Multi-objective optimization via equivariant deep hypervolume approximation

Optimizing multiple competing objectives is a common problem across science and industry. The inherent inextricable trade-off between those objectives leads one to the task of exploring their Pareto front. A meaningful quantity for the purpose of the latter is the hypervolume indicator, which is used in Bayesian Optimization (BO) and Evolutionary Algorithms (EAs). However, the computational complexity for the calculation of the hypervolume scales unfavorably with increasing number of objectives and data points, which restricts its use in those common multi-objective optimization frameworks. To overcome these restrictions we propose to approximate the hypervolume function with a deep neural network, which we call DeepHV. For better sample efficiency and generalization, we exploit the fact that the hypervolume is scale-equivariant in each of the objectives as well as permutation invariant w.r.t. both the objectives and the samples, by using a deep neural network that is equivariant w.r.t. the combined group of scalings and permutations. We evaluate our method against exact, and approximate hypervolume methods in terms of accuracy, computation time, and generalization. We also apply and compare our methods to state-of-the-art multi-objective BO methods and EAs on a range of synthetic benchmark test cases. The results show that our methods are promising for such multi-objective optimization tasks.Comment: Updated with camera-ready version. Accepted at ICLR 202

The segmentation issue: general stopping criteria and specific design considerations for practical application of evolutionary algorithms

