Minimal Width for Universal Property of Deep RNN

Deep learning, a type of machine learning that approximates a target function using a numerical model called an artificial neural network, has shown tremendous success in diverse fields, such as regression (Garnelo et al., 2018), image processing (Dai et al., 2021; Zhai et al., 2022), speech recognition (Baevski et al., 2020), and natural language processing (LeCun et al., 2015; Lavanya and Sasikala, 2021).

While the excellent performance of deep learning is attributed to a combination of various factors, it is reasonable to speculate that its notable success is partially based on the universal approximation theorem, which states that neural networks are capable of arbitrarily accurate approximations of the target function.

Formally, for any given function $f$ in a target class and $\epsilon>0$, there exists a network $\mathcal{N}$ such that $\|f-\mathcal{N}\|<\epsilon$. In topological language, the theorem states that a set of networks is dense in the class of the target function. In this sense, the closure of the network defines the range of functions it network can represent.

As universality provides the expressive power of a network structure, studies on the universal approximation theorem have attracted increasing attention. Examples include the universality of multi-layer perceptrons (MLPs), the most basic neural networks (Cybenko, 1989; Hornik et al., 1989; Leshno et al., 1993), and the universality of recurrent neural networks (RNNs) with the target class of open dynamical systems (Schäfer and Zimmermann, 2007). Recently, Zhou (2020) has demonstrated the universality of convolutional neural networks.

Classical universal approximation theorems specialize in the representation power of shallow wide networks with bounded depth and unbounded width. Based on mounting empirical evidence that deep networks demonstrate better performance than wide networks, the construction of deep networks instead of shallow networks has gained considerable attention in recent literature. Consequently, researchers have started to analyze the universal approximation property of deep networks (Cohen et al., 2016; Lin and Jegelka, 2018; Rolnick and Tegmark, 2018; Kidger and Lyons, 2020). Studies on MLP have shown that wide shallow networks require only two depths to have universality, while deep narrow networks require widths at least as their input dimension.

A wide network obtains universality by increasing its width even if the depth is only two (Cybenko, 1989; Hornik et al., 1989; Leshno et al., 1993). However, in the case of a deep network, there is a function for a narrow network that cannot be able to approximated, regardless of its depth (Lu et al., 2017; Park et al., 2021). Therefore, clarifying the minimum width to guarantee universality is crucial, and studies are underway to investigate its lower and upper bounds, narrowing the gap.

Recurrent neural networks (RNNs) (Rumelhart et al., 1986; Elman, 1990) have been crucial for modeling complex temporal dependencies in sequential data. They have various applications in diverse fields, such as language modeling (Mikolov et al., 2010; Jozefowicz et al., 2016), speech recognition (Graves and Jaitly, 2014; Bahdanau et al., 2016), recommendation systems (Hidasi et al., 2015; Wu et al., 2017), and machine translation (Bahdanau et al., 2014). Deep RNNs are widely used and have been successfully applied in practical applications. However, their theoretical understanding remains elusive despite their intensive use. This deficiency in existing studies motivated our work.

In this study, we prove the universal approximation theorem of deep narrow RNNs and discover the upper bound of their minimum width. The target class consists of a sequence-to-sequence function that depends solely on past information. We refer to such functions as past-dependent functions. We provide the upper bound of the minimum width of the RNN for universality in the space of the past-dependent functions. Surprisingly, the upper bound is independent of the length of the sequence. This theoretical result highlights the suitability of the recurrent structure for sequential data compared with other network structures. Furthermore, our results are not restricted to RNNs; they can be generalized to variants of RNNs, including long short-term memory (LSTM), gated recurrent units (GRU), and bidirectional RNNs (BRNN). As corollaries of our main theorem, LSTM and GRU are shown to have the same universality and target class as an RNN. We also prove that the BRNN can approximate any sequence-to-sequence function in a continuous or $L^{p}$ space under the respective norms. We also present the upper bound of the minimum width for these variants. Table 1 outlines our main results.

We briefly review some of the results of studies on the universal approximation property. Studies have been conducted to determine whether a neural network can learn a sufficiently wide range of functions, that is, whether it has universal properties. Cybenko (1989) and Hornik et al. (1989) first proved that the most basic network, a simple two-layered MLP, can approximate arbitrary continuous functions defined on a compact set. Some follow-up studies have investigated the universal properties of other structures for a specific task, such as a convolutional neural network for image processing (Zhou, 2020), an RNN for open dynamical systems (Schäfer and Zimmermann, 2007; Hanson and Raginsky, 2020), and transformer networks for translation and speech recognition (Yun et al., 2020). Particularly for RNNs, Schäfer and Zimmermann (2007) showed that an open dynamical system with continuous state transition and output function can be approximated by a network with a wide RNN cell and subsequent linear layer in finite time. Hanson and Raginsky (2020) showed that the trajectory of the dynamical system can be reproduced with arbitrarily small errors up to infinite time, assuming a stability condition on long-term behavior.

