Computational Limitations in Robust Classification and Win-Win Results
Computational Limitations in Robust Classification and Win-Win Results
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发表时间：
2019-02
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通讯作者：
Akshay Degwekar;V. Vaikuntanathan
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作者：
Akshay Degwekar;V. Vaikuntanathan
继 Bubeck、Lee、Price 和 Razenshteyn 最近的工作之后，我们继续研究学习鲁棒分类器的统计/计算权衡，他们展示了分类任务的示例，其中（a）在小扰动状态下存在有效的鲁棒分类器； (b) 可以有效地学习非鲁棒分类器；但是（c）假设分解大数的难度很大，学习鲁棒的分类器在计算上是困难的。在大扰动范围内是否存在适合其任务的鲁棒分类器的问题似乎与计算数论中重要的开放问题相关。在这项工作中，我们将他们的工作扩展到三个方向。首先，我们演示了分类任务，即使存在计算上无界的鲁棒分类器，计算上有效的鲁棒分类也是不可能的。为此，我们依赖于平均情况硬函数的存在。其次，我们展示了大扰动状态下难以鲁棒学习的分类任务。也就是说，我们表明，即使存在对大扰动具有鲁棒性的高效分类器，但学习任何非平凡的鲁棒分类器在计算上都是困难的。我们的第一个构造依赖于单向函数的存在，第二个构造依赖于噪声问题的学习奇偶性的难度。在后一种设置中，不仅存在非鲁棒分类器，而且还存在一种有效的算法，可以在访问多项式多个训练示例的情况下生成新的标记样本（Kearns 等人（1994）称为生成）。第三，我们证明任何这样的反例都意味着密码原语（例如单向函数）的存在。这给我们带来了一个双赢的场景：要么我们可以学习一个高效的鲁棒分类器，要么我们可以构造新的加密原语实例。
We continue the study of statistical/computational tradeoffs in learning robust classifiers, following the recent work of Bubeck, Lee, Price and Razenshteyn who showed examples of classification tasks where (a) an efficient robust classifier exists, in the small-perturbation regime; (b) a non-robust classifier can be learned efficiently; but (c) it is computationally hard to learn a robust classifier, assuming the hardness of factoring large numbers. The question of whether a robust classifier for their task exists in the large perturbation regime seems related to important open questions in computational number theory. In this work, we extend their work in three directions. First, we demonstrate classification tasks where computationally efficient robust classification is impossible, even when computationally unbounded robust classifiers exist. For this, we rely on the existence of average-case hard functions. Second, we show hard-to-robustly-learn classification tasks in the large-perturbation regime. Namely, we show that even though an efficient classifier that is robust to large perturbations exists, it is computationally hard to learn any non-trivial robust classifier. Our first construction relies on the existence of one-way functions, and the second on the hardness of the learning parity with noise problem. In the latter setting, not only does a non-robust classifier exist, but also an efficient algorithm that generates fresh new labeled samples given access to polynomially many training examples (termed as generation by Kearns et. al. (1994)). Third, we show that any such counterexample implies the existence of cryptographic primitives such as one-way functions. This leads us to a win-win scenario: either we can learn an efficient robust classifier, or we can construct new instances of cryptographic primitives.