Practical Considerations for Stochastic Optimization

There are important considerations when using stochastic optimization algorithms.

The stochastic nature of the procedure means that any single run of an algorithm will be different, given a different source of randomness used by the algorithm and, in turn, different starting points for the search and decisions made during the search.

The pseudorandom number generator used as the source of randomness can be seeded to ensure the same sequence of random numbers is provided each run of the algorithm. This is good for small demonstrations and tutorials, although it is fragile as it is working against the inherent randomness of the algorithm.

Instead, a given algorithm can be executed many times to control for the randomness of the procedure.

This idea of multiple runs of the algorithm can be used in two key situations:

  • Comparing Algorithms
  • Evaluating Final Result

Algorithms may be compared based on the relative quality of the result found, the number of function evaluations performed, or some combination or derivation of these considerations. The result of any one run will depend upon the randomness used by the algorithm and alone cannot meaningfully represent the capability of the algorithm. Instead, a strategy of repeated evaluation should be used.

Any comparison between stochastic optimization algorithms will require the repeated evaluation of each algorithm with a different source of randomness and the summarization of the probability distribution of best results found, such as the mean and standard deviation of objective values. The mean result from each algorithm can then be compared.

In cases where multiple local minima are likely to exist, it can be beneficial to incorporate random restarts after our terminiation conditions are met where we restart our local descent method from randomly selected initial points.

— Page 66, Algorithms for Optimization, 2019.

Similarly, any single run of a chosen optimization algorithm alone does not meaningfully represent the global optima of the objective function. Instead, a strategy of repeated evaluation should be used to develop a distribution of optimal solutions.

The maximum or minimum of the distribution can be taken as the final solution, and the distribution itself will provide a point of reference and confidence that the solution found is “relatively good” or “good enough” given the resources expended.

  • Multi-Restart: An approach for improving the likelihood of locating the global optima via the repeated application of a stochastic optimization algorithm to an optimization problem.

The repeated application of a stochastic optimization algorithm on an objective function is sometimes referred to as a multi-restart strategy and may be built in to the optimization algorithm itself or prescribed more generally as a procedure around the chosen stochastic optimization algorithm.