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Luo Long, Coralia Cartis, Paz Fink Shustin
Mathematical Institute, University of Oxford, Radcliffe Observatory, Andrew Wiles Building, Woodstock Rd, Oxford, OX2 6GG, United Kingdom.
Bayesian optimisation (BO) enables sample-efficient global optimisation of expensive black-box functions but remains challenging in high dimensions. We investigate nonlinear dimensionality reduction to a sequence of low-dimensional latent-space BO (LSBO) problems. Early LSBO used linear random and supervised embeddings; building on Grosnit et al., we employ variational autoencoders (VAEs), deep metric loss for structured latent manifolds, and retraining to adapt the encoder-decoder pair to newly sampled regions. We couple LSBO with sequential domain reduction (SDR) directly in latent space (SDR-LSBO), narrowing search domains as evidence accumulates. Implemented in GPU-accelerated BoTorch with Mat'ern-5/2 Gaussian-process surrogates, our methods improve benchmark optimisation quality, and retraining can enhance BO performance. Comparisons with adaptive supervised linear random embeddings demonstrate the effectiveness of VAE-based BO for nonlinear low-dimensional structures. We analyse BO-VAE with a fixed pretrained representation, decomposing ambient-space simple regret into latent BO error and a fixed VAE-induced representation gap. Under a PAC-Bayes-certified reconstruction condition and standard fixed-prior assumptions for expected improvement with a Mat'ern-5/2 kernel, latent BO error vanishes as the evaluation budget increases, whereas the representation gap remains fixed and may impose a non-vanishing error floor. Visualisations empirically assess accessibility of the ambient optimum through the learned decoder. To our knowledge, this is the first study combining SDR with VAE-based LSBO. Our analysis clarifies metric shaping and retraining choices critical for scalable latent-space BO. For reproducibility, source code is available at https://github.com/L-Lok/Nonlinear-Dimensionality-Reduction-Techniques-for-Bayesian-Optimization.git.