Treasure Search Optimization

We introduce Treasure Search Optimization (TSO), an interacting particle method for global optimization.

Most swarm methods balance exploration and exploitation within a single population, and typically switch between the two by degenerating the noise, annealing a temperature, or tuning a parameter.

TSO instead splits these tasks across two kinds of agents.

A swarm of explorers stays in exploration mode and a single treasure hunter performs exploitation.

The hunter drifts toward an objective-weighted average of the explorers and may teleport to it when the move lowers the objective.

The swarm then re-centers around the hunter, creating a feedback loop between search and capture.

We model the dynamics as coupled jump-diffusion stochastic differential equations (SDEs).

The hunter's jumps are shared by all explorers and act as a common noise.

The mean-field limit is therefore a conditional McKean-Vlasov jump-diffusion SDE, whose well-posedness we prove.

We also characterize the steady state and prove, via Laplace approximation techniques, that the hunter settles near the global minimum with error of order $1/\alpha$, where $\alpha$ is the weight parameter.

Linking the consensus drift to a smoothed free energy, we explain why the swarm ignores spurious local traps and demonstrate how to quantify uncertainty in inverse problems using post-processing Kalman steps after TSO iterations.

Numerical experiments on ODE-constrained problems and a low dimensional Bayesian inverse problem demonstrate the effectiveness of the TSO method.