The Sample Complexity of Smooth Boosting and the Tightness of the Hardcore Theorem

• URL: http://arxiv.org/abs/2409.11597v1
• Date: Tue, 17 Sep 2024 23:09:25 GMT
• Title: The Sample Complexity of Smooth Boosting and the Tightness of the Hardcore Theorem
• Authors: Guy Blanc, Alexandre Hayderi, Caleb Koch, Li-Yang Tan,
• Abstract summary: Smooth boosters generate distributions that do not place too much weight on any given example. Originally introduced for their noise-tolerant properties, such boosters have also found applications in differential privacy, mildly, and quantum learning theory.
• Score: 53.446980306786095
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Smooth boosters generate distributions that do not place too much weight on any given example. Originally introduced for their noise-tolerant properties, such boosters have also found applications in differential privacy, reproducibility, and quantum learning theory. We study and settle the sample complexity of smooth boosting: we exhibit a class that can be weak learned to $\gamma$-advantage over smooth distributions with $m$ samples, for which strong learning over the uniform distribution requires $\tilde{\Omega}(1/\gamma^2)\cdot m$ samples. This matches the overhead of existing smooth boosters and provides the first separation from the setting of distribution-independent boosting, for which the corresponding overhead is $O(1/\gamma)$. Our work also sheds new light on Impagliazzo's hardcore theorem from complexity theory, all known proofs of which can be cast in the framework of smooth boosting. For a function $f$ that is mildly hard against size-$s$ circuits, the hardcore theorem provides a set of inputs on which $f$ is extremely hard against size-$s'$ circuits. A downside of this important result is the loss in circuit size, i.e. that $s' \ll s$. Answering a question of Trevisan, we show that this size loss is necessary and in fact, the parameters achieved by known proofs are the best possible.

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