In my research on neural networks and learning theory I have recently come across the following problem dealing with gradient descent:
We consider a given column vector $ x=[x_1,x_2,...,x_{d}]^T \in \mathbb{R}^{d} $ and we define a general column vector $ w = [w_1,w_2,...,w_{d}]^T \in \mathbb{R}^{d} $, and we define the following function of w only as x is given $f(w)=\max{\{w^Tx,0\}}$ and we define the loss function in terms of the weight $ MSE(w) = (y-f(w))^2 $ where y is a constant know to be either 0 or 1. We denote by $ t \in \mathbb{N} $ the index of iteration and we optimize the weights via gradient descent: $ w^{(t+1)}=w^{(t)}-\frac{1}{t}(\nabla MSE(W))|_{W=W^{(t)}} $ where the gradient of the MSE function is with respect to the w vector which is the only variable in the problem, and we know of course that these iterations converge to a critical point where the gradient is defined and zero, or where the gradient is undefined when the max function is not differentiable where $w^Tx=0$
I was wondering if someone could please tell me if there is some way to determine via the initial setting of the $w$ vector if we converge to a gradient-zero point or a non-differentiable point? Is there a criterion of sorts on the initial choosing of $w$ to help us determine which regions in $ \mathbb{R}^{d} $ converge to which sort of points? Maybe starting with scalar w values may simplify the problem at hand? I thank all helpers.
