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This book examines the mismatch between discrete programs, which
lie at the center of modern applied mathematics, and the continuous
space phenomena they simulate. The author considers whether we can
imagine continuous spaces of programs, and asks what the structure
of such spaces would be and how they would be constituted. He
proposes a functional analysis of program spaces focused through
the lens of iterative optimization. The author begins with the
observation that optimization methods such as Genetic Algorithms,
Evolution Strategies, and Particle Swarm Optimization can be
analyzed as Estimation of Distributions Algorithms (EDAs) in that
they can be formulated as conditional probability distributions.
The probabilities themselves are mathematical objects that can be
compared and operated on, and thus many methods in Evolutionary
Computation can be placed in a shared vector space and analyzed
using techniques of functional analysis. The core ideas of this
book expand from that concept, eventually incorporating all
iterative stochastic search methods, including gradient-based
methods. Inspired by work on Randomized Search Heuristics, the
author covers all iterative optimization methods and not just
evolutionary methods. The No Free Lunch Theorem is viewed as a
useful introduction to the broader field of analysis that comes
from developing a shared mathematical space for optimization
algorithms. The author brings in intuitions from several branches
of mathematics such as topology, probability theory, and stochastic
processes and provides substantial background material to make the
work as self-contained as possible. The book will be valuable for
researchers in the areas of global optimization, machine learning,
evolutionary theory, and control theory.
This book examines the mismatch between discrete programs, which
lie at the center of modern applied mathematics, and the continuous
space phenomena they simulate. The author considers whether we can
imagine continuous spaces of programs, and asks what the structure
of such spaces would be and how they would be constituted. He
proposes a functional analysis of program spaces focused through
the lens of iterative optimization. The author begins with the
observation that optimization methods such as Genetic Algorithms,
Evolution Strategies, and Particle Swarm Optimization can be
analyzed as Estimation of Distributions Algorithms (EDAs) in that
they can be formulated as conditional probability distributions.
The probabilities themselves are mathematical objects that can be
compared and operated on, and thus many methods in Evolutionary
Computation can be placed in a shared vector space and analyzed
using techniques of functional analysis. The core ideas of this
book expand from that concept, eventually incorporating all
iterative stochastic search methods, including gradient-based
methods. Inspired by work on Randomized Search Heuristics, the
author covers all iterative optimization methods and not just
evolutionary methods. The No Free Lunch Theorem is viewed as a
useful introduction to the broader field of analysis that comes
from developing a shared mathematical space for optimization
algorithms. The author brings in intuitions from several branches
of mathematics such as topology, probability theory, and stochastic
processes and provides substantial background material to make the
work as self-contained as possible. The book will be valuable for
researchers in the areas of global optimization, machine learning,
evolutionary theory, and control theory.
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