202

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A
APPENDIX
CONTENTS
Transformer Neural Networks
16
A.2
Synthetic Data
16
A.3
Hyperparameters
17
A.4
Comparisons to Baselines
18
A.5
Detailed results for Section 6.
18
A.5.J
Results on linear data
18
A.5.2
Results on ANM data
19
A.3.3
Results on NN data
19
A.5.4
Results on MLP data
20
A.5.5
Results on Dirichlet data.
21
A.5.6
Results on scale- free graphs
22
A.5.7
Soft interventions
23
A.5.8
Unknown Interventions
23
A.6
Detailed results for Out-of Out-of-distributtion experiments
24
A.6.1
Varying graph density
24
A.6.2
Varying conditional distribution
24
A.6.3
Varying graph models
25
A.7
Ablation studies and analyses
25
A.7.I
Visualization of sampled graphs
25
A.7.2
Additional evaluations
26
A.7.3
Acyclicity of generated graphs
................................
27
A.7.4
Identifiability upon seeing intervention data.
27
A.7.5
Varying number of samples
28
A.7.6
Varying amount of training data
28
A.7.7
Interventions few variables
28
A.7.8
Ablation studies
29
A.7.9
Computation time
29
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A.1 TRANSFORMER NEURAL NETWORKS
The transformer architecture, introduced in Vaswani et al. (2017). is multi-layer neural network architecture using stacked self-attention and point-wise, fully connected, layers. The classic trans- former architecture has an encoder and a decoder, but the encoder and decoder do not necessarily have be used together.
Scaled dot-product attention. The attention mechanism lies at the core of the transformer archi- tecture. The transformer architecture uses a special form of attention, called the scaled dot- dot-product attention. The attention mechanism allows the model to flexibility learn to weigh the inputs depending on the context. The input to the QKV attention consists of a set of queries, keys and value vectors. The queries and keys have the same dimensionality of d3. and values often have a different dimensionality of d,. Transformers compute the dot products of the query with all keys, divide each by Vdk, and apply a softmax function to obtain the weights on the values. In practice, transformers compute the attention function and values are also packed together into matrices K and The matrix of outputs computed as: Attention Q, K, V) softmax OK"
set of queries simultaneously, packed together into matrix Q. The keys
Vdz )
The encoder is responsible for processing and summarizing the information in the inputs. The encoder is composed of a stack of N identical layers, where each layer has two sub-layers. The first sub-layer consists of a multi-head self-attention mechanism, and the second is simple, position-wise fully connected feed-forward network. Transformers employ residual connection (He
Encoder.
et al. 2016) around each of the two sub-layers, followed by layer normalization (Ba et al. 2016). That is, the output of each sub-layer LayerNorm(r Sublayer(z)). where Sublayer [ the function implemented by the sub-layer itself.
Decoder. The decoder is responsible for transforming the information summarized by the encoder into the outputs. The decoder also composes of stack of N identical layers, with small difference in the decoder transformer layer. In addition to the two sub-layers in each encoder layer, decoder layer consists of a third sub-layer. The third sub-layer performs a multi-head attention over the output of the encoder stack. Similar to the encoder, transformers employ residual connections around each of the sub-layers, followed by layer normalization. Transformers also modify the self-attention sub-layer in the decoder stack to prevent positions from attending to subsequent positions. This masking, combined with fact that the output embeddings are offset by one position, ensures that the predictions for position i can depend only on the known outputs at positions less than i.
A.2 SYNTHETIC DATA
In Section 4 we introduced the types of methods used to generate the conditional probability distribution. this section, we discuss the details of these methods.
We evaluate our model on datasets generated from 5 different methods, which covers both continuous and discrete valued data generated with a varying degree of difficulty. For continuous data, we generated the data using three different methods: linear, non-linear additive noise model (ANM) and non-linear non non-addirive noise neural network model (NN). For discrete data, we generated data using two different methods: MLP and Dirichlet. Let X be N x S matrix representing S samples of CBN with N nodes and weighted adjacency matrix A, and let Z be a random matrix of elements in (0 0.1). We describe each type of how different types of data is generated.
For linear data, we follow the setup /heng al. (2018) and Yu al. (019 Specifically, we generated data as X,.. = An, X + Zn.. The biases were initialized using U U[-0.5, 0.5) and the individual weights were initialized using a truncated normal distribution with standard deviation of 1.5. For nodes with interventions, values are sampled from the uniform distribution U 1].
For additive noise models (ANM), we follow the setup in (Brouillard et al. 2020 Lippe et al. 2021). We generated the data Xn.: (X) + 0.4 Zn.,, where F is fully connected neural network, the weights are randomly initialized from (0, 1). The neural network has one hidden layer with 10 hidden units, the activation function is leaky relu
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with slop of 0.25. The noise variables are sampled from 62). where o² ~ 2[1, 2]. For nodes with interventions, values are sampled from the uniform distribution (2 1).
For non additivee noise neural network (NN) models, we also follow the setup in (Brouillad et al. 2020 Lippe et al. 2021). We generate the data as X,,: (x, 7. such that F is fully connected neural network, the weights are randomly initialized from (000)) 1). The neural network has one hidden layer with 20 hidden units, the activation function is tanh function. The noise variables are sampled from A o²), where a² ~ 24[1, 2]. For nodes with interventions, values are sampled from the uniform distribution (22, 1).
For MLP data, the neural network has two fully connected layers of hidden dimensionality 32. Following past work (Ke al.) 2020a| Scherrer et al., 202T] Lippe et al., (02T), we used randomly initialized network. For nodes with interventions, values are randomly and independently sampled from ( K} where indicates the number of categories of the discrete variable.
For Dirichlet data, the generator filled in the rows of a conditional probability table by sampling a categorical distribution from a Dirichlet prior with symmetric parameters a. Values of a smaller than I encourage lower entropy distributions; values of a greater than I provide less information about the causal relationships among variables. Similar MLP data, for nodes with interventions, values are randomly and independently sampled from UI,...., K} where indicates the number of categories of the discrete variable.
A.3 HYPERPARAMETERS
Table 3: Hyperparameters used in all of our experiments.
Hyperparameter
Value
Hidden state dimension
64
Encoder transformer layers
8
Num. attention heads Optimizer
8
Adam 10-4
Learning rate
Number of random seeds S (number of samples) Training iterations
3
1500
500, 000
Num. training datasets
15, 000 (N z 20) 20, 000 (N > 20)
For all of our experiments (unless otherwise stated) our model was trained on / = 15, 000 (for graphs N < 20) and on / 20, 000 (for graphs N 20) pairs {(Di, A') where each dataset D contained S = 1500 observational and interventional samples. For experiments on discrete data, data-sample element x could take values in 21. 2, 3). Details of the data generating process can be found in Section For evaluation in Sections 6. and 6.2 our model was tested on I' = 128 (different for the training) pairs {(D",, .A' )} -, where each dataset D' contained S = 1500 observational and interventional samples. For the Asia, Sachs and Child benchmarks. our model was still tested on I' 128 (different for the training) pairs {(D", A " however, A" A since there is only a single adjacency matrix in each one of the benchmarks. We present test results averaging performance over the 128 datasets and random seeds and up to size N 80 graphs. The model was trained for 500, 000 iterations using the Adam optimizer (Kingma & Ba 2014) with learning rate 1e-4. Also, refer to Table 3 for the list of hyperparameters presented in a table.
