[论文解读] 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

Anne Auger, Nikolaus Hansen|arXiv (Cornell University)|Oct 31, 2013

Metaheuristic Optimization Algorithms Research参考文献 15被引用 21

一句话总结

本文证明了在正齐次函数（度数为 α）上，采用广义一分为五成功规则的（1+1）进化策略，通过基于比较的步长自适应机制，实现了全局线性收敛。该分析建立了在由严格递增函数与正齐次分量组成的单峰函数类上，几乎必然收敛与期望收敛的线性收敛性，包括非拟凸与非连续函数，前提是步长在直线函数上的增加满足充分条件。

ABSTRACT

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 $α$ (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).

研究动机与目标

• 在广泛的一类单峰函数上，建立基于比较的步长自适应随机搜索算法的全局线性收敛性。
• 解决长期悬而未决的难题：证明（1+1）ES使用广义一分为五成功规则时的线性收敛性。
• 将收敛性分析从凸二次函数与球形函数扩展至非连续与非拟凸函数。
• 形式化归一化马尔可夫链在基于比较算法收敛性分析中的应用。
• 验证该算法在以往分析仅限于简单类别的函数上仍能保持线性收敛。

提出的方法

• 该算法使用以当前解为中心的多变量正态分布，步长通过基于比较的成功率控制进行自适应调整。
• 步长自适应遵循广义一分为五成功规则，根据随时间变化的成功概率调整 σ。
• 收敛性分析利用不变性性质，将问题简化为在单位球面上研究归一化马尔可夫链（MC）的稳定性。
• 通过分析归一化MC的不可约性、常返性、正性与几何遍历性，证明线性收敛性。
• 证明依赖于一个几何漂移条件，确保归一化链快速收敛至其平稳分布。
• 收敛速率表示为 CR = −lnγ[(q+1)/q × PS − 1/q]，其中 PS 为渐近成功概率。

实验结果

研究问题

• RQ1 （1+1）ES使用广义一分为五成功规则时，是否在度数为 α 的正齐次函数上实现全局线性收敛？
• RQ2 能否证明基于比较的算法在非拟凸与非连续函数上的线性收敛性？
• RQ3 收敛速率是否与初始条件无关且几何快速？
• RQ4 步长自适应规则与线性收敛性在直线函数上的关系为何？
• RQ5 归一化马尔可夫链框架能否严格证明CB-SARS在凸二次函数之外的函数上实现线性收敛？

主要发现

• 对于由严格递增函数与度数为 α 的正齐次函数组成的单峰函数，（1+1）ES使用广义一分为五成功规则时，几乎必然收敛与期望收敛均成立。
• 该收敛性证明适用于比以往结果更广泛的函数类，包括非连续与非拟凸函数。
• 线性收敛的充分条件是步长在直线函数上增加，该条件由标准参数选择满足。
• 收敛速率由 CR = −lnγ[(q+1)/q × PS − 1/q] 给出，当收敛发生时，PS < 1/(q+1)。
• 与算法相关的归一化马尔可夫链是几何遍历的，确保收敛至平稳分布的速度与初始条件无关。
• 分析证实，CB-SARS可在球形函数与凸二次函数上实现线性收敛，扩展了先前结果。