# acv_independent_sampling

Sampling scheme within the approximate control variate (ACV) algorithm that employs independent samples (IS) across model pairings

**Specification**

*Alias:*acv_is*Arguments:*None

**Description**

This ACV-IS variant uses independent sample set definitions across
paired models. The pairings depend on the underlying directed acyclic
graph (DAG) definition of the model relationships, and in the case of
ACV without a `search_model_graphs`

specification (not generalized
ACV as in [BLWL22]), the default “peer DAG” is used for
which all approximation nodes point to the root node and the root node
identifies the reference “truth” model. This means that every
approximation participates in a sample set that is shared with the
truth model (the root allocation), and then each approximation
(each of the nodes connecting to the root) has its own independent
augmentation to this shared set.

**Additional Discussion**

In the case of a “hierarchical” DAG (not supported by standard ACV,
but one of the cases available within generalized ACV using
`search_model_graphs`

), each approximation node points to the next
approximation of higher fidelity, ending with the truth model at the
root node. For this DAG case, both ACV-IS and ACV-RD can be
considered to be a weighted multilevel Monte Carlo (MLMC) in terms of
structure, in that independent sample sets are defined for each level
and span a pair of consecutive models within a sequence. The key
difference from traditional MLMC is the control variate weights are
not assumed to be 1, and can better adapt to cases with non-ideal
correlation values. ACV-IS and ACV-RD then differ in these weight
definitions, as derived from the differing set definitions for
\(z^1\) and \(z^2\) (which are distinct sets in ACV-RD and
overlapping sets in ACV-IS; again see [GGEJ20]
and [BLWL22]).

**Theory**

Refer to [GGEJ20] for the original ACV formulation, including the \(z^1\) and \(z^2\) semantics used here. In this paper, the “W-RDiff” estimator is a special case that corresponds to ACV-RD with a hierarchical DAG (see also Additional Discussion above). Also, ACV-IS and ACV-MF both assume peer DAGs, and ACV-KL is a mix of hierarchical and peer.

Refer to [BLWL22] for understanding ACV generalizations for the different control variate pairings that are possible when codified into a more comprehensive set of potential DAGs. Both papers provide sample set diagrams that are instructive for understanding the different formulations.