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Coupled finite differences have a covariance-controlled variance

Statement

Let $Y^{+}$ and $Y^{-}$ be square-integrable real simulation outputs with any specified joint law, and let $h > 0$ . The central-difference estimator $D = (Y^{+} - Y^{-}) / (2 h)$ satisfies $Var (D) = \frac{Var (Y^{+}) + Var (Y^{-}) - 2 Cov (Y^{+}, Y^{-})}{4 h^{2}} .$ The formula applies in particular when the two outputs use common random numbers as in Coupled finite-difference simulation-gradient estimator. Positive covariance reduces this variance relative to independent outputs with the same marginals; coupling alone does not guarantee positive covariance.

Facts & Assumptions

Given: Square-integrable real outputs $Y^{+}, Y^{-}$ on one probability space and a scalar $h > 0$ .

[F1]

Moments, variance, and covariance on a probability space: variance and covariance are the expectations of centered squares and products.

[F2]

Variance and covariance identities for random variables: $Var (X) = E [X^{2}] - E [X]^{2}$ and $Cov (X, Y) = E [X Y] - E [X] E [Y]$ .

[F3]

Coupled finite-difference simulation-gradient estimator: the central difference is $(Y^{+} - Y^{-}) / (2 h)$ , with the joint law fixed by the simulation coupling.

[F4]

Expectations factor over finite products of independent random variables: independent integrable factors have product expectation equal to the product of their expectations.

Proof

1.1givenF1F3algebra

By [F3], $D = (Y^{+} - Y^{-}) / (2 h)$ . Centering gives $D - E D = \frac{(Y^{+} - E Y^{+}) - (Y^{-} - E Y^{-})}{2 h} .$ All terms are square-integrable by the hypotheses.

2.1step 1.1F1F2algebra

Squaring step 1.1, taking expectations, and expanding the cross term by [F1]–[F2] yields $Var (D) = \frac{Var (Y^{+}) + Var (Y^{-}) - 2 Cov (Y^{+}, Y^{-})}{4 h^{2}} .$

3.1F2F4step 2.1algebra∎

For independent outputs, [F4] gives $E [Y^{+} Y^{-}] = E [Y^{+}] E [Y^{-}]$ , so [F2] makes their covariance zero. Thus the right side in step 2.1 is no larger than the independent-output variance exactly when the chosen coupling has nonnegative covariance; no such sign is forced by sharing a random input.

Depends on

Used by

Common random numbers can remove finite-difference noise Example

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Sources

Michael C. Fu, Stochastic Gradient Estimation, preprint of Handbook of Simulation Optimization, §§3.1 and 3.3.4 (standard reference, not scraped)