Second-Order Approximation
We begin our derivation by considering the second order approximation of function f(x) at point x = xn that is given by:
[Tex]f(x) = f(x_n) + f'(x_n) (x-x_n) + \frac{1}{2!}f”(x_n) (x-x_n)^2[/Tex]
Now, rearranging the terms, we obtain
[Tex]f(x) = \frac{1}{2} f”(x_n) x^2 + [f'(x_n) – f”(x_n) x_n]x + [f(x_n) – f'(x_n) x_n + \frac{1}{2} f”(x_n) x_n^{2}][/Tex]
Next to find the value where the function is minimum, we compute the first derivative and equate it to zero to obtain the following:
[Tex]f”(x_n) x = f”(x_n) x_n – f'(x_n)[/Tex]
Finally, rearranging the terms, we obtain the update rule as
[Tex]x = x_n – \frac{f'(x_n)}{f”(x_n)}[/Tex]
Newton’s method in Machine Learning
Optimization algorithms are essential tools across various fields, ranging from engineering and computer science to economics and physics. Among these algorithms, Newton’s method holds a significant place due to its efficiency and effectiveness in finding the roots of equations and optimizing functions, here in this article we will study more about Newton’s method and it’s use in machine learning.
Table of Content
- Newton’s Method for Optimization
- Second-Order Approximation
- Newton’s Method for Finding Local Minima or Maxima in Python
- Convergence Properties of Newton’s Method
- Complexity of Newton’s Method
- Time Complexity of Newton’s Method
- Parameter Estimation in Logistic Regression using Newton’s Method
- Data Fitting with Newton’s Method
- Newton’s Method vs Other Optimization Algorithms
- Applications of Newton’s Method
Contact Us