# Beyond Stochastic Gradient Descent for Matrix Completion Based Indoor Localization
## 4 Simulation Results and Discussion
### Recovery Performance and Localization Accuracy
Adam converges at 790 iterations which is about the \(\frac{1}{18}\) of required iterations by Adadela. GD converges at 5910 iterations which is about the 0.41 of required iterations by Adadela and NAG converges at 3100 iterations. The performances of GD and NAG in terms of localization accuracy are very close. So that, their cumulative distribution functions (CDF) are confused for each sigma shadowing value. However, it accelerates the convergence of GD. Instead of converging at 5910, it converges at 3100 when working in a noiseless indoor environment. Adagrad exhibits the worst performances in terms of EDM reconstruction error and localization error. This is due to the fact that it accumulates the squared gradients in the denominator. So, the sum of positive terms keeps growing and the learning rate becomes very small, thus making the algorithm no longer able to ensure updates in order to reach a lower minimum. RMSProp, Adadelta and Adam aim to reduce decreasing learning rate. Instead of accumulating all past squared gradients, RMSProp and Adadelta use a window of size (\(\rho\)) of accumulated past gradients. RMSProp improves a little bit the EDM reconstruction error (Figure 6a) and the localization error compared to those introduced by Adagrad in a noiseless environment. However, the result is still worse than those obtained by GD and NAG. The performances of RMSProp and Adagrad are quite close in a noisy environment. They converge to close minimums (Figure 6c,e) and their CDF are almost confused (Figure 6d,f). Adadelta performs a bit better than GD and NAG for sigma shadowing = 0 and 2. However, it requires 2.4 times the number of iterations required by GD to converge and 4.5 times the number of iterations required by NAG to converge (in a noiseless environment) and it is more complex than GD and NAG, which increases the execution time of the algorithm. We notice that adadelta is more affected by noise than other algorithms. Its performances decrease more quickly than the others. In a noiseless indoor environment, Adadelta introduces the best localization accuracy. For sigma shadowing equal to 2, Adam performs better than it. Furthermore, for sigma shadowing equal to 5, Adam, GD and NAG are better than Adadelta in terms of EDM reconstruction error, localization accuracy and speed of convergence. The advantage of Adadelta is that we do not need to set a default learning rate since \(\alpha\) has been eliminated from the update rule. However, this can have a negative effect as we cannot control the learning rate. To resolve this flaw, Adam is used. Adam works well compared to other algorithms considering the compromise between the localization accuracy and the execution time. It requires the smallest number of iterations to converge. This is due to the fact that it uses an accelerator term by taking into account past updates. Furthermore, it achieves to the best localization mean error in a noisy environment. For sigma shadowing equal to 2, its localization mean error is 1.2 m and it reaches 2.6 m when sigma shadowing is equal to 5. As mentioned before, these results have been done on 10 simulations and the variance is about 0.1 m for each value of sigma shadowing. Therefore, Adam is the most adapted algorithm for indoor localization schemes.