  The feedback data of recommender systems are often subject to what was
exposed to the users; however, most learning and evaluation methods do not
account for the underlying exposure mechanism. We first show in theory that
applying supervised learning to detect user preferences may end up with
inconsistent results in the absence of exposure information. The counterfactual
propensity-weighting approach from causal inference can account for the
exposure mechanism; nevertheless, the partial-observation nature of the
feedback data can cause identifiability issues. We propose a principled
solution by introducing a minimax empirical risk formulation. We show that the
relaxation of the dual problem can be converted to an adversarial game between
two recommendation models, where the opponent of the candidate model
characterizes the underlying exposure mechanism. We provide learning bounds and
conduct extensive simulation studies to illustrate and justify the proposed
approach over a broad range of recommendation settings, which shed insights on
the various benefits of the proposed approach.
