Below is the pseudocode for minimizing the function ( f(x) = 0.5 \cdot |Ax – b|^2 ) using gradient-based optimization techniques, along with explanations for each line.

function gradient_descent(A, b, x_init, learning_rate, max_iter):
# Initialize x with the starting point
x = x_init

# Iterate for a maximum number of iterations
for i from 1 to max_iter:
# Compute the residual (error) vector
residual = A * x – b

# Compute the gradient of f(x)
gradient = A^T * residual # A^T is the transpose of A

# Update x using the gradient and learning rate
x = x – learning_rate * gradient

# Optionally: Check for convergence (not shown here)

return x # Return the optimized x

Explanation of Each Line

1. Function Declaration:

function gradient_descent(A, b, x_init, learning_rate, max_iter):

This line defines a function called gradient_descent that takes in the matrix ( A ), vector ( b ), an initial guess for ( x ) (x_init), a learning_rate, and a maximum number of iterations (max_iter).

2. Initialize x:

x = x_init

The variable x is initialized with the provided starting point x_init. This is where the optimization process begins.

3. Loop for Maximum Iterations:

for i from 1 to max_iter:

This line sets up a loop that will iterate a maximum number of times defined by max_iter, allowing the algorithm to refine its solution.

4. Compute Residual:

residual = A * x – b

The residual vector is computed as ( Ax – b ). This vector represents how far off our current estimate ( x ) is from the actual target defined by ( b ).

5. Compute Gradient:

gradient = A^T * residual # A^T is the transpose of A

The gradient of the function ( f(x) ) is calculated here. The gradient indicates the direction and rate of steepest ascent of the function. We use the transpose of ( A ) multiplied by the residual to compute this.

6. Update x:

x = x – learning_rate * gradient

The update rule for ( x ) is applied here. The current estimate of ( x ) is adjusted in the direction opposite to that of the gradient (indicating descent), scaled by the learning_rate. This step helps in reducing the function value iteratively.

7. Return Optimized x:

return x # Return the optimized x

After completing all iterations or achieving convergence, the function returns the optimized value of ( x ).

Additional Notes

– Convergence Check: In practice, it’s common to include a convergence check (not shown in this pseudocode) to stop iterations if changes in ( x ) are smaller than a certain threshold.
– Learning Rate: The choice of learning_rate is critical; if it’s too large, the algorithm may overshoot the minimum, while if it’s too small, convergence may be very slow.
– Matrix Operations: The pseudocode assumes that appropriate matrix-vector multiplications are defined in your programming environment.

This pseudocode provides a foundational understanding of how gradient-based optimization works for minimizing functions like the one presented in your question.