  In robust optimisation problems the objective function consists of an average
over (an ensemble of) uncertain parameters. Ensemble optimisation (EnOpt)
implements steepest descent by estimating the gradient using linear regression
on Monte-Carlo simulations of (an ensemble of) control parameters. Applying
EnOpt for robust optimisation is costly unless the evaluations over the two
ensembles are combined, i.e. 'paired'. Here, we provide a new and more rigorous
perspective on the stochastic simplex approximate gradient (StoSAG) used in
EnOpt, explaining how it addresses detrimental cross-correlations arising from
pairing by only capturing the variability due to the control vector, and not
the vector of uncertain parameters. A few minor variants are derived from a
generalised derivation, as well as a new approach using decorrelation. These
variants are tested on linear and non-linear toy gradient estimation problems,
where they achieve highly similar accuracy, but require a very large ensemble
size to outperform the non-robust approach when accounting for variance and not
just bias. Other original contributions include a discussion of the particular
robust control objectives for which EnOpt is suited, illustrations, a variance
reduction perspective, and a discussion on the centring in covariance and
gradient estimation.
