BoTier: Multi-Objective Bayesian Optimization with Tiered Composite Objectives

TL;DR

BoTier addresses hierarchical multi-objective optimization in scientific experimentation by introducing a differentiable composite objective $\\Xi$ that encodes tiered preferences over both inputs and outputs. The core idea is to compute $\\Xi = \\sum_{i=1}^N ( \\min(\\psi_i, t_i) \\cdot \\prod_{j=1}^{i-1} H(\\psi_j - t_j) )$, using smooth approximations for $\\min$ and $\\,H$ and Monte-Carlo evaluation to enable gradient-based optimization within BoTorch. In benchmarks on analytical surfaces and emulated chemistry problems, BoTier outperformed or matched alternatives such as Chimera and EHVI, with faster convergence to the hierarchical optimum, especially when used as a composite objective. The work provides an open-source, auto-differentiable toolkit that facilitates integration into self-driving laboratories and complex experimental planning.

Abstract

Scientific optimization problems are usually concerned with balancing multiple competing objectives, which come as preferences over both the outcomes of an experiment (e.g. maximize the reaction yield) and the corresponding input parameters (e.g. minimize the use of an expensive reagent). Typically, practical and economic considerations define a hierarchy over these objectives, which must be reflected in algorithms for sample-efficient experiment planning. Herein, we introduce BoTier, a composite objective that can flexibly represent a hierarchy of preferences over both experiment outcomes and input parameters. We provide systematic benchmarks on synthetic and real-life surfaces, demonstrating the robust applicability of BoTier across a number of use cases. Importantly, BoTier is implemented in an auto-differentiable fashion, enabling seamless integration with the BoTorch library, thereby facilitating adoption by the scientific community.

Figures (29)

• Figure 1: MOO with preferences over experiment inputs and outputs. A) Example from chemical reaction optimization. B) Pareto front for two competing objectives. C) Workflow of BO with multi-objective scalarization in a black-box (left) and composite manner (right, this work).
• Figure 2: Benchmarks of different MOO strategies on four analytical surfaces, each extended by an input-dependent objective (see SI for further details). Top panel: Best observed value of $\Xi$ as a function of the number of experimental evaluations. All statistics were calculated on 50 independent campaigns on each surface. Intervals are plotted as the standard error. Bottom panel: Number of experiments required to satisfy the first objective ($n=1$, green); the first two objectives ($n=2$, dark green); or all three objectives ($n=3$, blue).
• Figure 3: Evaluation of different MOO strategies for chemical reaction optimization. A) Optimization performance on different emulated reaction optimization problems. Number of experiments required to satisfy the first $n$ objectives. B) Case study of a Suzuki-Miyaura coupling. Plots show the trajectories of the respective objective values at the best experimental data point observed so far. See SI for further details.
• Figure S1: Objective density plots of the analytical BNH surface, as described in Tab. \ref{['tab:analytical_functions']} and implemented in BoTorch.Balandat2020 The function was evaluated on a grid of $5\cdot10^6$ points drawn from a Sobol sampler. Data points that satisfy all objectives are shown in red.
• Figure S2: Objective density plots of the modified analytical BNH surface with an additional input-dependent objective, as described in Tab. \ref{['tab:analytical_functions']}. The function was evaluated on a grid of $5\cdot10^6$ points drawn from a Sobol sampler. Data points that satisfy all objectives are shown in red.