Two-evaluation simultaneous-perturbation gradient estimator

Input a dimension $d\ge 1$, $\theta \in \mathrm{\Theta }\subseteq {\mathbb{R}}^d$, a scalar $c>0$, a payoff oracle from Simulation objectives and unbiased payoff oracles, and a random vector $\mathrm{\Delta }=({\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_d)$ whose coordinates are independent, symmetric about zero, and nonzero almost surely. Require $\theta +c\mathrm{\Delta },\theta -c\mathrm{\Delta }\in \mathrm{\Theta }$ almost surely. The perturbation and simulator inputs are realized on a common probability space. Require the paired random-parameter oracle outputs $Y^{+},Y^{-}$, extended by zero on the exceptional zero-output branch, to be measurable real random variables; fixed-parameter integrability alone does not impose this condition.

1. Draw one $\mathrm{\Delta }$. If any ${\mathrm{\Delta }}_i=0$ or either $\theta +c\mathrm{\Delta }$ or $\theta -c\mathrm{\Delta }$ is outside $\mathrm{\Theta }$, return the zero vector in ${\mathbb{R}}^d$ without calling the oracle.

2. Call the oracle once at $\theta +c\mathrm{\Delta }$ and once at $\theta -c\mathrm{\Delta }$, obtaining $Y^{+}$ and $Y^{-}$.

3. Return ${\widehat{G}}_i(\theta ):=\frac{Y^{+}-Y^{-}}{2c{\mathrm{\Delta }}_i},i=1,\dots ,d\mathrm{.}$

The same numerator supplies all $d$ coordinates. The zero-output branch has probability zero under the input assumptions, so one estimator uses exactly two oracle calls almost surely, independently of $d$. The definition does not assert unbiasedness, finite variance, or convergence of an update rule.