While such prior studies mainly focused on wide and shallow networks, several studies have determined whether a deep narrow network with bounded width can approximate arbitrary functions in some kinds of target function classes, such as a space of continuous functions or an $L^{p}$ space (Lu et al., 2017; Hanin and Sellke, 2017; Johnson, 2019; Kidger and Lyons, 2020; Park et al., 2021). Unlike the case of a wide network that requires only one hidden layer for universality, there exists a function that cannot be approximated by any network whose width is less than a certain threshold. Specifically, in a continuous function space $C\left(K,\mathbb{R}^{d_{y}}\right)$ with the supreme-norm $\left\|f\right\|_{\infty}\coloneqq\sup_{x\in K}\left\|f(x)\right\|$ on a compact subset $K\subset\mathbb{R}^{d_{x}}$, Hanin and Sellke (2017) showed negative and positive results that MLPs do not attain universality with width $d_{x}$. Still, width $d_{x}+d_{y}$ is sufficient for MLPs to achieve universality with the ReLU activation function. Johnson (2019); Kidger and Lyons (2020) generalized the condition on the activation and proved that $d_{x}$ is too narrow, but $d_{x}+d_{y}+2$ is sufficiently wide for the universality in $C\left(K,\mathbb{R}^{d_{y}}\right)$. On the other hand, in an $L^{p}$ space $L^{p}\left(\mathbb{R}^{d_{x}},\mathbb{R}^{d_{y}}\right)$ with $L^{p}$-norm $\left\|f\right\|_{p}\coloneqq\left(\int_{\mathbb{R}^{d_{x}}}f(x)\,dx\right)^{1/p}$ for $p\in[0,\infty)$, Lu et al. (2017) showed that the width $d_{x}$ is insufficient for MLPs to have universality, but $d_{x}+4$ is enough, with the ReLU activation when $p=1$ and $d_{y}=1$. Kidger and Lyons (2020) found that MLPs whose width is $d_{x}+d_{y}+1$ achieve universality in $L^{p}\left(\mathbb{R}^{d_{x}},\mathbb{R}^{d_{y}}\right)$ with the ReLU activation, and Park et al. (2021) determined the exact value $\max\left\{d_{x}+1,d_{y}\right\}$ required for universality in the $L^{p}$ space with ReLU.

As described earlier, studies on deep narrow MLP have been actively conducted, but the approximation ability of deep narrow RNNs remains unclear. This is because the process by which the input affects the result is complicated compared with that of an MLP. The RNN cell transfers information to both the next time step in the same layer and the same time step in the next layer, which makes it difficult to investigate the minimal width. In this regard, we examined the structure of the RNN to apply the methodology and results from the study of MLPs to deep narrow RNNs.

This section introduces the definition of network architecture and the notation used throughout this paper. $d_{x}$ and $d_{y}$ denote the dimension of input and output space, respectively. $\sigma$ is an activation function unless otherwise stated. Sometimes, $v$ indicates a vector with suitable dimensions.

First, we used square brackets, subscripts, and colon symbols to index a sequence of vectors. More precisely, for a given sequence of $d_{x}$-dimensional vectors $x:\mathbb{N}\rightarrow\mathbb{R}^{d_{x}}$, $x[t]_{j}$ or $x_{j}[t]$ denotes the $j$-th component of the $t$-th vector. The colon symbol $:$ is used to denote a continuous index, such as $x[a:b]=\left(x[i]\right)_{a\leq i\leq b}$ or $x[t]_{a:b}=\left(x[t]_{a},x[t]_{a+1},\ldots,x[t]_{b}\right)^{T}\in\mathbb{R}^{b-a+1}$. We call the sequential index $t$ by time and each $x[t]$ a token.

