A Meta-Transfer Objective for Learning to Disentangle Causal Mechanisms
Authors: Yoshua Bengio
Tristan Deleu
Nasim Rahaman
Nan Rosemary Ke
Sebastien Lachapelle
Olexa Bilaniuk
Anirudh Goyal
Christopher Pal
Venue:  (2020)
Keywords: meta-learning
transfer learning
structure learning
modularity
causality

Abstract: We propose to use a meta-learning objective that maximizes the speed of transfer on a modified distribution to learn how to modularize acquired knowledge. In particular, we focus on how to factor a joint distribution into appropriate conditionals, consistent with the causal directions. We explain when this can work, using the assumption that the changes in distributions are localized (e.g. to one of the marginals, for example due to an intervention on one of the variables). We prove that under this assumption of localized changes in causal mechanisms, the correct causal graph will tend to have only a few of its parameters with non-zero gradient, i.e. that need to be adapted (those of the modified variables). We argue and observe experimentally that this leads to faster adaptation, and use this property to define a meta-learning surrogate score which, in addition to a continuous parametrization of graphs, would favour correct causal graphs. Finally, motivated by the AI agent point of view (e.g. of a robot discovering its environment autonomously), we consider how the same objective can discover the causal variables themselves, as a transformation of observed low-level variables with no causal meaning. Experiments in the two-variable case validate the proposed ideas and theoretical results.

Published as a conference paper at ICLR 2020
A META- OBJECTIVE FOR LEARNING TO DISENTANGLE CAUSAL MECHANISMS
Yoshua Bengio 1.2.5 Tristan Deleu'
Nasim Rahaman' Anirudh Goyal'
Nan Rosemary Ke² Christopher Pal..5
Sébastien Lachapelle'
Olexa Bilaniuk' Mila Montreal, Quebec, Canada
ABSTRACT
We propose to use meta-learning objective that maximizes the speed of transfer on a modified distribution to learn how modularize acquired knowledge and discover causal dependencies. In particular, we focus on how to factor a joint distribution into appropriate conditionals, consistent with the causal directions. To replace the assumption that the test cases are of the same distribution as the training examples, this method exploits the assumption that the changes in distributions are localized (e.g. to one of the marginals, for example due to intervention on a cause). We prove that under this assumption of localized changes in causal mechanisms, the correct causal graph will tend to have only a few of its param- eters with non-zero gradient, i.c. that need to be adapted (those of the modified variables). We argue and observe experimentally that this leads faster adaptation, and use this property to define a meta-learning surrogate score which, in addition continuous parametrization of graphs, would favour correct causal graphs, making it possible to discover causal structure by gradient-based methods. Finally, motivated by the Al agent point of view (e.g. of a robot discovering its environ- ment autonomously), we consider how the same objective can discover the causal variables themselves, as transformation of observed low-level variables with no causal meaning. Experiments in the taovvari case validate the proposed ideas and theoretical results.
INTRODUCTION
The data used to train our models is often assumed to be independent and identically distributed (iid.), according to some unknown distribution. Likewise, the performance of a machine learning model is typically evaluated using test samples from the same distribution. assumed to be representative of the learned system's usage. While these assumptions are well analyzed from a statistical point of view, they are rarely satisfied in many real-world applications. For example, an accident on major highway could completely perturb the trajectories of cars, and a driving policy trained a static way might not be robust to such changes. Ideally, we would like our models to generalize well and adapt quickly to out-of distriibbbttonnn data.
However, this comes at price in order to successfully transfer to a novel distribution, one might need additional information about these distributions. In this paper, we are not considering assumptions the data distribution itself, but rather how it changes (e.g., when going from training distribution to a transfer distribution, possibly resulting from some agent's actions). We focus on the assumption that the changes are sparse when the knowledge is represented in an appropriately modularized way, with only one or few of the modules having changed. This is especially relevant when the distributional change is due to actions by one more agents, because agents intervene at particular place and time, and this is reflected in the form of the interventions discussed in the causality literature (Pearl) 2009) Peters et al., 2016), where a single causal variable clamped to particular value random variable. In general, it is difficult for agents to influence many underlying causal variables at a time, and although this paper not about agent learning as such, this is a property of the world that we propose to exploit here, to help discovering these variables
Université de Montréal, 2 CIFAR Senior Fellow, Fenle Polytechnique Montréal, Max-Planck Institute for Intelligent Systems, Tübingen, CCaadda CIFAR AT Chair

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and how they are causally related to each other. In this context, the causal graph is powerful tool because tells us how perturbations in the distribution of intervened variables will propagate to all other variables and affect their distributions.
As expected, it is often the case that the causal structure is not known in advance. The problem of causal discovery then entails obtaining the causal graph, feat which is in general achievable only with strong assumptions. One such assumption is that a learner that has learned to capture the correct structure of the true underlying data-generating process should still generalize the case where the structure has been perturbed in a certain, restrictive way. This can be illustrated by considering the example of temperature and altitude from Peters et al.) (2017): a learner that has learned capture the mechanisms of atmospheric physics by learning that it makes more sense to predict temperature from the altitude (rather than vice versa) given training data from (say) Switzerland, will still remain valid when tested on distiiiiitiiittion data from a less mountainous country like (say) the Netherlands. It has therefore been suggested that the out-o--distribution robustness of predictive models can be used to guide the inference of the true causal structure (Peters et al. 2016 2017).
How can we exploit the assumption of localized change? As we explain theoretically and verify experimentally here, if we have the right knowledge representation, then we should get fast adaptation to the transfer distribution when starting from a model that is well trained on the training distribution. This arises because of assumption that the ground truth data generative process is obtained as the composition of independent mechanisms, and that very few ground truth mechanisms and parameters need to change when going from the training distribution to the transfer distribution. A model capturing a corresponding factorization of knowledge would thus require just a few updates, a few examples, for this adaptation to the transfer distribution. As shown below, the expected gradient on the unchanged parameters would be near 0 (if the model was already well trained on the training distribution), so the effective search space during adaptation to the transfer distribution would be greatly reduced, which tends to produce faster adaptation, as found experimentally. Thus, based on the assumption of small change in the right knowledge representation space, we can define a meta-learning objective that measures the speed of online adaptation in order to optimize the way in which knowledge should be represented, factorized and structured. This is the core idea presented in this paper.
Returning to the example of temperature and altitude: when presented with -iitrii-dttiitttinnn data from the Netherlands, we expect the correct model to adapt faster given a few transfer samples of actual weather data collected in the Netherlands. Analogous to the case of robustness, the adaptation speed can then be used to guide the inference of the true causal structure of the problem hand, possibly along with other sources of signal about causal structure.
Contributions. We first verify on synthetic data that the model that correctly captures the underlying causal structure adapts faster when presented with data sampled after a performing certain interven- tions on the true two-variable causal graph (which unknown to the learner). This suggests that the adaptation speed can indeed function as a score to assess how well the learner fits the underlying causal graph. We then use a smooth parameterization of the considered causal graph to directly optimize this score in an end-to-end gradient-based manner. Finally, we show in simple setting that the score can be exploited to disentangle the correct causal variables given an unknown mixture of the said variables.
2 WHICH IS CAUSE AND WHICH IS EFFECT?
As an illustrative example of the proposed ideas, let us consider two discrete random variables A and B, each taking N possible values. We assume that A and B are correlated, without any hidden confounder. Our goal is to determine whether the underlying causal graph is A + B (A causes B). or B A. Note that this underlying causal graph cannot be identified from observational data from single (training) distribution p only, since both graphs are Markov equivalent for (Verma & Pearl 1991); ; see Appendix| A order to disambiguate between these two hypotheses, we will use samples from some transfer distribution ş in addition to our original samples from the training distribution p.
2.1 THE ADVANTAGE OF THE CORRECT CAUSAL MODEL
Without loss of generality, can fix the true causal graph be A + B, which is unknown to the learner. Moreover, to make the case stronger, we will consider a setting called covariate shift (Rojas-Carulla et al. 2018 Quiooero-Candela et al. 2009), where we assume that the change (again,
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whose nature is unknown to the learner) between the training and transfer distributions occurs after an intervention on the cause A. In other words, the marginal of A changes, while the conditional p(B B (A) does not, Le. p(B A) p(B | A)). Changes the cause will be most informative, since they will have direct effects on B. This is sufficient to fully identify the causal graph (Hauser & Bühlmann 2012))
In order to demonstrate the advantage of choosing the causal model - over the anti-causal B + A, we can compare how fast the two models can adapt to samples from the transfer distribution P. We quantify the speed of adaptation as the log-likelihoon after multiple steps of fine-tuning via (stochastic) gradient ascent, starting with both models trained on large amount of data from the training distribution. In Figure [ (see Section 3.3 for the experimental setup), we can see that the model corresponding to the underlying causal model adapts faster. Moreover, the difference is more significant when adapting on small amount of data, of the order of 10 to 30 samples from the transfer distribution. We will make use of this property as a noisy signal to infer the direction of causality, which here is equivalent choosing how modularize the joint distribution.
