Nonlinear Dimensionality Reduction Techniques for Bayesian Optimization

TL;DR

This is the first study to combine SDR with VAE-based LSBO, and the analysis clarifies design choices for metric shaping and retraining that are critical for scalable latent space BO.

Abstract

Bayesian optimisation (BO) is a standard approach for sample-efficient global optimisation of expensive black-box functions, yet its scalability to high dimensions remains challenging. Here, we investigate nonlinear dimensionality reduction techniques that reduce the problem to a sequence of low-dimensional Latent-Space BO (LSBO). While early LSBO methods used (linear) random projections (Wang et al., 2013), building on Grosnit et al. (2021), we employ Variational Autoencoders (VAEs) for LSBO, focusing on deep metric loss for structured latent manifolds and VAE retraining to adapt the encoder-decoder to newly sampled regions. We propose some changes in their implementation, originally designed for tasks such as molecule generation, and reformulate the algorithm for broader optimisation purposes. We then couple LSBO with Sequential Domain Reduction (SDR) directly in the latent space (SDR-LSBO), yielding an algorithm that narrows the latent search domains as evidence accumulates. Implemented in a GPU-accelerated BoTorch stack with Matern-5/2 Gaussian process surrogates, our numerical results show improved optimisation quality across benchmark tasks and that structured latent manifolds improve BO performance. Additionally, we compare random embeddings and VAEs as two mechanisms for dimensionality reduction, showing that the latter outperforms the former. To the best of our knowledge, this is the first study to combine SDR with VAE-based LSBO, and our analysis clarifies design choices for metric shaping and retraining that are critical for scalable latent space BO. For reproducibility, our source code is available at https://github.com/L-Lok/Nonlinear-Dimensionality-Reduction-Techniques-for-Bayesian-Optimization.git.

Figures (9)

• Figure 1: Effect of SDR on BO for the 10-dimensional Ackley function. SDR contracts the region of interest as evidence accumulates. The means and the standard deviations (shaded areas) of the minimum function values found at each iteration across $5$ repeated runs are plotted with $350$ iterations per run. In this example it yields improved performance relative to vanilla BO.
• Figure 2: Soft Triplet Loss. Given a latent triplet $\boldsymbol{z}_{ijk} = \langle \boldsymbol{z}_i, \boldsymbol{z}_j, \boldsymbol{z}_k \rangle$ with the anchor $\boldsymbol{z}_i$ (corresponding to the original anchor point $\boldsymbol{x}_i$). The right plot illustrates the discontinuous behaviour of $\mathcal{L}_{s-trip}(\cdot)$ when no weights are used. The discontinuity arises at the plane $|f(\boldsymbol{x}_i) - f(\boldsymbol{x}_k)| = \eta = 0.25$, marking the threshold beyond which $\boldsymbol{x}_k$ no longer belongs to the set of negative data points relative to the anchor $\boldsymbol{x}_i$. The left plot illustrates the a smooth transition as approaching the plane.
• Figure 3: BO–VAE (Alg. \ref{['BOVAE-SDR']}) with and without SDR on 10-D Ackley and Rosenbrock. he means and the standard deviations (shaded areas) of the minimum function values found are plotted across $5$ repeated runs. SDR updates the region of interest, improving sample efficiency and final quality.
• Figure 4: Comparison of Algorithm \ref{['BOVAE-SDR']} (Vanilla BO–VAE), Algorithm \ref{['BOVAE-retraining']} (Retrain BO–VAE), and Algorithm \ref{['BOVAE-with-DML']} (Retrain DML BO–VAE) on 100-D Ackley and Rosenbrock with latent dimensions $d\in\{2,10,50\}$ (VAE-4.3/4.4/4.5). The means and the standard deviations (shaded areas) of the minimum function values found are plotted across $5$ repeated runs.
• Figure 5: Performance Profiles on test problems in Table \ref{['tab:full-rank-test-problems']} when $\tau = 10^{-1}$ and $10^{-3}$