Chimera: enabling hierarchy based multi-objective optimization for self-driving laboratories

Chimera enables multi-target optimization for experimentation or expensive computations, where evaluations are the limiting factor.


S.1.1. Benchmark functions
The influence of the smoothing parameter τ on the shape of the achievement scalarizing function has been demonstrated on a set of three one-dimensional objectives, presented in Eq. 1.
Note, that all objectives were considered on the x ∈ [− 1,5]. The three objectives are also shown in Fig. 1 in the main text (see Sec. 3.1).
(1)      Chimera is constructed such that it is sensitive only to a single objective in any region of the parameter space, or mostly influenced by a single objective if the Heaviside function is replaced with the logistic function. For two particular objectives, f i and f j , we refer to R i as the region where Chimera is mostly sensitive to f i , and to R j the region where Chimera is mostly sensitive to f j . Furthermore, we denote with {x * } the set of points transitioning from R i to R j .
Without loss of generality we assume a hierarchy where f i is assigned a higher importance than f j . where f i monotonically increases along a curve directed towards x * . Based on our assumption, f j is monotonically decreasing along the same curve. Therefore, Chimera is also monotonic along 6 this curve.
Next, we consider the implication of f i and f j not competing with each other in proximity to a transition point x * if Chimera is monotonic along curves passing through x * . Again, Chimera assumes larger values in R i than in R j by construction. We define a curve going from R i to R j while passing through x * . By assumption, Chimera is monotonically decreasing along this curve. Since Chimera is dominated by f i in R i and dominated by f j in R j , both f i and f j are monotonically decreasing along this curve and thus do not compete.
This behavior of Chimera can be exploited when analyzing relations between different objectives. In particular, this behavior provides a qualitative tool to identify locally competing or noncompeting objectives. Analyzing the parameter region in which objectives are locally competing might reveal insights into fundamental underpinnings of the competition. Overall, we observe that Chimera enables the studied optimization algorithms to get closer to the Pareto optimal values faster from the very beginning of the optimization procedure. The fact, that tolerances are defined with respect to the current observed minima and maxima of the objectives therefore does not seem to significantly delay the optimization procedure.

S.1.5.2. Training a Bayesian neural network
We construct a test set by randomly sampling 10 % of all points in the dataset. From the remaining 90 % of the dataset, we select the most diverse 80 % for the training set based on principal component analysis (PCA) analysis following a procedure reported in the literature. [3] The remaining 10 % of the dataset are used as a validation set for early stopping.

S.1.6. Optimizations on periodic domains
The inverse-design problem of finding excitation energy transfer systems discussed in the main text (see Sec. 5.2) involves a total of ten independent parameters. Four of these parameters describe the orientation of transition dipoles with respect to a principal axis, expressed in terms of an angle ϕ ∈ [0, 2π]. The orientation of the transition dipoles is periodic, which imposes a constraint on the response surface of objectives with respect to these parameters. This constraint can be taken into account when constructing approximations to response surfaces during the optimization procedure. Indeed, by accounting for this periodicity constraint, a more accurate approximation to the response surface can be found, which has the potential to determine the location of the global minimum in fewer optimization iterations.
In this section, we demonstrate how Phoenics can be expanded to account for periodic boundary conditions on the parameter domain. Phoenics constructs approximations to an objective function by estimating the kernel density of observed parameter points and reweighting those by the corresponding observed objective function value. [6] Kernel densities p k are estimated via a Bayesian neural network as shown in Eq. 5. The Bayesian neural network is used to sample random variables φ 3 in the parameter domain based on previously observed parameter points x k . Ref. [6] provides a detailed description of the construction of kernel densities Importantly, the construction of the kernel densities p k at an arbitrary point x ∈ R d in the parameter domain depends on the distance d(x, φ 3 (θ; x k )) = x − φ 3 (θ; x k ) between this parameter point x and the random variable φ 3 (θ; x k ) sampled from the Bayesian neural network. We now consider a scenario where the objective f is periodic with periodicity P , i.e. f (x) = f (x + P ) for all x ∈ R d . A periodicity constraint on the parameter domain can be formulated by replacing Computing the periodic distance from all periodic images of the kernel density is computationally costly. As a compromise between the computational demand of the approach and accuracy of the periodicity constraint, we only account for nearest periodic images and neglect higher order periodic images. This approximation becomes more accurate with more optimization iterations, as the precision τ n increases.
We illustrate the construction of periodic objective function approximations one a onedimensional example. The considered objective function f consists of the product of two cosine functions, as shown in Eq. 6, with a period of P = 1. Phoenics was used to determine the location of the global minimum of this function within the x ∈ [0, 1] interval by constructing the approximation with and without periodicity support. Note, that the global minimum is located at x * = 0.05.
f (x) = − cos ((π(x − 0.05)) cos (3π(x − 0.05)) Fig. S.8 shows the approximations constructed to the objective function after two, five and eight optimization iterations with and without periodicity support. We find that the optimization run without periodicity support tend to sample the objective function at large values of x. Only after a few optimization iterations, the location of the global minimum at small values of x is discovered.
In contrast, the optimization procedure supporting periodicity in the objective function discovers the location of the global minimum within much fewer iterations, and needs fewer observations to construct reasonable approximations to the objective function.