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 find the optimal step size \(\alpha_k\) for the gradient descent scheme to minimize the function \( g(z) \), we start by analyzing the function and its gradient.

### Step 1: Define the function and compute the gradient
The function \( g(z) \) is defined as:
\[
g(z) = \frac{1}{2} \|Az - b\|_2^2 + \frac{1}{2} \|z\|_2^2
\]
Expanding the norm terms, we have:
\[
g(z) = \frac{1}{2} (Az - b)^\top (Az - b) + \frac{1}{2} z^\top z
\]

Now, let’s compute the gradient of \(g(z)\) with respect to \(z\):
\[
\nabla g(z) = \nabla \left( \frac{1}{2} \|Az - b\|_2^2 + \frac{1}{2} \|z\|_2^2 \right)
\]

For the first term:
\[
\nabla \left( \frac{1}{2} \|Az - b\|_2^2 \right) = \nabla \left( \frac{1}{2} (Az - b)^\top (Az - b) \right)
\]
Using the chain rule, we have:
\[
\nabla \left( \frac{1}{2} (Az - b)^\top (Az - b) \right) = A^\top (Az - b)
\]

For the second term:
\[
\nabla \left( \frac{1}{2} \|z\|_2^2 \right) = \nabla \left( \frac{1}{2} z^\top z \right) = z
\]

Thus, the gradient of \(g(z)\) is:
\[
\nabla g(z) = A^\top (Az - b) + z
\]

### Step 2: Gradient Descent Update Rule
The gradient descent update rule at iteration \(k\) is:
\[
z_k = z_{k-1} - \alpha_k \nabla g(z_{k-1})
\]
Substituting the gradient:
\[
z_k = z_{k-1} - \alpha_k \left( A^\top (A z_{k-1} - b) + z_{k-1} \right)
\]

### Step 3: Find the Optimal Step Size \(\alpha_k\)
To find the optimal step size \(\alpha_k\), we minimize \(g(z)\) along the direction of the negative gradient. Define the univariate function:
\[
h(\alpha) = g(z_{k-1} - \alpha \nabla g(z_{k-1}))
\]
Differentiate \(h(\alpha)\) with respect to \(\alpha\) and set it to zero to find the optimal \(\alpha_k\).

Substitute \(z = z_{k-1} - \alpha \nabla g(z_{k-1})\) into \(g(z)\):
\[
h(\alpha) = \frac{1}{2} \|A(z_{k-1} - \alpha \nabla g(z_{k-1})) - b\|_2^2 + \frac{1}{2} \|z_{k-1} - \alpha \nabla g(z_{k-1})\|_2^2
\]
Expanding the norm:
\[
h(\alpha) = \frac{1}{2} \left( Az_{k-1} - \alpha A \nabla g(z_{k-1}) - b \right)^\top \left( Az_{k-1} - \alpha A \nabla g(z_{k-1}) - b \right)
+ \frac{1}{2} \left( z_{k-1} - \alpha \nabla g(z_{k-1}) \right)^\top \left( z_{k-1} - \alpha \nabla g(z_{k-1}) \right)
\]
Taking the derivative with respect to \(\alpha\) and setting it to zero:
\[
\frac{d h(\alpha)}{d \alpha} = -\nabla g(z_{k-1})^\top \left( A^\top (Az_{k-1} - b) + z_{k-1} \right) + \alpha \nabla g(z_{k-1})^\top \left( A^\top A + I \right) \nabla g(z_{k-1}) = 0
\]

Thus, the optimal step size \(\alpha_k\) is:
\[
\alpha_k = \frac{\nabla g(z_{k-1})^\top \nabla g(z_{k-1})}{\nabla g(z_{k-1})^\top (A^\top A + I) \nabla g(z_{k-1})}
\]

This expression provides the optimal step size \(\alpha_k\) for each iteration in the gradient descent method.

Would you like more details on any part of the derivation, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How does the convergence rate of gradient descent change with different step sizes?
2. Can the optimal step size \(\alpha_k\) be constant for all iterations, and under what conditions?
3. How does the matrix \(A\) affect the optimization problem and the gradient descent convergence?
4. What are the implications of choosing a step size \(\alpha_k\) that is too large or too small?
5. How would you apply a line search method to determine the step size in a more general setting?

**Tip:** When implementing gradient descent, it’s important to monitor the objective function's value at each step to ensure the algorithm is converging as expected.

Explore how to find the optimal step size in gradient descent to minimize a quadratic function. This detailed solution covers matrix algebra, gradient descent principles, and vector space norms. Ideal for advanced undergraduate students and practitioners interested in optimization techniques.

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