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{{Short description|Concept in machine learning}}
{{Machine learning}}
[[File:BayesConsistentLosses2.jpg|thumb|Bayes consistent loss functions: Zero-one loss (gray), Savage loss (green), Logistic loss (orange), Exponential loss (purple), Tangent loss (brown), Square loss (blue)]]
{{Attention|reason=Discuss the difference compared to scoring rules|date=January 2024}}
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
:<math>I[f] = \displaystyle \int_{\mathcal{X} \times \mathcal{Y}} V(f(\vec{x}),y) \, p(\vec{x},y) \, d\vec{x} \, dy</math>
where <math>V(f(\vec{x}),y)</math> is
:<math>p(\vec{x},y)=p(y\mid\vec{x}) p(\vec{x}).</math>
In the case of binary classification, it is possible to simplify the calculation of expected risk from the integral specified above. Specifically,
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:<math>
\begin{align}
I[f] & = \int_{\mathcal{X} \times \mathcal{Y}} V(f(\vec{x}),y) \, p(\vec{x},y) \,d\vec{x} \,dy \\[6pt]
& = \
& = \
& = \
\end{align}
</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>{{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)
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
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where <math>H</math> indicates the [[Heaviside step function]].
However, this loss function is non-convex and non-smooth, and solving for the optimal solution is an [[NP-hard]] combinatorial optimization problem.<ref name="Utah">{{Citation | last= Piyush | first= Rai | title= Support Vector Machines (Contd.), Classification Loss Functions and Regularizers | publisher= Utah CS5350/6350: Machine Learning | date= 13 September 2011 | url=
In practice, the probability distribution <math>p(\vec{x},y)</math> is unknown. Consequently, utilizing a training set of <math>n</math> [[iid|independently and identically distributed]] sample points
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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)
:<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
{| class="wikitable"
|+Table-I
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==Proper loss functions, loss margin and regularization==
[[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|
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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The square loss function is both convex and smooth. However, the square loss function tends to penalize outliers excessively, leading to slower convergence rates (with regards to sample complexity) than for the logistic loss or hinge loss functions.<ref name="mit" /> In addition, functions which yield high values of <math>f(\vec{x})</math> for some <math>x \in X</math> will perform poorly with the square loss function, since high values of <math>yf(\vec{x})</math> will be penalized severely, regardless of whether the signs of <math>y</math> and <math>f(\vec{x})</math> match.
A benefit of the square loss function is that its structure lends itself to easy cross validation of regularization parameters. Specifically for [[Tikhonov regularization]], one can solve for the regularization parameter using leave-one-out [[cross-validation (statistics)|cross-validation]] in the same time as it would take to solve a single problem.<ref>{{Citation|
The minimizer of <math>I[f]</math> for the square loss function can be directly found from equation (1) as
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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
== Exponential loss ==
The exponential loss function can be generated using (2) and Table-I as follows
:<math>\phi(v)=C[f^{-1}(v)]+(1-f^{-1}(v))C'[f^{-1}(v)] = 2\sqrt{\left(\frac{e^{2v}}{1+e^{2v}}\right)\left(1-\frac{e^{2v}}{1+e^{2v}}\right)}+\left(1-\frac{e^{2v}}{1+e^{2v}}\right)\left(\frac{1-\frac{2e^{2v}}{1+e^{2v}}}{\sqrt{\frac{e^{2v}}{1+e^{2v}}(1-\frac{e^{2v}}{1+e^{2v}})}}\right) = e^{-v}</math>
The exponential loss is convex and grows exponentially for negative values which makes it more sensitive to outliers. The
The minimizer of <math>I[f]</math> for the exponential loss function can be directly found from equation (1) as
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The Savage loss<ref name=":0" /> can be generated using (2) and Table-I as follows
:<math>\phi(v)=C[f^{-1}(v)]+(1-f^{-1}(v))C'[f^{-1}(v)] = \left(\frac{e^v}{1+e^v}\right)\left(1-\frac{e^v}{1+e^v}\right)+\left(1-\frac{e^v}{1+e^v}\right)\left(1-\frac{2e^v}{1+e^v}\right) = \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 [[gradient boosting]] and the SavageBoost algorithm.
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== Tangent loss ==
The Tangent loss<ref>{{Cite
:<math>
\begin{align}
\phi(v) & = C[f^{-1}(v)]+\left( 1-f^{-1}(v)\right) C'[f^{-1}(v)]
\\ & = 4 \left( \arctan(v)+\frac{1}{2} \right) \left( 1- \left( \arctan(v)+\frac{1}{2} \right) \right) + \left( 1- \left( \arctan(v)+\frac{1}{2} \right) \right) \left( 4-8 \left( \arctan(v)+\frac{1}{2} \right) \right) \\
& = \left( 2\arctan(v)-1 \right) ^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-training on the data set. The Tangent loss has been used in [[gradient boosting]], the TangentBoost algorithm and Alternating Decision Forests.<ref>{{Cite
The minimizer of <math>I[f]</math> for the Tangent loss function can be directly found from equation (1) as
:<math>f^*_\text{Tangent}= \tan \left( \eta-\frac{1}{2} \right) =\tan \left( p \left( 1\mid x \right) -\frac{1}{2}\right) .</math>
== Hinge loss ==
{{main|Hinge loss}}
The hinge loss function is defined with <math>\phi(\upsilon) = \max(0, 1-\upsilon) = [1-\upsilon]_{+}</math>, where <math>[a]_{+} = \max(0,a)</math> is the [[positive part]] function.
:<math>V(f(\vec{x}),y) = \max(0, 1-yf(\vec{x})) =
The hinge loss provides a relatively tight, convex upper bound on the 0–1 [[indicator function]]. Specifically, the hinge loss equals the 0–1 [[indicator function]] when <math>\operatorname{sgn}(f(\vec{x})) = y</math> and <math>|yf(\vec{x})| \geq 1</math>. In addition, the empirical risk minimization of this loss is equivalent to the classical formulation for [[support vector machines]] (SVMs). Correctly classified points lying outside the margin boundaries of the support vectors are not penalized, whereas points within the margin boundaries or on the wrong side of the hyperplane are penalized in a linear fashion compared to their distance from the correct boundary.<ref name="Utah" />
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The generalized smooth hinge loss function with parameter <math>\alpha</math> is defined as
:<math>f^*_\alpha(z) \;=\; \begin{cases} \frac{\alpha}{\alpha + 1} - z & \text{if }z
where
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It is monotonically increasing and reaches 0 when <math>z = 1</math>.
== See also ==
*[[Differentiable programming]]
*[[Scoring function]]
== References ==
{{Reflist}}
{{Artificial intelligence navbox}}
[[Category:Machine learning algorithms]]
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