142,132 research outputs found
Objective Improvement in Information-Geometric Optimization
Information-Geometric Optimization (IGO) is a unified framework of stochastic
algorithms for optimization problems. Given a family of probability
distributions, IGO turns the original optimization problem into a new
maximization problem on the parameter space of the probability distributions.
IGO updates the parameter of the probability distribution along the natural
gradient, taken with respect to the Fisher metric on the parameter manifold,
aiming at maximizing an adaptive transform of the objective function. IGO
recovers several known algorithms as particular instances: for the family of
Bernoulli distributions IGO recovers PBIL, for the family of Gaussian
distributions the pure rank-mu CMA-ES update is recovered, and for exponential
families in expectation parametrization the cross-entropy/ML method is
recovered. This article provides a theoretical justification for the IGO
framework, by proving that any step size not greater than 1 guarantees monotone
improvement over the course of optimization, in terms of q-quantile values of
the objective function f. The range of admissible step sizes is independent of
f and its domain. We extend the result to cover the case of different step
sizes for blocks of the parameters in the IGO algorithm. Moreover, we prove
that expected fitness improves over time when fitness-proportional selection is
applied, in which case the RPP algorithm is recovered
Probability Metrics for Tropical Spaces of Different Dimensions
The problem of comparing probability distributions is at the heart of many
tasks in statistics and machine learning and the most classical comparison
methods assume that the distributions occur in spaces of the same dimension.
Recently, a new geometric solution has been proposed to address this problem
when the measures live in Euclidean spaces of differing dimensions. Here, we
study the same problem of comparing probability distributions of different
dimensions in the tropical geometric setting, which is becoming increasingly
relevant in computations and applications involving complex, geometric data
structures. Specifically, we construct a Wasserstein distance between measures
on different tropical projective tori - the focal metric spaces in both theory
and applications of tropical geometry - via tropical mappings between
probability measures. We prove equivalence of the directionality of the maps,
whether starting from the lower dimensional space and mapping to the higher
dimensional space or vice versa. As an important practical implication, our
work provides a framework for comparing probability distributions on the spaces
of phylogenetic trees with different leaf sets.Comment: 15 page
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