# Competing With Prediction Rules

Competing with prediction rules is a subfield of competitive on-line prediction in which the strategies in the benchmark class are functions of a "signal", a hint output by Reality at the beginning of each step (in typical machine-learning applications, this might be the object to be labelled). The basic protocol is:

**Players**: Forecaster, Reality**Protocol**:

Reality's *signal* $x_t$ is chosen from some *signal space* $X$. Forecaster's goal is to compete with all functions $f: X \to \Gamma$ that belong to a benchmark class $\mathcal{F}$; more formally, the strategies in the benchmark class are $\gamma_t:=f(x_t)$. An important special case is online linear regression, in which $\mathcal{F}$ is the class of all linear functions on $X$. In this paper, generalized linear regression is considered. More generally, $f$ can be allowed to range over various function classes, such as Banach or Hilbert spaces.

An important open problem in this area is Competing with Besov spaces.