We parameterized our architecture such that inputs to the encoder were embedded into 128- dimensional vectors. The encoder transformer has 10 layers and 8 attention- per layer. The final attention step for summarization has 8 attention heads. The decoder was smaller transformer with only 2 layers and 8 attention heads per layer. Discrete inputs were encoded using an embedding layer before passing into our model.
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A.4 COMPARISONS TO BASELINES
we compare CSIvA to four strong baselines in the literature, ranging from classic causal discovery baselines neural-network based causal discovery baselines. These baselines are DAG-GNN YYu et al., 2019), non-lineat ICP HHeizze-eem et al. 2018b), DCDI (Brouillard et al. 2020 and ENCO (Lippe et al. 2021).
In Section 6.1
Non-linear ICP, DCDI and ENCO can handle both observational and interventional data, while DAG-GNN can only use observational data. Non-linear ICP could not scale graphs larger than 20, therefore compare to DCDI (Brouillard et al. 2020) and ENCO (Lippe et al. (021]) on larger graphs. All baselines are unsupervised methods, I.e. they are not tuned to a particular training dataset but instead rely on general-purpose
purpose algorithm. We also compared to an all-absent model corresponding to zero adjacency matrix, which acts as a sanity check baseline. We also considered
other methods (Chickering 2002 Hauser & Bühlmann 2012 ZZhng al. 2012 Gamella & Heinze
Deml 2020). but only presented a comparison with DCDI, ENCO, non-linear ICP
and DAG-GNN as
these have shown be strong performing models in other works (Ke et al.
2020a
Lippe et al. 2021
Scherrer et al. 2021).
For Section 6.3
also compared to additional baselines from Chickering (2002); Hauser &
Bühlmann 2012); ZZeng et al. (2018); Gamella & Meinze-Deml (2020); Li et al. (2020). Note
that methods from Chickering 2002); Zheng al. (2018); : Yu et (2019); ; Li et (2020) take observational data only. DAG-GNN outputs several candidate graphs based on different scores, such as evidence lower bound or negative log likelihood, DCDI can also be run in two different settings (DCDI-G and DCDI-DSF), we chose the best result to compare to our model. Note that non- non-linear ICP does not work on discrete data, i.c. on the MLP and Dirichlet data, therefore a small amount of Gaussian noise (0, 0.1) was added to this data in order for the method to run.
A.S DETAILED RESULTS FOR SECTION
We present detailed results for experiments in Section 6.1 are described in the tables below.
A.5.1 RESULTS ON LINEAR DATA
ER
ER 2
Var
Var 10
Var 15
Var 20
Var
Var 10
Var 15
Var 20
Abs.
2.50
5.00
7.50
10.00
5.00
10.00
15.00
20.00
DAG-GNN
2.71
4.76
7.71
11.32
5.20
8.81
17.81
22.21
Non-lincar ICP
0.47
1.10
6.3
8.6
0.90
2.41
13.52
17.71
DCDI
0.23
0.78
4.51
6.21
0,92
2.01
8.12
12.59
ENCO
0.19
0.45
3.12
4.23
0.85
1.81
6.14
8.45
Our Model
0.12 0:0
0.35
2.01
2.21 + 0.07
0.81
1.73 0.04
5.32 0.19
5.86
Table 4: Results on Continuous linear data. Hamming distance H. for learned and ground-truth edges synthetic graphs, compared to other methods, averaged over 128 sampled graphs. The number of variables varies from 5 to 20, expected degree = or 2, and the value of variables are drawn from (O, 0.1). Note that for (Heinze- Deml et al.) 2018b), the method required nodes to be causally ordered, and 20 repeated samples taken per intervention, as interventions were continuously valued. "Abs" baselines All-Absent baselines, which is an baseline model that outputs all zero edges for the adjacency matrix. DCDI and ENCO are the strongest performing baselines. CSIvA outperforms all baselines (including DCDI and ENCO).
Results for comparions between model CSIvA and baselines non-linear ICP (Heinze-Dene ct al. 2018b), DAG-GNN (Yu et al. 2019). DCDI (Brouillard et al., 2020) and ENCO (Lippe et al. are shown in Table for smaller graphs with N < 20 and Table for larger graphs with
2021
20
< 80.
For smaller graphs with N < 20, all models that takes interventional data perform significantly better compared to DAG-GNN (Yu et al. 2019), which only takes observational data. CSIvA achieves Hamming distance H evaluated graphs up size 20. Similar to previous findings (Yu et al.)
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2019 Ke et al. 2020a), larger and denser graphs more challenging to learn. Non-linear ICP achieves fairly good performance on smaller graphs ( N Σ 10), however, the performance drops quickly as size of graphs increases (N >> 10). Also, note that Non-linear ICP can not scale to graphs larger than > 20. also required a modification Without this modification, the method achieved chance performance. to the dataset wherein multiple samples modified graph after a point intervention (20 samples per intervention), while other methods only sampled once per intervention.
collected from the same
For larger graphs of 20 < N < 280, we compare to strongest baselines: DCDI and ENCO, Results found in T CSIvA significantly outperforms both DCDI and ENCO. The difference because more upparent as the size of the graphs grow (N 2 50).
Dataset
All-absent
DCDI
(Brouillard et
2020
ENCO (pipe et al,
2021
CSIvA
30, ER
15.0
11.3
1.8
= 50, ER
25.0
18.1
27.5
3.5
N = 80, ER =
40.0
29.5
39.2
9.8
N 30. ER = 2
30.0
9.2
29.1
5.6
N 50, ER
50.0
23.7
73.2
11.7
= 80, ER
80.0
97.2
19.6
Table 5: Results Linear data larger graphs. Hamming distance H. for learned and ground- truth edges on synthetic graphs, compared to other methods. The number of variables varies from 30 80, expected degree
We compare DCDI (Brouillard et 2020) and ENCO (Lippe) et al. 202T) as they are the best performing baselines and they are able to handle larger graphs.
A.5.2 RESULTS ON ANM DATA
For additive noise non-linear model (ANM) data, we compare to the strongest baseline models: DCDI and ENCO N < 80 graphs. Results are found in Table 6 CSIvA achives hamming distance H 11 on all graphs size up to 80. Therefore, CSIvA significantly outperforms strong baseline models DCDI and ENCO all graphs. This difference becomes more apparent larger graphs of N 2 50.
Dataset
All-absent
DCDI (Brouillard
2020
ENCO (Lippe al.
2021
CSIvA
N 30, ER
15.0
3.1
14.7
0.7
N 50, ER
25.0
7.8
25.6
0.7
= 80, ER
40.0
16.9
39.7
4.3
30, ER
30.0
5.8
31.2
N = 50, ER =
50.0
9.7
48.2
3.2
N 80, ER = 80.0
37.1
78.2
10.6
Table 6: Results on continuous non-linear additive noise model (ANM) data with larger graphs. Hamming distance for learned and ground-truth edges on synthetic graphs, compared other methods. The number of variables varies from 30 to 80, expected degree or 2. We compare to DCDI (Brouillard al., (020) and ENCO (Lippe al. (02T) as they are the best performing baselines and they are able to handle larger graphs.
