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addex exercise week 37

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TITLE: Exercises week 36
TITLE: Exercises week 37
AUTHOR: Implementing gradient descent for Ridge and ordinary Least Squares Regression
DATE: September 8-12, 2025
@@ -11,7 +11,149 @@ o Be able to compare the analytical expressions for OLS and Rudge regression wit
o Explore the role of the learning rate in the gradient descent approach and the hyperparameter $\lambda$ in Ridge regression
o Scale the data properly
===== Ridge regression and a new Synthetic Dataset =====
===== Simple one-dimensional second-order polynomial =====
We start with a very simple function
!bt
\[
\f(x)= 2-x+5x^2,
\]
!et
defined for $x\in [-2,2]$. You can add noise if you wish.
We are going to fit this function with a polynomial ansatz. The easiest thing is to set up a second-order polynomial and see if you can fit the above function.
Feel free to play around with higher-order polynomials.
===== Exercise 1, scale your data =====
Before fitting a regression model, it is good practice to normalize or
standardize the features. This ensures all features are on a
comparable scale, which is especially important when using
regularization. Here we will perform standardization, scaling each
feature to have mean 0 and standard deviation 1.
=== 1a) ===
Compute the mean and standard deviation of each column (feature) in your design/feature matrix $\bm{X}$.
Subtract the mean and divide by the standard deviation for each feature.
We will also center the target $\bm{y}$ to mean $0$. Centering $\bm{y}$
(and each feature) means the model does not require a separate intercept
term, the data is shifted such that the intercept is effectively 0
. (In practice, one could include an intercept in the model and not
penalize it, but here we simplify by centering.)
Choose $n=100$ data points and set up $\bm{x}, $\bm{y} and the design matrix $\bm{X}$.
!bc pycod
# Standardize features (zero mean, unit variance for each feature)
X_mean = X.mean(axis=0)
X_std = X.std(axis=0)
X_std[X_std == 0] = 1 # safeguard to avoid division by zero for constant features
X_norm = (X - X_mean) / X_std
# Center the target to zero mean (optional, to simplify intercept handling)
y_mean = ?
y_centered = ?
!ec
Fill in the necessary details.
After this preprocessing, each column of $\bm{X}_{\mathrm{norm}}$ has mean zero and standard deviation $1$
and $\bm{y}_{\mathrm{centered}}$ has mean 0. This makes the optimization landscape
nicer and ensures the regularization penalty $\lambda \sum_j
\theta_j^2$ in Ridge regression treats each coefficient fairly (since features are on the
same scale).
===== Exercise 2, calculate the gradients =====
Find the gradients for OLS and Ridge regression using the mean-squared error as cost/loss function.
===== Exercise 3, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\bm{\theta}$ =====
!bc pycod
# Set regularization parameter, either a single value or a vector of values
lambda = ?
# Analytical form for OLS and Ridge solution: theta_Ridge = (X^T X + lambda * I)^{-1} X^T y and theta_OLS = (X^T X)^{-1} X^T y
I = np.eye(n_features)
theta_closed_formRidge = ?
theta_closed_formOLS = ?
print("Closed-form Ridge coefficients:", theta_closed_form)
print("Closed-form OLS coefficients:", theta_closed_form)
!ec
This computes the Ridge and OLS regression coefficients directly. The identity
matrix $I$ has the same size as $X^T X$. It adds $\lambda$ to the diagonal of $X^T X for Ridge regression. We
then invert this matrix and multiply by $X^T y$. The result
for $\bm{\theta}$ is a NumPy array of shape (n$\_$features,) containing the
fitted parameters $\bm{\theta}$..
=== 3a) ===
Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\bm{\theta}$.
=== 3b) ===
Explore the results as function of different values of the hyperparameter $\lambda$. See for example exercise 4 from week 36.
===== Exercise 4, Implementing the simplest form for gradient descent =====
Alternatively, we can fit the ridge regression model using gradient
descent. This is useful to visualize the iterative convergence and is
necessary if $n$ and $p$ are so large that the closed-form might be
too slow or memory-intensive. We derive the gradients from the cost
functions defined above. Use the gradients of the Ridge and OLS cost functions with respect to
the parameters $\bm{\theta}$ and set up (using the template below) your own gradient descent code for OLS and Ridge regression.
Below is a template code for gradient descent implementation of ridge:
!bc pycod
# Gradient descent parameters, learning rate eta first
eta = 0.1
# Then number of iterations
num_iters = 1000
# Initialize weights for gradient descent
theta = np.zeros(n_features)
# Arrays to store history for plotting
cost_history = np.zeros(num_iters)
# Gradient descent loop
m = n_samples # number of examples
for t in range(num_iters):
# Compute prediction error
error = X_norm.dot(theta) - y_centered
# Compute cost for OLS and Ridge (MSE + regularization for Ridge) for monitoring
cost_OLS = ?
cost_Ridge = ?
cost_history[t] = ?
# Compute gradients for OSL and Ridge
grad_OLS = ?
grad_Ridge = ?
# Update parameters theta
theta_gdOLS = ?
theta_gdRidge = ?
# After the loop, theta contains the fitted coefficients
theta_gdOLS = ?
theta_gdRidge = ?
print("Gradient Descent OLS coefficients:", theta_gdOLS)
print("Gradient Descent Ridge coefficients:", theta_gdRidge)
!ec
=== 4a) ===
Discuss the results as function of the learning rate parameters and the number of iterations.
