Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent
This work provides rigorous mathematical foundations for understanding fluctuations in mean-field limits of stochastic gradient descent, which is important for researchers in machine learning theory and stochastic analysis.
The paper establishes the convergence of stochastic interacting particle systems to conservative stochastic partial differential equations (SPDEs) with optimal rates, and derives a quantitative central limit theorem for these SPDEs, also with optimal rates. It applies these results to show that including fluctuations in the limiting SPDE improves convergence rates and retains fluctuation information for stochastic gradient descent in overparametrized, shallow neural networks.
The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a quantitative central limit theorem for such SPDEs is derived, again, with optimal rate of convergence. The results apply, in particular, to the convergence in the mean-field scaling of stochastic gradient descent dynamics in overparametrized, shallow neural networks to solutions of SPDEs. It is shown that the inclusion of fluctuations in the limiting SPDE improves the rate of convergence, and retains information about the fluctuations of stochastic gradient descent in the continuum limit.