A real time learning algorithm for recurrent analog neural networks

A new learning algorithm is described for a general class of recurrent analog neural networks which ultimately settle down to a steady state. Recently, Pineda (Pineda 1987; Almeida 1987; Ikeda et al. 1988) has introduced a learning rule for the recurrent net in which the connection weights are adjusted so that the distance between the stable outputs of the current system and the desired outputs will be maximally decreased. In this method, many cycles are needed in order to get a target system. In each cycle, the recurrent net is run until it reaches a stable state. After that, the weight change is calculated by using a linearized recurrent net which receives the current error of the system as a bias input. In the new algorithm the weights are changed so that the total error of neuron outputs over the entire trajectory is minimized. The weights are adjusted in real time when the network is running. In this method, the trajectory to the target system can be controlled, whereas Pineda's algorithm only controls the position of the fixed point. The relation to the back propagation method (Hinton et al. 1986) is also discussed.     

Transferring learning from external to internal weights in echo-state networks with sparse connectivity

Modifying weights within a recurrent network to improve performance on a task has proven to be difficult. Echo-state networks in which modification is restricted to the weights of connections onto network outputs provide an easier alternative, but at the expense of modifying the typically sparse architecture of the network by including feedback from the output back into the network. We derive methods for using the values of the output weights from a trained echo-state network to set recurrent weights within the network. The result of this "transfer of learning" is a recurrent network that performs the task without requiring the output feedback present in the original network. We also discuss a hybrid version in which online learning is applied to both output and recurrent weights. Both approaches provide efficient ways of training recurrent networks to perform complex tasks. Through an analysis of the conditions required to make transfer of learning work, we define the concept of a "self-sensing" network state, and we compare and contrast this with compressed sensing.     

A biologically inspired neural network for dynamic programming

An artificial neural network with a two-layer feedback topology and generalized recurrent neurons, for solving nonlinear discrete dynamic optimization problems, is developed. A direct method to assign the weights of neural networks is presented. The method is based on Bellmann's Optimality Principle and on the interchange of information which occurs during the synaptic chemical processing among neurons. The neural network based algorithm is an advantageous approach for dynamic programming due to the inherent parallelism of the neural networks; further it reduces the severity of computational problems that can occur in methods like conventional methods. Some illustrative application examples are presented to show how this approach works out including the shortest path and fuzzy decision making problems.     

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks IV

In neuronal networks, the changes of synaptic strength (or weight) performed by spike-timing-dependent plasticity (STDP) are hypothesized to give rise to functional network structure. This article investigates how this phenomenon occurs for the excitatory recurrent connections of a network with fixed input weights that is stimulated by external spike trains. We develop a theoretical framework based on the Poisson neuron model to analyze the interplay between the neuronal activity (firing rates and the spike-time correlations) and the learning dynamics, when the network is stimulated by correlated pools of homogeneous Poisson spike trains. STDP can lead to both a stabilization of all the neuron firing rates (homeostatic equilibrium) and a robust weight specialization. The pattern of specialization for the recurrent weights is determined by a relationship between the input firing-rate and correlation structures, the network topology, the STDP parameters and the synaptic response properties. We find conditions for feed-forward pathways or areas with strengthened self-feedback to emerge in an initially homogeneous recurrent network.     

Training simultaneous recurrent neural network with resilient propagation for static optimization

This paper proposes a non-recurrent training algorithm, resilient propagation, for the Simultaneous Recurrent Neural network operating in relaxation-mode for computing high quality solutions of static optimization problems. Implementation details related to adaptation of the recurrent neural network weights through the non-recurrent training algorithm, resilient backpropagation, are formulated through an algebraic approach. Performance of the proposed neuro-optimizer on a well-known static combinatorial optimization problem, the Traveling Salesman Problem, is evaluated on the basis of computational complexity measures and, subsequently, compared to performance of the Simultaneous Recurrent Neural network trained with the standard backpropagation, and recurrent backpropagation for the same static optimization problem. Simulation results indicate that the Simultaneous Recurrent Neural network trained with the resilient backpropagation algorithm is able to locate superior quality solutions through comparable amount of computational effort for the Traveling Salesman Problem.     

A local training-pruning approach for recurrent neural networks

The global extended Kalman filtering (EKF) algorithm for recurrent neural networks (RNNs) is plagued by the drawback of high computational cost and storage requirement. In this paper, we present a local EKF training-pruning approach that can solve this problem. In particular, the by-products, obtained along with the local EKF training, can be utilized to measure the importance of the network weights. Comparing with the original global approach, the proposed local approach results in much lower computational cost and storage requirement. Hence, it is more practical in solving real world problems. Simulation showed that our approach is an effective joint-training-pruning method for RNNs under online operation.     

