# 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:

Initialize $L_0:=0$
FOR $t=1,2,\dots$:
Reality announces $x_t \in X$
Forecaster announces $\gamma_t \in \Gamma$
Reality announces $y_t \in Y$
END

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.