Demonstrate that best-of-N works as claimed. Find or create a function that has more than one convergence point. Choose an optimizer. Run the optimizer many times (on the order of 10,000) from random...

Demonstrate that best-of-N works as claimed. Find or create a function that has more than one convergence point. Choose an optimizer. Run the optimizer many times (on the order of 10,000) from random initializations. This will give you 10,000 or so results, some of which may be “No Convergence.” Choose a value of f and c. (i) Use the best-of-N equation to determine N. (ii) Determine the OF value, OFf, which represents the value for which only f (fraction) have equal or lower values. (iii) Take your optimizer results, sequentially, in sets of N. If you used 10,000 trials and N is 25, then there will be 10,000/25 = 400 sets of N. Count the number of sets that found at least one OF value equal or lower than OFf. The fraction of sets with at least one OF value below OFf should be c. But this is a stochastic process, so the experimental number will not be exactly c. For example, flip a coin 100 times. You expect 50% will be heads, but if you get 53% heads, you will accept the ideal 50% calculation is accurate. (iv) Do the OF-less-than-OFf count for several f and c values to show that your result was not a happenstance of a fortuitous choice. You only need to generate the 10,000 optimization trials one time, on one function, for one optimizer. Use your programming skills to automate the generation of OF values from random starts and the assessment of results to validate the equation. But once the experiment is automated, it is easy to change function or optimizer and obtain results on other combinations.