A Statistical Difference between Single-Layer Learning and Hierarchical Learning in Wide Neural Networks

Hierarchical neural networks are widely used in artificial intelligence, yet their mathematical properties remain incompletely understood. In the infinite-width limit, two different theoretical frameworks have been proposed. One reduces deep learning to kernel regression with a fixed kernel by assuming that the parameters remain close to their initialization, whereas the other allows the parameters to move away from their initialization, requiring the kernel itself to be optimized. In this paper, we study a three-layer neural network with a finite but large number of hidden units. We show that training the input-to-hidden weights yields a smaller generalization error than keeping them fixed. Furthermore, the latter setting exhibits singularities in the parameter space, whereas the former does not. These findings indicate that singularities play an essential role even in wide neural networks.

Singular perturbations and hierarchical learning in two-layer neural networks

We study the population gradient flow of an infinitely wide two-layer neural network learning a misspecified single-index model in high dimension. The two layers are optimized jointly, with a perturbative parameter tuning the relative training speed between the first and second layer. This setting was considered by Berthier, Montanari and Zhou in \cite{berthier2024learning}, who conjectured a hierarchical learning scenario with explicit timescales as the second layer is trained faster than the first. In this paper, we prove that the constant and linear components of the hidden link function are indeed recovered within the predicted timescales, at sharp explicit thresholds. We then analyze the onset of learning of the quadratic component and show that the components learned at earlier stages continue to influence the dynamics in an essential way. Our proof is based on quantitative approximation results for singularly perturbed flows evolving near a manifold defined by integral constraints. At a phenomenological level, we also show that the empirical measure of the weights displays singular behaviour when reaching the quadratic component of the hidden link, with a small fraction of neurons growing significantly while the remaining ones rearrange to preserve the components already learned.

Hierarchical Scaffolding Enables Human-Like Cognitive Selectivity under Data Scarcity

Modern machine learning systems demand extensive datasets for visual recognition. Conversely, humans learn with high efficiency despite severe data limitations, often by acquiring broad categorical structures before refining finer distinctions. Inspired by this contrast, we introduce SCALA (Scaffolded Cognitive Architecture for Learning under limited dAta), a hierarchical learning framework grounded in cognitive psychology that guides models from coarse conceptual structures to fine-grained recognition. Our model exhibits human-like cognitive selectivity by effectively prioritizing task-relevant features while suppressing background distractors, a mechanism that induces a fundamental shift in representation learning. This shift is characterized by accelerated cluster formation, reduced intra-class dispersion, and enhanced semantic separability. Empirically, SCALA achieves significant accuracy improvements under severe data scarcity. Furthermore, this hierarchical scaffolding promotes robust generalization to unseen classes and accelerates the acquisition of novel categories. Collectively, our results establish SCALA as a powerful framework for achieving human-level sample efficiency and resilient category generalization in data-constrained environments.

The Multiscale Single-Index Model: A Stylized Model for Hierarchical Feature Learning

We consider the Multiscale Single-Index Model (MSIM), first introduced in \cite{oymak2021learning}, as a stylized model for hierarchical learning with \emph{scale separation}. Each layer extracts a shared single-index feature at one physical scale and passes it to the next, thus defining a tractable setting in which to study how deep architectures learn multiscale representations. Under non-degeneracy and delocalization assumptions on the link function and planted features respectively, for fixed depth $K$ and local scale $d$, the first Wiener chaos of the target behaves as a perturbed spiked tensor, where the perturbation of order $d^{-1/2}$ comes from the non-linearity -- revealing the MSIM as a natural non-linear analogue of the Tensor PCA model \cite{montanari2014statistical}. While this perturbative picture is sufficient to enable efficient spectral recovery based on Tensor unfolding (as already observed in \cite{oymak2021learning}), it is not precise enough for the analysis of backpropagation gradient-based methods. In this work, we address this limitation by performing a fine-grained analysis of the Wiener chaos using Edgeworth expansions. In the first chaos, this gives a finite-rank hierarchy at scales $d^{-q/2}$. In higher chaoses, balanced flattenings exhibit staircase singular-value plateaus of size $d^{-ρ/2}$ and multiplicity $d^ρ$ under a natural higher-chaos non-cancellation condition. Using this higher-chaos structure, and under an additional slow Hermite-energy tail condition, we first establish shallow-network approximation lower bounds, quantifying the benefit of depth in this model. Next, and most importantly, we prove that online SGD on the correlation objective, where all layers evolve in the same timescale, achieves $1 - o_d(1)$ recovery with $n = \widetilde{O}( d^{K-1})$ samples, recovering the same sample complexity as in the linear counterpart.

Self-Abstraction Learning for Effective and Stable Training of Deep Neural Networks

Training large-scale deep neural networks effectively and stably is essential for applying deep learning across various fields. However, conventional methods, which rely on training a single large network, often encounter challenges such as gradient vanishing, overfitting and unstable learning. To overcome these limitations, we introduce Self-Abstraction Learning (SAL), a hierarchical framework. In SAL, networks are arranged by structural complexity, where the simplest topmost network is trained first and its hidden and output layers serve as guidance for the successively more complex networks below. This top-down sequential guidance effectively mitigates optimization issues, enabling stable training of deep architectures. Various experiments across MLP, CNN, and RNN architectures demonstrate that SAL consistently outperforms conventional methods, ensuring robust generalization even in data-scarce and complex network regimes.