On-line Prediction Protocol

The simplest version of the on-line prediction protocol is where Nature outputs observations in the IID fashion. After Forecaster predicts an observation, it is added to the training set, which keeps growing. When each observation ⚠ $z=(x,y)\in\mathbf{Z}=\mathbf{X}\times\mathbf{Y}$ consists of an object ⚠ $x$ and its label ⚠ $y$, the protocol looks as follows:

for ⚠ $n=1,2,\ldots$:
Nature generates ⚠ $(x_n,y_n)$ from a probability distribution ⚠ $Q$ on ⚠ $\mathbf{Z}$
Forecaster is shown ⚠ $x_n$ and is asked to predict ⚠ $y_n$
Forecaster is shown the true label ⚠ $y_n$
end for

A modification is where ⚠ $z_1,z_2,\ldots$ are coming from an exchangeable probability distribution.

The online prediction protocol is used widely in conformal prediction. There are four important special cases:

(Besides, there are problems of unsupervised learning with the ⚠ $x_n$s missing.)