cs.LGOct 20, 2026

Do Neural Models Prefer Their Submodels?

Authors: Alvaro Ribot, Yulia Alexandr, Anna Seigal

Organizations: Harvard University · University of Melbourne

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Abstract

A neural model is the set of functions that can be expressed by a given neural network architecture. Neural models contain natural submodels, such as those obtained by setting some neurons to be inactive. We study the geometry of neural models and ask whether they prefer their submodels. By this we mean whether local minimizers of the loss function lie in a given submodel. We introduce a geometric framework to determine if this occurs. For shallow multilayer perceptrons, we show that subnetworks with inactive hidden neurons are never preferred. For deep multilayer perceptrons, we give conditions under which subnetworks with inactive hidden neurons are not preferred. For lightning self-attention, lower-rank submodels are usually not preferred, but we identify examples where they are because they lie on a boundary of the model. Our results show that, aside from a few exceptions, preference for submodels cannot be explained by the geometry of the neural model, and must instead come from other sources such as training dynamics.

Figures & tables

Figure 1: No proximity bias toward subnetworks of shallow multilayer perceptrons (MLPs) .
Figure 2: The presence or absence of proximity bias depends on local geometry. For curve F\mathcal{F} , there is proximity bias toward F′={0}\mathcal{F^{\prime}}=\{\mathbf{0}\} in (c) and (d), but not in (a) and (b), as shown by whether the local Voronoi cell Vor⁡Floc⁡(0)\operatorname{Vor}^{\operatorname{loc}}_{\mathcal{F}}(\mathbf{0}) (purple dashes) is full-dimensional. We also draw the tangent cone TF(0)T_{\mathcal{F}}(\mathbf{0}) , normal cone NF(0)N_{\mathcal{F}}(\mathbf{0}) , and Voronoi cell Vor⁡F(0)\operatorname{Vor}_{\mathcal{F}}(\mathbf{0}) for comparison. The blue dots in (d) are points in the Zariski closure of F\mathcal{F} . Details in G.2 .
Figure 3: No proximity bias toward the origin for single-neuron cones : \operatorname{Vor}^{\operatorname{loc}}_{{\color[rgb]{0.2,0.58,0.82}\mathcal{C}}}({\color[rgb]{0.72,0.16,0.46}0})=N_{{\color[rgb]{0.2,0.58,0.82}\mathcal{C}}}({\color[rgb]{0.72,0.16,0.46}0})=\{{\color[rgb]{0.72,0.16,0.46}0}\} .
Figure 4: No proximity bias toward submodels of multi-token self-attention mechanisms , except for a rank-one scalar output model (d′=a=1)(d^{\prime}=a=1) ; see Theorems 4.1 and 4.3 . Submodels are the magenta loci, and the dotted parts represent points in the Zariski closure of the respective neural models. Details in G.4 .
Figure 5: Implicit bias for softmax attention. Ratio σ2(St)/σ1(St)\sigma_{2}(S_{t})/\sigma_{1}(S_{t}) over 500500 steps. Both models use the same inputs, labels, and initialization. Thick curves show medians over 3030 runs; shaded regions show interquartile ranges; faint curves show 8 individual runs.
Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: No proximity bias toward subnetworks of polynomial CNNs.
Figure 7: Details for Figure 2 .
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