Second, we define the token-wise linear maps $\mathcal{P}:\mathbb{R}^{d_{x}\times N}\rightarrow\mathbb{R}^{d_{s}\times N}$ and $\mathcal{Q}:\mathbb{R}^{d_{s}\times N}\rightarrow\mathbb{R}^{{\color[rgb]{0,0,0}d_{y}}\times N}$ to connect the input, hidden state, and output space. As the dimension of the hidden state space $\mathbb{R}^{d_{s}}$ on which the RNN cells act is different from those of the input domain $\mathbb{R}^{d_{x}}$ and output domain $\mathbb{R}^{d_{y}}$, we need maps adjusting the dimensions of the spaces. For a given matrix $P\in\mathbb{R}^{d_{s}\times d_{x}}$, a lifting map $\mathcal{P}(x)[t]\coloneqq Px[t]$ lifts the input vector to the hidden state space. Similarly, for a given matrix $Q\in\mathbb{R}^{d_{y}\times d_{s}}$, a projection map $\mathcal{Q}(s)[t]\coloneqq Qs[t]$ projects a hidden state onto the output vector. As the first token defines a token-wise map, we sometimes represent token-wise maps without a time length, such as $\mathcal{P}:\mathbb{R}^{d_{x}}\rightarrow\mathbb{R}^{d_{s}}$ instead of $\mathcal{P}:\mathbb{R}^{d_{x}\times N}\rightarrow{\color[rgb]{0,0,0}\mathbb{R}^{d_{s}\times N}}$.

Subsequently, an RNN is constructed using a composition of basic recurrent cells between the lifting and projection maps. We considered four basic cells: RNN, LSTM, GRU, and BRNN.

• •

  RNN Cell A recurrent cell, recurrent layer, or RNN cell $\mathcal{R}$ maps an input sequence $x=\left(x[1],x[2],\ldots\right)=\left(x[t]\right)_{t\in\mathbb{N}}\in\mathbb{R}^{d_{s}\times\mathbb{N}}$ to an output sequence $y=\left(y[t]\right)_{t\in\mathbb{N}}\in\mathbb{R}^{d_{s}\times\mathbb{N}}$ using

  $$y[t+1]=\mathcal{R}(x)[t+1]=\sigma\left(A\mathcal{R}(x)[t]+Bx[t+1]+\theta\right),$$

  (1)

  where $\sigma$ is an activation function, $A,B\in\mathbb{R}^{d_{s}\times d_{s}}$ are the weight matrices, and $\theta\in\mathbb{R}^{d_{s}}$ is the bias vector. The initial state $\mathcal{R}(x)[0]$ can be an arbitrary constant vector, which is zero vector $\mathbf{0}$ in this setting.

• •

  LSTM cell An LSTM cell is an RNN cell variant with an internal cell state and gate structures. As an original version proposed in Hochreiter and Schmidhuber (1997), an LSTM cell includes only two gates: an input gate and an output gate. Still, the base of almost all networks used in modern deep learning is LSTM cells with an additional gate called the forget gate, introduced in Gers et al. (2000). Hence we present the LSTM cell with three gates instead of the original one.

  Mathematically, an LSTM cell $\mathcal{R}_{LSTM}$ is a process that computes an output $h\in\mathbb{R}^{d_{s}}$ and cell state $c\in\mathbb{R}^{d_{s}}$, defined by the following relation:

  	$\displaystyle i[t+1]$	$\displaystyle=\sigma_{\operatorname{sig}}\left(W_{i}x[t+1]+U_{i}h[t]+b_{i}\right)$		(2)
  	$\displaystyle f[t+1]$	$\displaystyle=\sigma_{\operatorname{sig}}\left(W_{f}x[t+1]+U_{f}h[t]+b_{f}\right)$
  	$\displaystyle g[t+1]$	$\displaystyle=\tanh\left(W_{g}x[t+1]+U_{g}h[t]+b_{g}\right)$
  	$\displaystyle o[t+1]$	$\displaystyle=\sigma_{\operatorname{sig}}\left(W_{o}x[t+1]+U_{o}h[t]+b_{o}\right)$
  	$\displaystyle c[t+1]$	$\displaystyle=f[t+1]\odot c[t]+i[t+1]\odot g[t+1]$
  	$\displaystyle h[t+1]$	$\displaystyle=o[t+1]\odot\tanh\left(c[t+1]\right)$

  where $W_{*}\in\mathbb{R}^{d_{s}\times d_{s}}$ and $U_{*}\in\mathbb{R}^{d_{s}\times d_{s}}$ are weight matrices; $b_{*}\in\mathbb{R}^{d_{s}}$ is the bias vector for each $*=i,f,g,o$; and $\sigma_{\operatorname{sig}}$ is the sigmoid activation function. $\odot$ denotes component-wise multiplication, and we set the initial state to zero in this study.

  Although some variants of the LSTM have additional connections, such as peephole connection, our result for LSTM extends naturally.

