Can we use gradient descent method for classifying the Multinomial target in logistic regression?
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Gradient descent method is used to optimize parameters (so optimization algorithm, you can use other algorithms) of your machine learning model (in this case logistic regression) so yes you can use gradient descent. If you want to have multiclass classification you need to use some strategy like:
OVO (one vs. one)
OVR (one vs rest)
etc.
You will need to train more than one model depending on strategy. To find more about those strategies try to Google them.

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