-44
4.6
-48
A4 8
- A
-5.0
10
11P 107
10
30
Nomber crampies
Figure I: Adaptation the transfer distribution (average log- log-likelihood of the model during fine- tuning adaptation to transfer examples, vertical axis), as more transfer examples are seen by the learner (horizontal axis). The curves are the median over 20,000 runs, with their 75t quantiles intervals. The dotted line is the asymptotic oogllikelhhod (here, that of the ground truth p). The red region corresponds to the range where the effect is the most significant (10-30 samples from transfer distribution).
2.2 PARAMETER COUNTING ARGUMENT
A simple parameter counting argument help understand what are observing in Figure I Since we are using gradient ascent for the adaptation, let's 's first inspect how the gradients of the log-likelihood wrt. each module behave under the transfer distribution.
Proposition 1. Let G be a causal graph, and (training) distribution that factorizes according to G. with parameters 6. Let P be a second (transfer) distribution that also factorizes according to G. If the training and transfer distributions have the same conditional probability distributions for all V; but a subset C (e.g. the transfer distribution is the result of an intervention on the nodes C):
pV1 (V)) d p(V. (ao(V
YV.
(1)
then the expected gradient w.e.t. the parameters 0g such that V. & C of the log-likelihood under the transfer distribution will be zero
P(V)
VV E C, Ev-p
0.
00;
Proposition (see proof in Appendix B.1) suggests that if both distributions factorize according to the correct causal graph, then only the parameters the mechanisms that changed between the training and transfer distributions need to be updated. This effectively reduces the number of parameters that need to be adapted compared to any other factorization over a different graph. also affects the number of examples necessary for the adaptation. since the sample complexity of model grows approximately linearly with the VC-dimension (Ehrenfeucht et al. 1989) Vapnik & Chervonenkis 1971), which itself also grows approximately linearly with the number of parameters (for linear
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models and neural networks; Shwarr-Shwartz & Ben- David 2014). Therefore argue that the performance on the transfer distribution (in terms of sog-liielihoo)) will tend to improve faster if factorizes according the correct causal graph, assertion which may not be true for every graph but that we can test by simulations.
Recall that in our example two discrete random variables (each taking say N values), we assumed that the underlying causal model is A - B, and the transfer distribution is the result of an intervention on the cause A. If the model we learn on the training distribution factorizes according the correct graph, then only N - 1 free parameters should be updated to adapt to the shifted distribution, accounting for the change in the marginal distribution p(4). since the conditional p A) = p(B A) stays invariant. On the other hand, if the model factorizes according to the anti-causal graph B - A, then the parameters for both the marginal ç B) and the conditional j(A B) must be adapted. Assuming there is a linear relationship between sample complexity and the number of free parameters, the sample complexity would be O(22) for the anti-causal graph, compared only O(N) for the true underlying causal graph A + B.
3 THE META TRANSFER OBJECTIVE
Since the speed of adaptation to some transfer distribution is closely related to the right modularization of knowledge, we propose to use
noisy signal to iteratively improve inference of the causal structure from data. Moreover, we saw in Figure that the gap between correct and incorrect models largest with a small amount of transfer data. In order to compare how fast some models adapt to a change in distribution, we can quantify the speed of adaptation based their accumulated online performance after fine-tuning with gradient ascent on few examples from the transfer distribution. More precisely, given small "intervention" dataset Dan likelihood as
{x, ) from p. define the online
T (Dit)) = II p(x:; 0
0 = GML (Dabs)
(3)
o = o + aV logp(x;; 0 G),
where 0(2 aggregates all the modules' parameters in G after steps of fine-tuning with gradient ascent, with learning rate a, starting from the maximum-likellhonn estimate (Dobs)))) ) on a large amount of data Dots from the training distribution p. Note that, in addition to its contribution to the update of the parameters, each data point x; is also used to evaluate the performance of our model so far; this is called prequential analysis (Dawid, 1984), also corresponding to sequential cross-valls vross-validation (Gingras et al. 1999). From a structure learning perspective, the online likelihood (or, equivalently, its logarithm) can be interpreted a score we would like maximize, in order to recover the correct causal graph.
3.1 CONNECTION TO THE BAYESIAN SCORE
We can draw an interesting connection between the online log- likeliiel,ood,, and a widely used score in structure learning called the Bayesian score (Heckerman et al.) 1993 Geiger & Heckerman 1994). The idea behind this score to treat the problem of learning the structure from fully Bayesian perspective. If we define prior over graphs p(G) and a prior p(fc G) over the parameters of each graph G, the Bayesian score defined as score (G: Dint) = log G) + logp(G), where p(Dine int
G) is the marginal likelihood
T
p(D (DDin int
G) = II p(x, Xi / G. 1=1
II 1=1 ec
p(x, fG. G)p(Oa G) dé
In the online likelihood, the adapted parameters 0(2) act as summary of past data XIæ-1- Eq. can be seen as an approximation of the marginal likelihood in Eq. where the posteriors over the parameters p(OG X1=-1, G) is approximated by the point estimate o(² Therefore, the online logelikel provides a simple way approximate the Bayesian score, which is often intractable.
3.2 A SMOOTH PARAMETRIZATION OF THE CAUSAL STRUCTURE
Due to the super- euper-exponential number of possible Directed Acyclic Graphs (DAGs) over n nodes, the problem of searching for causal structure that maximizes some score is, in general, NP-hard
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(Chickering 2002a). However, we can parametrize belief about causal graphs by keeping track of the probability for each directed edge to be present. This provides a smooth parametrization of graphs, which hinges gradually changing our belief in individual binary decisions associated with each edge of the causal graph. This allows us to define a fully differentiable meta-learning objective, with all the beliefs being updated at the same time by gradient descent.
In this section, we study the simplest version of this idea, applied to our example on two random variables from Section 2 Recall that here, we only have two hypotheses to choose from: either A + B or B + A. We represent our belief of having an edge connecting A to B with structural parameter y such that p(A + B) (1), where (1) 1/(1 exp( (-y) )) the sigmoid function. We propose, as meta-transfer objective, the negative log- log-likelhhood R (a form of regret) over the mixture of these two models, where the mixture parameter given by (7)):
(5) This meta-learning mixture combines the online adaptation likelihoods of each model over one meta-example or episode (specified by Dintt p), rather than considering and linearly mixing the per-example likelihoods as in ordinary mixtures.
R(Dint) - log
+ (1- (B)))))
In the experiments below, after each episode involving T examples Dint from the transfer distribution f. we update y by doing one step gradient descent, to reduce the regret R. Therefore, in order to update our belief about the edge A + B, the quantity of interest is the gradient of the objective R with respect to the structural parameter, 0R/00 0 This gradient is pushing (ry) towards the posterior probability that the correct model is A + B, given the evidence from the transfer data:
Proposition 2.
The gradient of the negative log- gellikelihood of the transfer data D int in Equation wrt. the structural parameter "y is given by 0R
0, - p(A
p(A Dint);
where p(A + B Dint) is the posterior probability of the hypothesis A - B (when the alternative B A). Furthermore, this can be equivalently written as
OR
0y = 0(7) - o +
(7)
where A = log CA-a(Dinn)) log L (Dint) ) is the difference between the online log- ligeliksi of the two hypotheses on the transfer data Dint-
The proof is given in Appendix B.2, Note how the posterior probability is basically measuring which hypothesis is better explaining the transfer data Dint overall, along the adaptation trajectory. This posterior depends on the difference in online log-likelihoods A, showing the close relation between minimizing the regret R and maximizing the online log- likeelidddoo score. The sign and magnitude of A have direct effect on the convergence of the meta- reta-transfer objective. We can show that the meta-transfer objective is guaranteed to converge to one of the two hypotheses.
Proposition 3. With stochastic gradient descent (and appropriately decreasing learning rate) the structural parameter
on ED.... (R(in))),, where the gradient steps are given by Proposition 2 converges towards
(77) + if
P..... (B
(8)
or (7)) 0 otherwise
This proposition (proved in Appendix B.3) shows that optimizing 7 is equivalent to picking the hypothesis that has the smallest regret (or fastest convergence), measured as the accumulated log- likelihood of the transfer dataset Dint during adaptation. The distribution over datasets Dint is similar distribution over tasks in meta-learning. This analogy with meta-learning also appears in our gradient-based adaptation procedure, which is linked to existing methods like the first-order approximation of MAMI. (Finn al. (217). and its related algorithms (Grant et al. 2018. Kim et al. Finn et al. 2018). The pscudo-code for the proposed algorithm given in Algorithm!]
2018
This smooth parametrization of the causal graph, along with the definition of the meta-transfer objective in Equation (5). can be extended to graphs with more than 2 variables. This general formulation builds on the bivariate case, where decisions are binary for each individual edge of the graph. See Appendix E for details and generalization of Proposition[] the structure of Algorithm [ remains unchanged. Experimentally, this generalization of the meta-transfer objective proved to be effective on larger graphs (Ke et al. (2019), in work following the initial release of this paper.
