Covariance Matrix Adaptation Research Articles

Optimal-Margin Evolutionary Classifier

We introduce a novel approach for discriminative classification using evolutionary algorithms. We first propose an algorithm to optimize the total loss value using a modified 0–1 loss function in a 1-D space for classification. We then extend this algorithm for multidimensional classification using an evolutionary algorithm. The proposed evolutionary algorithm aims to find a hyperplane that best classifies instances while minimizing the classification risk. We test particle swarm optimization, evolutionary strategy (ES), and covariance matrix adaptation ES for optimization purposes. After parameter selection, we compare our results with well-established and state-of-the-art classification algorithms, for both binary and multiclass classification, on 23 benchmark classification problems, with and without noise and outliers. We also compare these methods on a seizure detection task for 12 epileptic patients. Results show that the performance of the proposed algorithm is significantly (Wilcoxon test) better than all other methods in almost all problems tested. We also show that the proposed algorithm is significantly more robust against noise and outliers comparing to other methods. The running time of the algorithm is within a reasonable range for the solution of real-world classification problems.

Evolution Strategies Learning With Variable Impedance Control for Grasping Under Uncertainty

During a robot's interaction with the environment, it is necessary to ensure the safety and robustness of the robot's movements. To improve the safety and adaptiveness of robots in performing complex movement tasks, a novel method called covariance matrix adaptation-evolution strategies (CMA-ES) for learning complex and high-dimensional motor skills is presented. Considering the complex motion model of trajectories, dynamic movement primitives (DMPs), which is a generic method for trajectories modeling in attractor landscape based on differential dynamic systems, is used to represent the robot's trajectories. CMA-ES offers a theoretical rule for updating the parameters of DMPs and a variable impedance controller, which can reduce the impact of noisy environment on the robot's movement. In this paper, we propose two hierarchies for controlling the robot: the high-level neural-dynamic network optimization for redundancy resolution in task space and the low-level CMA-ES fusing with DMPs for learning trajectories in joint space. In this paper, CMA-ES method is explored to learn variable impedance control and the performance of the proposed method in learning the robot's movements is also tested.

OPTIMIZING THE THERMAL TRANSPORT PROPERTIES OF SINGLE LAYER (2D) TRANSITION METAL DICHALCOGENIDES (TMD)

In order to characterize thermal dependent physical properties of materials, potentially to be used in technological applications, an accurate interatomic-potential parameter set is a must. In general, conjugate-gradient methods and more recently, metaheuristics such as genetic algorithms are employed in determining these interatomic potentials, however, especially the use of metaheuristics specifically designed for optimization of real valued problems such as particle swarm and evaluation strategies are limited in the mentioned problem. In addition, some of these parameters are conflicting in nature, for which multi objective optimization procedures have a great potential for better understanding of these conflicts. In this respect, we aim to present a widely used interatomic potential parameter set, the Stillinger–Weber potential, obtained through three different optimization methods (particle swarm optimization, PSO, covariance matrix adaptation evolution strategies, CMA-ES, and non-dominated sorting genetic algorithm, NSGA-III) for two-dimensional materials MoS2, WS2, WSe2, and MoSe2. These two-dimensional transition metal dichalcogenides are considered as a case mainly due to their potential in a variety of promising technologies for next generation flexible and low-power nanoelectronics, (such as photonics, valleytronics, sensing, energy storage, and optoelectronic devices) as well as their excellent physical properties (such as electrical, mechanical, thermal, and optical properties) different from those of their bulk counterparts. The results show that the outputs of all optimization methods converge to ideal values with sufficiently long iterations and at different trials. However, when we consider the results of the statistical analyses of different trials under similar conditions, we observe that the method with the lowest error rate is the CMA-ES.

Multi-objective simulation optimization for complex urban mass rapid transit systems

In this paper, we present a multi-objective simulation-based headway optimization for complex urban mass rapid transit systems. Real-world applications often confront conflicting goals of cost versus service level. We propose a two-phase algorithm that combines the single-objective covariance matrix adaptation evolution strategy with a problem-specific multi-directional local search. With a computational study, we compare our proposed method against both a multi-objective covariance matrix adaptation evolution strategy and a non-dominated sorting genetic algorithm. The integrated discrete event simulation model has several stochastic elements. Fluctuating demand (i.e., creation of passengers) is driven by hourly origin-destination-matrices based on mobile phone and infrared count data. We also consider the passenger distribution along waiting platforms and within vehicles. Our two-phase optimization scheme outperforms the comparative approaches, in terms of both spread and the accuracy of the resulting Pareto front approximation.

