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[[File:BayesConsistentLosses2.jpg|thumb|Bayes
In [[machine learning]] and [[mathematical optimization]], '''loss functions for classification''' are computationally feasible [[loss functions]] representing the price paid for inaccuracy of predictions in [[statistical classification|classification problem]]s (problems of identifying which category a particular observation belongs to).<ref name="mit">{{Cite journal | last1 = Rosasco | first1 = L. | last2 = De Vito | first2 = E. D. | last3 = Caponnetto | first3 = A. | last4 = Piana | first4 = M. | last5 = Verri | first5 = A. | url = http://web.mit.edu/lrosasco/www/publications/loss.pdf| title = Are Loss Functions All the Same? | doi = 10.1162/089976604773135104 | journal = Neural Computation | volume = 16 | issue = 5 | pages = 1063–1076 | year = 2004 | pmid = 15070510| pmc = | citeseerx = 10.1.1.109.6786 }}</ref> Given <math>X</math> 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 <math>f: X \mapsto \mathbb{R}</math> which best maps <math>\vec{x}</math> to <math>y</math>.<ref name="penn">{{Citation | last= Shen | first= Yi | title= Loss Functions For Binary Classification and Class Probability Estimation | publisher= University of Pennsylvania | year= 2005 | url= http://stat.wharton.upenn.edu/~buja/PAPERS/yi-shen-dissertation.pdf | accessdate= 6 December 2014}}</ref> However, because of incomplete information, noise in the measurement, or probabilistic components in the underlying process, it is possible for the same <math>\vec{x}</math> to generate different <math>y</math>.<ref name="mitlec">{{Citation | last= Rosasco | first= Lorenzo | last2= Poggio | first2= Tomaso | title= A Regularization Tour of Machine Learning | series= MIT-9.520 Lectures Notes | volume= Manuscript | year= 2014}}</ref> As a result, the goal of the learning problem is to minimize expected risk, defined as
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impacts the optimal <math>f^{*}_\phi</math> which minimizes the expected risk. Loss functions in this form are known as ''margin losses''.
:<math>
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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>
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as a proxy for expected risk.<ref name="mitlec" /> (See [[statistical learning theory]] for a more detailed description.)
==Bayes
Utilizing [[Bayes' theorem]], it can be shown that the optimal <math>f^*_{0/1}</math>
:<math>f^*_{0/1}(\vec{x}) \;=\; \begin{cases} \;\;\;1& \text{if }p(1\mid\vec{x}) > p(-1\mid \vec{x}) \\ \;\;\;0 & \text{if }p(1\mid\vec{x}) = p(-1\mid\vec{x}) \\ -1 & \text{if }p(1\mid\vec{x}) < p(-1\mid\vec{x}) \end{cases}</math>.
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A loss function is said to be ''classification-calibrated or Bayes consistent'' if its optimal <math>f^*_{\phi}</math> is such that <math>f^*_{0/1}(\vec{x}) = \operatorname{sgn}(f^*_{\phi}(\vec{x}))</math>and is thus optimal under the Bayes decision rule. A Bayes consistent loss function allows us to find the Bayes optimal decision function <math>f^*_{\phi}</math> by directly minimizing the expected risk and without having to explicitly model the probability density functions.
For convex margin loss <math>\phi(\upsilon)</math>, it can be shown that <math>\phi(\upsilon)</math> is Bayes consistent if and only if it is differentiable at 0 and <math>\phi'(0)=0</math><ref>{{Cite journal|last=Bartlett|first=Peter L.|last2=Jordan|first2=Michael I.|last3=Mcauliffe|first3=Jon D.|date=2006|title=Convexity, Classification, and Risk Bounds|journal=Journal of the American Statistical Association|volume=101|issue=473|pages=138–156|issn=0162-1459|jstor=30047445|doi=10.1198/016214505000000907}}</ref><ref name="mit" />. Yet, this result does not exclude the existence of non-convex Bayes consistent loss functions. A more general result states that Bayes consistent loss functions can be generated using the following formulation <ref name=":0">{{Cite journal|last=Masnadi-Shirazi|first=Hamed|last2=Vasconcelos|first2=Nuno|date=2008|title=On the Design of Loss Functions for Classification: Theory, Robustness to Outliers, and SavageBoost|url=http://dl.acm.org/citation.cfm?id=2981780.2981911|journal=Proceedings of the 21st International Conference on Neural Information Processing Systems|series=NIPS'08|___location=USA|publisher=Curran Associates Inc.|pages=1049–1056|isbn=9781605609492}}</ref>
:<math>\phi(v)=C[f^{-1}(v)]+(1-f^{-1}(v))C'[f^{-1}(v)] \;\;\;\;\;(2)</math>,
where <math>f(\eta), (0\leq \eta \leq 1)</math> is any invertible function such that <math>f^{-1}(-v)=1-f^{-1}(v)</math> and <math>C(\eta)</math>is any differentiable strictly concave function such that <math>C(\eta)=C(1-\eta)</math>. Table-I shows the generated Bayes consistent loss functions for some example choices of <math>C(\eta)</math> and <math>f^{-1}(v)</math>. Note that the Savage and Tangent loss are not convex. Such non-convex loss functions have been shown to be useful in dealing with outliers in classification<ref name=":0" /><ref>{{Cite journal|last=Leistner|first=C.|last2=Saffari|first2=A.|last3=Roth|first3=P. M.|last4=Bischof|first4=H.|date=2009-9|title=On robustness of on-line boosting - a competitive study|url=https://ieeexplore.ieee.org/document/5457451|journal=2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops|pages=1362–1369|doi=10.1109/ICCVW.2009.5457451|isbn=978-1-4244-4442-7}}</ref>. For all loss functions generated from (2)
{| class="wikitable"
|+Table-I
!Loss
!<math>\phi(v)</math>
!<math>C(\eta)</math>
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|}<br />The sole minimizer of the expected risk, <math>f^*_{\phi}</math>, associated with the above generated loss functions can be directly found from equation (1) and shown to be equal to the corresponding <math>
f(\eta)
</math>. This holds even for the nonconvex loss functions, which means that gradient descent based algorithms such as [[
==Proper
[[File:LogitLossMarginWithMu.jpg|alt=|thumb|(Red) standard Logistic loss (<math>\gamma=1, \mu=2</math>) and (Blue) increased margin Logistic loss (<math>\gamma=0.2</math>).]]
