Learning Orthogonal Multi-Index Models Beyond Small Initialization: Incremental Learning, Competitive Dynamics and Symmetry — Paper

Abstract

Recent work has identified incremental learning in shallow networks trained on single-index and multi-index models. However, existing analyses often rely on simplifying settings, such as small initialization, correlation loss, or layer-wise training. These choices reduce neuron interactions and leave some feature learning dynamics under standard initialization unexplored. We study training dynamics for polynomial-width two-layer networks learning orthogonal multi-index targets under standard initialization using polynomially many samples. We first prove that incremental learning still occurs: the loss decreases sequentially according to the Hermite expansion of the target, with lower-order components learned before higher-order components recover the individual target directions. In this standard initialization regime, training also shows a competitive reallocation of parameter mass: after the total mass fits the target mean and stabilizes, mass shifts into the target subspace and then concentrates on aligned neurons. Our theoretical analysis uses slightly modified gradient flow, while vanilla gradient descent empirically exhibits the same qualitative dynamics. Technically, we introduce a symmetry-based finite-width approximation via symmetrized networks, rather than comparing directly with an infinite-width limit. This yields better control of approximation errors and may be of independent interest.

Authors 4

Mo Zhou, Weihang Xu, Simon S. Du, Maryam Fazel

Specification

Official page: arxiv.org/abs/2609.10879
Source:arXiv (Atom API + RSS)T1observed 1 h agohigh
Arxiv announce type: new
Source:arXiv (Atom API + RSS)T1observed 1 h agohigh
arXiv id: 2609.10879
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Categories: cs.LG, stat.ML
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PDF: arxiv.org/pdf/2609.10879
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Primary category: cs.LG
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Published: 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 1 h agohigh

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1 h ago

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Authors: Mo Zhou, Weihang Xu, Simon S. Du

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9 claims · 9 properties

Official pageofficial_url1

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https://arxiv.org/abs/2609.10879	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Abstractabstract1

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Recent work has identified incremental learning in shallow networks trained on single-index and multi-index models. However, existing analyses often rely on simplifying settings, such as small initialization, correlation loss, or layer-wise training. These choices reduce neuron interactions and leave some feature learning dynamics under standard initialization unexplored. We study training dynamics for polynomial-width two-layer networks learning orthogonal multi-index targets under standard initialization using polynomially many samples. We first prove that incremental learning still occurs: the loss decreases sequentially according to the Hermite expansion of the target, with lower-order components learned before higher-order components recover the individual target directions. In this standard initialization regime, training also shows a competitive reallocation of parameter mass: after the total mass fits the target mean and stabilizes, mass shifts into the target subspace and then concentrates on aligned neurons. Our theoretical analysis uses slightly modified gradient flow, while vanilla gradient descent empirically exhibits the same qualitative dynamics. Technically, we introduce a symmetry-based finite-width approximation via symmetrized networks, rather than comparing directly with an infinite-width limit. This yields better control of approximation errors and may be of independent interest.	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Arxiv announce typearxiv_announce_type1

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new	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

arXiv idarxiv_id1

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2609.10879	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Authorsauthors1

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Mo Zhou, Weihang Xu, Simon S. Du, Maryam Fazel	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Categoriescategories1

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cs.LG, stat.ML	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

PDFpdf_url1

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https://arxiv.org/pdf/2609.10879	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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cs.LG	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Publishedpublished_at1

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11 Sept 2026	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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11 Sept 2026
New paperPaperLearning Orthogonal Multi-Index Models Beyond Small Initialization: Incremental Learning, Competitive Dynamics and Symmetry
New paper: Learning Orthogonal Multi-Index Models Beyond Small Initialization: Incremental Learning, Competitive Dynamics and Symmetry
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arXiv (Atom API + RSS)	rss.arxiv.org/rss/cs.LG	feed	T1· Official	1 h ago	1

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