# A Geometrical Analysis of Global Stability in Trained Feedback Networks

ArticleinNeural Computation 31(6):1-43 · April 2019with 40 Reads
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Abstract
Recurrent neural networks have been extensively studied in the context of neuroscience and machine learning due to their ability to implement complex computations. While substantial progress in designing effective learning algorithms has been achieved, a full understanding of trained recurrent networks is still lacking. Specifically, the mechanisms that allow computations to emerge from the underlying recurrent dynamics are largely unknown. Here we focus on a simple yet underexplored computational setup: a feedback architecture trained to associate a stationary output to a stationary input. As a starting point, we derive an approximate analytical description of global dynamics in trained networks, which assumes uncorrelated connectivity weights in the feedback and in the random bulk. The resulting mean-field theory suggests that the task admits several classes of solutions, which imply different stability properties. Different classes are characterized in terms of the geometrical arrangement of the readout with respect to the input vectors, defined in the high-dimensional space spanned by the network population. We find that such an approximate theoretical approach can be used to understand how standard training techniques implement the input-output task in finite-size feedback networks. In particular, our simplified description captures the local and the global stability properties of the target solution, and thus predicts training performance.
• ... Our model in contrast exhibits an interplay between low rank structured connectivity implementing balance, and high rank disordered connectivity inducing chaos, each with independently adjustable strengths. In general, how computation emerges from an interplay between structured and random connectivity has been a subject of recent interest in theoretical neuroscience [18,23,24,25]. Here we show how structure and randomness interact by obtaining analytic insights into the efficacy of predictive coding, dissecting the individual contributions of balance, noise, weight disorder, chaos, delays and nonlinearity, in a model were all ingredients can coexist and be independently adjusted. ...
... We find: (1) strong balance is a key requirement for superclassical error scaling with network size; (2) without delays, increasing balance always suppresses errors via powers laws with different exponents (-1 for noise, -2 for chaos); (3) delays yield an oscillatory instability and a tradeoff between noise suppression and resonant amplification; (4) this tradeoff sets a maximal critical balance level which decreases with delay; (5) noise or chaos can increase this maximal level by promoting desynchronization; (6) the competition between noise suppression and resonant amplification sets an optimal balance level that is half the maximal level in the case of noise; (7) but is close to the maximal level in the case of chaos for small delays, because the slow chaos has small power at the high resonant frequency; (8) the optimal decoder error rises as a power law with delay (with exponent 1/2 for noise and 1 for chaos). Also, our model unifies a variety of perspectives in theoretical neuroscience, spanning classical synaptic balance [17,29,42,43,44,45], efficient coding in tight balance [7,46], the interplay of structured and random connectivity in computation [18,23,24,47,48], the relation between oscillations and delays in neural networks [49,50,51] and predictive coding [8,10]. Moreover, the mean-field theory developed here can be extended to spiking neurons with strong recurrent balance and delays [52], analytically explaining relations between delays, coding and oscillations observed in simulations but previously not understood [21,22] Acknowledgments JK thanks the Swartz Foundation for Theoretical Neuroscience for funding; JT thanks the National Science Foundation for funding. ...
... We now turn to compute the statistics of the fluctuations of a random network in its chaotic phase, when the variance of the weight distribution is above the critical transition point g > g c . The dynamic mean field theory for a chaotic neural network was first introduced by [15] and re-derived later by [35,38,36,53,24,54]. The connectivity in the subspace orthogonal to the readout direction is randomly distributed, thus the properties of the fluctuations in this subspace, δh ⊥ (t), are equivalent to previous studies of random neural networks. ...
