Loss functions for classification: Difference between revisions

Revision as of 19:40, 1 August 2019

Revision as of 02:24, 27 August 2019

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The logistic loss function can be generated using (2) and Table-I as follows

:<math>\begin{align}

:<math>\phi(v)=C[f^{-1}(v)]+(1-f^{-1}(v))C'[f^{-1}(v)] =\frac{1}{\log(2)}[\frac{-e^v}{1+e^v}\log(\frac{e^v}{1+e^v})-(1-\frac{e^v}{1+e^v})\log(1-\frac{e^v}{1+e^v}))]+(1-\frac{e^v}{1+e^v})[\frac{-1}{\log(2)}(\log(\frac{\frac{e^v}{1+e^v}}{1-\frac{e^v}{1+e^v}}))]=\frac{1}{\log(2)}\log(1+e^{-v}).</math>

\phi(v) &= C[f^{-1}(v)]+(1-f^{-1}(v))C'[f^{-1}(v)] \\

:<math>\phi(v)&=C[f^{-1}(v)]+(1-f^{-1}(v))C'[f^{-1}(v)] =\frac{1}{\log(2)}[\frac{-e^v}{1+e^v}\log(\frac{e^v}{1+e^v})-(1-\frac{e^v}{1+e^v})\log(1-\frac{e^v}{1+e^v}))]+(1-\frac{e^v}{1+e^v})[\frac{-1}{\log(2)}(\log(\frac{\frac{e^v}{1+e^v}}{1-\frac{e^v}{1+e^v}}))]=\frac{1}{\log(2)}\log(1+e^{-v}).</math>

\\ &=\frac{1}{\log(2)}\log(1+e^{-v}).

\end{align}</math>

The logistic loss is convex and grows linearly for negative values which make it less sensitive to outliers. The logistic loss is used in the [[LogitBoost|LogitBoost algorithm]].