This paper is concerned with functional learning by utilizing two-stage sampled distribution regression. We study a multi-penalty regularization algorithm for distribution regression under the framework of learning theory. The algorithm aims at regressing to real valued outputs from probability measures. The theoretical analysis on distribution regression is far from maturity and quite challenging, since only second stage samples are observable in practical setting. In the algorithm, to transform information from samples, we embed the distributions to a reproducing kernel Hilbert space $\mathcal{H}_K$ associated with Mercer kernel $K$ via mean embedding technique. The main contribution of the paper is to present a novel multi-penalty regularization algorithm to capture more features of distribution regression and derive optimal learning rates for the algorithm. The work also derives learning rates for distribution regression in the nonstandard setting $f_{\rho}\notin\mathcal{H}_K$, which is not explored in existing literature. Moreover, we propose a distribution regression-based distributed learning algorithm to face large-scale data or information challenge. The optimal learning rates are derived for the distributed learning algorithm. By providing new algorithms and showing their learning rates, we improve the existing work in different aspects in the literature.

翻译：本文关注利用两阶段采样分布回归进行函数学习。在学习理论框架下，我们研究了一种用于分布回归的多罚正则化算法。该算法旨在从概率测度中回归出实值输出。分布回归的理论分析远未成熟且极具挑战性，因为在实际设定中仅能观测到第二阶段样本。在算法中，为从样本中转换信息，我们通过均值嵌入技术将分布嵌入到与Mercer核$K$相关的再生核希尔伯特空间$\mathcal{H}_K$中。本文的主要贡献在于提出了一种新颖的多罚正则化算法，以捕捉分布回归的更多特征并推导出该算法的最优学习率。本文还推导了非标准设定$f_{\rho}\notin\mathcal{H}_K$下分布回归的学习率，这一情形在现有文献中尚未被探索。此外，我们提出了一种基于分布回归的分布式学习算法，以应对大规模数据或信息挑战，并推导出该分布式学习算法的最优学习率。通过提出新算法并展示其学习率，我们在文献的多个方面改进了现有工作。