Preprint
Biological neural networks face a formidable task: performing reliable computations in the face of intrinsic stochasticity in individual neurons, imprecisely specified synaptic connectivity, and nonnegligible delays in synaptic transmission. A common approach to combatting such biological heterogeneity involves averaging over large redundant networks of $N$ neurons resulting in coding errors that decrease classically as $1/\sqrt{N}$. Recent work demonstrated a novel mechanism whereby recurrent spiking networks could efficiently encode dynamic stimuli, achieving a superclassical scaling in which coding errors decrease as $1/N$. This specific mechanism involved two key ideas: predictive coding, and a tight balance, or cancellation between strong feedforward inputs and strong recurrent feedback. However, the theoretical principles governing the efficacy of balanced predictive coding and its robustness to noise, synaptic weight heterogeneity and communication delays remain poorly understood. To discover such principles, we introduce an analytically tractable model of balanced predictive coding, in which the degree of balance and the degree of weight disorder can be dissociated unlike in previous balanced network models, and we develop a mean field theory of coding accuracy. Overall, our work provides and solves a general theoretical framework for dissecting the differential contributions neural noise, synaptic disorder, chaos, synaptic delays, and balance to the fidelity of predictive neural codes, reveals the fundamental role that balance plays in achieving superclassical scaling, and unifies previously disparate models in theoretical neuroscience.
• ... From a system perspective, the feedforward neural net-work model has limited computing power, and the feedback dynamics of a feedback neural network more stronger computing power than a feedforward neural network, which is based on feedback to enhance global stability [88]. In feedback neural networks, all neurons have the same status and there is no hierarchical difference. ...
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Compared with von Neumann’s computer architecture, neuromorphic systems offer more unique and novel solutions to the artificial intelligence discipline. Inspired by biology, this novel system has implemented the theory of human brain modeling by connecting feigned neurons and synapses to reveal the new neuroscience concepts. Many researchers have vastly invested in neuro-inspired models, algorithms, learning approaches, operation systems for the exploration of the neuromorphic system and have implemented many corresponding applications. Recently, some researchers have demonstrated the capabilities of Hopfield algorithms in some large-scale notable hardware projects and seen significant progression. This paper presents a comprehensive review and focuses extensively on the Hopfield algorithm’s model and its potential advancement in new research applications. Towards the end, we conclude with a broad discussion and a viable plan for the latest application prospects to facilitate developers with a better understanding of the aforementioned model in accordance to build their own artificial intelligence projects.
• ... From a system perspective, the feedforward neural net-work model has limited computing power, and the feedback dynamics of a feedback neural network more stronger computing power than a feedforward neural network, which is based on feedback to enhance global stability [88]. In feedback neural networks, all neurons have the same status and there is no hierarchical difference. ...
Article
Full-text available
Compared with von Neumann's computer architecture, neuromorphic systems offer more unique and novel solutions to the artificial intelligence discipline. Inspired by biology, this novel system has implemented the theory of human brain modeling by connecting feigned neurons and synapses to reveal the new neuroscience concepts. Many researchers have vastly invested in neuro-inspired models, algorithms, learning approaches, operation systems for the exploration of the neuromorphic system and have implemented many corresponding applications. Recently, some researchers have demonstrated the capabilities of Hopfield algorithms in some large-scale notable hardware projects and seen significant progression. This paper presents a comprehensive review and focuses extensively on the Hopfield algorithm's model and its potential advancement in new research applications. Towards the end, we conclude with a broad discussion and a viable plan for the latest application prospects to facilitate developers with a better understanding of the aforementioned model in accordance to build their own artificial intelligence projects. INDEX TERMS Neuromorphic computing, Neuro-inspired model, Hopfield algorithm, Artificial intelligence.
• ... Maximizing memory does not necessarily lead to performance (e.g., prediction) maximization [11]. In recent years, a large effort has been devoted to tackle these problems, by studying the dynamical systems underlying RNNs [2,12,13,3,14,15]. ...
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Reservoir computing is a popular approach to design recurrent neural networks, due to its training simplicity and its approximation performance. The recurrent part of these networks is not trained (e.g. via gradient descent), making them appealing for analytical studies, raising the interest of a vast community of researcher spanning from dynamical systems to neuroscience. It emerges that, even in the simple linear case, the working principle of these networks is not fully understood and the applied research is usually driven by heuristics. A novel analysis of the dynamics of such networks is proposed, which allows one to express the state evolution using the controllability matrix. Such a matrix encodes salient characteristics of the network dynamics: in particular, its rank can be used as an input-indepedent measure of the memory of the network. Using the proposed approach, it is possible to compare different architectures and explain why a cyclic topology achieves favourable results.
• ... Our approach allows us to gain mechanistic insight into the computations underlying echo state and FORCE learning models which have the same connectivity structure as our model [20,21]. Here, the readout vector n is trained, which leads to correlations to the random part J [3,34]. Our results on multiple fixed points and oscillations show that these correlations are crucial for the rich functional repertoire. ...
