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Scaling Bayesian optimisation (BO) to high-dimensional search spaces is aactive and open research problems particularly when no assumptions are made onfunction structure. The main reason is that at each iteration, BO requires tofind global maximisation of acquisition function, which itself is a non-convexoptimization problem in the original search space. With growing dimensions, thecomputational budget for this maximisation gets increasingly short leading toinaccurate solution of the maximisation. This inaccuracy adversely affects boththe convergence and the efficiency of BO. We propose a novel approach where theacquisition function only requires maximisation on a discrete set of lowdimensional subspaces embedded in the original high-dimensional search space.Our method is free of any low dimensional structure assumption on the functionunlike many recent high-dimensional BO methods. Optimising acquisition functionin low dimensional subspaces allows our method to obtain accurate solutionswithin limited computational budget. We show that in spite of this convenience,our algorithm remains convergent. In particular, cumulative regret of ouralgorithm only grows sub-linearly with the number of iterations. Moreimportantly, as evident from our regret bounds, our algorithm provides a way totrade the convergence rate with the number of subspaces used in theoptimisation. Finally, when the number of subspaces is "sufficiently large",our algorithm's cumulative regret is at most$\mathcal{O}^{*}(\sqrt{T\gamma_T})$ as opposed to$\mathcal{O}^{*}(\sqrt{DT\gamma_T})$ for the GP-UCB of Srinivas et al. (2012),reducing a crucial factor $\sqrt{D}$ where $D$ being the dimensional number ofinput space.

Trading Convergence Rate with Computational Budget in High Dimensional Bayesian Optimization