JAREX: An Acquisition Function for Multi-Objective Algorithmic Process Characterization

AI summary

Overview

Research area: Bayesian optimization and active learning applied to pharmaceutical process characterization under the Quality-by-Design (QbD) paradigm, specifically multi-objective boundary (level set) estimation.

Technical level: Intermediate. The paper assumes familiarity with Gaussian process surrogate models, acquisition functions (UCB, Expected Improvement), and level set estimation, but explains each component it builds on.

Scope: The paper introduces JAREX, a Bayesian active-learning acquisition function that adaptively selects experiments to recover the joint pass region defined by simultaneous satisfaction of multiple quality thresholds, and benchmarks it against factorial DOE, space-filling designs, and greedy objective-wise strategies on simulated test problems and a four-dimensional kinetic model.

What This Paper Is About

Pharmaceutical process characterization aims to map where a manufacturing process meets its quality specifications and where it crosses the "edge of failure," but in practice it relies on factorial design of experiments, which samples the parameter space non-adaptively and scales poorly into higher dimensions. Most real processes involve several critical quality attributes at once, so the quantity that matters is the joint pass region — the set of conditions that satisfy every specification simultaneously — yet no acquisition function previously existed for this multi-objective characterization problem. The authors build one, targeting the joint boundary rather than any single optimum.

Key Contributions

1. JAREX (Joint Acceptable Region EXploration), a new acquisition function for multi-objective process characterization. It formulates characterization as a joint boundary-learning problem over the intersection of per-objective pass regions, rather than as optimization or as independent per-objective boundary learning.

2. An optimistic joint-feasibility mask with a differentiable relaxation. Each objective's pass region is widened using its Upper Confidence Bound, the joint optimistic pass region is their intersection, and a sech-based soft mask (with k = 1 by default) keeps the masking differentiable so gradient-based multi-start candidate optimization still works.

3. A multi-objective extension of randomized straddle using softmin-weighted aggregation. Per-objective randomized straddle scores are combined with a softmin-weighted sum (temperature τ = 0.5 by default), which rewards points where several objectives are simultaneously close to their thresholds — an AND-aggregation matching the intersection structure of the joint pass region.

4. Open-source implementation plus a benchmark study. The method is released in the obsidian package, built on BoTorch and PyTorch, with the JAREX object inheriting from BoTorch's MCAcquisitionFunction. The paper also reports that no formal convergence guarantee is claimed for the multi-objective case, and identifies in the Supplementary Information exactly which step of the single-objective proof fails.

Main Findings

• JAREX outperforms the baselines on the joint characterization task. In the benchmark study it gave more accurate and sample-efficient recovery of the joint pass region than factorial DOE, space-filling designs, and greedy objective-wise strategies across the full experimental budget range.

• Batching cuts experiment count substantially. For batched experimentation, JAREX reduced the number of iterative process characterization experiments by more than half while preserving high accuracy for the boundary-identification task. Batch sizes compared were serial (q = 1), moderate (q = 4), and high (q = 8).

• Rapid convergence within a modest number of iterations. On the four-dimensional, three-objective kinetic benchmark with 5% Gaussian noise, the joint Jaccard index rose rapidly within the first 30 to 40 iterations and continued to improve smoothly afterward with tighter error bars across the 10 campaigns.

• Boundary-focused sampling behaves differently from optimization. In the one-dimensional double-well illustration, randomized straddle selected high-uncertainty points early (step 5), then moved to the threshold region as uncertainty decreased (step 8), and by step 15 continued sampling at the threshold, causing the Jaccard index to plateau. It ignored regions far from the boundary even where the surrogate mean deviated from the truth.

• UCB is a poor characterization strategy beyond low dimensions. UCB, designed for optimization, performed poorly beyond 2D even with high exploration (β^{1/2} = 6), confirming that boundary exploration needs different acquisition strategies than optimum-seeking.

• Uninformed designs degrade with dimensionality. The response surface methodology (RSM) designs — a full two-level factorial and a central composite design (CCD) — and space-filling sampling were competitive in low-dimensional settings but fell off sharply as dimensionality and complexity increased; on the kinetic benchmark the RSM designs performed poorly and space-filling plateaued far below JAREX.

