Challenge of the day: come up with a real-life scenario where the utility function U(x) of setting a parameter to x looks like a cusp.
That is, the optimal value is "somewhere in the middle" and not an extreme, but the slope is higher the closer you get to that optimal point.
That is, the optimal value is "somewhere in the middle" and not an extreme, but the slope is higher the closer you get to that optimal point.
This particular cusp is y = y_max - (x - x_max) ^ (2/3), but yours can be different. The main property I'm looking for is that the closer x gets to x_max, the higher the additional gain from taking x one step closer to x_max.
One example: you are putting a small ball on top of a large ball, and want to maximize the time until the small ball slides down (the asymptote is not infinity because you don't care about time linearly, eventually you get bored). The param x would be where you put the small ball
Another example: you have N mice and 1000 sterile cats, where the mice can reproduce, and the cats eat the mice at a constant rate. You want to maximize the time that the ratio between mouse and cat population stays within some "healthy" range.
Both of the above are examples of "positive feedback" processes, where there is some unstable equilibrium at x*, and if you're not exactly at x*, then you move even further away from x* in the same direction. The time you stay within some range is usually -log(abs(x - x*)).
