Read carefully, section 4.3 Gradient-Based Optimization (pages 79 to 83) in "Goodfellow, I., Bengio, Y., Courville, A. (2016). Deep Learning. United Kingdom: MIT Press." Then:
Submit the algorithm in pseudocode (or any computer language) minimize vector x using gradient-based optimization: f(x) = 0.5 * ||A * x b||^2, where A, x, and b are some vectors.
Explain, in short, each line of the pseudocode.
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.