Probability Distribution Learning: A theoretical framework for Deep Learning
<|reference_start|>Probability Distribution Learning: A theoretical framework for Deep Learning: This paper introduces Probability Distribution Learning (PD learning), a novel theoretical learning framework designed to address a comprehensive range of machine learning and statistical tasks, including classification, regression, and parameter estimation. Departing from the traditional statistical learning framework, PD learning focuses on learning the underlying probability distribution of a dataset, which is modeled as a random variable within the probability simplex. In this framework, the learning error is decomposed into uncertainty and the model's fitting error to the optimal estimate. Uncertainty, which is non-optimizable and independent of both the model and optimization algorithm, depends solely on prior knowledge and sampling data, constituting the optimal bound of the learning error. Minimizing the fitting error represents a typical non-convex optimization problem. To address this, we initially demonstrate that under the conditions of unique optimum and sampling stability, the loss function exhibits a unified mathematical form, which we refer to as the standard loss function. Moreover, we prove that by employing the standard loss function, the optima of fitting error minimization can be approached by reducing the gradient norm and structural error. Subsequently, we demonstrate that with random parameter initialization, increasing network depth and the parameter count can reduce structural error. Consequently, from the perspective of structural error, techniques such as over-parameterization, non-convex optimization, and the flat minima in deep learning are beneficial in reducing structural error, thereby ensuring that gradient-based iterative algorithms can attain an approximate global optimum for fitting error minimization. Ultimately, the experimental results on various models have validated the effectiveness of the framework proposed in this paper.<|reference_end|>
@article{qi2024probability,
    title={Probability Distribution Learning: A theoretical framework for Deep
  Learning},
    author={Binchuan Qi, Wei Gong, Li Li},
    journal={arXiv preprint arXiv:2406.05666},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.05666},
    primaryClass={cs.LG cs.IR stat.ML}
}