Seems like some changes are more like Euclidean distance while others are more like turning a single knob. If I go visit my cousin for a week and a bunch of aspects of my lifestyles shift towards his, that is more Euclidean than if I change my lifestyle by adding a new habit of jogging each morning. (Although both are in between the extremes of pure Euclidean or purely a single knob—you could think of it in terms of the dimensionality of the subspace that you’re moving in.)
And something similar can apply to work habits, thinking styles, etc.
The relevant aspect which makes a Euclidean distance metric relevant is not “all the dimensions change a bit at once”. The relevant aspect is “cost of testing the change is a function of Euclidean distance”—e.g. in the e-coli case, the energy expended during straight-line swimming should be roughly proportional to the distance traveled.
I’m not sure about your visit-the-cousin example. My intuition is that O(1/n) is the default cost to e-coli-style optimization when there’s no special structure to the exploration cost which we can exploit, but I don’t have a strong mathematical argument for that which would apply in an obvious way to the cousin example. My weak argument would be “absent any special knowledge about the problem structure, we have to treat visiting-the-cousin as an independent knob to turn same as all the other knobs”—i.e. we don’t have a prior reason to believe that the effects of the visit are a combination of other (available) knobs.
Seems like some changes are more like Euclidean distance while others are more like turning a single knob. If I go visit my cousin for a week and a bunch of aspects of my lifestyles shift towards his, that is more Euclidean than if I change my lifestyle by adding a new habit of jogging each morning. (Although both are in between the extremes of pure Euclidean or purely a single knob—you could think of it in terms of the dimensionality of the subspace that you’re moving in.)
And something similar can apply to work habits, thinking styles, etc.
The relevant aspect which makes a Euclidean distance metric relevant is not “all the dimensions change a bit at once”. The relevant aspect is “cost of testing the change is a function of Euclidean distance”—e.g. in the e-coli case, the energy expended during straight-line swimming should be roughly proportional to the distance traveled.
I’m not sure about your visit-the-cousin example. My intuition is that O(1/n) is the default cost to e-coli-style optimization when there’s no special structure to the exploration cost which we can exploit, but I don’t have a strong mathematical argument for that which would apply in an obvious way to the cousin example. My weak argument would be “absent any special knowledge about the problem structure, we have to treat visiting-the-cousin as an independent knob to turn same as all the other knobs”—i.e. we don’t have a prior reason to believe that the effects of the visit are a combination of other (available) knobs.