Time Complexity of Newton’s Method
- Computing Gradient and Hessian: Computing the gradient typically requires O(n) operations for a function with n variables. Computing the Hessian involves O(n^2) operations for a function with n variables. However, if the Hessian has a specific structure (e.g., it’s sparse), specialized algorithms can reduce this complexity.
- Solving Linear System: In each iteration, Newton’s method involves solving a linear system, usually by methods like Gaussian elimination, LU decomposition, or iterative solvers like conjugate gradient descent. Solving a dense linear system typically requires O(n^3) operations, but this can be reduced to O(n^1.5) for certain specialized methods. If the Hessian is sparse, specialized solvers for sparse linear systems can be employed, potentially reducing the complexity significantly.
- Number of Iterations: The number of iterations required for convergence varies depending on factors such as the chosen optimization tolerance, the curvature of the function, and the choice of initial guess. In ideal conditions, Newton’s method converges quadratically.
So, the total time complexity of Newton’s method, after considering the cost per iteration and the number of iterations can be approximated as O(k⋅T), where k is the number of iterations and T is the complexity per iteration.
However, the actual time complexity can vary significantly based on the specific characteristics of the optimization problem, including the size of the problem (number of variables), the sparsity of the Hessian, and the computational efficiency of the algorithms used for gradient and Hessian computations and linear system solving.
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
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