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Empirical risk minimization for a classification problem with a 0-1 loss function is known to be an NP-hard problem even for a relatively simple class of functions such as linear classifiers. [5] Nevertheless, it can be solved efficiently when the minimal empirical risk is zero, i.e., data is linearly separable .
In machine learning, specifically empirical risk minimization, MSE may refer to the empirical risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator.
In others words, the sample complexity (,,) defines the rate of consistency of the algorithm: given a desired accuracy and confidence , one needs to sample (,,) data points to guarantee that the risk of the output function is within of the best possible, with probability at least .
The worst case empirical Rademacher complexity is ¯ = = {, …,} Let be a probability distribution over . The Rademacher complexity of the function class F {\displaystyle {\mathcal {F}}} with respect to P {\displaystyle P} for sample size m {\displaystyle m} is:
In words the VC inequality is saying that as the sample increases, provided that has a finite VC dimension, the empirical 0/1 risk becomes a good proxy for the expected 0/1 risk. Note that both RHS of the two inequalities will converge to 0, provided that S ( F , n ) {\displaystyle S({\mathcal {F}},n)} grows polynomially in n .
Neural networks are typically trained through empirical risk minimization.This method is based on the idea of optimizing the network's parameters to minimize the difference, or empirical risk, between the predicted output and the actual target values in a given dataset. [4]
The Bayes risk of ^ is defined as ((, ^)), where the expectation is taken over the probability distribution of : this defines the risk function as a function of ^. An estimator θ ^ {\displaystyle {\widehat {\theta }}} is said to be a Bayes estimator if it minimizes the Bayes risk among all estimators.
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