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.

Non-asymptotic implicit bias of logistic regression at early-stage gradient descent dynamics

Gradient descent has been of particular interest in modern machine learning beyond sole focus on optimization. Implicit bias emerging from optimization, though not being encoded by the learning objective, often prevents from overfitting to spurious patterns. A typical instance is the max-margin implicit bias of a linear classifier, widely established for exponentially tailed loss functions. Even after having a given dataset separated, the parameter vector continues to evolve towards the max-margin direction asymptotically along the gradient descent dynamics. This phenomenon corroborates a frequent empirical observation of "train longer, generalize better." However, the max-margin convergence is an asymptotic phenomenon, and what is worse, this asymptotic convergence rate is significantly slower than pure convex optimization. Even so, the parameter vector along gradient descent dynamics commonly correlates with the max-margin direction positively (though not exactly) within considerably fewer iterations than the asymptotic rate. By shedding another light on this classical problem, this work aims to understand the mechanism of this early-stage alignment phenomenon. Our theoretical results demonstrate that the parameter vector weakly aligns with the max-margin direction within $O(\exp(\exp(-δ)))$ iterations, where $δ>0$ is the permissible alignment error, which is shown to be tight. By tracking the radial and tangential flows, our proof operates on the alignment dynamics directly with dataset geometry and gets rid of the asymptotic expansion, which is a key insight to establishing faster weak alignment.

Quotient Dynamics, Effective Curvature, and Implicit Bias in Positive Quadratic Networks

Positive quadratic networks admit the low-rank representation f_U(x)=x^top UU^top x, where Uinmathbb{R}^{dtimes r} is identifiable only up to right orthogonal multiplication, representing a rank-r PSD matrix Q=UU^top. We study how this quotient structure governs training dynamics, curvature, recovery, and interpolation bias. On the full-column-rank stratum, we identify mathbb{R}^{dtimes r}_*/O(r) with the rank-r PSD manifold. For smooth objectives L(U)=ell(UU^top), the Euclidean factor gradient is horizontal. Thus, factor gradient flow projects exactly to quotient Riemannian gradient flow, while finite-step gradient descent induces an exact congruence recursion for the predictor. For quadratic regression, we derive the effective Hessian at interpolators as the empirical measurement Gram form restricted to the tangent space relative to the quotient metric. Under Gaussian rank-one measurements, we compute population curvature, prove uniform deviation bounds for the empirical normal operator, construct a spectral initializer, and establish local exponential convergence for gradient flow and linear convergence for small-step descent. Recovery guarantees are explicit but conservative due to reliance on full-space second-moment control. In underdetermined commuting regimes, factor gradient flow becomes an exact entropy mirror flow in joint spectral coordinates. Strictly positive initializations converge to Bregman projections onto the interpolation set. With isotropic initialization q(0)=varepsilon^2mathbf{1}, predictors approach the minimum-trace solution set as varepsilondownarrow0, resolving nonuniqueness via weighted entropy within the invariant joint spectral algebra. Finite-step descent selects interpolants differing from continuous-time Bregman projections by O(eta). Numerical experiments verify these quotient identities, curvature predictions, recovery behaviors, and selection laws.

Gradient Flow Dynamics and Implicit Bias of Diagonal Linear Networks under Infinitesimal Initialization

We study the gradient flow dynamics of diagonal linear networks for regression tasks under infinitesimal initialization. Extending Theorem 1 from Pesme & Flammarion (2023), we generalize the analysis to both deep diagonal linear networks and a broader class of two-layer diagonal linear networks (as defined in Definition 4.1). Specifically, we demonstrate that the training trajectories of these models can be equivalently characterized by the proposed Algorithm 1. We further prove that this algorithm converges to the solution of a modified $ \mathcal{l}_1 $ norm minimization problem. As a result, we establish that the implicit bias of both network architectures corresponds to a modified $ \mathcal{l}_1 $ norm in the regime of infinitesimal initialization. Additionally, we provide insights into the underlying mechanisms governing these dynamics by identifying the Structural Invariant Manifold (SIM) (Zhao et al., 2026) as the key geometric structure that shapes the learning process.

Stability Annealing Selects the Implicit Bias of Smoothed Sign Descent: A Rate-Indexed Barrier Path on Separable Data

Adaptive gradient methods can favor max-margin separators that differ from gradient descent, yet a fixed positive numerical stability constant eventually changes the update geometry again. This paper studies the rate-controlled middle case for full-batch linear classification on separable data. For memoryless stability-annealed smoothed-sign descent with weighted exponential loss, we prove that the normalized iterates converge to the minimizer of a convex Burg-type barrier over a margin slice. The proof rewrites the dynamics exactly as entropic mirror ascent on a concave dual objective, controls the dual gap by a KL recursion, and yields an explicit S_t^{-1/2} normalized-iterate envelope. The static barrier geometry is fully characterized, including KKT conditions and both endpoint limits. Experiments validate the exact dual identities to floating-point error, illustrate the predicted path and rate diagram, and show an empirical fixed-epsilon crossover scaling in cumulative time. We further report robustness and boundary diagnostics for logistic tails, fixed-epsilon crossover, and adaptive-method variants, delineating the scope of the proved smoothed-sign theory.

Implicit Bias of SGD in Multivariate ReLU Networks: Effective Width Collapse

We study the implicit bias of noisy stochastic gradient descent in training wide two-layer ReLU networks for multivariate regression. In a mean-field regime, the training dynamics are approximated by a Wasserstein gradient flow that converges to a unique stationary measure. We characterize the structure of this stationary measure and the predictor it represents. We show that, despite the network being infinitely overparameterized, the learned predictor admits an effectively finite representation: the input weights and biases align along finitely many directions, leading to an effective width collapse. In particular, the solution function is continuous piecewise affine, with affine regions determined by the cells of a finite hyperplane arrangement. The number of learned directions, and hence hyperplanes, is bounded above by $2\mathcal{P}-1$, where $\mathcal{P}$ denotes the number of linear dichotomies realizable on the training inputs. We further establish a non-redundancy property of the learned representation by proving that each learned direction induces a unique ternary activation pattern on the training data. Consequently, the complexity of the learned predictor is governed by the combinatorial geometry of the training data.

Singular Learning and Occam's Razor in Deep Monomial Networks

In the optimization of neural networks, gradient dynamics are influenced by critical points that arise from the model's architecture. These critical points occur where the Jacobian of the model's parametrization is rank-deficient, and are the most pronounced singularities studied in Singular Learning Theory. We investigate such points in deep fully-connected networks with monomial activations via tools from polynomial algebra such as Mason's Theorem. We show that, for sufficiently large activation degree, criticality occurs precisely at subnetworks, i.e., at parameter configurations where some neurons are inactive or redundant. This offers a mathematical perspective on the implicit bias in deep neural networks, explaining the tendency of these models to converge toward simpler functions.