Segmentation is a tool presented for representation and approximation of data, according to a set of appropriate models. These procedures have applications to many different domains, such as time series analysis, polygonal approximation, Air Traffic Control,... Different heuristic and metaheuristic proposals have been introduced to deal with this issue. This thesis provides a novel multiobjective evolutionary method, analyzing the required general tools for the application evolutionary algorithms to real problems and the specific modifications required over the different steps of general proposals to adapt them to the segmentation domain. An introduction to the domain is presented by means of the design of a specific heuristic for segmentation of Air Traffic Control (ATC) data. This domain has a series of characteristics which make it difficult to be faced with traditional techniques: noisy data and a large number of measurements. The proposal works on two phases, using a pre-segmentation which introduces available domain information and applying a standard technique over this initial technique's results. Its results according to the presented domain, tested with a set of eight different representative trajectories, show competitive advantages compared to general approaches, which oversegmentate noisy data and, in some cases, exhibit poor scalability. This heuristic proposal shows the costly process of adapting available approaches and designing specific ones, along with the multi-objective nature of the problem, which requires the use of quality indicators for a proper comparison process. Applying evolutionary algorithms to segmentation provides several advantages, highlighting the fact that the problem dependance of heuristics make it costly to adapt these heuristics to new domains, as introduced by the designed heuristic to ATC. However, the practical application of these algorithms requires the study of a topic which has received little research effort from the community: stopping criteria. An evolutionary approach should contain a dynamic procedure which can determine when stagnation has taken place and stop the algorithm accordingly (as opposed to a-priori cost budgets, either in function evaluations or generations, which are usually applied for test datasets). Stopping criteria have been faced for single and multi-objective cases in this thesis. Single-objective stopping criteria have been approached proposing an active role of the stopping criteria, actively increasing the diversity in the variable space while tracking the updates in the fitness function. Thus, the algorithm reuses the information obtained for the stopping decision and feeds it to a stopping prevention mechanism in order to prevent problematic situations such as early convergence. The presented algorithm has been tested according to a set of 27 different functions, with different characteristics regarding their dimensionality, search space, local minima... The results show that the introduced mechanisms enhance the robustness of the results, due to the improved exploration and the early convergence prevention. Multi-objective stopping criteria are faced with the use of progress indicators (comparison measures of the quality of the evolution results at different generations) and an associated data gathering tool. The final proposal uses three different progress indicators, (hypervolume, epsilon and Mutual Dominance Rate) and considers them jointly according to a decision fusion architecture. The stagnation analysis is based on the least squares regression parameters of the indicators values, including a normality analysis as well. The online nature of these algorithms is highlighted, preventing the recomputation of the indicators values which were present in other available alternatives, and also focusing on the simplicity of the final proposal, in order to reduce the cost of introducing it into available algorithms. The proposal has been tested with instances of the DTLZ algorithm family, obtaining satisfactory stops with a standard set of configuration values for the technique. However, there is a lack of quantitative measures to determine the objective quality of a stop and to properly compare its value to other alternatives. The multi-objective nature of the segmentation problem is analyzed to propose a multiobjective evolutionary algorithm (MOEA) to deal with it. This nature is analyzed according to a selection of available approaches, highlighting the difficulties which had to be faced in the parameter configuration in order to guide the processes to the desired solution values. A multi-objective a-posteriori approach such as the one presented allows the decision maker to choose from the front of possible final solutions the one which suits him best, simplifying this process. The presented approach chooses SPEA2 as its underlying MOEA, analyzing different representation and initialization proposals. The results have been validated against a representative set of heuristic and metaheuristic techniques, using three widely extended curves from the polygonal approximation domain (chromosome, leaf and semicircle), obtaining statistically better results for almost all the different test cases. This initial MOEA approach had unresolved issues, such as the archiving technique complexity order, and also lacked the proper specific design considerations to adapt it to the application domain. These issues have