Accelerating Multi-Objective Bayesian Optimisation via Predictive-Gradient Catalysts

Technical Summary: Accelerating Multi-Objective Bayesian Optimisation via Predictive-Gradient Catalysts

Problem Statement
Multi-objective Bayesian Optimisation (MOBO) is a framework used to optimize expensive, noisy, or black-box objective functions by balancing exploration and exploitation through probabilistic surrogate models, typically Gaussian Processes (GPs). While effective, MOBO often faces challenges when evaluation budgets are limited, particularly in navigating the trade-offs required to approximate the Pareto set. Although gradient-based stationarity concepts are well-established in deterministic multi-objective optimisation, their integration into MOBO has remained largely unexplored. Existing gradient-based acceleration methods are often restricted to single-objective settings or tightly coupled to specific acquisition functions, limiting their general applicability.

Methodology
The authors propose a general acceleration mechanism that leverages Gaussian process predictive gradients as auxiliary "catalyst" signals. Rather than replacing existing Pareto-compliant acquisition functions, this framework augments them with local stationarity information derived from surrogate gradients.

The core methodology involves:

1. Surrogate Modeling: The approach assumes conditional independence between objectives, modeling each with an individual GP. The predictive mean ($\mu_i$) and variance ($\sigma^2_i$) are computed, and the expected gradient of the predictive mean is analytically derived.
2. Catalytic Framework: The method formulates a bi-objective decision problem to select the next evaluation point:
   • Global Criterion ($g_1$): Maximize a standard Pareto-compliant acquisition function $\alpha(x)$ (e.g., EHVI, AugTch).
   • Local Criterion ($g_2$): Minimize the magnitude of the weighted predictive gradient, $\|\gamma \cdot \nabla \hat{f}(x)\|$, where $\gamma$ represents convex combination coefficients satisfying the Fritz–John (FJ) conditions for Pareto stationarity.
   • Scalarization: These two criteria are combined using an augmented Tchebycheff scalarisation to create a unified acquisition function $\alpha_{CAT}(x)$. This penalizes candidates that are inferior in either global performance or local stationarity.

Two specific instantiations of the catalyst are investigated:

• MGDA-Based Catalyst: Uses the Multiple-Gradient Descent Algorithm (MGDA) to adaptively determine the weight vector $\gamma$ at each point $x$. This finds the convex combination of gradients with the minimum Euclidean norm, satisfying FJ conditions locally.
• Predefined-Weight (W) Catalyst: Utilizes a fixed set of predefined weight vectors (after min-max normalization) to discretize the search directions. This approach sacrifices strict FJ condition satisfaction for robustness and focused exploration, which is beneficial under tight evaluation budgets.

Key Contributions

• Catalytic Framework: A novel framework that exploits GP predictive gradients to accelerate MOBO convergence without altering the underlying structure of standard acquisition functions.
• Dual Instantiations: The introduction of two catalyst strategies: an adaptive MGDA-based approach grounded in Pareto stationarity theory, and a predefined-weight approach for focused trade-off exploration.
• Principled Integration: A formal method for integrating catalytic signals with standard acquisition functions via augmented Tchebycheff scalarisation.
• Empirical Evaluation: Extensive experiments on the DTLZ benchmark suite (2 objectives, 10 decision variables) comparing the proposed methods against baselines (EHVI, AugTch, tMPoI, SAF).

Results and Analysis
The experiments were conducted on the DTLZ1–DTLZ7 benchmark problems with a budget of $30n$ evaluations. Key findings include:

• Surrogate Accuracy Dependency: The effectiveness of the acceleration is highly dependent on the accuracy of the surrogate models. When surrogates are accurate (indicated by low Standardised Log Loss, SLL), the catalytic approach yields significant improvements.
• Stationarity Correlation: The framework performs best on problems exhibiting stationary behavior (where local perturbations induce similar variations in the objective space).
  • DTLZ2 and DTLZ5: These problems showed strong stationarity and high surrogate accuracy. The catalytic methods consistently outperformed baselines (green cells in significance maps).
  • DTLZ1 and DTLZ3: These problems exhibited non-stationarity and inconsistent surrogate quality. The catalytic approach showed little to no improvement over baselines (grey cells).
  • DTLZ4 and DTLZ6: Mixed or poor surrogate quality led to performance degradation in several cases (red cells), highlighting the risk of using gradients when surrogates are unreliable.
• MGDA vs. Predefined Weights: Across pairwise comparisons, the Predefined-Weight (W) variant generally outperformed the adaptive MGDA variant. The authors suggest that for expensive optimisation with limited budgets, focusing the search on a coarse, finite set of promising directions is more effective than attempting to finely approximate the entire global Pareto set via adaptive local directions.

Significance and Claims
The paper claims that predictive-gradient catalysis offers a practical mechanism to improve efficiency in expensive multi-objective optimisation, provided the underlying surrogate models are accurate and the problem exhibits approximate stationarity. The authors emphasize that this approach does not replace existing acquisition functions but enhances them with local information, enabling faster convergence toward the global Pareto set.

The significance lies in bridging the gap between gradient-based stationarity theory and MOBO, demonstrating that even with limited budgets, leveraging the smoothness of GP predictive gradients can guide the search more effectively than standard acquisition functions alone. The authors modestly note that while the current evaluation is limited to the DTLZ suite with 2 objectives and 10 dimensions, the framework is designed to scale to higher dimensions due to the analytical derivation of gradients and the computational efficiency of the MGDA solver. Future work is suggested for constrained settings, noisy environments, and high-dimensional problems.