A.5.3 RESULTS ON NN DATA
Results for comparisons between CSIvA and strongest baselines DCDI and ENCO non-additive noise non-linear (NN) data are found Table 7 CSIvA achieves hamming distance < 11 on all graphs. Hence, CSIvA significantly outperforms DCDI and ENCO on all graphs. The differences grow larger as the size of the graph grows.
Additionally, generated non-additive noise -ineer (NN) data with scale-free (SF) graphs with degree of and for N < 50. We compare our model to strongest baselines DCDI and ENCO on these data. The results are found in Table
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Dataset
All-absent
DCDI (Brouillard al.
2020
ENCO (Li al.
2019)
CSIvA
N 30, ER
15.0
3,9
14.7
0.8
N E.R
25.0
9,4
23.5
1.3
N 80, ER
40.0
19.5
39.2
5.3
N 30, ER- 2
30.0
6.7
29.2
2.1
N 50, ER- 2
50.0
10.4
49.5
3.3
N S0, ER 2
80.0
39.8
80.6
10.5
Table 7: Results on continuous non-linear non-additive noise model (NN) data with larger Erdös-Rényi (ER) graphs. Hamming distance 'H. for learned and ground-truth edges synthetic graphs, compared to other methods. The number of variables varies from 30 to 80, expected degree 1 or 2. We compare DCDI (Brouillard al. 2020) and ENCO (Lippe et al. (021) as they are the best performing baselines and they are able to handle larger graphs.
Model
20
All-absent model
DNG-GNN
Hamming distance
Hamming distance
15
ER.1. Varw
ER-1. 10
ER11 Var=15
ER1, Var=20
ER-2. Vared
ER-2. Vane
ER-2. Vare15
ER-2 Var=20
ER.1 Vares
ER Vare
ER1, Vanw15 15
ER11 Varaz0
ER-2. Vared
ER-2, Var=
ER-2 Varw
ER-2 Van-20
Graphs
Graphs
(a) Results on Linear data on N S 20 graphs.
(b) Results on MLP data on S 20 graphs.
Figure 5: Hamming distance H between predicted and ground-truth adjacency matrices on the Linear and MLP data for 20 graphs, compared DAG-GNN (Yu et al. 2019) and non-linear ICP (Heinze- Demel et al. 2018b), averaged over 128 sampled graphs. CSIvA significantly outperforms all other baselines.
A.5.4 RESULTS ON MLP DATA
Results for comparisons between our model CSIvA and basclines non-linear ICP [Heinze-Dem et al., 2018b) and DAG-GNN YYu et al.) 2019) MLP data shown in Table 9 and hence more challenging compared to the continuous linear data. Our model CSIvA significantly outperforms non-linear ICP and DAG-GNN. The difference becomes more apparent the graph size grows larger and more dense.
MLP data is non-linear
Dataset
All-absent
DCDI (Brouillard et al.
2020
ENCO (Lippe et al.
2021
CSIvA
N 30, ER =
15.0
11.1
12.5
8.0
N = 50, E.R
25.0
27.3
28.3
17.6
N 80, E.
40.0
52.8
35.8
28.6
30, ER
17.4
16.9
= 50, FR
50.0
47.6
48.2
33.6
N = 80, F R =
80.0
75.3
61.5
52.3
Table Results on discrete MLP data with larger graphs. Hamming distance H for learned and ground-truth edges on synthetic graphs, compared to other methods. The number of variables varies from 30 to 80, expected degree I or 2. We compare to DCDI (Brouillard et al., (0200) and ENCO (Lippe al. 2021) they the best performing baselines and they are able to handle larger graphs.
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ER
ER
Var =
Var = 10
Var 15
Var 20
5
Var 10
Var = 15
Var 20
Abs.
2.50
5.00
7.50
10.00
5.00
10.00
15.00
20.00
DAG-GNN
2.52
7.30
9.74
12.72
4.33
12.78
16.73
22.33
Non-linear ICP
2.43
4.62
7.42
9.05
4.76
9.12
13.52
19.25
CSIvA
0.98 1: 0.16
2.25 +: 0.17
3.38 0.12
5.92 +: 0.19
1.47
5.12 +: 0.26
6.82 + 0.23
13.50 +: 0.35
Table 9: Results on MLP data. Hamming distance H. for learned and ground-truth edges synthetic graphs, compared to other methods, averaged over 128 sampled graphs (# standard deviation). The number of variables varies from 5 to 20, expected degree 1 or 2, and the dimensionality of the variables are fixed to 3. We compared to DAG-GNI (Yu et al. 2019), which a strong baseline that uses observational data. We also compare Non-linear ICP (Heinze-Demn Deml et al. (018b), which is strong baseline that uses interventional data. Note that for (Heinze- required nodes to be causally ordered, and Gaussian noise (0, 0.1) to be added. "Abs" baselines are All-Absent baselines, which is an baseline model that outputs all zero edges for the adjacency matrix.
et al. 2018b), the method
A.5.5 RESULTS ON DIRICILET DATA.
We run two sets of experiments on Dirichlet data. The first set of experiments aimed at understand- ing how different values impact the performance of our model. In the second set of experiments, we compare the performance of our model to four strong baselines: DCDI (Brouillard et al.) 22020). ENCO (Lippe et al. 2021), non-linear ICP (Heinze-Deml et al. 2018b) and DAG-GNN (Yu et al. 2019)
ER
ER 2
Var = 5
Var = 10
Var = 15
Var = 20
Var = 5
Var = 10
Var = 15
Var 20
a = 0.1
0.18 0.33
0.72+ 004
1.31 0.04
2.45 = 0.34
0.39 0.44
1.27 0.07
1.98 0.12
4.09 0.04
Q 0.25
0,14 0.03
0.77+ 0.05
1.62: 0.05
3,51 +: 0,05
0,29 0.01
1.27 0.07
3.04 0.20
6.41 0.12
0.5
0.14
0.94. 0.05
4.26
7.35 0.04
0.41
2.11 0.06
8.25
15,54:1
1.0
0.26 0.05
2.374 o.r
4.90 0.05
10.104: 0.07
0,68 0.03
4.32 0.07
10,24 0.07
21.81 0.07
a 5.0
1.27 0.12
a.9
14.73= 0.11
19.49 o.s
3.21 ons
9.99 + 0.03
24.19 0.05
37.03+ 0.24
Abs.
2.5
5.0
7.5
10.0
5.0
10.0
15.0
20.0
Table 10: Results Dirichlet data. Hamming distance H. (lower is better) for learned and ground- truth edges on synthetic graphs. averaged over 128 sampled graphs. Our model accomplished a hamming distance of less than 2.5 for Dirichlet data with 0.5. "Abs" baselines are All-Absent baselines, which is an baseline model that outputs all zero edges for the adjacency matrix.
The results for the first set of experiments are shown in Table I0 Our model performs well on all graphs where I 0.5, and the performance
10
Model
Ali-absent model
DAG GNN
starts to degrading as a = 1.0. When a = 5.0, our model is almost performing similarly to
Hammiing distance
Non linear ICP CSivA
the All-absent model (outputting all zero edges).
This is to be expected, larger alpha values is less informative of the causal relationships between variables.
