ERM: Empirical Risk Minimization
#Data #Loss
In a learning problem The Learning Problem The learning problem posed by Vapnik:1 Given a sample: $\{z_i\}$ in the probability space $Z$; Assuming a probability measure on the probability space $Z$; Assuming a set of functions $Q(z, \alpha)$ (e.g. loss functions), where $\alpha$ is a set of parameters; A risk functional to be minimized by tunning “the handles” $\alpha$, $R(\alpha)$. The risk functional is $$ R(\alpha) = \int Q(z, \alpha) \,\mathrm d F(z). $$ A learning problem is the minimization of this risk. Vapnik2000 … , empirical risk $R$ is a measurement the goodness of fit based on empirical information. Empirical risk minimization minimizes the empirical risk to select a good model $\hat f$ out of all possible models $f$ in our hypothesis space for a dataset $\mathcal D$,
$$ \hat f = \operatorname{argmin} R(f, \mathcal D). $$
Empirical Risk Example
For example, the emprical risk can be represented by the negative log likelihood.
A negative log likelihood (NLL) for a model $\theta$ of dataset $\mathcal D$
$$ NLL(\theta) = \log p(\mathcal D\mid\theta) = \sum_n \log (y_n \mid x_n, \theta). $$
An empirical risk loss function $\mathcal L$ is
$$ \mathcal L(\theta) = \frac{1}{N} \sum_n \mathscr l(y_n, \theta; x_n), $$
where $\mathscr l$ is a loss. For example, one could design a stepwise loss in classification
$$ \mathscr l = \begin{cases} 0, \qquad \text{if prediction matches data} \\ 1 \qquad \text{if prediction doesn't match data} \end{cases} $$
Another possibility is surrogate loss which is continuous.
Regularized Risk
However, ERM may lead to overfitting. One method to solve this is to add a penalty based on the complexity of the model $C(f)$,
$$ R_{Reg}(f, \mathcal D) = R(f, \mathcal D) + \lambda C(f). $$
L Ma (2021). 'ERM: Empirical Risk Minimization', Datumorphism, 02 April. Available at: https://datumorphism.leima.is/cards/machinelearning/learningtheories/empiricalriskminimization/.
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