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

TL;DR

This paper determines the sample complexity of smooth boosting, showing a separation from distribution-independent boosting, and clarifies the tightness of Impagliazzo's hardcore theorem regarding circuit size loss.

Contribution

It establishes the sample complexity bounds for smooth boosting and proves the optimality of the circuit size loss in the hardcore theorem, connecting boosting and complexity theory.

Findings

Smooth boosting can weakly learn classes with fewer samples than strong learning requires.

A separation is shown between smooth boosting and distribution-independent boosting in sample complexity.

The size loss in Impagliazzo's hardcore theorem is proven to be necessary and optimal.

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.