What is the difference between Gradient Descent and Newton’s Gradient Descent?

Gradient Descent vs Newton’s Method: A Comparative Study

In the realm of optimization, finding the minima (or maxima) of a function is a fundamental task. Gradient descent and Newton’s method are two widely used algorithms that tackle this problem, each with its strengths and weaknesses. This article delves into the differences between these methods, highlighting their key features and providing insights into their applications.

Gradient Descent

Gradient descent is an iterative optimization algorithm that starts with an initial guess for the minimum and iteratively moves towards the minimum by taking steps proportional to the negative of the gradient of the function.

Algorithm:

• Initialize a starting point x₀
• Repeat until convergence:
  • Calculate the gradient: ∇f(x_i)
  • Update the point: x_i+1 = x_i – α∇f(x_i)

Where:

• α is the learning rate, which controls the step size.
• ∇f(x) represents the gradient of the function f(x).

Advantages:

• Simple to implement.
• Can be applied to a wide range of functions.

Disadvantages:

• Can get stuck in local minima.
• Convergence can be slow for complex functions.
• Requires careful tuning of the learning rate.

Newton’s Method

Newton’s method, also known as Newton-Raphson method, is a second-order optimization algorithm that uses the Hessian matrix (matrix of second-order partial derivatives) to approximate the function’s curvature. This allows it to take larger and more informed steps towards the minimum.

Algorithm:

• Initialize a starting point x₀
• Repeat until convergence:
  • Calculate the gradient: ∇f(x_i)
  • Calculate the Hessian matrix: H(x_i)
  • Update the point: x_i+1 = x_i – H(x_i)^-1∇f(x_i)

Advantages:

• Often converges faster than gradient descent.
• Can escape local minima more effectively.

Disadvantages:

• More complex to implement than gradient descent.
• Requires the Hessian matrix to be invertible.
• Can be computationally expensive for high-dimensional problems.

Comparison

Example:

Let’s consider finding the minimum of the function f(x) = x².

Gradient Descent:

def gradient_descent(x, learning_rate):
  gradient = 2*x
  new_x = x - learning_rate*gradient
  return new_x

x = 10
learning_rate = 0.1

for i in range(10):
  x = gradient_descent(x, learning_rate)
  print(f'Iteration {i+1}: x = {x}')

Newton’s Method:

def newtons_method(x):
  gradient = 2*x
  hessian = 2
  new_x = x - (1/hessian)*gradient
  return new_x

x = 10

for i in range(10):
  x = newtons_method(x)
  print(f'Iteration {i+1}: x = {x}')

You can observe that Newton’s method converges much faster to the minimum (x=0) compared to gradient descent.

Conclusion

Both gradient descent and Newton’s method are powerful optimization techniques. Gradient descent is a good choice for its simplicity and applicability to a wide range of problems. Newton’s method excels in its faster convergence rate but comes with increased complexity. The choice between the two depends on factors such as the complexity of the function, computational resources, and the desired level of accuracy.