## Hint 1

Compute the two implicit rewards first. The margin of each pair is the preferred reward minus the rejected reward.

## Hint 2

$-\log \sigma(z) = \log(1 + e^{-z})$. The numpy function `np.logaddexp(0.0, -z)` computes this value without overflow for any $z$.

## Hint 3

The derivative of $\log(1 + e^{-z})$ with respect to $z$ is $-\sigma(-z)$. The margin changes by $\beta$ when the preferred policy log probability changes by 1, and by $-\beta$ for the rejected one. Divide by $N$ because the loss is a mean.

## Hint 4

For a stable sigmoid, use $\sigma(x) = \exp(-\log(1 + e^{-x}))$, which is `np.exp(-np.logaddexp(0.0, -x))`.