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We explore convergence of deep neural networks with the popular ReLU activation function, as the depth of the networks tends to infinity. To this end, we introduce the notion of activation domains and activation matrices of a ReLU network. By replacing applications of the ReLU activation function by multiplications with activation matrices on activation domains, we obtain an explicit expression of the ReLU network. We then identify the convergence of the ReLU networks as convergence of a class of infinite products of matrices. Sufficient and necessary conditions for convergence of these infinite products of matrices are studied. As a result, we establish necessary conditions for ReLU networks to converge that the sequence of weight matrices converges to the identity matrix and the sequence of the bias vectors converges to zero as the depth of ReLU networks increases to infinity. Moreover, we obtain sufficient conditions in terms of the weight matrices and bias vectors at hidden layers for pointwise convergence of deep ReLU networks. These results provide mathematical insights to convergence of deep neural networks. Experiments are conducted to mathematically verify the results and to illustrate their potential usefulness in initialization of deep neural networks.
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NeuralDivergence: Exploring and Understanding Neural Networks by Comparing Activation Distributions
As deep neural networks are increasingly used in solving highstake problems, there is a pressing need to understand their internal decision mechanisms. Visualization has helped address this problem by assisting with interpreting complex deep neural networks. However, current tools often support only single data instances, or visualize layers in isolation. We present NEURALDIVERGENCE, an interactive visualization system that uses activation distributions as a high-level summary of what a model has learned. NEURALDIVERGENCE enables users to interactively summarize and compare activation distributions across layers, classes, and instances (e.g., pairs of adversarial attacked and benign images), helping them gain better understanding of neural network models.
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- PAR ID:
- 10095875
- Date Published:
- Journal Name:
- PacificVis 2019
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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