We consider the effect of combining several least squares estimators on the expected performance of a regression problem. Computing the exact bias and variance curves as a function of the sample size we are able to quantitatively compare the effect of the combination on the bias and variance separately, and thus on the expected error which is the sum of the two. Our exact calculations, demonstrate that the combination of estimators is particularly useful in the case where the data set is small and noisy and the function to be learned is unrealizable. For large data sets the single estimator produces superior results. Finally, we show that by splitting the data set into several independent parts and training each estimator on a different subset, the performance can in some cases be significantly improved.

We consider the effect of combining several least squares estimators on the expected performance of a regression problem. Computing the exact bias and variance curves as a function of the sample size we are able to quantitatively compare the effect of the combination on the bias and variance separately, and thus on the expected error which is the sum of the two. Our exact calculations, demonstrate that the combination of estimators is particularly useful in the case where the data set is small and noisy and the function to be learned is unrealizable. For large data sets the single estimator produces superior results. Finally, we show that by splitting the data set into several independent parts and training each estimator on a different subset, the performance can in some cases be significantly improved.

Feedback connections are required so that the teacher signal on the output neurons can modify weights during supervised learning. Relaxation methods are needed for learning static patterns with full-time feedback connections. Feedback network learning techniques have not achieved wide popularity because of the still greater computational efficiency of back-propagation. We show by simulation that relaxation networks of the kind we are implementing in VLSI are capable of learning large problems just like back-propagation networks. A microchip incorporates deterministic mean-field theory learning as well as stochastic Boltzmann learning. A multiple-chip electronic system implementing these networks will make high-speed parallel learning in them feasible in the future.

Feedback connections are required so that the teacher signal on the output neurons can modify weights during supervised learning. Relaxation methods are needed for learning static patterns with full-time feedback connections. Feedback network learning techniques have not achieved wide popularity because of the still greater computational efficiency of back-propagation. We show by simulation that relaxation networks of the kind we are implementing in VLSI are capable of learning large problems just like back-propagation networks. A microchip incorporates deterministic mean-field theory learning as well as stochastic Boltzmann learning. A multiple-chip electronic system implementing these networks will make high-speed parallel learning in them feasible in the future.

We introduce a learning algorithm for multilayer neural networks composed of binary linear threshold elements. Whereas existing algorithms reduce the learning process to minimizing a cost function over the weights, our method treats the internal representations as the fundamental entities to be determined. Once a correct set of internal representations is arrived at, the weights are found by the local aild biologically plausible Perceptron Learning Rule (PLR). We tested our learning algorithm on four problems: adjacency, symmetry, parity and combined symmetry-parity.

We introduce a learning algorithm for multilayer neural networks composedof binary linear threshold elements. Whereas existing algorithms reduce the learning process to minimizing a cost function over the weights, our method treats the internal representations asthe fundamental entities to be determined. Once a correct set of internal representations is arrived at, the weights are found by the local aild biologically plausible Perceptron Learning Rule (PLR). We tested our learning algorithm on four problems: adjacency, symmetry, parity and combined symmetry-parity.

We introduce a learning algorithm for multilayer neural networks composed of binary linear threshold elements. Whereas existing algorithms reduce the learning process to minimizing a cost function over the weights, our method treats the internal representations as the fundamental entities to be determined. Once a correct set of internal representations is arrived at, the weights are found by the local aild biologically plausible Perceptron Learning Rule (PLR). We tested our learning algorithm on four problems: adjacency, symmetry, parity and combined symmetry-parity.