Supervised Classification Loss Function
A Supervised Classification Loss Function is a learning loss function for supervised classification.
- Example(s):
- Counter-Example(s):
- See: Statistical Classification, Empirical Risk Minimization, Loss Functions, Probability Density Function, Iid, Sample Space, Statistical Learning Theory, False Positives and False Negatives, Indicator Function.
References
2018
- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Loss_functions_for_classification Retrieved:2018-3-7.
- In machine learning and mathematical optimization, loss functions for classification are computationally feasible loss functions representing the price paid for inaccuracy of predictions in classification problems (problems of identifying which category a particular observation belongs to). Given [math]\displaystyle{ 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]\displaystyle{ f: X \mapsto \mathbb{R} }[/math] which best maps [math]\displaystyle{ \vec{x} }[/math] to [math]\displaystyle{ y }[/math] . However, because of incomplete information, noise in the measurement, or probabilistic components in the underlying process, it is possible for the same [math]\displaystyle{ \vec{x} }[/math] to generate different [math]\displaystyle{ y }[/math] . As a result, the goal of the learning problem is to minimize expected risk, defined as : [math]\displaystyle{ I[f] = \displaystyle \int_{X \times Y} V(f(\vec{x}),y) p(\vec{x},y) \, d\vec{x} \, dy }[/math] where [math]\displaystyle{ V(f(\vec{x}),y) }[/math] is the loss function, and [math]\displaystyle{ p(\vec{x},y) }[/math] is the probability density function of the process that generated the data, which can equivalently be written as : [math]\displaystyle{ p(\vec{x},y)=p(y\mid\vec{x}) p(\vec{x}). }[/math] In practice, the probability distribution [math]\displaystyle{ p(\vec{x},y) }[/math] is unknown. Consequently, utilizing a training set of [math]\displaystyle{ n }[/math] independently and identically distributed sample points : [math]\displaystyle{ S = \{(\vec{x}_1,y_1), \dots ,(\vec{x}_n,y_n)\} }[/math] drawn from the data sample space, one seeks to minimize empirical risk : [math]\displaystyle{ I_S[f] = \frac{1}{n} \sum_{i=1}^n V( f(\vec{x}_i),y_i) }[/math] as a proxy for expected risk. (See statistical learning theory for a more detailed description.)
For computational ease, it is standard practice to write loss functions as functions of only one variable. Within classification, loss functions are generally written solely in terms of the product of the true classifier [math]\displaystyle{ y }[/math] and the predicted value [math]\displaystyle{ f(\vec{x}) }[/math] . Selection of a loss function within this framework : [math]\displaystyle{ V(f(\vec{x}),y)=\phi(-yf(\vec{x})) }[/math] impacts the optimal [math]\displaystyle{ f^{*}_S }[/math] which minimizes empirical risk, as well as the computational complexity of the learning algorithm.
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 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 : [math]\displaystyle{ V(f(\vec{x}),y)=H(-yf(\vec{x})) }[/math] where [math]\displaystyle{ 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. As a result, it is better to substitute continuous, convex loss function surrogates which are tractable for commonly used learning algorithms. In addition to their computational tractability, one can show that the solutions to the learning problem using these loss surrogates allow for the recovery of the actual solution to the original classification problem. Some of these surrogates are described below.
- In machine learning and mathematical optimization, loss functions for classification are computationally feasible loss functions representing the price paid for inaccuracy of predictions in classification problems (problems of identifying which category a particular observation belongs to). Given [math]\displaystyle{ 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]\displaystyle{ f: X \mapsto \mathbb{R} }[/math] which best maps [math]\displaystyle{ \vec{x} }[/math] to [math]\displaystyle{ y }[/math] . However, because of incomplete information, noise in the measurement, or probabilistic components in the underlying process, it is possible for the same [math]\displaystyle{ \vec{x} }[/math] to generate different [math]\displaystyle{ y }[/math] . As a result, the goal of the learning problem is to minimize expected risk, defined as : [math]\displaystyle{ I[f] = \displaystyle \int_{X \times Y} V(f(\vec{x}),y) p(\vec{x},y) \, d\vec{x} \, dy }[/math] where [math]\displaystyle{ V(f(\vec{x}),y) }[/math] is the loss function, and [math]\displaystyle{ p(\vec{x},y) }[/math] is the probability density function of the process that generated the data, which can equivalently be written as : [math]\displaystyle{ p(\vec{x},y)=p(y\mid\vec{x}) p(\vec{x}). }[/math] In practice, the probability distribution [math]\displaystyle{ p(\vec{x},y) }[/math] is unknown. Consequently, utilizing a training set of [math]\displaystyle{ n }[/math] independently and identically distributed sample points : [math]\displaystyle{ S = \{(\vec{x}_1,y_1), \dots ,(\vec{x}_n,y_n)\} }[/math] drawn from the data sample space, one seeks to minimize empirical risk : [math]\displaystyle{ I_S[f] = \frac{1}{n} \sum_{i=1}^n V( f(\vec{x}_i),y_i) }[/math] as a proxy for expected risk. (See statistical learning theory for a more detailed description.)