Multiobjective Bayesian optimization for online accelerator tuning

Particle accelerators require constant tuning during operation to meet beam quality, total charge and particle energy requirements for use in a wide variety of physics, chemistry and biology experiments. Maximizing the performance of an accelerator facility often necessitates multiobjective optimization, where operators must balance trade-offs between multiple competing objectives simultaneously, often using limited, temporally expensive beam observations. Usually, accelerator optimization problems are solved off-line, prior to actual operation, with advanced beam line simulations and parallelized optimization methods (NSGA-II, swarm optimization). Unfortunately, it is not feasible to use these methods for online multiobjective optimization, since beam measurements can only be done in a serial fashion, and these optimization methods require a large number of measurements to converge to a useful solution. Here, we introduce a multiobjective Bayesian optimization scheme, which finds the full Pareto front of an accelerator optimization problem efficiently in a serialized manner and is thus a critical step towards practical online multiobjective optimization in accelerators. This method uses a set of Gaussian process surrogate models, along with a multiobjective acquisition function, to reduce the number of observations needed to converge by at least an order of magnitude over current methods. We demonstrate how this method can be modified to specifically solve optimization challenges posed by the tuning of accelerators. This includes the addition of optimization constraints, objective preferences and costs related to changing accelerator parameters.

Figure 1

Cartoon of multiobjective optimization where each objective is to be minimized. Multiobjective optimization attempts to find a set of points known as the Pareto front $\mathcal{P}$ that dominate over a reference point $\mathbf{r}$ and any other observed points in objective space. The Pareto front hypervolume $\mathcal{H}$ (shown in blue) is the axis-aligned volume enclosed by the Pareto front and a reference point $\mathbf{r}$. Making a new observation $\mathbf{y}$, which dominates over points in the current Pareto front, leads to an increase in hypervolume (shown in green), referred to as the hypervolume improvement ${\mathcal{H}}_I$.

Figure 2

Cartoons showing hypervolume improvement metrics used for MOBO. Blue regions denote the current Pareto front hypervolume defined by a reference point (upper right) and three observed points. Green regions denote the hypervolume improvement after adding an observation. (a) Each objective is modeled by an independent GP that predicts the function value and an uncertainty, which is used by the multiobjective acquisition function to determine the next observation. (b) Expected hypervolume improvement (EHVI) where the distribution of predicted objective values is given by a probability distribution (orange shading) centered at the black cross. (c) Upper confidence bound hypervolume improvement (UCB-HVI) where the hypervolume improvement is determined by an optimistic view of the predicted objective values. (d) Truncated UCB-HVI where we only consider hypervolume contributions within the unshaded subspace.

Figure 3

Optimization results after 15 function observations using the MOBO algorithm on the problem defined in Eq. (10). Functions plotted in (a)–(e) are normalized to the range 0 (blue) to 1 (yellow). (a),(b) Ground truth of $f_1$, $f_2$. The green line denotes the analytical location of Pareto optimal points. (c),(d) GP mean prediction for $f_1$, $f_2$ respectively. (e) UCB-HVI acquisition function with $\beta =0.01$. (f) Plot of observation values in objective space and the analytical Pareto front (green line). (c)–(f) Colored dots denote observation locations in input and output space, colored by observation number (blue to pink). Orange crosses denote the locations of five random, initial observation points.

Figure 4

(a) Probability map of satisfying the constraint $x_1\le 0.5$ after 50 observations, selected by constrained UCB-HVI, while optimizing Eq. (5). The red arrow points to where the constraint probability is most accurate, which is where the greatest number of observations are located. (b) Average fraction of points satisfying the constraint over 25 randomly initialized constrained MOBO optimization runs. Shading denotes one sigma spread.

Figure 5

Optimization trajectories in input space after 25 observations of objective functions in Eq. (5) using (a) the unmodified UCB-HVI acquisition function and (b) the P-UCB-HVI acquisition function. Insets: Distribution of distances ($L$) traveled per step in input space.

Figure 6

Cartoon of the AWA photoinjector and first linac cavity. Input and output parameters used in optimization are labeled. Reprinted with permission from A. Edelen et al., Phys. Rev. Accel. Beams 23, 044601 (2020). Copyright 2020, American Institute of Physics.

Figure 7

(a) Average Pareto front hypervolume $\mathcal{H}$ of ten multiobjective optimization runs of the AWA example using MOBO, NSGA-II and iterated neural network (I-NN) algorithms. Shading and error bars denote 1 sigma variance. (b) Projected hypervolume onto energy spread ($dE$) vs transverse emittance (${ϵ}_x$) subspace after 200 observations for each optimization algorithm shown in (a). (c) Projected hypervolume onto $dE$ vs ${ϵ}_x$ subspace after 100 (light blue), 200 (blue) and 500 (dark blue with orange outline) observations using the MOBO algorithm. Inset zoom (i) shows an increase in hypervolume due to increasing the Pareto front resolution, while inset zoom (ii) shows an increase in hypervolume due to finding new points that dominate old observations in projected space.

Figure 8

Plots showing energy spread ($dE$) and horizontal beam emittance (${ϵ}_x$) observations after 300 observations taken by MOBO algorithms. (a) MOBO with no constraints. (b) MOBO with an optimization preference of $dE<0.52\mathrm{MeV}$. (c) MOBO with an inequality constraint of $dE<0.52\mathrm{MeV}$. (d) Projected Pareto fronts for ${\sigma }_x$ vs ${ϵ}_x$ for each case above. The dotted lines in (b) and (c) denote the preference/constraint limit.

Figure 9

Comparison between UCB-HVI and the proximal UCB-HVI acquisition functions when used to perform optimization of the AWA photoinjector. (a) Solenoid 1 strength parameter over 300 observations using UCB-HVI (left) and corresponding distribution of $\mathrm{\Delta }{K}_1$ for each step (right). (b) Solenoid 1 strength parameter and travel distance distribution when P-UCB-HVI is used. (c) Average Pareto front hypervolume of ten optimization runs for UCB-HVI and P-UCB-HVI with identical random initialization sets. Shading denotes one sigma variance.