Deep ReLU Networks Preserve Expected Length

While neural networks are typically overparameterized and trained with little or no explicit regularization, the functions they learn in practice are often able to generalize to unseen data. This phenomenon indicates an implicit regularization that causes these learned functions to be surprisingly simple.

Why this occurs is not well-understood theoretically, but there is a high level intuition. In a nutshell, randomly initialized neural networks will compute functions of low complexity. Moreover, as optimization by first order methods is a kind of greedy local search, network training is attracted to minima of the loss that are not too far from the initialization and hence will still be well-behaved.

While this intuition is compelling, a key challenge in making it rigorous is to devise appropriate notions of complexity that are small throughout training. The present work is intended to provide a piece of the puzzle, making precise the idea that, at the start of training, neural networks compute tame functions. Specifically, we demonstrate that neural networks at initialization have low distortion of length and volume, as defined in §3.2.

An important aspect of our analysis is that we study networks in typical, average-case scenarios. A randomly initialized neural network could, in principle, compute any function expressible by a network of that architecture, since the weights might with low probability take on any set of values. Some settings of weights will lead to functions of high complexity, but these settings may be unlikely to occur in practice, depending on the distribution over weights that is under consideration. As prior work has emphasized [22, 23], atypical distributions of weights can lead to exponentially high length distortion. We show that, in contrast, for deep ReLU networks with the standard initialization, the functions computed have low distortion in expectation and with high probability.

Let $L\geq 1$ be a positive integer and fix positive integers $n_{0},\ldots,n_{L}$. We consider a fully connected feed-forward ReLU network $\mathcal{N}$ with input dimension $n_{0}$, output dimension $n_{L}$, and hidden layer widths $n_{1},\ldots,n_{L-1}$. Suppose that the weights and biases of $\mathcal{N}$ are independent and Gaussian with the weights $W_{ij}^{{}_{(\ell)}}$ between layers $\ell-1$ and $\ell$ and biases $b_{j}^{{}_{(\ell)}}$ in layer $\ell$ satisfying:

Here, $C_{b}>0$ is any fixed constant and $G(\mu,\sigma^{2})$ denotes a Gaussian with mean $0$ and variance $\sigma^{2}$. For any $M\subseteq\mathbb{R}^{n_{0}}$, we denote by $\mathcal{N}(M)\subseteq\mathbb{R}^{n_{L}}$ the (random) image of $M$ under the map $x\mapsto\mathcal{N}(x)$. Our primary object of the study will be the size of the output $\mathcal{N}(M)$ relative to that of $M$. Specifically, when $M$ is a $1$-dimensional curve, we define

Note that while a priori this random variable depends on $M$, we will find that its statistics depend only on the architecture of $\mathcal{N}$ (see Thms. 3.1 and 5.1).

We prove that the expected length distortion does not grow with depth – in fact, it slightly decreases. Our result may be informally stated thus:

For a formal statement (including the exact values of constants) and proof, see Thm. 5.1 and App. C.

Our experiments align with the theorem. In Figure 1, we compute the empirical mean of the length distortion for randomly initialized deep ReLU networks of various architectures with fixed input and output dimension. The figure confirms three of our key predictions: (1) the expected length distortion does not grow with depth, and actually decreases slightly; (2) the decrease happens faster for narrower networks (since $1/n_{\ell}$ is larger for smaller $n_{\ell}$); and (3) for equal input and output dimension $n_{0}=n_{L}$, there is an upper bound of 1 (in fact, the tighter upper bound of $C$ is approximately 0.9515 for $n_{L}=5$, as shown in the figure). The figure also shows the prior bounds in [23] to be quite loose. For further experimental details, see Appendix B.

In Fig. 2, we instead fix the set of hidden layer widths, but vary input and output dimensions. The results confirm that indeed the expected length distortion grows as the square root of the ratio of output to input dimension.

Note that our theorem also applies for initializations other than He normal; the result in such cases is simply less interesting. Suppose we initialize the weights in each layer $\ell$ as i.i.d. normal with variance $2c^{2}/n_{\ell-1}$ instead of $2/n_{\ell-1}$. This is equivalent to taking a He normal initialization and multiplying the weights by $c$. Multiplying the weights and biases in a depth-$L$ ReLU network by $c$ simply multiplies the output (and hence the length distortion) by $c^{L}$, so the expected distortion is $c^{L}$ times the value given in Thm. 3.1. This emphasizes why an exponentially large distortion necessarily occurs if one sets the weights too large.

