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Gradient-based learning in multi-agent systems is difficult because thegradient derives from a first-order model which does not account for theinteraction between agents' learning processes. LOLA (arXiv:1709.04326)accounts for this by differentiating through one step of optimization. Wepropose to judge joint policies by their long-term prospects as measured by themeta-value, a discounted sum over the returns of future optimization iterates.We apply a form of Q-learning to the meta-game of optimization, in a way thatavoids the need to explicitly represent the continuous action space of policyupdates. The resulting method, MeVa, is consistent and far-sighted, and doesnot require REINFORCE estimators. We analyze the behavior of our method on atoy game and compare to prior work on repeated matrix games.

Meta-Value Learning: a General Framework for Learning with Learning Awareness