Variance reduction with probing and Multilevel Monte Carlo in Lattice QCD

What the math gives to ML

The paper provides two constructive variance-reduction mechanisms for expensive stochastic trace estimation: an unbiased multilevel telescoping estimator built from a hierarchy of approximations, and graph-coloring probing vectors that suppress interactions between nearby matrix entries. Both mechanisms transfer naturally to neural-network workloads where traces or quadratic forms of Jacobians, Hessians, Fisher matrices, NTK kernels, or implicit-layer linear operators are estimated with Hutchinson probes. The most promising adaptation is to use probing on parameter- or activation-interaction graphs for localized operators, and multilevel control variates across model widths, resolutions, truncation ranks, or solver tolerances. These methods target estimator variance directly, potentially reducing the number of expensive forward/reverse-mode products needed by curvature-aware optimizers and regularizers.