NET-GD: Train a network by gradient descent in the error. Net-gd trains the parameters of a network by old-fashioned gradient descent in the error (which is minus the log likelihood, plus minus the log prior). The hyperparameters are not updated by this program. A scheme using differential stepsizes for weights out of different units can be used (in combination with "early stopping") to try to get some of the advantages of a hierarchical Bayesian model. Usage: net-gd log-file ["@"]iteration [ save-mod ] / stepsize { stepsize } [ method [ group { group } ] ] Gradient descent in the error is done starting with the last iteration saved in the log file and continuing up to the indicated iteration (or if the iteration is immediately preceded by "@", until a total of iteration minutes of cpu-time have been used, for all iterations). Results of the simulation are appended to the log file for iterations that are divisible by save-mod, which defaults to one (every iteration saved). If the log file does not contain a network to start with, a network is randomly created in which the parameters are drawn from the interval (-0.01,+0.01). The hyperparameters are set to the centres of their priors. Stepsizes for the gradient descent procedure must be specified. If just one stepsize is given, it applies to all parameters. If more than one is given, there must be one stepsize for each group of parameters, where a group corresponds to one "prior" argument of net-spec, and to one section of the output of net-display. The stepsizes are all scaled down by the number of training cases plus one before being used. The gradient descent method can be either "online" (the default) or "batch". For simple online gradient descent, each iteration consists of as many updates as there are training cases, with each update being based on the gradient due to one training case (taken in sequence), plus the prior gradient divided by the number of training cases. For simple batch gradient descent, each iteration consists of one update of the parameters, using the total gradient based on all training cases (and the prior gradient). For very small stepsizes, the batch and online methods should produce the same results. For larger stepsizes, the online method is usually faster, and it can also often use a larger stepsize without causing instability. On-line gradient descent does not find the exact minimum, however, except in the limit as the stepsize goes to zero. Differential stepsizes can be used for weights in specified groups that originate in different units - for example, in order to try to mimic the effect of the Bayesian "Automatic Relevance Determination" model. The groups for which this should be done are listed after the method (which must therefore not be left to default). Stepsizes for weights in these groups are found by computing or estimating the magnitude of the gradient for weights in this group that originate in each of the source units. The stepsize for the weights out of a unit is the stepsize specified for the group, times the magnitude of the gradient for weights out of that unit raised to the fourth power, divided by the maximum of this fourth-power magnitude over all units in the group. For batch gradient descent, the gradient magnitude for weights out of a unit is based on the total gradient for all training cases, plus the prior. For online gradient descent, the same is used for the first iteration, but thereafter, the total gradient for the previous pass (plus the prior) is used, even though this sum is based on different parameter values for different cases. Note that gradient descent learning with essentially no prior may be done by specifying the "priors" using just "+" and "-" (as described in net-spec.doc). However, if no prior is used, it may be necessary to use one of the networks found before convergence of the gradient descent procedure has been reached ("early stopping"), selected on the basis of the error on a validation set. The differential stepsizes are intended to improve the performance of early stopping, by stopping the weights out of some units from overfitting before the weights out of other units have had time to adjust to their proper values. Copyright (c) 1997, 1998 by Radford M. Neal