# Help interpreting formula for multi-class hinge loss

As I'm reading from wikipedia, and this Cross Validated question: Gradient for hinge loss multiclass, the gradient value for a training feature set is somewhat straightforward. However if I'm interpreting this correctly, this only gives a gradient value for the weight of the 'true' class. How do I find the gradients for the other weights? That is, if my total weight vector is [W1, W2, ... Wy... Wk], where y is the class of the training sample then what are the gradient/loss values for every weight that isn't Wy?

Since the classifier for the $j$ - th class is given by the row $j$ of $W$ (Which is notated at the answer as ${W}_{j}$) all you need on each iteration is to update ${W}_{j}$ according to ${\nabla}_{{W}_{j}} {L}_{i}$ according to train sample ${x}_{i}$.
Namely, ${W}_{j}^{\left( k + 1 \right)} = {W}_{j}^{\left( k \right)} - \eta {\nabla}_{ {W}_{j} } {L}_{i}$, Where $j$ is the index of the updated row, $k$ is the iteration counter and $\eta$ is the Step Size / Learning Rate..