$\alpha$ Annealing of Ant Colony Optimization in the infinite-range Ising model

TL;DR

This paper explores how gradually increasing the parameter alpha in ant colony optimization affects finding the ground state in the infinite-range Ising model, revealing a transition from trivial to optimal solutions and drawing parallels to quantum annealing.

Contribution

It introduces an alpha-annealing schedule for ACO in the infinite-range Ising model and derives the associated Fokker-Planck equation to analyze solution transitions.

Findings

Alpha-annealing causes the joint PDF to shift from mono-modal to multi-modal.

Gradually increasing alpha guides the search from trivial to ground state solutions.

The role of alpha in ACO is analogous to the transverse field in quantum annealing.

Abstract

Ant colony optimization (ACO) leverages the parameter $\alpha$ to modulate the decision function's sensitivity to pheromone levels, balancing the exploration of diverse solutions with the exploitation of promising areas. Identifying the optimal value for $\alpha$ and establishing an effective annealing schedule remain significant challenges, particularly in complex optimization scenarios. This study investigates the $\alpha$-annealing process of the linear Ant System within the infinite-range Ising model to address these challenges. Here, "linear" refers to the decision function employed by the ants. By systematically increasing $\alpha$, we explore its impact on enhancing the search for the ground state. We derive the Fokker-Planck equation for the pheromone ratios and obtain the joint probability density function (PDF) in stationary states. As $\alpha$ increases, the joint PDF transitions from a mono-modal to a multi-modal state. In the homogeneous fully connected Ising model, $\alpha$-annealing facilitates the transition from a trivial solution at $\alpha=0$ to the ground state. The parameter $\alpha$ in the annealing process plays a role analogous to the transverse field in quantum annealing. Our findings demonstrate the potential of $\alpha$-annealing in navigating complex optimization problems, suggesting its broader application beyond the infinite-range Ising model.