In this paper, we study the distributed learning algorithm and the distribution regression problem of coefficient regularization for Mercer kernels. By utilizing divided-and-conquer approach, we partition a data set into disjoint data subsets for different learning machines, and get the global estimator from local estimators. By using second order decomposition on the difference of operator inverse and properties of trace operator, we show that under some priori conditions of regression function, the result of distributed learning algorithm is as good as that in single batch data algorithm. On the other hand, we give a learning rate of distribution regression problem under the coefficient regularization scheme by using similar operator methods. We find that our learning scheme performs well when the regression function has stronger regularity. And we can see the deep relation of these two different problems.

本文研究了Mercer核的分布式学习算法和系数正则化的分布回归问题。通过使用分治法，我们将数据集划分为不同学习机的不相交数据子集，并从局部估计器获得全局估计器。通过对算子逆和跟踪算子性质的差异进行二阶分解，表明在一定的回归函数先验条件下，分布式学习算法的结果与单批数据算法的结果一样好。另一方面，我们使用相似的算子方法给出了系数正则化方案下分布回归问题的学习率。我们发现，当回归函数具有更强的规律性时，我们的学习方案表现良好。