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If the both do the same thing then which give us better accuracy?

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There is a key difference:

  • Softmax regression provides class probabilities for mutually exclusive classes.

  • Logistic regression treats class membership for each class separately. Classes do not need to be mutually exclusive.

The two are equivalent for a scenario with two mutually exclusive classes - e.g. a "positive" and "negative" class - where softmax would have two outputs summing to 1, and logistic regression would have one output giving probability of the "positive" class.

I am not certain of the results comparing logistic regression one-vs-all (taking max output) with softmax regression on the same multi-class problem. I would expect the performance to be quite similar. Neither model copes well with non-linear relationships between input and target classes.

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