been faced according to different improvements. First of all, an alternative representation is proposed, including partial fitness information and associated fitness-aware transformation operators (transformation operators which compute children fitness values according to their changes and the parents partial values). A novel archiving procedure is introduced according to the bi-objective nature of the domain, being one of them discrete. This leads to a relaxed Pareto dominance check, named epsilon glitches. Multi-objective local search versions of the traditional algorithms are proposed and tested for the initialization of the algorithm, along with the stopping criterion proposal which has also been adapted to the problem characteristics. The archive size in this case is big enough to contain all the different individuals in the optimal front, such that quality assessment is simplified and a simpler mechanism can be introduced to detect stagnation, according to the improvements in each of the possible individuals. The final evolutionary proposal is scalable, requires few configuration parameters and introduces an efficient dynamic stopping criterion. Its results have been tested against the original technique and the set of heuristic and metaheuristic techniques previously used, including the three original curves and also more complex versions of them (obtained with an introduced generation mechanism according to these original shapes). Even though the stopping results are very satisfactory, the obtained results are slightly worse than the original MOEA for the three simpler problem instances with the established configuration parameters (as was expected, due to the computational effort of the a-priori established number of generations and population size, based on the analysis of the algorithm's results). However, the comparison versus the alternative techniques stills shows the same statistically better results, and its reduced computational cost allows its application to a wider set of problems.La segmentación es una técnica creada para la representación y la aproximación de conjuntos de datos a través de un conjunto de modelos apropiados. Estos procedimientos tienen aplicaciones para múltiples dominios distintos, como el análisis de series temporales, la aproximación poligonal o el Control de Tráfico Aéreo. Se han hecho múltiples propuestas tanto de carácter heurístico como metaheurístico para lidiar con este problema. Esta tesis proporciona un nuevo método evolutivo multiobjetivo, analizando las herramientas generales necesarias para la aplicación de algoritmos evolutivos a problemas reales y las modificaciones específicas necesarias sobre los distintos pasos de las propuestas genéricas para adaptarlos al dominio de la segmentación. Se presenta una introducción al dominio mediante el diseño de una heurística específica para la segmentación de datos procedentes del Control de Tráfico Aéreo (CTA). Este dominio tiene una serie de características que dificultan la aplicación de técnicas tradicionales: datos con ruido y un gran número de muestras. La propuesta realizada funciona de acuerdo a dos fases, utilizando una presegmentación que introduce información del dominio disponible para posteriormente aplicar una técnica estándar sobre los resultados de esta técnica inicial. Sus resultados para el dominio presentado, probado con un conjunto de ocho trayectorias representativas distintas, presentan ventajas competitivas frente a los enfoques generales, que sobresegmentan los datos con ruido y, en algunos casos, presentan una mala escalabilidad. Esta propuesta heurística muestra el costoso proceso que implica adaptar los enfoques existentes o el diseño de otros nuevos, junto a la naturaleza multiobjectivo del problema, que precisa del uso de indicadores de calidad para realizar un proceso de comparación apropiado. La aplicación de algoritmos evolutivos a la segmentación tiene múltiples ventajas, destacando el hecho de la dependencia existente entre las heurísticas y el problema específico para el que han sido diseñadas, lo que hace que su adaptación a nuevos dominios sea costosa, como se ha introducido a través de la propuesta heurística para CTA. A pesar de ello, la aplicación práctica de estos algoritmos requiere el estudio de una faceta que ha recibido poca atención por parte de la comunidad desde el punto de vista de la investigación: los criterios de parada. Un enfoque evolutivo debería tener una técnica dinámica que pueda detectar cuando se ha producido el estancamiento del proceso, y parar el algoritmo de acuerdo a ello (de manera opuesta a los criterios a-priori que establecen un coste predeterminado, expresado como número de evaluaciones o de generaciones, y que son habitualmente aplicados para los conjuntos de datos de prueba). Los criterios de parada se han afrontado tanto desde el caso de un único objetivo como desde el caso multiobjectivo en esta tesis. Los criterios de parada para un único objetivo se han abordado proponiendo un rol activo para el criterio, aumentando la diversidad en el espacio de variables de una manera activa, mientras se monitorizan los cambios en la función objetivo. De esta manera, el algoritmo reutiliza la información obtenida para la decisión de parada y la inserta en un mecanismo de prevención de la parada con la finalidad de prevenir situaciones problemáticas como la convergencia temprana. El algoritmo presentado se ha probado sobre un conjunto