Deeper understanding of optimization dynamics

Develop a deeper theoretical understanding of the optimization dynamics for training single-head tied attention beyond the asymptotic characterization of minima provided in this work, clarifying the mechanisms that govern convergence and learning behavior.

Background

While the paper characterizes the asymptotics of the global minima and confirms empirical alignment with gradient-based training, it stops short of a full theory of the optimization process itself.

The authors explicitly identify the broader study of optimization dynamics as an open challenge beyond their present analysis.

References

Finally, a deeper understanding of optimization dynamics, beyond our asymptotic characterization of the minima, remains an important open challenge.

— Inductive Bias and Spectral Properties of Single-Head Attention in High Dimensions  (2509.24914 - Boncoraglio et al., 29 Sep 2025) in Section 6, Conclusion and limitations

We prove this for one step of gradient descent on the population loss restricted to the key-parameter space, i.e., we treat the per-token key vectors {\mathbf{k}_j(s)} as the trainable parameters and hold the remaining weights fixed, since tracking gradient dynamics over all transformer weights is substantially more complex; we conjecture an analogous (qualitative) effect under gradient descent on the full parameter set.

— Learn Your Own Thoughts: Abstract Token Curriculum  (2609.19717 - Gatmiry et al., 17 Sep 2026) in Section 3, immediately preceding Theorem 1 (Main Theoretical Result)

A second open direction concerns learning on a longer timescale. Corollary \ref{cor:untied_weak_recovery} characterizes the \Theta(d2\log d) sample scale. However, the numerical results suggest that recovery can occur on longer scales controlled by the information exponents. Analyzing these regimes requires tracking finite-d corrections over polynomially long training horizons, since higher-order terms that vanish at \Theta(1) time can accumulate over such scales. Developing such a theory would provide a more complete characterization of the parameterization-dependent sample complexity.

— High-Dimensional Learning Dynamics of Attention-Indexed Models  (2609.03858 - Xu et al., 3 Sep 2026) in Section “Conclusion and future work”

Without Assumption~\ref{assumption:init}, the asymmetric initialization of $\alpha_S$ and $\alpha_R$ can complicate the analysis of the dynamics $\beta_S$ and $\beta_R$ in the early training phase. We leave the study of the general initialization for future work.

— Muon Learns Facts Better: Understanding the Role of Spectral Orthogonalization  (2610.02798 - Li et al., 2 Oct 2026) in Section 4.2, immediately following the Spectral GF reduced dynamics

Can the exact denoiser be learned? Unlike in LDA, the sigmoidal term cannot be suppressed without losing the target itself, so the optimization problem remains genuinely nonlinear and nonconvex. We therefore examine it numerically at the population level.

— Softmax Attention on Gaussian Mixtures: Linear When It Can, Selective When It Must  (2610.11798 - Gabet et al., 8 Oct 2026) in Section 4.2, paragraph “Can the exact denoiser be learned?”