  Implicit feedback (e.g., click, dwell time) is an attractive source of
training data for Learning-to-Rank, but its naive use leads to learning results
that are distorted by presentation bias. For the special case of optimizing
average rank for linear ranking functions, however, the recently developed
SVM-PropRank method has shown that counterfactual inference techniques can be
used to provably overcome the distorting effect of presentation bias. Going
beyond this special case, this paper provides a general and theoretically
rigorous framework for counterfactual learning-to-rank that enables unbiased
training for a broad class of additive ranking metrics (e.g., Discounted
Cumulative Gain (DCG)) as well as a broad class of models (e.g., deep
networks). Specifically, we derive a relaxation for propensity-weighted
rank-based metrics which is subdifferentiable and thus suitable for
gradient-based optimization. We demonstrate the effectiveness of this general
approach by instantiating two new learning methods. One is a new type of
unbiased SVM that optimizes DCG -- called SVM PropDCG --, and we show how the
resulting optimization problem can be solved via the Convex Concave Procedure
(CCP). The other is Deep PropDCG, where the ranking function can be an
arbitrary deep network. In addition to the theoretical support, we empirically
find that SVM PropDCG significantly outperforms existing linear rankers in
terms of DCG. Moreover, the ability to train non-linear ranking functions via
Deep PropDCG further improves performance.
