The Sample Complexity of Smooth Boosting and the Tightness of the Hardcore Theorem
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arXiv
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| Format: | Preprint |
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2024
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| author | Blanc, Guy Hayderi, Alexandre Koch, Caleb Tan, Li-Yang |
| author_facet | Blanc, Guy Hayderi, Alexandre Koch, Caleb Tan, Li-Yang |
| contents | 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 $γ$-advantage over smooth distributions with $m$ samples, for which strong learning over the uniform distribution requires $\tildeΩ(1/γ^2)\cdot m$ 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 $O(1/γ)$.
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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_11597 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Sample Complexity of Smooth Boosting and the Tightness of the Hardcore Theorem Blanc, Guy Hayderi, Alexandre Koch, Caleb Tan, Li-Yang Computational Complexity Data Structures and Algorithms Machine Learning 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 $γ$-advantage over smooth distributions with $m$ samples, for which strong learning over the uniform distribution requires $\tildeΩ(1/γ^2)\cdot m$ 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 $O(1/γ)$. 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. |
| title | The Sample Complexity of Smooth Boosting and the Tightness of the Hardcore Theorem |
| topic | Computational Complexity Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2409.11597 |