Inspite ofitsgood (promising) performance, ratio-cut and other traditional spectral clustering methods (SC) suffer from the following drawbacks: (1) The timecomplexityoftraditional spectral clustering isO(n2c),which isoneofsignificant drawbacks of SC. Much effort has been devoted to accelerate the process.

Problem (1) has been comprehensively investigated in the literature [Duchi et al., 2011, Kingma and Ba, 2015, Loshchilov and Hutter, 2017], and it is well-known that the classical stochastic gradient descent (SGD) achieves a convergence rate of

Thatis,wedemonstrate that a finite variance isnot necessaryfor almost sure convergence of stochastic NPG, while controlling update aggressiveness is both necessary and sufficient.