Global existence of DMFT solution for nonlinear loss with positive noise

Establish global-in-time existence and uniqueness of the solution to the dynamical mean-field system \mathfrak{S} (the DMFT characterization of the stochastic gradient flow \de \btheta^t = -\big(h_t(\btheta^t) + \tfrac{1}{\delta}\bX^\top \ell_t(\br^t;\bz)\big)\,\de t + \sqrt{\tfrac{\tau}{\delta}}\sum_{i=1}^n \bx_i \ell_t(r_i^t;z_i)^\top\,\de B_i^t, with \br^t=\bX\btheta^t) for general nonlinear \ell_t and noise intensity \tau>0, under Assumptions 1 (data distribution) and 2 (function regularity), beyond the bounded time horizon T_* proved in Theorem 1.

Background

The paper proves existence and uniqueness of solutions to the DMFT system \mathfrak{S} over a bounded time horizon T_* and shows global existence in two special cases: the noiseless case (\tau=0) and when the loss gradient \ell_t is linear in its argument (\nabla_r2 \ell_t(r;z)=0).

For general nonlinear \ell_t with positive noise intensity \tau>0, the authors only establish a local-in-time result and provide a rough bound on the time horizon T_*. They conjecture that global existence should hold because the underlying SGF dynamics do not blow up in finite time under their assumptions, but they defer a rigorous proof.

References

We believe that the solution exists globally for general nonlinear $\ell_t$ with $\tau > 0$ since the SGF eq:sgf does not blow up in finite time under the assumptions considered in this work, but proving this requires further technical work, which we leave for future work.

— High-Dimensional Limit of Stochastic Gradient Flow via Dynamical Mean-Field Theory  (2602.06320 - Nishiyama et al., 6 Feb 2026) in Main results, following Theorem 1 (Theorem \ref{thm:dmft_sol})