All ways of modifying Y are only equivalent in a dense linear system. Sparsity (in a high-dimensional system) changes that. (That’s a fairly central concept behind this whole project: sparsity is one of the main ingredients necessary for the natural abstraction hypothesis.)
Suppose Y is unidimensional, and you have Y=f(g(X), h(X)). Suppose there are two perturbations i and j that X can emit, where g is only sensitive to i and h is only sensitive to j, i.e. g(j)=0, h(i)=0. Then because the system is linear, you can extract them from the rest:
This means that if X only cares about Y, it is free to choose whether to adjust a or to adjust b. In a nonlinear system, there might be all sorts of things like moderators, diminishing returns, etc., which would make it matter whether it tried to control Y using a or using b; but in a linear system, it can just do whatever.
Oh I see. Yeah, if either X or Y is unidimensional, then any linear model is really boring. They need to be high-dimensional to do anything interesting.
They need to be high-dimensional for the linear models themselves to do anything interesting, but I think adding a large number of low-dimensional linear models might, despite being boring, still change the dynamics of the graphs to be marginally more realistic for settings involving optimization. X turns into an estimate of Y, and tries to control this estimate towards zero; that’s a pattern that I assume would be rare in your graph, but common in reality, and it could lead to real graphs exhibiting certain “conspiracies” that the model graphs might lack (especially if there are many (X, Y) pairs, or many (individually unidimensional) Xs that all try to control a single common Y).
But there’s probably a lot of things that can be investigated about this. I should probably be working on getting my system for this working, or something. Gonna be exciting to see what else you figure out re natural abstractions.
All ways of modifying Y are only equivalent in a dense linear system. Sparsity (in a high-dimensional system) changes that. (That’s a fairly central concept behind this whole project: sparsity is one of the main ingredients necessary for the natural abstraction hypothesis.)
I think I phrased it wrong/in a confusing way.
Suppose Y is unidimensional, and you have Y=f(g(X), h(X)). Suppose there are two perturbations i and j that X can emit, where g is only sensitive to i and h is only sensitive to j, i.e. g(j)=0, h(i)=0. Then because the system is linear, you can extract them from the rest:
Y=f(g(X+ai+bj), h(X+ai+bj))=f(g(X), h(X))+af(g(i))+bf(h(j))
This means that if X only cares about Y, it is free to choose whether to adjust a or to adjust b. In a nonlinear system, there might be all sorts of things like moderators, diminishing returns, etc., which would make it matter whether it tried to control Y using a or using b; but in a linear system, it can just do whatever.
Oh I see. Yeah, if either X or Y is unidimensional, then any linear model is really boring. They need to be high-dimensional to do anything interesting.
They need to be high-dimensional for the linear models themselves to do anything interesting, but I think adding a large number of low-dimensional linear models might, despite being boring, still change the dynamics of the graphs to be marginally more realistic for settings involving optimization. X turns into an estimate of Y, and tries to control this estimate towards zero; that’s a pattern that I assume would be rare in your graph, but common in reality, and it could lead to real graphs exhibiting certain “conspiracies” that the model graphs might lack (especially if there are many (X, Y) pairs, or many (individually unidimensional) Xs that all try to control a single common Y).
But there’s probably a lot of things that can be investigated about this. I should probably be working on getting my system for this working, or something. Gonna be exciting to see what else you figure out re natural abstractions.