Background
Consider the following situation, which is a general setting of many supervised learning problems. We have two spaces of objects and and would like to learn a function (often called hypothesis) which outputs an object, given . To do so, we have at our disposal a training set of a few examples where is an input and is the corresponding response that we wish to get from .
To put it more formally, we assume that there is a joint probability distribution over and, and that the training set consists of instances drawn i.i.d. from . Note that the assumption of a joint probability distribution allows us to model uncertainty in predictions (e.g. from noise in data) because is not a deterministic function of, but rather a random variable with conditional distribution for a fixed .
We also assume that we are given a non-negative real-valued loss function which measures how different the prediction of a hypothesis is from the true outcome . The risk associated with hypothesis is then defined as the expectation of the loss function:
A loss function commonly used in theory is the 0-1 loss function:, where is the indicator notation.
The ultimate goal of a learning algorithm is to find a hypothesis among a fixed class of functions for which the risk is minimal:
Read more about this topic: Empirical Risk Minimization
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