Estimating Stochastic Linear Combination of Non-Linear Regressions

Authors

  • Di Wang State University of New York at Buffalo
  • Xiangyu Guo State University of New York at Buffalo
  • Chaowen Guan State University of New York at Buffalo
  • Shi Li State University of New York at Buffalo
  • Jinhui Xu State University of New York at Buffalo

DOI:

https://doi.org/10.1609/aaai.v34i04.6078

Abstract

In this paper we study the problem of estimating stochastic linear combination of non-linear regressions, which has a close connection with 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. 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. Then we extend our result to the case where x is 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 O(1/√p + √p/n) with high probability. 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.

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Published

2020-04-03

How to Cite

Wang, D., Guo, X., Guan, C., Li, S., & Xu, J. (2020). Estimating Stochastic Linear Combination of Non-Linear Regressions. Proceedings of the AAAI Conference on Artificial Intelligence, 34(04), 6137-6144. https://doi.org/10.1609/aaai.v34i04.6078

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Section

AAAI Technical Track: Machine Learning