On the Optimal Rate for the Convergence Problem in Mean Field Control
A talk in the SPDEs, optimal control and mean field games series by
François Delarue from Nizza
| Abstract: | The goal of this work is to obtain optimal rates for the convergence problem
in mean field control. Our analysis covers cases where the solutions to the
limiting problem may not be unique nor stable. Equivalently the value function
of the limiting problem might not be differentiable on the entire space. Our
main result is then to derive sharp rates of convergence in two distinct regimes.
When the data are sufficiently regular, we obtain rates proportional to N −1/2,
with N being the number of particles. When the data are merely Lipschitz and
semi-concave with respect to the first Wasserstein distance, we obtain rates
proportional to N −2/(3d+6). Noticeably, the exponent 2/(3d+6) is close to 1/d,
which is the optimal rate of convergence for uncontrolled particle systems driven
by data with a similar regularity. The key argument in our approach consists
in mollifying the value function of the limiting problem in order to produce
functions that are almost classical sub- solutions to the limiting Hamilton-Jacobi
equation (which is a PDE set on the space of probability measures). These sub-solutions
can be projected onto finite dimensional spaces and then compared with the value
functions associated with the particle systems. In the end, this comparison
is used to prove the most demanding bound in the estimates. The key challenge
therein is thus to exhibit a convenient form of mollification. We do so by employing
sup-convolution within a convenient functional Hilbert space. To make the whole
easier, we limit ourselves to the periodic setting. We also provide some examples
to show that our results are sharp up to some extent. Joint work with S. Daudin
(Nice, France) and J. Jackson (Austin, Texas, USA). |