Algorithm 5.2.

Optimizing the bins.

Choose an initial collection $\mathcal{B}$ of bins. Define an objective function on the bin space,

$$\mathcal{O}\left({\mathcal{B}}^{\prime }\right)=\sum _{u\in {\mathcal{B}}^{\prime }}\mathrm{Var}\left({\tilde{K}\tilde{h}|}_u\right),$$ (5.5)

where ${\tilde{K}\tilde{h}|}_u$ is the restriction of $\tilde{K}\tilde{h}$ to $\{p\in \mathcal{M}\mathcal{B}:p\in u\}$, and $\mathrm{Var}\left({\tilde{K}\tilde{h}|}_u\right)$ is the usual vector population variance. Choose an annealing parameter α > 0, set ${\mathcal{B}}_{opt}=\mathcal{B}$, and iterate the following for a user-prescribed number of steps:

1. Perturb $\mathcal{B}$ to get new bins ${\mathcal{B}}^{\prime }$.

(Say by moving a microbin from one bin to another bin.)

2. With probability $\min \left\{1,\exp \left[\alpha \left(\mathcal{O}(\mathcal{B})-\mathcal{O}\left({\mathcal{B}}^{\prime }\right)\right)\right]\right\}$, set $\mathcal{B}={\mathcal{B}}^{\prime }$.

3. If $\mathcal{O}(\mathcal{B})<\mathcal{O}\left({\mathcal{B}}_{opt}\right)$, then update ${\mathcal{B}}_{opt}=\mathcal{B}$. Return to step 1.

Once the bin search is complete, the output is $\mathcal{B}={\mathcal{B}}_{opt}$.