Self-selected modular recurrent neural networks with postural and inertial subnetworks applied to complex movements

 It has been shown that dynamic recurrent neural networks are successful in identifying the complex mapping relationship between full-wave-rectified electromyographic (EMG) signals and limb trajectories during complex movements. These connectionist models include two types of adaptive parameters: the interconnection weights between the units and the time constants associated to each neuron-like unit; they are governed by continuous-time equations. Due to their internal structure, these models are particularly appropriate to solve dynamical tasks (with time-varying input and output signals). We show in this paper that the introduction of a modular organization dedicated to different aspects of the dynamical mapping including privileged communication channels can refine the architecture of these recurrent networks. We first divide the initial individual network into two communicating subnetworks. These two modules receive the same EMG signals as input but are involved in different identification tasks related to position and acceleration. We then show that the introduction of an artificial distance in the model (using a Gaussian modulation factor of weights) induces a reduced modular architecture based on a self-elimination of null synaptic weights. Moreover, this self-selected reduced model based on two subnetworks performs the identification task better than the original single network while using fewer free parameters (better learning curve and better identification quality). We also show that this modular network exhibits several features that can be considered as biologically plausible after the learning process: self-selection of a specific inhibitory communicating path between both subnetworks after the learning process, appearance of tonic and phasic neurons, and coherent distribution of the values of the time constants within each subnetwork. Received: 17 September 2001 / Accepted in revised form: 15 January 2002     

Efficient and Exact Sampling of Simple Graphs with Given Arbitrary Degree Sequence

Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large , and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.     

DNA fragment assembly using neural prediction techniques

The paper describes an alternative approach to the fragment assembly problem. The key idea is to train a recurrent neural network to tracking the sequence of bases constituting a given fragment and to assign to a same cluster all the sequences which are well tracked by this network. We make use of a 3-layer Recurrent Perceptron and examine both edited sequences from a ftp site and artificial fragments from a common simulation software: the clusters we obtain exhibit interesting properties in terms of error filtering, stability and self consistency; we define as well, with a certain degree of approximation, a metric on the fragment set. The proposed assembly algorithm is susceptible to becoming an alternative method with the following properties: (i) high quality of the rebuilt genomic sequences, (ii) high parallelizability of the computing process with consequent drastic reduction of the running time.     

Recurrent neural networks of integrate-and-fire cells simulating short-term memory and wrist movement tasks derived from continuous dynamic networks

Dynamic recurrent neural networks composed of units with continuous activation functions provide a powerful tool for simulating a wide range of behaviors, since the requisite interconnections can be readily derived by gradient descent methods. However, it is not clear whether more realistic integrate-and-fire cells with comparable connection weights would perform the same functions. We therefore investigated methods to convert dynamic recurrent neural networks of continuous units into networks with integrate-and-fire cells. The transforms were tested on two recurrent networks derived by backpropagation. The first simulates a short-term memory task with units that mimic neural activity observed in cortex of monkeys performing instructed delay tasks. The network utilizes recurrent connections to generate sustained activity that codes the remembered value of a transient cue. The second network simulates patterns of neural activity observed in monkeys performing a step-tracking task with flexion/extension wrist movements. This more complicated network provides a working model of the interactions between multiple spinal and supraspinal centers controlling motoneurons. Our conversion algorithm replaced each continuous unit with multiple integrate-and-fire cells that interact through delayed "synaptic potentials". Successful transformation depends on obtaining an appropriate fit between the activation function of the continuous units and the input-output relation of the spiking cells. This fit can be achieved by adapting the parameters of the synaptic potentials to replicate the input-output behavior of a standard sigmoidal activation function (shown for the short-term memory network). Alternatively, a customized activation function can be derived from the input-output relation of the spiking cells for a chosen set of parameters (demonstrated for the wrist flexion/extension network). In both cases the resulting networks of spiking cells exhibited activity that replicated the activity of corresponding continuous units. This confirms that the network solutions obtained through backpropagation apply to spiking networks and provides a useful method for deriving recurrent spiking networks performing a wide range of functions.     

