On regression to minimize log distance rather than distance Suppose I have a lot of points $ x_i \in \mathbb{R}^N $ with corresponding non-negative labels $ y_i \in \mathbb{R} $ and I want to do regression and make a prediction on some new datapoint $ x^* \in \mathbb{R}^N $ for which I don't have a label.   Is there a name for the procedure of choosing a parametric model $ f_\theta : \mathbb{R}^N \rightarrow \mathbb{R} $ so as to minimize the cost function $ \sum_i {|\log(f_\theta(x_i)) - \log(y_i)|^2 } $ rather than $ \sum_i{|f_\theta(x_i) - y_i|^2} $?  It seems that minimizing the difference between logs has some nice properties, and I'm surprised I didn't see this discussed in Bishop's machine learning book, for example.  

I thought of this when I was considering a house pricing problem, where I figured I cared more about the percentage by which I was wrong than the pure difference.  After all, in my application (and I'm sure many others like it), being wrong by \$50,000 is terrible for a \$60,000 home, but it's okay for a $2.5M home.  

Any data science veterans reading this who have used a cost function like the one I suggested above with logs, or who can tell me what it's called (if it has a formal name)?
There is a loss called Root Mean Squared Log Error (RMSLE): $\sqrt[]{\frac{1}{n}\sum_{i=1}^n{(\log(y_i + 1) - \log(\hat{y_i} + 1))^2}}$ (do not forget the $+1$ as the $log$ is not defined at $0$)

You will find a brief explanation and discussion here. It has also been used in competitions as for example here.