Exact asymptotic characterisation of running time for approximate gradient descent on random graphs
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In this work we study the time complexity for the search of local minima in random graphs whose vertices have i.i.d. cost values. We show that, for Erdös-Rényi graphs with connection probability given by λ/n^α (with λ > 0 and 0 < α < 1), a family of local algorithms that approximate a gradient descent find local minima faster than the full gradient descent. Furthermore, we find a probabilistic representation for the running time of these algorithms leading to asymptotic estimates of the mean running times.
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