First published: 2016/05/20 (7 years ago) Abstract: In this work, we introduce a novel interpretation of residual networks
showing they are exponential ensembles. This observation is supported by a
large-scale lesion study that demonstrates they behave just like ensembles at
test time. Subsequently, we perform an analysis showing these ensembles mostly
consist of networks that are each relatively shallow. For example, contrary to
our expectations, most of the gradient in a residual network with 110 layers
comes from an ensemble of very short networks, i.e., only 10-34 layers deep.
This suggests that in addition to describing neural networks in terms of width
and depth, there is a third dimension: multiplicity, the size of the implicit
ensemble. Ultimately, residual networks do not resolve the vanishing gradient
problem by preserving gradient flow throughout the entire depth of the network
- rather, they avoid the problem simply by ensembling many short networks
together. This insight reveals that depth is still an open research question
and invites the exploration of the related notion of multiplicity.
This paper introduces an interpretation of deep residual networks as implicit ensembles of exponentially many shallow networks. For a residual block $i$, there are $2^{i-1}$ paths from input to $i$, and the input to $i$ is a mixture of $2^{i-1}$ different distributions. The interpretation is backed by a number of experiments such as removing or re-ordering residual blocks at test time and plotting norm of gradient v/s number of residual blocks the gradient signal passes through. Removing $k$ residual blocks (for k <= 20) from a network of depth n decreases the number of paths to $2^{n-k}$ but there are still sufficiently many valid paths to not hurt classification error, whereas sequential CNNs have a single viable path which gets corrupted. Plot of gradient at input v/s path length shows that almost all contributions to the gradient come from paths shorter than 20 residual blocks, which are the effective paths. The paper concludes by saying that network 'multiplicity', which is the number of paths, plays a key role in terms of the network's expressability.
## Strengths
- Extremely insightful set of experiments. These experiments nail down the intuitions as to why residual networks work, as well as clarify the connections with stochastic depth (sampling the network multiplicity during training i.e. ensemble by training) and highway networks (reduction in number of available paths by gating both skip connections and paths through residual blocks).
## Weaknesses / Notes
- Connections between effective paths and model compression.