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Enabling Long-range Exploration in Minimization of Multimodal Functions...

by Jiaxin Zhang, Hoang A Tran, Dan Lu, Guannan Zhang
Publication Type
Conference Paper
Journal Name
Proceedings of Machine Learning Research
Book Title
Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence, PMLR
Publication Date
Page Numbers
1639 to 1649
Publisher Location
Conference Name
37th Conference on Uncertainty in Artificial Intelligence (UAI 2021)
Conference Location
Virtual conference, California, United States of America
Conference Sponsor
Association for Uncertainty in Artificial Intelligence (AUAI)
Conference Date

We consider the problem of minimizing multi-modal loss functions with a large number of local optima. Since the local gradient points to the direction of the steepest slope in an infinitesimal neighborhood, an optimizer guided by the local gradient is often trapped in a local minimum. To address this issue, we develop a novel nonlocal gradient to skip small local minima by capturing major structures of the loss’s landscape in black-box optimization. The nonlocal gradient is defined by a directional Gaussian smoothing (DGS) approach. The key idea of DGS is to conducts 1D long-range exploration with a large smoothing radius along d orthogonal directions in Rd, each of which defines a nonlocal directional derivative as a 1D integral. Such long-range exploration enables the nonlocal gradient to skip small local minima. The d directional derivatives are then assembled to form the nonlocal gradient. We use the Gauss-Hermite quadrature rule to approximate the d 1D integrals to obtain an accurate estimator. The superior performance of our method is demonstrated in three sets of examples, including benchmark functions for global optimization, and two real-world scientific problems.