For the second set of experiments, we compared CSIvA to non-linear ICP and DAG-GNI on graphs of size up to 20. To limit the number of experiments to run, we set a 1.0, as this gives the least amount of prior information to CSIvA. As shown in Figure 6 our model sig- nificantly outperforms non-linear ICP and DAG- GNN. Our model achieves H. < 5 on size 10 graphs, almost half of the error rate compared
ER-1 Vands
BR.1 Varo?
BR VindI0
BR-2 Varad
6R-2, Vanw?
BR. Wan-10
Grapts
Figure 6: Results on Dirichlet data on N < 20 graphs. Hamming distance H between predicted and ground-truth adjacency matrices on Dirichlet data, averaged over 128 sampled graphs.
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ER
ER 2
Var 5
Var 7
Var = 10
Var 5
Var 7
Var = 10
All-absent baseline
2.5
3.5
5.0
5.0
7.0
10.0
Yu al. 2019) (Heinze- Deml et al. 2018b)
1.75
4.5
4.0
4.5
7.25
9.50
2.2
3.2
5.3
4.7
6.1
9.3
CSIvA
0.26 0.8
0.83
0.36
2.37
0.07
0.65
0.05
0.97
0.06
4.59 0.08
Table 11: Results on Dirichlet data. Hamming distance H. for learned and ground-truth edges on synthetic graphs, compared to other methods, averaged over 128 sampled graphs (+ standard deviation). The number of variables varies from 5 to 10, expected degree I or 2. the dimensionality of the variables are fixed to 3, and the a is fixed to 1.0. We compare to the strongest causal-inductoo methods that uses observational data Yu et al. 2019) and the strongest that uses interventional data (Heinze-Denn) et al. 2018b
non-linear ICP DAG-GNN, both achieving a significantly higher Hamming distance (H 9.3 and H = 9.5 respectively) on larger and denser graphs. Refer to Table 11 for complete sets of results.
For larger graphs (N 20), we compare CSIvA to DCDI (Brouillard al.; 2020) and ENCO (Lippe et al. (021) they could scale to larger graphs and are the strongest baselines. The results are illustrated Figure and detailed results are found in Table [2 the graphs get larger, the performance of baselines DCDI and ENCO drops significantly. for dense and largr graphs (ER-2, N = 80), the baseline models achieve almost near all-absent performance, while our model performs significantly better (achieving almost 30% performance gain in terms of structured hamming distance).
Dataset
All-absent
DCDI (Brouillard et al. 2020
ENCO (Lippe et al.
2021
CSIvA
= 30, FR
15.0
4.9
2.9
40, ER
20.0
16.2
9.9
5.3
N 50, ER
25.0
26.4
17.7
8.9
N 60, ER
30.0
28.4
25.6
12.6
N 70, ER
35,0
34.1
29.1
18.4
= 80, ER =
40.0
54.1
39.1
23.1
= 30, ER
12.3
8.1
6.5
40, ER =
40.0
21.6
14.3
9.6
N = 50, ER
50.0
47.6
28.6
17.3
N 60, BR 2
60.0
57.6
38.2
26.6
N 70, ER
70.0
71.2
53.4
40.9
80, ER
80.0
78.6
72.6
56.3
Table 12: Results Dirichlet data with larger graphs. Hamming distance H for learned and ground-truth edges on synthetic graphs, compared to other methods. The number variables varies from 30 to 80, expected degree 1 or and the a is fixed 0.1. We compare the approaches from (Brouillard et al.) (0200 and (Lippe et al. 202T) as they are the strongest baselines and they are able to handle larger graphs.
A.5.6 RESULTS ON SCALE-FREE GRAPHS
We further evaluated our model's performance on scale-free (SF) graphs. We run experiments for both discrete and continuous variables. For discrete variables. We train and test our model on Dirichlet data with
0.1 on scale-free (SF) graphs. We run them SF graphs with degree 4 and similar to the setup in Zheng et al. (2018). Results on Dirichlet data is reported in Table 14 and results on NN data is found in Table 15 Our model significantly outperforms strong baseline models DCDI (Brouillar et al. 2020) and ENCO (Lippe et al. 202T) in all cases, regardless the type of data, size of the graph and the sparsity of the graphs.
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ER 1
ER
Var =
Var = 15
Var = 20
Var 10
Var 15
Var 20
All-absent baseline
5.0
7.5
10.0
10.0
15.0
20.0
0.1
0.5
1.2
2.4
0.9
2.3
3.4
0.25
0.5
1.7
3.3
1.7
3.5
4.6
Table 13: Results on imperfect interventions. Hamming distance H. for learned and ground-truth edges on synthetic graphs, compared other methods, averaged over 128 sampled graphs (# standard deviation). The number of variables varies from 10 to 20, expected degree or 2, the dimensionality of the variables are fixed to and the a fixed to 0.1 or 0.25.
Dataset
All-absent
(CDI(Brouillard al. 2020)
ENCO et
2021
CSIvA
N 10, SF
20.0
5.1
3.1
0.7
10.
30,0
5.2
1.7
= 20, SF =
40.0
11.7
8.1
3.2
N = 20, SF =
60.0
13.1
9.3
3.7
N 30, SF 4
60.0
17.4
12.7
4,9
30, SF
90.0
21.2
17.1
9.1
= 50.
100.0
43.7
25.8
N = 50, SF
150.0
70.6
47.1
30.3
Table Results on Dirichlet data on Scale-Free graphs. Hamming distance for learned and ground-truth edges on synthetic graphs, compared to other methods. The number of variables varies from 30 to 80, expected degree 4 or 6, and the a is fixed to 0.1. We compare to the approaches from Brouillard et al. (2020) and Lippe et al. 2021 they are the strongest baselines and they able to handle larger graphs.
A.5.7 SOFT INTERVENTIONS
addition to hard interventions, we also consider soft interventions. Hence, we further evaluate our model on imperfect interventions as discussed in Section 6.1 To limit the number of experiments to run, we focus on Dirichlet data with a = 0.1 and a 0.25.
Soft interventions on Dirichlet data is generated as follows. For intervention variable X,, first sample the a; for the Dirichlet distribution from a Uniform distribution U 70.11, 0.3, 0.5, 0.7, 0.9}. then using the sampled a;, sample the conditional probabilities from the Dirichlet distribution with the new a;.
The results for CSIvA soft interventions shown in Table 13 CSIvA was able to achieve hamming distance H < 5 on all graphs of size H < 20. These results are strong indications that that our model still works well the imperfect interventions.
A.5.8 UNKNOWN INTERVENTIONS
Until now, have considered cases where the target of the intervention is known, we now consider the case when the target of the intervention is unknown, this is also referred to as unknown interven- tions. Again, to limit the number of experiments to run, focus Dirichlet data with = 0.25. The model does not know the target of the intervention and all other training procedures remains exactly the same. The results are shown in Table 16 We compare how well CSIvAperforms on known and unknown interventions. We can see that the performance for the known and unknown interventions are almost the same for sparser graphs (ER-1). The differences increases slightly for denser graphs with higher values of a, for example ER 2 graphs with expected, as denser graphs with higher a values are more challenging to learn. The biggest differences for the performances of known and unknown interventions is less than H. < 1.0 in hamming distance. These results shown clear indications that our model performs well even when the intervention target unknown.
