markh@csd4.csd.uwm.edu (Mark William Hopkins) (11/16/90)
In article <GREENBA.90Nov13115837@gambia.crd.ge.com> greenba@gambia.crd.ge.com (ben a green) writes: >Backpropagation is nothing more than the application of the chain rule >of differentiation to the task of calculating the gradient with >respect to weights and biases of a cost function for a layered, >feedforward net. It can also be applied to neural nets that have feedback loops, not in the relatively roundabout way described in the McClelland & Rumelhart PDP book, but in a very direct way that involves iterating both the forward activation and error propagation without breaking up the feedback loops into an infinite series of identical layers. Hopfield nets can be embedded in the larger picture as a special case... The resulting error propagation is guaranteed to settle down in a finite number of steps below any prespecified tolerance IF the forward activation does. Experience has proven that they both settle down in the same number of steps, and usually a small number. About the only time I found any instability is where I put it there on purpose (running a simulated flip-flop in an undefined mode). Which brings up another point: these kind of nets with feedback can exist in multiple states that not only depend on the current input but also the prior history of inputs. You MAY be able to use backpropagation here to get the net to *learn* to become bi-stable, like a flip-flop. Then backpropagation provides a natural way for training finite state machines. The only problem I've observed with using backpropagation in this way so far is that it causes the net to converge, but to converge by first passing through an instable region (which takes forever do to). So it never quite crosses the threshold, as it were... >It's amazing that it took so many years after Minsky and Paepert's >denunciation of the perceptron for people to think of BP as a solution >to the training problem... (Maybe the discovery really was the use of >differentiable node functions instead of flipflops, not >backpropagation.) > Note the double irony here...