Memorization is worst-case generalization. Based on MacKay's information theoretic model of supervised machine learning, this article discusses how to practically estimate the maximum size of a neural network given a training data set. First, we present four easily applicable rules to analytically determine the capacity of neural network architectures. This allows the comparison of the efficiency of different network architectures independently of a task. Second, we introduce and experimentally validate a heuristic method to estimate the neural network capacity requirement for a given dataset and labeling. This allows an estimate of the required size of a neural network for a given problem. We conclude the article with a discussion on the consequences of sizing the network wrongly, which includes both increased computation effort for training as well as reduced generalization capability.
The perceptron was introduced in 1958 @cite_13 and since then, it has been extended in many variants, including, but not limited to, the structures described in @cite_8 @cite_15 @cite_10 @cite_31 . The perceptron uses a @math -dimensional input and generates the output by applying a linear function to the input, followed by a gating function. The gating function is typically the identity function, the sign function, a sigmoid function, or the rectified linear unit (ReLU) @cite_23 @cite_36 . Motivated by brain research @cite_32 , perceptrons are stacked together to networks and they are usually trained by a chain rule known as backpropagation @cite_1 @cite_7 .
@cite_13 @cite_7 @cite_15 @cite_8 @cite_36 @cite_1 @cite_32 @cite_23 @cite_31 @cite_10
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We describe a new learning procedure, back-propagation, for networks of neurone-like units. The procedure repeatedly adjusts the weights of the connections in the network so as to minimize a measure of the difference between the actual output vector of the net and the desired output vector. As a result of the weight adjustments, internal ‘hidden’ units which are not part of the input or output come to represent important features of the task domain, and the regularities in the task are captured by the interactions of these units. The ability to create useful new features distinguishes back-propagation from earlier, simpler methods such as the perceptron-convergence procedure1. The Perceptron algorithm, despite its simplicity, often performs well in online classification tasks. The Perceptron becomes especially effective when it is used in conjunction with kernel functions. However, a common difficulty encountered when implementing kernel-based online algorithms is the amount of memory required to store the online hypothesis, which may grow unboundedly as the algorithm progresses. Moreover, the running time of each online round grows linearly with the amount of memory used to store the hypothesis. In this paper, we present the Forgetron family of kernel-based online classification algorithms, which overcome this problem by restricting themselves to a predefined memory budget. We obtain different members of this family by modifying the kernel-based Perceptron in various ways. We also prove a unified mistake bound for all of the Forgetron algorithms. To our knowledge, this is the first online kernel-based learning paradigm which, on one hand, maintains a strict limit on the amount of memory it uses and, on the other hand, entertains a relative mistake bound. We conclude with experiments using real datasets, which underscore the merits of our approach. This chapter contains sections titled: The Problem, The Generalized Delta Rule, Simulation Results, Some Further Generalizations, Conclusion Massively parallel (neural-like) networks are receiving increasing attention as a mechanism for expressing information processing models. By exploiting powerful primitive units and stability-preserving construction rules, various workers have been able to construct and test quite complex models, particularly in vision research. But all of the detailed technical work was concerned with the structure and behavior offixed networks. The purpose of this paper is to extend the methodology to cover several aspects of change and memory. Rectified activation units (rectifiers) are essential for state-of-the-art neural networks. In this work, we study rectifier neural networks for image classification from two aspects. First, we propose a Parametric Rectified Linear Unit (PReLU) that generalizes the traditional rectified unit. PReLU improves model fitting with nearly zero extra computational cost and little overfitting risk. Second, we derive a robust initialization method that particularly considers the rectifier nonlinearities. This method enables us to train extremely deep rectified models directly from scratch and to investigate deeper or wider network architectures. Based on the learnable activation and advanced initialization, we achieve 4.94 top-5 test error on the ImageNet 2012 classification dataset. This is a 26 relative improvement over the ILSVRC 2014 winner (GoogLeNet, 6.66 [33]). To our knowledge, our result is the first to surpass the reported human-level performance (5.1 , [26]) on this dataset.