Log-space target can be used to encode proportional error price across the target distribution

This is because applying a $lo{g}_{10}$ function over the target allows, for example, the error of 1 and 10 be the same as 1,000,000 and 10,000,000 (in both cases it’s 1). This is desired when the same error on the low-end y and the tail-end y has a different importance. For example, an error of 10 for $y=10$ should be more expensive than the error of 10 for $y=10,000,000$. If the error of 10 has the same importance for both y’s, we should avoid using this transformation.

It’s just another angle of looking at how to decide how we want the model to behave when encountering the same error at a different magnitude

Some clarifying notes:

• Log function is applied to target variable at preprocessing time
• We’re describing the absolute difference between data points
• Loss function is applied after target transformation process