The Convergence of Incremental Neural Networks

The investigation of neural network convergence represents a pivotal and indispensable area of research, as it plays a crucial role in unraveling the universal approximation capability and the intricate structural complexity inherent in these systems. In this study, we delve into an innovative and generalized convex incremental iteration method, which surpasses previous studies by offering a more expansive formulation capable of encompassing a broader range of weight parameters. Moreover, we rigorously and systematically demonstrate the convergence rate of this convex iteration technique, shedding light on its reliability and effectiveness. Furthermore, we adopt a discrete statistical perspective to effectively tackle the challenges arising from the non-compactness of input data and the inherent unknowability of the objective function in practical settings, thereby enhancing the robustness and applicability of our research. To support our conclusions, we introduce two implementation algorithms, namely back propagation and random search. The latter algorithm plays a vital role in preventing the neural network from becoming entrapped in suboptimal local minima during the training process. Finally, we present comprehensive results obtained from a variety of regression problems, which not only serve as empirical evidence of the superior performance of our algorithms but also validate their alignment with our theoretical predictions. These results contribute significantly to the advancement of our understanding of neural network convergence and its profound implications for the universal approximation capability inherent in these complex systems.