TY - JOUR

T1 - Estimating stochastic linear combination of non-linear regressions efficiently and scalably

AU - Wang, Di

AU - Guo, Xiangyu

AU - Guan, Chaowen

AU - Li, Shi

AU - Xu, Jinhui

N1 - Generated from Scopus record by KAUST IRTS on 2022-09-15

PY - 2020/7/25

Y1 - 2020/7/25

N2 - Recently, many machine learning and statistical models such as non-linear regressions, the Single Index, Multi-index, Varying Coefficient Index Models and Two-layer Neural Networks can be reduced to or be seen as a special case of a new model which is called the Stochastic Linear Combination of Non-linear Regressions model. However, due to the high non-convexity of the problem, there is no previous work study how to estimate the model. In this paper, we provide the first study on how to estimate the model efficiently and scalably. Specifically, we first show that with some mild assumptions, if the variate vector x is multivariate Gaussian, then there is an algorithm whose output vectors have ℓ2-norm estimation errors of O(p/n) with high probability, where p is the dimension of x and n is the number of samples. The key idea of the proof is based on an observation motived by the Stein's lemma. Then we extend our result to the case where x is bounded and sub-Gaussian using the zero-bias transformation, which could be seen as a generalization of the classic Stein's lemma. We also show that with some additional assumptions there is an algorithm whose output vectors have ℓ∞-norm estimation errors of [Formula presented] with high probability. We also provide a concrete example to show that there exists some link function which satisfies the previous assumptions. Finally, for both Gaussian and sub-Gaussian cases we propose a faster sub-sampling based algorithm and show that when the sub-sample sizes are large enough then the estimation errors will not be sacrificed by too much. Experiments for both cases support our theoretical results. To the best of our knowledge, this is the first work that studies and provides theoretical guarantees for the stochastic linear combination of non-linear regressions model.

AB - Recently, many machine learning and statistical models such as non-linear regressions, the Single Index, Multi-index, Varying Coefficient Index Models and Two-layer Neural Networks can be reduced to or be seen as a special case of a new model which is called the Stochastic Linear Combination of Non-linear Regressions model. However, due to the high non-convexity of the problem, there is no previous work study how to estimate the model. In this paper, we provide the first study on how to estimate the model efficiently and scalably. Specifically, we first show that with some mild assumptions, if the variate vector x is multivariate Gaussian, then there is an algorithm whose output vectors have ℓ2-norm estimation errors of O(p/n) with high probability, where p is the dimension of x and n is the number of samples. The key idea of the proof is based on an observation motived by the Stein's lemma. Then we extend our result to the case where x is bounded and sub-Gaussian using the zero-bias transformation, which could be seen as a generalization of the classic Stein's lemma. We also show that with some additional assumptions there is an algorithm whose output vectors have ℓ∞-norm estimation errors of [Formula presented] with high probability. We also provide a concrete example to show that there exists some link function which satisfies the previous assumptions. Finally, for both Gaussian and sub-Gaussian cases we propose a faster sub-sampling based algorithm and show that when the sub-sample sizes are large enough then the estimation errors will not be sacrificed by too much. Experiments for both cases support our theoretical results. To the best of our knowledge, this is the first work that studies and provides theoretical guarantees for the stochastic linear combination of non-linear regressions model.

UR - https://linkinghub.elsevier.com/retrieve/pii/S0925231220302642

UR - http://www.scopus.com/inward/record.url?scp=85081249507&partnerID=8YFLogxK

U2 - 10.1016/j.neucom.2020.02.074

DO - 10.1016/j.neucom.2020.02.074

M3 - Article

VL - 399

SP - 129

EP - 140

JO - Neurocomputing

JF - Neurocomputing

SN - 1872-8286

ER -