针对有界区分器测试分布
Testing Distributions Against Bounded Distinguishers
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中文总结 AI 辅助
研究高维或连续域分布测试，针对有界区分器类。核心方法是基于欺骗距离测试，揭示了可测试学习、学习算法验证和结构化分布测试间联系，产生了如可测试学习者、验证下界及恒等测试器等新结果。
中文摘要 AI 辅助
受高维或连续域上分布测试挑战的推动，我们研究了针对有界区分器类的分布测试。一个代表性任务是使用来自非常大域上未知分布\(P\)的样本，在两种情况之间做出决定：对于固定参考分布\(P_{\mathsf{ref}}\)，\(P = P_{\mathsf{ref}}\)；或者在有界类\(\mathcal{F}\)中存在一个区分器\(f\)，证明分离\(|\mathbf{E}_P[f] - \mathbf{E}_{P_{\mathsf{ref}}}[f]| > \epsilon\)。这是关于欺骗距离的恒等测试任务，其名称源于与伪随机性的概念联系。我们表明，关于欺骗距离的测试不仅是一个自然的计算问题，即使在高维设置中也允许样本高效算法，而且揭示并奠定了三个看似不相关研究领域之间的联系：可测试学习、学习算法验证和结构化分布测试（我们的框架扩展了其“\(\mathcal{A}_k\)测试”模型）。这些联系为所有这些模型产生了新结果，包括：1. 使用成员查询对半空间和决策树的可测试适当学习者。2. 基于拉德马赫复杂度的可测试 PAC 验证的下界，以及\(k\)个多维矩形不相交并集的数据无关验证协议。3. 在布尔和连续超立方体域上针对决策树分布和低阶多项式密度分布的（关于总变差距离的）恒等测试器。
英文摘要
Motivated by the challenge of testing distributions over high-dimensional or continuous domains, we study distribution testing with respect to bounded classes of distinguishers. A representative task is to use samples from an unknown distribution $P$ over a very large domain to decide between two cases: $P = P_{\mathsf{ref}}$ for a fixed reference distribution $P_{\mathsf{ref}}$, or there exists a distinguisher $f$ in a bounded class $\mathcal{F}$ which witnesses the separation $|\mathbf{E}_P[f] - \mathbf{E}_{P_{\mathsf{ref}}}[f]| > ε$. This is the task of identity testing with respect to fooling distance, a name inspired by the conceptual connection with pseudorandomness. (Formally, our model instantiates integral probability metrics from Boolean classes of bounded expressivity.) We show that testing with respect to fooling distance is not only a natural computational problem that admits sample-efficient algorithms even in high-dimensional settings, but also one that reveals and underlies connections between three seemingly unrelated areas of study: testable learning, verification of learning algorithms, and testing of structured distributions (whose "$\mathcal{A}_k$-testing" model our framework extends). These connections yield new results for all of these models, including: 1. Testable proper learners using membership queries for halfspaces and decision trees. 2. A lower bound for testable PAC verification in terms of Rademacher complexity, and a distribution-free verification protocol for disjoint unions of $k$ multidimensional rectangles. 3. Identity testers (with respect to total variation distance) for decision tree distributions and distributions with low-degree polynomial densities, over Boolean and continuous hypercube domains.
发表机构
- Boston University（波士顿大学）
- Columbia University（哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。
