Empirical risk minimization

Empirical risk minimization (ERM) is a principle in statistical learning theory which defines a family of learning algorithms and is used to give theoretical bounds on the performance of learning algorithms.

Consider the following situation, which is a general setting of many supervised learning problems. We have two spaces of objects ${\displaystyle X}$ and ${\displaystyle Y}$ and would like to learn a function ${\displaystyle \ h:X\to Y}$ (often called hypothesis) which outputs an object ${\displaystyle y\in Y}$, given ${\displaystyle x\in X}$. To do so, we have at our disposal a training set of a few examples ${\displaystyle \ (x_{1},y_{1}),\ldots ,(x_{m},y_{m})}$ where ${\displaystyle x_{i}\in X}$ is an input and ${\displaystyle y_{i}\in Y}$ is the corresponding response that we wish to get from ${\displaystyle \ h(x_{i})}$.

...
Wikipedia