Generative Adversarial Networks (GANs) are a machine learning approach capable of generating novel example outputs across a space of provided training examples. Procedural Content Generation (PCG) of levels for video games could benefit from such models, especially for games where there is a pre-existing corpus of levels to emulate. This paper trains a GAN to generate levels for Super Mario Bros using a level from the Video Game Level Corpus. The approach successfully generates a variety of levels similar to one in the original corpus, but is further improved by application of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). Specifically, various fitness functions are used to discover levels within the latent space of the GAN that maximize desired properties. Simple static properties are optimized, such as a given distribution of tile types. Additionally, the champion A* agent from the 2009 Mario AI competition is used to assess whether a level is playable, and how many jumping actions are required to beat it. These fitness functions allow for the discovery of levels that exist within the space of examples designed by experts, and also guide the search towards levels that fulfill one or more specified objectives.
Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) @cite_2 is a powerful and widely used evolutionary algorithm that is well suited for evolving vectors of real numbers. The CMA-ES is a second-order method using the covariance matrix estimated iteratively by finite differences. It has been demonstrated to be efficient for optimizing non-linear non-convex problems in the continuous domain without a-priori domain knowledge, and it does not rely on the assumption of a smooth fitness landscape.
@cite_2
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This paper presents a novel evolutionary optimization strategy based on the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). This new approach is intended to reduce the number of generations required for convergence to the optimum. Reducing the number of generations, i.e., the time complexity of the algorithm, is important if a large population size is desired: (1) to reduce the effect of noise; (2) to improve global search properties; and (3) to implement the algorithm on (highly) parallel machines. Our method results in a highly parallel algorithm which scales favorably with large numbers of processors. This is accomplished by efficiently incorporating the available information from a large population, thus significantly reducing the number of generations needed to adapt the covariance matrix. The original version of the CMA-ES was designed to reliably adapt the covariance matrix in small populations but it cannot exploit large populations efficiently. Our modifications scale up the efficiency to population sizes of up to 10n, where n is the problem dimension. This method has been applied to a large number of test problems, demonstrating that in many cases the CMA-ES can be advanced from quadratic to linear time complexity. CMA-ES is one of the most popular stochastic search algorithms. It performs favourably in many tasks without the need of extensive parameter tuning. The algorithm has many beneficial properties, including automatic step-size adaptation, efficient covariance updates that incorporates the current samples as well as the evolution path and its invariance properties. Its update rules are composed of well established heuristics where the theoretical foundations of some of these rules are also well understood. In this paper we will fully derive all CMA-ES update rules within the framework of expectation-maximisation-based stochastic search algorithms using information-geometric trust regions. We show that the use of the trust region results in similar updates to CMA-ES for the mean and the covariance matrix while it allows for the derivation of an improved update rule for the step-size. Our new algorithm, Trust-Region Co-variance Matrix Adaptation Evolution Strategy (TR-CMA-ES) is fully derived from first order optimization principles and performs favourably in compare to standard CMA-ES algorithm. Hyperparameters of deep neural networks are often optimized by grid search, random search or Bayesian optimization. As an alternative, we propose to use the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), which is known for its state-of-the-art performance in derivative-free optimization. CMA-ES has some useful invariance properties and is friendly to parallel evaluations of solutions. We provide a toy example comparing CMA-ES and state-of-the-art Bayesian optimization algorithms for tuning the hyperparameters of a convolutional neural network for the MNIST dataset on 30 GPUs in parallel. Derived from the concept of self-adaptation in evolution strategies, the CMA (Covariance Matrix Adaptation) adapts the covariance matrix of a multi-variate normal search distribution. The CMA was originally designed to perform well with small populations. In this review, the argument starts out with large population sizes, reflecting recent extensions of the CMA algorithm. Commonalities and differences to continuous Estimation of Distribution Algorithms are analyzed. The aspects of reliability of the estimation, overall step size control, and independence from the coordinate system (invariance) become particularly important in small populations sizes. Consequently, performing the adaptation task with small populations is more intricate. © 1997-2012 IEEE. This paper presents a method for generating complex problems that allow multiple nonobvious solutions for the