0.1 and
0.25. This is to be
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Dataset
All-absent
DCDI Brouillard et al.
2020)
ENCO al.
2021
CSIvA
N 10
20.0
4.7
2.7
0.9
N 10, SF
30.0
6.8
4.4
1.0
N 20, SF =
40.0
9.2
6,5
1.5
N 20,SF 6
60.0
13.1
8.7
3,4
N 30 1
60.0
13.7
9.1
1.8
N 30,
90.0
19.4
14.7
6.4
N 50, SF
100.0
25.5
19.1
6.9
N = 50,SF SF =
150.0
40.1
29.3
19.2
Table 15: Results continuous non-linear non-additive noise model (NN) data Scale-Free (SF) graphs. Hamming distance H. for learned and ground-trum edges on synthetic graphs, compared other methods. The number of variables varies from 30 80, expected degree 4 6, and the fixed to 0.1. We compare to the approaches from Brouillard et al. (2020) and Lippe et al. (2021) as they are the strongest baselines and they are able to handle larger graphs
ER
ER
Var = 15
Var 20
Var = 15
Var 20
α 0.1
a
0.25
a
0.1
& = 0.25
a 0.1
a 0.25
a 0.1
a = 0.25
Known
1.7
2.5
2.3
3.0
6.3
Unknown
1.4
1.9
2.5
3.3
2.3
3.5
4.1
6.6
Table 16: Results on unknown interventions compared to known interventions. Hamming distance H. from learned and ground-trun -ruth edges synthetic graphs, for known vs unknown inter- ventions, averaged over 128 sampled graphs (#: standard deviation). The number variables varies from 15 to 20, expected degree = or 2, 0.1 or 0.25, and the dimensionality of the variables are fixed to 3.
A.6 DETAILED RESULTS FOR OUT-OF-DISTRIBUTNE EXPERIMENTS
This section contains detailed results the out-of-distribution experiments experiments Section
6.2
A.6.1
VARYING GRAPH DENSITY
Results for the experiments on varying graph density in Section 6.2 are shown in Table 17 and Table
18
Our model generalizes well to OOD test distribution where the graphs vary in terms of density.
A.6.2 VARYING CONDITIONAL DISTRIBUTION
The results on varying a values in Section 6.2 are found Table 19 Our model generalizes well to OOD test distributions, where the conditional probability vary in terms of values for the Dirichlet distributions.
Train
ER-1
ER-2
ER-3
Train
ER-1
ER-2
ER-3
ER-1
1.2
0.9
1.3
ER-1
0.19
0.21
0.28
Test
ER-2
3.3
1.8
2.1
Test
0.86
0.29
0.25
ER-3
5.0
2.8
2.8
ER-3
1.61
0.60
0.23
Table 17: Results on varying graph density for MLP data: Hamming distance H be- tween predicted and ground-truth adjacency matrices.
Table Results on graph sparsity for Dirichlet data (a 1): Hamming distance H between predicted and ground -truth adja- cency matrices.
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Train
a 0.1
a 0.25
a a= 0.5
0.1
0.31
0.33
0.52
Test
a 0.25
0.72
0.40
0.41
0.5
1.8
0.71
0.35
Table 19: Results varying a values for Dirichlet data: Hamming distance H. between predicted and ground-truth adjacency matrices.
Train
N=10
ER-2
ER-3
ER-4
Train
N=20
ER-2
ER-3
ER-4
SF-2
1.2
0.9
1.3
SF-2
0.19
0.21
0.28
Test
SF-3
3.3
1.8
2.1
Test
SF-3
0.86
0.29
0.25
SF-4
5.0
2.8
2.8
SF-4
1.61
0.60
0.23
Table 20: Results on varying graph mod- els for Dirichlet data for N = 10 graphs: Hamming distance between predicted and ground-trute truth adjacency matrices for Dirichlet data trained on Frdös-Rényi (HR) graphs and tested scale-free (SF) graphs.
Table 21: Results on varying graph mod- els for Dirichlet data for N = 20 graphs: Hamming distance H. between predicted and ground-truth adjacency matrices for Dirichlet data trained on Erdōs-Rényi (ER) graphs and tested scale-free (SF) graphs.
A.6.3 VARYING GRAPII MODELS
We also evaluate how well model generalizes when the model used for generating the graphs during training and test differs. We trained and tested our model on Erdós Rényi (ER) and Scale- Freee-Free (SF) graph models. To limit the number experiments run, evaluate our model on Dirichlet data with a = 0.1. We run experiments both on N 10 and N = 20 graphs. The models are trained on Erdös-Rényy Rényi (ER) graphs and tested on Scale-Free (SF) graphs. The results are shown Table 20 and Table 21
A.7 ABLATION STUDIES AND ANALYSES
In order to better understand the performance of our model, we performed further analysis and ablations. This section contains full results for Section 66.4 In this section, we aim to answer the following questions:
What does visualizations of the generated graphs look like?
Additional evaluation metrics to measure structural accuracy.
Are these generated graphs from our model acyclic?
Does intervention data improve identifiability in our model?
How does varying the number of samples impact the performance of our model?
How does varying the amount of training datasets impact the peformance our model?
How does different components of our model (such as sample-level attention and auxiliary loss) impact the performance of our model?
A.7.1 VISUALIZATION OF SAMPLED GRAPHS
We visualized samples that our model generated on the test data. The samples are shown in Figure and Figure 7 The samples are randomly chosen, each subplot is a sample from a distinct test data. The edges in the graph shown in 3 colors, they each represent the following: (a) Green edges indicate that our model has generated the correct edge. (b) red edge indicates a missing edge, that model did not generate the edge, which exist in the groundtruth graph. (c) A blue edge indicates
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Figure 7:
This figures visualizes samples that our model generated on test data. The model was trained and tested on MLP data of size with ER-1 graphs. The samples are randomly chosen. The green edges indicate that our model has generated the correct edges; red edges indicate edges that our model had missed; and blue edges are the ones that our model generated, which were not in the groundtruth graph. As shown above, our model able to generate the correct graph almost all the times, while only occasionally generating or incorrect edges in graph.
a redundant edge, such that our model generated an edge that does not exist in the groundtruth graph. As shown in Figure 8 and 7 our model able generate the correct graph almost all of the times.
A.7.2 ADDITIONAL EVALUATIONS
We evaluated our model on additional evaluation metrics to better understand the performance of the model. In particular, evaluated our model on the balanced scoring function (BSF), Fl score, precision, recall and area under roc (AUROC). We evaluated our model both on discrete and continous data. For discrete data, used Dirichlet data with 0.1 and for continous data, we used non- linear non-additive noise (NN) data. Results on Dirichlet data is in Table 22 results on NN data is in Table 23
For Dirichlet data, our model achieved strong results. Again, the area under ROC (AUROC) is greater than 0.95 for all cases. The score is also always above 0.90. For continuous data, our model achieved even stronger results. Our model scored between 0.98 and 0.99 for all scores. This an indication that the edges were almost always correct and the correct edges also have strong probability.
