Near-optimal stochastic approximation for online principal component estimation

Chris Junchi Li, Mengdi Wang, Han Liu, Tong Zhang

Research output: Contribution to journalArticlepeer-review

31 Scopus citations


Principal component analysis (PCA) has been a prominent tool for high-dimensional data analysis. Online algorithms that estimate the principal component by processing streaming data are of tremendous practical and theoretical interests. Despite its rich applications, theoretical convergence analysis remains largely open. In this paper, we cast online PCA into a stochastic nonconvex optimization problem, and we analyze the online PCA algorithm as a stochastic approximation iteration. The stochastic approximation iteration processes data points incrementally and maintains a running estimate of the principal component. We prove for the first time a nearly optimal finite-sample error bound for the online PCA algorithm. Under the subgaussian assumption, we show that the finite-sample error bound closely matches the minimax information lower bound.

Original languageEnglish (US)
Pages (from-to)75-97
Number of pages23
JournalMathematical Programming
Issue number1
StatePublished - Jan 1 2018

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics


  • Finite-sample analysis
  • High-dimensional data
  • Nonconvex optimization
  • Online algorithm
  • Principal component analysis
  • Stochastic approximation
  • Stochastic gradient method


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