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A new method of numerical computation of the universal QED contributions to the electron anomalous magnetic moment is presented. The method consists of two parts. The first part is the forest-like subtraction formula that removes all ultraviolet and infrared divergences in each Feynman graph point-by-point in Feynman-parametric space. The second part is an importance sampling Monte-Carlo algorithm with the probability density function that is constructed for each Feynman graph individually. The algorithm of probability density construction is developed only for Feynman graphs without lepton loops. The computation method is fully automated at any order of the perturbation series. The results of applying the method to 2-loop, 3-loop, 4-loop Feynman graphs and to some individual 5-loop graphs are presented. The comparison of this method with the other ones with respect to Monte Carlo convergence speed is adduced.