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Algorithm 1 Meta-learning algorithm for learning the structural parameter
Require: Two graph candidates G A B and B A
Require: A training distribution that factorizes over the correct causal graph
Set the initial structural parameter Y
equal belief for both hypotheses
2: Sample a large dataset Dobs from the training distribution
3: Pretrain the parameters of both models with maximum likelihood on Dobs
4: for each episode do
5: Draw a transfer distribution P (via an intervention)
6: Sample (small) transfer dataset Dintt {x+[=1 from
7: for = I.... I do
8:
Accumulate the online log-likelihood for both models LA- and LB-AA as they adapt
Do one step of gradient ascent for both models: 0(++1) = @ + a a loglogplee p(x 0 G)
9:
10:
Compute the regret (Diine)
11:
Compute the gradient of the regret wIt. y (see Proposition
12:
Do one step of gradient descent the regret w.r.t
13:
Reset the models' parameters to the maximum likelihood estimate on Dobs
3.3
EXPERIMENTAL RESULTS
To illustrate the convergence result from Proposition B] we experiment with learning the structural parameter in a bivariate model. Following the setting presented in Section 2.1 assume in all our experiments that and are two correlated random variables, and the underlying causal model (unknown to the algorithm) is fixed to A - B. Recall that both variables are observed, and there is no hidden confounding factor. Since the correct causal model + B, the structural parameter should converge correctly, with ((>) 1. The details of the experimental setups, as well as details about the models, can be found in Appendix
We first experiment with the case where both A and B are discrete random variables, taking N possible values. this setting, we explored how two different parametrizations of the conditional probability distributions (CPDs) might influence the convergence of the structural parameter. In the first experiment, we parametrized the CPDs as multinomial logistic CPDs (Koller & Friedman, (2009), maintaining tabular representation of the conditional probabilities. For example, the conditional distribution p(
() is represented as
exp(Org)
p( = --iAA=1 = Σ. cxp(6
where the parameter 0 is N N matrix. We used similar representation for the other marginal and conditional distributions pl ()), p( (3) and p( B). In a second experiment, we used structured CPDs, parametrized with multi-layer perceptrons (MLPs) with softmax nonlinearity the output layer. The advantage over tabular representation is the ability to share parameters for similar contexts, and reduces the overall number of parameters required for each module. This would be crucial if either the number of categories N, or the number of variables, increased significantly.
B
0.8
B
0.0
0.
B
0.0
0.0
100
300
300 Nambur spisodos
Numbbs spisodss
Figure 2: Evolution of the belief that A + B is the correct causal model, the number of episodes increases, starting with an equal belief for both hypotheses. (Left) multinomial logistic CPDs, (right) MLP parametrization.

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In Figure 2 we show the evolution of (7), which is the model's belief of A + B being the correct causal model, as the number of episodes increases, for different values of N. As expected, the struc- tural parameter converges correctly to o(x) + 1, within few hundreds episodes. This observation is consistent in both experiments, regardless of the parametrization of the CPDs. Interestingly, the structural parameter tends converge faster with larger value of N and a tabular representation, illustrating the effect of the parameter counting argument described in Section 2.2 which is stronger as N increases. Precisely when generalization is more difficult (too many parameters and too few examples), we get a stronger signal about the better modularization.
We also experimented with A and B being continuous random variables, where they follow either multimodal distributions, or they are Gaussiann Similar parameter (7)) consistently converges to the correct causal model as well. See Appendix C.3 and Appendix for details about these experiments.
to Figure we found that the structural
REPRESENTATION LEARNING
So far, we have assumed that all the variables in the causal graph are fully observed. However, in many realistic scenarios for learning agents, the learner might only have access to low-level observations (e.g. sensory-level data, like pixels acoustic samples), which are very unlikely to be individually meaningful as causal variables. In that case, our assumption that the changes in distributions are localized might not hold at this level of observed data. To tackle this, we propose to follow the deep learning objective of disentangling the underlying causal variables (Bengio al., 2013), and learn representation in which the variables can be meaningfully cause or effect each other. Our approach is to jointly learn this representation, well the causal graph over the latent variables.
consider the simplest setting where the learner maps raw observations to a hidden representation space with two causal variables, via encoder È. The encoder is trained such that this latent space helps to optimize the meta- weta-transfer objective described in Section 3 We consider the parameters of the encoder, well as y (see Section 3.2 as part of the set of structural meta-parameters to be optimized. We assume that we have two raw observed variables (X, Y), generated from the true causal variables (A, B) via the action of ground truth decoder (or generator network), that the learner is not aware of. This allows us to still have the ability to intervene on the underlying causal variables (e.g. to shift from training transfer distributions) for the purpose of conducting experiments, while the leamerr only sees data from (X, Y).
Duñt generation (unknown the leanner)
V
W
O
B
R(Op)
R(0s)
or
O+O
V
(A, B)
Decoder D
(X,Y)
Encoder E
(U, V)
Figure The complete experimental setup. The ground-truth variables (A, B) are assumed to originate from the true underlying causal model, but the observations available to the learner are samples from (X The observed variables (X, Y) are derived from (A, B) via the action of decoder D. The encoder & must be learned to undo this action of the decoder, and thereby recover the true causal variables up to symmetries. The components of the data generation on the left are hidden to the model.
In this experiment, we only want to validate the proposed meta-objective as way to recover good encoder, and we assume that both the decoder D and the encoder are rotations, whose angles are Op and Og respectively. The encoder maps the raw observed variables (X, Y) to the latent variables (U, V). over which we want to infer the causal graph. Similar to our experiments Section 3.3 we assume that the underlying causal graph is A + B, and the transfer distribution 5 (now over (X, Y)) the result of an intervention over A. Therefore, the encoder should ideally recover the structure U + V in the learned latent space, along with the angle of the encoder 0g =-Op. However, since the encoder is not uniquely defined, V + U might also be a valid solution, if the encoder is 0g = -m/2 OD. Details about the experimental setup provided in Appendix [) In Figure
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consider that the learner succeeds, since both structural parameters converge one of the two options. This shows how minimizing the meta meta-transfer objective can disentangle (here in a very simple setting) the ground-truth variables.
0.S
0G-
0.4
Soltiion
0.2
V
Sutniion's 2
Skit
1000
Numaar
Nunler ophoodss
Figure 4: Evolution of structural parameters 0g and 7, as number of episodes increases. Angle the rotation for the decoder is set to 0D - /4, there are two valid solutions for the angle 0g of the encoder: either 0g = т. 7/4, or 0g /4; /4; the model converges to the former solution.
RELATED WORK
As stated already by [Bengio et al. (013), and clearly demonstrated by Locatello al. (2019).
assumptions, priors, or inductive biases are necessary to identify the underlying explanatory variables. The latter paper (Locatello et al. 2019) also reviews and evaluates recent work disentangling, and discusses different metrics that have been proposed. Chalupka et al. (2015) 2017) recognize the potential and the challenges underlying causal representation learning. Closely related to our efforts is (Chalupka et al. 2017), which places strong focus on the coalescence of low (e.g. sensory) level observations (microvariables) to higher level causal variables (macrovariables), albeit more observational setting.
There also exists an extensive literature on learning the structure Bayesian networks from (observa- tional) data, via score-based methods (Koller & Friedman, 2009). [Heckerman et al. (1995): Daly et al. (2011) provide comprehensive review these methods. Many of these algorithms are based on greedy-search with local changes to the graphs (Chickering) 2002b), whereas propose continuous and fully-differentiable alternative. While most these approaches only rely on observational data, it is sometimes possible to extend the definition of these scores to interventional data (Hauser & Bühlmann 2012). The poline-likelihen score presented here supports interventional data as its main feature.
Some identifiability results exist for causal models with purely observational data though (Peters et al. 2017). based specific assumptions the underlying causal graph. However, causal discovery is more natural under local changes in distributions (Tian & Pearl 2001), similar to the setting used
in this paper. Pearl's seminal work do-caleulus (Pearl 1995] 2009 Bareinboim Pearl 2016
lays the foundation for expressing the impact of interventions on causal graphical models. Here we are proposing a meta-learning objective function for learning the causal structure (without hidden variables), requiring mild assumptions such as localized changes in distributions and faithfulness of the causal graph, in contrast the stronger assumptions necessary for these identifiability results.
Our work is also related to other recent advances in causation, domain adaptation, and transfer learning. MMgliacane et al. (2018) have sought to identify a subset of features that leads to the best predictions for variable of interest in source domain, such that the conditional distribution of that variable given these features the same in the target domain. Zhang et al.) (2017) also examine non-stationarity and find that it makes causal discovery easier. Our adaptation procedure, using methods in meta- meta-learning (Finn et al.) [0077]
gradient ascent, is also closely related to gradicnt- Finn 2018). Alet al. (2018) proposed a meta-learning algorithm recover modules, but did not establish any connections to causal mechanisms. More recently, (Daguppta et al. (2019) adopted meta-learning approach perform causal inference on purely observational data.
of specialized
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6 DISCUSSION & FUTURE WORK
We have established, in very simple bivariate settings, that the rate which learner adapts to sparse changes in the distribution of observed data can be exploited to infer the causal structure, and disentangle the causal variables. This relies the assumption that with the correct causal structure, those distributional changes are localized. We have demonstrated these ideas through some theoretical results, as well as experimental validation. The source code for the experiments is available here: bttps://bit.o 1y/ 2M6X1al
This work is only a first step the direction of causal structure learning based on the speed of adaptation to modified distributions. On the experimental side, many settings other than those studied here should be considered, with different kinds of parametrizations, richer and larger causal graphs (see already Ke et al. (2019), based on first version of this paper), different kinds of optimization procedures. On the theoretical side, much more needs to be done to formally link the locality of interventions needs to be done in exploring how the proposed ideas can be used learn good representations in which the causal variables towards improving the way learning agents deal with non non-staiionarities, and thus improving sample complexity and robustness of these agents.
faster adaptation, to clarify the conditions for this to work. Also, more work
disentangled. Scaling up these ideas would permit their application
An extreme view of disentangling is that the explanatory variables should be marginally independent, and many deep generative models (Goodfellow al., 2016), and Independent Component Analysis models (Hyvärinen et al. [001;] Hyvääinen et al. (2018), 8), are built on this assumption. However, the kinds of high-level variables that we manipulate with natural language are marginally independent: they are related to each other through statements that are usually expressed in sentences (e.g. sentence in natural language, or a classical symbolic AI fact or rule), involving only few concepts at a time. This kind of assumption has been proposed to help discover relevant high-level representations from raw observations, such as the consciousness prior (Bengio) 2017). with the idea that humans focus at any particular time just a few concepts that are present to our consciousness. The work presented here could provide an interesting meta-lcarning approach to help learn such encoders outputting causal variables, well as figure out how the resulting variables In that case, one should distinguish two important assumptions: the first one is that the causal graph is sparse, which common assumption in structure learning (Schmidt et al. (007); the second is that the changes in distributions are sparse, which is the focus of this work.
related to each other.
