PolytopeWalk: Sparse MCMC Sampling over Polytopes

High dimensional sampling is an important computational tool in statistics and other computational disciplines, with applications ranging from Bayesian statistical uncertainty quantification, metabolic modeling in systems biology to volume computation. We present $\textsf{PolytopeWalk}$, a new scalable Python library designed for uniform sampling over polytopes. The library provides an end-to-end solution, which includes preprocessing algorithms such as facial reduction and initialization methods. Six state-of-the-art MCMC algorithms on polytopes are implemented, including the Dikin, Vaidya, and John Walk. Additionally, we introduce novel sparse constrained formulations of these algorithms, enabling efficient sampling from sparse polytopes of the form $K_2 = \{x \in \mathbb{R}^d \ | \ Ax = b, x \succeq_k 0\}$. This implementation maintains sparsity in $A$, ensuring scalability to high dimensional settings $(d > 10^5)$. We demonstrate the improved sampling efficiency and per-iteration cost on both Netlib datasets and structured polytopes. $\textsf{PolytopeWalk}$ is available at github.com/ethz-randomwalk/polytopewalk with documentation at polytopewalk.readthedocs.io .

Diverse Expected Improvement (DEI): Diverse Bayesian Optimization of Expensive Computer Simulators

The optimization of expensive black-box simulators arises in a myriad of modern scientific and engineering applications. Bayesian optimization provides an appealing solution, by leveraging a fitted surrogate model to guide the selection of subsequent simulator evaluations. In practice, however, the objective is often not to obtain a single good solution, but rather a ''basket'' of good solutions from which users can choose for downstream decision-making. This need arises in our motivating application for real-time control of internal combustion engines for flight propulsion, where a diverse set of control strategies is essential for stable flight control. There has been little work on this front for Bayesian optimization. We thus propose a new Diverse Expected Improvement (DEI) method that searches for diverse ''$\epsilon$-optimal'' solutions: locally-optimal solutions within a tolerance level $\epsilon > 0$ from a global optimum. We show that DEI yields a closed-form acquisition function under a Gaussian process surrogate model, which facilitates efficient sequential queries via automatic differentiation. This closed form further reveals a novel exploration-exploitation-diversity trade-off, which incorporates the desired diversity property within the well-known exploration-exploitation trade-off. We demonstrate the improvement of DEI over existing methods in a suite of numerical experiments, then explore the DEI in two applications on rover trajectory optimization and engine control for flight propulsion.