Stochastic Reconfiguration

Algorithm, Optimization

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$$\begin{aligned} \partial_{W}\langle E\rangle = \partial_{W} \frac{\langle\Psi|H|\Psi\rangle}{\langle\Psi|\Psi\rangle} &= \frac{\partial}{\partial W} (\frac{1}{Z}\cdot \sum_{ss'} \overline{\Psi}_s\Psi_{s'}H_{ss'})\\ & = -\frac{1}{Z^2}\cdot \frac{\partial Z}{\partial W}\sum_{ss'}\overline{\Psi}_s\Psi_{s'}H_{ss'} + \frac{1}{Z}\sum_{ss'}\frac{\partial}{\partial W}(\overline{\Psi}_s\Psi_{s'}H_{ss'})\\ &= -\frac{1}{Z^2}\cdot \sum_{s}(\frac{\partial \overline{\Psi}_s}{\partial W}\Psi_s + \overline{\Psi}_s\frac{\partial \Psi_s}{\partial W})\sum_{ss'}\overline{\Psi}_s\Psi_{s'}H_{ss'} +\\ & \frac{1}{Z}\sum_{ss'}\frac{\partial \overline{\Psi}_{s}}{\partial W}\Psi_{s'} H_{ss'} + \overline{\Psi}_s\frac{\partial \Psi_{s'}}{\partial W}H_{ss'}\\ \end{aligned}$$

let $O_{s} = \frac{1}{\Psi_s}\frac{\partial \Psi_s}{\partial W}$, $E_{s} = \frac{\sum_{s'} H_{ss'}\Psi_{s'}}{\Psi_s}$

$$\begin{aligned} & -\frac{1}{Z^2}\cdot \sum_{s}(\frac{\partial \overline{\Psi}_s}{\partial W}\Psi_s + \overline{\Psi}_s\frac{\partial \Psi_s}{\partial W})\sum_{ss'}\overline{\Psi}_s\Psi_{s'}H_{ss'} +\\ & \frac{1}{Z}\sum_{ss'}\frac{\partial \overline{\Psi}_{s}}{\partial W}\Psi_{s'} H_{ss'} + \overline{\Psi}_s\frac{\partial \Psi_{s'}}{\partial W}H_{ss'}\\ & = -\frac{1}{Z^2}\cdot \sum_{s}|\Psi_s|^2(\overline{O}_s + O_{s}) \sum_{ss'}\overline{\Psi}_s\Psi_{s'}H_{ss'} + \\ & \frac{1}{Z}\sum_{ss'}\overline{O}_{s}\overline{\Psi}_s \Psi_{s'} H_{ss'} + \overline{\Psi}_s \Psi_{s'} O_{s'} H_{ss'}\\ & = -\frac{1}{Z^2}\cdot \sum_{s}|\Psi_s|^2(\overline{O}_s + O_{s}) \sum_{ss'}\overline{\Psi}_s\Psi_{s'}H_{ss'} + \\ & \frac{1}{Z}\sum_{ss'}\overline{O}_{s}\overline{\Psi}_s \Psi_{s'} H_{ss'} + \frac{1}{Z}\sum_{ss'}\overline{\Psi}_{s'} \Psi_{s} O_{s} H_{ss'}\\ & = -\frac{1}{Z^2}\cdot \sum_{s}|\Psi_s|^2(\overline{O}_s + O_{s}) \sum_{ss'}\overline{\Psi}_s\Psi_{s'}H_{ss'} + \\ & \frac{1}{Z}\sum_{s}\overline{O}_{s}|\Psi_s|^2 \frac{\sum_{s'}\Psi_{s'} H_{ss'}}{\Psi_{s}}+\\ & \frac{1}{Z}\sum_{s}O_{s}|\Psi_s|^2\frac{\sum_{s'}\overline{\Psi}_{s'}H_{ss'}}{\overline{\Psi}_s}\\ & = -\langle \overline{O} + O\rangle \langle E_{loc} \rangle + \langle\overline{O}E_{loc} +O\overline{E}_{loc}\rangle \end{aligned}$$

Therefore,

$$\vec{F} = \langle \overline{O} E_{loc}\rangle - \langle\overline{O}\rangle\langle E_{loc}\rangle + \overline{\langle\overline{O}E_{loc}\rangle - \langle\overline{O}\rangle\langle E_{loc}\rangle}$$

This equation meets previous one in MPS.

conclusion:

It seems that the science paper's derivatives should be defined as:

$$O_k (S) = \frac{1}{\Psi(S)}\partial_{W_k}\Psi(S) + \frac{1}{\Psi^*(S)}\partial_{W_k}\Psi^*(S)$$

and the covariance matrix then should be (or it won't work)

$$S_{kk'} = \langle \overline{O}_{k} O_{k'}\rangle - \langle \overline{O}_k\rangle\langle O_{k'}\rangle$$

Numerical Results

3 spins for $h=0.1$ in TFI system with 200 train steps, does not work for $h=0.0$