Number of variables
Graph sparsity
SHD
BSF
F1
Precision
Recall
AUROC
N 10
FR
0.5
0.91
0.95
0.97
0.93
0.96
N 10
F.R
1.0
0.93
0.95
0.96
0.93
0.98
20
2.4
0.91
0.90
0.88
0.92
0.99
N 20
ER 2
3.3
0.94
0.94
0.91
0.95
0.99
Table 22: Results Dirichlet data evaluated with additional evaluation metrics Hamming distance 7/ for learned and ground-trui -uuth edges on synthetic graphs, compared to other methods. The number of variables varies from 10 to 20, expected degree = 2, and the is fixed to 0.1.
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Figure 8:
This figures visualizes samples that our model generated on test data. The model was trained and tested on MLP data of size with ER-2 graphs. The samples are randomly chosen. The green edges indicate that our model has generated the correct edges; red edges indicate edges that our model had missed; and blue edges are the ones that our model generated, which were not in the groundtruth graph. As shown above, our model able to generate the correct graph almost all the times, while only occasionally generating or incorrect edges in graph.
Number of variables
Graph sparsity
SHD
BSF
Fl score
Precision
Recall
AUROC
N= 10
ER ER 2
0.17
0.98
0.99
0.99
0.98
0.93
N 10
0.30
0.98
0.99
0.99
0.98
0.93
20
0.12
0.99
0.99
0.99
0.99
0.92
N 20
ER
0.62
0.99
0.99
0.99
0.99
0.95
Table 23: Results NN data evaluated with additional evaluation metrics Hamming distance
H. for learned and ground-trutte trouh edges on synthetic graphs, compared to other methods. The number of variables varies from 10 to 20, expected degree 1 or 2.
A.7.3 ACYCLICITY OF GENERATED GRAPHS
analyzed the generated graphs for acyclicity on Dirichlet data with a 0.1 and
As discussed in Section 6.4
Hamming distance
0.25 for graphs of size up to N < 20. We evaluated the model on 128 test datasets per setting, and found that none of the generated graphs contains cycles, all graphs are acyclic. This clear indication showing that although we do not have acyclicity reg- ularizer, our model was still able to learn to generated DAGs.
0%
20%
40%
60%
100
100%
Percentage of interventional data in test
Figure 9: Results varying amount of interventions in data. Hamming distance H for learned and ground- truth edges on synthetic Dirichlet graphs (N 15 and a 0.25). Pure observational data (0% interventions) performs the worst, which is to be expect, since inter- vention data is need for causal identifiability. The per- formance of our model improves as observes more interventions, this suggest that our model is able to ex- tract useful information from interventions in order to Prediit the causal structure.
A.7.4 IDENTIHABILITY UPON SEEING INTERVENTION DATA.
Our previous experiments were all performed using a fixed amount of interventions (80%) in

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the training and test sets. To investigate how changing the proportion of interventional data impacts the performance of our model, we train the model with varying amounts of interventions in the training set and evaluate it using different amount of interventions during test time.
To be specific, during training, the model is trained data with varying amount of interventions, which is randomly sampled from the uniform distribution U/0 0.1, 0.3, 0.5, 0.7, 0.9, 1.0]. During test time, we evaluate the model on different amount of interventions, and we report the performance in hamming distance. We trained the model on Dirichlet data with 15 nodes and 0.25. The model trained and tested on 1000 samples per dateset. The results are found Figure
As shown in Figure 9 our model's performance is worst if it only receives observational data (0% interventions), and the performance of our model improves as the amount of interventional data increases. This is a clear indication that our model is able to extract information from interventional data for predicting the graph structure.
A.7.5 VARYING NUMBER OF SAMPLES
We evaluated CSIvA on different amount of samples (100, 200, 500, 1000, 1500) per CBNs. To limit the number of experiments to run, we focus Dirichlet data sampled from N 10 graphs. During the ith iteration in train- ing, the model takes in as input dataset with 4 samples, where 1 is sampled from U[100, 200, 500, 1000, 1500]. During test time, we run separate evaluations on the model for test datasets with different number of samples and the results are reported in Figure I0 We can see that the model performance improves as it observes up 1000 samples for ER-1 graphs, whereas having 1500 samples gives slightly bet- results compared to 1000 samples for ER-2 graphs.
100
500
1000
1500
Famming
Graphs
Figure 10: Results on varying number of sam- ples. Hamming distance between predicted and ground-truth adjacency matrices for synthetic data. Results for CSIvA trained on Dirichlet data with N = 10 and 0.5 with different numbers of samples per CBNs. The model performance in- creases the sample size increases.
A.7.6 VARYING
AMOUNT TRAINING DATA
In this section, we evaluate how the amount of training data impacts the performance our model (CSIvA) ). To limit the number of experiments to run, we focus on Dirichlet data with = 0.1. The model receives M amount of training datasets, where M is sampled from U55000, 10000, 15000, 20000).
Dataset
I 5000
10000
I
15000
20000
= 10,
1.5
0.8
10, ER =
3.2
2.2
1.7
1.4
N = 25, ER =
6.9
2.8
1.7
1.4
N 25, ER
13.5
7.2
5.8
5.2
N 50. ER
32.0
12.2
11.3
8.9
= 50, ER
42.0
30.3
23.3
17.3
Table 24: Results on Dirichlet data trained with varying amount of training datasets. Hamming distance H for learned and ground-truth edges on synthetic graphs, compared other methods. The number of variables varies from 10 to 50, expected degree = or 2, and the a is fixed to 0.1.
A.7.7 INTERVENTIONS ON A VARIABLES
We performed additional experiments to understand the performance of our model when only a few variables intervened on. We varied the percentage of variables intervene from 10% 100%.
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We used Dirichlet data with alpha 0.1 for both training and test data. Our model's s performance improves consistently as more nodes are intervened on.
Results reported in Table 25
Dataset
10%
30%
50%
70%
90%
100%
N 10, ER
6.4
5.0
3.2
2.2
0.9
0.4
N 10, ER
11.9
9.5
7.6
4.8
3.0
1.7
Table 23: Results on Dirichlet data trained with varying percentage of intervention nodes. Hamming distance H for learned and ground-truth edges synthetic graphs, compared other methods. The percentage of node that the model can intervene on varies from 10% 100%. The performance of the model improves consistently as it more nodes are intervened on.
A.7.8 ABLATION STUDIES
We investigate the role that different components play in our model, for example, what are the effects of having auxillary loss and sample-level attention for our model. The details for each experiment described below.
Sample-level attention We first evaluated the performance of our with and without sample-level attention. In the case without sample-level attention, our model will only use node-level attention for all 10 layers in the encoder transformer. The results are reported in Table 26 The models with sample-attention is vanilla CSIvA model; and the model without sample-attention is CSIvA without sample-level attention. As one could see that the performance of the model drops significantly sample-attention is not used. Hereby making sample-level attention crucial component in model.
ER =1
ER =
Var 15
Var
Var
0.1
0.25
0.1
0.25
0.1
0.25
0.1
0.25
Sample
attention
1.3
1.7
2.5
3.4
2.3
3.0
4.1
6.3
No sample
attention
15.3
15.5
21.8
22.4
29.3
29.7
39.4
40.0
Table 26: Results for CSIvA with and without sample-level attention. Hamming distance H. from learned and ground-truth edges synthetic graphs, for known unknown interventions, averaged over 128 sampled graphs (#: standard deviation). The number of variables varies from 15 to 20, expected degree or 2, 0.1 0.25, and the dimensionality of the variables are fixed to 3.
