Abstract

We study finite sample expressivity, i.e., memorization power of ReLU networks. Recent results require N hidden nodes to memorize/interpolate arbitrary N data points.  In contrast, by exploiting depth, we show that 3-layer ReLU networks with hidden nodes  can perfectly memorize most datasets with N points. We also prove that width is necessary and sufficient for memorizing N data points, proving tight bounds on memorization capacity. The sufficiency result can be extended to deeper networks; we show that an L-layer network with W parameters in the hidden layers can memorize N data points if W . Combined with a recent upper bound O(W L log W ) on VC dimension, our construction is nearly tight for any fixed L. Subsequently, we analyze memorization capacity of residual networks under a general position assumption; we prove results that substantially reduce the known requirement of N hidden nodes. Finally, we study the dynamics of stochastic gradient descent (SGD), and show that when initialized near a memorizing global minimum of the empirical risk, SGD quickly finds a nearby point with much smaller empirical risk.