REFERENCES
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A RESULTS ON NON-IDENTIHABILITY OF THE CAUSAL STRUCTURE
Suppose that A and B are two discrete random variables, each taking N possible values. We show here that the maximum likelihood estimation of both models A + B and B A yields the same estimated distribution over and B. The joint likelihood on the training distribution is not sufficient to distinguish the causal model between the two hypotheses. If p is the training distribution, let
0: = p(A = 1) T = p(B =)
A= Th1 = = B
f) = p( B
(10)
(11)
Let Dobe be a training dataset. If (V(A) is the number of samples in Dobe where A = i, ( the number of samples where B j. and N₁ the number of samples where and = j. then the maximum likelihood estimator for each parameter is
= NV(4) /N
0ز11 = Ny//N(4)
(12)
f /N /NN
î No//N
(13)
The estimated distributions for each model A + B and B A. under the maximum likelihood estimator, will be equal:
p - B j: A - B) = 0.0 No/NN
(14)
iA j; B + A) = f). = Nu/N
(15)
To illustrate this result, we also experiment with maximizing the likelihood for each modules for both models A - B and B -> A with SGD. In Figure I. between these two models, evaluated on training and test data sampled from the same distribution, during training. We see that while the model A - B fits the data faster than the other model (corresponding to a positive difference in the figure), both models achieve the same log-likelihoods convergence. This shows that the two models are indistinguishable, in the limit, based on data sampled from the same distribution, even on test data.
show the difference in logliikelssoods
Training
0:301
0.175
20
01333
X 30
0.135
4. 125-
0.0m0
8.000-0
0.00
0.83
21001
0803
2.0
413
3033
30C
500
1000
Numbbr cxampke
Numlen cousspple
Figure A.1: Difference in log-likelihoods between the two models - B and B ~ A on training and test data from the same distribution on discrete data, for different values of N, the number of discrete values per variable. Once fully trained, both models become indistinguishable from their log- ligllikels only, even on test data. The solid curves represent the median values over 100 different runs, and the shaded areas their 25-75 quantiles.
B PROOFS
B. ZERO-GRADIENT UNDER MECHANISM CHANGE
Let us restate Proposition here for convenience:
Proposition 1. Let G be a causal graph, and a (training) distribution that factorizes according to G. with parameters 6. Let & be a second (transfer) distribution that also factorizes according to G. If the training and transfer distributions have the same conditional probability distributions for all V but a subset C (e.g. the transfer distribution is the result of an intervention on the nodes in C);
p(V. (V)()))) d p[V] Pac((V))
AV, $ C
(16)
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then the expected gradient w.r.t. the parameters 0, such that V1 & C of the log log-likelihood under the transfer distribution will be zero
Ölog (V)
AV, $ C, Ev~p
= 0.
(17)
00;
Proof. For V; & C, we can simplify the expected gradient as follows:
O Olog((V)
i
Ev-p
Evep
log (V)) 0z)
(18)
00;
00;
/=1
EV~D
log p(Vi
Pac(Vi); 0.)
(19)
00,
Ev~D
log [(V;
Pac(VV); 0:)
(20)
00,
= Ey-p
log (V)(( 07)
(21)
0e;
3=1
log p(V)
Ev
(22)
08;
where Equation (20) arises from our assumption that the conditional distribution of V2 given its parents in G does not change between the training distribution and the transfer distribution p.
Moreover, the last equality arises from the marginalization
Ø((v)=
(23)
B.2 GRADIENT OF THE STRUCTURAL PARAMETER
Let us restate Proposition here for convenience:
Proposition 2. The gradient of the negative log-likelihood of the transfer data Dint in Equation wrt. the structural parameter "i is given by
0R 07
P(A +
B Dina),
(24)
where p(A B Dint) is the posterior probability of the hypothesis B (when the alternative B + A). Furthermore, this can be equivalently written as
0R
= o(7) - + (17))
AA)
(25)
where A = log L log CB+ (Dine) is the difference between the online log- logllikelssoods of the two hypotheses on the transfer data D snt-
Proof. First note that, using Bayes rule,
((Dint A B)p(A A - B)
p( B Dine) D(Dine A B)p(AA A - B) + p( Dint B- A)p(B + A)
(26)
LA-AB( (Dint)
(27)
CA (Dine) ) L LB (D ins (1 (1-((7) o(7))
CA-B( (DisB (1))
(28)
M
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where M = LLA-B(Dmm))t LB-A (D - (n)) is the online likelihood of the transfer data under the mixture, so that the regret is i(D log M. For Equation (27) note that Dint {a,, bt
T II p(a,, by A + B, (as, b===1)
p(int A B)
(29)
II pla,, by A B; 0BBB) LL
(30)
where 044 encapsulates the information about the previous datapoints {a,, ) in the graph A + B3 through some adaptation procedure. Since we only consider the two hypotheses A + B and B -> A, we also have
p(B + A Dim))= = 1-A11A------ B Dint) CB LB-+A(Di))(L o(1))
(31)
M
Therefore, the gradient of the regret wrt. the structural parameter 7 is
0R
0y M I ( (r0
(32)
- A - -1- (1)))p( B Dint)
(33)
= <(1) - ( - Dint) p(4 - B Dime) + (7))p(A + B
B Dine)
(34)
= =a(1) - B (int)
(35)
= -P(A- + B) -p(A | Dint)
(36)
which concludes the first part of the proof. Moreover, given Equation (35). it is sufficient to show
that p(A + B D Dint) o(7 + A) to prove the equivalent formulation Equation (25). Using the
logit function (z) = log --g" and the expression in Equation 28 we have
(())CA-B(Din))
0 (p(A + B Dint)) = log M A-
(37)
= log
(38)
(1-o(7))CBB) (Dine)
0(7)
log
llog CA-B((Din)) log LB-AA (Dint)
(39)
1-0(7)
=7
AA
(40)
B.3 CONVERGENCE POINT OF GRADIENT DESCENT ON THE STRUCTURAL PARAMETER
Let us restate Proposition here for convenience:
Proposition 3. With stochastic gradient descent (and an appropriately decreasing learning rate) on ED,,.. (R(Din))), where the gradient steps are given by Proposition 2 converges towards
the structural parameter
o(7) +1 if or (77) + otherwise
(41)
Proof. We are going consider the fixed point of gradient descent (a point where the gradient is
zero), since we already know that SGD converges with an appropriately decreasing learning rate.
Let us introduce some notations to simplify the algebra: let p o()). M = pCA-B B (Pint) + (1
BB-AA (Dint). so that the regret is R(DD..)) = log M. We define P1 and P2 as (see also the proof in Appendix(..) B.2)
P PCA-Bi (D = p(A + B Dine)
(1 11 A = Pi (42)
M
M
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Framing the stationary point in terms of p rather than 7 gives constrained optimization problem, with inequality constraints -p I and p 1 I 0, and no equality constraint.
min D... (R(D
(43)
s.t. - <
(44)
p-150
(45)
Applying the KKT conditions to this problem, with constraint functions -p and 1, gives us
0R
EDi..
0p
in
112
(46)
A4 2 0
for 1,2
(47)
A1P 0
(48)
P2 1) 0
(49)
We already see from equations (88 & 44) that p E (0, 1) (i.e. excluding and 1), must have 141 = 122 O, that is
OR
E D
(50)
0p
Let us study that case first, and show that it leads to an inconsistent set of equations (thus, forcing the solution to be either p or p 1). Let us rewrite the gradient to highlight p in it (using Proposition 2
OR 1
(p- p(A + B Dint))
(51)
Op
p(1 p)
1
p pCA-B((Din
(52)
p(1 p)
M
1
p(pLA- (Dint) (1 p) (Dint)) - A.A-
n (Dine)
(53)
p(1 p) p)
M
LB (Dit)( - CA-sB(Pint)
(54)
M
This derivation valid since we assume that p E (0, 1). Suppose that p * multiplying both sides of Equation (50) by gives
p(CB->A(Din) - CA (Dint))
0 EDEnt
(55)
M
pC p 13
EDiie
(56)
M.
