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In this work we study the time complexity for the search of local minima inrandom graphs whose vertices have i.i.d. cost values. We show that, forErd\"os-R\'enyi graphs with connection probability given by $\lambda/n^\alpha$(with $\lambda > 0$ and $0 < \alpha < 1$), a family of local algorithms thatapproximate a gradient descent find local minima faster than the full gradientdescent. Furthermore, we find a probabilistic representation for the runningtime of these algorithms leading to asymptotic estimates of the mean runningtimes.

Exact asymptotic characterisation of running time for approximate gradient descent on random graphs