Auxiliary loss We also conducted ablation studies understand the effect of the auxiliary loss in the objective function, results are reported in Table 27 The model with auxiliary loss is the vanilla CSIvA model, where the one without CSIvA trained without the auxiliary loss objective. Experiments are conducted on Dirichlet data with a = 0.1, the model does gain small amount of performance (HK 1.0) by having the auxiliary loss, the difference becomes more apparent as the size of the graph grows, indicating that auxiliary loss plays an important role part of the objective function.
A.7.9 COMPUTATION TIME
All the models with the proposed method are trained for 500k iterations and all models have converged by then (in fact they have all converged by 400k). Once the model trained, evaluation/ inference is fast. All models are evaluated on 128 test datasets (graphs).
The focus of causal discovery is on high-stakes test cases where accuracy matters more than compute. All models (baselines and model) are trained on GPUs. For baseline methods like DCDI, each test
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ER
ER
Var 15
Var 20
Var 15
Var 20
Aux loss No aux loss
1.2
2.4
2.3
3.4 4.5
1.8
3.1
2.8
Table 27: Results for CSIvA trained with and without auxiliary loss. Hamming distance H. from learned and ground-truth edges on synthetic graphs, for known unknown interventions, averaged over 1 128 sampled graphs (+ standard deviation). The number of variables varies from 15 to 20, expected degree or a 0.1, and the dimensionality the variables are fixed to 3.
can take up 48 hours compute; baseline model ENCO takes up 30 minutes to compute. In contrast, our method can take up to hours to train for large graphs, and then inference takes minutes. (More details are provided below.) The training time of model be amortized if the model can be used on multiple test cases; in contrast, training time for ENCO and DCDI is proportional to the number of test cases.
For our model CSIvA, training on smaller graphs (N S 20) converges within 50k, which takes to 45 minutes. Medium sized graphs (20 < < 50) converge within 250k and take 4 hours. Large graphs (50 < N < 80) converge within 400k and take 8 hours. Note that once the model is trained, is very fast to evaluate on different test cases. Our model is evaluated on 128 test cases.
For the baseline model DCDI takes up to 4 hours on single graph of size up to 20 (N < 20). The model takes up to 12 hours on a medium size graph (N < 50) and 48 hours for larger graphs 50 < N < 80). For the baseline model ENCO. Smaller graphs (N < 20) take 20 minutes per graph and larger graphs (N < 80) take 30 minutes per graph. To compute 128 test cases, ENCO takes 2.5 days of computation time.
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Reviews:
[{"review_id": "mPs42ooB2Z", "reviewer": "Reviewer_UgAy", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors propose a method for learning the structure of a directed graphical model using supervised learning from synthetically generated data. They perform extensive evaluation of the proposed method, with impressive results.", "summary_of_the_review": "The empirical results from applying the proposed method are very impressive and should not be ignored. Clearly, there is more work to do, but the existing work presented here should be published so that others can build on these results.", "strength_and_weaknesses": "– Of the four contributions listed at the end of Section 1, only the last two are contributions. Introducing approaches and architectures is easy, but showing whether, how, and why they work is difficult.\n\n– The authors use the term \"causal Bayesian network\". While the terminology of causal graphical models is hardly unified, a more widely used and accurate term would be either \"causal graphical model,\" \"structural causal model,\" or \"directed graphical model.\" The models targeted in this paper are not necessarily Bayesian, so \"causal Bayesian network\" isn't really accurate.\n\n– The current version of the experiments uses structural Hamming distance as a measure of accuracy. The authors should strongly consider using more useful measures of structural accuracy.  Structural Hamming distance has been shown to be a poor measure of model quality. Instead, consider SID (Peters & Bühlmann 2015) or BSF (Constantinou et al. 2021). \n\n+ Even given the use of SHD, the experimental methodology and results are impressive. The authors perform experiments under a wide range of circumstances, far wider than the typical paper. The use of a variety of methods (ablation studies, visualization, effect of interventional data, etc.) is particularly impressive. In all cases, the proposed method performs much better than existing baselines.\n\n– Even given the impressive results, readers would benefit from a more through analysis showing what sorts of errors were corrected in CSIvA’s results and explaining why this happened. Were the edges (that were correctly added or omitted) weak or strong? Was it primarily about edge orientation or edge existence? Was it primarily about adding missing edges or omitting incorrect edges?\n\nReferences\n\nPeters, J., & Bühlmann, P. (2015). Structural intervention distance for evaluating causal graphs. Neural computation, 27(3), 771-799.\n\nConstantinou, A. C., Liu, Y., Chobtham, K., Guo, Z., & Kitson, N. K. (2021). Large-scale empirical validation of Bayesian Network structure learning algorithms with noisy data. International Journal of Approximate Reasoning, 131, 151-188.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "fnSgJC1OyAG", "reviewer": "Reviewer_s1nT", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper provides an algorithm for learning the structure of a directed causal model from a combination of observational and interventional data, where interventions consist of setting one node to a specific value.\n", "summary_of_the_review": "UPDATE: My score has been increased to 8.\n\n----\n\nThe simulations are thorough and the results do seem promising, but I would have to hear a convincing answer to the question about runtimes, and have a link to the authors' code before I were to revise my score upwards.\n\n### Minor Comments\n\n- Figure 4 doesn't seem well presented.  It would seem more sensible (to me) to group the graphs by ER-1 vs ER-2, and then have separate line plots over the increasing number of nodes.  This would make it easier to visualize the best methods.\n- Table 1 seems limited in terms of the methods listed.  What about the others you use in your comparisons?  (i.e. unsupervised approaches)\n- page 8: \"BnLearn\" should be \"bnlearn\"\n- Your spelling is inconsistent.  For example, on page 3 in the top paragraph you use both \"modeling\" and \"modelling\", and then on page 6 you use both \"modelling\" (not North America) and \"analyzing\" (only North America).  \n- The capitalization of \"Hamming\" is also not consistent, even within A.5.  I would suggest always capitalizing, as it is someone's name.\n- Please ensure that the capitalization of your references is correct (use {} around such words in article titles in BibTeX).  e.g. \"dag\", \"markov\", \"bayesian\", \"kaggle\", \"Chalearn\", \"Innovations in machine learning\" etc.\n- Caption for Table 11: \"<=\" should be \"\\leq\", and the opening quote on 'Abs' is not inverted.\n- Tables 5-8 and 12 - again, it would seem better to group the ER=1 and ER=2 results together.", "strength_and_weaknesses": "UPDATE: The reviewers have answered my concerns, so I'm upping my score to 8.\n\n----\n\nThe empirical results look impressive, and it seems to beat all existing methods at this task.  \n\nThe main weakness is that there is no discussion of how much training is required for CSIvA to obtain the results it does.  How much synthetic data must be generated, and how long does the computation take?  How does this compare to the other methods you consider?\n\nAnother weakness is that the method does not enforce acyclicity of the resulting graph, though the authors report that empirically this is never a problem.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "XiLo7H91e8", "reviewer": "Reviewer_uyeG", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper presents a novel deep-learning model capable of inferring the underlying causal structure. The method does this by treating the process as a black-box function which is estimated through a network that maps observational and interventional data to a causal diagram (the output). They demonstrate their approach in a number of experimental settings.", "summary_of_the_review": "The authors have contributed a novel deep-learning model which has some interesting properties when it comes to causal discovery. They demonstrate impressive empirical results on a range of datasets but ultimately contribute very little methodological or theoretical novelty. Their contribution is empirical which is examined in detail in the review.", "strength_and_weaknesses": "This is the main review. It is organised according to the headings of your paper.