Let (X, G), (Y, G) be two G-systems, where G is an infinite countable discrete amenable group and X, Y are compact metric spaces. Suppose that \({\mathcal {U}}\) is a cover of X. We first introduce the conditional local topological intricacy \(\text {Int}_\text {top} (G,{\mathcal {U}}|Y)\) and average sample complexity \(\text {Asc}_\text {top} (G,{\mathcal {U}}|Y)\) . Given an invariant measure \(\mu \) of X, we study the conditional local measure-theoretical intricacy \(\text {Int}_\mu ^\pm (G,{\mathcal {U}}|Y)\) and average sample complexity \(\text {Asc}_\mu ^\pm (G,{\mathcal {U}}|Y)\) . For any Følner sequence \(\{F_n\}_{n\in \mathbb {N}}\) , we take \(\{c^{F_n}_S\}_{S\subseteq F_n}\) to be the uniform system of coefficients. We establish the equivalence of \(\text {Asc}_\mu ^-(G,{\mathcal {U}}|Y)\) and \(\text {Asc}_\mu ^+(G,{\mathcal {U}}|Y)\) when \(G=\mathbb {Z}\) . Furthermore, we verified that \(\text {Asc}_\mu ^-(G,{\mathcal {U}})\) is equal to \(\text {Asc}_\mu ^+(G,{\mathcal {U}})\) in general case. Finally, we give a local variational principle of average sample complexity.
