How to choose? Which is the preferred loss for ANN regression?

Mean Squared Error Loss or MSE, Quadratic loss, L2 Loss

It is a default and most commonly used regression loss function. MSE is the sum of squared distances between our actual/target variables and predicted values. It is very acceptable under the theory of maximum likelihood, if the distribution of actual/target variables is Gaussian(Normal Distribution).

Under maximum likelihood, a loss fn. estimates how closely the distribution of predictions made by a model matches the distribution of actual/target variables in the training data.

Mean Squared Logarithmic Error Loss or MSLE

Use when the target value has scatteredness and encountered with large values. In MSLE, natural logarithm is calculated followed by a mean squared error.

It has the effect of relaxing the punishing effect of large differences in large predicted values and you don’t want large errors to be significantly more penalized than small ones.

Mean Absolute Error Loss

When the distribution of the target variable may be mostly Gaussian but may have outliers, e.g. large or small values far from the mean value.

The MAE is more robust to outliers. It is calculated as the average of the absolute difference between the actual and predicted values.