# Notation and Terminology

Throughout this discussion we will be talking about fully-connected neural networks (multilayer perceptron networks, which I abbreviate as MLP) which have $H$ hidden layers, i.e., which are the composition of $H+1$ linear transformations, where each of the hidden layers has the same number of units, which is also the same as the number of input units. By an $l$-layer neural network I mean a network consisting of $l$ linear transformations, i.e., $l = H+1$. Concretely, an $l$-layer MLP is the following function:

\begin{align*} f(x) = W_l \cdots W_1 x, \end{align*}

where $W_{l-1},\ldots,W_1 \in \mathbb{R}^{d \times d}$ where $d$ is the number of features in the input, and $W_l \in \mathbb{R}^{k \times d}$ where $k$ is the number of output dimensions.

We will always be minimizing the expected squared error loss given some labels $y$:

\begin{align*} \min_{W_1,\ldots,W_l} \mathbb{E}\left[\|y - f(x)\|_2^2 \right]. \end{align*}

Any variable with a single capital letter name, including $W_i$ for any integer $i$, is a matrix. I write the spectral norm (largest singular value) of a matrix $W_i$ as $\|W_i\|$.

Under this setting there is no network that an $l$-layer neural network can learn which an $(l+1)$-layer neural network can’t. However, the authors of the ResNet paper noted that in practice it becomes harder and harder to train MLP networks as depth increases. Thus even though deeper MLPs can represent a superset (indeed a strict superset with ReLU activations) of the functions that can be represented by shallower ones, it becomes harder to train the MLP networks as depth increases, leading to poorer convergence and thus poorer performance. They designed residual neural networks (ResNets) to fix this problem.

# The Big Problem and a Logical Solution

Before we learn about the solution, let’s articulate the problem. The reason an $(l+1)$-layer MLP can always represent any function that an $l$-layer MLP can represent is that it is always possible for the “extra” layer to learn the identity transformation. However, something being possible does not equate to it being easy. Indeed if we initialize the MLP to a zero-centered distribution with small variance, which is necessary for gradient descent to not blow up, then it is extremely hard for any layer to learn the identity transformation, since as the number of layers increases, the gradient to any one layer gets attenuated even more, slowing down learning.

This hints at a solution: making it easy for any layer to model the identity transformation. Doing so (in hindsight) is no rocket science at all. We simply need to replace $f: \mathbb{R}^d \to \mathbb{R}^k$ from above with $g: \mathbb{R}^d \to \mathbb{R}^k$ defined as:

\begin{align*} g(x) = (W_l + I) \cdots (W_1 + I) x. \end{align*}

Now when we initialize the $W_i$ matrices near zero, the network represents a transformation close to the identity transformation instead of the zero transformation. This lets gradients flow much better through the network, and more importantly, this makes it easy for any layer to learn the identity transformation. Empirically this has led practitioners to train previously unthinkably deep neural networks, even one with 1200 layers with “no optimization difficulty” (see page 8 of this paper). This solution carries over quite well to convolutional neural networks as well, which Kaiming He et al. discuss at length in this paper.

# Theoretically Awesome

Moritz Hardt and Tengyu Ma prove two amazing theorems about deep residual neural networks in their paper “Identity Matters in Deep Learning” restricted to the regime where all activation functions are linear:

1. For a deep residual neural network with enough layers $l$ that being trained to minimize the squared error loss, there exists a global minimum such that the spectral norm of all of the matrices $W_i$ is bounded above by $\mathcal{O}(1/l)$. This means none of the matrices will need to learn a value far away from its near-zero initialization.

2. In the ball $\mathcal{B}_{\tau} := \{ (W_1,\ldots,W_l) : \max_{1\le i\le l} \|W_i\| \le \tau \}$ where the spectral norm of all the matrices $W_i$ is bounded by $\tau$, if $\tau < 1$, then any critical point of the loss function must be a global minimum.

The second point is stronger than any claim we can make about MLPs with linear activations. While there are no sub-optimal local minima, there are loci of saddle points with zero curvature whenever $l \ge 3$, as proved by Kenji Kawaguchi in “Deep Learning Without Poor Local Minima”. This means gradient-based optimization algorithms like batch gradient descent will not get stuck at any points other than the global minimizer when training deep (linear) residual neural networks with initialization suitably close to zero, while the same cannot be said about deep linear MLPs.

For proofs of the claims, please: