Control variates

The control variates method is a variance reduction technique used in Monte Carlo methods. It exploits information about the errors in estimates of known quantities to reduce the error of an estimate of an unknown quantity.^{[1] [2] [3]}

Let the unknown parameter of interest be ${\displaystyle \mu }${\displaystyle \mu }, and assume we have a statistic ${\displaystyle m}${\displaystyle m} such that the expected value of m is μ: ${\displaystyle \mathbb {E} \left[m\right]=\mu }${\displaystyle \mathbb {E} \left[m\right]=\mu }, i.e. m is an unbiased estimator for μ. Suppose we calculate another statistic ${\displaystyle t}${\displaystyle t} such that ${\displaystyle \mathbb {E} \left[t\right]=\tau }${\displaystyle \mathbb {E} \left[t\right]=\tau } is a known value. Then

${\displaystyle m^{\star }=m+c\left(t-\tau \right),円}${\displaystyle m^{\star }=m+c\left(t-\tau \right),円}

is also an unbiased estimator for ${\displaystyle \mu }${\displaystyle \mu } for any choice of the coefficient ${\displaystyle c}${\displaystyle c}. The variance of the resulting estimator ${\displaystyle m^{\star }}${\displaystyle m^{\star }} is

${\displaystyle {\textrm {Var}}\left(m^{\star }\right)={\textrm {Var}}\left(m\right)+c^{2},円{\textrm {Var}}\left(t\right)+2c,円{\textrm {Cov}}\left(m,t\right).}${\displaystyle {\textrm {Var}}\left(m^{\star }\right)={\textrm {Var}}\left(m\right)+c^{2},円{\textrm {Var}}\left(t\right)+2c,円{\textrm {Cov}}\left(m,t\right).}

By differentiating the above expression with respect to ${\displaystyle c}${\displaystyle c}, it can be shown that choosing the optimal coefficient

${\displaystyle c^{\star }=-{\frac {{\textrm {Cov}}\left(m,t\right)}{{\textrm {Var}}\left(t\right)}}}${\displaystyle c^{\star }=-{\frac {{\textrm {Cov}}\left(m,t\right)}{{\textrm {Var}}\left(t\right)}}}

minimizes the variance of ${\displaystyle m^{\star }}${\displaystyle m^{\star }}. (Note that this coefficient is the same as the coefficient obtained from a linear regression.) With this choice,

${\displaystyle {\begin{aligned}{\textrm {Var}}\left(m^{\star }\right)&={\textrm {Var}}\left(m\right)-{\frac {\left[{\textrm {Cov}}\left(m,t\right)\right]^{2}}{{\textrm {Var}}\left(t\right)}}\\&=\left(1-\rho _{m,t}^{2}\right){\textrm {Var}}\left(m\right)\end{aligned}}}${\displaystyle {\begin{aligned}{\textrm {Var}}\left(m^{\star }\right)&={\textrm {Var}}\left(m\right)-{\frac {\left[{\textrm {Cov}}\left(m,t\right)\right]^{2}}{{\textrm {Var}}\left(t\right)}}\\&=\left(1-\rho _{m,t}^{2}\right){\textrm {Var}}\left(m\right)\end{aligned}}}

where

${\displaystyle \rho _{m,t}={\textrm {Corr}}\left(m,t\right),円}${\displaystyle \rho _{m,t}={\textrm {Corr}}\left(m,t\right),円}

is the correlation coefficient of ${\displaystyle m}${\displaystyle m} and ${\displaystyle t}${\displaystyle t}. The greater the value of ${\displaystyle \vert \rho _{m,t}\vert }${\displaystyle \vert \rho _{m,t}\vert }, the greater the variance reduction achieved.

In the case that ${\displaystyle {\textrm {Cov}}\left(m,t\right)}${\displaystyle {\textrm {Cov}}\left(m,t\right)}, ${\displaystyle {\textrm {Var}}\left(t\right)}${\displaystyle {\textrm {Var}}\left(t\right)}, and/or ${\displaystyle \rho _{m,t}\;}${\displaystyle \rho _{m,t}\;} are unknown, they can be estimated across the Monte Carlo replicates. This is equivalent to solving a certain least squares system; therefore this technique is also known as regression sampling.

When the expectation of the control variable, ${\displaystyle \mathbb {E} \left[t\right]=\tau }${\displaystyle \mathbb {E} \left[t\right]=\tau }, is not known analytically, it is still possible to increase the precision in estimating ${\displaystyle \mu }${\displaystyle \mu } (for a given fixed simulation budget), provided that the two conditions are met: 1) evaluating ${\displaystyle t}${\displaystyle t} is significantly cheaper than computing ${\displaystyle m}${\displaystyle m}; 2) the magnitude of the correlation coefficient ${\displaystyle |\rho _{m,t}|}${\displaystyle |\rho _{m,t}|} is close to unity. ^[3]

We would like to estimate

${\displaystyle I=\int _{0}^{1}{\frac {1}{1+x}},円\mathrm {d} x}${\displaystyle I=\int _{0}^{1}{\frac {1}{1+x}},円\mathrm {d} x}

using Monte Carlo integration. This integral is the expected value of ${\displaystyle f(U)}${\displaystyle f(U)}, where

${\displaystyle f(U)={\frac {1}{1+U}}}${\displaystyle f(U)={\frac {1}{1+U}}}

and U follows a uniform distribution [0, 1]. Using a sample of size n denote the points in the sample as ${\displaystyle u_{1},\cdots ,u_{n}}${\displaystyle u_{1},\cdots ,u_{n}}. Then the estimate is given by

${\displaystyle I\approx {\frac {1}{n}}\sum _{i}f(u_{i}).}${\displaystyle I\approx {\frac {1}{n}}\sum _{i}f(u_{i}).}

Now we introduce ${\displaystyle g(U)=1+U}${\displaystyle g(U)=1+U} as a control variate with a known expected value ${\displaystyle \mathbb {E} \left[g\left(U\right)\right]=\int _{0}^{1}(1+x),円\mathrm {d} x={\tfrac {3}{2}}}${\displaystyle \mathbb {E} \left[g\left(U\right)\right]=\int _{0}^{1}(1+x),円\mathrm {d} x={\tfrac {3}{2}}} and combine the two into a new estimate

${\displaystyle I\approx {\frac {1}{n}}\sum _{i}f(u_{i})+c\left({\frac {1}{n}}\sum _{i}g(u_{i})-3/2\right).}${\displaystyle I\approx {\frac {1}{n}}\sum _{i}f(u_{i})+c\left({\frac {1}{n}}\sum _{i}g(u_{i})-3/2\right).}

Using ${\displaystyle n=1500}${\displaystyle n=1500} realizations and an estimated optimal coefficient ${\displaystyle c^{\star }\approx 0.4773}${\displaystyle c^{\star }\approx 0.4773} we obtain the following results

The variance was significantly reduced after using the control variates technique. (The exact result is ${\displaystyle I=\ln 2\approx 0.69314718}${\displaystyle I=\ln 2\approx 0.69314718}.)

• Antithetic variates
• Importance sampling

• Ross, Sheldon M. (2002) Simulation 3rd edition ISBN 978-0-12-598053-1
• Averill M. Law & W. David Kelton (2000), Simulation Modeling and Analysis, 3rd edition. ISBN 0-07-116537-1
• S. P. Meyn (2007) Control Techniques for Complex Networks, Cambridge University Press. ISBN 978-0-521-88441-9. Downloadable draft (Section 11.4: Control variates and shadow functions)

• Monte Carlo methods
• Statistical randomness
• Computational statistics
• Variance reduction

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