LB A
-Epint
P2
(57)
M
LB-A((Dnnl
EDii
(58)
M
For this equation to be satisfied, we need LB-AA = M almost surely, since (D=a M by construction. This would, however, correspond p which contradicts our assumption. Similarly, assuming that * 1, we can also multiply both sides of Equation (50) by - p and get
- (D L LA (Dine)
0 E
(59)
M
Pg (1
E
(60)
M
L (Dint)
E
P₂ + Pj-
(61)
M
LA (Dins)
ED..
(62)
M
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Again, this can only be true if CA-B M almost surely, meaning that p 1, contradicting assumption. We conclude that the solutions p E (0, 1) are not possible because they would lead to inconsistent conclusions, which leaves only p 0 or p = 1.
C RESULTS ON LEARNING WHICH CAUSE AND WHICH IS EFFECT
In order to assess the performance of meta-learning algorithm, applied it on generated data from three different domains: discrete random variables, multimodal continuous random variables and multivariate Gaussian-distribunn experiments, along with additional results to complement the results descrbed in Section [33] Note that in all these experiments, we fix the ground-truth structure A B, and only perform interventions on the cause A.
variables. In this section, we describe the setups for all three
C.I DISCRETE VARIABLES WITH TABULAR REPRESENTATION
We consider bivariate model, where both random variables are sampled from a categorical distribu- tion. The underlying ground-truth model can be described as
A Categorical (T)
(63)
B A
(ategorical(TT))))
(64)
with ग probability vector of size N, and Bia probability vector of size N, which depends on the value of the variable A. In our experiment, each random variable can take one of N 10 or N = 100 values. Since we are working with only two variables, the only two possible models are:
Model + B: p (A, B) = (B A)
Model > A: p( A, B) = (B))(AA))))) B)
We build 4 different modules, corresponding to every possible marginal and conditional distributions. Here, we use multinomial logistic Conditional Probability Distributions (Koller & Friedman, (009).
The modules' definition, and their corresponding parameters, are shown in Table C.1
Table C.I:
Description of the 2 models, with the parametrization of each module, for bivariate model with discrete random variables. Model + B and Model > A both have the same number of parameters N N.
Distribution
Module
Parameters
Dimension
Model A(A)
p(IA- = i; 0A) (softmax(OA)]
0a
N
A A B
p(B A)
p(un = =j A i; OBIA) = /softmaxx (f))
Obja
N2
Model
p(B)
p(an =j;
j; 0n) [softmax(0n))
On
N
B + A
p(A B)
p(zA= = i JB = =j: (AAB) = (o
A
N2
In order to get set of initial parameters, we first train all 4 modules on a training distribution (p in the main text). This distribution corresponds to fixed choice of A (1) and Bia (for all N possible
values of a). The superseript in ( emphasizes the fact that this defines the distribution prior to an intervention, with the mechanism p(B A) being unchanged by the intervention. These probability vectors sampled randomly from uniform Dirichlet distribution:
π A
Dirichlet(1x))
(65)
TB a
Dirichlet(1w)
Va [1,
(66)
Given this training distribution, we can sample a large dataset of samples Dane {a;, by )m for the ground truth model, using ancestral sampling. Using Dots. we can train all modules using gradient ascent the log- likelihood (or any other advanced first-order optimizer, like RMSprop). The parameters A,, B.JA. 9 & 9 Aa of the maximum likelihood estimate will be used as the initial parameters for the adaptation the transfer distribution.
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Similar to the way defined the training distribution, can define transfer distribution (p in the main text) as an intervention on the random variable A. In this experiment, this accounts for changing
(2) the distribution of A, that is with new probability vector A also sampled from uniform Dirichlet distribution
Dirichlet(1x).
(67)
To perform adaptation on the transfer distribution, we also sample a smaller transfer dataset Dint {ae, ) with 7 < m. In our experiment, we used T = 20 datapoints, following the observation from Section 2.1
C.2 DISCRETE VARIABLES WITH MLP PARAMETRIZATION
We consider a bivariate model, similar to the one defined in Appendix C.I where each random variable is sampled from categorical distribution. Instead of expressing the CPDs in tabular form, we use structured CPDs, parametrized with multi-layer perceptrons (MLPs). In our experiment, all the MLPs have only one hidden layer with H 8 hidden units, with ReLU non-linearity, and the output layer has a softmax non-linearity. To avoid any modeling bias, we assume that the ground-truth model is also parametrized by MLPs, such that
~~ Categorical(MLP(0 Wa)) B A = a Categorical WB))
(68)
(69)
where O is a vector of size N will all zeros, and 1 [a] is a one-hot vector of size N. WA and WB summarize the parameters of the ground truth model, with the weights and biases for the layers. Similar to the tabular representation, we define 4 different modules, this time using MLPs. Their definition, as well as their corresponding parameters, are shown Table C.2
Table C.2:
Description of the 2 models, with the parametrization of each module, for a bivariate model with discrete random variables, and MLP parametrization. Model + B and Model B A both have the same number of parameters 3NH + 2(N + H).
Distribution
Module
Parameters
Dimension
Model p(A)
p = i; 0A) = [MLP(0; 0A)]:
0 paja
NH + H + N 2N H N
A A+ B
p(B] A)
p(zB = j IAA: BR)A) = [MLP (1 x A] b)))
Model p(B)
p(æB = j: 0B)) = [MLP(0: 8B) ),,
θB
NH + H N 2NH + H +
B + A
p(A B)
P(zA = FB (alS) [MLP [TB) AA)B)
O B
Again, to define the training distribution, first fix the parameters A and W n- We randomly initialized networks for the training distribution, with the parameters sampled using the He initializa- large dataset of training samples Doher
tion. We train all the modules using maximum likelihood to get the initial set of parameters for the adaptation on the transfer distribution.
We also define a transfer distribution the result of an intervention A. In this experiment, this means sampling a new set of parameters (2( still as randomly initialized network. We sample a
A transfer dataset Dint = {ae, be be with T = 20 datapoints.
C.3 CONTINUOUS MULTIMODAL VARIABLES
Consider a family of joint distributions py. (A, B) over the causal variables and B, defined by the following structural causal model (SCM):
A (A) = 0²
(70)
B = S(A) NB
NB ~ ~( 1):
(71)
where f is a randomly generated spline, and the noise NB is sampled iid. from the unit Gaussian distribution. To obtain the spline, we sample K points {= uniformly spaced from the interval [-8, 8), and another K points {y uniformly randomly from the interval [-8, 8). This yields K
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pairs {aR. yk Ti=1 which make the knots of second-order spline. We choose K points in experiments.
The conditional distributions p(B A) and p(A B) parametrized as 2-layer Mixture Den- sity Networks (MDNs; [Bshhop 1994), with 32 hidden units and 10 components. The marginal distributions p(A) and pl p(B) are parametrized Gaussian Mixture Models (GMMs), also with 10 components. The definition of the different modules, as well as their corresponding parameters, are shown in Table C33
Table C.3:
Description of the 2 models, with the parametrization of each module, for bivariate model with continuous multimodal variables. Model A + B and Model B - A both have the same number parameters 2,140.
Distribution
Module
Parameters
Dimension
Model
p( p(A)
p( 0A) GMM(EA: 0A) p(ZB a 0BJA) = MDN(ZB, ZA; 0
0
30
A + B
p( ( A)
f
2,110
Model
p(B)
p(#B: 0g) GMM(ZB 0B)
0g
30
+
p(A B)
p(zA zB (AIB) MDN( (
AalB
2,110
We select po(A, B) as the training distribution, from which we sample large dataset Dobs using ancestral sampling. Similar to earlier experiments, this dataset used to get the initial set of parameters for the adaptation on the transfer distribution. The MDNs are fitted with gradient descent, while the GMMs are learned via Expectation Maximization. The transfer distribution the result of intervention
A, where shift the distribution Pa(A) with sampled uniformly in 1]. Figure C.I I we plot samples from the training distribution (p = 0), as well as two transfer
distributions (1
5.0
25-
0.0
-5.0
-7.5
Training Tranefer (J -1)
-10.0
Transfer (j +1)
-12.5
-10
10
Figure C.I: Samples from the training (blue) and transfer (red and green) distributions, from an SCM generated with the procedure described above. The red datapoints sampled from p p-, B), the green datapoints from pa (A, B), and the blue datapoints from Po(A, B).
The structural regret R(y) is now minimized with respect to y for 500 iterations (updates of y). In the notation Algorithm [ these are the iterations over the number of episodes. Figure C.2 shows the evolution of (<r) training progresses. This is expected, given that we expect the causal model to perform better on the transfer distributions, i.c. we expect CA-B > LB-A in expectation. Consequently, assigning larger weight to LA-BB optimizes the objective (see Proposition B
Finally as a sanity check, we test the experimental set-up described above on linear SCM with additive Gaussian noise. In this setting, is well known that the causal structure cannot be discovered from observations alone Peters et al. (2017) and one must rely on the transfer distribution tell cause from effect.
To that end, we repeat the experiment in Figure C.2 with the following amendments: (a) we replace the non-lrnear spline with linear curve (Figure C.3) and (b) in addition to training the structural
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1.0
0.6
0.1
B
0
100
200
300
400
500
Number of episodes
Figure C.2: Evolution of the sigmoid of the structural parameter (77), with the number of episodes (meta-training iterations). The belief of + being the correct causal model increases as the number of episodes increases.
parameter by adapting the A + B and B + A models to multiple interventional distributions, we train it by "adapting" the said model the train distribution, where the latter serves as baseline.
