Paper:
Machine learning models, especially deep neural networks have been shown to reveal membership information of inputs in the training data. Such membership inference attacks are a serious privacy concern, for example, patients providing medical records to build a model that detects HIV would not want their identity to be leaked. Further, we show that the attack accuracy amplifies when the model is used to predict samples that come from a different distribution than the training set, which is often the case in real world applications. Therefore, we propose the use of causal learning approaches where a model learns the causal relationship between the input features and the outcome. An ideal causal model is known to be invariant to the training distribution and hence generalizes well to shifts between samples from the same distribution and across different distributions. First, we prove that models learned using causal structure provide stronger differential privacy guarantees than associational models under reasonable assumptions. Next, we show that causal models trained on sufficiently large samples are robust to membership inference attacks across different distributions of datasets and those trained on smaller sample sizes always have lower attack accuracy than corresponding associational models. Finally, we confirm our theoretical claims with experimental evaluation on 4 datasets with moderately complex Bayesian networks. We observe that neural network-based associational models exhibit upto 80% attack accuracy under different test distributions and sample sizes whereas causal models exhibit attack accuracy close to a random guess. Our results confirm the value of the generalizability of causal models in reducing susceptibility to privacy attacks.

Review:
Summary:
   The authors consider a transfer learning problem where the source distribution is P(X,Y) while the target distribution is P*(X,Y) and classifier is trained on data from the source distribution. They also assume that the causal graph generating the data (X and Y) is identical while the conditional probabilities (mechanisms) could change between the source and the target. Further, they assume that if X_C is the Markov Blanket for variable Y in P and P*,  then P(Y|X_C) = P*(Y|X_C). Therefore the best predictor in terms of cross entropy loss for both distributions is identical if it focuses on the variables in the Markov Blanket. Authors define "causal hypothesis" as the one that uses only variables in the Markov Blanket (X_C) to predict Y.
 In this setting, the authors show two sets of results: a) Out of Distribution Generalization Error is less for the optimal causal predictors from any class of causal hypotheses under any loss function. b) If we tweak the definition of differential privacy where neighboring datasets are defined by replacing one of the variables in the training set by a sample from the test distribution (I have lots of questions about this definition later on), then the authors show that causal classifiers that depend only on the Markov Blanket has lower sensitivity to the change than that of associative classifiers (that use all features). This is used to prove that the causal ones have tighter differential privacy guarantees than the associative ones
 c) Using the differential privacy results, they also show that optimal causal classifiers are more resistant to membership attacks.
The authors demonstrate the results through membership attack accuracies on causally trained and associative models on 4 datasets where the causal graph and the parameters (conditional probabilities) of the Bayesian Network are known apriori.
Major Issues:
    I have lots of issues with the theory in the paper. Thats the main reason for my recommendation.
  1. Why is h_{c,P}^{OPT} = h_{c,P*}^{OPT} ?? (Equation 23 in Page 11) - Authors say that since the markov blanket is the only thing used to predict Y for causal predictors and P (Y|X_C) = P *(Y|X_C), this should be true. But I have problems with this statement/argument. First of all, this is claimed for for any loss function L (not just Cross Entropy Loss) and particularly for a generic hypothesis class H_C (that depends only on Markov Blanket).  
Consider the following counter example - Suppose all the features are in the Markov Blanket of Y (even a simpler case where all features are causal parents of Y). Suppose the true P (Y|X) is a logistic model with weights w_1 on one part of the domain D_1 and with weights w_2 for another part of the domain D_2. Suppose P is a mixture distribution of D_1 and D_2. P* is another mixture distribution (mixed differently) of D_1 and D_2.
Suppose I consider the class of logistic classifiers (but a single logistic model with one weight w) to be my hypothesis class and I use the standard logistic loss (logistic regression) on P, since a single logistic model with one slope cannot match the different slopes in different parts of the domain, it will result in some weight vector w^{opt}_1. Now, since the mixtures of D_1 and D_2 are different in the P* (but P (Y|X ) is identical), the optimal w^{opt}_2 for the P* will be different. 
So for arbitrary hypothesis classes (that do not capture the true P(Y|X_c)) and for a non cross entropy loss - clearly this does not hold at all !! Covariate shifts amongst X_C alone with create a different classifier for an arbitrary loss (even if P (Y|X_C) is the same across both). In fact, the only way I see to salvage this is to assume Cross Entropy loss and talk about all soft classifiers (without restrictions to hypothesis class). But even if thats the case, then the best associational model will be the one that uses the Markov Blanket too ! .
This claim about h_C is crucially used in proof of theorem 1 (Page 11) and Proof of Corollary 1 (Page 13). This is fundamental to all theorems later. That brings into question the validity of many (if not all) the theoretical results in the paper. Authors must address this. 
2. Issues regarding definition of certain quantities.  
a) In equation 5 a quantity max_{x,x'} L_{x sampled from P} (h_c,f) - L{x' sampled from P*} (h_c,f) is defined - the inner quantity is a random quantity that depends on the samples x and x', Then what is the max operator over ?? - What does it mean to have worst case over samples from a distribution ??? Does it mean samples from two different domains ?? 
Even the quantity does not seem to be well defined. 
b) Similar issue occurs in Lemma 1 - Neighboring datasets S and S' are created by first sampling S from P and then S' is obtained by replacing an arbitrary point in S by a random point from P*. Then sensitivity is defines as a max over pair of neighboring datasets - again S and S' are random samples, so what is the max over ?? If it is the worst case - why is the sampling coming in there ? Since it uses Corollary 1 - the main result inherits the same fundamental issues the has been pointed out above.
c) In theorem 2, {cal F}_a is an algorithm. What does it mean to add noise ? - Does it mean you add noise to the model parameters ?? - This is confusing at best.
Minor Issues:
1. Authors claim that connection between causality and privacy has not been explored (Page 2). Pls refer to https://arxiv.org/pdf/1710.05899.pdf where differential privacy itself is related to interventional effects in a system. This connection is very different from the scope of the current paper. However, the statement by the authors is strictly not true.
2. Why is the ground truth function f:X->Y (Section 2.2.) relevant when clearly you have distribution P (X,Y) and P*(X,Y) ?? We might as well define Loss with L (h(x), y) where (x,y) is drawn according to P. What f is never defined anywhere. Authors seems to mean the suggestion I just said in the paper. Authors could clarify. This confuses stuff in the proofs too.
3. Markov Blanket is not causal by any means in my opinion. It is just a minimal set of features conditioned on which Y does not depend on anything else. This only requires conditional independence tests to determine - a purely observational notion - in fact the markov blanket only depend on the moralized graph which does not change across the members of the equivalence class. So calling it causal is a bit confusing. If the features referred to least causal Parents - then still it would be consistent with the invariance in the Invariant Causal Prediction Literature (Peters et al 2016.) and would be causal.