Difference between revisions of "Quasi-Newton methods"

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According to (1.12), (1.13) and (1.14), update the <math>B_{k+1}^{-1} </math> with <br /> <math> B_{k+1}^{-1}=(I-\rho s_ky_k^T)B_k^{-1}(I-\rho y_ks_k^T)+\rho s_ks_k^T</math> ,  <math>\rho=\frac{1}{y_k^Ts_k}</math>
 
According to (1.12), (1.13) and (1.14), update the <math>B_{k+1}^{-1} </math> with <br /> <math> B_{k+1}^{-1}=(I-\rho s_ky_k^T)B_k^{-1}(I-\rho y_ks_k^T)+\rho s_ks_k^T</math> ,  <math>\rho=\frac{1}{y_k^Ts_k}</math>
 +
 +
We skip procedures of solving the minimization problem (1.10) and here is the unique solution of (1.10):
 +
 +
<math> B_{k+1}=(I-\rho y_ks_k^T)B_k(I-\rho s_ky_k^T)+\rho y_ky_k^T</math> (1.12)
 +
 +
where <math>\rho=\frac{1}{y_k^Ts_k}</math> (1.13)
 +
 +
Finally, we get the updated <math>B_{k+1}</math>. However, according to (1.2) and (1.3), we also need the inverse of <math>B_{k+1}</math> in next iterate.
 +
 +
To get the inverse of <math>B_{k+1}</math>, we can apply the Sherman-Morrison formula to avoid complicated calculation of inverse.
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 +
Set <math>M_k=B_k^{-1} </math>, with Sherman-Morrison formula we can get:
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<math>M_{k+1}=M_k+\frac{s_k s_k^T}{s_k^T y_k}-\frac{M_k y_k y_k^T M_k}{y_k^T M_k y_k} </math> (1.14)
  
 
== Numerical Example ==
 
== Numerical Example ==

Revision as of 15:31, 13 December 2020

Author: Jianmin Su (ChemE 6800 Fall 2020)

Steward: Allen Yang, Fengqi You

Quasi-Newton Methods are a kind of methods used to solve nonlinear optimization problems. They are based on Newton's method yet can be an alternative to Newton's method when the objective function is not twice-differentiable, which means the Hessian matrix is unavailable, or it is too expensive to calculate the Hessian matrix and its inverse.

Introduction

The first quasi-Newton algorithm was developed by W.C. Davidon in the mid1950s and it turned out to be a milestone in nonlinear optimization problems. He was trying to solve a long optimization calculation but he failed to get the result with the original method due to the low performance of computers. Thus he managed to build the quasi-Newton method to solve it. Later, Fletcher and Powell proved that the new algorithm was more efficient and more reliable than the other existing methods.

During the following years, numerous variants were proposed, include Broyden's method (1965), the SR1 formula (Davidon 1959, Broyden 1967), the DFP method (Davidon, 1959; Fletcher and Powell, 1963), and the BFGS method (Broyden, 1969; Fletcher, 1970; Goldfarb, 1970; Shanno, 1970)[1].

In optimization problems, Newton's method uses first and second derivatives, gradient and the Hessian in multivariate scenarios, to find the optimal point, it is applied to a twice-differentiable function to find the roots of the first derivative (solutions to ), also known as the stationary points of [2].

The iteration of Newton's method is usually written as: , where is the iteration number, is the Hessian matrix and

Iteraton would stop when it satisfies the convergence criteria like

Though we can solve an optimization problem quickly with Newton's method, it has two obvious disadvantages:

  1. The objective function must be twice-differentiable, and the Hessian matrix must be positive definite.
  2. The calculation is costly because it requires to compute the Jacobian matrix, Hessian matrix, and its inverse, which is time-consuming when dealing with a large-scale optimization problem.

However, we can use Quasi-Newton methods to avoid these two disadvantages.

Quasi-Newton methods are similar to Newton's method, but with one key idea that is different, they don't calculate the Hessian matrix. They introduce a matrix to estimate the Hessian matrix instead so that they can avoid the time-consuming calculations of the Hessian matrix and its inverse. And there are many variants of quasi-Newton methods that simply depend on the exact methods they use to estimate the Hessian matrix.

