基于负相关性的小型核心集：DPPs、线性统计量与集中性
Small coresets via negative dependence: DPPs, linear statistics, and concentration
AI总结：
本文针对DPP基核心集基数是否优于独立采样核心集的开放问题，将核心集损失视为线性统计量，推导了涵盖非对称核的更优集中不等式，证明DPP核心集效果更优，还拓展了向量值目标的核心集研究。
摘要：
行列式点过程（Determinantal point processes, DPPs）是具有可调负相关性的点随机构型。由于采样易于处理，DPPs是子采样任务（如小批量选择或核心集构建）的天然候选方案。核心集（coreset）是（大型）训练集的一个子集，使得在核心集上最小化平均经验损失，可作为原经验损失难以处理的最小化问题的受控替代方案。通常，这种控制表现为一项保证：核心集上的平均损失在整个参数空间上一致逼近总损失。近期研究为使用DPPs构建随机核心集提供了重要的实证支持，同时也给出了具有启发性但仍遗留若干关键问题未解决的理论结果。尤其是，基于DPP的核心集的基数是否从根本上小于基于独立采样的核心集这一核心问题仍悬而未决。在本文中，我们对该问题给出了肯定回答，证明了DPPs可证地优于独立抽取的核心集。在此方向上，我们提出了将核心集损失视为（随机）核心集的线性统计量的概念性理解。我们利用这一结构观察，将核心集问题与更广泛的DPPs线性统计量集中现象问题联系起来，由此得到了远超当前最优水平的有效集中不等式，涵盖了一般的非投影核，甚至非对称核。后者近期已被发现在核心集之外的机器学习领域具有研究价值，但相关理论工具有限，我们的结果对其拓展做出了贡献。最后，我们还能够解决向量值目标函数的核心集问题，这是核心集文献中的一项创新。
英文摘要：
Determinantal point processes (DPPs) are random configurations of points with tunable negative dependence. Because sampling is tractable, DPPs are natural candidates for subsampling tasks, such as minibatch selection or coreset construction. A \emph{coreset} is a subset of a (large) training set, such that minimizing an empirical loss averaged over the coreset is a controlled replacement for the intractable minimization of the original empirical loss. Typically, the control takes the form of a guarantee that the average loss over the coreset approximates the total loss uniformly across the parameter space. Recent work has provided significant empirical support in favor of using DPPs to build randomized coresets, coupled with interesting theoretical results that are suggestive but leave some key questions unanswered. In particular, the central question of whether the cardinality of a DPP-based coreset is fundamentally smaller than one based on independent sampling remained open. In this paper, we answer this question in the affirmative, demonstrating that \emph{DPPs can provably outperform independently drawn coresets}. In this vein, we contribute a conceptual understanding of coreset loss as a \emph{linear statistic} of the (random) coreset. We leverage this structural observation to connect the coresets problem to a more general problem of concentration phenomena for linear statistics of DPPs, wherein we obtain \emph{effective concentration inequalities that extend well-beyond the state-of-the-art}, encompassing general non-projection, even non-symmetric kernels. The latter have been recently shown to be of interest in machine learning beyond coresets, but come with a limited theoretical toolbox, to the extension of which our result contributes. Finally, we are also able to address the coresets problem for vector-valued objective functions, a novelty in the coresets literature.
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