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## 2.1. SELF-REFERENTIAL' DYNAMICS AND OBJECTIVE FUNCTION

I assume that the input sequence observed by the network has length (where ) and can be divided into equal-sized blocks of length during which the input pattern does not change. This does not imply a loss of generality -- it just means speeding up the network's hardware such that each input pattern is presented for time-steps before the next pattern can be observed. This gives the architecture time-steps to do some sequential processing (including immediate weight changes) before seeing a new pattern of the input sequence.

In what follows, unquantized variables are assumed to take on their maximal range. The network dynamics are specified as follows:  (1)

The network can quickly read information about its current weights into the special input unit according to (2)

where denotes Euclidean length, and is a differentiable function emitting values between 0 and 1 that determines how close a connection address has to be to the activations of the analyzing units in order for its weight to contribute to at that time. Such a function might have a narrow peak at 1 around the origin and be zero (or nearly zero) everywhere else. This essentially allows the network to pick out a single connection at a time and obtain its current weight value without receiving cross-talk' from other weights.

The network can quickly modify its current weights using and according to (3)

Again, if has a narrow peak at 1 around the origin and is zero (or nearly zero) everywhere else, the network will be able to pick out a single connection at a time and change its weight without affecting other weights.

Objective function and dynamics of the eval units. As with typical supervised sequence-learning tasks, we want to minimize where (4)

Here may be a desired target value for the -th output unit at time step .   Next: 3. INITIAL LEARNING ALGORITHM Up: 2. THE INTROSPECTIVE' NETWORK Previous: 2. THE INTROSPECTIVE' NETWORK
Juergen Schmidhuber 2003-02-21

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