In this work we introduce a differentiable version of the Compositional Pattern Producing Network, called the DPPN. Unlike a standard CPPN, the topology of a DPPN is evolved but the weights are learned. A Lamarckian algorithm, that combines evolution and learning, produces DPPNs to reconstruct an image. Our main result is that DPPNs can be evolved trained to compress the weights of a denoising autoencoder from 157684 to roughly 200 parameters, while achieving a reconstruction accuracy comparable to a fully connected network with more than two orders of magnitude more parameters. The regularization ability of the DPPN allows it to rediscover (approximate) convolutional network architectures embedded within a fully connected architecture. Such convolutional architectures are the current state of the art for many computer vision applications, so it is satisfying that DPPNs are capable of discovering this structure rather than having to build it in by design. DPPNs exhibit better generalization when tested on the Omniglot dataset after being trained on MNIST, than directly encoded fully connected autoencoders. DPPNs are therefore a new framework for integrating learning and evolution.
In HyperNEAT @cite_19 , CPPNs are used as indirect compressed encodings of the weights of a larger neural network. The inputs to the CPPN are the coordinates of the presynaptic and postsynaptic neuron, and the output is the weight joining those two neurons. If a single CPPN must encode multiple layers of a deeper neural network then there are two possibilities, either an extra input is given signaling which layer of weights the CPPN is outputting @cite_10 , or the CPPN is constrained to always have multiple output nodes, with a specific node outputting the weight for its assigned layer @cite_11 .
@cite_19 @cite_10 @cite_11
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Research in neuroevolution---that is, evolving artificial neural networks (ANNs) through evolutionary algorithms---is inspired by the evolution of biological brains, which can contain trillions of connections. Yet while neuroevolution has produced successful results, the scale of natural brains remains far beyond reach. This article presents a method called hypercube-based NeuroEvolution of Augmenting Topologies (HyperNEAT) that aims to narrow this gap. HyperNEAT employs an indirect encoding called connective compositional pattern-producing networks (CPPNs) that can produce connectivity patterns with symmetries and repeating motifs by interpreting spatial patterns generated within a hypercube as connectivity patterns in a lower-dimensional space. This approach can exploit the geometry of the task by mapping its regularities onto the topology of the network, thereby shifting problem difficulty away from dimensionality to the underlying problem structure. Furthermore, connective CPPNs can represent the same connectivity pattern at any resolution, allowing ANNs to scale to new numbers of inputs and outputs without further evolution. HyperNEAT is demonstrated through visual discrimination and food-gathering tasks, including successful visual discrimination networks containing over eight million connections. The main conclusion is that the ability to explore the space of regular connectivity patterns opens up a new class of complex high-dimensional tasks to neuroevolution. An important issue in the design and implementation of a neural network is the sensitivity of its output to input and weight perturbations. In this paper, we discuss the sensitivity of the most popular and general feedforward neural networks-multilayer perceptron (MLP). The sensitivity is defined as the mathematical expectation of the output errors of the MLP due to input and weight perturbations with respect to all input and weight values in a given continuous interval. The sensitivity for a single neuron is discussed first and an analytical expression that is a function of the absolute values of input and weight perturbations is approximately derived. Then an algorithm is given to compute the sensitivity for the entire MLP. As intuitively expected, the sensitivity increases with input and weight perturbations, but the increase has an upper bound that is determined by the structural configuration of the MLP, namely the number of neurons per layer and the number of layers. There exists an optimal value for the number of neurons in a layer, which yields the highest sensitivity value. The effect caused by the number of layers is quite unexpected. The sensitivity of a neural network may decrease at first and then almost keeps constant while the number increases. Natural DNA can encode complexity on an enormous scale. Researchers are attempting to achieve the same representational efficiency in computers by implementing developmental encodings, i.e. encodings that map the genotype to the phenotype through a process of growth from a small starting point to a mature form. A major challenge in in this effort is to find the right level of abstraction of biological development to capture its essential properties without introducing unnecessary inefficiencies. In this paper, a novel abstraction of natural development, called Compositional Pattern Producing Networks (CPPNs), is proposed. Unlike currently accepted abstractions such as iterative rewrite systems and cellular growth simulations, CPPNs map to the phenotype without local interaction, that is, each individual component of the phenotype is determined independently of every other component. Results produced with CPPNs through interactive evolution of two-dimensional images show that such an encoding can nevertheless produce structural motifs often attributed to more conventional developmental abstractions, suggesting that local interaction may not be essential to the desirable properties of natural encoding in the way that is usually assumed.