Theory and Algorithm

To illustrate the basic idea behind quasi-Newton methods, we start with building a quadratic model of the objective function at the current iterate :

(1.1),

where is an symmetric positive definite matrix that will be updated at every iteration.

The minimizer of this convex quadratic model is:

(1.2),

which is also used as the search direction.

Then the new iterate could be written as: (1.3),

where is the step length that should satisfy the Wolfe conditions. The iteration is similar to Newton's method, but we use the approximate Hessian instead of the true Hessian.

To maintain the curve information we got from the previous iteration in , we generate a new iterate and new quadratic modelto in the form of:

(1.4).

To construct the relationship between 1.1 and 1.4, we require that in 1.1 at the function value and gradient match and , and the gradient of should match the gradient of the objective function at the latest two iterates and , then we can get:

(1.5)

and with some arrangements:

(1.6)

Define:

, (1.7)

So that (1.6) becomes: (1.8), which is the secant equation.

To make sure is still a symmetric positive definite matrix, we need (1.9).

To further preserve properties of and determine uniquely, we assume that among all symmetric matrices satisfying secant equation, is closest to the current matrix , which leads to a minimization problem:

(1.10) s.t. , ,

where and satisfy (1.9) and is symmetric and positive definite.

Different matrix norms applied in (1.10) results in different quasi-Newton methods. The weighted Frobenius norm can help us get an easy solution to the minimization problem: (1.11).

The weighted matrix can be any matrix that satisfies the relation .

We skip procedures of solving the minimization problem (1.10) and here is the unique solution of (1.10):

(1.12)

where (1.13)

Finally, we get the updated . However, according to (1.2) and (1.3), we also need the inverse of in next iterate.

To get the inverse of , we can apply the Sherman-Morrison formula to avoid complicated calculation of inverse.

Set , with Sherman-Morrison formula we can get:

(1.14)

With the derivation[3] above, we can now understand how do quasi-Newton methods get rid of calculating the Hessian matrix and its inverse. We can directly estimate the inverse of Hessian, and we can use (1.14) to update the approximation of the inverse of Hessian, which leads to the DFP method, or we can directly estimate the Hessian matrix, and this is the main idea in the BFGS method.


DFP method

The DFP method, which is also known as the Davidon–Fletcher–Powell formula, is named after W.C. Davidon, Roger Fletcher, and Michael J.D. Powell. It was proposed by Davidon in 1959 first and then improved by Fletched and Powell. DFP method uses an symmetric positive definite matrix to estimate the inverse of Hessian matrix and its algorithm is shown below[4].

DFP Algorithm

To avoid confusion, we use to represent the approximation of the inverse of the Hessian matrix.

  1. Given the starting point ; convergence tolerance ; the initial estimation of inverse Hessian matrix ; .
  2. Compute the search direction .
  3. Compute the step length with a line search procedure that satisfies Wolfe conditions. And then set
    ,
  4. If , then end of the iteration, otherwise continue step5.
  5. Computing .
  6. Update the with
  7. Update with and go back to step2.


BFGS method

BFGS method is named for its four discoverers Broyden, Fletcher, Goldfarb, and Shanno. It is considered the most effective quasi-Newton algorithm. Unlike the DFP method, the BFGS method uses an symmetric positive definite matrix to estimate the Hessian matrix[5].

BFGS Algorithm

  1. Given the starting point ; convergence tolerance ; the initial estimation of Hessian matrix ; .
  2. Compute the search direction .
  3. Compute the step length with a line search procedure that satisfies Wolfe conditions. And then set
    ,
  4. If , then end of the iteration, otherwise continue step5.
  5. Computing .
  6. According to (1.12), (1.13) and (1.14), update the with
    ,
  7. Update with and go back to step2.


According to (1.12), (1.13) and (1.14), update the with
,

We skip procedures of solving the minimization problem (1.10) and here is the unique solution of (1.10):

(1.12)

where (1.13)

Finally, we get the updated . However, according to (1.2) and (1.3), we also need the inverse of in next iterate.

