# How to find a cost function with only a statistical measure of success?

Using the U.S.A. as a loose analogy, we have search algorithms that find the names and number of States adjacent to a given State (containing a selected city). The goal is to minimize the number of "probes" and maximize the number of "hits". Constraints:

• we know only the coordinates of the selected City and
• the mean and stdev of the areas of the States and
• that the overall distribution of the States is fairly random.
• using the coordinates of a probe, a query returns the probed State's name (like the game Battleship, fire a missile and get feedback of the result)
• so far, the measure of success is how close the probe's results come to the statistically expected number of adjacent States.

For starters, we're using the "generalized" Archimedes Spiral (gives pretty good results), to get the polar coordinates of each probe: $$r = c + \alpha\theta^{1/n},\; n > 1,\; c = 0, \; \alpha = 1$$ where $\theta$ is incremented by some fraction of $\pi$ for each iteration. The x and y coordinates of each probe are then given by $$x = r\,cos(\theta),\; y = r\,sin(\theta)$$ We then query the blackbox with each (x,y) and get the name of the State we hit.

For our training data, we have a several dozen cities and their exact lists of adjacent States, but without coordinate data, we don't know how to find a cost function and are requesting guidance.

• Someone suggested using the centroid of each State for training - we might be able to get that data. – belwood May 26 '15 at 0:38