= Solution
A <control variate> is a random variable $C$ correlated with the estimator and with known mean $c_0$. For a fixed real coefficient $a$, define $Y_a=Y-a(C-c_0)$. Then $\mathbb EY_a=\theta$, while
$$
\operatorname{Var}(Y_a)=\operatorname{Var}(Y)-2a\operatorname{Cov}(Y,C)+a^2\operatorname{Var}(C).
$$
Assuming finite second moments and positive control <variance>, complete the square to find
$$
\boxed{a_* =\frac{\operatorname{Cov}(Y,C)}{\operatorname{Var}(C)},\qquad
\min_a\operatorname{Var}(Y_a)=\operatorname{Var}(Y)-\frac{\operatorname{Cov}(Y,C)^2}{\operatorname{Var}(C)}.}
$$
When $\operatorname{Var}(Y)>0$, the minimum is $\operatorname{Var}(Y)(1-\rho_{Y,C}^2)$. A constant control cannot reduce <variance>. A strongly correlated control, including a negatively correlated one with a negative coefficient, can substantially reduce it.
For <multivariate control variates>, write $Y_\beta=Y-\beta^T(C-\mathbb EC)$, $\Sigma=\operatorname{Cov}(C)$ and $c=\operatorname{Cov}(C,Y)$. Completing the multivariate square gives
$$
\boxed{\beta_* =\Sigma^{-1}c,\qquad
\min_\beta\operatorname{Var}(Y_\beta)=\operatorname{Var}(Y)-c^T\Sigma^{-1}c.}
$$
If $\Sigma$ is singular, use its <Moore-Penrose inverse>. Indeed any null direction has a constant centered control combination and therefore zero <covariance> with $Y$, so $c$ lies in the range of $\Sigma$. The elementary unbiasedness statement assumes a deterministic coefficient; estimating it from the same simulation may introduce finite-sample bias, whereas an independently trained coefficient keeps the conditional unbiasedness argument valid.
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