Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.04533

Weight Decay and Neuron Condensation: A Three-Stage Analysis of Two-Layer ReLU Networks

Cheng Xu·Pengxiao Lin·Zhangchen Zhou·Zhi-Qin John Xu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Weight decay is widely used as a regularization technique in neural network training, yet its role in neuron condensation (parameter direction alignment) remains unclear. Starting from a parameter initialization in the neural tangent kernel regime, we characterize training dynamics under weight decay through three stages: rapid fitting, amplitude compression, and neuron condensation. Using a two-layer ReLU network, we analyze a residual correlation field that governs both neuron amplitudes and directions. During rapid fitting, the residual approaches a quasi-static equilibrium maintained by weight decay while the tangent kernel remains nearly unchanged. In the early stage of amplitude compression, kernel decay amplifies the residual correlation field, whose isolated attracting extrema provide candidates for condensation directions. As neuron amplitudes stabilize, we bound the drift of attracting extrema and demonstrate contraction of neuron directions around them, leading to neuron condensation. This staged analysis provides a dynamical understanding of how weight decay promotes a condensed representation, beyond reducing parameter norms.

Explore related subjects

cs.LGcs.AI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Cheng Xu, Pengxiao Lin, Zhangchen Zhou, Zhi-Qin John Xu. 2026-10-03. Weight Decay and Neuron Condensation: A Three-Stage Analysis of Two-Layer ReLU Networks. https://arxiv.org/abs/2610.04533

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Probabilistic Truly Unordered Rule Sets

Rule set learning has recently been frequently revisited because of its interpretability. Existing methods have several shortcomings though. First, most existing methods impose orders among rules, either explicitly or implicitly, which makes the models less comprehensible. Second, due to the difficulty of handling conflicts caused by overlaps (i.e., instances covered by multiple rules), existing methods often do not consider probabilistic rules. Third, learning classification rules for multi-class target is understudied, as most existing methods focus on binary classification or multi-class classification via the ``one-versus-rest" approach. To address these shortcomings, we propose TURS, for Truly Unordered Rule Sets. To resolve conflicts caused by overlapping rules, we propose a novel model that exploits the probabilistic properties of our rule sets, with the intuition of only allowing rules to overlap if they have similar probabilistic outputs. We next formalize the problem of learning a TURS model based on the MDL principle and develop a carefully designed heuristic algorithm. We benchmark against a wide range of rule-based methods and demonstrate that our method learns rule sets that have lower model complexity and highly competitive predictive performance. In addition, we empirically show that rules in our model are empirically ``independent" and hence truly unordered.

cs.LG↗
arXiv2026

FreDF: Learning to Forecast in the Frequency Domain

Time series modeling presents unique challenges due to autocorrelation in both historical data and future sequences. While current research predominantly addresses autocorrelation within historical data, the correlations among future labels are often overlooked. Specifically, modern forecasting models primarily adhere to the Direct Forecast (DF) paradigm, generating multi-step forecasts independently and disregarding label autocorrelation over time. In this work, we demonstrate that the learning objective of DF is biased in the presence of label autocorrelation. To address this issue, we propose the Frequency-enhanced Direct Forecast (FreDF), which mitigates label autocorrelation by learning to forecast in the frequency domain, thereby reducing estimation bias. Our experiments show that FreDF significantly outperforms existing state-of-the-art methods and is compatible with a variety of forecast models. Code is available at https://github.com/Master-PLC/FreDF.

cs.LG↗
arXiv2026

Convergence of Sharpness-Aware Minimization Algorithms using Increasing Batch Size and Decaying Learning Rate

The sharpness-aware minimization (SAM) algorithm and its variants, including gap guided SAM (GSAM), have been successful at improving the generalization capability of deep neural network models by finding flat local minima of the empirical loss in training. Meanwhile, it has been shown theoretically and practically that increasing the batch size or decaying the learning rate avoids sharp local minima of the empirical loss. In this paper, we consider the GSAM algorithm with increasing batch sizes or decaying learning rates, such as cosine annealing or linear learning rate, and theoretically show its convergence. Moreover, we numerically compare SAM (GSAM) with and without an increasing batch size and conclude that using an increasing batch size { achieves a lower worst-case $\ell_\infty$ adaptive sharpness} than compared with using a constant batch size and learning rate.

cs.LG↗