Newton's method in optimization

In mathematics, Newton's method is a well-known algorithm for finding roots of equations in one or more dimensions. It can also be used to find local maxima and local minima of functions by noticing that if a real number $x*$ is a stationary point of a function $f(x)$, then $x*$ is a root of the derivative $f'(x)$, and therefore one can solve for $x*$ by applying Newton's method to $f'(x)$.

Thus, provided that $f(x)$ is a twice-differentiable function and the initial guess $x_{0}$ is chosen close enough to $x*$, the sequence $(x_{n})$ defined by

$x_{n+1}=x_{n}-{\frac {f'(x_{n})}{f''(x_{n})}},\ n\geq 0$

will converge towards $x*$.

This iterative scheme can be generalized to several dimensions by replacing the derivative with the gradient, $\nabla f(\mathbf {x} )$, and the reciprocal of the second derivative with the inverse of the Hessian matrix, $Hf(\mathbf {x} )$. One obtains the iterative scheme

$\mathbf {x} _{n+1}=\mathbf {x} _{n}-[Hf(\mathbf {x} _{n})]^{-1}\nabla f(\mathbf {x} _{n}),\ n\geq 0.$

Usually Newton's method is modified to include a small step size $\gamma >0$ instead of $\gamma =1$

$\mathbf {x} _{n+1}=\mathbf {x} _{n}-\gamma [Hf(\mathbf {x} _{n})]^{-1}\nabla f(\mathbf {x} _{n}).$

This is often done to ensure that the Wolfe conditions are satisfied at each step $\mathbf {x} _{n}\to \mathbf {x} _{n+1}$ of the iteration.

The geometric interpretation of Newton's method is that at each iteration one approximates $f(\mathbf {x} )$ by a quadratic function around $\mathbf {x} _{n}$, and then takes a step towards the maximum/minimum of that quadratic function.

Newton's method converges much faster towards a local maximum or minimum than gradient descent. In fact, every local minimum has a neighborhood $N$ such that, if we start with $\mathbf {x} _{0}\in N,$ Newton's method with step size $\gamma =1$ converges quadratically (if the Hessian is invertible in that neighborhood).

Finding the inverse of the Hessian is an expensive operation, so the linear equation

$\mathbf {p} _{n}=\mathbf {x} _{n+1}-\mathbf {x} _{n}=[Hf(\mathbf {x} _{n})]^{-1}\nabla f(\mathbf {x} _{n}),\ n\geq 0.$.

is often solved approximately (but to great accuracy) using a method such as conjugate gradient. There also exist various quasi-Newton methods, where an approximation for the Hessian is used instead.

If the Hessian is close to a non-invertible matrix, the inverted Hessian can be numerically unstable and the solution may diverge. In this case, certain workarounds have been tried in the past, which have varied success with certain problems. One can, for example, modify the Hessian by adding a correction matrix $B_{n}$ so as to make $H_{f}(\mathbf {x} _{n})+B_{n}$ positive definite. One approach is to diagonalize $H_{f}$ and choose $B_{n}$ so that $H_{f}(\mathbf {x} _{n})+B_{n}$ has the same eigenvectors as $H_{f}$, but with each negative eigenvalue replaced by $\epsilon >0.$

Proof

Assuming that the function is quadratic, Newton's method can find the solution in a single iteration, otherwise it will take further iterations.

A quadratic and its first and second derivatives are defined as:

$f(x)=ax^{2}+bx+c$

$f'(x)=2ax+b$

$f''(x)=2a$

Starting with an initial guess $x$, the optimal solution (where the derivative is 0) will be at $x+\triangle x$:

$0=f'(x+\triangle x)$

$0=2a(x+\triangle x)+b$

$0=2ax+b+2a\triangle x$

$0=f'(x)+2a\triangle x$

$0=f'(x)+f''(x)\triangle x$

$\triangle x=-{\frac {f'(x)}{f''(x)}}$

$\therefore x_{n+1}=x_{n}-{\frac {f'(x)}{f''(x)}}$

See also

• Gradient descent
• Gauss-Newton algorithm
• Levenberg-Marquardt algorithm
• Optimization

References

• Avriel, Mordecai (2003). Nonlinear Programming: Analysis and Methods. Dover Publishing. ISBN 0-486-43227-0.
• Nocedal, Jorge & Wright, Stephen J. (1999). Numerical Optimization. Springer-Verlag. ISBN 0-387-98793-2.