  Recent advances in unbiased learning to rank (LTR) count on Inverse
Propensity Scoring (IPS) to eliminate bias in implicit feedback. Though
theoretically sound in correcting the bias introduced by treating clicked
documents as relevant, IPS ignores the bias caused by (implicitly) treating
non-clicked ones as irrelevant. In this work, we first rigorously prove that
such use of click data leads to unnecessary pairwise comparisons between
relevant documents, which prevent unbiased ranker optimization. Based on the
proof, we derive a simple yet well justified new weighting scheme, called
Propensity Ratio Scoring (PRS), which provides treatments on both clicks and
non-clicks. Besides correcting the bias in clicks, PRS avoids relevant-relevant
document comparisons in LTR training and enjoys a lower variability. Our
extensive empirical evaluations confirm that PRS ensures a more effective use
of click data and improved performance in both synthetic data from a set of LTR
benchmarks, as well as in the real-world large-scale data from GMail search.
