In this paper, we study the Karush-Kuhn-Tucker (KKT) points of the associated maximum-margin problem in homogeneous neural networks, including fullyconnected and convolutional neural networks. In particular, We investigates the relationship between such KKT points across networks of different widths generated. We introduce and formalize the KKT point embedding principle, establishing that KKT points of a homogeneous network's max-margin problem (PΦ) can be embedded into the KKT points of a larger network's problem (P Φ) via specific linear isometric transformations. We rigorously prove this principle holds for neuron splitting in fully-connected networks and channel splitting in convolutional neural networks. Furthermore, we connect this static embedding to the dynamics of gradient flow training with smooth losses. We demonstrate that trajectories initiated from appropriately mapped points remain mapped throughout training and that the resulting ω-limit sets of directions are correspondingly mapped, thereby preserving the alignment with KKT directions dynamically when directional convergence occurs. We conduct several experiments to justify that trajectories are preserved. Our findings offer insights into the effects of network width, parameter redundancy, and the structural connections between solutions found via optimization in homogeneous networks of varying sizes.

In this paper, we study the Karush-Kuhn-Tucker (KKT) points of the associated maximum-margin problem in homogeneous neural networks, including fully-connected and convolutional neural networks. In particular, We investigates the relationship between such KKT points across networks of different widths generated. We introduce and formalize the \textbf{KKT point embedding principle}, establishing that KKT points of a homogeneous network's max-margin problem ($P_{\Phi}$) can be embedded into the KKT points of a larger network's problem ($P_{\tilde{\Phi}}$) via specific linear isometric transformations. We rigorously prove this principle holds for neuron splitting in fully-connected networks and channel splitting in convolutional neural networks. Furthermore, we connect this static embedding to the dynamics of gradient flow training with smooth losses. We demonstrate that trajectories initiated from appropriately mapped points remain mapped throughout training and that the resulting $\omega$-limit sets of directions are correspondingly mapped, thereby preserving the alignment with KKT directions dynamically when directional convergence occurs. We conduct several experiments to justify that trajectories are preserved. Our findings offer insights into the effects of network width, parameter redundancy, and the structural connections between solutions found via optimization in homogeneous networks of varying sizes.

In this work, we study the implications of the implicit bias of gradient flow on generalization and adversarial robustness in ReLU networks. We focus on a setting where the data consists of clusters and the correlations between cluster means are small, and show that in two-layer ReLU networks gradient flow is biased towards solutions that generalize well, but are vulnerable to adversarial examples. Our results hold even in cases where the network is highly overparameterized. Despite the potential for harmful overfitting in such settings, we prove that the implicit bias of gradient flow prevents it. However, the implicit bias also leads to non-robust solutions (susceptible to small adversarial ℓ2-perturbations), even though robust networks that fit the data exist.

Multi-channel learning has gained significant attention in recent applications, where neural networks with t-product layers (t-NNs) have shown promising performance through novel feature mapping in the transformed domain. However, despite the practical success of t-NNs, the theoretical analysis of their generalization remains unexplored. We address this gap by deriving upper bounds on the generalization error of t-NNs in both standard and adversarial settings. Notably, it reveals that t-NNs compressed with exact transformed low-rank parameterization can achieve tighter adversarial generalization bounds compared to non-compressed models. While exact transformed low-rank weights are rare in practice, the analysis demonstrates that through adversarial training with gradient flow, highly over-parameterized t-NNs with the ReLU activation can be implicitly regularized towards a transformed low-rank parameterization under certain conditions. Moreover, this paper establishes sharp adversarial generalization bounds for t-NNs with approximately transformed low-rank weights. Our analysis highlights the potential of transformed low-rank parameterization in enhancing the robust generalization of t-NNs, offering valuable insights for further research and development.

We study the implicit bias of momentum-based optimizers on homogeneous models. We first extend existing results on the implicit bias of steepest descent in homogeneous models to normalized steepest descent with an optional learning rate schedule. We then show that for smooth homogeneous models, momentum steepest descent algorithms like Muon (spectral norm), MomentumGD ($\ell_2$ norm), and Signum ($\ell_\infty$ norm) are approximate steepest descent trajectories under a decaying learning rate schedule, proving that these algorithms too have a bias towards KKT points of the corresponding margin maximization problem. We extend the analysis to Adam (without the stability constant), which maximizes the $\ell_\infty$ margin, and to Muon-Signum and Muon-Adam, which maximize a hybrid norm. Our experiments corroborate the theory and show that the identity of the margin maximized depends on the choice of optimizer. Overall, our results extend earlier lines of work on steepest descent in homogeneous models and momentum-based optimizers in linear models.