= Algorithm parameter space
{title2=$\Theta$}
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.<ParamILS: An Automatic Algorithm Configuration Framework by Frank Hutter and others>[Section 2, conditional parameters; Definition 1]{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 $2\times3=6$ tuples, but only four distinct effective configurations:
$$
\Theta=\{(A,0),(A,1),(A,2),(B,\ast)\}.
$$
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.
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