Stochastic Reservoir Computers

TL;DR

This paper investigates the universality of stochastic reservoir computers which use the probabilities of each stochastic reservoir state as the readout instead of the states themselves, and proves that classes of stochastic echo state networks form universal approximating classes.

Abstract

Reservoir computing is a form of machine learning that utilizes nonlinear dynamical systems to perform complex tasks in a cost-effective manner when compared to typical neural networks. Many recent advancements in reservoir computing, in particular quantum reservoir computing, make use of reservoirs that are inherently stochastic. However, the theoretical justification for using these systems has not yet been well established. In this paper, we investigate the universality of stochastic reservoir computers, in which we use a stochastic system for reservoir computing using the probabilities of each reservoir state as the readout instead of the states themselves. In stochastic reservoir computing, the number of distinct states of the entire reservoir computer can potentially scale exponentially with the size of the reservoir hardware, offering the advantage of compact device size. We prove that classes of stochastic echo state networks, and therefore the class of all stochastic reservoir computers, are universal approximating classes. We also investigate the performance of two practical examples of stochastic reservoir computers in classification and chaotic time series prediction. While shot noise is a limiting factor in the performance of stochastic reservoir computing, we show significantly improved performance compared to a deterministic reservoir computer with similar hardware in cases where the effects of noise are small.

Figures (8)

• Figure 1: Diagram of the design for the qubit reservoir network.
• Figure 2: The left plot shows average NMSE values for the Lorenz $X$ task, while the right plot shows the average error percentage for the Sine-Square wave identification task, both as a function of the number of detectors using the qubit reservoir network. The blue dots correspond to the deterministic ESN, the orange dots correspond to the stochastic ESN averaged over 100000 runs for the Lorenz $X$ task and 10000 runs for the Sine-Square wave task, and the green dots correspond to the stochastic ESN results in the theoretical limit of infinite statistics, using the exact probabilities. Each data point represents the average over 100 random choices of $A$ and $B$ for the Lorenz $X$ task and 1000 choices for the Sine-Square wave task, and the error bars give the standard deviations of each data point over the samples of ESNs with different choices of $A$ and $B$.
• Figure 3: Diagram of the design for the stochastic optical network.
• Figure 4: The left plot shows average NMSE values for the Lorenz $X$ task, while the right plot shows the average error percentage for the Sine-Square wave identification task, both as a function of the number of detectors using the stochastic optical network. The blue dots correspond to the deterministic ESN, the orange dots correspond to the stochastic ESN averaged over 100000 runs for the Lorenz $X$ task and 10000 runs for the Sine-Square wave task, and the green dots correspond to the stochastic ESN results in the theoretical limit of infinite statistics, using the exact probabilities. Each data point represents the average over 100 random choices of $A$ and $B$ for the Lorenz $X$ task and 1000 choices for the Sine-Square wave task, and the error bars give the standard deviations of each data point over the samples of ESNs with different choices of $A$ and $B$.
• Figure 5: The left plot shows NMSE values for the Lorenz $X$ task, while the right plot shows the error percentage for the Sine-Square wave identification task, both as a function of the number of runs for the qubit reservoir network. Each data point represents the relevant error measure of the stochastic ESN with fixed $A$ and $B$ using 2 detectors.

Theorems & Definitions (16)

• Theorem 1: Universality
• Theorem 2: Uniform Convergence and Fading Memory
• Theorem 3
• Lemma 1
• Definition 1
• Theorem 4: Separation
• proof
• Corollary 1
• proof
• proof : Proof of Theorem \ref{['thm:universal']}