Given the problem of maximization of log-likelihood in logistic regression.
$log(P(D)) = \sum_{n = 1}^ N y_{n}log(σ(w^{T} x_{n})) + (1 − y_{n})log(1 − σ(w^{T} x_{n}))$
Where $x_{n}$ are the observations, $y_{n} ∈ {0, 1}$ are their true labels
I'm trying to see what is the condition for which there exists a unique solution for logistic regression.
I tried the following based on the minimal machine learning knowledge I have.
I estimate the gradient of log-likelihood function with respect to W. which is :
$\sum_{n = 1}^{N}y_{n}x_{n} - \sum_{n = 1}^{N} \sigma(w^Tx_n)x_n$
Similarly, I found the Hessain.
$ - \sum_{n = 1}^{N} \sigma(w^Tx_n).(1 - \sigma(w^Tx_n))x_n.x^T_n$
So, to have a unique maximizer I believe that $x_n.x^T_n$ should be PSD.
I'm not sure whether I'm correct here. Please, can someone help me with this? Thank you.