A Closed-form Solution for Weight Optimization in Fully-connected Feed-forward Neural Networks

This work addresses weight optimization problem for fully-connected feed-forward neural networks. Unlike existing approaches that are based on back-propagation (BP) and chain rule gradient-based optimization (which implies iterative execution, potentially burdensome and time-consuming in some cases), the proposed approach offers the solution for weight optimization in closed-form by means of least squares (LS) methodology. In the case where the input-to-output mapping is injective, the new approach optimizes the weights in a back-propagating fashion in a single iteration by jointly optimizing a set of weights in each layer for each neuron. In the case where the input-to-output mapping is not injective (e.g., in classification problems), the proposed solution is easily adapted to obtain its final solution in a few iterations. An important advantage over the existing solutions is that these computations (for all neurons in a layer) are independent from each other; thus, they can be carried out in parallel to optimize all weights in a given layer simultaneously. Furthermore, its running time is deterministic in the sense that one can obtain the exact number of computations necessary to optimize the weights in all network layers (per iteration, in the case of non-injective mapping). Our simulation and empirical results show that the proposed scheme, BPLS, works well and is competitive with existing ones in terms of accuracy, but significantly surpasses them in terms of running time. To summarize, the new method is straightforward to implement, is competitive and computationally more efficient than the existing ones, and is well-tailored for parallel implementation.