For a simple HJB control example, MLP approximations have L2 error growing like d^{n/2}/(c^n sqrt(n!)) - 1, so no polynomial-in-dimension error bound can hold uniformly as the number of levels grows.
Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
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Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem
For a simple HJB control example, MLP approximations have L2 error growing like d^{n/2}/(c^n sqrt(n!)) - 1, so no polynomial-in-dimension error bound can hold uniformly as the number of levels grows.