Algorithm parameter space ($\Theta$)

An algorithm parameter space, commonly written $\Theta$, is the set of allowed settings for a parameterized algorithm. It is the search space of algorithm configuration, rather than the set of problem instances that the target algorithm must solve. ParamILS explicitly accounts for parameters whose relevance depends on other settings.^[ref]

For an original finite example, let a mode be either $A$ or $B$. Mode $A$ has an extra setting $h\in \{0,1,2\}$, while mode $B$ ignores $h$. A rectangular encoding has $2\times 3=6$ tuples, but only four distinct effective configurations:The star means that the conditional algorithm parameter is inactive. Treating the three encodings of $B$ as three distinct experiments spends effort revisiting the same behavior. If $(A,2)$ is a forbidden parameter combination, the valid effective space has three members.

The representation also determines what a nearby configuration means. A search that changes one active parameter can move among the $A$ settings or change mode; changing an inactive value alone does not change behavior. Thus the number of encoded tuples, number of meaningful settings and neighborhood used by local search are separate design choices.

Continuous settings require declared ranges and a search method that supports them; replacing a range with a finite grid changes the problem being searched. Whatever the representation, a returned setting must be valid in $\Theta$ before its measured cost can be compared with other candidates.