Mutation, Sexual Reproduction and Survival in Dynamic Environments

A new approach to understanding evolution [Valiant, JACM 2009], namely viewing it through the lens of computation, has already started yielding new insights, e.g., natural selection under sexual reproduction can be interpreted as the Multiplicative Weight Update (MWU) Algorithm in coordination games played among genes [Chastain, Livnat, Papadimitriou, Vazirani, PNAS 2014]. Using this machinery, we study the role of mutation in changing environments in the presence of sexual reproduction. Following [Wolf, Vazirani, Arkin, J. Theor. Biology], we model changing environments via a Markov chain, with the states representing environments, each with its own fitness matrix. In this setting, we show that in the absence of mutation, the population goes extinct, but in the presence of mutation, the population survives with positive probability. On the way to proving the above theorem, we need to establish some facts about dynamics in games. We provide the first, to our knowledge, polynomial convergence bound for noisy MWU in a coordination game. Finally, we also show that in static environments, sexual evolution with mutation converges, for any level of mutation

Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions

Given a non-convex twice differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue has (Lebesgue) measure zero, even for cost functions f with non-isolated critical points, answering an open question in [Lee, Simchowitz, Jordan, Recht, COLT2016]. Moreover, this result extends to forward-invariant convex subspaces, allowing for weak (non-globally Lipschitz) smoothness assumptions. Finally, we produce an upper bound on the allowable step-size.Comment: 2 figure