5.2 Quasi-Newton methods

Quasi-Newton methods are powerful optimization algorithms that balance speed and efficiency. They approximate the Hessian matrix using gradient information, avoiding costly computations while maintaining fast convergence.

These methods, including BFGS and L-BFGS, are widely used in machine learning, data fitting, and scientific computing. They offer superlinear convergence and can handle large-scale problems, making them valuable tools for solving complex optimization tasks.

Quasi-Newton methods overview

• Quasi-Newton methods are iterative optimization algorithms used to solve nonlinear optimization problems
• These methods approximate the Hessian matrix or its inverse using the gradient information from previous iterations
• Quasi-Newton methods strike a balance between the fast convergence of Newton's method and the low computational cost of gradient descent methods

Motivation for Quasi-Newton methods

• Newton's method requires the computation and inversion of the Hessian matrix at each iteration, which can be computationally expensive for high-dimensional problems
• Quasi-Newton methods aim to reduce the computational cost by approximating the Hessian matrix or its inverse using gradient information
• These methods maintain the superlinear convergence rate of Newton's method while avoiding the explicit computation and inversion of the Hessian matrix

Secant method in one dimension

• The secant method is a root-finding algorithm that uses a linear approximation to find the root of a function
• It requires two initial points and iteratively updates the approximation based on the secant line passing through the two most recent points
• The secant method has a superlinear convergence rate, typically around 1.618 (the golden ratio)

Secant method vs Newton's method

Top images from around the web for Secant method vs Newton's method

• 

  Secant method - Wikipedia View original

  Is this image relevant?

• 

  Newton’s Method · Calculus View original

  Is this image relevant?

• 

  SecantMethodFindRoot | Wolfram Function Repository View original

  Is this image relevant?

• 

  Secant method - Wikipedia View original

  Is this image relevant?

• 

  Newton’s Method · Calculus View original

  Is this image relevant?

1 of 3

Top images from around the web for Secant method vs Newton's method

• 

  Secant method - Wikipedia View original

  Is this image relevant?

• 

  Newton’s Method · Calculus View original

  Is this image relevant?

• 

  SecantMethodFindRoot | Wolfram Function Repository View original

  Is this image relevant?

• 

  Secant method - Wikipedia View original

  Is this image relevant?

• 

  Newton’s Method · Calculus View original

  Is this image relevant?

1 of 3

• The secant method approximates the derivative using finite differences, while Newton's method requires the explicit computation of the derivative
• The secant method uses two points to approximate the derivative, while Newton's method uses a single point and the derivative at that point
• The secant method has a slightly slower convergence rate compared to Newton's method but avoids the need for explicit derivative calculations

Broyden's method for multiple dimensions

• Broyden's method is a generalization of the secant method for solving systems of nonlinear equations in multiple dimensions
• It updates the approximate Jacobian matrix using the secant equation, which relates the change in the function values to the change in the variables
• Broyden's method maintains an approximation of the Jacobian matrix, avoiding the need for explicit computation of the Jacobian at each iteration

Broyden's method algorithm

1. Choose an initial point $x_0$ and an initial approximation of the Jacobian matrix $B_0$
2. For $k = 0, 1, 2, \ldots$ until convergence:
   • Solve the linear system $B_k s_k = -F(x_k)$ for $s_k$
   • Set $x_{k+1} = x_k + s_k$
   • Update the approximate Jacobian matrix $B_{k+1}$ using the secant equation
3. Return the final approximation $x_{k+1}$

Broyden's method convergence

• Broyden's method has a superlinear convergence rate under certain conditions
• The convergence rate depends on the accuracy of the initial Jacobian approximation and the smoothness of the function
• Broyden's method may not converge if the initial approximation is too far from the solution or if the function is not sufficiently smooth

BFGS method

• The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is a popular Quasi-Newton method for unconstrained optimization
• It updates an approximation of the inverse Hessian matrix using the gradient information from the previous iteration
• The BFGS method has a superlinear convergence rate and is considered one of the most efficient Quasi-Newton methods

BFGS method algorithm

1. Choose an initial point $x_0$ and an initial approximation of the inverse Hessian matrix $H_0$ (usually the identity matrix)
2. For $k = 0, 1, 2, \ldots$ until convergence:
   • Compute the search direction $p_k = -H_k \nabla f(x_k)$
   • Perform a line search to find a step size $\alpha_k$ that satisfies the Wolfe conditions
   • Set $x_{k+1} = x_k + \alpha_k p_k$
   • Update the approximate inverse Hessian matrix $H_{k+1}$ using the BFGS update formula
3. Return the final approximation $x_{k+1}$

BFGS method convergence

• The BFGS method has a superlinear convergence rate, typically converging faster than the steepest descent and conjugate gradient methods
• The convergence rate is influenced by the accuracy of the initial inverse Hessian approximation and the conditioning of the problem
• The BFGS method may not converge if the function is not sufficiently smooth or if the initial approximation is too far from the solution

BFGS method vs Broyden's method

• The BFGS method is specifically designed for optimization problems, while Broyden's method is a general-purpose method for solving systems of nonlinear equations
• The BFGS method updates an approximation of the inverse Hessian matrix, while Broyden's method updates an approximation of the Jacobian matrix
• The BFGS method typically converges faster than Broyden's method for optimization problems due to its specialized update formula

Limited-memory BFGS (L-BFGS) method

• The L-BFGS method is a variant of the BFGS method that is designed for large-scale optimization problems
• It stores a limited number of vectors from previous iterations to approximate the inverse Hessian matrix implicitly
• The L-BFGS method has a lower memory footprint compared to the standard BFGS method, making it suitable for problems with a large number of variables

