Online Gradient Descent Research Articles

Specialization processes in on-line unsupervised learning

Abstract From the recent analysis of supervised learning by on-line gradient descent in multilayered neural networks it is known that the necessary process of student specialization can be delayed significantly. We demonstrate that this phenomenon also occurs in various models of unsupervised learning. A solvable model of competitive learning is presented, which identifies prototype vectors suitable for the representation of high-dimensional data. The specific case of two overlapping clusters of data and a matching number of prototype vectors exhibits non-trivial behaviour like almost stationary plateau configurations. As a second example scenario we investigate the application of Sanger's algorithm for principal component analysis in the presence of two relevant directions in input space. Here, the fast learning of the first principal component may lead to an almost complete loss of initial knowledge about the second one.

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.

Online Learning in Radial Basis Function Networks

An analytic investigation of the average case learning and generalization properties of radial basis function (RBFs) networks is presented, utilizing online 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 some 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 that strongly confirm the analytic results.

Worst-case quadratic loss bounds for prediction using linear functions and gradient descent

Studies the performance of gradient descent (GD) when applied to the problem of online linear prediction in arbitrary inner product spaces. We prove worst-case bounds on the sum of the squared prediction errors under various assumptions concerning the amount of a priori information about the sequence to predict. The algorithms we use are variants and extensions of online GD. Whereas our algorithms always predict using linear functions as hypotheses, none of our results requires the data to be linearly related. In fact, the bounds proved on the total prediction loss are typically expressed as a function of the total loss of the best fixed linear predictor with bounded norm. All the upper bounds are tight to within constants. Matching lower bounds are provided in some cases. Finally, we apply our results to the problem of online prediction for classes of smooth functions.

On-line backpropagation in two-layered neural networks

We present an exact analysis of learning a rule by on-line gradient descent in a two-layered neural network with adjustable hidden-to-output weights (backpropagation of error). Results are compared with the training of networks having the same architecture but fixed weights in the second layer.

Learning by on-line gradient descent

We study on-line gradient-descent learning in multilayer networks analytically and numerically. The training is based on randomly drawn inputs and their corresponding outputs as defined by a target rule. In the thermodynamic limit we derive deterministic differential equations for the order parameters of the problem which allow an exact calculation of the evolution of the generalization error. First we consider a single-layer perceptron with sigmoidal activation function learning a target rule defined by a network of the same architecture. For this model the generalization error decays exponentially with the number of training examples if the learning rate is sufficiently small. However, if the learning rate is increased above a critical value, perfect learning is no longer possible. For architectures with hidden layers and fixed hidden-to-output weights, such as the parity and the committee machine, we find additional effects related to the existence of symmetries in these problems.