Numerical Solution to MLE

The method that you applied in the previous two examples was very effective at finding a solution quickly - but that is not always the case. In more complicated problems, finding the analytical solution may involve lengthy computations.

Luckily there is an alternative - numerical solutions to maximum likelihood problems can be found in a fraction of the time. We will explore what a numerical solution to the previous example would look like.

Numerical solution

The graph of the error function from the previous example is seen below. In this example, it is very easy to see where the global minimum is located. However, in more complicated examples with multiple dimensions this is not as trivial.

This MLE can be solved numerically by applying an optimization algorithm. The goal of an optimization algorithm is to speedily find the optimal solution - in this case, the local minimum. There are several different algorithms that can tackle this problem; in SLAM, the gradient descent, Levenberg-Marquardt, and conjugate gradient algorithms are quite common. The gradient-descent algorithm was introduced in term 1, if you need a refresher - please revisit the gradient descent lesson. A very brief summary is also provided below.

Quick Refresher on Gradient Descent

Recall that the gradient of a function is a vector that points in the direction of the greatest rate of change; or in the case of an extrema, is equal to zero.

In gradient descent - you make an initial guess, and then adjust it incrementally in the direction opposite the gradient. Eventually, you should reach a minimum of the function.

This algorithm does have a shortcoming - in complex distributions, the initial guess can change the end result significantly. Depending on the initial guess, the algorithm converges on two different local minima. The algorithm has no way to determine where the global minimum is - it very naively moves down the steepest slope, and when it reaches a local minima, it considers its task complete. One solution to this problem is to use stochastic gradient descent (SGD). If you need a deeper refresh on SGD, please revisit the stochastic gradient descent lesson from term 1.