Efficient Algorithms for Learning from Coarse Labels

TL;DR

This paper introduces algorithms for learning from coarse labels, showing that many problems solvable with fine labels can also be learned efficiently from coarser, less detailed data through a generic reduction method.

Contribution

The work formalizes learning from coarse labels and provides a reduction approach that enables efficient learning comparable to fine label scenarios, including Gaussian mean estimation.

Findings

Efficient algorithms for learning from coarse labels via a generic reduction.

Polynomial dependence of coarse label requirements on information distortion.

NP-hardness results for non-convex set partitions in Gaussian estimation.

Abstract

For many learning problems one may not have access to fine grained label information; e.g., an image can be labeled as husky, dog, or even animal depending on the expertise of the annotator. In this work, we formalize these settings and study the problem of learning from such coarse data. Instead of observing the actual labels from a set $\mathcal{Z}$, we observe coarse labels corresponding to a partition of $\mathcal{Z}$ (or a mixture of partitions). Our main algorithmic result is that essentially any problem learnable from fine grained labels can also be learned efficiently when the coarse data are sufficiently informative. We obtain our result through a generic reduction for answering Statistical Queries (SQ) over fine grained labels given only coarse labels. The number of coarse labels required depends polynomially on the information distortion due to coarsening and the number of fine labels $|\mathcal{Z}|$. We also investigate the case of (infinitely many) real valued labels focusing on a central problem in censored and truncated statistics: Gaussian mean estimation from coarse data. We provide an efficient algorithm when the sets in the partition are convex and establish that the problem is NP-hard even for very simple non-convex sets.