Loss functions for classification: Difference between revisions

Revision as of 06:24, 8 April 2024

Revision as of 02:51, 11 July 2024

Line 30:

One can solve for the minimizer of <math>I[f]</math> by taking the functional derivative of the last equality with respect to <math>f</math> and setting the derivative equal to 0. This will result in the following equation

:<math>\frac{\partial \phi(f)}{\partial f}\eta + \frac{\partial \phi(-f)}{\partial f}(1-\eta)=0, \;\;\;\;\;(1)</math>{{Citation needed|date=February 2023}}{{Clarify|reason=What is η?|date=February 2023}}

:<math>

\frac{\partial \phi(f)}{\partial f}\eta + \frac{\partial \phi(-f)}{\partial f}(1-\eta)=0 \;\;\;\;\;(1)

</math>{{Citation needed|date=February 2023}}{{Clarify|reason=What is η?|date=February 2023}}

:where <math>

which is also equivalent to setting the derivative of the conditional risk equal to zero.

\eta=p(y=1|\vec{x})

</math>, which is also equivalent to setting the derivative of the conditional risk equal to zero.

Given the binary nature of classification, a natural selection for a loss function (assuming equal cost for [[false positives and false negatives]]) would be the [[0-1 loss function]] (0–1 [[indicator function]]), which takes the value of 0 if the predicted classification equals that of the true class or a 1 if the predicted classification does not match the true class. This selection is modeled by