physical traveling salesman problem (PTSP). PTSP is a single-player game adaptation of the classical traveling salesman problem that makes use of a simple physics model: the player has to visit a number of waypoints as quickly as possible by navigating a ship in real time across an obstacle-filled 2-D map. The difficulty of this game depends on the distribution of waypoints and obstacles across the 2-D plane. Due to the physics of the game, the shortest route is not necessarily the fastest, as the ship's momentum makes it difficult to turn sharply at high speed. This paper proposes an evolutionary approach to obtaining maps where the optimal solution is not immediately obvious. In particular, any optimal route for these maps should differ distinctively from: 1) the optimal distance-based TSP route and 2) the route that corresponds to always approaching the nearest waypoint first. To achieve this, the evolutionary algorithm covariance matrix adaptation-evolutionary strategy (CMA-ES) is employed, where maps, indirectly represented as vectors of real numbers, are evolved to differentiate maximally between a game-playing agent that follows two or more different routes. The results presented in this paper show that CMA-ES is able to generate maps that fulfil the desired conditions. The problem of model selection for support vector machines (SVMs) is considered. We propose an evolutionary approach to determine multiple SVM hyperparameters: The covariance matrix adaptation evolution strategy (CMA-ES) is used to determine the kernel from a parameterized kernel space and to control the regularization. Our method is applicable to optimize non-differentiable kernel functions and arbitrary model selection criteria. We demonstrate on benchmark datasets that the CMA-ES improves the results achieved by grid search already when applied to few hyperparameters. Further, we show that the CMA-ES is able to handle much more kernel parameters compared to grid-search and that tuning of the scaling and the rotation of Gaussian kernels can lead to better results in comparison to standard Gaussian kernels with a single bandwidth parameter. In particular, more flexibility of the kernel can reduce the number of support vectors. This paper investigates the relation between the covariance matrix adaptation evolution strategy and the natural evolution strategy, the latter of which is recently proposed and is formulated as a natural gradient based method on the expected fitness under the mutation distribution. To enable to compare these algorithms, we derive the explicit form of the natural gradient of the expected fitness and transform it into the forms corresponding to the mean vector and the covariance matrix of the mutation distribution. We show that the natural evolution strategy can be viewed as a variant of covariance matrix adaptation evolution strategies using Cholesky update and also that the covariance matrix adaptation evolution strategy can be formulated as a variant of natural evolution strategies. Uncertainty arises in reinforcement learning from various sources, and therefore it is necessary to consider statistics based on several roll-outs for evaluating behavioral policies. We add an adaptive uncertainty handling based on Hoeffding and empirical Bernstein races to the CMA-ES, a variable metric evolution strategy proposed for direct policy search. The uncertainty handling adjusts individually the number of episodes considered for the evaluation of a policy. The performance estimation is kept just accurate enough for a sufficiently good ranking of candidate policies, which is in turn sufficient for the CMA-ES to find better solutions. This increases the learning speed as well as the robustness of the algorithm. This paper puts forward two useful methods for self-adaptation of the mutation distribution - the concepts of derandomization and cumulation. Principle shortcomings of the concept of mutative strategy parameter control and two levels of derandomization are reviewed. Basic demands on the self-adaptation of arbitrary (normal) mutation distributions are developed. Applying arbitrary, normal mutation distributions is equivalent to applying a general, linear problem encoding. The underlying objective of mutative strategy parameter control is roughly to favor previously selected mutation steps in the future. If this objective is pursued rigorously, a completely derandomized self-adaptation scheme results, which adapts arbitrary normal mutation distributions. This scheme, called covariance matrix adaptation (CMA), meets the previously stated demands. It can still be considerably improved by cumulation - utilizing an evolution path rather than single search steps. Simulations on various test functions reveal local and global search properties of the evolution strategy with and without covariance matrix adaptation. Their performances are comparable only on perfectly scaled functions. On badly scaled, non-separable functions usually a speed up factor of several orders of magnitude is observed. On moderately mis-scaled functions a speed up factor of three to ten can be expected. Many state-of-the-art evolutionary vector optimization algorithms compute the contributing hypervolume for ranking candidate solutions. However, with an increasing number of objectives, calculating the volumes becomes intractable. Therefore, although hypervolume-based algorithms are often the method of choice for bi-criteria optimization, they are regarded as not suitable for many-objective optimization. Recently, Monte Carlo methods have been derived and analyzed for