Article
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A given neural network in the brain is involved in many different tasks. This implies that, when considering a specific task, the network's connectivity contains a component which is related to the task and another component which can be considered random. Understanding the interplay between the structured and random components and their effect on network dynamics and functionality is an important open question. Recent studies addressed the coexistence of random and structured connectivity but considered the two parts to be uncorrelated. This constraint limits the dynamics and leaves the random connectivity nonfunctional. Algorithms that train networks to perform specific tasks typically generate correlations between structure and random connectivity. Here we study nonlinear networks with correlated structured and random components, assuming the structure to have a low rank. We develop an analytic framework to establish the precise effect of the correlations on the eigenvalue spectrum of the joint connectivity. We find that the spectrum consists of a bulk and multiple outliers, whose location is predicted by our theory. Using mean-field theory, we show that these outliers directly determine both the fixed points of the system and their stability. Taken together, our analysis elucidates how correlations allow structured and random connectivity to synergistically extend the range of computations available to networks.
• ... This simple training protocol is not sufficient in many applications, e.g. when it is required to learn memory states. To this end, training mechanisms based on output feedback [24,25] and online training [10,26] have been proposed, with successful applications in physics [27,28], complex systems modeling [29,30], and neuroscience [31], just to name a few. ...
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A recurrent neural network (RNN) possesses the echo state property (ESP) if, for a given input sequence, it "forgets" any internal states of the driven (nonautonomous) system and asymptotically follows a unique, possibly complex trajectory. The lack of ESP is conventionally understood as a lack of reliable behaviour in RNNs. Here, we show that RNNs can reliably perform computations under a more general principle that accounts only for their local behaviour in phase space. To this end, we formulate a generalisation of the ESP and introduce an echo index to characterise the number of simultaneously stable responses of a driven RNN. We show that it is possible for the echo index to change with inputs, highlighting a potential source of computational errors in RNNs due to characteristics of the inputs driving the dynamics.
• ... Our approach allows us to gain mechanistic insight into the computations underlying echo state and FORCE learning models which have the same connectivity structure as our model [20,21]. Here, the readout vector n is trained, which leads to correlations to the random part J [3,31]. Our results on multiple fixed points and oscillations show that these correlations are crucial for the rich functional repertoire. ...
Preprint
A given neural network in the brain is involved in many different tasks. This implies that, when considering a specific task, the network's connectivity contains a component which is related to the task and another component which can be considered random. Understanding the interplay between the structured and random components, and their effect on network dynamics and functionality is an important open question. Recent studies addressed the co-existence of random and structured connectivity, but considered the two parts to be uncorrelated. This constraint limits the dynamics and leaves the random connectivity non-functional. Algorithms that train networks to perform specific tasks typically generate correlations between structure and random connectivity. Here we study nonlinear networks with correlated structured and random components, assuming the structure to have a low rank. We develop an analytic framework to establish the precise effect of the correlations on the eigenvalue spectrum of the joint connectivity. We find that the spectrum consists of a bulk and multiple outliers, whose location is predicted by our theory. Using mean-field theory, we show that these outliers directly determine both the fixed points of the system and their stability. Taken together, our analysis elucidates how correlations allow structured and random connectivity to synergistically extend the range of computations available to networks.
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Output feedback is crucial for autonomous and parameterized pattern generation with reservoir networks. Read-out learning can lead to error amplification in these settings and therefore regularization is important for both generalization and reduction of error amplification. We show that regularization of the inner reservoir network mitigates parameter dependencies and boosts the task-specific performance. 1
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It is known that if one perturbs a large iid random matrix by a bounded rank error, then the majority of the eigenvalues will remain distributed according to the circular law. However, the bounded rank perturbation may also create one or more outlier eigenvalues. We show that if the perturbation is small, then the outlier eigenvalues are created next to the outlier eigenvalues of the bounded rank perturbation; but if the perturbation is large, then many more outliers can be created, and their law is governed by the zeroes of a random Laurent series with Gaussian coefficients. On the other hand, these outliers may be eliminated by enforcing a row sum condition on the final matrix.
