Are Random Decompositions all we need in High Dimensional Bayesian Optimisation? (2301.12844v2)

Abstract: Learning decompositions of expensive-to-evaluate black-box functions promises to scale Bayesian optimisation (BO) to high-dimensional problems. However, the success of these techniques depends on finding proper decompositions that accurately represent the black-box. While previous works learn those decompositions based on data, we investigate data-independent decomposition sampling rules in this paper. We find that data-driven learners of decompositions can be easily misled towards local decompositions that do not hold globally across the search space. Then, we formally show that a random tree-based decomposition sampler exhibits favourable theoretical guarantees that effectively trade off maximal information gain and functional mismatch between the actual black-box and its surrogate as provided by the decomposition. Those results motivate the development of the random decomposition upper-confidence bound algorithm (RDUCB) that is straightforward to implement - (almost) plug-and-play - and, surprisingly, yields significant empirical gains compared to the previous state-of-the-art on a comprehensive set of benchmarks. We also confirm the plug-and-play nature of our modelling component by integrating our method with HEBO, showing improved practical gains in the highest dimensional tasks from Bayesmark.

• The paper introduces RDUCB, a random tree-based strategy that balances maximal information gain with minimized functional mismatch through rigorous theoretical guarantees.
• The paper presents RDUCB as a simple, plug-and-play algorithm that integrates easily with existing Bayesian optimization frameworks while outperforming state-of-the-art methods.
• Empirical benchmarks confirm that RDUCB scales exceptionally well in high-dimensional tasks, notably enhancing performance when used with frameworks like HEBO.

Overview of Random Decompositions in High Dimensional Bayesian Optimization

This essay provides an expert analysis of the paper titled "Are Random Decompositions all we need in High Dimensional Bayesian Optimisation?" which investigates the potential for using random decompositions in scaling Bayesian optimization (BO) to high-dimensional problems. Traditionally, BO methods have struggled in high-dimensional settings, primarily due to the challenge of learning effective decompositions of expensive-to-evaluate black-box functions. The paper explores a departure from data-driven decomposition learning to data-independent rules, demonstrated to be theoretically sound and empirically successful.

Key Findings

The authors propose a novel approach leveraging random tree-based decomposition samplers, termed RDUCB (Random Decomposition Upper-Confidence Bound). The core idea is that random decompositions avoid the pitfalls of data-driven models that may become biased by local information, failing to generalize globally.

1. Theoretical Guarantees: The paper provides rigorous theoretical guarantees, specifically focusing on the balance between maximizing information gain and minimizing functional mismatch. Through a series of theoretical propositions and theorems, it demonstrates that random decomposition sampling strategies bound the maximal information gain favorably, keeping the complexity in check.
2. Algorithmic Simplicity: RDUCB is presented as an implementation-friendly algorithm requiring minimal adjustments to existing BO frameworks. The method is essentially plug-and-play, demonstrating significant improvements over state-of-the-art techniques without requiring intricate modelling.
3. Empirical Validation: Benchmark experiments solidify the paper's contributions, with RDUCB achieving superior empirical performance across a comprehensive set of high-dimensional tasks, particularly excelling as the dimensionality increases. The integration of this method with HEBO—an existing BO framework—shows notable improvements on the highest dimensional tasks from the Bayesmark problem suite.

Implications and Future Directions

Practical Implications: The simplicity and effectiveness of RDUCB suggest a wide range of practical applications. In real-world scenarios requiring the optimization of complex systems with numerous interdependencies, this method provides a robust, scalable solution.

Theoretical Implications: The shift from data-driven decomposition learning to data-independent strategies invites further exploration into other facets of BO where such methodologies might be advantageous. The promising theoretical guarantees of RDUCB open avenues for extending this approach to other domains of machine learning optimization.

Future Developments: The paper points toward several directions for future research. Among them, handling non-numerical or structured inputs like graphs and sequences poses a natural extension. Additionally, as the methodology scales well with dimensionality, investigations into distributed or parallel implementations could yield further performance enhancements.

Conclusion

The paper makes a significant contribution to the field of high-dimensional Bayesian optimization by introducing and validating a random decomposition strategy. RDUCB stands out for its theoretical soundness, practical relevance, and empirical strength, offering a new avenue for tackling high-dimensional optimization challenges efficiently. The work paves the way for further research into data-independent approaches within AI and machine learning, potentially influencing a broad spectrum of optimization problems.