Towards large‐scale stochastic refraction tomography: a comparison of three evolutionary algorithms

ABSTRACTThe main goal of this study is to assess the potential of evolutionary algorithms to solve highly non‐linear and multi‐modal tomography problems (such as first arrival traveltime tomography) and their abilities to estimate reliable uncertainties. Classical tomography methods apply derivative‐based optimization algorithms that require the user to determine the value of several parameters (such as regularization level and initial model) prior to the inversion as they strongly affect the final inverted model. In addition, derivative‐based methods only perform a local search dependent on the chosen starting model. Global optimization methods based on Markov chain Monte Carlo that thoroughly sample the model parameter space are theoretically insensitive to the initial model but turn out to be computationally expensive. Evolutionary algorithms are population‐based global optimization methods and are thus intrinsically parallel, allowing these algorithms to fully handle available computer resources. We apply three evolutionary algorithms to solve a refraction traveltime tomography problem, namely the differential evolution, the competitive particle swarm optimization and the covariance matrix adaptation–evolution strategy. We apply these methodologies on a smoothed version of the Marmousi velocity model and compare their performances in terms of optimization and estimates of uncertainty. By performing scalability and statistical analysis over the results obtained with several runs, we assess the benefits and shortcomings of each algorithm.

The spherical search algorithm for bound-constrained global optimization problems

In this paper, a new optimization algorithm called Spherical Search (SS) is proposed to solve the bound-constrained non-linear global optimization problems. The main operations of SS are the calculation of spherical boundary and generation of new trial solution on the surface of the spherical boundary. These operations are mathematically modeled with some more basic level operators: Initialization of solution, greedy selection and parameter adaptation, and are employed on the 30 black-box bound constrained global optimization problems. This study also analyzes the applicability of the proposed algorithm on a set of real-life optimization problems. Meanwhile, to show the robustness and proficiency of SS, the obtained results of the proposed algorithm are compared with the results of other well-known optimization algorithms and their advanced variants: Particle Swarm Optimization (PSO), Differential Evolution (DE), and Covariance Matrix Adapted Evolution Strategy (CMA-ES). The comparative analysis reveals that the performance of SS is quite competitive with respect to the other peer algorithms.

CMA-ES based fuzzy clustering approach for MRI images segmentation

One of emerging challenges in Medical image analysis is clustering. Fuzzy C-means (FCM) algorithm is one of the most popular clustering algorithms because it is efficient, straightforward, and easy to implement. However, FCM is sensitive to initialization and is easily trapped in local optima, such a drawback could be overcome by evolutionary algorithms. This paper is dedicated to implement a fuzzy strategy evolutionary approach based to optimize the centers of the clusters by minimizing the objective function of the FCM algorithm. This approach is based on the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) optimization algorithm to find the optimum values of the centers of the clusters in order to classify Magnetic Resonance Imaging (MRI) brain images. The proposed approach has been validated against both simulated and clinical MRI and it has yield competitive results when compared to FCM algorithms. Results show that the proposed algorithm has obtained reasonable segmentation of white matter, gray matter, and cerebrospinal fluid from MRI data.

An Experimental Method to Estimate Running Time of Evolutionary Algorithms for Continuous Optimization

Running time analysis is a fundamental problem of critical importance in evolutionary computation. However, the analysis results have rarely been applied to advanced evolutionary algorithms (EAs) in practice, let alone their variants for continuous optimization. In this paper, an experimental method is proposed for analyzing the running time of EAs that are widely used for solving continuous optimization problems. Based on Glivenko–Cantelli theorem, the proposed method simulates the distribution of gain, which is introduced by average gain model to characterize progress during the optimization process. Data fitting techniques are subsequently adopted to obtain a desired function for further analyses. To verify the validity of the proposed method, experiments were conducted to estimate the upper bounds on expected first hitting time of various evolutionary strategies, such as (1, $\lambda $ ) evolution strategy, standard evolution strategy, covariance matrix adaptation evolution strategy, and its improved variants. The results suggest that all estimated upper bounds are correct. Backed up by the proposed method, state-of-the-art EAs for continuous optimization will have identical results about the running time as simplified schemes, which will bridge the gap between theoretical foundation and applications of evolutionary computation.

