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488 Solutions to the XOR Problem
 In M.C. Mozer et. al. (Eds.) Neural Information Processing Systems  Natural and Synthetic
, 1996
"... A globally convergent homotopy method is defined that is capable of sequentially producing large numbers of stationary points of the multilayer perceptron meansquared error surface. Using this algorithm large subsets of the stationary points of two test problems are found. It is shown empirical ..."
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A globally convergent homotopy method is defined that is capable of sequentially producing large numbers of stationary points of the multilayer perceptron meansquared error surface. Using this algorithm large subsets of the stationary points of two test problems are found. It is shown empirically that the MLP neural network appears to have an extreme ratio of saddle points compared to local minima, and that even small neural network problems have extremely large numbers of solutions. 1 Introduction Knowledge of the number and type of stationary points of the error surface provide insight of the difficulty of finding the optimal parameters of the network, since these determine the degree of the system[1]. Unfortunately, even for the small canonical test problems commonly used in neural network studies, it is still unknown how many solutions there are, where they are, and how these are divided into minima, maxima and saddle points. Since solving the neural equations explicitly...
Development of Advanced Verification and Validation Procedures and Tools for the Certification of Learning Systems in Aerospace Applications
"... Adaptive control technologies that incorporate learning algorithms have been proposed to enable automatic flight control and vehicle recovery, autonomous flight, and to maintain vehicle performance in the face of unknown, changing, or poorly defined operating environments. In order for adaptive cont ..."
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Adaptive control technologies that incorporate learning algorithms have been proposed to enable automatic flight control and vehicle recovery, autonomous flight, and to maintain vehicle performance in the face of unknown, changing, or poorly defined operating environments. In order for adaptive control systems to be used in safetycritical aerospace applications, they must be proven to be highly safe and reliable. Rigorous methods for adaptive software verification and validation must be developed to ensure that control system software failures will not occur. Of central importance in this regard is the need to establish reliable methods that guarantee convergent learning, rapid convergence (learning) rate, and algorithm stability. This paper presents the major problems of adaptive control systems that use learning to improve performance. The paper then presents the major procedures and tools presently developed or currently being developed to enable the verification, validation, and ultimate certification of these adaptive control systems. These technologies include the application of automated program analysis methods, techniques to improve the learning process, analytical methods to verify stability, methods to automatically synthesize code, simulation and test methods, and tools to provide online software assurance.
On Producing Multiple Solutions Using Repeated Trials
"... . The number of repeated trials that is required by an algorithm to produce a given fraction of the problem solutions with a specified level of confidence is analyzed. The analysis indicates that the number of trials required to find a large fraction of the solutions rapidly decreases as the number ..."
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. The number of repeated trials that is required by an algorithm to produce a given fraction of the problem solutions with a specified level of confidence is analyzed. The analysis indicates that the number of trials required to find a large fraction of the solutions rapidly decreases as the number of solutions obtained on each trial by an algorithm increases. In applications where multiple solutions are sought, this decrease in the number of trials could potentially offset the additional computational cost of algorithms that produce multiple solutions on a single trial. The analysis framework presented is used to compare the efficiency of a homotopy algorithm to that of a Newton method by measuring both the number of trials and the number of calculations required to obtain a specified fraction of the solutions. Key words: Repeated Trials, Exhaustive Solution Methods, Homotopy Methods 1 Introduction Frequently a numerical solution procedure can be considered to be successful only if...