Swarm-Based Gradient Descent Method for Non-Convex Optimization

Abstract: We introduce a new Swarm-Based Gradient Descent (SBGD) method for non-convex optimization. The swarm consists of agents, each is identified with a position, ${\mathbf x}$, and mass, $m$. The key to their dynamics is communication: masses are being transferred from agents at high ground to low(-est) ground. At the same time, agents change positions with step size, $h=h({\mathbf x},m)$, adjusted to their relative mass: heavier agents proceed with small time-steps in the direction of local gradient, while lighter agents take larger time-steps based on a backtracking protocol. Accordingly, the crowd of agents is dynamically divided between heavier' leaders, expected to approach local minima, andlighter' explorers. With their large-step protocol, explorers are expected to encounter improved position for the swarm; if they do, then they assume the role of `heavy' swarm leaders and so on. Convergence analysis and numerical simulations in one-, two-, and 20-dimensional benchmarks demonstrate the effectiveness of SBGD as a global optimizer.