Transferring learning from external to internal weights in echo-state networks with sparse connectivity

Abstract

Modifying weights within a recurrent network to improve performance on a task has proven to be difficult. Echo-state networks in which modification is restricted to the weights of connections onto network outputs provide an easier alternative, but at the expense of modifying the typically sparse architecture of the network by including feedback from the output back into the network. We derive methods for using the values of the output weights from a trained echo-state network to set recurrent weights within the network. The result of this "transfer of learning" is a recurrent network that performs the task without requiring the output feedback present in the original network. We also discuss a hybrid version in which online learning is applied to both output and recurrent weights. Both approaches provide efficient ways of training recurrent networks to perform complex tasks. Through an analysis of the conditions required to make transfer of learning work, we define the concept of a "self-sensing" network state, and we compare and contrast this with compressed sensing.

Figure 1. The two recurrent network architectures being considered.

The nets are shown with non-modifiable connections shown in black and modifiable connections in red. Both networks receives input , contain units that interact through a sparse weight matrix , and produce an output , obtained by summing activity from the entire network weighted by the modifiable components of the vector . (A) The output unit sends feedback to all of the network units through connections of fixed weight . Learning affects only the output weights . (B) The same network as in A, but without output feedback. Learning takes place both in the network through the modification , to implement the effect of the feedback loop, and at the output weights , to correctly learn .

Figure 2. Comparison of network simulations and analytical results.

The network simulations (filled circles) and analytic results (solid lines) for sparse (red) and PC (blue) reconstruction errors as a function of for different values. The “error” here is either (red points and curve) or (blue points and curve). The input was with and  = 0, 0.4, 0.6 in the three panels, from left to right. The value of was adjusted by changing . Inserts show the PC eigenvalues (blue) and the exponential fits to them (red), using the value of indicated. Logarithms are base 10.

Figure 3. An example input-output task implemented in a network with feedback (A) and then transferred to a network without feedback using

equation 21 . The upper row shows the input to the network, consisting of two pulses separate by less than 1 s (left columns of A and B) or more than 1 s (right columns of A and B). The red traces show the output of the two networks correctly responding only to the input pulses separated by less than 1 s. The blue traces show 5 sample network units. The green traces show in A and in B for the five sample units. The similarity in these traces shows that the transfer was successful at getting the recurrent input in B to approximate well the feedback input in A for each unit.

Figure 4. The distribution of the elements of , for equally spaced values of .

The eigenvectors for a correlation matrix from simulations similar to those in figure 2 used to demonstrate the approximately Gaussian distribution for the elements of . The red distribution in the front is for , and the black distribution in the back is for , with intermediate layers corresponding to intermediate values. The matrix was randomly initialized for each value of .