Extended convexity and smoothness and their applications in deep learning

The underlying mechanism by which simple gradient-based iterative algorithms can effectively handle the non-convex problem of deep model training remains incompletely understood within the traditional convex and non-convex analysis frameworks, which often require the Lipschitz smoothness of the gradient and strong convexity. In this paper, we introduce $\mathcal{H}(\phi)$-convexity and $\mathcal{H}(\Phi)$-smoothness, which broaden the existing concepts of smoothness and convexity, and delineate their fundamental properties. Building on these concepts, we introduce the high-order gradient descent and high-order stochastic gradient descent methods, which serve as extensions to the traditional gradient descent and stochastic gradient descent methods, respectively. Furthermore, we establish descent lemmas for the $\mathcal{H}(\phi)$-convex and $\mathcal{H}(\Phi)$-smooth objective functions when utilizing these four methods. On the basis of these findings, we develop the gradient structure control algorithm to address non-convex optimization objectives, encompassing both the functions represented by machine learning models and common loss functions in deep learning. The effectiveness of the proposed methodology is empirically validated through experiments.

General Distribution Learning: A theoretical framework for Deep Learning

There remain numerous unanswered research questions on deep learning (DL) within the classical learning theory framework. These include the remarkable generalization capabilities of overparametrized neural networks (NNs), the efficient optimization performance despite non-convexity of objectives, the mechanism of flat minima for generalization, and the exceptional performance of deep architectures in solving physical problems. This paper introduces General Distribution Learning (GD 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 traditional statistical machine learning, GD Learning focuses on the true underlying distribution. In GD Learning, learning error, corresponding to the expected error in classical statistical learning framework, is divided into fitting errors due to models and algorithms, as well as sampling errors introduced by limited sampling data. The framework significantly incorporates prior knowledge, especially in scenarios characterized by data scarcity, thereby enhancing performance. Within the GD Learning framework, we demonstrate that the global optimal solutions in non-convex optimization can be approached by minimizing the gradient norm and the non-uniformity of the eigenvalues of the model's Jacobian matrix. This insight leads to the development of the gradient structure control algorithm. GD Learning also offers fresh insights into the questions on deep learning, including overparameterization and non-convex optimization, bias-variance trade-off, and the mechanism of flat minima.

Error Bounds of Supervised Classification from Information-Theoretic Perspective

There remains a list of unanswered research questions on deep learning (DL), including the remarkable generalization power of overparametrized neural networks, the efficient optimization performance despite the non-convexity, and the mechanisms behind flat minima in generalization. In this paper, we adopt an information-theoretic perspective to explore the theoretical foundations of supervised classification using deep neural networks (DNNs). Our analysis introduces the concepts of fitting error and model risk, which, together with generalization error, constitute an upper bound on the expected risk. We demonstrate that the generalization errors are bounded by the complexity, influenced by both the smoothness of distribution and the sample size. Consequently, task complexity serves as a reliable indicator of the dataset's quality, guiding the setting of regularization hyperparameters. Furthermore, the derived upper bound fitting error links the back-propagated gradient, Neural Tangent Kernel (NTK), and the model's parameter count with the fitting error. Utilizing the triangle inequality, we establish an upper bound on the expected risk. This bound offers valuable insights into the effects of overparameterization, non-convex optimization, and the flat minima in DNNs.Finally, empirical verification confirms a significant positive correlation between the derived theoretical bounds and the practical expected risk, confirming the practical relevance of the theoretical findings.