  Learning from implicit user feedback is challenging as we can only observe
positive samples but never access negative ones. Most conventional methods cope
with this issue by adopting a pairwise ranking approach with negative sampling.
However, the pairwise ranking approach has a severe disadvantage in the
convergence time owing to the quadratically increasing computational cost with
respect to the sample size; it is problematic, particularly for large-scale
datasets and complex models such as neural networks. By contrast, a pointwise
approach does not directly solve a ranking problem, and is therefore inferior
to a pairwise counterpart in top-K ranking tasks; however, it is generally
advantageous in regards to the convergence time. This study aims to establish
an approach to learn personalised ranking from implicit feedback, which
reconciles the training efficiency of the pointwise approach and ranking
effectiveness of the pairwise counterpart. The key idea is to estimate the
ranking of items in a pointwise manner; we first reformulate the conventional
pointwise approach based on density ratio estimation and then incorporate the
essence of ranking-oriented approaches (e.g. the pairwise approach) into our
formulation. Through experiments on three real-world datasets, we demonstrate
that our approach not only dramatically reduces the convergence time (one to
two orders of magnitude faster) but also significantly improving the ranking
performance.
