Nonlinear Dynamics In Optimization Landscape of Shallow Neural Networks with Tunable Leaky ReLU

• URL: http://arxiv.org/abs/2510.25060v1
• Date: Wed, 29 Oct 2025 01:00:07 GMT
• Title: Nonlinear Dynamics In Optimization Landscape of Shallow Neural Networks with Tunable Leaky ReLU
• Authors: Jingzhou Liu,
• Abstract summary: We study the nonlinear dynamics of a shallow neural network trained with mean-squared loss and leaky ReLU activation.<n>We detect bifurcation of critical points with associated symmetries from global minimum as leaky parameter $alpha$ varies.
• Score: 3.2063364612188416
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work, we study the nonlinear dynamics of a shallow neural network trained with mean-squared loss and leaky ReLU activation. Under Gaussian inputs and equal layer width k, (1) we establish, based on the equivariant gradient degree, a theoretical framework, applicable to any number of neurons k>= 4, to detect bifurcation of critical points with associated symmetries from global minimum as leaky parameter $\alpha$ varies. Typically, our analysis reveals that a multi-mode degeneracy consistently occurs at the critical number 0, independent of k. (2) As a by-product, we further show that such bifurcations are width-independent, arise only for nonnegative $\alpha$ and that the global minimum undergoes no further symmetry-breaking instability throughout the engineering regime $\alpha$ in range (0,1). An explicit example with k=5 is presented to illustrate the framework and exhibit the resulting bifurcation together with their symmetries.

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