A Combinatorial Theory of Dropout: Subnetworks, Graph Geometry, and Generalization

We propose a combinatorial and graph-theoretic theory of dropout by modeling training as a random walk over a high-dimensional graph of binary subnetworks. Each node represents a masked version of the network, and dropout induces stochastic traversal across this space. We define a subnetwork contribution score that quantifies generalization and show that it varies smoothly over the graph. Using tools from spectral graph theory, PAC-Bayes analysis, and combinatorics, we prove that generalizing subnetworks form large, connected, low-resistance clusters, and that their number grows exponentially with network width. This reveals dropout as a mechanism for sampling from a robust, structured ensemble of well-generalizing subnetworks with built-in redundancy. Extensive experiments validate every theoretical claim across diverse architectures. Together, our results offer a unified foundation for understanding dropout and suggest new directions for mask-guided regularization and subnetwork optimization.