Computer Science/Discrete Mathematics Seminar I
The Sample Complexity of Smooth Boosting and the Tightness of the Hardcore Theorem
This talk will be about two related results, one in complexity theory and one in learning.
On the learning side, we investigate the sample complexity of smooth boosters - These are boosting algorithms that do not place too much weight on any given example. They have a variety of applications including robustness, privacy, and reproducibility. We show that the sample complexity overhead of existing smooth boosters is optimal, separating smooth boosting from distribution-independent boosting.
On the complexity side, 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′≪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.