Abstract | We survey learning algorithms for recurrent neural networks with hidden units, and put the various techniques into a common framework. We discuss xedpoint learning algorithms, namely recurrent backpropagation and deterministic Boltzmann Machines, and non- xedpoint algorithms, namely backpropagation through time, Elman's history cuto, and Jordan's output feedback architecture. Forward propagation, an online technique that uses adjoint equations, and variations thereof, are also discussed. In many cases, the uni ed presentation leads to generalizations of various sorts. We discuss advantages and disadvantages of temporally continuous neural networks in contrast to clocked ones, continue with some \tricks of the trade" for training, using, and simulating continuous time and recurrent neural networks. We present somesimulations, and at the end, address issues of computational complexity and learning speed.

On-line first order backpropagation is sufficiently fast and effective for many large-scale classification problems but for very high precision mappings, batch processing may be the method of choice. This paper reviews first- and second-order optimization methods for learning in feedforward neural networks. The viewpoint is that of optimization: many methods can be cast in the language of optimization techniques, allowing the transfer to neural nets of detailed results about computational complexity and safety procedures to ensure convergence and to avoid numerical problems. The review is not intended to deliver detailed prescriptions for the most appropriate methods in specific applications, but to illustrate the main characteristics of the different methods and their mutual relations.

I present a general taxonomy of neural net architectures for processing time-varying patterns. This taxonomy subsumes many existing architectures in the literature, and points to several promising architectures that have yet to be examined. Any architecture that processes timevarying patterns requires two conceptually distinct components: a short-term memory that holds on to relevant past events and an associator that uses the short-term memory to classify or predict. My taxonomy is based on a characterization of short-term memory models along the dimensions of form, content, and adaptability. Experiments on predicting future values of a financial time series (US dollar--Swiss franc exchange rates) are presented using several alternative memory models. The results of these experiments serve as a baseline against which more sophisticated architectures can be compared. Neural networks have proven to be a promising alternative to traditional techniques for nonlinear temporal prediction t...

HINE Parameters X0, X1, ....Xp Output E0, E1,....Ep Error Desired Output D0, D1,...Dp Y0, Y1,...Yp Input w w0 w1 AT&T Laboratories (c) COST FUNCTION Output E0, E1,....Ep Error Desired Output D0, D1,...Dp Y0, Y1,...Yp X0, X1, ....Xp Input Parameters w B R A COMPUTING THE GRADIENT WITH BACKPROPAGATION O = A(I1, I2) dI1 = dO ¶ A ¶ I1 dI2 = dO ¶ A ¶ I2 - The learning machine is composed of modules (e.g. layers) - Each module can do two things: 1- compute its outputs from its inputs (FPROP) 2- compute gradient vectors at its inputs from gradient vectors at its outputs (BPROP) A O, dO I1, dI1 I2, dI2 AT&T Laboratories (c) AN INTERESTING SPECIAL CASE: MULTILAYER NETWORKS X0, X1, ....Xp Output Desired Output D0, D1,...Dp Y0, Y1,...Yp Input || D - Y || 2 2 1 WX F() WX F() Mean Square Error Parameters (weights + biases) w Weight matrix E0, E1,....Ep Sigmoids + Biase

Just storing the Hessian H (the matrix of second derivatives d^2 E/dw_i dw_j of the error E with respect to each pair of weights) of a large neural network is difficult. Since a common use of a large matrix like H is to compute its product with various vectors, we derive a technique that directly calculates Hv, where v is an arbitrary vector. This allows H to be treated as a generalized sparse matrix. To calculate Hv, we first define a differential operator R{f(w)} = (d/dr)f(w + rv)|_{r=0}, note that R{grad_w} = Hv and R{w} = v, and then apply R{} to the equations used to compute grad_w. The result is an exact and numerically stable procedure for computing Hv, which takes about as much computation, and is about as local, as a gradient evaluation. We then apply the technique to backpropagation networks, recurrent backpropagation, and stochastic Boltzmann Machines. Finally, we show that this technique can be used at the heart of many iterative techniques for computing various properties of H, obviating the need for direct methods.

Many neural net learning algorithms aim at finding "simple" nets to explain training data. The expectation is: the "simpler" the networks, the better the generalization on test data (! Occam's razor). Previous implementations, however, use measures for "simplicity" that lack the power, universality and elegance of those based on Kolmogorov complexity and Solomonoff's algorithmic probability. Likewise, most previous approaches (especially those of the "Bayesian" kind) suffer from the problem of choosing appropriate priors. This paper addresses both issues. It first reviews some basic concepts of algorithmic complexity theory relevant to machine learning, and how the Solomonoff-Levin distribution (or universal prior) deals with the prior problem. The universal prior leads to a probabilistic method for finding "algorithmically simple" problem solutions with high generalization capability. The method is based on Levin complexity (a time-bounded generalization of Kolmogorov comple...

. The calculation of second derivatives is required by recent training and analyses techniques of connectionist networks, such as the elimination of superfluous weights, and the estimation of confidence intervals both for weights and network outputs. We here review and develop exact and approximate algorithms for calculating second derivatives. For networks with jwj weights, simply writing the full matrix of second derivatives requires O(jwj 2 ) operations. For networks of radial basis units or sigmoid units, exact calculation of the necessary intermediate terms requires of the order of 2h + 2 backward/forward-propagation passes where h is the number of hidden units in the network. We also review and compare three approximations (ignoring some components of the second derivative, numerical differentiation, and scoring). Our algorithms apply to arbitrary activation functions, networks, and error functions (for instance, with connections that skip layers, or radial basis functions, or ...

Many machine learning algorithms aim at finding "simple" rules to explain training data. The expectation is: the "simpler" the rules, the better the generalization on test data (! Occam's razor). Most practical implementations, however, use measures for "simplicity" that lack the power, universality and elegance of those based on Kolmogorov complexity and Solomonoff's algorithmic probability. Likewise, most previous approaches (especially those of the "Bayesian" kind) suffer from the problem of choosing appropriate priors. This paper addresses both issues. It first reviews some basic concepts of algorithmic complexity theory relevant to machine learning, and how the Solomonoff-Levin distribution (or universal prior) deals with the prior problem. The universal prior leads to a probabilistic method for finding "algorithmically simple" problem solutions with high generalization capability. The method is based on Levin complexity (a time-bounded generalization of Kolmogorov complexity) and...

Backpropagation learning algorithms typically collapse the network's structure into a single vector of weight parameters to be optimized. We suggest that their performance may be improved by utilizing the structural information instead of discarding it, and introduce a framework for "tempering" each weight accordingly.

In this paper a general class of fast learning algorithms for feedforward neural networks is introduced and described. The approach exploits the separability of each layer into linear and nonlinear blocks and consists of two steps. The first step is the descent of the error functional in the space of the outputs of the linear blocks (descent in the neuron space), which can be performed using any preferred optimization strategy. In the second step, each linear block is optimized separately by using a Least Squares (LS) criterion. To demonstrate the effectiveness of the new approach, a detailed treatment of a gradient descent in the neuron space is conducted. The main properties of this approach are the higher speed of convergence with respect to methods that employ an ordinary gradient descent in the weight space (Backpropagation, BP), better numerical conditioning and lower computational cost compared to techniques based on the Hessian matrix. The numerical stability is assured by the use of robust LS linear system solvers, operating directly on the input data of each layer. Experimental results obtained in three problems are described, which confirm the effectiveness of the new method.