• •

  GRU cell Another import variation of an RNN is GRU, proposed by Cho et al. (2014). Mathematically, a GRU cell $\mathcal{R}_{GRU}$ is a process that computes $h{\color[rgb]{0,0,0}\in\mathbb{R}^{d_{s}}}$ defined by

  	$\displaystyle r[t+1]$	$\displaystyle=\sigma_{\operatorname{sig}}\left(W_{r}x[t+1]+U_{r}h[t]+b_{r}\right),$		(3)
  	$\displaystyle\tilde{h}[t+1]$	$\displaystyle=\tanh\left(W_{h}x[t+1]+U_{h}\left(r[t+1]\odot h[t]\right)+b_{h}\right),$
  	$\displaystyle z[t+1]$	$\displaystyle=\sigma_{\operatorname{sig}}\left(W_{z}x[t+1]+U_{z}h[t]+b_{z}\right),$
  	$\displaystyle h[t+1]$	$\displaystyle=\left(1-z[t+1]\right){\color[rgb]{0,0,0}\odot}h[t]+z[t+1]{\color[rgb]{0,0,0}\odot}\tilde{h}[t+1],$

  where $W_{*}\in\mathbb{R}^{d_{s}\times{\color[rgb]{0,0,0}d_{s}}}$ and $U_{*}\in\mathbb{R}^{d_{s}\times d_{s}}$ are weight matrices, $b_{*}\in\mathbb{R}^{d_{s}}$ is the bias vector for each $*=r,z,h$, and $\sigma_{\operatorname{sig}}$ is the sigmoid activation function. $\odot$ denotes component-wise multiplication, and we set the initial state to zero in this study.

• •

  BRNN cell A BRNN cell $\mathcal{B}\mathcal{R}$ consists of a pair of RNN cells and a token-wise linear map that follows the cells (Schuster and Paliwal, 1997). An RNN cell $\mathcal{R}_{1}$ in the BRNN cell $\mathcal{B}\mathcal{R}$ receives input from $x[1]$ to $x[N]$ and the other $\mathcal{R}_{2}$ receives input from $x[N]$ to $x[1]$ in reverse order. Then, the linear map $\mathcal{L}$ in $\mathcal{B}\mathcal{R}$ combines the two outputs from the RNN cells. Specifically, a BRNN cell $\mathcal{B}\mathcal{R}$ is defined as follows:

  	$\displaystyle\mathcal{R}(x)[t+1]$	$\displaystyle\coloneqq\sigma\left(A\mathcal{R}_{1}(x)[t]+Bx[t+1]+\theta\right),$		(4)
  	$\displaystyle\bar{\mathcal{R}}(x)[t-1]$	$\displaystyle\coloneqq\sigma\left(\bar{A}\bar{\mathcal{R}}(x)[t]+\bar{B}x[t-1]+\bar{\theta}\right),$
  	$\displaystyle\mathcal{B}\mathcal{R}(x)[t]$	$\displaystyle\coloneqq\mathcal{L}\left(\mathcal{R}(x)[t],\bar{\mathcal{R}}(x)[t]\right)$
  		$\displaystyle\coloneqq W\mathcal{R}(x)[t]+\bar{W}\bar{\mathcal{R}}(x)[t].$

  where $A$, $B$, $\bar{A}$, $\bar{B}$, $W$, and $\bar{W}$ are weight matrices; $\theta$ and $\bar{\theta}$ are bias vectors.

• •

  Network architecture An RNN $\mathcal{N}$ comprises a lifting map $\mathcal{P}$, projection map $\mathcal{Q}$, and $L$ recurrent cells $\mathcal{R}_{1},\ldots,\mathcal{R}_{L}$;

  $$\mathcal{N}\coloneqq\mathcal{Q}\circ\mathcal{R}_{L}\circ\cdots\circ\mathcal{R}_{1}\circ\mathcal{P}.$$

  (5)

  We denote the network as a stack RNN or deep RNN when $L\geq 2$, and each output of the cell $\mathcal{R}_{i}$ as the $i$-th hidden state. $d_{s}$ indicates the width of the network. As a special case, a recurrent cell $\mathcal{R}$ in (1) is a composition of activation and a token-wise affine map with $A=0$; A token-wise MLP is a specific example of RNN. If LSTM, GRU, or BRNN cells replace recurrent cells, the network is called an LSTM, a GRU, or a BRNN.

In addition to the type of cell, the activation function $\sigma$ affects universality. We focus on the case of ReLU or $\tanh$ while also considering the general activation function satisfying the condition proposed by (Kidger and Lyons, 2020). $\sigma$ is a continuous non-polynomial function that is continuously differentiable at some $z_{0}$ with $\sigma^{\prime}(z_{0})\neq 0$. We refer to the condition as a non-degenerate condition and $z_{0}$ as a non-degenerating point.

