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If you have stochastic gradient descent and you want to update the weights $w_t$ with the L2 norm regularization penalty, how is this math done? I am using sklearn for this and am just wondering how the weight $w_m$ is updated according to $l- \lambda||w||_{2}^{2}$. I read through the sklearn docs but they didn't go into to much detail about how this is actually performed mathematically.

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It's the same procedure as SGD with any other loss function. The only difference is that the loss function now has a penalty term added for $\ell_2$ regularization.

The standard SGD iteration for loss function $L(w)$ and step size $\alpha$ is: $$w_{t+1} = w_t - \alpha \nabla_w L(w_t)$$

Say the original loss function was $L_0$, and you add a penalty term for $\ell_2$ regularization. The new loss function is: $$L(w) = L_0(w) + \lambda \|w\|_2^2$$

The gradient is: $$\nabla_w L(w) = \nabla_w \left [ L_0(w) + \lambda \|w\|_2^2 \right ] = \nabla_w L_0(w) + 2 \lambda w$$

Plug this into the SGD iteration above.

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