Linear Convergence of Comparison-based Step-size Adaptive Randomized Search via Stability of Markov Chains

In this paper, we consider comparison-based adaptive stochastic algorithms for solving numerical optimisation problems. We consider a specific subclass of algorithms that we call comparison-based step-size adaptive randomized search (CB-SARS), where the state variables at a given iteration are a vector of the search space and a positive parameter, the step-size, typically controlling the overall standard deviation of the underlying search distribution.We investigate the linear convergence of CB-SARS on\emph{scaling-invariant} objective functions. Scaling-invariantfunctions preserve the ordering of points with respect to their functionvalue when the points are scaled with the same positive parameter (thescaling is done w.r.t. a fixed reference point). This class offunctions includes norms composed with strictly increasing functions aswell as many non quasi-convex and non-continuousfunctions. On scaling-invariant functions, we show the existence of ahomogeneous Markov chain, as a consequence of natural invarianceproperties of CB-SARS (essentially scale-invariance and invariance tostrictly increasing transformation of the objective function). We thenderive sufficient conditions for \emph{global linear convergence} ofCB-SARS, expressed in terms of different stability conditions of thenormalised homogeneous Markov chain (irreducibility, positivity, Harrisrecurrence, geometric ergodicity) and thus define a general methodologyfor proving global linear convergence of CB-SARS algorithms onscaling-invariant functions. As a by-product we provide aconnexion between comparison-based adaptive stochasticalgorithms and Markov chain Monte Carlo algorithms.Comment: SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 201

Evolutionary Tournament-Based Comparison of Learning and Non-Learning Algorithms for Iterated Games

Evolutionary tournaments have been used effectively as a tool for comparing game-playing algorithms. For instance, in the late 1970's, Axelrod organized tournaments to compare algorithms for playing the iterated prisoner's dilemma (PD) game. These tournaments capture the dynamics in a population of agents that periodically adopt relatively successful algorithms in the environment. While these tournaments have provided us with a better understanding of the relative merits of algorithms for iterated PD, our understanding is less clear about algorithms for playing iterated versions of arbitrary single-stage games in an environment of heterogeneous agents. While the Nash equilibrium solution concept has been used to recommend using Nash equilibrium strategies for rational players playing general-sum games, learning algorithms like fictitious play may be preferred for playing against sub-rational players. In this paper, we study the relative performance of learning and non-learning algorithms in an evolutionary tournament where agents periodically adopt relatively successful algorithms in the population. The tournament is played over a testbed composed of all possible structurally distinct 2×2 conflicted games with ordinal payoffs: a baseline, neutral testbed for comparing algorithms. Before analyzing results from the evolutionary tournament, we discuss the testbed, our choice of representative learning and non-learning algorithms and relative rankings of these algorithms in a round-robin competition. The results from the tournament highlight the advantage of learning algorithms over players using static equilibrium strategies for repeated plays of arbitrary single-stage games. The results are likely to be of more benefit compared to work on static analysis of equilibrium strategies for choosing decision procedures for open, adapting agent society consisting of a variety of competitors.Repeated Games, Evolution, Simulation

Linear Convergence on Positively Homogeneous Functions of a Comparison Based Step-Size Adaptive Randomized Search: the (1+1) ES with Generalized One-fifth Success Rule

In the context of unconstraint numerical optimization, this paper investigates the global linear convergence of a simple probabilistic derivative-free optimization algorithm (DFO). The algorithm samples a candidate solution from a standard multivariate normal distribution scaled by a step-size and centered in the current solution. This solution is accepted if it has a better objective function value than the current one. Crucial to the algorithm is the adaptation of the step-size that is done in order to maintain a certain probability of success. The algorithm, already proposed in the 60's, is a generalization of the well-known Rechenberg's $(1+1)$ Evolution Strategy (ES) with one-fifth success rule which was also proposed by Devroye under the name compound random search or by Schumer and Steiglitz under the name step-size adaptive random search. In addition to be derivative-free, the algorithm is function-value-free: it exploits the objective function only through comparisons. It belongs to the class of comparison-based step-size adaptive randomized search (CB-SARS). For the convergence analysis, we follow the methodology developed in a companion paper for investigating linear convergence of CB-SARS: by exploiting invariance properties of the algorithm, we turn the study of global linear convergence on scaling-invariant functions into the study of the stability of an underlying normalized Markov chain (MC). We hence prove global linear convergence by studying the stability (irreducibility, recurrence, positivity, geometric ergodicity) of the normalized MC associated to the $(1+1)$-ES. More precisely, we prove that starting from any initial solution and any step-size, linear convergence with probability one and in expectation occurs. Our proof holds on unimodal functions that are the composite of strictly increasing functions by positively homogeneous functions with degree $\alpha$ (assumed also to be continuously differentiable). This function class includes composite of norm functions but also non-quasi convex functions. Because of the composition by a strictly increasing function, it includes non continuous functions. We find that a sufficient condition for global linear convergence is the step-size increase on linear functions, a condition typically satisfied for standard parameter choices. While introduced more than 40 years ago, we provide here the first proof of global linear convergence for the $(1+1)$-ES with generalized one-fifth success rule and the first proof of linear convergence for a CB-SARS on such a class of functions that includes non-quasi convex and non-continuous functions. Our proof also holds on functions where linear convergence of some CB-SARS was previously proven, namely convex-quadratic functions (including the well-know sphere function)

Genetic Optimization Using Derivatives: The rgenoud Package for R

genoud is an R function that combines evolutionary algorithm methods with a derivative-based (quasi-Newton) method to solve difficult optimization problems. genoud may also be used for optimization problems for which derivatives do not exist. genoud solves problems that are nonlinear or perhaps even discontinuous in the parameters of the function to be optimized. When the function to be optimized (for example, a log-likelihood) is nonlinear in the model's parameters, the function will generally not be globally concave and may have irregularities such as saddlepoints or discontinuities. Optimization methods that rely on derivatives of the objective function may be unable to find any optimum at all. Multiple local optima may exist, so that there is no guarantee that a derivative-based method will converge to the global optimum. On the other hand, algorithms that do not use derivative information (such as pure genetic algorithms) are for many problems needlessly poor at local hill climbing. Most statistical problems are regular in a neighborhood of the solution. Therefore, for some portion of the search space, derivative information is useful. The function supports parallel processing on multiple CPUs on a single machine or a cluster of computers.