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Register a new userNon-Archimedean Replicator Dynamics and Eigens Paradox
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Added by
W. A. Zuniga-Galindo
Publication date
2018
fields
Biology
and research's language is
English
Authors
W. A. Zu~niga-Galindo
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We present a new non-Archimedean model of evolutionary dynamics, in which the genomes are represented by p-adic numbers. In this model the genomes have a variable length, not necessarily bounded, in contrast with the classical models where the length is fixed. The time evolution of the concentration of a given genome is controlled by a p-adic evolution equation. This equation depends on a fitness function f and on mutation measure Q. By choosing a mutation measure of Gibbs type, and by using a p-adic version of the Maynard Smith Ansatz, we show the existence of threshold function M_{c}(f,Q), such that the long term survival of a genome requires that its length grows faster than M_{c}(f,Q). This implies that Eigens paradox does not occur if the complexity of genomes grows at the right pace. About twenty years ago, Scheuring and Poole, Jeffares, Penny proposed a hypothesis to explain Eigens paradox. Our mathematical model shows that this biological hypothesis is feasible, but it requires p-adic analysis instead of real analysis. More exactly, the Darwin-Eigen cycle proposed by Poole et al. takes place if the length of the genomes exceeds M_{c}(f,Q).
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Policy gradient and actor-critic algorithms form the basis of many commonly used training techniques in deep reinforcement learning. Using these algorithms in multiagent environments poses problems such as nonstationarity and instability. In this paper, we first demonstrate that standard softmax-based policy gradient can be prone to poor performance in the presence of even the most benign nonstationarity. By contrast, it is known that the replicator dynamics, a well-studied model from evolutionary game theory, eliminates dominated strategies and exhibits convergence of the time-averaged trajectories to interior Nash equilibria in zero-sum games. Thus, using the replicator dynamics as a foundation, we derive an elegant one-line change to policy gradient methods that simply bypasses the gradient step through the softmax, yielding a new algorithm titled Neural Replicator Dynamics (NeuRD). NeuRD reduces to the exponential weights/Hedge algorithm in the single-state all-actions case. Additionally, NeuRD has formal equivalence to softmax counterfactual regret minimization, which guarantees convergence in the sequential tabular case. Importantly, our algorithm provides a straightforward way of extending the replicator dynamics to the function approximation setting. Empirical results show that NeuRD quickly adapts to nonstationarities, outperforming policy gradient significantly in both tabular and function approximation settings, when evaluated on the standard imperfect information benchmarks of Kuhn Poker, Leduc Poker, and Goofspiel.
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