=== 4b) ===
Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?
===== Exercise 5, Ridge regression and a new Synthetic Dataset =====
We create a synthetic linear regression dataset with a sparse
@@ -62,128 +204,8 @@ y \approx 5 \times x_0 \;-\; 3 \times x_1 \;+\; 2 \times x_6 \;+\; \text{noise}.
!et
You can remove the noise if you wish to.
===== Exercise 1, scale your data =====
Before fitting a regression model, it is good practice to normalize or
standardize the features. This ensures all features are on a
comparable scale, which is especially important when using
regularization. Here we will perform standardization, scaling each
feature to have mean 0 and standard deviation 1.
=== 1a) ===
Compute the mean and standard deviation of each column (feature) in $\bm{X}$.
Subtract the mean and divide by the standard deviation for each feature.
We will also center the target $\bm{y}$ to mean $0$. Centering $\bm{y}$
(and each feature) means the model does not require a separate intercept
term, the data is shifted such that the intercept is effectively 0
. (In practice, one could include an intercept in the model and not
penalize it, but here we simplify by centering.)
!bc pycod
# Standardize features (zero mean, unit variance for each feature)
X_mean = X.mean(axis=0)
X_std = X.std(axis=0)
X_std[X_std == 0] = 1 # safeguard to avoid division by zero for constant features
X_norm = (X - X_mean) / X_std
# Center the target to zero mean (optional, to simplify intercept handling)
y_mean = ?
y_centered = ?
!ec
Fill in the necessary details.
After this preprocessing, each column of $\bm{X}_{\mathrm{norm}}$ has mean zero and standard deviation $1$
and $\bm{y}_{\mathrm{centered}}$ has mean 0. This makes the optimization landscape
nicer and ensures the regularization penalty $\lambda \sum_j
\theta_j^2$ in Ridge regression treats each coefficient fairly (since features are on the
same scale).
===== Exercise 2, use the analytical formulae for OLS and Ridge regression to find the optimal paramters $\bm{\theta}$ =====
!bc pycod
# Set regularization parameter, either a single value or a vector of values
lambda = ?
# Analytical form for OLS and Ridge solution: theta_Ridge = (X^T X + lambda * I)^{-1} X^T y and theta_OLS = (X^T X)^{-1} X^T y
I = np.eye(n_features)
theta_closed_formRidge = ?
theta_closed_formOLS = ?
print("Closed-form Ridge coefficients:", theta_closed_form)
print("Closed-form OLS coefficients:", theta_closed_form)
!ec
This computes the Ridge and OLS regression coefficients directly. The identity
matrix $I$ has the same size as $X^T X$. It adds $\lambda$ to the diagonal of $X^T X for Ridge regression. We
then invert this matrix and multiply by $X^T y$. The result
for $\bm{\theta}$ is a NumPy array of shape (n$\_$features,) containing the
fitted parameters $\bm{\theta}$..
=== 2a) ===
Finalize, in the above code, the OLS and Ridge regression determination of the optimal parameters $\bm{\theta}$.
=== 2b) ===
Explore the results as function of different values of the hyperparameter $\lambda$. See for example exercise 4 from week 36.
===== Exercise 3, Implementing the simplest form for gradient descent =====
Alternatively, we can fit the ridge regression model using gradient
descent. This is useful to visualize the iterative convergence and is
necessary if $n$ and $p$ are so large that the closed-form might be
too slow or memory-intensive. We derive the gradients from the cost
functions defined above. Use the gradients of the Ridge and OLS cost functions with respect to
the parameters $\bm{\theta}$ and set up (using the template below) your own gradient descent code for OLS and Ridge regression.
Below is a template code for gradient descent implementation of ridge:
!bc pycod
# Gradient descent parameters, learning rate eta first
eta = 0.1
# Then number of iterations
num_iters = 1000
# Initialize weights for gradient descent
theta = np.zeros(n_features)
# Arrays to store history for plotting
cost_history = np.zeros(num_iters)
# Gradient descent loop
m = n_samples # number of examples
for t in range(num_iters):
# Compute prediction error
error = X_norm.dot(theta) - y_centered
# Compute cost for OLS and Ridge (MSE + regularization for Ridge) for monitoring
cost_OLS = ?
cost_Ridge = ?
cost_history[t] = ?
# Compute gradients for OSL and Ridge
grad_OLS = ?
grad_Ridge = ?
# Update parameters theta
theta_gdOLS = ?
theta_gdRidge = ?
# After the loop, theta contains the fitted coefficients
theta_gdOLS = ?
theta_gdRidge = ?
print("Gradient Descent OLS coefficients:", theta_gdOLS)
print("Gradient Descent Ridge coefficients:", theta_gdRidge)
!ec
=== 3a) ===
Discuss the results as function of the learning rate parameters and the number of iterations.
=== 3b) ===
Try to add a stopping parameter as function of the number iterations. How would you define a stopping criterion?
Try to fit the above data set using OLS and Ridge regression with the analytical expressions and your own gradient descent codes.
If everything worked correctly, the learned coefficients should be
close to the true values [5.0, -3.0, 0.0, …, 2.0, …] that we used to