Figure C.4 shows that using multiple transfer (i.e. interventional) distributions ("With Interventions") enables causal discovery, opposed to the model trained with single observational distribution. This confirms that our method indeed relies on the interventional distributions to discover the causal structure.
SCM Parameter -4
10
SCM Parameter
SCM Parameter
5
0
-5
-10
-7.5
-5.0
-2.5
0.0
2.5
5.0
7.5
10.0
X
Figure C.3: Samples from linear SCM, showing training (orange) and two transfer distributions (blue and green).
C.4 LINEAR GAUSSIAN MODEL
In this experiment, the two variables A and B are vector-valued, taking values in Rd. The ground-truth causal model is given by
(W, Σ)
(72)
B = 3,A + 30 + Ng
No N)O, E),
(73)
where E Rdd 30 E d and B E gdxd Σ and E are two covariance matrices. our experiment, d = 100. Once again, want identify the correct causal direction between A and B.
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1.0
0.9
d(y) [Mean StdDev]
0.8
0.7
0.6
0.5
0.4
Without Interventions
0.3
With Interventions
0.2
0
20
40
60
80
100
Number of episodes
Figure C.4: Evolution of the sigmoid (meta-training iterations) in case of linear model with additive Gaussian noise. The blue curve corresponds to the setting where we make use of interventions, whereas the orange curve corresponds to one where do not (i.e. use single distribution). The shaded bands show the standard deviation over 40 runs (of both pre- and teta--raaining. We find (as expected) that causal discovery fails without interventions but succeeds when transfer distributions available.
structural parameter (77), with number of episodes
To do so, we consider two models The details of the modules' definitions, as well as their parameters, is given in Table C.4 Note that each covariance matrix is parametrized using the Cholesky decomposition.
B and > A parametrized with Gaussian distributions.
Table C.4:
Description of the 2 models, with the parametrization of each module, for a bivariate model with linear Gaussian variables. Model A - B and Model > A both have the same number of parameters 2d + 3d.
Distribution
Module
Parameters
Dimension
Model
p(A)
p(ZA;00 = N(xx MA, ΣA) p(an WA; 0B a) )) = N Wi + Wo. EBJA)
HA EA Wi, Wo, ΣBIA
d(d + 1)/2 3d(d+1))
A B
p(B A)
Model
p(B)
p(an: 0n) = M(xn un, En) p(aA FB; : OAIB) NN VE + Vo, EAIB
μAB Σn V1, Vo, EAIB
d(d ((d+1)/2 1)/2 + d 3d(d+1)/2
B A
p(A B)
To build the training distribution, we draw (1) 30 and 81 from a Gaussian distribution, and [() and Σ from inverse Wishart distribution. The transfer distribution is the result of intervention on A, meaning that the marginal 5(A) changes. To do so, we sample new parameters (2( from Gaussian distribution, and (2) from an inverse Wishart distribution as well.
Unlike the previous experiments, training distribution. Instead, we fix the parameters of both models to their exact values, according to the ground truth distribution. For Model A B, this be done easily. For the Model B - A, we compute the exact parameters analytically using Bayes rule. This can be seen as the maximum likelihood estimate in the limit of infinite data. In Figure C.5 show that, after 200 episodes, ( converges to I, indicating the success of the method on this particular task.
not conduction any pre- pre-training actual data from the
C.5 EXPERIMENTS WITH SOFT INTERVENTION
In this section, we describe experimental setting where the conditional p(B A) is perturbed while the distribution of the cause, P(A), is left unchanged. To that end, consider a set-up similar to that in Section X
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1.0
0.8
0.6
otv
0.0
25
125
150
175
Number of episodes
Figure C.5: Convergence of the causal belief (to the correct answer) function of the number of meta-lcarning episodes, for the linear Gaussian experiments.
A ~ (A) = (((8,88) = fo(A) + NB
(74)
NB ~ NO 1),
(75)
where fo is randomly generated spline and NB is sampled iid. from the unit Gaussian distribution and the cause variable A is sampled from the uniform distribution supported on [-8,8]. 8].
To induce soft soft-interventions, the order 3 spline fo: we obtain a new spline fint by randomly perturbing the b-coordinate of the knots, where the perturbations are sampled from another uniform distribution Using the perturbed spline fint instead of fo in Equation (74) results in a new SCM, from which we generate a single transfer distribution (i.e. for single episode). In Figure C.6 distributions.
modify the SCM follows. Consider the knots {a, b of
we plot samples from three such transfer
10
Training Transfer SCM
Training Transfer SCM
Training Transfer SCM
5
B
-5
Figure C.6: Samples from different training (blue) and transfer (orange) distributions, from SCMs generated with the procedure described above, namely: all transfer SCMs (orange) are obtained by soft-intervening of the underlying training SCM (bluc).
The models used are identical to those detailed in Appendix C.3 and are trained on the training SCM (corresponding base-spline fo) with large amount of samples (>V 3,000k). The meta-training procedure differs in that (a) every transfer episode, we create new spline fint and sample transfer distribution Dine from the corresponding SCM, and (b) we use the following measure of adaptation:
Co exp (log (0 - log 0(00)
(76)
where G is one of + B or + The meta-transfer objective in Equation (5) remains the same.
Figure C.7 shows the evolution of the (77) the training progresses, and we find that the structural parameter correctly converges to representing the correct causal graph A + B.
Thee scale of the perturbation 0.5 times that of the original knots.
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1.0
oly) [Mean + StdDev]
0.9
0.8
0.7
0.6
0 20
40
60
80
100
Number of episodes
Figure C.7: Convergence of the causal belief (to the correct answer) as function of the number of meta-learning episodes, for the the experiments with soft interventions. The error band is over different runs.
Failure case In addition to the result above, also observed that using soft interventions on the effect B instead on changes on the marginal p(A) was sometimes failing to recover the correct causal graph. Instead, the anti-causal graph (here B A) was found, with high confidence. We describe here one such experiment where using the meta-transfer objective failed at recovering the correct causal graph.
Our experimental setting is similar to the one described in Appendix C.I changing the marginal p(A), the conditional distribution p( B A) changes and p(A) remains unchanged. Following the notations in Appendix C.I. we have
However instead of
A Dirichlet(1N)
(77)
(1)
Dirichlet(1w) &
(2) Bla
Dirichlet(1N))
Va E N1,
(78)
Bla
where " BBaa are the parameters of the conditional distribution before intervention, and π TB its parameters after intervention. We again sample data from both the training and transfer distributions to get datasets Dans and Dint- The different modules and their corresponding parameters are defined
(1)
(2)
Table C.I
10
0.0
B
300 Numbur
Figure C.8: Evolution of the belief that A + B is the correct causal model, as the number of episodes increases, starting with an equal belief for both hypotheses, under soft interventions the effect B.
In Figure C.8
we show the evolution of the structural parameter ( the model's belief that + B the correct causal model, the number of episodes increases. Unlike our previous experiments in Section 3.3 the structural parameter now converges to (7)) - 0, corresponding to strong belief' that the model is B A. We are therefore unable to recover the correct causal graph here under the
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assumption that p(B A) changes. Note that here the parameter counting argument from Section 2.2 clearly does not hold anymore, since the modules all use tabular representation. and both models require the same order ((N2) of updates to adapt to transfer distribution.
D RESULTS ON REPRESENTATION LEARNING
The true latent causal variables (A, B) are sampled from the distribution described in Appendix C.3 (Equations (74) & (75).. These variables are then mapped to observations (X,Y) ~ Pu (X, Y) via hidden (and unknown the learner) decoder D = Ro where Ro is rotation of angle 0. The observations are then mapped to the hidden state (U, V) ~ Pa (U, V) via the encoder & Ro. in this experiment, the angle 0g the only additional ieeaa-paaaettet besides the structural parameter 7. The computational graph is depicted in Figure 3 In our experiment, OD = - is fixed for all our observation and intervention datasets. Interventional data acquired by intervening on the latent variables (A, B), following the process described in Appendix C.3 and then mapping the data through the decoder D.
Since the underlying latent causal variables (A, B) are unobserved, we need to define the online likelihood over the recovered variables (U, V) instead. Analogous to how we defined the online likelihood in the fully observable case in Section 3 this is defined as
T
0 = L.L (Rog (Doba)) 0(++1) of + aV log p (xt); ,),
Ca(Dint O0) = II pl Rac (xx); 8 G)
(79)
where Ro: (Dobe) = {Reg (x) | x E Do Note that here the online likelihood depends on the parameters of the encoder & (here, Og). Using this definition of the online likelihood that takes into account the encoder, the meta-transfer objective is also similar to the one defined Equation (5)
RDDint 7,0g) log (v ; 0E) + Cv (Dine ; 0g)]-
(80)
On the one hand, the gradient of R(Din 7, 0g) with respect to the structural parameter y can be computed using Proposition 2 similar the fully observable case. On the other hand, the gradient of the meta-transfer objective with respect to the meta-paramete fg is computed using backpropagation through the T updates of the parameters fG of the modules in Equation (79): this process similar to backpropagation through time. In experiment, we did not observe any degenerate behaviour like vanishing gradients, due to the limited amount of interventional data (T' = 5).