To get the inverse of , we can apply the Sherman-Morrison formula to avoid complicated calculation of inverse.

Set , with Sherman-Morrison formula we can get:

(1.14)

Numerical Example

The following is an example to show how to solve an unconstrained nonlinear optimization problem with the DFP method.

Step 1:

Usually, we set the approximation of the inverse of the Hessian matrix as an identity matrix with the same dimension as the Hessian matrix. In this case, is a identity matrix.

:

:

For convenience, we can set .

Step 2:

Step 3:

Step 4:

Since is not less than , we need to continue.

Step 5:

Step 6:

And then go back to Step 2 with the update to start a new iterate until .

We continue the rest of the steps in python and the results are listed below:

Iteration times: 0 Result:[-1. 0.]

Iteration times: 1 Result:[ 0.06666667 -0.13333333]

Iteration times: 2 Result:[0.00083175 0.01330805]

Iteration times: 3 Result:[-0.00018037 -0.00016196]

Iteration times: 4 Result:[ 3.74e-06 -5.60e-07]

After four times of iteration, we finally get the optimal solution, which can be assumed as and the minimum of the objective function is 3.

As we can see from the calculation in Step 6, though the updated formula for looks complicated, it's actually not. We can see results of and are constant numbers and results of and are matrix that with the same dimension as . Therefore, the calculation of quasi-Newton methods is faster and simpler since it's related to some basic matrix calculations like inner product and outer product.

Application

Quasi-newton methods are applied to various areas such as physics, biology, engineering, geophysics, chemistry, and industry to solve the nonlinear systems of equations because of their faster calculation. The ICUM (Inverse Column-Updating Method), one type of quasi-Newton methods, is not only efficient in solving large scale sparse nonlinear systems but also perfumes well in not necessarily large-scale systems in real applications. It is used to solve the Two-pint ray tracing problem in geophysics, to estimate the transmission coefficients for AIDS and for Tuberculosis in Biology, and in Multiple target 3D location airborne ultrasonic system. [6] Moreover, they can be applied and developed into the Deep Learning area as sampled quasi-Newton methods to help make use of more reliable information.[7]

Besides, to make quasi-Newton methods more available, they are integrated into programming languages so that people can use them to solve nonlinear optimization problems conveniently, for example, Mathematic (quasi-Newton solvers)[8], MATLAB (Optimization Toolbox)[9], R[10], SciPy extension to Python[11].

Conclusion

Quasi-Newton methods are a milestone in solving nonlinear optimization problems, they are more efficient than Newton's method in large-scale optimization problems because they don't need to compute second derivatives, which makes calculation less costly. Because of their efficiency, they can be applied to different areas and remain appealing.

References

  1. Hennig, Philipp, and Martin Kiefel. "Quasi-Newton method: A new direction." Journal of Machine Learning Research 14.Mar (2013): 843-865.
  2. Newton’s Method. Retrieved from: https://en.wikipedia.org/wiki/Quasi-Newton_method
  3. Nocedal, Jorge, and Stephen Wright. Numerical optimization. Springer Science & Business Media, 2006.
  4. Davidon–Fletcher–Powell formula. Retrieved from: https://en.wikipedia.org/wiki/Davidon%E2%80%93Fletcher%E2%80%93Powell_formula
  5. Broyden–Fletcher–Goldfarb–Shanno algorithm. Retrieved from: https://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm
  6. Pérez, Rosana, and Véra Lucia Rocha Lopes. "Recent applications and numerical implementation of quasi-Newton methods for solving nonlinear systems of equations." Numerical Algorithms 35.2-4 (2004): 261-285.
  7. Berahas, Albert S., Majid Jahani, and Martin Takáč. "Quasi-newton methods for deep learning: Forget the past, just sample." arXiv preprint arXiv:1901.09997 (2019).
  8. http://reference.wolfram.com/mathematica/tutorial/UnconstrainedOptimizationQuasiNewtonMethods.html
  9. http://www.mathworks.com/help/toolbox/optim/ug/fminunc.html
  10. [1]
  11. http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html