Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs

Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a quantum setting, whereby classical functions can be approximated by parameterised quantum circuits. We provide here precise error bounds for specific classes of functions and extend these results to the interesting new setup of randomised quantum circuits, mimicking classical reservoir neural networks. Our results show in particular that a quantum neural network with $\mathcal{O}(\varepsilon^{-2})$ weights and $\mathcal{O} (\lceil \log_2(\varepsilon^{-1}) \rceil)$ qubits suffices to achieve accuracy $\varepsilon>0$ when approximating functions with integrable Fourier transform.

Infinite-dimensional reservoir computing

Reservoir computing approximation and generalization bounds are proved for a new concept class of input/output systems that extends the so-called generalized Barron functionals to a dynamic context. This new class is characterized by the readouts with a certain integral representation built on infinite-dimensional state-space systems. It is shown that this class is very rich and possesses useful features and universal approximation properties. The reservoir architectures used for the approximation and estimation of elements in the new class are randomly generated echo state networks with either linear or ReLU activation functions. Their readouts are built using randomly generated neural networks in which only the output layer is trained (extreme learning machines or random feature neural networks). The results in the paper yield a fully implementable recurrent neural network-based learning algorithm with provable convergence guarantees that do not suffer from the curse of dimensionality.

The necessity of depth for artificial neural networks to approximate certain classes of smooth and bounded functions without the curse of dimensionality

In this article we study high-dimensional approximation capacities of shallow and deep artificial neural networks (ANNs) with the rectified linear unit (ReLU) activation. In particular, it is a key contribution of this work to reveal that for all $a,b\in\mathbb{R}$ with $b-a\geq 7$ we have that the functions $[a,b]^d\ni x=(x_1,\dots,x_d)\mapsto\prod_{i=1}^d x_i\in\mathbb{R}$ for $d\in\mathbb{N}$ as well as the functions $[a,b]^d\ni x =(x_1,\dots, x_d)\mapsto\sin(\prod_{i=1}^d x_i) \in \mathbb{R} $ for $ d \in \mathbb{N} $ can neither be approximated without the curse of dimensionality by means of shallow ANNs nor insufficiently deep ANNs with ReLU activation but can be approximated without the curse of dimensionality by sufficiently deep ANNs with ReLU activation. We show that the product functions and the sine of the product functions are polynomially tractable approximation problems among the approximating class of deep ReLU ANNs with the number of hidden layers being allowed to grow in the dimension $ d \in \mathbb{N} $. We establish the above outlined statements not only for the product functions and the sine of the product functions but also for other classes of target functions, in particular, for classes of uniformly globally bounded $ C^{ \infty } $-functions with compact support on any $[a,b]^d$ with $a\in\mathbb{R}$, $b\in(a,\infty)$. Roughly speaking, in this work we lay open that simple approximation problems such as approximating the sine or cosine of products cannot be solved in standard implementation frameworks by shallow or insufficiently deep ANNs with ReLU activation in polynomial time, but can be approximated by sufficiently deep ReLU ANNs with the number of parameters growing at most polynomially.

Reservoir kernels and Volterra series

A universal kernel is constructed whose sections approximate any causal and time-invariant filter in the fading memory category with inputs and outputs in a finite-dimensional Euclidean space. This kernel is built using the reservoir functional associated with a state-space representation of the Volterra series expansion available for any analytic fading memory filter. It is hence called the Volterra reservoir kernel. Even though the state-space representation and the corresponding reservoir feature map are defined on an infinite-dimensional tensor algebra space, the kernel map is characterized by explicit recursions that are readily computable for specific data sets when employed in estimation problems using the representer theorem. We showcase the performance of the Volterra reservoir kernel in a popular data science application in relation to bitcoin price prediction.

