Hardness of Learning Boolean Functions from Aggregated Labels

Core Concepts

It is NP-hard to efficiently learn OR functions and parities from aggregated label proportions, in contrast to the efficient PAC learnability of these functions.

Abstract

The paper studies the computational learning aspects of the learning from label proportions (LLP) framework, where the training examples are aggregated into subsets or bags and only the average label per bag is available for learning an example-level predictor.
The key findings are:

For bags of size at most 2 that are consistent with an OR function, it is NP-hard to find a constant-clause CNF formula that satisfies a constant fraction of the bags. This separates the learnability of ORs using constant-clause CNFs versus halfspaces in the LLP setting.

It is NP-hard to satisfy more than a (1/2 + o(1)) fraction of such bags using a t-DNF formula for any constant t. This hardness was previously known only for learning noisy ORs in the standard PAC setting.

For parities, it is NP-hard to satisfy more than a (q/2^(q-1) + o(1)) fraction of q-sized bags that are consistent with a parity, while a random parity-based algorithm achieves a (1/2^(q-2))-approximation.

The hardness results demonstrate a qualitative difference between the learnability of simple Boolean functions like ORs and parities in the LLP setting versus the standard PAC learning framework, where they are efficiently learnable.

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Key Insights Distilled From

Hardness of Learning Boolean Functions from Label Proportions

by Venkatesan G... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.19401.pdf

Deeper Inquiries

What are some real-world applications where the LLP setting is more relevant than the standard PAC learning framework

In real-world applications where privacy is a concern, the LLP setting is more relevant than the standard PAC learning framework. For example, in medical image classification, patient data needs to be protected to comply with privacy regulations. By aggregating labels into bags and only providing average label proportions, the LLP framework allows for learning without exposing individual patient data. This is crucial in scenarios where maintaining the privacy of sensitive information is a top priority.

Can the hardness results be extended to other Boolean function classes beyond ORs and parities

The hardness results obtained for learning ORs and parities in the LLP setting can potentially be extended to other Boolean function classes. By constructing reductions from known hard problems, it may be possible to show the intractability of learning more complex Boolean functions using label proportions. For example, by adapting the techniques used in the hardness proofs for ORs and parities, similar results could potentially be established for other classes of Boolean functions such as AND, NAND, XOR, and more.

Are there any positive results on the learnability of Boolean functions in the LLP setting, perhaps by considering more expressive hypothesis classes or relaxed notions of learnability

While the hardness results demonstrate the challenges of learning Boolean functions from label proportions, there is ongoing research on the learnability of Boolean functions in the LLP setting. One approach to improving learnability could involve considering more expressive hypothesis classes. By exploring different types of functions or incorporating additional features into the hypothesis space, it may be possible to achieve better approximations or develop algorithms with improved performance. Additionally, relaxing the strict bag satisfaction requirement to allow for partial satisfaction or probabilistic guarantees could also lead to positive results in the learnability of Boolean functions in the LLP setting.