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In this paper, we prove that an Adam-type algorithm with smooth clippingapproaches the global minimizer of the regularized non-convex loss function.Adding smooth clipping and taking the state space as the set of alltrajectories, we can apply the ergodic theory of Markov semigroups for thisalgorithm and investigate its asymptotic behavior. The ergodic theory weestablish in this paper reduces the problem of evaluating the convergence,generalization error and discretization error of this algorithm to the problemof evaluating the difference between two functional stochastic differentialequations (SDEs) with different drift coefficients. As a result of ouranalysis, we have shown that this algorithm minimizes the the regularizednon-convex loss function with errors of the form $n^{-1/2}$, $\eta^{1/4}$,$\beta^{-1} \log (\beta + 1)$ and $e^{- c t}$. Here, $c$ is a constant and $n$,$\eta$, $\beta$ and $t$ denote the size of the training dataset, learning rate,inverse temperature and time, respectively.

Adam-like Algorithm with Smooth Clipping Attains Global Minima: Analysis Based on Ergodicity of Functional SDEs