Finally, the target class must be set as a subset of the sequence-to-sequence function space, from $\mathbb{R}^{d_{x}}$ to $\mathbb{R}^{d_{y}}$. Given an RNN $\mathcal{N}$, each token $y[t]$ of the output sequence $y=\mathcal{N}(x)$ depends only on $x[1\mathbin{:}t]\coloneqq\left(x[1],x[2],\ldots,x[t]\right)$ for the input sequence $x$. We define this property as past dependency and a function with this property as a past-dependent function. More precisely, if all the output tokens of a sequence-to-sequence function are given by $f[t]\left(x[1\mathbin{:}t]\right)$ for functions $f[t]:\mathbb{R}^{d_{x}\times t}\rightarrow\mathbb{R}^{d_{y}}$, we say that the function is past-dependent. Meanwhile, we must fix the finite length or terminal time $N<\infty$ of the input and output sequence. Without additional assumptions such as in (Hanson and Raginsky, 2020), errors generally accumulate over time, making it impossible to approximate implicit dynamics up to infinite time regardless of past dependency. Therefore we set the target function class as a class of past-dependent sequence-to-sequence functions with sequence length $N$.

Sometimes, only the last value $\mathcal{N}(x)[N]$ is required considering an RNN $\mathcal{N}$ as a sequence-to-vector function $\mathcal{N}:\mathbb{R}^{d_{x}\times N}\rightarrow\mathbb{R}^{d_{y}}$. We freely use the terminology RNN for sequence-to-sequence and sequence-to-vector functions because there is no confusion when the output domain is evident.

We have described all the concepts necessary to set a problem, but we end this section with an introduction to the concepts used in the proof of the main theorem. For the convenience of the proof, we slightly modify the activation $\sigma$ to act only on some components, instead of all components. With activation $\sigma$ and index set $I\subseteq\mathbb{N}$, the modified activation $\sigma_{I}$ is defined as

Using the modified activation function $\sigma_{I}$, the basic cells of the network are modified in (1). For example, a modified recurrent cell can be defined as

Similarly, modified RNN, LSTM, GRU, or BRNN is defined using modified cells in (1). This concept is similar to the enhanced neuron of Kidger and Lyons (2020) in that activation can be selectively applied, but is different in that activation can be applied to partial components.

As activation leads to the non-linearity of a network, modifying the activation can affect the minimum width of the network. Fortunately, the following lemma shows that the minimum width increases by at most one owing to the modification. We briefly introduce the ideas here, with a detailed proof provided in the appendix.

The lemma implies that a modified RNN can be approximated by an RNN with at most one additional width. For a given modified RNN $\bar{\mathcal{Q}}\circ\bar{\mathcal{R}}_{L}\circ\cdots\circ\bar{\mathcal{R}}_{1}\circ{\color[rgb]{0,0,0}\bar{\mathcal{P}}}$ of width $d$ and $\epsilon>0$, we can find RNN $\mathcal{R}_{1},\ldots,\mathcal{R}_{2L}$ and linear maps $\mathcal{P}_{1},\ldots,\mathcal{P}_{L}$, $\mathcal{Q}_{1},\ldots,\mathcal{Q}_{L}$ such that

The composition $\mathcal{R}\circ\mathcal{P}$ of an RNN cell $\mathcal{R}$ and token-wise linear map $\mathcal{P}$ can be substituted by another RNN cell $\mathcal{R}^{\prime}$. More concretely, for $\mathcal{R}$ and $\mathcal{P}$ defined by

Thus, $\mathcal{R}_{2l+1}\left(\mathcal{P}_{l+1}\mathcal{Q}_{l}\right)$ becomes a recurrent cell, and $\left(\mathcal{Q}_{L}\mathcal{R}_{2L}\mathcal{R}_{2L-1}\mathcal{P}_{L}\right)\circ\cdots\circ\left(\mathcal{Q}_{1}\mathcal{R}_{2}\mathcal{R}_{1}\mathcal{P}_{1}\right)(x)$ defines a network of form (5).

This section introduces the universal approximation theorem of deep RNNs in continuous function space.

To prove the above theorem, we deal with the case of the sequence-to-vector function $\mathcal{N}:\mathbb{R}^{d_{x}\times N}\rightarrow\mathbb{R}^{d_{y}}$ first, in subsection 4.1. Then, we extend our idea to a sequence-to-sequence function using bias terms to separate the input vectors at different times in subsection 4.2, and the proof of Theorem 5 is presented at the end of this section. As the number of additional components required in each step in the subsections depends on the activation function, we use $\alpha(\sigma)$ to state the theorem briefly.