de 27 funciones distintas, con diferentes características respecto a su dimensionalidad, espacio de búsqueda, mínimos locales... Los resultados muestran que los mecanismos introducidos mejoran la robustez de los resultados, haciendo uso de la exploración mejorada y la prevención de la convergencia temprana. Los criterios de parada multiobjetivo se han planteado con el uso de indicadores de avance (medidas comparativas de la calidad de los resultados de la evolución en diferentes generaciones) y una herramienta de recolección de datos asociada. La propuesta final utiliza tres indicadores de avance distintos (hypervolumen, epsilon y ratio de dominancia mutua) y los considera de una manera conjunta de acuerdo a una arquitectura de fusión de decisiones. El análisis del estancamiento se basa en los parámetros de una regresión de mínimos cuadrados sobre los valores de los indicadores, incluyendo asimismo un análisis de normalidad. Se recalca la naturaleza online de estos algoritmos, evitando el recálculo de los valores de los indicadores que estaba presente en otras alternativas disponibles, y también focalizándose en la simplicidad de la propuesta final, de manera que se facilite el proceso de introducir el criterio en los algoritmos existentes. La propuesta ha sido probada con instancias de la familia de algoritmos DTLZ, obteniendo resultados de parada satisfactorios con un conjunto de valores de configuración estándar para la técnica. Sin embargo, existe una falta de medidas cuantitativas para determinar la calidad objetiva de una parada, así como para comparar de manera apropiada su valor frente al de otras alternativas. La naturaleza multiobjetivo del problema de segmentación se ha analizado para proponer un algoritmo evolutivo multiobjetivo (AEMO) para resolverlo. Esta naturaleza ha sido analizada de acuerdo a una selección de los enfoques disponibles, destacando las dificultades que se tienen que afrontar en la configuración de los parámetros de cara a guiar el proceso hacia los valores de solución deseados. Un enfoque multiobjetivo a-posteriori como el que se ha presentado permite al responsable elegir del frente de posibles soluciones finales aquella que encaja mejor, simplificando este proceso. El enfoque presentado ha elegido SPEA2 como algoritmo de base, analizando diferentes propuestas de inicialización y representación. Los resultados se han validado frente a un conjunto significativo de técnicas heurísticas y metaheurísticas, utilizando tres curvas ampliamente extendidas en el dominio de la segmentación poligonal (cromosoma, hoja y semicírculo), obteniendo resultados estadísticamente mejores para la casi totatilidad de los casos de prueba. Esta propuesta inicial de AEMO presentaba una serie de problemas sin resolver, como el orden de complejidad de la técnica de almacenaje, y además carecía de las consideraciones específicas de diseño para su adaptación al dominio de aplicación. Estos problemas se han afrontado de acuerdo a diferentes mejoras. Por un lado, se ha propuesto una representación alternativa, incluyendo información parcial de la función objetivo y operadores de transformación informados (operadores de transformación que calculan los valores de la función objetivo de los hijos de acuerdo a los cambios realizados y los valores parciales de los padres). Una nueva técnica de almacenaje se ha introducido de acuerdo a la naturaleza biobjetivo del dominio, siendo uno de ellos además discreto. Esta naturaleza ha llevado a la aplicación de una forma relajada de dominancia de Pareto, que hemos denominado pulsos épsilon. Versiones multiobjetivo de los algoritmos tradicionales de búsqueda local han sido propuestas y probadas para la inicialización del algoritmo, junto con la propuesta de criterio de parada, que también ha sido adaptada a las características del problema. En este caso, el tamaño del almacén es suficientemente grande como para almacenar todos los individuos del frente óptimo, de manera que las técnicas de análisis de calidad de los frentes se simplifican, y un mecanismo más sencillo puede ser introducido para detectar el estancamiento, de acuerdo a las mejoras en cada uno de los individuos posibles. La propuesta evolutiva final es escalable, requiere pocos parámetros de configuración e introduce un criterio de parada dinámico y eficiente. Sus resultados se han probado frente a la técnica original y el conjunto de técnicas heurísticas y metaheurísticas previamente utilizadas, incluyendo las tres curvas originales y versiones más complejas de las mismas (obtenidas con un mecanismo de generación incluido de acuerdo a estas tres formas originales). A pesar de que los resultados de parada son muy satisfactorios, los resultados obtenidos son ligeramente peores que el AEMO original para las tres instancias del problema más simples, utilizando el conjunto de parámetros de configuración establecidos (como cabía esperar, dado el coste computacional del número de generaciones y tamaño de la población establecidos a priori, basados en el análisis de los resultados del algoritmo). En cualquier caso, la comparación frente a las técnicas alternativas todavía presenta los mismos resultados estadísticamente mejores, y las mejoras en el coste computacional permiten su aplicación a un mayor conjunto de problemas.Programa Oficial de Doctorado en Ciencia y Tecnología InformáticaPresidente: Pedro Isasi Viñuela.- Secretario: Rafael Martínez Tomás.- Vocal: Javier Segovia Pére

