Motivations for the Definition of an Optimisation Space
An intuitive conception of optimisation is navigating from a “larger” subset of the configuration space (the “basin of attraction”) to a “smaller” subset (the “set of target configurations”/“attractor”).
There are however issues with this.
For one, the notion of the “attractor” being smaller than the basin of attraction is not very sensible for infinte configuration spaces. For example, a square root calculation algorithm, may converge to a neighbourhood around the square root(s) that is the same size as its basin of attraction (the set of positive real numbers), but intuitively, it seems that the algorithm still did optimisation work on the configuration space.
Furthermore, even for finite spaces, I don’t think the notion of the size of the configuration space is quite right. If the basin of attraction is larger than the attractor, but has a lower probability, then moving from a configuration in the basin of attraction to a configuration in the attractor isn’t optimisation; there’s a sense in which that was likely to happen anyway.
I think the intuition behind the “larger” basin of attraction and “smaller” target configuration set is implicitly assuming a uniform probability distribution over configuration space. That is, the basin of attraction is indeed “bigger” than the attractor, but the relevant measure isn’t cardinality, but probability.
If a probability measure is required to sensibly define optimisation, then when talking about we’re a probability space.
The set of outcomes of our probability space is obviously our set of configurations. We have a set of events (for discrete configuration spaces this will just be the power set of our configuration space) and the probability measure.
To that probability space I added a basin of attraction, attractor and the objective function(s).
I think the optimisation space notion I have is sufficient to rigorously define an optimisation process and precisely quantify the work done by optimisation, but I’m not sure it’s necessary.
Perhaps I could remove some of the above apparatus without any loss of generality? Alas, I don’t see it. I could remove either:
The collection of objective functions
Infer from the definition of the basin of attraction and the attractor OR:
The basin of attraction and attractor
Infer from the collection of objective functions
And still be able to rigorously define an optimisation process, but at the cost of an intuitively sensible quantification of the work done by optimisation.
As is, the notion of an optimisation space is the minimal construct I see to satisfy both objectives.
Previously: Optimisation Space
Motivations for the Definition of an Optimisation Space
An intuitive conception of optimisation is navigating from a “larger” subset of the configuration space (the “basin of attraction”) to a “smaller” subset (the “set of target configurations”/“attractor”).
There are however issues with this.
For one, the notion of the “attractor” being smaller than the basin of attraction is not very sensible for infinte configuration spaces. For example, a square root calculation algorithm, may converge to a neighbourhood around the square root(s) that is the same size as its basin of attraction (the set of positive real numbers), but intuitively, it seems that the algorithm still did optimisation work on the configuration space.
Furthermore, even for finite spaces, I don’t think the notion of the size of the configuration space is quite right. If the basin of attraction is larger than the attractor, but has a lower probability, then moving from a configuration in the basin of attraction to a configuration in the attractor isn’t optimisation; there’s a sense in which that was likely to happen anyway.
I think the intuition behind the “larger” basin of attraction and “smaller” target configuration set is implicitly assuming a uniform probability distribution over configuration space. That is, the basin of attraction is indeed “bigger” than the attractor, but the relevant measure isn’t cardinality, but probability.
If a probability measure is required to sensibly define optimisation, then when talking about we’re a probability space.
The set of outcomes of our probability space is obviously our set of configurations. We have a set of events (for discrete configuration spaces this will just be the power set of our configuration space) and the probability measure.
To that probability space I added a basin of attraction, attractor and the objective function(s).
I think the optimisation space notion I have is sufficient to rigorously define an optimisation process and precisely quantify the work done by optimisation, but I’m not sure it’s necessary.
Perhaps I could remove some of the above apparatus without any loss of generality? Alas, I don’t see it. I could remove either:
The collection of objective functions
Infer from the definition of the basin of attraction and the attractor OR:
The basin of attraction and attractor
Infer from the collection of objective functions
And still be able to rigorously define an optimisation process, but at the cost of an intuitively sensible quantification of the work done by optimisation.
As is, the notion of an optimisation space is the minimal construct I see to satisfy both objectives.
Next: Preliminary Thoughts on Quantifying Optimisation: “Work”
The notion of numerocity is good try to for infinite things where cardinality is unexpectedly equal.