With the growth of machine learning techniques, privacy of data of users has become a major concern. Most of the machine learning algorithms rely heavily on large amount of data which may be collected from various sources. Collecting these data yet maintaining privacy policies has become one of the most challenging tasks for the researchers. To combat this issue, researchers have introduced federated learning, where a prediction model is learnt by ensuring the privacy of data of clients data. However, the prevalent federated learning algorithms possess an accuracy and efficiency trade-off, especially for non-IID data. In this research, we propose a centralized, neural network-based federated learning system. The centralized algorithm incorporates micro-level parallel processing inspired by the traditional mini-batch algorithm where the client devices and the server handle the forward and backward propagation respectively. We also devise a semi-centralized version of our proposed algorithm. This algorithm takes advantage of edge computing for minimizing the load from the central server, where clients handle both the forward and backward propagation while sacrificing the overall train time to some extent. We evaluate our proposed systems on five well-known benchmark datasets and achieve satisfactory performance in a reasonable time across various data distribution settings as compared to some existing benchmark algorithms.