  A growing number of machine learning architectures, such as Generative
Adversarial Networks, rely on the design of games which implement a desired
functionality via a Nash equilibrium. In practice these games have an implicit
complexity (e.g. from underlying datasets and the deep networks used) that
makes directly computing a Nash equilibrium impractical or impossible. For this
reason, numerous learning algorithms have been developed with the goal of
iteratively converging to a Nash equilibrium. Unfortunately, the dynamics
generated by the learning process can be very intricate and instances of
training failure hard to interpret. In this paper we show that, in a strong
sense, this dynamic complexity is inherent to games. Specifically, we prove
that replicator dynamics, the continuous-time analogue of Multiplicative
Weights Update, even when applied in a very restricted class of games -- known
as finite matrix games -- is rich enough to be able to approximate arbitrary
dynamical systems. Our results are positive in the sense that they show the
nearly boundless dynamic modelling capabilities of current machine learning
practices, but also negative in implying that these capabilities may come at
the cost of interpretability. As a concrete example, we show how replicator
dynamics can effectively reproduce the well-known strange attractor of Lonrenz
dynamics (the "butterfly effect") while achieving no regret.
