An Algebraically Converging Stochastic Gradient Descent Algorithm for Global Optimization
This addresses the challenge of global optimization in nonconvex settings for researchers and practitioners, representing a significant but incremental improvement over existing stochastic methods.
The authors tackled the problem of finding global optimizers in nonconvex optimization by proposing a stochastic gradient descent algorithm with adaptive, state-dependent randomness, proving global convergence with an algebraic rate, which improves over classical methods.
We propose a new gradient descent algorithm with added stochastic terms for finding the global optimizers of nonconvex optimization problems. 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 a significant improvement over the classical rate from using a more straightforward control of the noise term. The convergence proof is based on the actual discrete setup of the algorithm, not just its continuous limit as often done in the literature. We also present several numerical examples to demonstrate the efficiency and robustness of the algorithm for reasonably complex objective functions.