Loss functions for classification

This sandbox is in the article namespace. Either move this page into your userspace, or remove the {{User sandbox}} template. Loss function surrogates for classification are computationally feasible loss functions representing the price we will pay for inaccuracy in our predictions in classification problems. ^[1] Specifically, if ${\displaystyle g:X\mapsto }$ {-1,1} represents the mapping of a vector ${\displaystyle {\vec {x}}\in X}$ to a class label ${\displaystyle y\in }$ {-1,1}, we wish to find a function ${\displaystyle f:X\mapsto \mathbb {R} }$ which best approximates the true mapping ${\displaystyle g}$. (citation needed) Given that loss functions are always true functions of only one variable, it is standard practice to define loss functions for classification solely in terms of the product of the true classifier ${\displaystyle y}$ and the predicted value ${\displaystyle f({\vec {x}})}$. (citation needed) Selection of this ${\displaystyle \phi (yf({\vec {x}}))}$ impacts the optimal ${\displaystyle f^{*}}$ which minimizes empirical risk, as well as the computational complexity of the learning algorithm.

Square Loss

Hinge Loss

Main article: Hinge loss

Logistic Loss

References

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