Negative-log-likelihood dimensions in logistic regression

I'm starting to attempt to learn how regularized multi-class logistic regression classifiers work, but I'm stuck at the very beginning. The negative log-likelihood function of logistic regression for $m$ classes and its gradients are given by:

If I have a feature matrix $\bf X$ that is size 800K x 50, and a $\bf Y$ that is size 800K x 1 with 30K unique values, what are the dimensions of $\bf W$, $\bf w_{k}$, $\bf x_{j}$, and what do $n$ and $m$ equal?

I thought that $m=50$, $n=800K$, $\bf W$ is also a 800K x 50 matrix, $\bf w_{k}$ is a column of $\bf W$ of size 800K x 1, and $\bf x_{j}$ is a row of $\bf X$ of size 50 x 1. However obviously I'm wrong because I can't take the dot product $\bf w_{k}^{T} \bf x_{j}$ if these vectors have unequal lengths. What part(s) am I misunderstanding?

$n=800K$ and $m = 300K$. (Recall $m$ is the number of classes.) So, $\mathbf{W}$ is $50 \times 300K$ with columns $\mathbf{w}_k$. Note that $k \in \{1,\ldots,m\}$.