Optimization

Parallel Evaluations Performing evaluations in parallel using multiple computing resources allow obtaining multiple function evaluations in the time that would ordinarily be required to obtain just one with sequential evaluations. For this reason, parallel function evaluations is a conceptually appealing way to solve optimization problems in less time. EI, KG, ES, and PES can all be extended in a straightforward way to allow parallel function evaluations. For example, EI becomes

^EIn^{(x(1:q)) = E}n ^{[ max}
i=1,...,q
f (x⁽ⁱ⁾) − f_n^∗]⁺ , (14)

where x^(1:q) = (x⁽¹⁾, . . . , x^(q)) is a collection of points at which we are proposing to evaluate (Ginsbourger et al., 2007). Parallel EI (also called multipoints EI by Ginsbourger et al. (2007)) then proposes to evaluate the set of points that jointly maximize this criteria. This approach can also be used asynchronously, where we hold fixed those x⁽ⁱ⁾ currently being evaluated and we allocate our idle computational resources by optimizing over their corresponding x^(j).

Parallel EI (14) and other parallel acquisition functions are more challenging to optimize than their original sequential versions from Section 4. One innovation is the Constant Liar approximation to the parallel EI acquisition function (Ginsbourger et al., 2010), which chooses x⁽ⁱ⁾ sequentially by assuming that f (x^(j)) for j < i have been already observed, and have values equal to a constant (usually the expected value of f (x^(j))) under the posterior. This substantially speeds up computation. Expanding on this, Wang et al. (2016a) showed that infinitesimal perturbation analysis can produce random stochastic ^{gradients that are unbiased estimates of ∇EI}n^{(x(1:q)), which can then be used in multistart stochastic} gradient ascent to optimize (14). This method has been used to implement the parallel EI procedure for as many as q = 128 parallel evaluations. Computational methods for parallel KG were developed by Wu and Frazier (2016), and are implemented in the Cornell MOE software package discussed in Section 6. That article follows the stochastic gradient ascent approach described above in Section 4.2, which generalizes well to the parallel setting.