Why Lottery Ticket Wins? A Theoretical Perspective of Sample Complexity on Sparse Neural Networks

The $\textit{lottery ticket hypothesis}$ (LTH) states that learning on a properly pruned network (the $\textit{winning ticket}$) has improved test accuracy over the originally unpruned network. Although LTH has been justified empirically in a broad range of deep neural network (DNN) involved applications like computer vision and natural language processing, the theoretical validation of the improved generalization of a winning ticket remains elusive. To the best of our knowledge, our work, for the first time, characterizes the performance of training a sparse neural network by analyzing the geometric structure of the objective function and the sample complexity to achieve zero generalization error. We show that the convex region near a desirable model with guaranteed generalization enlarges as the neural network model is pruned, indicating the structural importance of a winning ticket. Moreover, as the algorithm for training a sparse neural network is specified as (accelerated) stochastic gradient descent algorithm, we theoretically show that the number of samples required for achieving zero generalization error is proportional to the number of the non-pruned model weights in the hidden layer. With a fixed number of samples, training a pruned neural network enjoys a faster convergence rate to the desirable model than training the original unpruned one, providing a formal justification of the improved generalization of the winning ticket. Our theoretical results are acquired from learning a sparse neural network of one hidden layer, while experimental results are further provided to justify the implications in pruning multi-layer neural networks.