寻宝优化
Treasure Search Optimization
AI总结：
本文提出寻宝优化方法，将探索与利用任务分配给不同智能体，通过耦合跳跃扩散随机微分方程建模动力学，证明了平均场极限的适定性，刻画了稳态，展示了该方法在相关问题上的有效性及对反问题不确定性的量化作用。
AI中文摘要：
我们介绍了寻宝优化（TSO），一种用于全局优化的相互作用粒子方法。大多数群体方法在单个群体内平衡探索和利用，通常通过降低噪声、退火温度或调整参数在两者之间切换。TSO则将这些任务分配给两种智能体。一群探索者保持探索模式，单个寻宝者进行利用。寻宝者向探索者的目标加权平均值漂移，当移动降低目标时可能瞬移到该平均值。群体然后围绕寻宝者重新居中，在搜索和捕获之间创建反馈回路。我们将动力学建模为耦合跳跃扩散随机微分方程（SDEs）。寻宝者的跳跃由所有探索者共享并充当共同噪声。因此，平均场极限是一个条件McKean-Vlasov跳跃扩散SDE，我们证明了其适定性。我们还刻画了稳态，并通过拉普拉斯近似技术证明，寻宝者以$1/\alpha$阶的误差在全局最小值附近稳定，其中$\alpha$是权重参数。将共识漂移与平滑自由能联系起来，我们解释了群体为何忽略虚假局部陷阱，并展示了如何在TSO迭代后使用后处理卡尔曼步骤量化反问题中的不确定性。关于ODE约束问题和低维贝叶斯反问题的数值实验证明了TSO方法的有效性。
英文摘要：
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 $1/α$, where $α$ 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.
