Handling Expensive Optimization with Large Noise

Rémi Coulom (1) , Philippe Rolet (2, 3) , Nataliya Sokolovska (2, 3) , Olivier Teytaud (2, 3)

1 SEQUEL - Sequential Learning
2 TAO - Machine Learning and Optimisation
3 LRI - Laboratoire de Recherche en Informatique

This paper exhibits lower and upper bounds on runtimes for expensive noisy optimization problems. Runtimes are expressed in terms of number of fitness evaluations. Fitnesses considered are monotonic transformations of the {\em sphere} function. The analysis focuses on the common case of fitness functions quadratic in the distance to the optimum in the neighborhood of this optimum---it is nonetheless also valid for any monotonic polynomial of degree p>2. Upper bounds are derived via a bandit-based estimation of distribution algorithm that relies on Bernstein races called R-EDA. It is known that the algorithm is consistent even in non-differentiable cases. Here we show that: (i) if the variance of the noise decreases to 0 around the optimum, it can perform optimally for quadratic transformations of the norm to the optimum, (ii) otherwise, it provides a slower convergence rate than the one exhibited empirically by an algorithm called Quadratic Logistic Regression based on surrogate models---although QLR requires a probabilistic prior on the fitness class.

Keywords: Noisy optimization; Bernstein races

Domains: Machine Learning [cs.LG]; Optimization and Control [math.OC]

Rémi Coulom, Philippe Rolet, Nataliya Sokolovska, Olivier Teytaud. Handling Expensive Optimization with Large Noise. Foundations of Genetic Algorithms, Jan 2011, Austria. pp.TBA. ⟨hal-00517157v2⟩