Newton's method in optimization

Geometric interpretation

The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of ${\displaystyle f(x)}$ at the trial value ${\displaystyle x_k}$, having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below. Note that if ${\displaystyle f}$ happens to be a quadratic function, then the exact extremum is found in one step.

Higher dimensions

The above iterative scheme can be generalized to ${\displaystyle d>1}$ dimensions by replacing the derivative with the gradient (different authors use different notation for the gradient, including ${\displaystyle f^{\prime }(x)=\mathrm{\nabla }f(x)=g_f(x)\in {\mathbb{R}}^d}$), and the reciprocal of the second derivative with the inverse of the Hessian matrix (different authors use different notation for the Hessian, including ${\displaystyle f^{″}(x)={\mathrm{\nabla }}^{2}f(x)=H_f(x)\in {\mathbb{R}}^{d\times d}}$). One thus obtains the iterative scheme

${\displaystyle x_{k+1}=x_k-[f^{″}(x_k){]}^{-1}{f}^{\prime }(x_k),k\ge 0.}$

Often Newton's method is modified to include a small step size ${\displaystyle 0<\gamma \le 1}$ instead of ${\displaystyle \gamma =1}$:

${\displaystyle x_{k+1}=x_k-\gamma [f^{″}(x_k){]}^{-1}{f}^{\prime }(x_k).}$

This is often done to ensure that the Wolfe conditions, or much simpler and efficient Armijo's condition, are satisfied at each step of the method. For step sizes other than 1, the method is often referred to as the relaxed or damped Newton's method.

Convergence

If f is a strongly convex function with Lipschitz Hessian, then provided that ${\displaystyle x_0}$ is close enough to ${\displaystyle x_{\ast }=\arg minf(x)}$, the sequence ${\displaystyle x_0,x_1,x_2,\dots }$ generated by Newton's method will converge to the (necessarily unique) minimizer ${\displaystyle x_{\ast }}$ of ${\displaystyle f}$ quadratically fast. That is,

${\displaystyle \Vert x_{k+1}-x_{\ast }\Vert \le \frac{1}{2}\Vert x_k-x_{\ast }{\Vert }^2,\mathrm{\forall }k\ge 0.}$

Computing the Newton direction

Finding the inverse of the Hessian in high dimensions to compute the Newton direction ${\displaystyle h=-(f^{″}(x_k){)}^{-1}{f}^{\prime }(x_k)}$ can be an expensive operation. In such cases, instead of directly inverting the Hessian, it is better to calculate the vector ${\displaystyle h}$ as the solution to the system of linear equations

${\displaystyle [f^{″}(x_k)]h=-f^{\prime }(x_k)}$

which may be solved by various factorizations or approximately (but to great accuracy) using iterative methods. Many of these methods are only applicable to certain types of equations, for example the Cholesky factorization and conjugate gradient will only work if ${\displaystyle f^{″}(x_k)}$ is a positive definite matrix. While this may seem like a limitation, it is often a useful indicator of something gone wrong; for example if a minimization problem is being approached and ${\displaystyle f^{″}(x_k)}$ is not positive definite, then the iterations are converging to a saddle point and not a minimum.

On the other hand, if a constrained optimization is done (for example, with Lagrange multipliers), the problem may become one of saddle point finding, in which case the Hessian will be symmetric indefinite and the solution of ${\displaystyle x_{k+1}}$ will need to be done with a method that will work for such, such as the ${\displaystyle LD{L}^{\mathrm{\top }}}$ variant of Cholesky factorization or the conjugate residual method.

There also exist various quasi-Newton methods, where an approximation for the Hessian (or its inverse directly) is built up from changes in the gradient.

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 ${\displaystyle B_k}$ so as to make ${\displaystyle f^{″}(x_k)+B_k}$ positive definite. One approach is to diagonalize the Hessian and choose ${\displaystyle B_k}$ so that ${\displaystyle f^{″}(x_k)+B_k}$ has the same eigenvectors as the Hessian, but with each negative eigenvalue replaced by ${\displaystyle ϵ>0}$.

An approach exploited in the Levenberg–Marquardt algorithm (which uses an approximate Hessian) is to add a scaled identity matrix to the Hessian, ${\displaystyle \mu I}$, with the scale adjusted at every iteration as needed. For large ${\displaystyle \mu }$ and small Hessian, the iterations will behave like gradient descent with step size ${\displaystyle 1/\mu }$. This results in slower but more reliable convergence where the Hessian doesn't provide useful information.

Some caveats

Newton's method, in its original version, has several caveats:

1. It does not work if the Hessian is not invertible. This is clear from the very definition of Newton's method, which requires taking the inverse of the Hessian.
2. It may not converge at all, but can enter a cycle having more than 1 point. See the Newton's method § Failure analysis.
3. It can converge to a saddle point instead of to a local minimum, see the section "Geometric interpretation" in this article.

The popular modifications of Newton's method, such as quasi-Newton methods or Levenberg-Marquardt algorithm mentioned above, also have caveats:

For example, it is usually required that the cost function is (strongly) convex and the Hessian is globally bounded or Lipschitz continuous, for example this is mentioned in the section "Convergence" in this article. If one looks at the papers by Levenberg and Marquardt in the reference for Levenberg–Marquardt algorithm, which are the original sources for the mentioned method, one can see that there is basically no theoretical analysis in the paper by Levenberg, while the paper by Marquardt only analyses a local situation and does not prove a global convergence result. One can compare with Backtracking line search method for Gradient descent, which has good theoretical guarantee under more general assumptions, and can be implemented and works well in practical large scale problems such as Deep Neural Networks.