Solving LASSO ($ {L}_{1} $ Regularized Least Squares) with Gradient Descent To the best of my knowledge, state of the art methods for optimizing the LASSO objective function include the LARS algorithm and proximal gradient methods.

I was wondering however, if the LASSO objective function 
$$ ||y-Ax||_2^2 + \lambda ||x||_1$$ can be optimized using (vanilla) gradient descent?

I know that the 1-norm is not differentiable at zero, but numerically one is never exactly at zero. Therefore the gradient exists at each iteration point $x^{(k)}$ and I do not really need to resort to subgradients. 

Unlike for the L2-penalized least squares, the gradient descent step would of course need a decreasing step size and would not yield exact zeros. 

Can this method be called gradient descent or would it be better to call it subgradient descent or something else?

Many thanks!
It can easily solved by the Gradient Descent Framework with one adjustment in order to take care of the $ {L}_{1} $ norm term.

Since the $ {L}_{1} $ norm isn't smooth you need to use the concept of Sub Gradient / Sub Derivative.
When you integrate Sub Gradient instead of Gradient into the Gradient Descent Method it becomes the Sub Gradient Method.

In the case of the $ {L}_{1} $ norm the Sub Gradient is given by the function $ \operatorname{sign} \left( \cdot \right) $.
In this context it works on vector in Element Wise manner.

So the Pseudo Code (MATLAB) would look something like:

for ii = 1:numIterations
    vG = (mA.' * (mA * vX - vY)) + (paramLambda * sign(vX)); %&lt;! Sub Gradient
    vX = vX - (stepSize * vG); %&lt;! Sub Gradient Update
end


Yet, As mentioned by @David, this is not a modern way to solve this problem.
Iteration wise it seems that Coordinate Descent is the fastest method for solving the LASSO ($ {L}_{1} $ Regularized Least Square) problem.

You may look at a project I created (${L}_{1}$ Regularized Least Squares - Solvers Analysis) which compares many method for that optimization problem:



Implementation is included so you can see how it works.