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Reservoir computing is a best-in-class machine learning algorithm for processing information generated by dynamical systems using observed time-series data. Importantly, it requires very small training data sets, uses linear optimization, and thus requires minimal computing resources. However, the algorithm uses randomly sampled matrices to define the underlying recurrent neural network and has a multitude of metaparameters that must be optimized. Recent results demonstrate the equivalence of reservoir computing to nonlinear vector autoregression, which requires no random matrices, fewer metaparameters, and provides interpretable results. Here, we demonstrate that nonlinear vector autoregression excels at reservoir computing benchmark tasks and requires even shorter training data sets and training time, heralding the next generation of reservoir computing.
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We demonstrate reservoir computing with a physical system using a single autonomous Boolean logic element with time-delay feedback. The system generates a chaotic transient with a window of consistency lasting between 30 and 300 ns, which we show is sufficient for reservoir computing. We then characterize the dependence of computational performance on system parameters to find the best operating point of the reservoir. When the best parameters are chosen, the reservoir is able to classify short input patterns with performance that decreases over time. In particular, we show that four distinct input patterns can be classified for 70 ns, even though the inputs are only provided to the reservoir for 7.5 ns.
We analyze the practices of reservoir computing in the framework of statistical learning theory. In particular, we derive finite sample upper bounds for the generalization error committed by specific families of reservoir computing systems when processing discrete-time inputs under various hypotheses on their dependence structure. Non-asymptotic bounds are explicitly written down in terms of the multivariate Rademacher complexities of the reservoir systems and the weak dependence structure of the signals that are being handled. This allows, in particular, to determine the minimal number of observations needed in order to guarantee a prescribed estimation accuracy with high probability for a given reservoir family. At the same time, the asymptotic behavior of the devised bounds guarantees the consistency of the empirical risk minimization procedure for various hypothesis classes of reservoir functionals.
Reservoir Computing (RC) refers to a Recurrent Neural Networks (RNNs) framework, frequently used for sequence learning and time series prediction. The RC system consists of a random fixed-weight RNN (the input-hidden reservoir layer) and a classifier (the hidden-output readout layer). Here we focus on the sequence learning problem, and we explore a different approach to RC. More specifically, we remove the non-linear neural activation function, and we consider an orthogonal reservoir acting on normalized states on the unit hypersphere. Surprisingly, our numerical results show that the systems memory capacity exceeds the dimensionality of the reservoir, which is the upper bound for the typical RC approach based on Echo State Networks (ESNs). We also show how the proposed system can be applied to symmetric cryptography problems, and we include a numerical implementation.
Computing has dramatically changed nearly every aspect of our lives, from business and agriculture to communication and entertainment. As a nation, we rely on computing in the design of systems for energy, transportation and defense; and computing fuels scientific discoveries that will improve our fundamental understanding of the world and help develop solutions to major challenges in health and the environment. Computing has changed our world, in part, because our innovations can run on computers whose performance and cost-performance has improved a million-fold over the last few decades. A driving force behind this has been a repeated doubling of the transistors per chip, dubbed Moores Law. A concomitant enabler has been Dennard Scaling that has permitted these performance doublings at roughly constant power, but, as we will see, both trends face challenges. Consider for a moment the impact of these two trends over the past 30 years. A 1980s supercomputer (e.g. a Cray 2) was rated at nearly 2 Gflops and consumed nearly 200 KW of power. At the time, it was used for high performance and national-scale applications ranging from weather forecasting to nuclear weapons research. A computer of similar performance now fits in our pocket and consumes less than 10 watts. What would be the implications of a similar computing/power reduction over the next 30 years - that is, taking a petaflop-scale machine (e.g. the Cray XK7 which requires about 500 KW for 1 Pflop (=1015 operations/sec) performance) and repeating that process? What is possible with such a computer in your pocket? How would it change the landscape of high capacity computing? In the remainder of this paper, we articulate some opportunities and challenges for dramatic performance improvements of both personal to national scale computing, and discuss some out of the box possibilities for achieving computing at this scale.
There is a wave of interest in using unsupervised neural networks for solving differential equations. The existing methods are based on feed-forward networks, {while} recurrent neural network differential equation solvers have not yet been reported. We introduce an unsupervised reservoir computing (RC), an echo-state recurrent neural network capable of discovering approximate solutions that satisfy ordinary differential equations (ODEs). We suggest an approach to calculate time derivatives of recurrent neural network outputs without using backpropagation. The internal weights of an RC are fixed, while only a linear output layer is trained, yielding efficient training. However, RC performance strongly depends on finding the optimal hyper-parameters, which is a computationally expensive process. We use Bayesian optimization to efficiently discover optimal sets in a high-dimensional hyper-parameter space and numerically show that one set is robust and can be used to solve an ODE for different initial conditions and time ranges. A closed-form formula for the optimal output weights is derived to solve first order linear equations in a backpropagation-free learning process. We extend the RC approach by solving nonlinear system of ODEs using a hybrid optimization method consisting of gradient descent and Bayesian optimization. Evaluation of linear and nonlinear systems of equations demonstrates the efficiency of the RC ODE solver.
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