E MORE THAN TWO CAUSAL HYPOTHESES
we defined the meta-transfer objective only in the context of bivariate models. The challenge with learning the structure of graphs on variables is that there is super-exponential number of DAGs on n variables, making the problem of structure learning NP-hard (Chickering) 2002a If we were naively extend the meta- meta-transfer objective to graphs would require adaptation of (0(2)) different models (hypotheses), which is intractable.
In Section 3.2
>> 2 variables, this
Instead, we can decouple the optimization of the graph from the acyclicity constraint, since causal extra penalty the meta-transfer objective (Zheng et al. 2018). We consider the problem of optimization on the graph as O(n²) independent binary decisions on whether V, a parent (or direct cause) of V.. Motivated by the mechanism independence assumption (Parascandolo et al. 2017). we propose heuristic to learn the causal graph, in which we independently parametrize the binary probability Piy that V, a parent of V, We can then define distribution over graphs (or more precisely, their adjacency matrix B) as:
graphs can have cycles (Peters al. 2017). This constraint
be enforced
By ~ Bernoulli(p.))
(81)
p(B) II p(B.))
(82)
where Pij = =71g)) We denote (V)) as the parent set of V; in the graph defined by the adjacency matrix B (that is the nodes V, such that Bij 1). We slightly rewrite the definition of the
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as in Section 3 to show the dependence on B:
T p (0)
(Dint)) II p(x; 0 B)
(0) (V) (
(83)
where the second equality uses the factorization of in the graph defined by B. Note that since the graph defined by B can contain cycles, the definition in Equation (83) involves the pseudolikelihood instead of the joint likelihood (which is defined as the product of individual conditional distributions only if the graph is a DAG). The pseudolikelihood was shown to be a reasonable approximation the true joint likelihood when maximizing the joint likelihood (which is performed here for adaptation; Koller & Friedman (2009).. Similar to the bivariate case, we want to consider a mixture over all possible graph structures, but where each component must explain the whole adaptation sequence. We can generalize our definition of the regret as
sm)s = log E (P(Le((Pme)]
(84)
Note, however, that this expectation is over the O(2")) possible values of B, which is intractable. We can rewrite the regret more convenient form:
Proposition 4. The regret (Diin)) defined in Equation
84 can be decomposed as
R(Dint) = L
log EB, (Din))s
(85)
i=1
where B, is a row of the matrix B, and LB, (Dina) appears in the factorization of L (Dins) in Equation
Ca. II (( (1)
(c) 3 0
(86)
Proof. Recall that LB (Dint) = II; LB. (Dint). so that we can rewrite the regret as follows:
R(Dint) = log FB (L (Dine)
(87)
log _
(88)
log
Σ
Ln.
(89)
B,.
log
II
Σ pl CB, (Dint)
(90)
n
log
Σ CB, (Diit)
(91)
log EB 1L, (Dint)]
(92)
The structural parameters, here, are the (nn²) scalars Yij- Regardless of the intractability of the regret, we can still derive its gradient wrt. each Tig. The following proposition provides a direct extension of Proposition too the case of multiple variables:
Proposition 5. The gradient of the regret R(Dint) wrt. the structural parameter "lej is given by
0R
0h -(()) - a(ns + Aij)
(93)
where Aij is the difference in log-likelihoods of two mixture candidates, conditioning the variable V₁
Au log [CB, ( (Dint) V, E (Va)(V))) log(EB, (CB, VV, & Pag (V)))
(94)
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Proof. To simplify the notation, remove the explicit dependence the transfer distribution Dint in this proof. Recall from Proposition that the regret can be written as
R =
Σ
log En, [Cp,]-
(95)
Using conditional expectation, it follows that for any i,
BB, [CB.] = Ep(B.)CB.
(96)
= pl Big =) E pBB Bi = 1)CB, + (B., =0) E p(B:) Big 0).
BrlVGEPag(V)
BelVGEPag(V))
(97)
=o(()) E (BB Bi = 1)LB, + +1- Σ p(B;) Bij = 0) B.IV)EPag(Vi)
(98)
To simplify the notation, let us define Fig and F the two conditional expectations of LB. conditioned on whether or not Vj is parent of V; in B
Σ (BB, Bi 0)C EBB[LB, I VV Pa)(V))
(99)
B. (V.
Σ P(B, Bij 1)Ln; = Lnn, | VV E Pa (Vi),,
(100)
EP
so that Equation 98
be written
EB, [CB.] ( +1 d(H)))
(101)
Note that neither (0)
Fij depend on the structural parameter 7/y- Therefore we can easily
(1)
compute the gradient of R wrt. Ni only
8R
θ log (o(16) + (1- (11)))))))
(102)
0nu
87H
(0(ks) (1i - =
(103)
EB, [C..]
If we substract (71y) from this expression gives us
0R = EB, [B.] (o(1)) 2 + +1()))) - - (7
a(1))) Oyj
+ +00(i)) o((s)) E - =E,O
(104)
a(1s)) E
(105)
Ep, [p.]
Denoting the previous expression v, we can also easily compute 1 xt:
(7k5) (
(106)
ER, Pp,]]
Using the logit function 1xx) log I-zx we can conclude that
a (1Hg) F
(a(7))) 8ris
8R
= log
(107)
(1 - ( 1:10
(0)
a(vi))
+ log E. - log E (0)
-log
(108)
1 (76)))
= = + Asj
(109)
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While Proposition gives analytic form for the gradient of the regret wrt. the structural parameters, computing it still intractable, due to Aij. However, we can still get an effecient stochastic gradient estimator from Proposition 4 which can be computed separately for each node of the graph (with samples arising only out of B;, the incoming edges V.):
Proposition 6. we consider multiple samples of B in parallel, a biased but asymptotically unbiased (as the number K of these samples B(k) increases to infinity) estimator of the gradien of the overall regret with respect to the meta parameters can be defined as:
9ij Ex(o(re) Big L
(k) (k)
(110)
Σx (()
where the index (R) indicates the values obtained for the k-th draw of B.
Proof. The gradient of the regret with respect to the meta patampparaeterrs ys node i
0R
ΣB, Pl CB)
(l..p(B.)
8%
0%
ΣB. P(B.)Cn.
EB, [CB,
(logp(B:1
0%
(111)
EB, [CB.]
Note that with the sigmoidal parametrization of p(B:),
log p( B:) B.y log a(N)) + (1 B,z) log(1 ((a)))
(112)
as in the cross-entropy loss. Its gradient similarly be simplified to
0 log b(ggg(B))) Bi (1))))(1111)))))))))))))))) (1o()) (1 - o (7:5)) (76s) (76)))
(1 B3,)
Oni
((1)))
Bú ((ig).
(113)
biased, but asymptotically unbiased, estimator of aR/0ms is thus obtained by sampling graphs (over which the means below are run):
(k)
L
9* -((())) - ((())
(114)
Σn
&
BB
where index (k) indicates the k-th draw of B, and we obtain weighted sum of the individual binomial gradients weighted by the relative regret of each draw B((a) of B,, leading Equation (10)..
We can therefore adapt Algorithm [ using the gradient estimate in Proposition 6 to update the structural parameters "Kj without having to explicitly compute the full regret int) addition to the gradient estimate provided by Proposition 6 we can also derive a Bao-Blackwelllizer (Rao) 1992, Blackwell, 1947) estimate of the gradient of the regret, based on the formulation derived in
Proposition 5
k= be K binary matrices (corresponding to sample graphs), sampled from independent Bernoulli distributions depending the structural parameters "is
Proposition 7. Let {B(
B(k) rdd (ernoulli((((ss)))
(115)
and their corresponding likelihoods ((k) Monte- Carlo estimate of the log-likelihood difference Aij in Equation (94) is given by
(K) = log
(k)
log
(())
(116)
|K!p]
B:
REX!!
REX:
where K:D) = (k; B 0} and xcl) (k: BBAa)
1} are (disjoint) of indices k, depending
on the value of B
Based on this Monte-Carie estimate of Aig, we can define an estimate of the gradient of the regret R. wrt. the structural parameter Tii by
8R
)(Tig) - o + 1/
(K)
(117)
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F THE EFFECT OF ADAPTATION ON THE ONLINE LIKELIHOOD
Since we are using the online likelihood defined in Equation as a measure of adaptation in our meta-transfer objective, it is reasonable to know if this measure sound. To validate this assumption we are running an experiment similar to the one described in Section 2.1 and Figure 1 using the same experimental setup on discrete variables described in Section 3.3 However, instead of measuring the raw log likelihood on validation set, we report the online likelihood (C(Dint) for both models in Figure(F] F.I The online likelihoods are scaled by the number of transfer examples seen for visualization. Similar to Figure 1 we can see that the difference in online likelihoods for both models is most significant on small amount data.
