Substitution Function

This is a function that is used in Strong and Weak Aggregating Algorithms to make their prediction. They mix losses of the experts using their weights, and receive the so-called generalized prediction. It is not always a real prediction, and the substitution function is needed to make it. The substitution function $S$ is such a function that for $\gamma=S(g)$ the $\lambda(\omega,\gamma) \le g(\omega)$ for all $\omega \in \Omega$. For various loss functions substitution functions are different.