The purpose of this section is to give an intuitive but technical explanation for why, in Theorem 3.1, we find that the distortion $\operatorname{len}(\mathcal{N}(M))/\operatorname{len}(M)$ of the length of a $1D$ curve $M$ under the neural network $\mathcal{N}$ is typically not much larger than $1$, even for deep networks.

Our starting point is that, near a given point $x\in M$, the length of the image under $\mathcal{N}$ of a small portion of $M$ with length $dx$ is approximately given by $\left|\left|J_{x}u\right|\right|dx$, where $J_{x}$ is the input-output Jacobian of $\mathcal{N}$ at the input $x$ and $u$ is the unit tangent vector to $M$ at $x$. In fact, in Lemma C.1 of the Supp. Material, we use a simple argument to give upper bounds on the moments of $\operatorname{len}(\mathcal{N}(M))/\operatorname{len}(M)$ in terms of the moments of the norm $\left|\left|J_{x}u\right|\right|$ of the Jacobian-vector product $J_{x}u$.

Thus, upper bounds on the length distortion reduce to studying the Jacobian $J_{x}$, which in a network with $L$ hidden layers can be written as a product $J_{x}=J_{L,x}\cdots J_{1,x}$ of the layer $\ell-1$ to layer $\ell$ Jacobians $J_{\ell,x}$. In a worst-case analysis, the left singular vectors of $J_{\ell,x}$ would align with the right singular vectors of $J_{\ell+1,x}$, causing the resulting product $J_{x}$ to have a largest singular value that grows exponentially with $L$. However, at initialization, this is provably not the case with high probability. Indeed, the Jacobians $J_{\ell,x}$ are independent (see Lemma C.3) and their singular vectors are therefore incoherent. We find in particular (Lemma C.2) the following equality in distribution:

where $e_{1}$ is the first standard unit vector and the terms in the product are independent. On average each term in this product is close to $1$:

This is a consequence of initializing weights to have variance $2/\text{fan-in}$. Put together, the two preceding displayed equations suggest writing

The terms being summed in the exponent are independent and the argument of each logarithm scales like $1+O(n_{\ell}^{-1})$. With probability $1-\delta$ we have for all $\ell$ that $\log(\left|\left|J_{\ell,x}e_{1}\right|\right|)\leq cn_{\ell}^{-1}$, where $c$ is a constant depending only on $\delta$. Thus, all together,

with high probability. In particular, in the simple case where $n_{\ell}$ are proportional to a single large constant $n$, we find the typical size of $\left|\left|J_{x}u\right|\right|$ is exponential in $L/n$ rather than exponential in $L$, as in the worst case. If $\mathcal{N}$ is wider than it is deep, so that $L/n$ is bounded above, the typical size of $\left|\left|J_{x}u\right|\right|$ and hence of $\operatorname{len}(\mathcal{N}(M))/\operatorname{len}(M)$ remains bounded. Theorem 5.1 makes this argument precise. Moreover, let us note that articles like [7, 10, 21] show low distortion of angles and volumes in very wide networks. Our results, however, apply directly to more realistic network widths.

We have seen that the mean length distortion is upper-bounded by 1, and indeed by a function of the architecture that decreases with the depth of the network. However, a small mean is insufficient by itself to ensure that typical networks have low length distortion. For this, we now show that in fact all moments of the length distortion are well-controlled. Specifically, the variance is bounded above by the ratio of output to input dimension, and higher moments are upper-bounded by a function that grows very slowly with depth. Our results may be informally stated thus:

A formal statement and proof of this result are given in Theorem 5.1 and Appendix C.

We consider the mean and variance of length distortion for a wide range of network architectures in Figure 3. The figure confirms that variance is modest for all architectures and does not increase with depth, and that mean length distortion is consistently decreasing with the sum of layer width reciprocals, as predicted by Theorem 4.1.

Another natural generalization to consider is how ReLU networks distort higher-dimensional volumes: Given a $d$-dimensional input $M$, the output $\mathcal{N}(M)$ will in general also be $d$-dimensional, and we can therefore consider the volume distortion $\operatorname{vol}_{d}(\mathcal{N}(M))/\operatorname{vol}_{d}(M)$. The theorems presented in §3.3 when $d=1$ can be extended to all $d$, and the results may be informally stated thus:

A formal statement and proof of this result are given in Theorem 5.2 and Appendix D.

