Probability Distribution Learning: A theoretical framework for Deep Learning

This paper introduces probability distribution learning (PD learning), a novel theoretical learning framework. Departing from the traditional statistical learning framework, PD learning focuses on learning the underlying probability distribution, which is modeled as a random variable within the probability simplex. Within this framework, the learning error is decomposed into uncertainty, estimation error, and the model's fitting error. Subsequently, we present the methodology for calculating uncertainty, along with optimization strategies for both estimation error and fitting error. Given that minimizing the fitting error typically constitutes a non-convex optimization problem, we introduce a standard loss function and the gradient structural control (GSC) algorithm, and demonstrate that by employing this function, the optima of fitting error minimization can be approached by reducing the gradient norm and structural error. Furthermore, we apply the PD learning framework to deep learning, elucidating the mechanisms by which techniques such as random parameter initialization, over-parameterization, bias-variance trade-off, and dropout influence deep model training. Finally, experimental results on various models validate the effectiveness of the proposed framework.
Comment: arXiv admin note: text overlap with arXiv:2105.04026 by other authors. arXiv admin note: text overlap with arXiv:2105.04026 by other authors