Using stochastic gradient descent procedures in AGRS, as a part of contribution of this study, has to tolerate a lot of iterations in high dimensional problems such as spectrometry. Also, impracticable nature of quasi-Newton approaches, which was inherited from excessive computational cost of Hessian inverses extraction, extremely increases the demand of selecting optimization algorithm as it is proposed by this paper, ERAS. The novel approach, which is proposed in this study, is based on the introduction of a combination of Hessian approximation matrix computed from finite gradient differences.
Using Hessian-vector products, ERAS technique controls the quality of the curvature estimates in spite of the classical quasi-Newton based approaches such as Polak–Ribière \cite{Ghani_2017} , Fletcher–Reeves \cite{Wanto_2017} , Hestenes–Stiefel \cite{Gazi_Sharee_2014} and Dai–Yuan \cite{Dai_2013} techniques. The results showed that, this training approach is much successive in compare with former quasi-Newton methods such as LMBP \cite{Sapna_2012} , S-CGBP \cite{Nayak_2017} , RBP \cite{Saputra_2017} , BFGS \cite{Silaban_2017} , CGBP-PR \cite{Ghani_2017} , CGBP-FR \cite{Wanto_2017} , CGBP-HS \cite{Gazi_Sharee_2014} , CGBP-DY \cite{Dai_2013} .
The evaluation results of ANN for AGRS in all channels of Cs137 were also depicted in Figs. 4-6. These evaluations were examined for different thickness of lead absorbers which were used to attenuate the 662 KeV gamma photon. The evaluated results were compared with published experimental data and they were represented in various color and shapes for “no absorber” and for lead absorbers of different thicknesses 2.651 g/cm2, 4.451 g/cm2, 7.194 g/cm2 and 9.845 g/cm2. Figs. 4 shows that the evaluations of proposed updating formula are successful in ANN with different altitudes and the evaluated function was well fitted when the ERAS was used. The outcome of radiation instrumentation, AGRS, was improved based on data reconstruction and analysis using proposed ERAS updating formula.
Conclusion
In this study, several optimization methods have been implemented for training of ANNs with AGRS data. Experimental results, based on an AGRS task, showed that all implemented algorithms may be used to train the proposed ANN. They are also indicated that the new ERAS training update of ANN has better performance than other quasi-Newton algorithm such as LMBP \cite{Sapna_2012} , S-CGBP \cite{Nayak_2017} , RBP \cite{Saputra_2017} , BFGS, CGBP-PR \cite{Ghani_2017} , CGBP-FR \cite{Wanto_2017} , CGBP-HS \cite{Gazi_Sharee_2014} , CGBP-DY \cite{Dai_2013} . The new ERAS updating formula is very suitable for training networks with large number of free parameters such as AGRS. In general, when the weights of the ANN for AGRS are further adapted with the ERAS updating formula, there are improvements in both accuracy and convergence speed of training. The proposed algorithm, CGBP- ERAS method, increases the quality of curvature information of AGRS data while cost of the algorithm is reduced in each iteration.