Multigrade Neural Network Approximation
This addresses the problem of unstable optimization in deep learning for researchers and practitioners, offering a novel theoretical foundation for structured training.
The paper tackles the challenge of training deep neural networks by introducing Multigrade Deep Learning (MGDL), a framework that trains networks grade by grade to reduce approximation error, and proves that this approach yields provable vanishing error for continuous functions.
We study multigrade deep learning (MGDL) as a principled framework for structured error refinement in deep neural networks. While the approximation power of neural networks is now relatively well understood, training very deep architectures remains challenging due to highly non-convex and often ill-conditioned optimization landscapes. In contrast, for relatively shallow networks, most notably one-hidden-layer $\texttt{ReLU}$ models, training admits convex reformulations with global guarantees, motivating learning paradigms that improve stability while scaling to depth. MGDL builds upon this insight by training deep networks grade by grade: previously learned grades are frozen, and each new residual block is trained solely to reduce the remaining approximation error, yielding an interpretable and stable hierarchical refinement process. We develop an operator-theoretic foundation for MGDL and prove that, for any continuous target function, there exists a fixed-width multigrade $\texttt{ReLU}$ scheme whose residuals decrease strictly across grades and converge uniformly to zero. To the best of our knowledge, this work provides the first rigorous theoretical guarantee that grade-wise training yields provable vanishing approximation error in deep networks. Numerical experiments further illustrate the theoretical results.