Are deep ResNets provably better than linear predictors?
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Recently, a residual network (ResNet) with a single residual block has been shown to outperform linear predictors, in the sense that all its local minima are at least as good as the best linear predictor. We take a step towards extending this result to deep ResNets. As motivation, we first show that there exist datasets for which all local minima of a fully-connected ReLU network are no better than the best linear predictor, while a ResNet can have strictly better local minima. Second, we show that even at its global minimum, the representation obtained from the residual blocks of a 2-block ResNet does not necessarily improve monotonically as more blocks are added, highlighting a fundamental difficulty in analyzing deep ResNets. Our main result on deep ResNets shows that (under some geometric conditions) any critical point is either (i) at least as good as the best linear predictor; or (ii) the Hessian at this critical point has a strictly negative eigenvalue. Finally, we complement our results by analyzing near-identity regions of deep ResNets, obtaining size-independent upper bounds for the risk attained at critical points as well as the Rademacher complexity.
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