# What is Quasi-Newton methods

##### Quasi-Newton Methods in Optimization

Introduction

Quasi-Newton methods are an important class of optimization algorithms that are widely used in machine learning and artificial intelligence. These methods are particularly useful in situations where it is difficult or impossible to calculate the exact gradient of a function. In this article, we will explore the key features of Quasi-Newton methods and their applications in optimization.

What are Quasi-Newton methods?

Quasi-Newton methods are iterative optimization algorithms that use an approximation of the Hessian matrix to update the estimate of the minimum of a function. The Hessian matrix is the second derivative matrix of a function, and it can be used to determine the curvature of the function at a given point. Quasi-Newton methods are based on the idea of approximating the Hessian matrix using gradient information to make step-size decisions. In contrast, Newton's method is an exact optimization algorithm that calculates the Hessian matrix at each step.

How do Quasi-Newton methods work?

Quasi-Newton methods work by using an approximation of the Hessian matrix to update the estimate of the minimum of a function. The approximation takes the form of an inverse Hessian matrix, which is iteratively updated using gradient information. The basic steps of Quasi-Newton methods are as follows:

• Choose an initial guess for the minimum of the function.
• Calculate the gradient of the function at the initial guess.
• Approximate the Hessian matrix using the gradient information.
• Use the inverse of the approximated Hessian matrix to update the estimate of the minimum of the function.
• Repeat steps 2-4 until convergence.

Advantages of Quasi-Newton methods

Quasi-Newton methods have several advantages over other optimization algorithms:

• They do not require the calculation of second derivatives, which can be computationally expensive.
• They can handle large-scale optimization problems.
• They tend to converge quickly.
• They are robust to noise in the gradient information.

Applications of Quasi-Newton methods

Quasi-Newton methods are widely used in machine learning and deep learning for a variety of applications, including:

• Training neural networks.
• Optimizing loss functions.
• Finding the minimum of probability distributions.
• Fitting statistical models.

Types of Quasi-Newton methods

There are several types of Quasi-Newton methods, including:

• The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.
• The Davidon-Fletcher-Powell (DFP) method.
• The limited-memory BFGS (L-BFGS) method.

Broyden-Fletcher-Goldfarb-Shanno (BFGS) method

The BFGS method is one of the most popular Quasi-Newton methods. It is an iterative algorithm that uses an approximation of the Hessian matrix to update the estimate of the minimum of a function. The approximation takes the form of an inverse Hessian matrix, which is iteratively updated using gradient information. The BFGS method has several advantages over other optimization algorithms:

• It does not require the calculation of second derivatives, which can be computationally expensive.
• It can handle large-scale optimization problems.
• It tends to converge quickly.
• It is robust to noise in the gradient information.

Davidon-Fletcher-Powell (DFP) method

The DFP method is another popular Quasi-Newton method. It is an iterative algorithm that uses an approximation of the Hessian matrix to update the estimate of the minimum of a function. The approximation takes the form of an inverse Hessian matrix, which is iteratively updated using gradient information. The DFP method has several advantages over other optimization algorithms:

• It does not require the calculation of second derivatives, which can be computationally expensive.
• It can handle large-scale optimization problems.
• It tends to converge quickly.
• It is robust to noise in the gradient information.

Limited-memory BFGS (L-BFGS) method

The L-BFGS method is a variant of the BFGS method that is specifically designed for large-scale problems. It is an iterative algorithm that uses an approximation of the Hessian matrix to update the estimate of the minimum of a function. The approximation takes the form of an inverse Hessian matrix, which is iteratively updated using gradient information. The L-BFGS method has several advantages over other optimization algorithms:

• It can handle large-scale optimization problems.
• It requires less memory than the BFGS method.
• It tends to converge quickly.
• It is robust to noise in the gradient information.

Conclusion

Quasi-Newton methods are an important class of optimization algorithms that are widely used in machine learning and artificial intelligence. These methods have several advantages over other optimization algorithms, including the ability to handle large-scale optimization problems and to converge quickly. The BFGS, DFP, and L-BFGS methods are popular types of Quasi-Newton methods that are used in a variety of applications, including training neural networks, optimizing loss functions, finding the minimum of probability distributions, and fitting statistical models.