Two highly efficient second-order algorithms for training feedforward networks.

We present two highly efficient second-order algorithms for the training of multilayer feedforward neural networks. The algorithms are based on iterations of the form employed in the Levenberg-Marquardt (LM) method for nonlinear least squares problems with the inclusion of an additional adaptive momentum term arising from the formulation of the training task as a constrained optimization problem. Their implementation requires minimal additional computations compared to a standard LM iteration. Simulations of large scale classical neural-network benchmarks are presented which reveal the power of the two methods to obtain solutions in difficult problems, whereas other standard second-order techniques (including LM) fail to converge.

A dynamical model for the analysis and acceleration of learning in feedforward networks.

A dynamical system model is derived for feedforward neural networks with one layer of hidden nodes. The model is valid in the vicinity of flat minima of the cost function that rise due to the formation of clusters of redundant hidden nodes with nearly identical outputs. The derivation is carried out for networks with an arbitrary number of hidden and output nodes and is, therefore, a generalization of previous work valid for networks with only two hidden nodes and one output node. The Jacobian matrix of the system is obtained, whose eigenvalues characterize the evolution of learning. Flat minima correspond to critical points of the phase plane trajectories and the bifurcation of the eigenvalues signifies their abandonment. Following the derivation of the dynamical model, we show that identification of the hidden nodes clusters using unsupervised learning techniques enables the application of a constrained application (Dynamically Constrained Back Propagation-DCBP) whose purpose is to facilitate prompt bifurcation of the eigenvalues of the Jacobian matrix and, thus, accelerate learning. DCBP is applied to standard benchmark tasks either autonomously or as an aid to other standard learning algorithms in the vicinity of flat minima. Its application leads to significant reduction in the number of required epochs for convergence.

Efficient perceptron learning using constrained steepest descent.

An algorithm is proposed for training the single-layered perceptron. The algorithm follows successive steepest descent directions with respect to the perceptron cost function, taking care not to increase the number of misclassified patterns. The problem of finding these directions is stated as a quadratic programming task, to which a fast and effective solution is proposed. The resulting algorithm has no free parameters and therefore no heuristics are involved in its application. It is proved that the algorithm always converges in a finite number of steps. For linearly separable problems, it always finds a hyperplane that completely separates patterns belonging to different categories. Termination of the algorithm without separating all given patterns means that the presented set of patterns is indeed linearly inseparable. Thus the algorithm provides a natural criterion for linear separability. Compared to other state of the art algorithms, the proposed method exhibits substantially improved speed, as demonstrated in a number of demanding benchmark classification tasks.

Dynamics of multilayer networks in the vicinity of temporary minima.

A dynamical system model is derived for a single-output, two-layer neural network, which learns according to the back-propagation algorithm. Particular emphasis is placed on the analysis of the occurrence of temporary minima. The Jacobian matrix of the system is derived, whose eigenvalues characterize the evolution of learning. Temporary minima correspond to critical points of the phase plane trajectories, and the bifurcation of the Jacobian matrix eigenvalues signifies their abandonment. Following this analysis, we show that the employment of constrained optimization methods can decrease the time spent in the vicinity of this type of minima. A number of numerical results illustrates the analytical conclusions.

An efficient constrained training algorithm for feedforward networks.

A novel algorithm is presented which supplements the training phase in feedforward networks with various forms of information about desired learning properties. This information is represented by conditions which must be satisfied in addition to the demand for minimization of the usual mean square error cost function. The purpose of these conditions is to improve convergence, learning speed, and generalization properties through prompt activation of the hidden units, optimal alignment of successive weight vector offsets, elimination of excessive hidden nodes, and regulation of the magnitude of search steps in the weight space. The algorithm is applied to several small- and large-scale binary benchmark training tasks, to test its convergence ability and learning speed, as well as to a large-scale OCR problem, to test its generalization capability. Its performance in terms of percentage of local minima, learning speed, and generalization ability is evaluated and found superior to the performance of the backpropagation algorithm and variants thereof taking especially into account the statistical significance of the results.

Translation, rotation, and scale invariant pattern recognition by high-order neural networks and moment classifiers.

The classification and recognition of two-dimensional patterns independently of their position, orientation, and size by using high-order networks are discussed. A method is introduced for reducing and controlling the number of weights of a third-order network used for invariant pattern recognition. The method leads to economical networks that exhibit high recognition rates for translated, rotated, and scaled, as well as locally distorted, patterns. The performance of these networks at recognizing types and handwritten numerals independently of their position, size, and orientation is compared with and found superior to the performance of a layered feedforward network to which image features extracted by the method of moments are presented as input.