Loss functions for classification

In machine learning and mathematical optimization, loss functions for classification are computationally feasible loss functions representing the price paid for inaccuracy of predictions in classification problems. Given ${\displaystyle X}$ as the vector space of all possible inputs, and Y = {–1,1} as the vector space of all possible outputs, we wish to find a function ${\displaystyle f:X\mapsto \mathbb {R} }$ which best maps ${\displaystyle {\vec {x}}}$ to ${\displaystyle y}$. However, because of incomplete information, noise in the measurement, or probabilistic components in the underlying process, it is possible for the same ${\displaystyle {\vec {x}}}$ to generate different ${\displaystyle y}$. As a result, the goal of the learning problem is to minimize expected risk, defined as

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