Are deep ResNets provably better than linear predictors?
This work addresses a theoretical gap in understanding the optimization landscape of deep ResNets for machine learning researchers, though it is incremental as it builds on existing results for shallow networks.
The paper tackles the problem of whether deep residual networks (ResNets) are provably better than linear predictors by extending prior results from shallow to deep architectures, showing that under certain conditions, critical points are either as good as linear predictors or have negative Hessian eigenvalues, and providing depth-independent bounds on risk and complexity.
Recent results in the literature indicate that a residual network (ResNet) composed of a single residual block outperforms linear predictors, in the sense that all local minima in its optimization landscape are at least as good as the best linear predictor. However, these results are limited to a single residual block (i.e., shallow ResNets), instead of the deep ResNets composed of multiple residual blocks. We take a step towards extending this result to deep ResNets. We start by two motivating examples. First, we show that there exist datasets for which all local minima of a fully-connected ReLU network are no better than the best linear predictor, whereas a ResNet has strictly better local minima. Second, we show that even at the global minimum, the representation obtained from the residual block outputs of a 2-block ResNet do not necessarily improve monotonically over subsequent blocks, which highlights a fundamental difficulty in analyzing deep ResNets. Our main theorem on deep ResNets shows under simple geometric conditions that, any critical point in the optimization landscape 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. Notably, our theorem shows that a chain of multiple skip-connections can improve the optimization landscape, whereas existing results study direct skip-connections to the last hidden layer or output layer. Finally, we complement our results by showing benign properties of the "near-identity regions" of deep ResNets, showing depth-independent upper bounds for the risk attained at critical points as well as the Rademacher complexity.