High-Dimensional Learning Dynamics of Attention-Indexed Models

Attention mechanisms are central to modern foundation models, yet their training dynamics remain poorly understood, especially when the attention matrices have extensive rank. In this work, we study attention-indexed models, a broad framework that can represent multi-layer and multi-head attention architectures. First, we show that, in a suitable high-dimensional limit, the population-loss landscape is characterized by a finite set of trace order parameters. In contrast, online stochastic gradient descent (SGD) is governed by an infinite hierarchy of matrix moments, which we show can be exponentially well-approximated by a finite truncated system. Second, this framework reveals that attention parameterization itself can act as an architectural implicit bias. Direct optimization of an attention matrix $S\in\mathbb{R}^{d\times d}$ can remain trapped in an uninformative state. Tied attention ($S=WW^\top$) induces an automatic symmetry-breaking mechanism and yields weak recovery in $Θ(d^2\log d)$ samples. For untied attention, $S=UV^\top$, we uncover a fast-slow mechanism: the pre-activation mean first evolves on a fast timescale, while the overlaps evolve on a slower one. Weak recovery on the $Θ(d^2\log d)$ scale occurs when the state selected by the fast dynamics breaks the initial symmetry.

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.

How AI settled the complexity of the oldest SGD algorithm

In 1937, Stefan Kaczmarz proposed a simple algorithm for solving systems of linear equations. This algorithm turned out to be the earliest known example of stochastic gradient descent, a ubiquitous computing paradigm that drives the training of modern AI models such as ChatGPT and Gemini. Now, those AI models have joined forces to discover the worst-case complexity of the Kaczmarz algorithm. This paper tells the story of how it happened.