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The problem seems to be focused on Ridge Regression with a regularization term involving a positive definite matrix \( S \). Let me break down the mathematical notation and steps to solve this problem.

### Given:
The loss function for ridge regression with the regularization term is:

\[
L(w) = \| y - Xw \|^2 + \lambda \| S w \|^2
\]

Where:
- \( w \) is the vector of weights,
- \( X \) is the matrix of input features,
- \( y \) is the vector of target values,
- \( \lambda \) is the regularization parameter,
- \( S \) is a fixed symmetric positive definite matrix.

### Steps to Solve:

1. **Expand the loss function**:
   The first term \( \| y - Xw \|^2 \) is the squared error, and the second term \( \| S w \|^2 \) is the regularization. Expanding both:
   
   \[
   L(w) = (y - Xw)^\top (y - Xw) + \lambda (Sw)^\top (Sw)
   \]
   
   \[
   L(w) = y^\top y - 2 y^\top Xw + w^\top X^\top Xw + \lambda w^\top S^\top S w
   \]

2. **Gradient Calculation**:
   To find the optimal weights \( w \), you need to compute the gradient \( \nabla_w L(w) \) and set it to zero. The gradient is:

   \[
   \nabla_w L(w) = -2X^\top y + 2X^\top Xw + 2\lambda S^\top S w
   \]

   Set the gradient to zero for minimization:

   \[
   -2X^\top y + 2X^\top Xw + 2\lambda S^\top S w = 0
   \]

3. **Solve for \( w \)**:
   Simplify the equation:

   \[
   X^\top Xw + \lambda S^\top S w = X^\top y
   \]

   \[
   (X^\top X + \lambda S^\top S)w = X^\top y
   \]

   The solution for \( w \) is:

   \[
   w = (X^\top X + \lambda S^\top S)^{-1} X^\top y
   \]

4. **Optimal Solution**:
   The optimal weights are given by the above expression. To ensure that this is a minimum, we need to check the second derivative, which involves verifying the Hessian matrix. Since the regularization term involves \( S \), which is positive definite, and the matrix \( X^\top X \) is symmetric and positive semi-definite, the function is convex, meaning the solution is indeed a minimum.

Let me know if you need further details on any step or explanation!

---

Here are five related questions you might explore further:
1. How does changing the matrix \( S \) affect the solution for \( w \)?
2. What happens to the solution when \( \lambda \to 0 \)?
3. How would you modify this solution for a LASSO regression?
4. How does the regularization term \( \| S w \|^2 \) differ from standard \( \| w \|^2 \)?
5. Can you derive the solution when \( X \) is not full-rank?

**Tip**: Always check the condition of the matrix \( X^\top X + \lambda S^\top S \) before inverting it. If it's ill-conditioned, consider numerical stability techniques.

We are training a ridge regression model with a modified regularization term involving a symmetric positive definite matrix. The loss function is given as L(w) = ||y - Xw||^2 + λ ||S w||^2. Find the gradient ∇w L(w) and determine the optimal value for w. Confirm that this solution is a true minimum.

Learn how to solve a ridge regression problem where the regularization term involves a symmetric positive definite matrix. This advanced problem covers optimization techniques, gradient computation, and matrix algebra, suitable for university-level studies.

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