Predictive models that generalize well under distributional shift are often desirable and sometimes crucial to machine learning applications. One example is the estimation of treatment effects from observational data, where a subtask is to predict the effect of a treatment on subjects that are systematically different from those who received the treatment in the data. A related kind of distributional shift appears in unsupervised domain adaptation, where we are tasked with generalizing to a distribution of inputs that is different from the one in which we observe labels. We pose both of these problems as prediction under a shift in design. Popular methods for overcoming distributional shift are often heuristic or rely on assumptions that are rarely true in practice, such as having a well-specified model or knowing the policy that gave rise to the observed data. Other methods are hindered by their need for a pre-specified metric for comparing observations, or by poor asymptotic properties. In this work, we devise a bound on the generalization error under design shift, based on integral probability metrics and sample re-weighting. We combine this idea with representation learning, generalizing and tightening existing results in this space. Finally, we propose an algorithmic framework inspired by our bound and verify is effectiveness in causal effect estimation.
The paper proposes a novel way of causal inference in situations where in causal SEM notation the outcome Y = f(T,X) is a function of a treatment T and covariates X. The goal is to infer the treatment effect E(Y|T=1,X=x) - E(Y|T=0,X=x) for binary treatments at every location x. If the treatment effect can be learned, then forecasts of Y under new policies that assign treatment conditional on X will still "work" and the distribution of X can also change without affecting the accuracy of the predictions. 
What is proposed seems to be twofold:
- instead of using a standard inverse probability weighting, the authors construct a bound for the prediction performance under new distributions of X and new policies and learn the weights by optimizing this bound. The goal is to avoid issues that arise if the ratio between source and target densities become very large or small and the weights in a standard approach would become very sparse, thus leading to a small effective sample size.
- as an additional ingredient the authors also propose "representation learning" by mapping x to some representation Phi(x). 
The goal is to learn the mapping Phi (and its inverse) and the weighting function simultaneously by optimizing the derived bound on the prediction performance. 
Pros: 
- The problem is relevant and also appears in similar form in domain adaptation and transfer learning. 
- The derived bounds and procedures are interesting and nontrivial, even if there is some overlap with earlier work of Shalit et al. 
Cons:
- I am not sure if ICLR is the optimal venue for this manuscript but will leave this decision to others. 
- The manuscript is written in a very compact style and I wish some passages would have been explained in more depth and detail. Especially the second half of page 5 is at times very hard to understand as it is so dense. 
- The implications of the assumptions in Theorem 1 are not easy to understand, especially relating to the quantities B_\Phi, C^\mathcal{F}_{n,\delta} and D^{\Phi,\mathcal{H}}_\delta. Why would we expect these quantities to be small or bounded? How does that compare to the assumptions needed for standard inverse probability weighting? 
- I appreciate that it is difficult to find good test datasets for evaluating causal estimator.  The experiment on the semi-synthetic IHDP dataset is ok, even though there is very little information about its structure in the manuscript (even basic information like number of instances or dimensions seems missing?). The example does not provide much insight into the main ideas and when we would expect the procedure to work more generally.