Convergence analysis of cascade error projection--an efficient learning algorithm for hardware implementation

In this paper, we present a mathematical foundation, including a convergence analysis, for cascading architecture neural network. Our analysis also shows that the convergence of the cascade architecture neural network is assured because it satisfies Liapunov criteria, in an added hidden unit domain rather than in the time domain. From this analysis, a mathematical foundation for the cascade correlation learning algorithm can be found. Furthermore, it becomes apparent that the cascade correlation scheme is a special case from mathematical analysis in which an efficient hardware learning algorithm called Cascade Error Projection(CEP) is proposed. The CEP provides efficient learning in hardware and it is faster to train, because part of the weights are deterministically obtained, and the learning of the remaining weights from the inputs to the hidden unit is performed as a single-layer perceptron learning with previously determined weights kept frozen. In addition, one can start out with zero weight values (rather than random finite weight values) when the learning of each layer is commenced. Further, unlike cascade correlation algorithm (where a pool of candidate hidden units is added), only a single hidden unit is added at a time. Therefore, the simplicity in hardware implementation is also achieved. Finally, 5- to 8-bit parity and chaotic time series prediction problems are investigated; the simulation results demonstrate that 4-bit or more weight quantization is sufficient for learning neural network using CEP. In addition, it is demonstrated that this technique is able to compensate for less bit weight resolution by incorporating additional hidden units. However, generation result may suffer somewhat with lower bit weight quantization.     

Radical pruning: a method to construct skeleton radial basis function networks

Trained radial basis function networks are well-suited for use in extracting rules and explanations because they contain a set of locally tuned units. However, for rule extraction to be useful, these networks must first be pruned to eliminate unnecessary weights. The pruning algorithm cannot search the network exhaustively because of the computational effort involved. It is shown that using multiple pruning methods with smart ordering of the pruning candidates, the number of weights in a radial basis function network can be reduced to a small fraction of the original number. The complexity of the pruning algorithm is quadratic (instead of exponential) in the number of network weights. Pruning performance is shown using a variety of benchmark problems from the University of California, Irvine machine learning database.     

Short-term prediction of chaotic time series by using RBF network with regression weights

We propose a framework for constructing and training a radial basis function (RBF) neural network. The structure of the gaussian functions is modified using a pseudo-gaussian function (PG) in which two scaling parameters sigma are introduced, which eliminates the symmetry restriction and provides the neurons in the hidden layer with greater flexibility with respect to function approximation. We propose a modified PG-BF (pseudo-gaussian basis function) network in which the regression weights are used to replace the constant weights in the output layer. For this purpose, a sequential learning algorithm is presented to adapt the structure of the network, in which it is possible to create a new hidden unit and also to detect and remove inactive units. A salient feature of the network systems is that the method used for calculating the overall output is the weighted average of the output associated with each receptive field. The superior performance of the proposed PG-BF system over the standard RBF are illustrated using the problem of short-term prediction of chaotic time series.     

Software reliability prediction using recurrent neural network with Bayesian regularization

A recurrent neural network modeling approach for software reliability prediction with respect to cumulative failure time is proposed. Our proposed network structure has the capability of learning and recognizing the inherent internal temporal property of cumulative failure time sequence. Further, by adding a penalty term of sum of network connection weights, Bayesian regularization is applied to our network training scheme to improve the generalization capability and lower the susceptibility of overfitting. The performance of our proposed approach has been tested using four real-time control and flight dynamic application data sets. Numerical results show that our proposed approach is robust across different software projects, and has a better performance with respect to both goodness-of-fit and next-step-predictability compared to existing neural network models for failure time prediction.     

A hierarchical method for finding optimal architecture and weights using evolutionary least square based learning

In this paper, we present a novel approach of implementing a combination methodology to find appropriate neural network architecture and weights using an evolutionary least square based algorithm (GALS).1 This paper focuses on aspects such as the heuristics of updating weights using an evolutionary least square based algorithm, finding the number of hidden neurons for a two layer feed forward neural network, the stopping criterion for the algorithm and finally some comparisons of the results with other existing methods for searching optimal or near optimal solution in the multidimensional complex search space comprising the architecture and the weight variables. We explain how the weight updating algorithm using evolutionary least square based approach can be combined with the growing architecture model to find the optimum number of hidden neurons. We also discuss the issues of finding a probabilistic solution space as a starting point for the least square method and address the problems involving fitness breaking. We apply the proposed approach to XOR problem, 10 bit odd parity problem and many real-world benchmark data sets such as handwriting data set from CEDAR, breast cancer and heart disease data sets from UCI ML repository. The comparative results based on classification accuracy and the time complexity are discussed.     