-4.00
Online Lilnhhord c. (D)
-4.70
-5.00
4
-5.00
0 -
10
10
Number exomples
Figure F.1: Adaptation the transfer distribution (online likelihood on transfer examples, vertical axis), more transfer examples are seen by the learner (horizontal axis). The curves are the median over 20,000 runs, with their 25-75th quantiles intervals. The dotted line is the asymptotic online likelihood
27


Reviews:
[{"review_id": "rkl6smfDtB", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "\nSummary: \nThe paper first shows that, in a very simple two-variable task, the model with the correct underlying structure will adapt faster to a causal intervention than the model with the incorrect structure. This idea is used to develop a “meta-transfer” objective function for which gradient ascent on a continuous representation of the model structure allows learning of that structure. The paper shows that optimizing with respect to this objective with a simple model is guaranteed to converge to the correct structure, and also presents experimental results on toy problems to demonstrate.\n\nOverall: Accept.\nI really enjoyed reading this paper. It is clear, well-motivated, well-written, does a good job of connecting to related work, and presents an interesting method for structure learning. While the experiments are quite toy and questions about how well this will work in more complex models with many variables remain largely unaddressed, these do not detract much from the paper for me. Instead, the paper does a good job of motivating its contribution and exploring its effect in simple intelligible tasks, and I feel I got more out of this paper than most SOTA papers.\n\n\nClarity: Very clear.\nSignificance: Potentially quite significant as this is starting to bring causal structure learning into the realm of tensorflow and pytorch.\n\nQuestions and comments:\n- All else being equal, the speed of adaptation between two very similar models will serve as a good proxy, as shown in this paper. However, I can easily imagine scenarios where the two models one wants to differentiate between are quite different, and have very different optimization landscapes. Here, the speed of adaptation will be quite dependent on these landscapes and not just on the underlying model structure. Do you have thoughts about how this can be extended to such a scenario?\n\n- The parameter counting argument is not nearly so strong if what actually changes is the conditional p(A|B). In that case, the sample complexity for the correct model would be N^2 = O(N^2) and for the incorrect model would be N + N^2 = O(N^2). Does the objective still work here? Would be great to add an additional experiment showing the results in this case.\n\n- Doing an intervention and drawing a new D_int for each step of gradient descent seems quite prohibitive in a lot of domains. Are there ways to decrease this burden?\n\n- In Figure 2, can you speak to why the N=100 curve for the MLP parameterization converges more slowly than the N=10 curve? I would still expect more data to be beneficial here.\n", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper."}, "extra_text": {}}, {"review_id": "r1lca6O85S", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "In this work, the authors proposed a general and systematic framework of meta-transfer objective incorporating the causal structure learning under unknown interventions. Under the assumption of small change (out-of-distribution data), the work mainly focuses on the theoretical and empirical analysis of relations on two random variables in causal graphs (causal and anti-causal directions), so that a differentiable regret function using the joint distribution of the small \"intervention\" dataset can be built. \n \nThe motivation is to adapt or transfer quickly by discovering the correct causal direction and learning representation based on it. The idea of disentangling the marginal and conditional factors to reduce the sample complexity and thus achieve fast adaptation is novel and insightful. Proposition 1 and its proof provide the theoretical supports on this point very well. The structure causal model is parametrized and then optimized in a meta-learning procedure. Experiments on simulated data under categorical or continuous distributions can verify the efficiency of inferring causal graphs. \n \nHere are some concerns about the proposed algorithm:\n \n1). When the authors discussed small change, there is no formal (mathematical) definition on it. For instance, an invertible function could be one of the properties given for the out-of-distribution data. The example (rotation) in Fig. 3 works to some extent because the small transformation is invertible. Also, the intervention seems simple in the work, for example, the rotate angle (a value) in Fig. 3 only involves one parameter dimension. In this case, learning an encoder to infer the correct causal relation is not that difficult. Is there possible that the encoder cannot learn a good enough theta to find the correct causal direction? It would be nice if the limitations of using causal graphs are discussed.\n \n2). Given a direction A causes B, the experiments are conducted by performing interventions on the cause A. How about to put an intervention on the effect B? According to the algorithm analysis (Table D.1), for the discrete bivariate model, the parameter dimension of a correct structure becomes N^2, while the one of an incorrect structure becomes N + N^2. Compared to intervention on cause, the reduction of sample complexity here is not that obvious. A general discussion on the effect intervention for bivariate models would be helpful. \n \n3). The work opens a new direction of inferring causal relationships together with representation learning, which has the potential for more out-of-distribution scenarios. While the authors claim that it is the first step, the current empirical studies for structure models use synthetic data with relatively constraint assumptions. It is highly recommended for the authors to provide discussions about real-data tasks with neural causal models in future work.      \n", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I carefully checked the derivations and theory."}, "extra_text": {}}, {"review_id": "SygYFAKF9S", "reviewer": "AnonReviewer5", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposes a method of discovering causal mechanisms through meta-learning, by assuming that models transfer faster if their causal graph is correct.\n\nFor a possible causal graph, for adaptation to one interventional dataset, the samples are iteratively revealed and the log likelihood of the next sample is measured, after which the parameters are updated using one step of gradient descent on that sample. The sum of these log likelihoods is the ‘online likelihood’ and a measure of speed of adaptation. The parameters are initialised using maximum likelihood estimation on the train dataset.\nThe meta learning procedure then at each episode samples an interventional distribution and an interventional dataset from that distribution. It then performs a gradient based update of the belief over graphs based on the difference between the online likelihoods of each graph on that dataset.\n\nThe meta-learning approach appears to be a novel contribution. The authors provide a theoretical argument by counting ‘effective parameters’ to suggest why models using the right causal model obtain a higher online likelihood. Additionally, they prove that the gradient updates to the graph belief are easy to compute and converge. The method is validated with several synthetic experiments which discover the direction of the arrow between two random variables, each either continuous or discrete. Furthermore, they successfully experiment with the combination of learning a representation of a raw data to two random variables with learning the causal direction. The paper is very well written and most claims are carefully proven.\n\nI recommend a weak rejection for this paper, because:\n1) The empirical validation is not strong enough, as no real dataset is used, only toy datasets. The toy experiments themselves could also be more extensive.\n2) I am unconvinced of two of the theoretical claims made: (A) the fact that the expected gradients in Prop 1 are 0, implies that the right causal graph has better online likelihood and (B) that the method is easily extensible to more than two random variables [Appendix E].\n\nSupporting arguments:\n1.1) Fig D.1 suggests that, in the experiments using continuous random variables, the training dataset alone is sufficient to discover the true causal model, under the assumption of independent additive noise, as is done in e.g. [1]. I find it plausible to believe that the training curve on the training dataset alone already makes it possible to disambiguate the causal from anti-causal model. A similar pattern is shown by the authors themselves in Appendix B on discrete variables.\nFor finite training data, the models are distinguishable, while for infinite training data they are not [Fig B.1]. The exact same holds for finite and infinite interventional data [Fig 1].\nAre these experiments then good benchmarks for causal discovery based on intervention when the causal model can already be inferred from the non-interventional training data?\n1.2) The simplicity of the representation learning setup doesn’t convince that the method is applicable to more real-world settings with more complicated encoders. Additionally, some important details on this experiment are missing (see below).\n1.3) All experiments show the effect of intervention on the cause p(A). No experiments are given for intervention on p(B|A). Does the method then still work?\n1.4) No experiments for more than two random variables are performed in this paper.\n2.A) Prop 1 shows that the expected maximum likelihood gradient for one conditional probability distribution is zero if the graph is correct and that CPD is not intervened on. Subsequently a claim is made that this effectively reduces the number of parameters and thus that adaptation is faster. However, the zero-expectation gradient may still be non-zero and even large on the small intervention sample. It is unclear to me why they therefore can be excluded in the number of parameters. Furthermore, whether the online likelihood is large will depend not only on generalisation, but also on the training convergence, since not empirical risk minimization is used, but SGD with a fixed number of steps. Thus, even though the authors prove that the method will converge to the causal graph with lowest online likelihood, it is unclear why this is necessarily the correct causal graph.\n2.B) In appendix E it is mentioned that cycles can occur in causal models. However, it is unclear why factorization (76) is still correct in the cyclic case. Perhaps I am misunderstanding, but it seems to me that p(x) = \\prod_i p(x_i | x_{Pa_i}) only makes sense for a DAG. Hence, how would the online likelihood be computed for cyclic models?\n\nSuggestions for improvement:\n- Could you explain in what realistic settings we would have access to data from a large number of different interventional distributions?\n- Could you show a plot similar to Fig 1, but with online likelihoods? Such a plot may be more indicative of the ideal episode length than Fig 1.\n- Could you provide details on the representation learning experiment? In particular: (1) is \\theta_D different in the train dataset and each interventional dataset? (2) How do the gradients of theta flow through the meta-update steps?\n- A reference to Appendix B appears missing in main text.\n\n[1] Nonlinear causal discovery with additive noise models. Hoyer et al 2009", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes to discover causal mechanisms through meta-learning, and suggests an approach for doing so. The reviewers raised concerns about the key hypothesis (that the right causal model implies higher expected online likelihood) not being sufficiently backed up through theory or through experiments on real data. The authors pointed to a recent paper that builds upon this work and tests on a more realistic problem setting. However, the newer paper measures not the online likelihood of adaptation, but just the training error during adaptation, suggesting that the approach in this paper may be worse. Despite the concerns, the reviewers generally agreed that the paper included novel and interesting ideas, and addressed a number of the reviewers' other concerns about the clarity, references, and experiments. Hence, it makes a worthwhile contribution to ICLR.
Decision: Accept (Poster)