  We propose a method to obtain the optimal weight function of 9 paths in
(3+1)D space-time whose length is less than or equal to $2\times (6+2)$ lattice
units. The factor 2 comes from inclusion of opposite direction path or time
reversed paths. There are $2\times 2$ time shifts, which we assume that they
can be regarded as stochastic Markov processes. We prepare the input 9D vector
${\bf X}$ and a $9\times 9$ matrix ${\bf W}$ and a bias vector ${\bf b}$, and
consider affine transformations ${\bf Z}^{(h)}={\bf X}^{(in)}{\bf
W}^{(h)T}+{\bf b}^{(h)}$ and ${\bf A}^{(h)}=\sigma({\bf Z}^{(h)})$ from an
input layer to a hidden layer, the hidden layer to another hidden layer and
from the hidden layer to an output layer, using the transformation ${\bf
Z}^{(x)}={\bf A}^{(h)}{\bf W}^{(out) T} +{\bf b}^{(x)}$ and ${\bf
A}^{(x)}=\sigma({\bf Z}^{(x)})$.
  By choosing the matrix ${\bf W}$ a diagonal matrix, and introducing the
information of action of the 9 paths, a simple Monte Carlo simulation yields
actions on a 2D plane spanned by $e_1, e_2$ for a fixed $u_2=j_2 e_2$ as a
function of $u_1=j_1 e_1$. The action at high momentum region has small
fluctuations, but at small momentum region, has large fluctuation. Generalizing
$\bf W$ including mixing of paths, we search the optimal weight function using
the Machine Learning (ML) techniques. For fixed point actions, actions of the
output layer are defined by the output of final hidden layer