approximating the contributing hypervolume. Turning theory into practice, we employ these results in the ranking procedure of the multi-objective covariance matrix adaptation evolution strategy (MO-CMA-ES) as an example of a state-of-the-art method for vector optimization. It is empirically shown that the approximation does not impair the quality of the obtained solutions given a budget of objective function evaluations, while considerably reducing the computation time in the case of multiple objectives. These results are obtained on common benchmark functions as well as on two design optimization tasks. Thus, employing Monte Carlo approximations makes hypervolume-based algorithms applicable to many-objective optimization. A new formulation for coordinate system independent adaptation of arbitrary normal mutation distributions with zero mean is presented. This enables the evolution strategy (ES) to adapt the correct scaling of a given problem and also ensures invariance with respect to any rotation of the fitness function (or the coordinate system). Especially rotation invariance, here resulting directly from the coordinate system independent adaptation of the mutation distribution, is an essential feature of the ES with regard to its general applicability to complex fitness functions. Compared to previous work on this subject, the introduced formulation facilitates an interpretation of the resulting mutation distribution, making sensible manipulation by the user possible (if desired). Furthermore it enables a more effective control of the overall mutation variance (expected step length). Many different algorithms have been developed in the last few decades for solving complex real-world search and optimization problems. The main focus in this research has been on the development of a single universal genetic operator for population evolution that is always efficient for a diverse set of optimization problems. In this paper, we argue that significant advances to the field of evolutionary computation can be made if we embrace a concept of self-adaptive multimethod optimization in which multiple different search algorithms are run concurrently, and learn from each other through information exchange using a common population of points. We present an evolutionary algorithm, entitled A Multialgorithm Genetically Adaptive Method for Single Objective Optimization (AMALGAM-SO), that implements this concept of self adaptive multimethod search. This method simultaneously merges the strengths of the covariance matrix adaptation (CMA) evolution strategy, genetic algorithm (GA), and particle swarm optimizer (PSO) for population evolution and implements a self-adaptive learning strategy to automatically tune the number of offspring these three individual algorithms are allowed to contribute during each generation. Benchmark results in 10, 30, and 50 dimensions using synthetic functions from the special session on real-parameter optimization of CEC 2005 show that AMALGAM-SO obtains similar efficiencies as existing algorithms on relatively simple unimodal problems, but is superior for more complex higher dimensional multimodal optimization problems. The new search method scales well with increasing number of dimensions, converges in the close proximity of the global minimum for functions with noise induced multimodality, and is designed to take full advantage of the power of distributed computer networks. In this paper we introduce a restart-CMA-evolution strategy, where the population size is increased for each restart (IPOP). By increasing the population size the search characteristic becomes more global after each restart. The IPOP-CMA-ES is evaluated on the test suit of 25 functions designed for the special session on real-parameter optimization of CEC 2005. Its performance is compared to a local restart strategy with constant small population size. On unimodal functions the performance is similar. On multi-modal functions the local restart strategy significantly outperforms IPOP in 4 test cases whereas IPOP performs significantly better in 29 out of 60 tested cases. We explore the use of Evolution Strategies (ES), a class of black box optimization algorithms, as an alternative to popular MDP-based RL techniques such as Q-learning and Policy Gradients. Experiments on MuJoCo and Atari show that ES is a viable solution strategy that scales extremely well with the number of CPUs available: By using a novel communication strategy based on common random numbers, our ES implementation only needs to communicate scalars, making it possible to scale to over a thousand parallel workers. This allows us to solve 3D humanoid walking in 10 minutes and obtain competitive results on most Atari games after one hour of training. In addition, we highlight several advantages of ES as a black box optimization technique: it is invariant to action frequency and delayed rewards, tolerant of extremely long horizons, and does not need temporal discounting or value function approximation. This paper introduces an application of Multi-Objective Evolution Algorithm (MOEA) to design Q and R weighting matrices in Linear Quadratic regulators (LQR). Considering the difficulty of designing weighting matrices for a linear quadratic regulator, a multi-objective evolutionary algorithm based approach is proposed. The LQR weighting matrices, state feedback control rate and consequently the optimal controller are obtained by means of establishing the multi-objective optimization model of LQR weighting matrices and applying MOEA to it, which makes control system meet multiple performance