L-BFGS method algorithm

1. Choose an initial point $x_0$, an initial approximation of the inverse Hessian matrix $H_0$ (usually the identity matrix), and the number of stored vectors $m$
2. For $k = 0, 1, 2, \ldots$ until convergence:
   • Compute the search direction $p_k$ using the L-BFGS two-loop recursion
   • Perform a line search to find a step size $\alpha_k$ that satisfies the Wolfe conditions
   • Set $x_{k+1} = x_k + \alpha_k p_k$
   • Update the stored vectors using the gradient and variable differences
3. Return the final approximation $x_{k+1}$

L-BFGS method vs BFGS method

• The L-BFGS method uses a limited amount of memory to approximate the inverse Hessian matrix, while the BFGS method stores a full approximation
• The L-BFGS method is more memory-efficient and suitable for large-scale problems, while the BFGS method may be faster for smaller problems
• The convergence rate of the L-BFGS method is similar to that of the BFGS method, but it may require more iterations due to the limited-memory approximation

Quasi-Newton methods in optimization

• Quasi-Newton methods are widely used in optimization due to their fast convergence and ability to handle large-scale problems
• They are particularly effective for unconstrained optimization problems where the objective function is smooth and well-conditioned
• Quasi-Newton methods can also be adapted for constrained optimization problems by incorporating constraint handling techniques

Quasi-Newton methods for unconstrained optimization

• Unconstrained optimization problems involve minimizing or maximizing an objective function without any constraints on the variables
• Quasi-Newton methods, such as the BFGS and L-BFGS methods, are highly effective for unconstrained optimization
• These methods approximate the curvature information of the objective function using gradient differences, leading to faster convergence compared to first-order methods

Quasi-Newton methods for constrained optimization

• Constrained optimization problems involve minimizing or maximizing an objective function subject to equality and/or inequality constraints
• Quasi-Newton methods can be extended to handle constrained optimization problems by incorporating constraint handling techniques
• Popular approaches include the sequential quadratic programming (SQP) method and the interior-point method, which use Quasi-Newton approximations of the Hessian matrix

Convergence analysis of Quasi-Newton methods

• Convergence analysis studies the theoretical properties and convergence rates of Quasi-Newton methods
• It provides insights into the conditions under which these methods converge and the factors that influence their convergence speed
• Convergence analysis helps in understanding the strengths and limitations of Quasi-Newton methods and guides the selection of appropriate methods for specific problems

Local convergence of Quasi-Newton methods

• Local convergence analysis focuses on the behavior of Quasi-Newton methods in the vicinity of a solution
• Under certain assumptions, such as sufficient smoothness of the objective function and positive definiteness of the Hessian matrix, Quasi-Newton methods exhibit superlinear convergence
• The local convergence rate depends on the accuracy of the Hessian approximation and the conditioning of the problem

Global convergence of Quasi-Newton methods

• Global convergence analysis studies the convergence of Quasi-Newton methods from any starting point
• Quasi-Newton methods are not guaranteed to converge globally without additional safeguards
• Techniques such as line search methods and trust-region methods can be used to ensure global convergence
• Global convergence proofs often rely on the assumption of a bounded level set and the satisfaction of the Wolfe conditions

Advantages and disadvantages of Quasi-Newton methods

• Quasi-Newton methods offer several advantages over other optimization methods, but they also have some limitations
• Understanding the strengths and weaknesses of Quasi-Newton methods helps in selecting the appropriate method for a given problem and interpreting the results

Advantages of Quasi-Newton methods

• Fast convergence: Quasi-Newton methods exhibit superlinear convergence, often converging faster than first-order methods
• Avoidance of Hessian computation: Quasi-Newton methods approximate the Hessian matrix using gradient information, avoiding the need for explicit Hessian computation and inversion
• Robustness: Quasi-Newton methods are generally more robust than Newton's method, as they can handle ill-conditioned problems and inaccurate initial approximations
• Scalability: Methods like L-BFGS are suitable for large-scale problems due to their memory-efficient approximation of the inverse Hessian matrix

Disadvantages of Quasi-Newton methods

• Sensitivity to initial approximation: The convergence of Quasi-Newton methods can be sensitive to the choice of the initial approximation of the Hessian or its inverse
• Lack of global convergence guarantees: Quasi-Newton methods may not converge globally without additional safeguards, such as line search or trust-region methods
• Ineffectiveness for highly nonlinear problems: Quasi-Newton methods may struggle with highly nonlinear problems where the Hessian matrix varies significantly across the domain
• Overhead of Hessian approximation: The cost of updating and storing the Hessian approximation can be significant for high-dimensional problems, even with limited-memory methods

Applications of Quasi-Newton methods

• Quasi-Newton methods find applications in various fields where optimization problems arise
• They are particularly useful in machine learning, data fitting, and parameter estimation tasks
• Quasi-Newton methods are also employed in scientific computing, engineering optimization, and operations research

Quasi-Newton methods in machine learning

• Machine learning often involves optimizing objective functions, such as loss functions or likelihood functions
• Quasi-Newton methods, particularly L-BFGS, are widely used for training machine learning models, including logistic regression, support vector machines, and neural networks
• These methods provide faster convergence compared to first-order methods, especially when the number of variables is large

Quasi-Newton methods in data fitting

• Data fitting problems involve finding the best parameters of a model to fit observed data
• Quasi-Newton methods are effective for nonlinear least squares problems, where the objective is to minimize the sum of squared residuals
• These methods can handle large-scale data fitting problems and provide accurate parameter estimates
• Examples include curve fitting, parameter estimation in differential equations, and calibration of complex models