• Greedy multi-objective straddle is unstable. The greedy strategy — cycling through objectives one at a time — was highly sensitive to objective ordering, producing erratic convergence histories. The authors describe this circular dependency as fundamental: choosing which objective to prioritize requires prior knowledge of the response surface, but acquiring that knowledge is the point of characterization. Greedy methods also refine each objective boundary separately without accounting for their interplay.

• Randomized straddle is the strongest single-objective characterization method tested. It was comparable to feasibility EI in simple cases and showed superior convergence speed and numerical stability in 6D, substantially outperforming all other approaches there.

• Traditional characterization outputs are recoverable. Proven acceptable ranges (PARs) and multi-factor interaction plots can be recovered from the fitted surrogate via post-processing.

Methodology in Plain English

The setup is a parameter space of d process parameters and m quality attributes, each with a threshold h_i encoding the pass/fail specification. A point "passes" objective i if O_i(x) ≥ h_i. The target is the joint pass region: the intersection of all individual pass regions, whose boundary is the edge of failure.

Existing theory offers two framings. The feasibility framing asks, at each candidate point, how likely the objective is to pass and samples where that likelihood is most informative; the authors cite feasibility EI as an example. The boundary framing treats the surface where O_i(x) = h_i as the object to recover — the level set estimation problem — and the authors build on the straddle family within it.

The method assembles three pieces. First, an optimistic view of each pass region: instead of applying the threshold to the predicted mean alone, it is applied to the Upper Confidence Bound μ_i(x) + β_i^{1/2} σ_i(x), so that uncertain points near the boundary are not excluded prematurely. The intersection of these per-objective optimistic regions is the joint optimistic pass region, which serves as a mask: only candidates inside it are eligible.

Second, within that region, each point gets a per-objective randomized straddle score, max[-|μ_i(x) − h_i| + β_i^{1/2} σ_i(x), 0], where β is redrawn each iteration from a chi-squared distribution with two degrees of freedom rather than fixed at the conventional β^{1/2} = 1.96. These scores are combined into a single acquisition value using a softmin-weighted sum with temperature τ = 0.5, so that points informative for several objectives at once are favored over points that matter for only one.

Third, because the joint optimistic pass region is irregular and generally non-convex and a hard indicator mask would create a flat-zero cliff that breaks gradient-based multi-start search, the mask is relaxed. The authors define d(x) = min_i [UCB_i(x) − h_i], which is non-negative exactly inside the joint optimistic region, and set the mask to 1 when d ≥ 0 and to sech(k d) when d < 0, with k = 1 by default. The next experiment maximizes mask times acquisition value; the surrogate is updated and the loop repeats.

Evaluation uses the Jaccard index between predicted and true pass regions, a set-based metric focused on boundary classification rather than the magnitude of prediction error. Baselines are RSM designs (full two-level factorial, CCD) and space-filling sampling. Each test function was run with 10 independent campaigns using fixed random seeds, all methods sharing the same Latin hypercube sampling initial observations. Gaussian process surrogates with Matérn kernels were used throughout, with hyperparameters tuned by multi-start optimization, and Jaccard indices were computed on Sobol-discretized parameter spaces.

The multi-objective benchmark is a four-dimensional simulated reaction kinetic model with a main catalytic pathway producing desired product P and a competing reversible side reaction forming impurity X. The four inputs are initial reactant concentration [Z_0] in [0.8, 3.0] M, initial co-reactant concentration [Y_0] in [0.8, 6.0] M, catalyst loading [Cat_0] in [0.005, 0.2] M, and temperature T in [273, 333] K. The three objectives are Z conversion ≥ 90%, purity ≥ 99.3% (molar ratio [P]/([P]+[X]) at the final time point), and cost per unit product ≤ $570.

Why This Matters

The work moves process characterization from static, non-adaptive experimental grids toward adaptive sampling that spends experiments where the pass/fail decision actually changes. Because the method targets the joint boundary across multiple quality attributes rather than treating each attribute in isolation, it addresses a gap the authors state was previously unfilled: no acquisition function existed for multi-objective process characterization.

