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# Learning in Matrix Games can be Arbitrarily Complex

*Proceedings of Thirty Fourth Conference on Learning Theory*, PMLR 134:159-185, 2021.

#### Abstract

Many multi-agent systems with strategic interactions have their desired functionality encoded as the Nash equilibrium of a game, e.g. machine learning architectures such as Generative Adversarial Networks. Directly computing a Nash equilibrium of these games is often impractical or impossible in practice, which has led to the development of numerous learning algorithms with the goal of iteratively converging on a Nash equilibrium. Unfortunately, the dynamics generated by the learning process can be very intricate and instances failing to converge become 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. In the context of machine learning, 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.