学习划分的近最优查询复杂度的随机化算法：常数轮设置

中文摘要 AI 辅助

我们研究在常数轮次下，利用PAIR查询学习n元论域上隐藏划分$\boldsymbol{\textit{P}}$的轮次复杂度：PAIR($x,y$)返回$x$与$y$是否属于同一划分块。基础算法用$n|\boldsymbol{\textit{P}}|$次查询即可完成学习，该查询复杂度是最优的，但此算法高度串行。Black、Mazumdar与Saha[COLT 2025]近期给出当划分块数已知时，确定型算法的轮次/查询复杂度紧折衷关系，证明要将查询数限制为$n|\boldsymbol{\textit{P}}|$，$\boldsymbol{\theta}(\boldsymbol{\textit{loglog}}n)$轮次是充分且必要的，他们将随机化下界的证明留作开放方向。我们发现随机化会显著改变该问题的特性：当划分块数$k=|\boldsymbol{\textit{P}}|$已知时，我们给出一个简单的3轮随机化算法，高概率下仅需$O(nk\boldsymbol{\textit{log}}n)$次查询，并证明2轮次需要$\boldsymbol{\textit{\u03a9}}(n^{4/3}k^{2/3})$次查询，与确定型算法的下界一致；当划分块数未知时，我们给出一个4轮随机化算法，高概率下仅需$O(n|\boldsymbol{\textit{P}}|\boldsymbol{\textit{log}}^2n)$次查询，并证明3轮次无法达到近最优查询复杂度，且此场景下随机化与确定型算法存在更大差距：确定型算法要达到近最优查询复杂度，需要$\boldsymbol{\theta}(\boldsymbol{\textit{log}}n/\boldsymbol{\textit{loglog}}n)$轮次。

英文摘要

We study the round complexity of learning a hidden partition $\mathcal{P}$ of an $n$-element universe using PAIR queries: PAIR($x,y$) tells us whether $x$ and $y$ belong to the same part of the partition or not. While it is easy to learn using $n|\mathcal{P}|$ queries using a basic algorithm and this query complexity is optimal, this basic algorithm is highly sequential. Black, Mazumdar, and Saha [COLT 2025] recently gave tight deterministic round/query tradeoffs when the number of parts of $\mathcal{P}$ is known. In particular they prove $Θ(\log\log n)$ rounds are sufficient and necessary to limit the number of queries to $n|\mathcal{P}|$. They leave proving a randomized lower bound as an open direction. We show that randomization dramatically changes the picture. When the number of parts $k = |\mathcal{P}|$ is known, we give a simple 3-round randomized algorithm using $O(nk\log n)$ queries with high probability, and prove that 2 rounds require $Ω(n^{4/3}k^{2/3})$ queries -- the same as deterministic algorithms. We also study a more general setting where the number of parts is unknown. In this case, we give a 4-round randomized algorithm using $O(n|\mathcal P|\log^2 n)$ queries with high probability, and prove that 3-rounds cannot achieve near-optimal query complexity. Furthermore, we show an even bigger separation in this regime between randomized and deterministic algorithms: for the latter, $Θ(\log n/\log\log n)$ rounds are necessary and sufficient to obtain near-optimal query complexity.

发表机构

• Dartmouth College（达特茅斯学院）
• The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））
• CNRS（法国国家科学研究中心）
• Université Paris Cité（巴黎西岱大学）

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