截断布尔乘积分布参数的高效估计
Efficient Parameter Estimation of Truncated Boolean Product Distributions
AI总结：
研究如何在截断样本下高效估计布尔乘积分布参数，提出截断集的肥厚性概念，证明肥厚截断集可使样本生成和统计任务高效完成，同时探讨了截断学习的必要条件。
摘要：
我们研究了在d维中，当样本被一个可通过成员Oracle访问的集合S⊆{0,1}^d截断时，估计布尔乘积分布参数的问题。这是首次在离散设置中考虑从截断样本学习的计算和统计复杂性。我们引入了截断集S的肥厚性概念，表明肥厚的截断集能提供足够的信息。我们证明，如果截断集足够肥厚，真实分布的样本可从截断样本生成。有趣的结果是，几乎任何统计任务（如总变差距离学习、参数估计、均匀性或身份测试）若能高效完成布尔乘积分布，也可从截断样本完成，仅需略微增加样本复杂性。我们还将方法推广到d种替代物的排名分布，展示了肥厚性如何使Mallows模型的参数估计高效。探索从截断样本学习离散模型的极限，我们识别了三个必要条件：(i)截断集S应足够丰富；(ii)S应可通过成员查询访问；(iii)截断应保留所有方向上的足够随机性。通过仔细调整(Daskalakis等，FOCS 2018)的随机梯度下降方法，我们证明这些条件也是高效学习截断布尔乘积分布的充分条件。
英文摘要：
We study the problem of estimating the parameters of a Boolean product distribution in $d$ dimensions, when the samples are truncated by a set $S \subset \{0, 1\}^d$ accessible through a membership oracle. This is the first time that the computational and statistical complexity of learning from truncated samples is considered in a discrete setting. We introduce a natural notion of fatness of the truncation set $S$, under which truncated samples reveal enough information about the true distribution. We show that if the truncation set is sufficiently fat, samples from the true distribution can be generated from truncated samples. A stunning consequence is that virtually any statistical task (e.g., learning in total variation distance, parameter estimation, uniformity or identity testing) that can be performed efficiently for Boolean product distributions, can also be performed from truncated samples, with a small increase in sample complexity. We generalize our approach to ranking distributions over $d$ alternatives, where we show how fatness implies efficient parameter estimation of Mallows models from truncated samples. Exploring the limits of learning discrete models from truncated samples, we identify three natural conditions that are necessary for efficient identifiability: (i) the truncation set $S$ should be rich enough; (ii) $S$ should be accessible through membership queries; and (iii) the truncation by $S$ should leave enough randomness in all directions. By carefully adapting the Stochastic Gradient Descent approach of (Daskalakis et al., FOCS 2018), we show that these conditions are also sufficient for efficient learning of truncated Boolean product distributions.
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