An Algebraically Converging Stochastic Gradient Descent Algorithm for Global Optimization

We propose a new gradient descent algorithm with added stochastic terms for finding the global optimizers of nonconvex optimization problems, referred to as ``AdaVar'' here. A key component in the algorithm is the adaptive tuning of the randomness based on the value of the objective function. In the language of simulated annealing, the temperature is state-dependent. With this, we prove the global convergence of the algorithm with an algebraic rate both in probability and in the parameter space. This is an improvement over the classical rate from using a simpler control of the noise term. The convergence proof is based on the actual discrete setup of the algorithm. We also present several numerical examples to demonstrate the efficiency and robustness of the algorithm for reasonably complex objective functions.