For proper loss functions, the ''loss margin'' can be defined as <math>\mu_{\phi}=-\frac{\phi'(0)}{\phi''(0)}</math> and shown to be directly related to the regularization properties of the classifier<ref>{{Cite journal|last=Vasconcelos|first=Nuno|last2=Masnadi-Shirazi|first2=Hamed|date=2015|title=A View of Margin Losses as Regularizers of Probability Estimates|url=http://jmlr.org/papers/v16/masnadi15a.html|journal=Journal of Machine Learning Research|volume=16|issue=85|pages=2751–2795|issn=1533-7928}}</ref>. Specifically a loss function of larger margin increases regularization and produces better estimates of the posterior probability. For example, the loss margin can be increased for the logistic loss by introducing a <math>\gamma</math> parameter and writing the logistic loss as <math>\frac{1}{\gamma}\log(1+e^{-\gamma v})</math> where smaller <math>0<\gamma<1</math> increases the margin of the loss. It is shown that this is directly equivalent to decreasing the learning rate in [[
In conclusion, by choosing a loss function with larger margin (smaller <math>\gamma</math>) we increase regularization and improve our estimates of the posterior probability which in turn improves the ROC curve of the final classifier.
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This function is undefined when <math>p(1\mid x)=1</math> or <math>p(1\mid x)=0</math> (tending toward ∞ and −∞ respectively), but predicts a smooth curve which grows when <math>p(1\mid x)</math> increases and equals 0 when <math>p(1\mid x)= 0.5</math>.<ref name="mitlec" />
It's easy to check that the logistic loss and binary [[cross entropy]] loss (Log loss) are in fact the same (up to a multiplicative constant <math>\frac{1}{\log(2)}</math>).The cross entropy loss is closely related to the [[
== Exponential loss ==
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:<math>\phi(v)=C[f^{-1}(v)]+(1-f^{-1}(v))C'[f^{-1}(v)] = (\frac{e^v}{1+e^v})(1-\frac{e^v}{1+e^v})+(1-\frac{e^v}{1+e^v})(1-\frac{2e^v}{1+e^v}) = \frac{1}{(1+e^v)^2}.</math>
The Savage loss is quasi-convex and is bounded for large negative values which makes it less sensitive to outliers. The Savage loss has been used in [[
The minimizer of <math>I[f]</math> for the Savage loss function can be directly found from equation (1) as
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The Tangent loss<ref>{{Cite journal|last=Masnadi-Shirazi|first=H.|last2=Mahadevan|first2=V.|last3=Vasconcelos|first3=N.|date=2010-6|title=On the design of robust classifiers for computer vision|url=https://ieeexplore.ieee.org/document/5540136|journal=2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition|pages=779–786|doi=10.1109/CVPR.2010.5540136|isbn=978-1-4244-6984-0}}</ref> can be generated using (2) and Table-I as follows
:<math>
:<math>\phi(v)=C[f^{-1}(v)]+(1-f^{-1}(v))C'[f^{-1}(v)] = 4(\arctan(v)+\frac{1}{2})(1-(\arctan(v)+\frac{1}{2}))+(1-(\arctan(v)+\frac{1}{2}))(4-8(\arctan(v)+\frac{1}{2})) = (2\arctan(v)-1)^2.</math>▼
\begin{align}
▲
& = (2\arctan(v)-1)^2.
\end{align}
</math>
The Tangent loss is quasi-convex and is bounded for large negative values which makes it less sensitive to outliers. Interestingly, the Tangent loss also assigns a bounded penalty to data points that have been classified "too correctly". This can help prevent over
The minimizer of <math>I[f]</math> for the Tangent loss function can be directly found from equation (1) as
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when <math>p(1\mid x) \ne 0.5</math>, which matches that of the 0–1 indicator function. This conclusion makes the hinge loss quite attractive, as bounds can be placed on the difference between expected risk and the sign of hinge loss function.<ref name="mit" /> The Hinge loss cannot be derived from (2) since <math>f^*_{\text{Hinge}}</math> is not invertible.
== Generalized
The generalized smooth hinge loss function with parameter <math>\alpha</math> is defined as
:<math>f^*_\alpha(z) \;=\; \begin{cases} \frac{\alpha}{\alpha + 1}& \text{if }z< 0 \\ \frac{1}{\alpha + 1}z^{\alpha + 1} - z + \frac{\alpha}{\alpha + 1} & \text{if } 0<z<1 \\ 0 & \text{if } z \geq 1 \end{cases}
where
:<math>z = yf(\vec{x}).</math>
It is monotonically increasing and reaches 0 when
== References ==
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