基于不完美建议的乘积分布学习
Product distribution learning with imperfect advice
AI总结：
针对布尔超立方体上乘积分布学习问题，提出利用不完美建议分布$Q$的高效算法，在$\\|\mathbf{p}-\mathbf{q}\\|_1$满足一定条件时，将样本复杂度从$Ω(d/ε²)$降至$\tilde{O}(d^{1-η}/ε²)$。
摘要：
给定来自未知分布$P$的独立同分布样本，分布学习的目标是恢复与$P$接近的分布参数。当$P$属于布尔超立方体$\{0,1\}^d$上的乘积分布类时，已知需要$Ω(d/\varepsilon^2)$个样本才能在总变差（TV）距离$\varepsilon$内学习$P$。我们重新研究该问题，此时学习者还获得乘积分布$Q$的参数作为建议。我们证明存在一种高效算法，若$\\|\mathbf{p} - \mathbf{q}\\|_1 < \varepsilon d^{0.5 - Ω(η)}$，则能在TV距离$\varepsilon$内学习$P$，其样本复杂度为$\tilde{O}(d^{1-η}/\varepsilon^2)$。其中$\mathbf{p}$和$\mathbf{q}$分别是$P$和$Q$的均值向量，且算法预先不知道$\\|\mathbf{p} - \mathbf{q}\\|_1$的界。
英文摘要：
Given i.i.d.~samples from an unknown distribution $P$, the goal of distribution learning is to recover the parameters of a distribution that is close to $P$. When $P$ belongs to the class of product distributions on the Boolean hypercube $\{0,1\}^d$, it is known that $Ω(d/\varepsilon^2)$ samples are necessary to learn $P$ within total variation (TV) distance $\varepsilon$. We revisit this problem when the learner is also given as advice the parameters of a product distribution $Q$. We show that there is an efficient algorithm to learn $P$ within TV distance $\varepsilon$ that has sample complexity $\tilde{O}(d^{1-η}/\varepsilon^2)$, if $\|\mathbf{p} - \mathbf{q}\|_1 < \varepsilon d^{0.5 - Ω(η)}$. Here, $\mathbf{p}$ and $\mathbf{q}$ are the mean vectors of $P$ and $Q$ respectively, and no bound on $\|\mathbf{p} - \mathbf{q}\|_1$ is known to the algorithm a priori.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
