In the Generalized Mastermind problem, there is an unknown subset $H$ of the hypercube $\{0,1\}^d$ containing $n$ points. The goal is to learn $H$ by making a few queries to an oracle, which, given a point $q$ in $\{0,1\}^d$, returns the point in $H$ nearest to $q$. We give a two-round adaptive algorithm for this problem that learns $H$ while making at most $\exp(\tilde{O}(\sqrt{d \log n}))$ queries. Furthermore, we show that any $r$-round adaptive randomized algorithm that learns $H$ with constant probability must make $\exp(\Omega(d^{3^{-(r-1)}}))$ queries even when the input has $\text{poly}(d)$ points; thus, any $\text{poly}(d)$ query algorithm must necessarily use $\Omega(\log \log d)$ rounds of adaptivity. We give optimal query complexity bounds for the variant of the problem where queries are allowed to be from $\{0,1,2\}^d$. We also study a continuous variant of the problem in which $H$ is a subset of unit vectors in $\mathbb{R}^d$, and one can query unit vectors in $\mathbb{R}^d$. For this setting, we give an $O(n^{d/2})$ query deterministic algorithm to learn the hidden set of points.

翻译：在广义Mastermind问题中，超立方体$\{0,1\}^d$内存在一个包含$n$个点的未知子集$H$。目标是通过向预言机进行少量查询来学习$H$：给定$\{0,1\}^d$中的点$q$，预言机返回$H$中与$q$最近的点。我们针对该问题提出一种两轮自适应算法，该算法最多使用$\exp(\tilde{O}(\sqrt{d \log n}))$次查询即可学习$H$。进一步证明，任何能以恒定概率学习$H$的$r$轮自适应随机算法，即使在输入仅包含$\text{poly}(d)$个点时也必须进行$\exp(\Omega(d^{3^{-(r-1)}}))$次查询；因此，任何$\text{poly}(d)$查询算法必须使用$\Omega(\log \log d)$轮自适应性。我们针对允许从$\{0,1,2\}^d$进行查询的变体问题给出了最优查询复杂度边界。同时研究该问题的连续变体：$H$是$\mathbb{R}^d$中单位向量的子集，且允许查询$\mathbb{R}^d$中的单位向量。针对此设定，我们提出一种$O(n^{d/2})$查询的确定性算法来学习隐藏的点集。