Dynamics of on-line gradient descent learning for multilayer neural networks

We consider the problem of on-line gradient descent learning for general two-layer neural networks. An analytic solution is presented and used to investigate the role of the learning rate in controlling the evolution and convergence of the learning process.

Computationally efficient natural gradient descent

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On-line learning in multilayer neural networks

We present an analytic solution to the problem of on-line gradient-descent learning for two-layer neural networks with an arbitrary number of hidden units in both teacher and student networks. The technique, demonstrated here for the case of adaptive input-to-hidden weights, becomes exact as the dimensionality of the input space increases.

Globally optimal parameters for on-line learning in multilayer neural networks

We present a framework for calculating globally optimal parameters, within a given time frame, for on-line learning in multilayer neural networks. We demonstrate the capability of this method by computing optimal learning rates in typical learning scenarios. A similar treatment allows one to determine the relevance of related training algorithms based on modifications to the basic gradient descent rule as well as to compare different training methods.

On-line learning with adaptive back-propagation in two-layer networks

An adaptive back-propagation algorithm parameterized by an inverse temperature 1/T is studied and compared with gradient descent (standard back-propagation) for on-line learning in two-layer neural networks with an arbitrary number of hidden units. Within a statistical mechanics framework, we analyse these learning algorithms in both the symmetric and the convergence phase for finite learning rates in the case of uncorrelated teachers of similar but arbitrary length T. These analyses show that adaptive back-propagation results generally in faster training by breaking the symmetry between hidden units more efficiently and by providing faster convergence to optimal generalization than gradient descent.

Globally optimal learning rates in multilayer neural networks

A method for calculating the globally optimal learning rate in on-line gradient-descent training of multilayer neural networks is presented. The method is based on a variational approach which maximizes the decrease in generalization error over a given time frame. We demonstrate the method by computing optimal learning rates in typical learning scenarios. The method can also be employed when different learning rates are allowed for different parameter vectors as well as to determine the relevance of related training algorithms based on modifications to the basic gradient descent rule.

Incorporating curvature information into on-line learning

We analyse the dynamics of a number of second order on-line learning algorithms training multi-layer neural networks, using the methods of statistical mechanics. We first consider on-line Newton's method, which is known to provide optimal asymptotic performance. We determine the asymptotic generalization error decay for a soft committee machine, which is shown to compare favourably with the result for standard gradient descent. Matrix momentum provides a practical approximation to this method by allowing an efficient inversion of the Hessian. We consider an idealized matrix momentum algorithm which requires access to the Hessian and find close correspondence with the dynamics of on-line Newton's method. In practice, the Hessian will not be known on-line and we therefore consider matrix momentum using a single example approximation to the Hessian. In this case good asymptotic performance may still be achieved, but the algorithm is now sensitive to parameter choice because of noise in the Hessian estimate. On-line Newton's method is not appropriate during the transient learning phase, since a suboptimal unstable fixed point of the gradient descent dynamics becomes stable for this algorithm. A principled alternative is to use Amari's natural gradient learning algorithm and we show how this method provides a significant reduction in learning time when compared to gradient descent, while retaining the asymptotic performance of on-line Newton's method.

Transients and asymptotics of natural gradient learning

We analyse natural gradient learning in a two-layer feed-forward neural network using a statistical mechanics framework which is appropriate for large input dimension. We find significant improvement over standard gradient descent in both the transient and asymptotic phases of learning.

Natural gradient matrix momentum

Natural gradient learning is an efficient and principled method for improving on-line learning. In practical applications there will be an increased cost required in estimating and inverting the Fisher information matrix. We propose to use the matrix momentum algorithm in order to carry out efficient inversion and study the efficacy of a single step estimation of the Fisher information matrix. We analyse the proposed algorithm in a two-layer network, using a statistical mechanics framework which allows us to describe analytically the learning dynamics, and compare performance with true natural gradient learning and standard gradient descent.

Adaptive back-propagation in on-line learning of multilayer networks

An adaptive back-propagation algorithm is studied and compared with gradient descent (standard back-propagation) for on-line learning in two-layer neural networks with an arbitrary number of hidden units. Within a statistical mechanics framework, both numerical studies and a rigorous analysis show that the adaptive back-propagation method results in faster training by breaking the symmetry between hidden units more efficiently and by providing faster convergence to optimal generalization than gradient descent.

Learning with noise and regularizers in multilayer neural networks

We study the effect of two types of noise, data noise and model noise, in an on-line gradient-descent learning scenario for general two-layer student network with an arbitrary number of hidden units. Training examples are randomly drawn input vectors labeled by a two-layer teacher network with an arbitrary number of hidden units. Data is then corrupted by Gaussian noise affecting either the output or the model itself. We examine the effect of both types of noise on the evolution of order parameters and the generalization error in various phases of the learning process.

Finite-size effects in on-line learning of multilayer neural networks

We complement recent advances in thermodynamic limit analyses of mean on-line gradient descent learning dynamics in multi-layer networks by calculating fluctuations possessed by finite dimensional systems. Fluctuations from the mean dynamics are largest at the onset of specialisation as student hidden unit weight vectors begin to imitate specific teacher vectors, increasing with the degree of symmetry of the initial conditions. In light of this, we include a term to stimulate asymmetry in the learning process, which typically also leads to a significant decrease in training time.

Learning with regularizers in multilayer neural networks

We study the effect of regularization in an on-line gradient-descent learning scenario for a general two-layer student network with an arbitrary number of hidden units. Training examples are randomly drawn input vectors labelled by a two-layer teacher network with an arbitrary number of hidden units which may be corrupted by Gaussian output noise. We examine the effect of weight decay regularization on the dynamical evolution of the order parameters and generalization error in various phases of the learning process, in both noiseless and noisy scenarios.

The role of biases in on-line learning of two-layer networks

The influence of biases on the learning dynamics of a two-layer neural network, a normalized soft-committee machine, is studied for on-line gradient descent learning. Within a statistical mechanics framework, numerical studies show that the inclusion of adjustable biases dramatically alters the learning dynamics found previously. The symmetric phase which has often been predominant in the original model all but disappears for a non-degenerate bias task. The extended model furthermore exhibits a much richer dynamical behavior, e.g. attractive suboptimal symmetric phases even for realizable cases and noiseless data.

On-line learning in radial basis functions networks

An analytic investigation of the average case learning and generalization properties of Radial Basis Function Networks (RBFs) is presented, utilising on-line gradient descent as the learning rule. The analytic method employed allows both the calculation of generalization error and the examination of the internal dynamics of the network. The generalization error and internal dynamics are then used to examine the role of the learning rate and the specialization of the hidden units, which gives insight into decreasing the time required for training. The realizable and over-realizable cases are studied in detail; the phase of learning in which the hidden units are unspecialized (symmetric phase) and the phase in which asymptotic convergence occurs are analyzed, and their typical properties found. Finally, simulations are performed which strongly confirm the analytic results.