Re-Examining Linear Embeddings for High-Dimensional Bayesian Optimization (2001.11659v2)

Abstract: Bayesian optimization (BO) is a popular approach to optimize expensive-to-evaluate black-box functions. A significant challenge in BO is to scale to high-dimensional parameter spaces while retaining sample efficiency. A solution considered in existing literature is to embed the high-dimensional space in a lower-dimensional manifold, often via a random linear embedding. In this paper, we identify several crucial issues and misconceptions about the use of linear embeddings for BO. We study the properties of linear embeddings from the literature and show that some of the design choices in current approaches adversely impact their performance. We show empirically that properly addressing these issues significantly improves the efficacy of linear embeddings for BO on a range of problems, including learning a gait policy for robot locomotion.

Overview of "Re-Examining Linear Embeddings for High-Dimensional Bayesian Optimization"

The paper "Re-Examining Linear Embeddings for High-Dimensional Bayesian Optimization" revisits the use of linear embedding techniques within the context of Bayesian optimization (BO) in high-dimensional parameter spaces. It addresses the challenge of maintaining sample efficiency while performing optimization with Bayesian methods on expensive-to-evaluate black-box functions, which becomes complex with increasing dimensionality. The authors critically evaluate existing techniques and introduce enhancements to overcome identified limitations.

At the core of this paper is the concept of embedding high-dimensional spaces into lower-dimensional manifolds. Existing techniques predominantly use random linear projections, such as the REMBO (Random EMbeddings for BO) framework, leveraging the Johnson-Lindenstrauss lemma for distance-preserving embeddings. However, the authors demonstrate that these random linear embeddings often produce suboptimal results due to certain inherent design limitations and misconceptions.

Key Contributions

1. Analysis of Existing Approaches: The paper identifies specific issues in the current usage of linear embeddings in BO. The previous approaches often result in embeddings that are inadequately represented by Gaussian processes (GP), or even worse, fail to incorporate an optimum within their bounds.
2. Enhanced Embedding Representation: They propose improved methods for creating representations using Mahalanobis kernels and polytope bounds on embeddings. These methods address non-stationarity and better preserve the true structure of the data, which traditional product kernels like the ARD kernel fail to achieve.
3. Adaptive Linear Embedding Method: They introduce a new method called ALEBO (Adaptive Linear Embedding BO) which constructs embeddings using tailored linear constraints, thereby avoiding nonlinear distortions which could arise from clipping projections. ALEBO also optimizes over a suitably defined objective function while incorporating probabilistic methods to ensure that the embeddings are more likely to contain optima.

Empirical Results and Discussion

The authors empirically validate the ALEBO method on a suite of benchmark problems commonly used in BO. They compare it against a broad array of methods, demonstrating ALEBO's superiority in terms of average performance on these tasks. Notably, ALEBO shows robust performance on linear subspace benchmark tasks where traditional methods would otherwise struggle.

From their experiments, it is revealed that ALEBO considerably reduces high-dimensional optimization issues discovered in REMBO and similar methods—particularly when addressing the allocation and maximization of optima within the embedding. The improvements are statistically significant across different datasets and dimensions, pointing towards ALEBO's effectiveness. For instance, on tasks involving learning policy for robot locomotion and constrained neural architecture search, ALEBO not only retains rigour in identifying optimal parameters but also enhances sample efficiency.

Theoretical and Practical Implications

Theoretically, this work advances the understanding of embedding strategies' impact on BO's efficacy. By introducing adaptive strategies such as Mahalanobis kernels and posterior sampling methods, the paper makes a significant theoretical contribution to maintaining modelability in BO embeddings.

Practically, the insights offer beneficial implications for fields where BO is extensively applied—spanning domains from automated machine learning to robotics—by allowing practitioners to efficiently exploit high-dimensional parameter spaces with reduced resource consumption. The paper's findings open pathways for further investigation into alternative embedding approaches, including nonlinear embeddings and data-driven dimensionality reduction, paving the way for enhanced efficiency in future high-dimensional optimization tasks.

In conclusion, the paper presents a thorough critique, adjustment, and empirical backing for adaptive linear embeddings in high-dimensional BO, showcasing potential for future research and industrial applications that rely heavily on BO methodologies.