Convergence of backward-error-propagation learning in photorefractive crystals.

We analytically determine that the backward-error-propagation learning algorithm has a well-defined region of convergence in neural learning-parameter space for two classes of photorefractive-based optical neural-network architectures. The first class uses electric-field amplitude encoding of signals and weights in a fully coherent system, whereas the second class uses intensity encoding of signals and weights in an incoherent/coherent system. Under typical assumptions on the grating formation in photorefractive materials used in adaptive optical interconnections, we compute weight updates for both classes of architectures. Using these weight updates, we derive a set of conditions that are sufficient for such a network to operate within the region of convergence. The results are verified empirically by simulations of the xor sample problem. The computed weight updates for both classes of architectures contain two neural learning parameters: a learning-rate coefficient and a weight-decay coefficient. We show that these learning parameters are directly related to two important design parameters: system gain and exposure energy. The system gain determines the ratio of the learning-rate parameter to decay-rate parameter, and the exposure energy determines the size of the decay-rate parameter. We conclude that convergence is guaranteed (assuming no spurious local minima in the error function) by using a sufficiently high gain and a sufficiently low exposure energy per weight update.

Gain and exposure scheduling to compensate for photorefractive neural-network weight decay.

A gain and exposure schedule that theoretically eliminates the effect of photorefractive weight decay for the general class of outer-product neural-network learning algorithms (e.g., backpropagation, Widrow-Hoff, perceptron) is presented. This schedule compensates for photorefractive diffraction-efficiency decay by iteratively increasing the spatial-light-modulator transfer function gain and decreasing the weight-update exposure time. Simulation results for the scheduling procedure, as applied to backpropagation learning for the exclusive-OR problem, show improved learning performance compared with results for networks trained without scheduling.