Real-world applications:

• Pharmaceutical design space and proven acceptable range determination. A converged estimate of the joint pass region bounds the largest region consistent with all specifications, giving a principled basis for choosing a conservative, regulator-facing design space inside it.

• Reaction and process development with competing objectives. The benchmark model involves coupled trade-offs among conversion, purity, and cost per unit product, mirroring real process development where high conversion requires high reagent loadings while high purity requires suppressing side reactions that reduce conversion.

• Reducing experimental cost in costly or slow measurement settings. Because JAREX reduces the number of iterative experiments by more than half in batched mode while maintaining accuracy, it is relevant where each run is expensive, slow, or material-limited.

• Establishing multi-factor interactions and operating windows. Multi-factor interaction plots and PARs are recovered from the same fitted surrogate, so a single adaptive campaign can feed conventional characterization outputs.

Industry relevance: The work comes from Merck & Co., Inc. (Pharmaceutical Analysis & Digital Technologies, Rahway, New Jersey), and is framed explicitly around ICH Q8(R2) concepts — the edge of failure and the design space. It is released in an open-source package built on widely used tooling (BoTorch, PyTorch), which lowers the barrier to adoption in industrial process development.

Future Directions

• A formal convergence guarantee for the joint problem. The softmin combination and the soft mask break a step in the convergence proof of the single-objective randomized straddle method. Only a much weaker statement can be recovered, and only under an assumption — that every objective is queried sufficiently often — that contradicts JAREX's deliberate deprioritization of regions confidently outside the joint optimistic pass region. The authors defer a full analysis.

• Understanding the trade-off between theoretical guarantees and empirical performance. The paper frames the missing bound as a trade-off judged acceptable based on empirical results, leaving open whether a guarantee can be obtained without abandoning the joint-boundary focus.

• Extending beyond the tested dimensionality and setting. The reported benchmarks cover noise-free analytical test functions up to 6D for the single-objective case and a 4D, three-objective kinetic model with 5% Gaussian noise. How the method scales to higher-dimensional and more objective-rich industrial problems is not established in the content available.

• Design space selection under uncertainty. The paper notes that any estimate of the joint pass region from finite data carries residual model uncertainty and that a real design space is deliberately chosen as a conservative subset strictly inside it. How to set those margins given the surrogate's uncertainty is left as a practical question rather than resolved.

Target Audience

Pharmaceutical and chemical process engineers and statisticians who design characterization and design-of-experiments campaigns and need to justify design spaces and proven acceptable ranges. Also relevant to method developers working on Bayesian optimization, active learning, and level set estimation who are interested in multi-objective boundary-recovery problems and in the trade-offs involved in relaxing theoretical guarantees for practical acquisition functions. Readers should be comfortable with Gaussian process surrogates and standard acquisition functions; the paper is approachable for practitioners without a deep theoretical background, since no formal convergence claim is made.

Authors’ abstract

Pharmaceutical process characterization is central to Quality by Design because it defines how variations in process parameters affect the ability to meet product quality specifications, thereby supporting proven acceptable ranges and robust manufacturing. In practice, however, characterization still relies largely on factorial design of experiments (DOE) approaches, which are inefficient for resolving multivariate pass/fail boundaries in higher-dimensional spaces. While Bayesian optimization has transformed process optimization, adaptive methods for multi-objective process characterization remain lacking. Here, we introduce JAREX (Joint Acceptable Region EXploration), a Bayesian active-learning acquisition function for multi-objective process characterization. JAREX formulates characterization as a joint boundary-learning problem and adaptively selects experiments to recover the joint pass region defined by simultaneous satisfaction of threshold criteria across multiple objectives. JAREX combines an optimistic joint-feasibility mask with a multi-objective extension of randomized straddle, focusing sampling on the joint edge of failure. Our benchmark study suggests that JAREX provides more accurate and sample-efficient recovery of the joint pass region than factorial DOE, space-filling designs, and greedy objective-wise strategies over the full experimental budget range. For batched experimentation, it reduces the number of iterative process characterization experiments by more than half while preserving high accuracy for the boundary-identification task. Implemented in the open-source obsidian package, JAREX provides a modular framework for adaptive, data-efficient multi-objective algorithmic process characterization, supporting sample-efficient range finding in high-dimensional spaces.