Inverse design of charged colloidal particle interactions for self assembly into specified crystal structures.

We study the inverse problem of tuning interaction parameters between charged colloidal particles interacting with a hard-core repulsive Yukawa potential, so that they assemble into specified crystal structures. Here, we target the body-centered-cubic (bcc) structure which is only stable in a small region in the phase diagram of charged colloids and is, therefore, challenging to find. In order to achieve this goal, we use the statistical fluctuations in the bond orientational order parameters to tune the interaction parameters for the bcc structure, while initializing the system in the fluid phase, using the Statistical Physics-inspired Inverse Design algorithm. We also find that this optimization algorithm correctly senses the fluid-solid phase boundaries for charged colloids. Finally, we repeat the procedure employing the covariance matrix adaptation-evolution strategy, a cutting edge optimization technique, and compare the relative efficacy of the two methods.

A covariance matrix adaptation evolution strategy variant and its engineering application

This paper proposes a novel covariance matrix adaptation evolution strategy (CMA-ES) variant, named AEALSCE, for single-objective numerical optimization problems in the continuous domain. To avoid premature convergence and strengthen the exploration capacity of the basic CMA-ES, AEALSCE is obtained by integrating the CMA-ES with two strategies that can adjust the evolutionary directions and enrich the population diversity. The first strategy is named the anisotropic eigenvalue adaptation (AEA) technique, which adapts the search scope towards the optimal evolutionary directions. It scales the eigenvalues of the covariance matrix anisotropically based on local fitness landscape detection. The other strategy is named the local search (LS) strategy, which is executed under the eigen coordinate system and can be subdivided into two parts. In the first part, the new candidates of superior solutions are sampled around the best solution to perform local exploration. In the other part, the new candidates of inferior solutions are generated using a modified mean point along the fitness descent direction. The proposed AEALSCE algorithm is compared with other top competitors, including the CEC 2014 champion, L-SHADE, and the promising NBIPOP-aCMA-ES, by benchmarking the CEC 2014 testbed. Moreover, AEALSCE is applied in solving three constrained engineering design problems and parameter estimation of photovoltaic (PV) models. According to the statistical results of the experiments, our proposed AEALSCE is competitive with other algorithms in convergence efficiency and accuracy. AEALSCE benefits from a good balance of exploration and exploitation, and it exhibits a potential to address real-world optimization problems.

Model complex control CMA-ES

Abstract Covariance Matrix Adaptation Evolution Strategy (CMA-ES) has shown great performance on nonseparable optimization problems largely due to its rotation-invariant feature. However, as the computational cost of the self-adaption operation is sensitive to the scale of problems, the performance of CMA-ES heavily suffers from the well-known curse of dimensionality, which makes it impractical to many Large Scale Global Optimization (LSGO) problems. In this paper, a correlation coefficient based grouping (CCG) strategy is proposed to detect the correlations between variables in a simple yet efficient way. Then coupled with a model complexity control (MCC) framework, a new variant of CMA-ES, named MCC-CCG-CMAES, is presented for LSGO problems, which suffers less from curse of dimensionality and significantly reduces the computational cost compared with the standard CMA-ES. To the best of our knowledge, this work is the first attempt at enhancing CMA-ES with the MCC framework rather than the cooperative coevolution (CC) framework. Experimental results on the CEC′2010 large-scale global optimization (LSGO) benchmark functions show that the performance of MCC-CCG-CMAES outperforms the state-of-the-art counterparts.

Mapping Neural Networks to FPGA-Based IoT Devices for Ultra-Low Latency Processing.

Internet of things (IoT) infrastructure, fast access to knowledge becomes critical. In some application domains, such as robotics, autonomous driving, predictive maintenance, and anomaly detection, the response time of the system is more critical to ensure Quality of Service than the quality of the answer. In this paper, we propose a methodology, a set of predefined steps to be taken in order to map the models to hardware, especially field programmable gate arrays (FPGAs), with the main focus on latency reduction. Multi-objective covariance matrix adaptation evolution strategy (MO-CMA-ES) was employed along with custom scores for sparsity, bit-width of the representation and quality of the model. Furthermore, we created a framework which enables mapping of neural models to FPGAs. The proposed solution is validated using three case studies and Xilinx Zynq UltraScale+ MPSoC 285 XCZU15EG as a platform. The results show a compression ratio for quantization and pruning in different scenarios with and without retraining procedures. Using our publicly available framework, we achieved 210 ns of latency for a single processing step for a model composed of two long short-term memory (LSTM) and a single dense layer.

