Newton’s Method for Optimization
Newton’s method can be extended to solve optimization problems by finding the minima or maxima of a real-valued function f(x). The goal of optimization is to find the value of x that minimizes or maximizes the function f(x). We are interested in finding critical points of the function where the first derivative is zero (for minima or maxima). Newton’s method utilizes the first and second derivatives of the function to iteratively refine the solution.
The iterative formula for Newton’s method is given by:
[Tex]x_{n+1} = x_n – \frac{f'(x_n)}{f”(x_n)}[/Tex]
where [Tex]x_{n+1}[/Tex] is the next approximation of the critical point, [Tex]x_n[/Tex] is the current approximation, [Tex]f'(x_n) [/Tex]is the first derivative of the function at [Tex]x_n[/Tex] and [Tex]f”(x_n) [/Tex]is the second order derivative (Hessian) of the function at [Tex]x_n[/Tex].
Intuitive Understanding of Newton’s Method
Intuitively, Newton’s method can be understood as follows:
- At each iteration, the method updates the current approximation [Tex]x_n [/Tex]by subtracting the ratio of the gradient [Tex]f'(x_n) [/Tex]and the curvature[Tex] f”(x_n).[/Tex]
- If the curvature is positive (convex function), this ratio decreases the value of x, bringing it closer to the minimum.
- If the curvature is negative (concave function), this ratio increases the value of x, bringing it closer to the maximum.
The process continues iteratively until a stopping criterion is met or a desired convergence is achieved.
We must keep in mind that Newton’s method may not always converge, especially if the initial guess is far from the true root (or critical point) or if the function has complex behavior (e.g., oscillations, multiple roots). We must be careful when dealing with functions with singularities or regions where the derivative approaches zero.
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