Evolving a Vector Space with any Generating Set

Abstract: In Valiant's model of evolution, a class of representations is evolvable iff a polynomial-time process of random mutations guided by selection converges with high probability to a representation as $\epsilon$-close as desired from the optimal one, for any required $\epsilon>0$. Several previous positive results exist that can be related to evolving a vector space, but each former result imposes disproportionate representations or restrictions on (re)initialisations, distributions, performance functions and/or the mutator. In this paper, we show that all it takes to evolve a normed vector space is merely a set that generates the space. Furthermore, it takes only $\tilde{O}(1/\epsilon^2)$ steps and it is essentially stable, agnostic and handles target drifts that rival some proven in fairly restricted settings. Our algorithm can be viewed as a close relative to a popular fifty-years old gradient-free optimization method for which little is still known from the convergence standpoint: Nelder-Mead simplex method.