# Performance metric in recommender systems with implicit feedback

This paper describes a technique for making recommendations when the feedback is implicit, that is, $r_{ui}$ is only a guess. The recommendation problems boils down to the following optimisation problem: $$\min_{x^*, y^*}\ \ \sum_{u,i}c_{ui}(p_{ui}-x_i^Ty_i)^2+\lambda \left(\sum_u\|x_u\|^2+\|y_u\|^2 \right)$$ where $c_{ui} = 1+\alpha r_{ui}$.

I have the following concerns:

1. The algorithm above approximates the binary indicator function $p_{ui}$, weighted by the term $c_{ui}$. However, most people use RMSE between $r_{ui}$ and $x_i^Ty_i$ for assessing performance. I am not sure why that would make sense.
2. Why using RMSE at all? With implicit feedback, it would make more sense to me to use rank-based metrics. I would imagine RMSE is more suitable in machine learning competition where the objective metric is usually RMSE.