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. II. Input selectivity—symmetry breaking

Spike-timing-dependent plasticity (STDP) is believed to structure neuronal networks by slowly changing the strengths (or weights) of the synaptic connections between neurons depending upon their spiking activity, which in turn modifies the neuronal firing dynamics. In this paper, we investigate the change in synaptic weights induced by STDP in a recurrently connected network in which the input weights are plastic but the recurrent weights are fixed. The inputs are divided into two pools with identical constant firing rates and equal within-pool spike-time correlations, but with no between-pool correlations. Our analysis uses the Poisson neuron model in order to predict the evolution of the input synaptic weights and focuses on the asymptotic weight distribution that emerges due to STDP. The learning dynamics induces a symmetry breaking for the individual neurons, namely for sufficiently strong within-pool spike-time correlation each neuron specializes to one of the input pools. We show that the presence of fixed excitatory recurrent connections between neurons induces a group symmetry-breaking effect, in which neurons tend to specialize to the same input pool. Consequently STDP generates a functional structure on the input connections of the network.     

Analysis of interval‐censored recurrent event processes subject to resolution

Interval‐censored recurrent event data arise when the event of interest is not readily observed but the cumulative event count can be recorded at periodic assessment times. In some settings, chronic disease processes may resolve, and individuals will cease to be at risk of events at the time of disease resolution. We develop an expectation‐maximization algorithm for fitting a dynamic mover‐stayer model to interval‐censored recurrent event data under a Markov model with a piecewise‐constant baseline rate function given a latent process. The model is motivated by settings in which the event times and the resolution time of the disease process are unobserved. The likelihood and algorithm are shown to yield estimators with small empirical bias in simulation studies. Data are analyzed on the cumulative number of damaged joints in patients with psoriatic arthritis where individuals experience disease remission.     

Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks III: Partially connected neurons driven by spontaneous activity

In contrast to a feed-forward architecture, the weight dynamics induced by spike-timing-dependent plasticity (STDP) in a recurrent neuronal network is not yet well understood. In this article, we extend a previous study of the impact of additive STDP in a recurrent network that is driven by spontaneous activity (no external stimulating inputs) from a fully connected network to one that is only partially connected. The asymptotic state of the network is analyzed, and it is found that the equilibrium and stability conditions for the firing rates are similar for both full and partial connectivity: STDP causes the firing rates to converge toward the same value and remain quasi-homogeneous. However, when STDP induces strong weight competition, the connectivity affects the weight dynamics in that the distribution of the weights disperses more quickly for lower density than for higher density. The asymptotic weight distribution strongly depends upon that at the beginning of the learning epoch; consequently, homogeneous connectivity alone is not sufficient to obtain homogeneous neuronal activity. In the absence of external inputs, STDP can nevertheless generate structure in the network through autocorrelation effects, for example, by introducing asymmetry in network topology.     

Fine-tuning and the stability of recurrent neural networks

A central criticism of standard theoretical approaches to constructing stable, recurrent model networks is that the synaptic connection weights need to be finely-tuned. This criticism is severe because proposed rules for learning these weights have been shown to have various limitations to their biological plausibility. Hence it is unlikely that such rules are used to continuously fine-tune the network in vivo. We describe a learning rule that is able to tune synaptic weights in a biologically plausible manner. We demonstrate and test this rule in the context of the oculomotor integrator, showing that only known neural signals are needed to tune the weights. We demonstrate that the rule appropriately accounts for a wide variety of experimental results, and is robust under several kinds of perturbation. Furthermore, we show that the rule is able to achieve stability as good as or better than that provided by the linearly optimal weights often used in recurrent models of the integrator. Finally, we discuss how this rule can be generalized to tune a wide variety of recurrent attractor networks, such as those found in head direction and path integration systems, suggesting that it may be used to tune a wide variety of stable neural systems.     

The simultaneous recurrent neural network for addressing the scaling problem in static optimization.

A trainable recurrent neural network, Simultaneous Recurrent Neural network, is proposed to address the scaling problem faced by neural network algorithms in static optimization. The proposed algorithm derives its computational power to address the scaling problem through its ability to "learn" compared to existing recurrent neural algorithms, which are not trainable. Recurrent backpropagation algorithm is employed to train the recurrent, relaxation-based neural network in order to associate fixed points of the network dynamics with locally optimal solutions of the static optimization problems. Performance of the algorithm is tested on the NP-hard Traveling Salesman Problem in the range of 100 to 600 cities. Simulation results indicate that the proposed algorithm is able to consistently locate high-quality solutions for all problem sizes tested. In other words, the proposed algorithm scales demonstrably well with the problem size with respect to quality of solutions and at the expense of increased computational cost for large problem sizes.