Randomized symmetric finite-difference derivative estimator
= Randomized symmetric finite-difference derivative estimator
Given noisy evaluations at $x_0+hZ_i$, where the $Z_i$ are independent <Rademacher random variables>, the estimator
$$
\widehat g'_N(x_0)=\frac1N\sum_{i=1}^N\frac{Z_i(Y_i-g(x_0))}{h}
$$
averages one-sided finite differences from both directions. If $|g''|\leq M$ and the noise variance is $\sigma^2$, its <mean squared error> is at most $h^2M^2/4+\sigma^2/(Nh^2)$.