Exact Count of Boundary Pieces of ReLU Classifiers: Towards the Proper Complexity Measure for Classification

Classic learning theory suggests that proper regularization is the key to good generalization and robustness. In classification, current training schemes only target the complexity of the classifier itself, which can be misleading and ineffective. Instead, we advocate directly measuring the complexity of the decision boundary. Existing literature is limited in this area with few well-established definitions of boundary complexity. As a proof of concept, we start by analyzing ReLU neural networks, whose boundary complexity can be conveniently characterized by the number of affine pieces. With the help of tropical geometry, we develop a novel method that can explicitly count the exact number of boundary pieces, and as a by-product, the exact number of total affine pieces. Numerical experiments are conducted and distinctive properties of our boundary complexity are uncovered. First, the boundary piece count appears largely independent of other measures, e.g., total piece count, and $l_2$ norm of weights, during the training process. Second, the boundary piece count is negatively correlated with robustness, where popular robust training techniques, e.g., adversarial training or random noise injection, are found to reduce the number of boundary pieces.