Component-based Sketching for Deep ReLU Nets
This addresses a fundamental optimization-generalization trade-off in deep learning, offering a novel approach that is incremental but with strong theoretical and empirical gains.
The paper tackles the inconsistency between optimization and generalization in deep learning by developing a component-based sketching scheme that transforms deep net training into a linear empirical risk minimization problem, achieving almost optimal generalization error bounds and superior generalization performance with reduced training costs compared to gradient-based methods.
Deep learning has made profound impacts in the domains of data mining and AI, distinguished by the groundbreaking achievements in numerous real-world applications and the innovative algorithm design philosophy. However, it suffers from the inconsistency issue between optimization and generalization, as achieving good generalization, guided by the bias-variance trade-off principle, favors under-parameterized networks, whereas ensuring effective convergence of gradient-based algorithms demands over-parameterized networks. To address this issue, we develop a novel sketching scheme based on deep net components for various tasks. Specifically, we use deep net components with specific efficacy to build a sketching basis that embodies the advantages of deep networks. Subsequently, we transform deep net training into a linear empirical risk minimization problem based on the constructed basis, successfully avoiding the complicated convergence analysis of iterative algorithms. The efficacy of the proposed component-based sketching is validated through both theoretical analysis and numerical experiments. Theoretically, we show that the proposed component-based sketching provides almost optimal rates in approximating saturated functions for shallow nets and also achieves almost optimal generalization error bounds. Numerically, we demonstrate that, compared with the existing gradient-based training methods, component-based sketching possesses superior generalization performance with reduced training costs.