  Sequential decision making techniques hold great promise to improve the
performance of many real-world systems, but computational complexity hampers
their principled application. Influence-based abstraction aims to gain leverage
by modeling local subproblems together with the 'influence' that the rest of
the system exerts on them. While computing exact representations of such
influence might be intractable, learning approximate representations offers a
promising approach to enable scalable solutions. This paper investigates the
performance of such approaches from a theoretical perspective. The primary
contribution is the derivation of sufficient conditions on approximate
influence representations that can guarantee solutions with small value loss.
In particular we show that neural networks trained with cross entropy are well
suited to learn approximate influence representations. Moreover, we provide a
sample based formulation of the bounds, which reduces the gap to applications.
Finally, driven by our theoretical insights, we propose approximation error
estimators, which empirically reveal to correlate well with the value loss.
