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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?

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The answer at Gradient for Hinge Loss Multiclass gives you the Gradient per class.
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..

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