On regression to minimize log distance rather than distance
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.