This paper is concerned with functional learning by utilizing two-stage sampled distribution regression. We study a multi-penalty regularization algorithm for distribution regression in the framework of learning theory. The algorithm aims at regressing to real-valued outputs from probability measures. The theoretical analysis of distribution regression is far from maturity and quite challenging since only second-stage samples are observable in practical settings. In our algorithm, to transform information of distribution samples, we embed the distributions to a reproducing kernel Hilbert space ${\mathcal{H}}_K$ associated with Mercer kernel K via mean embedding technique. One of the primary contributions of this work is the introduction of a novel multi-penalty regularization algorithm, which is able to capture more potential features of distribution regression. Optimal learning rates of the algorithm are obtained under mild conditions. The work also derives learning rates for distribution regression in the hard learning scenario $f_{\rho }\notin {\mathcal{H}}_K$, which has not been explored in the existing literature. Moreover, we propose a new distribution-regression-based distributed learning algorithm to face large-scale data or information challenges arising from distribution data. The optimal learning rates are derived for the distributed learning algorithm. By providing new algorithms and showing their learning rates, the work improves the existing literature in various aspects.

本文关注利用两阶段采样分布回归的函数学习。我们在学习理论的框架内研究了一种用于分布回归的多重惩罚正则化算法。该算法旨在从概率度量回归到实值输出。分布回归的理论分析还远未成熟，而且相当具有挑战性，因为在实际情况中只能观察到第二阶段的样本。在我们的算法中，为了转换分布样本的信息，我们将分布嵌入到再生核希尔伯特空间中${\mathcal{H}}_K$通过平均嵌入技术与 Mercer 核K相关联。这项工作的主要贡献之一是引入了一种新颖的多重惩罚正则化算法，该算法能够捕获分布回归的更多潜在特征。在温和的条件下获得算法的最佳学习率。该工作还推导了硬学习场景中分布回归的学习率$F_{\rho }\notin {\mathcal{H}}_K$，现有文献中尚未对此进行探讨。此外，我们提出了一种新的基于分布回归的分布式学习算法，以应对分布数据带来的大规模数据或信息挑战。导出分布式学习算法的最佳学习率。通过提供新算法并展示其学习率，该工作在各个方面改进了现有文献。