indexes simultaneously. Controller of double inverted pendulum system is designed using the proposed approach. Simulation results show that it has shorter adjusting time and smaller amplitude value deviating from steady-state than a Non-dominated Sorting Genetic Algorithm LQR ( NSGA- LQR )weighting matrices design approach. A key search mechanism in Evolutionary Algorithms is the mixing or juxtaposing of partial solutions present in the parent solutions. In this paper we look at the efficiency of mixing in genetic algorithms (GAs) and estimation-of-distribution algorithms (EDAs). We compute the mixing probabilities of two partial solutions and discuss the effect of the covariance build-up in GAs and EDas. Moreover, we propose two new Evolutionary Algorithms that maximize the juxtaposing of the partial solutions present in the parents: the Recombinative Optimal Mixing Evolutionary Algorithm (ROMEA) and the Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA). Given a high-dimensional data matrix @math , Approximate Message Passing (AMP) algorithms construct sequences of vectors @math , @math , indexed by @math by iteratively applying @math or @math , and suitable non-linear functions, which depend on the specific application. Special instances of this approach have been developed --among other applications-- for compressed sensing reconstruction, robust regression, Bayesian estimation, low-rank matrix recovery, phase retrieval, and community detection in graphs. For certain classes of random matrices @math , AMP admits an asymptotically exact description in the high-dimensional limit @math , which goes under the name of state evolution.' Earlier work established state evolution for separable non-linearities (under certain regularity conditions). Nevertheless, empirical work demonstrated several important applications that require non-separable functions. In this paper we generalize state evolution to Lipschitz continuous non-separable nonlinearities, for Gaussian matrices @math . Our proof makes use of Bolthausen's conditioning technique along with several approximation arguments. In particular, we introduce a modified algorithm (called LAMP for Long AMP) which is of independent interest. An evolution strategy (ES) variant recently attracted significant attention due to its surprisingly good performance at optimizing neural networks in challenging deep reinforcement learning domains. It searches directly in the parameter space of neural networks by generating perturbations to the current set of parameters, checking their performance, and moving in the direction of higher reward. The resemblance of this algorithm to a traditional finite-difference approximation of the reward gradient in parameter space naturally leads to the assumption that it is just that. However, this assumption is incorrect. The aim of this paper is to definitively demonstrate this point empirically. ES is a gradient approximator, but optimizes for a different gradient than just reward (especially when the magnitude of candidate perturbations is high). Instead, it optimizes for the average reward of the entire population, often also promoting parameters that are robust to perturbation. This difference can channel ES into significantly different areas of the search space than gradient descent in parameter space, and also consequently to networks with significantly different properties. This unique robustness-seeking property, and its consequences for optimization, are demonstrated in several domains. They include humanoid locomotion, where networks from policy gradient-based reinforcement learning are far less robust to parameter perturbation than ES-based policies that solve the same task. While the implications of such robustness and robustness-seeking remain open to further study, the main contribution of this work is to highlight that such differences indeed exist and deserve attention. CMF is a technique for simultaneously learning low-rank representations based on a collection of matrices with shared entities. A typical example is the joint modeling of user-item, item-property, and user-feature matrices in a recommender system. The key idea in CMF is that the embeddings are shared across the matrices, which enables transferring information between them. The existing solutions, however, break down when the individual matrices have low-rank structure not shared with others. In this work we present a novel CMF solution that allows each of the matrices to have a separate low-rank structure that is independent of the other matrices, as well as structures that are shared only by a subset of them. We compare MAP and variational Bayesian solutions based on alternating optimization algorithms and show that the model automatically infers the nature of each factor using group-wise sparsity. Our approach supports in a principled way continuous, binary and count observations and is efficient for sparse matrices involving missing data. We illustrate the solution on a number of examples, focusing in particular on an interesting use-case of augmented multi-view learning. Given a covariance matrix, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis (PCA), and has wide applications ranging from biology to finance. We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming-based relaxation for our problem. We also discuss Nesterov's smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse PCA problem. The method has complexity @math , where @math is the size of the underlying covariance matrix and @math is the desired absolute accuracy on the optimal value of the problem.