Yes, this is really possible to use RHC in NN. This algorithm is simple.
Let me set a few variables:
$W_{i}$ - current weight in i-th layer.
$Wnew_{i}$ - the new weight which we will compute in new epoch.
$CD$ - Some matrix with Cauchy distributed values in $(0, 1)$
$E(x)$ - Network error function. For example it can be MSE (Mean square error).
$a$ and $b$ - some small variables between $0$ and $1$ which correct your learning step.
$epoch$ - current trainig epoch
$countOfEpoch$ - total count of training epochs
$epochError$ - previous epoch error
$newEpochError$ - current epoch error
$errorLimit$ - this is some small parameter which tell you when you can stop your trainig process. For example $errorLimit = 0.001$
Algorithm:
1) Choose random matrix and assign it to $W$. (It can me normal distribution)
2) Compute your $epochError = E(W_{i})$ with random weights.
3) If your $epochError < errorLimit$ or $epoch < countOfEpoch$ - then stop training process. If not - go to the next step.
4) Update all your network weights. Fir of all you must compute
$Wnew_{i} = W_{i} + CD * (CD * (a * min(epochError, 1) + b)$
5) Compute $newEpochError = E(Wnew_{i})$ with new weights.
6) Check epoch error difference. If $newEpochError - epochError < 0$ than you need save your update ($W_{i} = Wnew_{i}$) and update your current epoch error ($epochError = newEpochError$)
7) Increment variable $epoch$ and go back to step 3.
So that's it.