Exponential Natural Evolution Strategies

The family of natural evolution strategies (NES) offers a principled approach to real-valued evolutionary optimization by following the natural gradient of the expected fitness. Like the well-known CMA-ES, the most competitive algorithm in the field, NES comes with important invariance properties. In this paper, we introduce a number of elegant and efficient improvements of the basic NES algorithm. First, we propose to parameterize the positive definite covariance matrix using the exponential map, which allows the covariance matrix to be updated in a vector space. This new technique makes the algorithm completely invariant under linear transformations of the underlying search space, which was previously achieved only in the limit of small step sizes. Second, we compute all updates in the natural coordinate system, such that the natural gradient coincides with the vanilla gradient. This way we avoid the computation of the inverse Fisher information matrix, which is the main computational bottleneck of the original NES algorithm. Our new algorithm, exponential NES (xNES), is significantly simpler than its predecessors. We show that the various update rules in CMA-ES are closely related to the natural gradient updates of xNES. However, xNES is more principled than CMA-ES, as all the update rules needed for covariance matrix adaptation are derived from a single principle. We empirically assess the performance of the new algorithm on standard benchmark function

On the exponential stability of uniformly damped wave equations

We study damped wave propagation problems phrased as abstract evolution equations in Hilbert spaces. Under some general assumptions, including a natural compatibility condition for initial values, we establish exponential decay estimates for all mild solutions using the language and tools of Hilbert complexes. This framework turns out strong enough to conduct our analysis but also general enough to include a number of interesting examples. Some of these are briefly discussed. By a slight modification of the main arguments, we also obtain corresponding decay results for numerical approximations obtained by compatible discretization strategies

Intertemporal accounting of climate change - Harmonizing economic efficiency and climate stewardship

Continuing a discussion on the intertemporal accounting of climate-change damages initiated by Nordhaus, Heal and Brown in response to the recent demonstration of Hasselmann et al. that standard exponential discounting applied uniformly to all goods and services invariably leads to a `climate catastrophe' in cost-benefit analyses, it is argued that (1) there exists no economically satisfactory alternative to cost-benefit analysis for the determination of optimal climate protection strategies, and (2) it is essential to allow for the different long-term evolution of climate damage costs relative to mitigation costs in determining the optimal cost-benefit solution. A climate catastrophe can be avoided only if it is assumed that climate damage costs increase significantly in the long term relative to mitigation costs. Cost-benefit analysis is regarded here in the generalized sense of optimizing a social welfare function that incorporates all relevant `quality-of-life' factors, including not only consumption and the value of the environment, but also the ethical values of equitable intertemporal and intrasocietal distribution. Thus, economic efficiency and climate stewardship are not regarded as conflicting goals, but as synonyms for a single encompassing economic optimization exercise. The same reasoning applies generally to the problem of sustainable development. To quantify the concept of sustainable development in cost-benefit analyses, the projected time evolution of the future values of natural resources and the environment (judged by the present generation, acting as representative agents of future generations) must be related to the time-evolution of all other relevant quality-of-life factors. Different ethical interpretations of the concept of sustainable development can be readily operationalized by incorporation in a generalized cost-benefit analysis in which the evolution paths of all relevant material and ethical values are explicitly specified

Information-Geometric Optimization Algorithms: A Unifying Picture via Invariance Principles

We present a canonical way to turn any smooth parametric family of probability distributions on an arbitrary search space $X$ into a continuous-time black-box optimization method on $X$, the \emph{information-geometric optimization} (IGO) method. Invariance as a design principle minimizes the number of arbitrary choices. The resulting \emph{IGO flow} conducts the natural gradient ascent of an adaptive, time-dependent, quantile-based transformation of the objective function. It makes no assumptions on the objective function to be optimized. The IGO method produces explicit IGO algorithms through time discretization. It naturally recovers versions of known algorithms and offers a systematic way to derive new ones. The cross-entropy method is recovered in a particular case, and can be extended into a smoothed, parametrization-independent maximum likelihood update (IGO-ML). For Gaussian distributions on $\mathbb{R}^d$, IGO is related to natural evolution strategies (NES) and recovers a version of the CMA-ES algorithm. For Bernoulli distributions on $\{0,1\}^d$, we recover the PBIL algorithm. From restricted Boltzmann machines, we obtain a novel algorithm for optimization on $\{0,1\}^d$. All these algorithms are unified under a single information-geometric optimization framework. Thanks to its intrinsic formulation, the IGO method achieves invariance under reparametrization of the search space $X$, under a change of parameters of the probability distributions, and under increasing transformations of the objective function. Theory strongly suggests that IGO algorithms have minimal loss in diversity during optimization, provided the initial diversity is high. First experiments using restricted Boltzmann machines confirm this insight. Thus IGO seems to provide, from information theory, an elegant way to spontaneously explore several valleys of a fitness landscape in a single run.Comment: Final published versio