Learning Neural Causal Models From Unknown Interventions Summary - - PowerPoint PPT Presentation

Learning Neural Causal Models From Unknown Interventions

Summary

• The relationship between each variable and its parents is

modeled by a neural network, modulated by structural meta-parameters which capture the overall topology of a directed graphical model.

• Assume Random intervention on a single unknown

variable of an unknown ground truth causal model.

• To disentangle the slow-changing aspects of each

conditional from the fast-changing adaptations to each intervention, the neural network is parameterized into fast parameters and slow meta-parameters.

Summary

• meta-learning objective that favors solutions robust to

frequent but sparse interventional distribution change.

• Challenging aspect of this setting is to not only learn the

causal graph structure, but also predict the intervention accurately.

Task Description

• At most one intervention is concurrently performed.
• Soft/imperfect Interventions: the conditional distribution of the variable on which

the intervention is performed is changed.

• Provided data:
• data from the original ground-truth model.
• data from a modified ground-truth model with a random intervention applied.
• Learner aware of the intervention, but not aware of the node. (each run an

episode)

• The learner, over a large number of episodes, will experience all nodes being

intervened upon, and should be able to infer the SCM from these interventions.

Objectives

• Avoid an exponential search over all possible DAGs
• Handle unknown interventions
• Model the effect of interventions
• Model the underlying causal structure.

• Function approximation: 1 NN/var
• Belief over

: drop-out probability for i-th input of network j

• Represents all

possible graphs.

• Learning drop-out probability
• Prevents discrete search

i → j 2M2

Model

• SCM (

Categorical Random Variables and Categories):

• Configuration
• counts the number of length-n walks from node to node
• f the graph in element

.

• counts the number of length-n cycles in the graph
• M

N C ∈ {0,1}M×M Cn i j cij Tr(Cn) C = ?

Model

• Consider two node graph

.

• :
• r

= 1 (versus 0)

• and
• ,
• Problem becomes simultaneously learning the structural meta-parameters

and the functional meta-parameters .

• : Easily learned by ML (Back Propagation)
• : More Difficult (Bengio et al. 2019)

A, B M = 2 cAB cBA P(cAB = 1) = σ(γAB) P(cBA = 1) = σ(γBA) γ θ θ γ

Model

Problems

• An M-variable SCM over random variables

can induce a super-exponential number of adjacency matrices .

• The super-exponential explosion in the number of

potential graph connectivity patterns

• The super-exponentially growing storage requirements of

their defining conditional probability tables make CPT- based parametrization of the structural assignments increasingly unwieldy as M scales.

Xi C fi

Solution

• Neural networks with
• masked inputs can provide a

more manageable parametrization.

cij

Proposed Method

• Disentangle :
• : The slow-changing meta-parameters, which

reflect the stationary properties discovered by the learner.

• : The fast-changing parameters, which adapt in

response to interventional changes in distribution.

θ θslow θfast

Proposed Method

• Two kinds of meta-parameters: the causal graph structure

and the model’s slow weights .

• Model’s fast weights

.

• , the sum of the slow, stationary meta-

parameters and the fast, adaptational parameters.

γ θslow θfast θ = θslow + θfast

Optimization Problem

• The strategy of considering all the possible structural graphs as

separate hypotheses is not feasible because it would require maintaining O( ) models of the data.

• Sampling

independently using Bernoulli Distribution.

• We only need to learn the

coefficients .

• a slight dependency between the

is induced if we require the causal graph to be acyclic.

• A regularizer acting on the .

2M2 cij M2 γij cij γ

Optimization Problem

• Each random variable

.

• is a neural network (MLP) with parameters
• And
• ptimized to maximize the likelihood of data under the model
• ptimized with respect to a meta-learning objective arising from changes in

distribution because of interventions.

• Analogous to an ensemble of neural nets differing by their binary input

dropout masks, which select what variables are used as predictors of another variable.

