QUESTION:

We often use squared error as a loss function in regression. Can you think of a situation in which that might not be a good idea or other loss functions would be better? (What would be a good loss function if you were trying to throw a ball of radius r into a circular hole of radius R?)

ANSWER:

"Squared error is a commonly used loss function in regression because it is differentiable, convex and has a closed-form solution. However, in some cases it might not be the best choice. For example, if the data has outliers, the squared error loss function will be sensitive to them and will give more weight to them, potentially leading to a model that is not robust to these outliers. In these cases, other loss functions such as the Huber loss function, which is more robust to outliers, or the quantile loss function, which minimizes the quantile of the errors, may be more appropriate.

Another example is when we try to throw a ball of radius r into a circular hole of radius R. If we used the squared error as the loss function, it would give the same weight to errors in both the distance and the direction. However, in this problem the distance from the center of the hole is likely more important than the direction, so a loss function based on the distance, such as the mean absolute error, would be more appropriate."

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