- TL;DR: Whether or not a sparse subnetwork trains to the same accuracy of the full network depends on whether two runs of SGD on the subnetwork land in the same convex levelset.
- Abstract: We uncover a connection between two seemingly unrelated empirical phenomena: mode connectivity and sparsity. On the one hand, there is growing catalog of situations where, across multiple runs, SGD learns weights that fall into minima that are connected (mode connectivity). A striking example is described by Nagarajan & Kolter (2019). They observe that test error on MNIST does not change along the linear path connecting the end points of two independent SGD runs, starting from the same random initialization. On the other hand, there is the lottery ticket hypothesis of Frankle & Carbin (2019), where dense, randomly initialized networks have sparse subnetworks capable of training in isolation to full accuracy. However, neither phenomenon scales beyond small vision networks. We start by proposing a technique to find sparse subnetworks after initialization. We observe that these subnetworks match the accuracy of the full network only when two SGD runs for the same subnetwork are connected by linear paths with the no change in test error. Our findings connect the existence of sparse subnetworks that train to high accuracy with the dynamics of optimization via mode connectivity. In doing so, we identify analogues of the phenomena uncovered by Nagarajan & Kolter and Frankle & Carbin in ImageNet-scale architectures at state-of-the-art sparsity levels.
- Keywords: sparsity, mode connectivity, lottery ticket, optimization landscape