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# Dilemma: Avoiding gradient decay prevents long-term latching

In Bengio et al.'s paper , the analysis of the problem of gradient decays is generalized to parameterized dynamical systems (hence including second order and other recurrent architectures). The main theorem shows that a sufficient condition to obtain gradient decay is also a necessary condition for the system to robustly store discrete state information for the long-term. In other words, when the weights and the state trajectory are such that the network can latch'' on information in its hidden units (i.e., represent long-term dependencies), the problem of gradient decay is obtained. When the long-term gradients decay exponentially, it is very difficult to learn such long-term dependencies because the total gradient is the sum of long-term and short-term influences and the short-term influences then completely dominate the gradient. This result is based on a decomposition of the state-space of hidden units in two types of regions: one where gradients decay and one where it is not possible to robustly latch information. Let denote the -dimensional state vector at time (for example, the vector when considering a standard first-order recurrent network) and let be the map from the state at time to for the autonomous (without inputs) dynamical system. The above decomposition is expressed in terms of the condition (no robust latching possible) or (gradient decay), where is the norm of the Jacobian (matrix of partial derivatives) of the map . The analysis focuses on the basins of attraction of attractors of in the domain of (or manifolds within that domain). In particular, the analysis is concerned with so-called hyperbolic attractors, which are locally stable (but need not be fixed points) and where the eigenvalues of are less than 1 in absolute value. If the state (or a function of it) remains within a certain region of space (versus another region) even in the presence of perturbations (such as noise in the inputs) then it is possible to store at least one bit of information for arbitrary durations.

In regions where it can be shown that arbitrarily small perturbations (for example due to the inputs) can eventually kick the state out of a basin of attraction  (see the sample trajectory on the right of Figure 1). In regions where there is a level of perturbation (depending on ) below which the state will remain in the basin of attraction (and will gradually get closer to a certain volume around the attractor -- see left of Figure 1). For this reason we call this condition information latching,'' since it allows to store discrete information for arbitrary duration in the state variable . Unfortunately, in the regions where (where one can latch information) one can also show that gradients decay. The argument is similar to the one developed in the previous section. The partial derivative of with respect to with is simply the product of the map derivatives between and :

When the norm of each of the factors on the right hand side is less than 1, the left hand side converges exponentially fast to zero as increases. The effect of this decay of gradients can be made explicit as follows:

Hence for a term of the sum with , we have

This term tends to become very small in comparison to terms for which is close to . This means that even though there might exist a change in that would allow to jump to another (better) basin of attraction, the gradient of the cost with respect to does not clearly reflect that possibility. The explanation is that the effect of a small change in would be felt mostly on the near past ( close to ).   Next: Remedies Up: Gradient Flow in Recurrent Previous: Weak upper bound for
Juergen Schmidhuber 2003-02-19