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Neuronal activity arises from an interaction between ongoing firing generated spontaneously by neural circuits and responses driven by external stimuli. Using mean-field analysis, we ask how a neural network that intrinsically generates chaotic patterns of activity can remain sensitive to extrinsic input. We find that inputs not only drive network responses, but they also actively suppress ongoing activity, ultimately leading to a phase transition in which chaos is completely eliminated. The critical input intensity at the phase transition is a nonmonotonic function of stimulus frequency, revealing a "resonant" frequency at which the input is most effective at suppressing chaos even though the power spectrum of the spontaneous activity peaks at zero and falls exponentially. A prediction of our analysis is that the variance of neural responses should be most strongly suppressed at frequencies matching the range over which many sensory systems operate.
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Neural circuits display complex activity patterns both spontaneously and when responding to a stimulus or generating a motor output. How are these two forms of activity related? We develop a procedure called FORCE learning for modifying synaptic strengths either external to or within a model neural network to change chaotic spontaneous activity into a wide variety of desired activity patterns. FORCE learning works even though the networks we train are spontaneously chaotic and we leave feedback loops intact and unclamped during learning. Using this approach, we construct networks that produce a wide variety of complex output patterns, input-output transformations that require memory, multiple outputs that can be switched by control inputs, and motor patterns matching human motion capture data. Our results reproduce data on premovement activity in motor and premotor cortex, and suggest that synaptic plasticity may be a more rapid and powerful modulator of network activity than generally appreciated.
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Computational properties of use of biological organisms or to the construction of computers can emerge as collective properties of systems having a large number of simple equivalent components (or neurons). The physical meaning of content-addressable memory is described by an appropriate phase space flow of the state of a system. A model of such a system is given, based on aspects of neurobiology but readily adapted to integrated circuits. The collective properties of this model produce a content-addressable memory which correctly yields an entire memory from any subpart of sufficient size. The algorithm for the time evolution of the state of the system is based on asynchronous parallel processing. Additional emergent collective properties include some capacity for generalization, familiarity recognition, categorization, error correction, and time sequence retention. The collective properties are only weakly sensitive to details of the modeling or the failure of individual devices.
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A continuous-time dynamic model of a network of N nonlinear elements interacting via random asymmetric couplings is studied. A self-consistent mean-field theory, exact in the N-->∞ limit, predicts a transition from a stationary phase to a chaotic phase occurring at a critical value of the gain parameter. The autocorrelations of the chaotic flow as well as the maximal Lyapunov exponent are calculated.
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We study discrete parallel dynamics of a fully connected network of nonlinear elements interacting via long-range random asymmetric couplings under the influence of external noise. Using dynamical mean-field equations, which become exact in the thermodynamical limit, we calculate the activity and the maximal Lyapunov exponent of the network in dependence of a nonlinearity (gain) parameter and the noise intensity.
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We present a method for learning nonlinear systems, echo state networks (ESNs). ESNs employ artificial recurrent neural networks in a way that has recently been proposed independently as a learning mechanism in biological brains. The learning method is computationally efficient and easy to use. On a benchmark task of predicting a chaotic time series, accuracy is improved by a factor of 2400 over previous techniques. The potential for engineering applications is illustrated by equalizing a communication channel, where the signal error rate is improved by two orders of magnitude.
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The dynamics of neural networks is influenced strongly by the spectrum of eigenvalues of the matrix describing their synaptic connectivity. In large networks, elements of the synaptic connectivity matrix can be chosen randomly from appropriate distributions, making results from random matrix theory highly relevant. Unfortunately, classic results on the eigenvalue spectra of random matrices do not apply to synaptic connectivity matrices because of the constraint that individual neurons are either excitatory or inhibitory. Therefore, we compute eigenvalue spectra of large random matrices with excitatory and inhibitory columns drawn from distributions with different means and equal or different variances.
