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Oja's rule [Oja, Journal of mathematical biology 1982] is a well-knownbiologically-plausible algorithm using a Hebbian-type synaptic update rule tosolve streaming principal component analysis (PCA). Computationalneuroscientists have known that this biological version of Oja's rule convergesto 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 rateguarantee.  In this work, we give the first convergence rate analysis for the biologicalversion of Oja's rule in solving streaming PCA. Moreover, our convergence ratematches the information theoretical lower bound up to logarithmic factors andoutperforms the state-of-the-art upper bound for streaming PCA. Furthermore, wedevelop a novel framework inspired by ordinary differential equations (ODE) toanalyze general stochastic dynamics. The framework abandons the traditionalstep-by-step analysis and instead analyzes a stochastic dynamic in one-shot bygiving a closed-form solution to the entire dynamic. The one-shot frameworkallows us to apply stopping time and martingale techniques to have a flexibleand precise control on the dynamic. We believe that this general framework ispowerful and should lead to effective yet simple analysis for a large class ofproblems with stochastic dynamics.

ODE-Inspired Analysis for the Biological Version of Oja's Rule in Solving Streaming PCA