Well most system don’t have a tendency to evolve towards any small set of target states despite perturbations. Most systems, if you perturb then, just go off in some different direction. For example, if you perturb most running computer programs by modifying some variable with a debugger, they do not self-correct. Same with the satellite and billiard balls example. Most systems just don’t have this “attractor” dynamic.
Hmm, I see what you’re saying, but there still seems to be an analogy to me here with arbitrary utility functions, where you need the set of target states to be small (as you do say). Otherwise I could just say that the set of target states is all the directions the system might fly off in if you perturb it.
So you might say that, for this version of optimization to be meaningful, the set of target states has to be small (however that’s quantified), and for the utility maximization version to be meaningful, you need the utility function to be simple (however that’s quantified).
EDIT: And actually, maybe the two concepts are sort of dual to each other. If you have an agent with a simple utility function, then you could consider all its local optima to be a (small) set of target states for an optimizing system. And if you have an optimizing system with a small set of target states, then you could easily convert that into a simple utility function with a gradient towards those states.
And if your utility function isn’t simple, maybe you wouldn’t get a small set of target states when you do the conversion, and vice versa?
I’d say the utility function needs to contain one or more local optima with large basins of attraction that contain the initial state, not that the utility function needs to be simple. The simplest possible utility function is a constant function, which allows the system to wander aimlessly and certainly not “correct” in any way for perturbations.
Well most system don’t have a tendency to evolve towards any small set of target states despite perturbations. Most systems, if you perturb then, just go off in some different direction. For example, if you perturb most running computer programs by modifying some variable with a debugger, they do not self-correct. Same with the satellite and billiard balls example. Most systems just don’t have this “attractor” dynamic.
Hmm, I see what you’re saying, but there still seems to be an analogy to me here with arbitrary utility functions, where you need the set of target states to be small (as you do say). Otherwise I could just say that the set of target states is all the directions the system might fly off in if you perturb it.
So you might say that, for this version of optimization to be meaningful, the set of target states has to be small (however that’s quantified), and for the utility maximization version to be meaningful, you need the utility function to be simple (however that’s quantified).
EDIT: And actually, maybe the two concepts are sort of dual to each other. If you have an agent with a simple utility function, then you could consider all its local optima to be a (small) set of target states for an optimizing system. And if you have an optimizing system with a small set of target states, then you could easily convert that into a simple utility function with a gradient towards those states.
And if your utility function isn’t simple, maybe you wouldn’t get a small set of target states when you do the conversion, and vice versa?
I’d say the utility function needs to contain one or more local optima with large basins of attraction that contain the initial state, not that the utility function needs to be simple. The simplest possible utility function is a constant function, which allows the system to wander aimlessly and certainly not “correct” in any way for perturbations.
Ah, good points!