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It has previously been shown that generic cortical microcircuit models can perform complex real-time computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate the computational capability of such circuits in the more realistic case where not only readout neurons, but in addition a few neurons within the circuit, have been trained for specific tasks. This is essentially equivalent to the case where the output of trained readout neurons is fed back into the circuit. We show that this new model overcomes the limitation of a rapidly fading memory. In fact, we prove that in the idealized case without noise it can carry out any conceivable digital or analog computation on time-varying inputs. But even with noise, the resulting computational model can perform a large class of biologically relevant real-time computations that require a nonfading memory. We demonstrate these computational implications of feedback both theoretically, and through computer simulations of detailed cortical microcircuit models that are subject to noise and have complex inherent dynamics. We show that the application of simple learning procedures (such as linear regression or perceptron learning) to a few neurons enables such circuits to represent time over behaviorally relevant long time spans, to integrate evidence from incoming spike trains over longer periods of time, and to process new information contained in such spike trains in diverse ways according to the current internal state of the circuit. In particular we show that such generic cortical microcircuits with feedback provide a new model for working memory that is consistent with a large set of biological constraints. Although this article examines primarily the computational role of feedback in circuits of neurons, the mathematical principles on which its analysis is based apply to a variety of dynamical systems. Hence they may also throw new light on the computational role of feedback in other complex biological dynamical systems, such as, for example, genetic regulatory networks.
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How to efficiently train recurrent networks remains a challenging and active research topic. Most of the proposed training approaches are based on computational ways to efficiently obtain the gradient of the error function, and can be generally grouped into five major groups. In this study we present a derivation that unifies these approaches. We demonstrate that the approaches are only five different ways of solving a particular matrix equation. The second goal of this paper is develop a new algorithm based on the insights gained from the novel formulation. The new algorithm, which is based on approximating the error gradient, has lower computational complexity in computing the weight update than the competing techniques for most typical problems. In addition, it reaches the error minimum in a much smaller number of iterations. A desirable characteristic of recurrent network training algorithms is to be able to update the weights in an on-line fashion. We have also developed an on-line version of the proposed algorithm, that is based on updating the error gradient approximation in a recursive manner.
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Recurrent neural networks can be used to map input sequences to output sequences, such as for recognition, production or prediction problems. However, practical difficulties have been reported in training recurrent neural networks to perform tasks in which the temporal contingencies present in the input/output sequences span long intervals. We show why gradient based learning algorithms face an increasingly difficult problem as the duration of the dependencies to be captured increases. These results expose a trade-off between efficient learning by gradient descent and latching on information for long periods. Based on an understanding of this problem, alternatives to standard gradient descent are considered.
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The convergence properties of a fairly general class of adaptive recursive least-squares algorithms are studied under the assumption that the data generation mechanism is deterministic and time invariant. First, the (open-loop) identification case is considered. By a suitable notion of excitation subspace, the convergence analysis of the identification algorithm is carried out with no persistent excitation hypothesis, i.e. it is proven that the projection of the parameter error on the excitation subspace tends to zero, while the orthogonal component of the error remains bounded. The convergence of an adaptive control scheme based on the minimum variance control law is then dealt with. It is shown that under the standard minimum-phase assumption, the tracking error converges to zero whenever the reference signal is bounded. Furthermore, the control variable turns out to be bounded
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Echo state networks (ESN) are a novel approach to recurrent neural network training. An ESN consists of a large, fixed, recurrent "reservoir" network, from which the desired output is obtained by training suitable output connection weights. Determination of optimal output weights becomes a linear, uniquely solvable task of MSE minimization. This article reviews the basic ideas and describes an online adaptation scheme based on the RLS algorithm known from adaptive linear systems. As an example, a 10-th order NARMA system is adaptively identified. The known benefits of the RLS algorithms carry over from linear systems to nonlinear ones; specifically, the convergence rate and misadjustment can be determined at design time.
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Gradient descent algorithms in recurrent neural networks can have problems when the network dynamics experience bifurcations in the course of learning. The possible hazards caused by the bifurcations of the network dynamics and the learning equations are investigated. The roles of teacher forcing, preprogramming of network structures, and the approximate learning algorithms are discussed. 1 Introduction Supervised learning in recurrent neural networks has been extensively applied to speech recognition, language processing [2, 5, 6], and the modeling of biological neural networks [1, 11, 16, 18]. Although gradient descent algorithms for recurrent networks are considered as a simple extension to the back-propagation learning for feed-forward networks, there is an essential difference between the learning processes in feed-forward and recurrent networks. The output of a feed-forward network is a continuous function of the weights if each unit has a smooth output function, such as a sig...