  We extend our study of the swarm-based gradient descent method for non-convex
optimization, [Lu, Tadmor & Zenginoglu, arXiv:2211.17157], to allow random
descent directions. We recall that the swarm-based approach consists of a swarm
of agents, each identified with a position, ${\mathbf x}$, and mass, $m$. The
key is the transfer of mass from high ground to low(-est) ground. The mass of
an agent dictates its step size: lighter agents take larger steps. In this
paper, the essential new feature is the choice of direction: rather than
restricting the swarm to march in the steepest gradient descent, we let agents
proceed in randomly chosen directions centered around -- but otherwise
different from -- the gradient direction. The random search secures the descent
property while at the same time, enabling greater exploration of ambient space.
Convergence analysis and benchmark optimizations demonstrate the effectiveness
of the swarm-based random descent method as a multi-dimensional global
optimizer.