Fix a smooth one-dimensional manifold $M\subseteq\mathbb{R}^{n_{0}}$ (i.e. a curve) with unit length. Each point on $M$ represents a possible input to our ReLU network $\mathcal{N}$, which we recall has input dimension $n_{0}$, output dimension $n_{L}$, and hidden layer widths $n_{1},\ldots,n_{L-1}$. The function $x\mapsto\mathcal{N}(x)$ computed by $\mathcal{N}$ is continuous and piecewise linear. Thus, the image $\mathcal{N}(M)\subseteq\mathbb{R}^{n_{L}}$ of $M$ under this map is a piecewise smooth curve in $\mathbb{R}^{n_{L}}$ with a finite number of pieces, and its length $\operatorname{len}(\mathcal{N}(M))$ is a random variable. Our first result states that, provided $\mathcal{N}$ is wider than it is deep, this distortion is small with high probability at initialization.

Theorem 5.1 is proved in §C. Several comments are in order. First, we have assumed for simplicity that $M$ has unit length. For curves of arbitrary length, both the equality (5.1) and the bounds (5.2) and (5.3) all hold and are derived in the same way provided $\operatorname{len}(\mathcal{N}(M))$ is replaced by the distortion $\operatorname{len}(\mathcal{N}(M))/\operatorname{len}(M)$ per unit length.

Second, in the expression (5.1), note that the exponent tends to $0$ as $n_{\ell}\rightarrow\infty$ for any fixed $L$. More precisely, when $n_{\ell}=n$ is large and constant, it scales as $-5L/8n$. This shows that, for fixed $n_{0},n_{L},n$, the mean $\mathbb{E}\left[\operatorname{len}(\mathcal{N}(M))\right]$ is actually decreasing with $L$. This somewhat surprising phenomenon is borne out in Fig. 1 and is a consequence of the fact that wide fully connected ReLU nets are strongly contracting (see e.g. Thms. 3 and 4 in [10]).

Third, the pre-factor $(n_{L}/n_{0})^{1/2}$ encodes the fact that, on average over initialization, for any vector of inputs $x\in\mathbb{R}^{n_{0}}$, we have (see e.g. Cor. 1 in [14])

where the approximate equality becomes exact if the bias variance $C_{b}$ is set to $0$ (see (3.1)). In other words, at the start of training, the network $\mathcal{N}$ rescales inputs by an average factor $(n_{L}/n_{0})^{1/2}$. This overall scaling factor also dilates $\operatorname{len}(\mathcal{N}(M))$ relative to $\operatorname{len}(M)$.

Fourth, we stated Theorem 5.1 for Gaussian weights and biases (see (3.1)), but the result holds for any weight distribution that is symmetric around $0$, has variance $2/\text{fan-in},$ and has finite higher moments. The only difference is that the constant $5/8$ may need to be slightly enlarged (e.g. around (C.6)).

Finally, all the estimates (5.1), (5.2), and (5.3) are consistent with the statement that, up to the scaling factor $(n_{L}/n_{0})^{1/2}$, the random variable $\operatorname{len}(\mathcal{N}(M))$ is bounded above by a log-normal distribution:

where $\beta=c\sum_{\ell=1}^{L}n_{\ell}^{-1}$ and $c$ is a fixed constant.

The results in the previous section generalize to the case when $M$ has higher dimension. To consider this case, fix a positive integer $d\leq\min\left\{n_{0},\ldots,n_{L}\right\}$ and a smooth $d-$dimensional manifold $M\subseteq\mathbb{R}^{n_{0}}$ of unit volume $\operatorname{vol}_{d}(M)=1$. Note that if $d>\min\left\{n_{0},\ldots,n_{L}\right\}$, then $\mathcal{N}(M)$ is at most $(d-1)$-dimensional and its $d$-dimensional volume vanishes. Its image $\mathcal{N}(M)\subseteq\mathbb{R}^{n_{L}}$ is a piecewise smooth manifold of dimension at most $d$. The following result, proved in §D, gives an upper bound on the typical value of the $d-$dimensional volume of $\mathcal{N}(M).$