Savo G. Glisic - Artificial Intelligence and Quantum Computing for Advanced Wireless Networks

For small learning rates, the total accumulated weight change is approximately equal to the true gradient. This training algorithm is termed temporal backpropagation .

To complete the algorithm, recall the summing junction is defined as

(3.22)

where intermediate variable are defined for convenience. The partial derivative in Eq. (3.21)is easily evaluated as

(3.23)

This holds for all layers in the network. Defining allows us to rewrite Eq. (3.21)as

(3.24)

We now show that a simple recursive formula exists for finding . Starting with the output layer, we observe that influences only the instantaneous output node error e j( k ). Thus, we have

(3.25)

For a hidden layer, has an impact on the error indirectly through all node values in the subsequent layer. Due to the tap delay lines, also has an impact on the error across time. Therefore, the chain rule now becomes

(3.26)

where by definition . Continuing with the remaining term

(3.27)

Now

(3.27a)

since the only influence has on is via the synapse connecting unit j in layer l to unit m in layer l + 1. The definition of the synapse is explicitly given as

(3.28)

Thus

(3.29)

(3.30)

Making all substitutions into Eq. (3.26), we get

(3.31)

where we have defined the vector

(3.32)

Figure 3.7 Temporal backpropagation.

Each term within the sum corresponds to a reverse FIR filter. This is illustrated in Figure 3.7. The filter is drawn in such a way to emphasize the reversal of signal propagation through the FIR. Representing the forward propagation of states and the backward propagation of error terms requires simply reversing the direction of signal flow. In this process, unit delay operators q −1should be replaced with unit advances q +1. The complete adaptation algorithm can be summarized as follows:

(3.33)

(3.34)

The bias weight may again be adapted by letting in Eq. (3.33). Observe the similarities between these equations and those for standard backpropagation. In fact, by replacing the vectors a, w, and δ by scalars, the previous equations reduce to precisely the backpropagation algorithm for static networks. Differences in the temporal version are due to implicit time relations. To find , we filter the δ’s from the next layer backward through the FIR (see Figure 3.7). In other words, δ’s are created not only by taking weighted sums, but also by backward filtering. For each x(k) and desired vector d(k), the forward filters are incremented one time step, producing the current output y(k) and corresponding error e(k). Next, the backward filters are incremented one time step, advancing the δ(k) terms and allowing the filter coefficients to be updated. The process is then repeated for a new input at time k + 1.