S. Thrun and A. Linden. Inversion in time. In Proceedings of the EURASIP Workshop on Neural Networks, Sesimbra, Portugal, February 1990. EURASIP.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is predicted by a chain of model networks. Then we define a differentiable reinforcement energy function which measures the deviation of some predefined optimal reinforcement and the predicted one. The plan is now updated with gradients of this energy with respect to the plan [KL90, LK89, Sch90, TL90] The procedure is repeated several times which leads to a progressively optimization of reinforcement. First we consider the second step. 3.1 Plan Optimization by Gradient Descent Let us assume we have already an initial plan, i.e. a sequence of N actions. If, as it is discussed later, we use ....

S. Thrun and A. Linden. Inversion in time. In Proceedings of the EURASIP Workshop on Neural Networks, Sesimbra, Portugal, February 1990. EURASIP.

....the consistency of rules with a neural network. It does this by an iterative process of propagating intervals of valid activations through the network. Unlike the approaches summarized above, VI 1 which are discussed in more detail in Section 5 2 VI Analysis was originally proposed in [Thrun and Linden, 1990] [Thrun, 1993] Extracting Symbolic Knowledge from Artificial Neural Networks 3 Analysis analyzes networks as a whole by examining the relation between input and output of the network. Symbolic rules are extracted by a generate and test procedure. Candidate rules are generated by either of two ....

....known to take worst case exponential time in the number of variables (i.e. units in a layer) we did not yet observe this algorithm to be too slow in practice. It should be noted that there exist more complex algorithms for linear programming which are worst case polynomial [Karmarkar, 1984] In [Thrun and Linden, 1990] , a related refinement algorithm for VI Analysis is described that works much faster, but is not as powerful. Thus far, we have described the refinement procedure for the units connected to a single weight layer. VI Analysis iteratively refines all intervals in the network. As in the initial AND ....

S. Thrun and A. Linden. Inversion in time. In Proceedings of the EURASIP Workshop on Neural Networks, Sesimbra, Portugal, Feb 1990. EURASIP.

....N 1 states and reinforcements (c.f. figure 2) Thus we obtain a prediction for all reinforcements r pred (t 1) r pred (t 2) r pred (t N) in this look ahead window. For improving the actions with respect to the predicted reinforcement we use a gradient descent algorithm in action space [7, 8, 19], which will be derived in detail in the next section. It computes the gradients of the reinforcement with respect to the plan, which give us the information, how to change the plan in small steps in order to improve its performance. The whole procedure described above is to be repeated. After a ....

....tell us how to change the actions of the plan in order to improve the predicted reinforcement, thus how to optimize the plan with respect to our world model. One way of computing the gradients of E r with respect to the actions is using backpropagation through the spatial unfolded time structure [16, 19]. Since no dynamical determination of the plan length N is possible by using this backpropagation in time technique, we will derive a pure feed forward algorithm for computing these gradients. Let us define the gradient of each activation x k with respect to the external input I i (t s) by k is ....

S. Thrun and A. Linden. Inversion in time. In Proceedings of the EURASIP Workshop on Neural Networks, Sesimbra, Portugal, February 15-17. EURASIP, 1990.

....unknown a priori, they are often specified by uniform don t know values. But they can be optimized by gradient descent as well. In addition, a feed forward inversion algorithm is presented. This algorithm is used for finding input sequences for a given output target for an already adapted network [47, 24, 22, 42]. Having described all this, we proceed by comparing complexities of serial and parallel implementations of the feed forward algorithm with the unfolding in time method. Since we will not describe this method formally in this paper, see [36] for further details. Lastly, we present some simulation ....

....= 1 Gamma Deltat T k fl k i (t) Deltat T k oe 0 (net k (t) 0 B B X l2N X m2S mt Gammat 0 w klm fl l i (t Gamma m) 1 C C A 8t 2 ft 0 ; t 0 Deltat; t 1 Gamma 2 Deltat; t 1 Gamma Deltatg (60) 7. 2 Inversion : Adaption of the External Input Inversion [24, 22, 42] turned out to be a useful tool for finding spurious input attractors of a trained connectionist network. Inversion is the gradient search of an external input (I i (t) which is mapped by the network to a specific given target trajectory ( t i (t) 8i 2 O; t 2 [t 0 ; t 1 ] except a small ....

[Article contains additional citation context not shown here]

S. Thrun and A. Linden. Inversion in time. In Proceedings of the EURASIP Workshop on Neural Networks, Sesimbra, Portugal, February 1990. EURASIP.

....are provably correct, making minimal assumptions concerning the type of network, and no assumption as to the particular training routine employed. This technique is based on a generic tool for analyzing dependencies within neural networks, called Validity Interval Analysis (VI Analysis or VIA) [Thrun and Linden, 1990] . VIAnalysis iteratively analyzes the input output functionality of an artificial neural network by propagating sets of intervals through the network. The basic notion of VI Analysis is presented in Section 2, followed in Section 3 by a demonstration using the simple boolean function XOR. ....

....of the network and to demonstrate VI Analysis. Figure 7 displays the results. Each row summarizes a single experiment, and each diagram shows the refinement of validity intervals over time. Each of the five columns corresponds to a particular unit in the network. 7 In an earlier paper [Thrun and Linden, 1990] we proposed a more straightforward algorithm which works much faster by sacrificing some solutions to the interval refinement problem. Extracting Provably Correct Rules from Artificial Neural Networks 14 1 2 3 4 5 (a) b) to node from node 3 (hidden) 4 (hidden) 5 (output) 1 (input) 4.60248 ....

S. Thrun and A. Linden. Inversion in time. In Proceedings of the EURASIP Workshop on Neural Networks, Sesimbra, Portugal, Feb 1990. EURASIP.

No context found.

S. Thrun and A. Linden. Inversion in time. In Proceedings of the EURASIP Workshop on Neural Networks, Sesimbra, Portugal, February 1990. EURASIP.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC