Leveraging Trust for Joint Multi-Objective and Multi-Fidelity Optimization

TL;DR

This work tackles expensive black-box optimization with multiple competing objectives and heterogeneous fidelity sources. It introduces trust-based MOMF Bayesian optimization, unifying input and fidelity decisions by pairing the objective with a trust metric and optimizing via Pareto-based acquisition (EHVI) while penalizing cost. Two strategies are proposed: a holistic, joint Trust-MOMF method and a sequential MOMF benchmark, both demonstrated on synthetic MF MO functions and a high-cost laser-plasma PIC application. Results indicate substantial cost savings (often an order of magnitude) and faster convergence to the Pareto front, with the joint approach consistently outperforming the sequential one. The methods are designed for straightforward integration into existing Bayesian frameworks (e.g., BoTorch) and show strong potential for complex simulations in physics and engineering.

Abstract

In the pursuit of efficient optimization of expensive-to-evaluate systems, this paper investigates a novel approach to Bayesian multi-objective and multi-fidelity (MOMF) optimization. Traditional optimization methods, while effective, often encounter prohibitively high costs in multi-dimensional optimizations of one or more objectives. Multi-fidelity approaches offer potential remedies by utilizing multiple, less costly information sources, such as low-resolution simulations. However, integrating these two strategies presents a significant challenge. We suggest the innovative use of a trust metric to support simultaneous optimization of multiple objectives and data sources. Our method modifies a multi-objective optimization policy to incorporate the trust gain per evaluation cost as one objective in a Pareto optimization problem, enabling simultaneous MOMF at lower costs. We present and compare two MOMF optimization methods: a holistic approach selecting both the input parameters and the trust parameter jointly, and a sequential approach for benchmarking. Through benchmarks on synthetic test functions, our approach is shown to yield significant cost reductions - up to an order of magnitude compared to pure multi-objective optimization. Furthermore, we find that joint optimization of the trust and objective domains outperforms addressing them in sequential manner. We validate our results using the use case of optimizing laser-plasma acceleration simulations, demonstrating our method's potential in Pareto optimization of high-cost black-box functions. Implementing these methods in existing Bayesian frameworks is simple, and they can be readily extended to batch optimization. With their capability to handle various continuous or discrete fidelity dimensions, our techniques offer broad applicability in solving simulation problems in fields such as plasma physics and fluid dynamics.

Figures (7)

• Figure 1: Consecutive iterations of Bayesian optimization (maximization) of the Forrester test function. Top: The dashed black line represents the true function that is being optimized. The blue line is the GP regression mean with a shaded region around it representing the standard deviation. The blue dots are the points that were evaluated. Arrows indicate the next measurement point suggested by the respective acquisition functions from the bottom plot, showing the different prioritization towards exploration and exploitation. Bottom: Evaluations of three different metrics, expected improvement (EI, Eq. \ref{['EI']}), max-value entropy search (MES, Eq.\ref{['MES']}) and upper confidence bound (UCB, Eq. \ref{['UCB']} with $\kappa=2$) for each step in the upper plot.
• Figure 2: Pareto front. Illustration how a multi-objective function $\bm f(\bm x)=\bm y$ acts on a two-dimensional input space $\bm x = (x_1,x_2)$ and transforms it to the objective space $\bm y = (y_1,y_2)$ on the right. The entirety of possible input positions is uniquely color-coded on the left and the resulting position in the objective space is shown in the same color on the right. The Pareto front is the ensemble of points that dominate others, meaning points that give the highest combination of $y_1$ and $y_2$. The corresponding set of coordinates in the input space is called the Pareto set. Note that both the Pareto front and the Pareto set may be continuously defined locally, but can also contain discontinuities when local maxima get involved. In this example, $f$ is a modified version of the Branin-Currin function from dixon1978currin1991 that exhibits a single, global maximum in $y_2$ but multiple local maxima in $y_1$, see also illustration in the center.
• Figure 3: Multi-fidelity optimization via hypervolume improvement of a modified Forrester function. Left: Mean and variance (shaded curve) of the fitted Gaussian process after 40 iterations of the optimizer. The true values are indicated as dashed lines. Note the small variance close to the maximum at all fidelity values. Center: Map of the objective values $f_1(x,s)$ and sampled points colored according to the iteration number. This illustrates how the optimizer first explores at low fidelity and then moves along the Pareto set. Right: Same optimization in the objective space, showing how the optimizer tries to increase the hypervolume, which is simply the area in 2D, below the Pareto front.
• Figure 4: Benchmark with 2-D-Branin-Currin problem. Top Left: Hypervolume of a single representative trial expressed as a percentage of the total hypervolume versus total cost for both MOMF versions and single-fidelity MO. Top Right: The number of points taken at different fidelites for the representative trials shown on the top left. The Trust-MOMF takes more iterations at intermediate fidelities when compared with sequential MOMF which has relatively high peaks at fidelity 0 and 1. Bottom Left: The Pareto front for each of the three algorithms for the same trial at a cost of 500 (indicated by the dashed line in the figure on the top left). The dashed black line represents an estimated Pareto front calculated from 10000 random points. The Trust-MOMF has already found values in the trade-off region and close to the individual maxima of the Branin and Currin functions. The sequential MOMF is approaching the maximum of the Branin function but both sequential MOMF and MO algorithms have yet to explore the trade-off points. Bottom Right: The Pareto front for a cost of 1000 cost for a single representative trial, clearly showing how the Trust-MOMF has reached an accurate representation of the estimated Pareto front while the conventional, single-fidelity MO algorithm has only discovered a small region of the Pareto front at this cost.
• Figure 5: Benchmark with 4-D-Park$\bm{_{1,2}}$ problem. Top Left: Hypervolume of a single representative trial expressed as a percentage of the total hypervolume versus total cost for both MOMF versions and single-fidelity MO. Top Right: The number of points taken at different fidelites for the representative trials shown on the left. The Trust-MOMF in this case has a higher number of points taken at the highest fidelity. This is because once it has converged it takes 6 points at the highest fidelity to increase hypervolume. The sequential MOMF as seen in Branin-Currin case takes less intermediate fidelity points when compared to the Trust-MOMF. Bottom Left: The Pareto front for each of the three algorithms for the same trial at a cost of 750 (indicated by dashed line in the figure on the top left). The area represents the amount of Pareto front covered by each algorithm. The Trust-MOMF has already converged to almost $95\%$ of the total hypervolume. The sequential MOMF also converged but to a lower overall hypervolume. The MO optimization at this cost has only found maximum of Park 2 function.Bottom Right: The Pareto front for a cost of 1500 cost showing little changes in both MOMF Pareto fronts, but a better coverage for the MO Pareto front. At this cost the MO optimization still has not reached the hypervolume that the Trust-MOMF reached at a cost of 750.

Theorems & Definitions (1)

• Definition 1