Quality-diversity optimization: a novel branch of stochastic optimization

Traditional optimization algorithms search for a single global optimum that maximizes (or minimizes) the objective function. Multimodal optimization algorithms search for the highest peaks in the search space that can be more than one. Quality-Diversity algorithms are a recent addition to the evolutionary computation toolbox that do not only search for a single set of local optima, but instead try to illuminate the search space. In effect, they provide a holistic view of how high-performing solutions are distributed throughout a search space. The main differences with multimodal optimization algorithms are that (1) Quality-Diversity typically works in the behavioral space (or feature space), and not in the genotypic (or parameter) space, and (2) Quality-Diversity attempts to fill the whole behavior space, even if the niche is not a peak in the fitness landscape. In this chapter, we provide a gentle introduction to Quality-Diversity optimization, discuss the main representative algorithms, and the main current topics under consideration in the community. Throughout the chapter, we also discuss several successful applications of Quality-Diversity algorithms, including deep learning, robotics, and reinforcement learning

Quality-diversity optimization: a novel branch of stochastic optimization

Traditional optimization algorithms search for a single global optimum that maximizes (or minimizes) the objective function. Multimodal optimization algorithms search for the highest peaks in the search space that can be more than one. Quality-Diversity algorithms are a recent addition to the evolutionary computation toolbox that do not only search for a single set of local optima, but instead try to illuminate the search space. In effect, they provide a holistic view of how high-performing solutions are distributed throughout a search space. The main differences with multimodal optimization algorithms are that (1) Quality-Diversity typically works in the behavioral space (or feature space), and not in the genotypic (or parameter) space, and (2) Quality-Diversity attempts to fill the whole behavior space, even if the niche is not a peak in the fitness landscape. In this chapter, we provide a gentle introduction to Quality-Diversity optimization, discuss the main representative algorithms, and the main current topics under consideration in the community. Throughout the chapter, we also discuss several successful applications of Quality-Diversity algorithms, including deep learning, robotics, and reinforcement learning

ALOHA: A Unified Platform-Aware Evaluation Method for CNNs Execution on Heterogeneous Systems at the Edge

CNN design and deployment on embedded edge-processing systems is an error-prone and effort-hungry process, that poses the need for accurate and effective automated assisting tools. In such tools, pre-evaluating the platform-aware CNN metrics such as latency, energy cost, and throughput is a key requirement for successfully reaching the implementation goals imposed by use-case constraints. Especially when more complex parallel and heterogeneous computing platforms are considered, currently utilized estimation methods are inaccurate or require a lot of characterization experiments and efforts. In this paper, we propose an alternative method, designed to be flexible, easy to use, and accurate at the same time. Considering a modular platform and execution model that adequately describes the details of the platform and the scheduling of different CNN operators on different platform processing elements, our method captures precisely operations and data transfers and their deployment on computing and communication resources, significantly improving the evaluation accuracy. We have tested our method on more than 2000 CNN layers, targeting an FPGA-based accelerator and a GPU platform as reference example architectures. Results have shown that our evaluation method increases the estimation precision by up to 5× for execution time, and by 2\times for energy, compared to other widely used analytical methods. Moreover, we assessed the impact of the improved platform-awareness on a set of neural architecture search experiments, targeting both hardware platforms, and enforcing 2 sets of latency constraints, performing 5 trials on each search space, for a total number of 20 experiments. The predictability is improved by 4\times , reaching, with respect to alternatives, selection results clearly more similar to those obtained with on-hardware measurements

Evolutionary Algorithms in Engineering Design Optimization

Evolutionary algorithms (EAs) are population-based global optimizers, which, due to their characteristics, have allowed us to solve, in a straightforward way, many real world optimization problems in the last three decades, particularly in engineering fields. Their main advantages are the following: they do not require any requisite to the objective/fitness evaluation function (continuity, derivability, convexity, etc.); they are not limited by the appearance of discrete and/or mixed variables or by the requirement of uncertainty quantification in the search. Moreover, they can deal with more than one objective function simultaneously through the use of evolutionary multi-objective optimization algorithms. This set of advantages, and the continuously increased computing capability of modern computers, has enhanced their application in research and industry. From the application point of view, in this Special Issue, all engineering fields are welcomed, such as aerospace and aeronautical, biomedical, civil, chemical and materials science, electronic and telecommunications, energy and electrical, manufacturing, logistics and transportation, mechanical, naval architecture, reliability, robotics, structural, etc. Within the EA field, the integration of innovative and improvement aspects in the algorithms for solving real world engineering design problems, in the abovementioned application fields, are welcomed and encouraged, such as the following: parallel EAs, surrogate modelling, hybridization with other optimization techniques, multi-objective and many-objective optimization, etc