Evolving Local Plasticity Rules for Synergistic Learning in Echo State Networks.

Existing synaptic plasticity rules for optimizing the connections between neurons within the reservoir of echo state networks (ESNs) remain to be global in that the same type of plasticity rule with the same parameters is applied to all neurons. However, this is biologically implausible and practically inflexible for learning the structures in the input signals, thereby limiting the learning performance of ESNs. In this paper, we propose to use local plasticity rules that allow different neurons to use different types of plasticity rules and different parameters, which are achieved by optimizing the parameters of the local plasticity rules using the evolution strategy (ES) with covariance matrix adaptation (CMA-ES). We show that evolving neural plasticity will result in a synergistic learning of different plasticity rules, which plays an important role in improving the learning performance. Meanwhile, we show that the local plasticity rules can effectively alleviate synaptic interferences in learning the structure in sensory inputs. The proposed local plasticity rules are compared with a number of the state-of-the-art ESN models and the canonical ESN using a global plasticity rule on a set of widely used prediction and classification benchmark problems to demonstrate its competitive learning performance.

Comparing optimization algorithms for conventional and freeform optical design.

Five search algorithms from the literature of black-box optimization were implemented and applied to optical design problems. These algorithms are the Particle Swarm Optimization, Gravity Search Algorithm, Cuckoo Search, Covariance Matrix Adaptation Evolution Strategy and Nelder&Mead Simplex search. The performance of these search algorithms' implementations was assessed using the BBOB2009 (Black Box Optimization Benchmark) benchmark suite. These algorithms were compared in the context of two optical case studies, one with conventional rotationally symmetric optics and one with freeform optics. A comparison was performed against a commercial optical design software. Finally we provided a simple restart scheme applicable in the workflow of an optical designer. To the best of our knowledge, this is the first in-depth quantitative comparison of optimization algorithms for optical design.

A covariance matrix adaptation evolution strategy in reproducing kernel Hilbert space

The covariance matrix adaptation evolution strategy (CMA-ES) is an efficient derivative-free optimization algorithm. It optimizes a black-box objective function over a well-defined parameter space in which feature functions are often defined manually. Therefore, the performance of those techniques strongly depends on the quality of the chosen features or the underlying parametric function space. Hence, enabling CMA-ES to optimize on a more complex and general function class has long been desired. In this paper, we consider modeling the input spaces in black-box optimization non-parametrically in reproducing kernel Hilbert spaces (RKHS). This modeling leads to a functional optimisation problem whose domain is a RKHS function space that enables optimisation in a very rich function class. We propose CMA-ES-RKHS, a generalized CMA-ES framework that is able to carry out black-box functional optimisation in RKHS. A search distribution on non-parametric function spaces, represented as a Gaussian process, is adapted by updating both its mean function and covariance operator. Adaptive and sparse representation of the mean function and the covariance operator can be retained for efficient computation in the updates and evaluations of CMA-ES-RKHS by resorting to sparsification. We will also show how to apply our new black-box framework to search for an optimum policy in reinforcement learning in which policies are represented as functions in a RKHS. CMA-ES-RKHS is evaluated on two functional optimization problems and two bench-marking reinforcement learning domains.

Metamodel assisted evolution strategies for global optimization of solder joints reliability in embedded mechatronic devices

This paper proposes a Kriging assisted covariance matrix adaptation evolution strategy (AK-CMA-ES) for optimization of mechatronics device structures. The objective of the proposed method is to design reliable structures with reduced computational cost of the multiphysics finite element simulation. The Finite element model intends to analyze the sequence of failure events in mechatronic devices. This numerical model is used to estimate the thermal cycles to failure. Subsequently, the AK-CMA-ES optimization process is performed in order to improve the performance of structural design of mechatronic systems and to find the best designs with safety and reasonable computational costs. The proposed method couple the kriging metamodel with the covariance matrix adaptation evolution strategy (CMA-ES). Kriging metamodel is used to replace the finite element simulation in order to overcome the computational cost of finite element model simulation. Kriging is used together with CMA-ES and sequentially updated and its quality is measured according to its ability in the ranking of the population through approximate ranking procedure (ARP). The application of this method in the optimization of mechatronic systems proves its efficiency and ability to improve the performance of mechatronic systems with avoiding the problem of tedious computation.

