# How to inference the dual form of perceptron?

The model of perceptron is a linear binary classifier, which is $$f(x)=\mathbb{sign}(w^Tx+b)$$. $$x$$ is the datapoint as $$w$$ as well as $$b$$ are the parameters.

The cost function of Primal Perceptron is $$\min\limits_{w,b}-\sum_{x_i\in M}{y_i\left(w^Tx_i+b\right)}$$.

Where $$M$$ means the datapoints set where some points is misclassified that satisfied $$y_i(w^Tx_i+b)\le0$$. And $$y_i \in \{-1,+1\}$$ is the label(target) of $$x_i$$, which is in the output spaces.

We can use SGD algorithm to update the parameters $$w$$ and $$b$$ with $$w \leftarrow w+\eta y_ix_i,b\leftarrow \eta y_i$$. Finally, we could get $$w=\sum_{i=1}^N\alpha_iy_ix_i,b=\sum_{i=1}^N\alpha_iy_i$$, if we let $$\alpha_i = n_i\eta$$ and the original value of $$(w,b)$$ is $$(0,0)$$. And $$n_i$$ means the counts where datapoint $$(x_i,y_i)$$ is misclassified.

So we can get the dual form of perceptron. $$f(x)=sign\left(\sum_{j=1}^N\alpha_jy_jx_j+b\right)$$.

Using that $$\alpha_i \leftarrow \alpha_i+\eta,b\leftarrow b+\eta y_i$$ to update.

Generally, We can get the dual form of an constrained optimization problem with Lagrange duality. But I failed to inferene the dual form of perceptron using Lagrange duality. It seems that this form is obtained directly through computing the final optimal solution. As a noob to operation reaserch, I am confused about this process. So could you guys explian the theoretical basis behind this process to me? Thanks!