Learning Structured Distributions From Untrusted Batches: Faster and Simpler

• URL: http://arxiv.org/abs/2002.10435v2
• Date: Sun, 7 Jun 2020 17:50:33 GMT
• Title: Learning Structured Distributions From Untrusted Batches: Faster and Simpler
• Authors: Sitan Chen, Jerry Li, Ankur Moitra
• Abstract summary: We revisit the problem of learning from untrusted batches introduced by Qiao and Valiant [QV17]. 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 monotone time and can exploit structure in the underlying distribution to achieve sublinear sample.
• Score: 26.57313255285721
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\mu$ 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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