# A Layer Separation Optimization Framework for Cross-Entropy Training in Deep Learning Source: [A Layer Separation Optimization Framework for Cross-Entropy Training in Deep Learning]() Collection index: [llms.txt]() Data updated: 2026-04-29T05:00:05.003Z Authors: Yaru Liu, Michael K. Ng, Yiqi Gu Published: 2026-04-25T09:33:24.000Z ## Abstract This paper investigates the deep learning optimization problem with softmax cross-entropy loss. We propose a layer separation strategy to alleviate the strong nonconvexity encountered during training deep networks. For cross-entropy models with fully connected and convolutional neural networks, we introduce auxiliary variables associated with hidden layer outputs and construct corresponding layer separation models, which decompose the original deeply nested optimization problem into a sequence of more manageable subproblems. We also conduct theoretical analyses, proving that the new layer separation loss provides an upper bound for the original cross-entropy loss. Moreover, we design alternating minimization algorithms and prove that, under appropriate conditions, these algorithms exhibit decreasing properties of the loss function. Numerical experiments validate the effectiveness of the proposed methods and indicate improved optimization behavior, especially for fully connected and convolutional neural networks. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Digit Classification on MNIST (train)](): Multi-class digit classification; MNIST (train); Cross-Entropy Loss: 0.028. ## Original sources and implementation [Original paper]() [PDF]()