Xi = fθi(ci0 × X0, ci1 × X1, …, cim × Xm, ϵi) fθi θi cij ∼ Bin (σ(γij)) θ γ

Learning

• To disentangle:
• Environment’s stable, unchanging properties (the

causal structure)

• From unstable, changing properties (the effects of an

intervention)

• MLP:

, where .

Pi(Xi|Xpa(i); θi) θ = θslow + θfast

Learning

• are reset after each episode of transfer distribution adaptation
• Since an intervention is generally not persistent from one transfer

distribution to another.

• meta-parameters (

) are preserved, then updated after each episode.

• The meta-objective for each meta-example over some intervention

distribution is the following:

θfast θslow, γ Dint

Learning

• The meta-objective for each meta-example over some intervention distribution

is the following meta-transfer loss:

• is an example sampled from the intervention distribution

, is an adjacency matrix drawn from our belief distribution (parametrized by ) about graph structure configurations.

• The likelihood of the -th variable
• f the sample when predicting it under

the configuration from the set of its putative parents:

Dint X Dint C γ i Xi X C

Learning

• A discrete Bernoulli random sampling process is used to produce the

configurations under which the log-likelihood of data samples is obtained.

• A gradient estimator is required to propagate gradient through to the

structural meta-parameters.

• superscript indicates the values obtained for the -th draw of

.

• This gradient is estimated solely with

because estimates employing have much greater variance.

γ

(k)

k C θslow θ

Acyclic Constraint

Acyclic Constraint

Predicting Interventions

• After an intervention on

, the gradients into and the slow weights for the -th conditional are false, because they do not bear the blame for ’s outcome (which lies with the intervener).

• The conditional likelihood of the intervened variable tends

to have a poorer relative likelihood under .

• Hence, the variable with the greatest deterioration in

likelihood is picked as a good guess.

Xi γi i Xi Dint

Model Description

• The MLPs are identical in shape but do not share any

parameters, since they are modeling independent causal

• mechanisms. (M one-hot vectors (nodes) of length N each)

cij ∼ Bin (σ(γij))

Xi = fθi(ci0 × X0, ci1 × X1, …, cim × Xm, ϵi)

Stability of Training

• Simultaneous training of both the structural and the

functional meta-parameters.

• These are not independent and do influence each other,

which leads to instability in training.

• Pre-train the model under observational data (from the

distribution of the data before interventions) using dropout on the inputs.

• functional meta-parameters

are not too biased towards certain configurations of the meta-parameters .

θslow γ

Regularizers

• DAG Constraint
• Sparsity:
• sparse representation of edges in the causal graph.
• L1 regularizer
• Slightly faster convergence

Temperature

• A temperature hyperparameter to encourage the

groundtruth model to generate some very rare events in the conditional probability tables (CPTs) more frequently.

• The near-ground-truth MLP model’s logit outputs are divided

by the temperature before being used for sampling.

All in all: Algorithm

Pre-train on Observational Data Predict Intervened Node

Simulations: Synthetic Data

The results are, however, sensitive to some hyperparameters, notably the DAG penalty and the sparsity penalty.

Simulations: Real-World Data

• BNLearn: Earthquake, Cancer, Asia
• variables respectively, maximum 2 parents per node.
• Learn a near-ground-truth MLP from the dataset’s CPT and use it as the ground-truth data generator.
• In spite of same causal graphs, CPTs were different; hence different SCMs.

M = 5,5,8

Simulations: Comparison

• Peters et al., (2016): ICP
• Eaton & Murphy (2007a): uncertain interventions
• Peters et al. (2016): unknown interventions
• However, neither attempt to predict the intervention.

Importance of Dropout

• Used for initial training on observational data.
• Fully connected off-diagonal (the most DOF).
• Pre-training cannot be carried out this way.

Importance of Intervention Prediction

• After the intervention has been performed, the learner

draws data samples from the intervention distribution and computes the per-variable average log-probability under sampled adjacency matrices.

• The variable consistently producing the least-likely
• utputs is predicted to be the intervention node.

Importance of Intervention Prediction