Earlier quoted context omitted.
Abstractions are useful because they simplify and if you don't allow error you don't allow maximization of simplification. You can formally relate this to learning problem formulation complexity. When you do you encounter things like branching factors which effect solution times which lead to natural results like fast but finishes being better than optimal but never terminates. This can hold even despite error in the…
I don't think this constitutes a formal proof on any way at all.
Obviously this only gets worse when we impose reality - we don't actually have infinite space on our computers. They don't compute for an infinite amount of time either. But notice that before we searched for infinite space and time and we failed? When we move down to finite space and finite time we still have the property of completing after full enumeration. We have a finite amount of computing capacity. We have a finite amount of computational storage. Yet the growth rate for unabstracted game trees is exponential. Let c be our constraints.
An abstracted game maps n states to one state. So it has log_n(X) where X is your state count. An abstracted game has X states. Since log_n(x) This is actually much less than the real world gains. Since in learning we get the policy expectations multiple times over the game graph and the convergence guarantees relate to the complexity of the graph you get a much more worthwhile window than just the difference of log_n(X) versus X. For much tighter bounds check out game theory research. They get error bounds on the abstraction error too by choosing clustering solutions with provable properties. So it really has a much stronger formal treatment than you might imagine.