Let A \in \mathbb{R}^{n \times n} be a constant matrix and b \in \mathbb{R}^{n} be a constant vector. Let z \in \mathbb{R}^{n}. Consider the function g(z) defined as
g(z)=\frac{1}{2}\|A z-b\|_{2}^{2}+\frac{1}{2}\|z\|_{2}^{2}
Consider the problem of finding the minimum of g(z) using gradient descent. Derive the optimal step size \alpha_{k} to be used at iteration number k of the gradient descent scheme assuming estimate z_{k-1} from the previous iteration.

To solve the optimization problem using gradient descent, we start by computing the gradient of the function \( g(z) \) with respect to \( z \).

### Step 1: Compute the gradient \( \nabla g(z) \)

Given the function:

\[
g(z) = \frac{1}{2}\|A z - b\|_{2}^{2} + \frac{1}{2}\|z\|_{2}^{2}
\]

We first need to express the norm terms explicitly:

\[
\|A z - b\|_{2}^{2} = (A z - b)^\top (A z - b)
\]

\[
\|z\|_{2}^{2} = z^\top z
\]

Thus, the function \( g(z) \) becomes:

\[
g(z) = \frac{1}{2}(A z - b)^\top (A z - b) + \frac{1}{2}z^\top z
\]

Now, we compute the gradient \( \nabla g(z) \):

\[
\nabla g(z) = \frac{\partial}{\partial z} \left[ \frac{1}{2}(A z - b)^\top (A z - b) + \frac{1}{2}z^\top z \right]
\]

### Step 2: Gradient calculation

Let's compute the gradient term by term.

**First term**: 

\[
\frac{\partial}{\partial z} \left[ \frac{1}{2}(A z - b)^\top (A z - b) \right]
\]

Expanding the quadratic form:

\[
(A z - b)^\top (A z - b) = z^\top A^\top A z - 2b^\top A z + b^\top b
\]

Taking the derivative with respect to \( z \):

\[
\frac{\partial}{\partial z} \left( z^\top A^\top A z - 2b^\top A z + b^\top b \right) = 2A^\top A z - 2A^\top b
\]

The factor of \( \frac{1}{2} \) cancels the 2, so:

\[
\frac{\partial}{\partial z} \left[ \frac{1}{2}(A z - b)^\top (A z - b) \right] = A^\top A z - A^\top b
\]

**Second term**:

\[
\frac{\partial}{\partial z} \left( \frac{1}{2}z^\top z \right) = z
\]

So the total gradient is:

\[
\nabla g(z) = A^\top A z - A^\top b + z
\]

### Step 3: Gradient descent update

The gradient descent update rule is:

\[
z_k = z_{k-1} - \alpha_k \nabla g(z_{k-1})
\]

Substitute the expression for the gradient:

\[
z_k = z_{k-1} - \alpha_k \left( A^\top A z_{k-1} - A^\top b + z_{k-1} \right)
\]

### Step 4: Derive the optimal step size \( \alpha_k \)

To find the optimal step size \( \alpha_k \), minimize \( g(z) \) along the direction of the negative gradient:

\[
\alpha_k = \arg\min_\alpha g(z_{k-1} - \alpha \nabla g(z_{k-1}))
\]

Substituting \( z = z_{k-1} - \alpha \nabla g(z_{k-1}) \) into \( g(z) \), we get:

\[
g(z_{k-1} - \alpha \nabla g(z_{k-1})) = \frac{1}{

Learn how to derive the optimal step size \( \alpha_k \) for minimizing a quadratic function using gradient descent. This problem involves computing gradients, matrix operations, and optimizing iterative solutions.

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