Reliability Assessment Model for Industrial Control System Based on Belief Rule Base

This paper establishes a novel reliability assessment method for industrial control system (ICS). Firstly, the qualitative and quantitative information were integrated by evidential reasoning(ER) rule. Then, an ICS reliability assessment model was constructed based on belief rule base (BRB). In this way, both expert experience and historical data were fully utilized in the assessment. The model consists of two parts, a fault assessment model and a security assessment model. In addition, the initial parameters were optimized by covariance matrix adaptation evolution strategy (CMA-ES) algorithm, making the proposed model in line with the actual situation. Finally, the proposed model was compared with two other popular prediction methods through case study. The results show that the proposed method is reliable, efficient and accurate, laying a solid basis for reliability assessment of complex ICSs.

Diagonal Acceleration for Covariance Matrix Adaptation Evolution Strategies.

We introduce an acceleration for covariance matrix adaptation evolution strategies (CMA-ES) by means of adaptive diagonal decoding (dd-CMA). This diagonal acceleration endows the default CMA-ES with the advantages of separable CMA-ES without inheriting its drawbacks. Technically, we introduce a diagonal matrix that expresses coordinate-wise variances of the sampling distribution in DCD form. The diagonal matrix can learn a rescaling of the problem in the coordinates within a linear number of function evaluations. Diagonal decoding can also exploit separability of the problem, but, crucially, does not compromise the performance on nonseparable problems. The latter is accomplished by modulating the learning rate for the diagonal matrix based on the condition number of the underlying correlation matrix. dd-CMA-ES not only combines the advantages of default and separable CMA-ES, but may achieve overadditive speedup: it improves the performance, and even the scaling, of the better of default and separable CMA-ES on classes of nonseparable test functions that reflect, arguably, a landscape feature commonly observed in practice. The article makes two further secondary contributions: we introduce two different approaches to guarantee positive definiteness of the covariance matrix with active CMA, which is valuable in particular with large population size; we revise the default parameter setting in CMA-ES, proposing accelerated settings in particular for large dimension. All our contributions can be viewed as independent improvements of CMA-ES, yet they are also complementary and can be seamlessly combined. In numerical experiments with dd-CMA-ES up to dimension 5120, we observe remarkable improvements over the original covariance matrix adaptation on functions with coordinate-wise ill-conditioning. The improvement is observed also for large population sizes up to about dimension squared.

Adaptive covariance estimation of LiDAR‐based positioning errors for UAVs

Outdoor positioning for unmanned aerial vehicles (UAVs) commonly relies on global navigation satellite system (GNSS) signals, which might be reflected or blocked in urban areas. Thus, additional on-board sensors such as light detection and ranging (LiDAR) are desirable to aid positioning. To fuse measurements from different sensors, it is important to accurately characterize the error covariance matrices of individual sensor measurements. We propose a novel method for adaptively estimating the LiDAR-based positioning error covariance matrix based on the point cloud features surrounding the UAV. We model the position error as a multivariate Gaussian distribution and estimate its covariance matrix from individual surface and edge feature points. Simulations show that our model is more accurate than a distance-based covariance matrix model. Furthermore, we conduct an outdoor experiment that fuses global positioning system (GPS) signals and LiDAR position measurements. We demonstrate a clear improvement in the UAV's global position estimation using our adaptive covariance matrix for LiDAR-based measurements.

Population-based simulation optimization for urban mass rapid transit networks

In this paper, we present a simulation-based headway optimization for urban mass rapid transit networks. The underlying discrete event simulation model contains several stochastic elements, including time-dependent demand and turning maneuver times as well as direction-dependent vehicle travel and passenger transfer times. Passenger creation is a Poisson process that uses hourly origin–destination-matrices based on anonymous mobile phone and infrared count data. The numbers of passengers on platforms and within vehicles are subject to capacity restrictions. As a microscopic element, passenger distribution along platforms and within vehicles is considered. The bi-objective problem, involving cost reduction and service level improvement, is transformed into a single-objective optimization problem by normalization and scalarization. Population-based evolutionary algorithms and different solution encoding variants are applied. Computational experience is gained from test instances based on real-world data (i.e., the Viennese subway network). A covariance matrix adaptation evolution strategy performs best in most cases, and a newly developed encoding helps accelerate the optimization process by producing better short-term results.