Deep neural network expressivity for optimal stopping problems

This article studies deep neural network expression rates for optimal stopping problems of discrete-time Markov processes on high-dimensional state spaces. A general framework is established in which the value function and continuation value of an optimal stopping problem can be approximated with error at most $\varepsilon$ by a deep ReLU neural network of size at most $\kappa d^{\mathfrak{q}} \varepsilon^{-\mathfrak{r}}$. The constants $\kappa,\mathfrak{q},\mathfrak{r} \geq 0$ do not depend on the dimension $d$ of the state space or the approximation accuracy $\varepsilon$. This proves that deep neural networks do not suffer from the curse of dimensionality when employed to solve optimal stopping problems. The framework covers, for example, exponential L\'evy models, discrete diffusion processes and their running minima and maxima. These results mathematically justify the use of deep neural networks for numerically solving optimal stopping problems and pricing American options in high dimensions.

Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality

This article investigates the use of random feature neural networks for learning Kolmogorov partial (integro-)differential equations associated to Black-Scholes and more general exponential L\'evy models. Random feature neural networks are single-hidden-layer feedforward neural networks in which only the output weights are trainable. This makes training particularly simple, but (a priori) reduces expressivity. Interestingly, this is not the case for Black-Scholes type PDEs, as we show here. We derive bounds for the prediction error of random neural networks for learning sufficiently non-degenerate Black-Scholes type models. A full error analysis is provided and it is shown that the derived bounds do not suffer from the curse of dimensionality. We also investigate an application of these results to basket options and validate the bounds numerically. These results prove that neural networks are able to \textit{learn} solutions to Black-Scholes type PDEs without the curse of dimensionality. In addition, this provides an example of a relevant learning problem in which random feature neural networks are provably efficient.

Fading memory echo state networks are universal

Echo state networks (ESNs) have been recently proved to be universal approximants for input/output systems with respect to various $L ^p$-type criteria. When $1\leq p< \infty$, only $p$-integrability hypotheses need to be imposed, while in the case $p=\infty$ a uniform boundedness hypotheses on the inputs is required. This note shows that, in the last case, a universal family of ESNs can be constructed that contains exclusively elements that have the echo state and the fading memory properties. This conclusion could not be drawn with the results and methods available so far in the literature.

Weak error analysis for stochastic gradient descent optimization algorithms

Stochastic gradient descent (SGD) type optimization schemes are fundamental ingredients in a large number of machine learning based algorithms. In particular, SGD type optimization schemes are frequently employed in applications involving natural language processing, object and face recognition, fraud detection, computational advertisement, and numerical approximations of partial differential equations. In mathematical convergence results for SGD type optimization schemes there are usually two types of error criteria studied in the scientific literature, that is, the error in the strong sense and the error with respect to the objective function. In applications one is often not only interested in the size of the error with respect to the objective function but also in the size of the error with respect to a test function which is possibly different from the objective function. The analysis of the size of this error is the subject of this article. In particular, the main result of this article proves under suitable assumptions that the size of this error decays at the same speed as in the special case where the test function coincides with the objective function.

Memory and forecasting capacities of nonlinear recurrent networks

The notion of memory capacity, originally introduced for echo state and linear networks with independent inputs, is generalized to nonlinear recurrent networks with stationary but dependent inputs. The presence of dependence in the inputs makes natural the introduction of the network forecasting capacity, that measures the possibility of forecasting time series values using network states. Generic bounds for memory and forecasting capacities are formulated in terms of the number of neurons of the network and the autocovariance function of the input. These bounds generalize well-known estimates in the literature to a dependent inputs setup. Finally, for linear recurrent networks and independent inputs it is proved that the memory capacity is given by the rank of the associated controllability matrix.

Approximation Bounds for Random Neural Networks and Reservoir Systems

This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hyperparameters are optimized, have been successfully applied in a wide range of static and dynamic learning problems. Despite the popularity of this approach in empirical tasks, important theoretical questions regarding the relation between the unknown function, the weight distribution, and the approximation rate have remained open. In this work it is proved that, as long as the unknown function, functional, or dynamical system is sufficiently regular, it is possible to draw the internal weights of the random (recurrent) neural network from a generic distribution (not depending on the unknown object) and quantify the error in terms of the number of neurons and the hyperparameters. In particular, this proves that echo state networks with randomly generated weights are capable of approximating a wide class of dynamical systems arbitrarily well and thus provides the first mathematical explanation for their empirically observed success at learning dynamical systems.