# Evolving a Vector Space with any Generating Set

## TL;DR

This paper demonstrates that any normed vector space can be evolved efficiently using a simple generating set, requiring only polynomial steps, and remains stable under target changes, extending Valiant's evolution model.

## Contribution

It introduces a minimalistic evolutionary process for vector spaces based solely on generating sets, broadening the scope of evolvability in Valiant's framework.

## Key findings

- Evolves a vector space with only a generating set
- Requires O(1/^2) steps for convergence
- Handles target drifts effectively

## 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.