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Typical support vector classifier uses the following optimization procedure:

$$\min ||w||^2 + C\sum_{i=1}^N \zeta_i$$ $$y_i(w^Tx_i+b) \geq 1 - \zeta_i$$ $$\zeta_i \geq 0$$

This hinge loss setup slightly penalizes the correctly classified data points within the margin. Now if we gently modify the constraint the result will be a learning machinery with a regularized perceptron loss and it will only penalize misclassified data points.

$$y_i(w^Tx_i+b) \geq - \zeta_i$$

I understand there are historical reasons things are the way they are (maximizing margin rhetoric, etc.) But is there a particular theoretical reason for not implementing a support vector classifier in this manner? What will be the pros and cons?

Typical support vector classifier uses the following optimization procedure:

$$\min \dfrac{1}{2}||w||^2 + C\sum_{i=1}^N \zeta_i$$ $$y_i(w^Tx_i+b) \geq 1 - \zeta_i$$ $$\zeta_i \geq 0$$

This hinge loss setup slightly penalizes the correctly classified data points within the margin. Now if we gently modify the constraint the result will be a learning machinery with a regularized perceptron loss and it will only penalize misclassified data points.

$$y_i(w^Tx_i+b) \geq - \zeta_i$$

I understand there are historical reasons things are the way they are (maximizing margin rhetoric, etc.) But is there a particular theoretical reason for not implementing a support vector classifier in this manner? What will be the pros and cons?

Typical support vector classifier uses the following optimization procedure:

$$\min ||w||^2 + C\sum_{i=1}^N \zeta_i$$ $$y_i(w^Tx_i+b) \geq 1 - \zeta_i$$ $$\zeta_i \geq 0$$

This hinge loss setup slightly penalizes the correctly classified data points within the margin. Now if we gently modify the constraint the result will be a learning machinery with a regularized perceptron loss.

$$y_i(w^Tx_i+b) \geq - \zeta_i$$

I understand there are historical reasons things are the way they are (maximizing margin rhetoric, etc.) But is there a particular theoretical reason for not implementing a support vector classifier in this manner? What will be the pros and cons?

Typical support vector classifier uses the following optimization procedure:

$$\min ||w||^2 + C\sum_{i=1}^N \zeta_i$$ $$y_i(w^Tx_i+b) \geq 1 - \zeta_i$$ $$\zeta_i \geq 0$$

This hinge loss setup slightly penalizes the correctly classified data points within the margin. Now if we gently modify the constraint the result will be a learning machinery with a regularized perceptron loss and it will only penalize misclassified data points.

$$y_i(w^Tx_i+b) \geq - \zeta_i$$

I understand there are historical reasons things are the way they are (maximizing margin rhetoric, etc.) But is there a particular theoretical reason for not implementing a support vector classifier in this manner? What will be the pros and cons?

Typical support vector classifier uses the following optimization procedure:

$$\min ||w||^2 + C\sum_{i=1}^N \zeta_i$$ $$y_i(w^Tx_i+b) \geq 1 - \zeta_i$$ $$\zeta_i \geq 0$$

This hinge loss setup slightly penalizes the correctly classified data points within the margin. Now if we gently modify the constraint the result will be a learning machinery with a regularized perceptron loss.

$$y_i(w^Tx_i+b) \geq - \zeta_i$$

I understand there are historical reasons things are the way they are (maximizing margin rhetoric, etc.) But is there a particular theoretical reason for not implementing a support vector classifier in this manner? What will be the pros and cons?

Typical support vector classifier uses the following optimization procedure:

$$\min ||w||^2 + C\sum_{i=1}^N \zeta_i$$ $$y_i(w^Tx_i+b) \geq 1 - \zeta_i$$ $$\zeta_i \geq 0$$

This hinge loss setup slightly penalizes the correctly classified data points within the margin. Now if we gently modify the constraint the result will be a learning machinery with a regularized perceptron loss.

$$y_i(w^Tx_i+b) \geq - \zeta_i$$

I understand there are historical reasons things are the way they are (maximizing margin rhetoric, etc.) But is there a particular theoretical reason for not implementing a support vector classifier in this manner? What will be the pros and cons?