1 research outputs found
ODE-Inspired Analysis for the Biological Version of Oja's Rule in Solving Streaming PCA
Oja's rule [Oja, Journal of mathematical biology 1982] is a well-known
biologically-plausible algorithm using a Hebbian-type synaptic update rule to
solve streaming principal component analysis (PCA). Computational
neuroscientists have known that this biological version of Oja's rule converges
to the top eigenvector of the covariance matrix of the input in the limit.
However, prior to this work, it was open to prove any convergence rate
guarantee.
In this work, we give the first convergence rate analysis for the biological
version of Oja's rule in solving streaming PCA. Moreover, our convergence rate
matches the information theoretical lower bound up to logarithmic factors and
outperforms the state-of-the-art upper bound for streaming PCA. Furthermore, we
develop a novel framework inspired by ordinary differential equations (ODE) to
analyze general stochastic dynamics. The framework abandons the traditional
step-by-step analysis and instead analyzes a stochastic dynamic in one-shot by
giving a closed-form solution to the entire dynamic. The one-shot framework
allows us to apply stopping time and martingale techniques to have a flexible
and precise control on the dynamic. We believe that this general framework is
powerful and should lead to effective yet simple analysis for a large class of
problems with stochastic dynamics.Comment: Accepted for presentation at the Conference on Learning Theory (COLT)
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