Statistical Optimality of Divide and Conquer Kernel-based Functional Linear Regression

被引:0
作者
Liu, Jiading [1 ]
Shi, Lei [1 ,2 ,3 ]
机构
[1] Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China
[2] Fudan Univ, Shanghai Key Lab Contemporary Appl Math, Shanghai 200433, Peoples R China
[3] Shanghai Artificial Intelligence Lab, 701 Yunjin Rd, Shanghai 200232, Peoples R China
基金
中国国家自然科学基金;
关键词
functional linear regression; reproducing kernel Hilbert space; divide-and- conquer estimator; model misspecification; mini-max optimal rates; MINIMAX; RATES; CONSISTENCY; PREDICTION;
D O I
暂无
中图分类号
TP [自动化技术、计算机技术];
学科分类号
0812 ;
摘要
Previous analysis of regularized functional linear regression in a reproducing kernel Hilbert space (RKHS) typically requires the target function to be contained in this kernel space. This paper studies the convergence performance of divide-and-conquer estimators in the scenario that the target function does not necessarily reside in the underlying RKHS. As a decomposition-based scalable approach, the divide-and-conquer estimators of functional linear regression can substantially reduce the algorithmic complexities in time and memory. We develop an integral operator approach to establish sharp finite sample upper bounds for prediction with divide-and-conquer estimators under various regularity conditions of explanatory variables and target function. We also prove the asymptotic optimality of the derived rates by building the mini -max lower bounds. Finally, we consider the convergence of noiseless estimators and show that the rates can be arbitrarily fast under mild conditions.
引用
收藏
页码:1 / 56
页数:56
相关论文
共 54 条
[1]   THEORY OF REPRODUCING KERNELS [J].
ARONSZAJN, N .
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1950, 68 (MAY) :337-404
[2]  
Ash R. B., 2014, Topics in Stochastic Processes
[3]  
Bach FR, 2008, J MACH LEARN RES, V9, P1179
[4]   On regularization algorithms in learning theory [J].
Bauer, Frank ;
Pereverzev, Sergei ;
Rosasco, Lorenzo .
JOURNAL OF COMPLEXITY, 2007, 23 (01) :52-72
[5]  
Berthier R, 2020, ADV NEUR IN, V33
[6]   Optimal Rates for Regularization of Statistical Inverse Learning Problems [J].
Blanchard, Gilles ;
Muecke, Nicole .
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS, 2018, 18 (04) :971-1013
[7]   Prediction in functional linear regression [J].
Cai, T. Tony ;
Hall, Peter .
ANNALS OF STATISTICS, 2006, 34 (05) :2159-2179
[8]   Minimax and Adaptive Prediction for Functional Linear Regression [J].
Cai, T. Tony ;
Yuan, Ming .
JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION, 2012, 107 (499) :1201-1216
[9]   How well can we estimate a sparse vector? [J].
Candes, Emmanuel J. ;
Davenport, Mark A. .
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS, 2013, 34 (02) :317-323
[10]  
Caponnetto A, 2007, FOUND COMPUT MATH, V7, P331, DOI [10.1007/s10208-006-0196-8, 10.1007/S10208-006-0196-8]