Loss functions for classification: Difference between revisions

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[[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)]]

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 | s2cid = 11845688 }}</ref> Given <math>\mathcal{X}</math> as the space of all possible inputs (usually <math>\mathcal{X} \subset \mathbb{R}^d</math>), and <math>\mathcal{Y} = \{ -1,1 \}</math> as the set of labels (possible outputs), a typical goal of classification algorithms is to find a function <math>f: \mathcal{X} \mapsto \mathbb{R}</math> which best predicts a label <math>y</math> for a given input <math>\vec{x}</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 | lastlast1= Rosasco | firstfirst1= 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 loss (also known as the risk), defined as

:<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 a given loss function, and <math>p(\vec{x},y)</math> is the [[probability density function]] of the process that generated the data, which can equivalently be written as

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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|lastlast1=Bartlett|firstfirst1=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|s2cid=2833811}}</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|lastlast1=Masnadi-Shirazi|firstfirst1=Hamed|last2=Vasconcelos|first2=Nuno|date=2008|title=On the Design of Loss Functions for Classification: Theory, Robustness to Outliers, and SavageBoost|url=https://papers.nips.cc/paper/3591-on-the-design-of-loss-functions-for-classification-theory-robustness-to-outliers-and-savageboost.pdf|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|lastlast1=Leistner|firstfirst1=C.|last2=Saffari|first2=A.|last3=Roth|first3=P. M.|last4=Bischof|first4=H.|date=September 2009|title=On robustness of on-line boosting - a competitive study|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|s2cid=6032045}}</ref> For all loss functions generated from (2), the posterior probability <math>p(y=1|\vec{x})</math> can be found using the invertible ''link function'' as <math>p(y=1|\vec{x})=\eta=f^{-1}(v)</math>. Such loss functions where the posterior probability can be recovered using the invertible link are called ''proper loss functions''.

{| 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|lastlast1=Vasconcelos|firstfirst1=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 [[gradient boosting]] <math>F_m(x) = F_{m-1}(x) + \gamma h_m(x),</math> where decreasing <math>\gamma</math> improves the regularization of the boosted classifier. The theory makes it clear that when a learning rate of <math>\gamma</math> is used, the correct formula for retrieving the posterior probability is now <math>\eta=f^{-1}(\gamma F(x))</math>.

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| lastlast1= Rifkin| firstfirst1= Ryan M.| last2= Lippert| first2= Ross A.| title= Notes on Regularized Least Squares| publisher= MIT Computer Science and Artificial Intelligence Laboratory| date= 1 May 2007|url=https://dspace.mit.edu/bitstream/handle/1721.1/37318/MIT-CSAIL-TR-2007-025.pdf?sequence=1}}</ref>

The minimizer of <math>I[f]</math> for the square loss function can be directly found from equation (1) as

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== Tangent loss ==

The Tangent loss<ref>{{Cite journal|lastlast1=Masnadi-Shirazi|firstfirst1=H.|last2=Mahadevan|first2=V.|last3=Vasconcelos|first3=N.|date=June 2010|title=On the design of robust classifiers for computer vision|journal=2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition|pages=779–786|doi=10.1109/CVPR.2010.5540136|citeseerx=10.1.1.172.6416|isbn=978-1-4244-6984-0|s2cid=632758}}</ref> can be generated using (2) and Table-I as follows

:<math>

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</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 journal|lastlast1=Schulter|firstfirst1=S.|last2=Wohlhart|first2=P.|last3=Leistner|first3=C.|last4=Saffari|first4=A.|last5=Roth|first5=P. M.|last6=Bischof|first6=H.|date=June 2013|title=Alternating Decision Forests|journal=2013 IEEE Conference on Computer Vision and Pattern Recognition|pages=508–515|doi=10.1109/CVPR.2013.72|citeseerx=10.1.1.301.1305|isbn=978-0-7695-4989-7|s2cid=6557162}}</ref>

The minimizer of <math>I[f]</math> for the Tangent loss function can be directly found from equation (1) as