# Optimization / personalization within clusters

I have the following optimization problem:

I have a (random and very noisy) objective function f(A, P), where A is a vector of "observable" parameters of the input and P is the parameters that I can control.

I'd like to find P(A) for every A, such that f(A, P(A)) is maximized.

For example, I'm writing a game; I know a bunch of facts about every user (their age, gender, level, various in-game statistics), and I can control the difficulty of monsters they face and the scarcity of resources they find on their journey. The objective function "f" is, for example, how much money they spend.

I think a reasonable approach to this would be to:

1) cluster the users by A in some way, such that within a single cluster, the shape of "f" with respect to "P" is approximately the same;

2) run some guided optimization/experimentation algorithm within every cluster, taking advantage of the fact that a single user's data point gives me information about the whole cluster, but avoiding meaningless comparisons between dissimilar users.

The problem here is that I cannot use a conventional clustering method "as is" to cluster by A, because I don't know which dimensions of "A" are important - and perhaps some dimensions are important only in some regions of the "A" space. So it's not clear how to formulate the distance function D(A,A).

Is this indeed a reasonable approach? Does it have a name? What's some existing research in this area?

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Why not optimize $f$ directly instead of clustering $A$ first? –  Victor May Jan 20 at 19:34
The reason for that is explained in #2 : clustering, or some measure of similarity between points of A, would mean that I could optimize P within a whole cluster (at least I could get a very good approximation to the optimum), rather than at each individual point. –  jkff Jan 22 at 16:37
So it's an of issue of runtime optimization? –  Victor May Jan 22 at 16:54
Not necessarily runtime, but every measurement has a high cost (and still higher cost if I pick a bad value of P), so I'd like to minimize the number of measurements - naturally, reusing measurements between similar species of A would help greatly. –  jkff Jan 23 at 16:13
What do you mean by a measurement? Evaluating $f$ for a fixed value of A and different values of P? –  Victor May Jan 24 at 17:01