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I have two loss functions $\mathcal{L}_1$ and $\mathcal{L}_2$ to train my model. The model is predominantly a classification model. Both $\mathcal{L}_1$ and $\mathcal{L}_2$ takes are two variants of the focal losses. $\mathcal{L}_1$ and $\mathcal{L}_2$ takes as input the same class probabilities and a hyper-parameter $\gamma$. The formulation of $\mathcal{L}_1$ and $\mathcal{L}_2$ are distinct. It is possible to mathematically show that

\begin{equation} \mathcal{L}_1\geq\mathcal{L}_2. \end{equation}

Based on this information can we comment on the useful-ness of $\mathcal{L}_1$ and $\mathcal{L}_2$? i.e. which of $\mathcal{L}_1$ and $\mathcal{L}_2$ is more useful to train the model?

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A loss being less than another loss has nothing to do with its usefulness. You can simply subtract a constant from a loss to obtain another loss (i.e. $\mathcal{L}_2 = \mathcal{L}_1 - \alpha$) that is always less than the former but nonetheless, it is exactly as useful as the other one was.

Also, a loss is as good as the model it can train so you normally judge the usefulness of the loss by comparing the performance of the models themselves.

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