Revisiting the Lottery Ticket Hypothesis: A Ramanujan Graph Perspective

BITHIKA PAL, Arindam Biswas, Pabitra Mitra, BISWAJIT BASU

Keywords: Deep Neural Networks, Network Pruning, Ramanujan Graphs, Eigenvalue bounds, Spectral Gap

Abstract: Neural networks often yield to weight pruning resulting in a sparse subnetwork that is adequate for a given task. Retraining these `lottery ticket' subnetworks from their initialization minimizes the computational burden while preserving the test set accuracy of the original network. Based on our knowledge, the existing literature only confirms that pruning is needed and it can be achieved up to certain sparsity. We analyze the pruned network in the context of the properties of Ramanujan expander graphs. We consider the feed-forward network (both multi-layer perceptron and convolutional network) as a series of bipartite graphs which establish the connection from input to output. Now, as the fraction of remaining weights reduce with increasingly aggressive pruning two distinct regimes are observed: initially, no significant decrease in accuracy is demonstrated, and then the accuracy starts dropping rapidly. We empirically show that in the first regime the pruned lottery ticket sub-network remains a Ramanujan graph. Subsequently, with the loss of Ramanujan graph property, accuracy begins to reduce sharply. This characterizes an absence of resilient connectivity in the pruned sub-network. We also propose a new magnitude-based pruning algorithm to preserve the above property. We perform experiments on MNIST and CIFAR10 datasets using different established feed-forward architectures and show that the winning ticket obtained from the proposed algorithm is much more robust.

One-sentence Summary: A Ramanujan graph perspective to explain the lottery ticket hypothesis