Work-Normalized Efficiency of Variance Reduction

Techniques like antithetic variates, control variates, or importance sampling can cut the variance of a Monte Carlo estimate, so you need fewer simulated paths to hit a target precision. But each of these techniques also costs extra computing time per path — generating an antithetic pair, evaluating a control variate, or reweighting under a different sampling distribution all take longer than a plain simulation step. Work-normalized efficiency compares methods on the basis of variance reduction achieved per unit of computing time, not per sample, because a method that needs three times the work per sample to halve the variance is actually a net loss.

The standard measure is the product of the per-sample runtime and the resulting variance: a smaller value means the method reaches a given precision faster in wall-clock terms. A technique can look impressive on a per-sample variance chart and still be the wrong choice once its extra computational overhead is priced in.

A worked example

Suppose plain simulation needs 10,000 paths, each taking 1 unit of time, to hit a target precision — 10,000 units of total work. A control-variate version cuts the required number of paths to 3,000 but each path takes 2 units of time because of the extra bookkeeping, for 6,000 units total: still a genuine improvement. A more elaborate importance-sampling scheme might cut paths to 1,000 but cost 8 units of time each, for 8,000 units total — less impressive than it first looked, and possibly not worth the added implementation complexity.