\n\n# Abstract\n\n- First sentence, perhaps we can be a bit less specific and say that graph learning is _one_ fundamental problem in causal inference (there are many).\n- First question; if yours is a supervised approach, what happens when you have a setting for which you have no data? Learning directed acyclic graphs (DAGs) from data is an NP-hard problem after all.\n- It would help if you in the appendix stated if your method assumed causal sufficiency or not.\n\n# Introduction\n\n- Excellent first paragraph.\n- You say: \"Our approach can be viewed as a form of meta-learning, as the model learns about the relationship between datasets and structures underlying them\" - what precisely is \"meta\" about this?\n- What is a \"naturalistic CBNs\"? A commonly occurring CBN, something we would find in nature?\n- Please make the subfigures larger in figure 1; even with an exceptionally large screen, it is difficult to read the text in your images.\n- I am curious: what do you do about data scarcity and imbalance (the question relates even more to the latter)? There is very little interventional data knocking about, and when there is, there is usually not very much of it compared to the amount of observational data in the same setting which would suggest that your approach has to deal with the data imbalance problem too. How do you do that?\n- Again... what is a \"naturalistic\" CBN?\n\n# BACKGROUND\n\n- If you are tight on space, you can reduce both of these sections and just link to the relevant literature. You've given the main sources for CBNs and a few sentences are enough IF you are short on space. If not, leave as is, it is good.\n\n# CAUSAL STRUCTURE INDUCTION VIA ATTENTION (CSIVA)\n\n- The last paragraph on page three could do with a few equation blocks; it is currently very dense and hard to follow.\n- I cannot help but think that section 3.2 could have been considerably shorter by simply employing a depiction (image) of the encoder and decoder and simple explanations to go alongside it. This long and verbose text to explain a model seems far too verbose when there is no novelty in the individual parts, simply just how they are put together - which could be more easily shown with an image.\n\n# SYNTHETIC DATA\n\n- Careful not to confuse your reader when you discuss identifiability, as it has two meanings within the field of causal inference (and I have not seen it used the way you are using it). See Pearl, 2009 for a formal take - essentially we say that an intervention is 'identifiable' if it can be composed of observational distributions i.e. $p(y| do(x)) = p(x|y)p(x)$. The way Eberhardt et al. use the word is not common and not standard and clashes with a much more central domain within causal inference as noted.\n\n# EXPERIMENTS\n\n- Again, I think you are being far too verbose in your writing. Figure 3 clearly tells us what we need to know regarding the in-distribution experiments, you do not need separate paragraphs for each type of data.\n- It would be helpful if you had included some graph topologies which CSiVA actually learned compared to the other methods. \n- When did you stop learning? Table 3 only tells us the hyperparameters but not when you consider a graph 'learned' from CSiVA's perspective.\n\n# DISCUSSION\n\n- It would have been more useful if you had shown that this method is useful in an RL setting in which an RL agent interacts with the environment via observing low-level pixel data.\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "T0sDB99h7P4", "reviewer": "Reviewer_REpm", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes an interesting supervised learning setting for causal discovery. Firstly, it creates synthetic training data ( observational and international data ) with various simulators and settings. And then it designs a neural network that outputs the causal graphs. Based on such a supervised learning manner, it claims that the method can recover the ground-truth graph (which lacks analysis and proofs, see details in the following comments) and can handle the distribution-shift / out-of-distribution problem (which can be over-claiming the contribution, see details in the following comments).", "summary_of_the_review": "The new and interesting idea of causal discovery deserves support and encouragement, but the paper requires more consideration towards causal discovery. ", "strength_and_weaknesses": "__Strength__\n\nIt deserves to encourage different thoughts and problem formulations for causal discovery. The idea of leveraging synthetic data,  supervised learning, and meta-learning for solving causal discovery problems is interesting and new. If there are related works with similar ideas that I may not realize, then the novelty should be deducted; otherwise, I would like to support such new ideas in causal discovery. However, the work still requires more rigorous considerations. \n\n\n__Weakness__\n\nThe work proposes a framework for causal discovery. However, besides the accuracy, causal discovery also cares about what are the learned results, what are their properties, and what guarantee we have. For example, we know that supervised learning can suffer from \n covariate shift, poor transfer learning performance, the lack of uncertainty, overfitting, etc al. Then how would they bother the proposed method? Maybe the paper could discuss such points. It mentions OOD performance of which I will provide detailed comments in the clarification part. \n\nMoreover, some other aspects can be interesting for causal discovery is that \n1. Can you show the identifiability and consistency of the method? (I will further explain this point in the clarification part.)\n2. Are the results in Markov equivalent class of the underlying causal graph? \n3. How are the results related to the existing results about causal discovery, e.g., the identifiability of different functional causal models (ANMs, PNLs, etc al.)?\n4. Try to explain how the discrepancy between synthetic/simulation and real-world data is or is not a problem for the method. And what can be the problem? Because it is well-known that the evaluation of causal discovery on synthetic data is too simple compared with real-world data. Then why such a gap between synthetic data and real-world data is (or not) a problem for the proposed method?\n\n__Some minor concerns:__\n\nIn Sec, 1, it mentions that the work is related to meta-learning. However, the method doesn't seem to have any meta-learning module to deal with the distribution shift or OOD problem.\nIn Sec. 3.1\n1. Why distributions are denoted by t, not the convention p? \n2. Why there are $N^2$ numbers of $A_l$, for DAGs should it be less ?\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "K56WIbJz6_0", "reviewer": "Reviewer_iFSS", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes a transformer based neural network architecture to learn the mapping from both observational and interventional data to graph structures in a supervised manner. The learned model is shown to be able to generalize to new synthetic graphs and is robust to some train-test distribution shifts. \n\n", "summary_of_the_review": "I am not convinced that there is a transferrable quantity across various causal discovery tasks. I look forward to author response on this point and would like to change score if this concern is addressed.\n\n==== after reading author response====\n\nI am accepting the transferable part. Based on my understanding of the transferability and the listed examples, I feel that the application scope is weakened. For instance, what if we are given a totally new dataset that we do not have such prior knowledge, or that the associated prior knowledge is far different from the training datasets? I suggest authors to add more examples where the proposed method could be applied.\n\nDue to the above point, I decide to increase my score to WA, but I also feel OK if the paper is not accepted. ", "strength_and_weaknesses": "Pros:\n\n- a new design of transformer model to enable supervised learning for structure learning\n- extensive empirical results\n- writing is good and clear.\n\nCons:\n\n- It is not clear how one can trust the transferability result. That is, if we can give a new setting, how accurate can we say about the output? Unlike NLP or CV tasks, I don't see what could be transferred across causal discovery tasks.\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster