从不受信任的批次中学习结构化分布：更快，更简单
Learning Structured Distributions From Untrusted Batches: Faster and Simpler
摘要：
我们重新讨论了由Qiao和Valiant [QV 17]引入的从不可信批次中学习的问题。最近，Jain和Orlitsky [JO 19]给出了一个简单的半定规划方法，该方法基于割范数，在多项式时间内实现了基本上信息理论上的最优误差。同时，Chen等人[CLM 19]考虑了问题的一个变体，其中假设$μ$是结构化的，例如对数凹、单调风险率、$t$ -模态等。在这种情况下，可以在$n$中实现样本复杂度次线性的相同误差，并且他们使用Haar小波展示了一种准多项式时间算法。在本文中，我们找到了一种有吸引力的方法来综合[JO 19]和[CLM 19]的技术，以提供两全其美的效果：一种在多项式时间内运行的算法，可以利用底层分布中的结构来实现次线性样本复杂度。一路上，我们简化了[JO 19]的方法，避免了SDP舍入的需要，并通过软滤波的镜头对其进行了更直接的解释，软滤波是一种强大的高维鲁棒估计技术。我们验证了我们的算法在初步的实验评估的有用性。
英文摘要：
We revisit the problem of learning from untrusted batches introduced by Qiao and Valiant [QV17]. Recently, Jain and Orlitsky [JO19] gave a simple semidefinite programming approach based on the cut-norm that achieves essentially information-theoretically optimal error in polynomial time. Concurrently, Chen et al. [CLM19] considered a variant of the problem where $μ$ is assumed to be structured, e.g. log-concave, monotone hazard rate, $t$-modal, etc. In this case, it is possible to achieve the same error with sample complexity sublinear in $n$, and they exhibited a quasi-polynomial time algorithm for doing so using Haar wavelets. In this paper, we find an appealing way to synthesize the techniques of [JO19] and [CLM19] to give the best of both worlds: an algorithm which runs in polynomial time and can exploit structure in the underlying distribution to achieve sublinear sample complexity. Along the way, we simplify the approach of [JO19] by avoiding the need for SDP rounding and giving a more direct interpretation of it through the lens of soft filtering, a powerful recent technique in high-dimensional robust estimation. We validate the usefulness of our algorithms in preliminary experimental evaluations.
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