We here study whether training biases can make hidden neurons specialize in minimal one-hidden-layer MLPs, and whether such specialization improves prototype-based reconstruction of the training dataset from the learned weights. We consider Gaussianactivation MLPs of width equal to dataset size and compare three structural losses that respectively encourage coverage of the training samples, separation between neuron-induced prototypes, and low overlap of hidden responses, against the standard fitting baseline. Experiments on uniformly sampled one-dimensional datasets show a stable pattern from N = 3 to N = 100 across 480 controlled runs. Coverage regularization gives the lowest mean reconstruction error at every tested size and raises the prototype-usage specialization ratio relative to the standard baseline, while separation has mixed effects and overlap penalties are systematically harmful. We show that the harm is not an optimization failure: overlap-active approaches fit the data as well as overlap-free ones but route the optimizer to a degenerate equilibrium in which prototype centers are pushed outside the convex hull of the training inputs. Coverage cannot reward this expulsion and acts as an attractor: separation admits it only at large temperature and overlap admits it at the nominal hyperparameter choice. A direct τ-sweep on the separation-only mask and a prototype-position visualization at N = 100 confirm the mechanism. The findings yield a simple design principle for prototype-recoverability-aware training: every repulsive structural loss must be compensated by a compatible attractor, or it will collapse the latent geometry it was meant to refine.
Understanding how neural networks learn and organize features is central to understanding their behavior. Much existing theory of feature learning has focused on the emergence of a global low-dimensional representation. We show that this picture is incomplete. In regression problems with clustered data, we demonstrate that multilayer perceptrons (MLPs) naturally develop monosemantic specialized neurons: individual neurons become strongly aligned with a specific predictive feature relevant to a particular region of the input space. Rather than learning a single global low-dimensional representation, MLPs learn a collection of local low-dimensional representations. We show that this ability to specialize gives MLPs a provable data-efficiency advantage over feature-learning methods based on a global low-dimensional representation.
Training changes a network's predictions while allocating task-relevant structure across its internal units. In an overparameterized ReLU network, several neurons can begin with exactly the same functional role, yet one may acquire a teacher feature while the others become redundant. We call the identity of that neuron feature ownership and ask whether it can be controlled by a parameter choice invisible to the initial predictor. In a tractable Gaussian teacher--student model, we fix the complete initial function and vary only a positive-homogeneous scaling gauge. Opposite gauges produce distinct feature trajectories and a sharp Θ(D2) separation in specialization time that no global change of clock can explain. Among any fixed number of initially duplicate students, assigning the favorable gauge to one neuron deterministically selects it as the owner and drives the remaining functional contribution to zero. An exact reaction--transport decomposition attributes the effect to different mobilities for changing a feature's coefficient and direction. We prove global selection and functional pruning, extend finite-time selection to visible perturbations and small-step full-batch gradient descent, and verify the predicted loss, alignment, pruning, and dissipation trajectories in population and finite-sample training. The initial predictor therefore determines neither when the feature is learned nor which neuron learns it.
The optimization of neural networks under weight decay remains poorly understood from a theoretical standpoint. While weight decay is standard practice in modern training procedures, most theoretical analyses focus on unregularized settings. In this work, we investigate the loss landscape of the ℓ2-regularized training loss for two-layer ReLU networks. We show that the landscape becomes benign -- i.e., free of spurious local minima -- under large overparametrization, specifically when the network width m satisfies m≳min(nd,2n), where n is the number of data points and d the input dimension. More precisely in this regime, almost all constant activation regions contain a global minimum and no spurious local minima. We further show that this level of overparametrization is not only sufficient but also necessary via the example of orthogonal data. Finally, we demonstrate that such loss landscape results primarily hold relevance in the large initialization regime. In contrast, for small initializations -- corresponding to the feature learning regime -- optimization can still converge to spurious local minima, despite the global benignity of the landscape.
Etienne Boursier, Matthew Bowditch, Matthias Englert +1