Swarm-Based Optimization with Random Descent

• Published in: Acta applicandae mathematicae Vol. 190; no. 1; p. 2
• Main Authors: Tadmor, Eitan, Zenginoğlu, Anil
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Computational Mathematics and Numerical Analysis; Convergence; Convex analysis; Convexity; Mathematics; Mathematics and Statistics; Optimization; Partial Differential Equations; Particle swarm optimization; Probability Theory and Stochastic Processes; Gradient descent; Backtracking; 92D25; 65K10; 90C26; Swarming; Optimization; Convergence analysis
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-024-00639-0

We extend our study of the swarm-based gradient descent method for non-convex optimization, (Lu et al., Swarm-based gradient descent method for non-convex optimization, 2022 , 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, 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.