Cosmin G. Petra

Jingyi Wang, Haowei Wang, Nai-Yuan Chiang et al.

Efficient global optimization (EGO) is one of the most widely used noise-free Bayesian optimization algorithms.It comprises the Gaussian process (GP) surrogate model and expected improvement (EI) acquisition function. In practice, when EGO is applied, a scalar matrix of a small positive value (also called a nugget or jitter) is usually added to the covariance matrix of the deterministic GP to improve numerical stability. We refer to this EGO with a positive nugget as the practical EGO. Despite its wide adoption and empirical success, to date, cumulative regret bounds for practical EGO have yet to be established. In this paper, we present for the first time the cumulative regret upper bound of practical EGO. In particular, we show that practical EGO has sublinear cumulative regret bounds and thus is a no-regret algorithm for commonly used kernels including the squared exponential (SE) and Matérn kernels ($ν>\frac{1}{2}$). Moreover, we analyze the effect of the nugget on the regret bound and discuss the theoretical implication on its choice. Numerical experiments are conducted to support and validate our findings.

Slaven Peles, Kalyan S. Perumalla, Maksudul Alam et al.

While interior point methods have been the centerpiece of nonlinear programming tools used in science and engineering, their reliance on linear solvers that can tackle sparse symmetric indefinite and highly ill-conditioned problems made it difficult to implement them effectively on hardware accelerators. At this time, there are few sparse linear solvers that can be used in this context. Here, we present a novel formulation of an interior point method implemented in our HiOp library, which is designed to be able to run entirely on hardware accelerators. This formulation avoids dependence on sparse solvers altogether, which is achieved by compressing the underlying sparse linear problem into a dense one of manageable size. We demonstrate feasibility of this approach and provide a baseline for future interior point method implementations on hardware accelerators. Our investigation is motivated by problems arising in optimal power flow analysis in power systems engineering and our approach is tailored to the broad class of problems arising in that important domain. We also demonstrate utility of modern programming models based on performance portability libraries, namely, Umpire and RAJA. We discuss trade-offs between performance, portability and development cost in the solution space for this non-linear optimization problem. As a result of this research, we demonstrate for the first time that interior point methods for sparse problems can be efficiently realized on modern computing systems where more than 90% of processing power is in GPUs.

Jingyi Wang, Haowei Wang, Nai-Yuan Chiang et al.

Expected improvement (EI) is one of the most widely used acquisition functions in Bayesian optimization (BO). Despite its proven success in applications for decades, important open questions remain on the theoretical convergence behaviors and rates for EI. In this paper, we contribute to the convergence theory of EI in three novel and critical areas. First, we consider objective functions that fit under the Gaussian process (GP) prior assumption, whereas existing works mostly focus on functions in the reproducing kernel Hilbert space (RKHS). Second, we establish for the first time the asymptotic error bound and its corresponding rate for GP-EI with noisy observations under the GP prior assumption. Third, by investigating the exploration and exploitation properties of the non-convex EI function, we establish improved error bounds of GP-EI for both the noise-free and noisy cases.

Jingyi Wang, Haowei Wang, Cosmin G. Petra et al.

Bayesian optimization (BO) with Gaussian process (GP) surrogate models is a powerful black-box optimization method. Acquisition functions are a critical part of a BO algorithm as they determine how the new samples are selected. Some of the most widely used acquisition functions include upper confidence bound (UCB) and Thompson sampling (TS). The convergence analysis of BO algorithms has focused on the cumulative regret under both the Bayesian and frequentist settings for the objective. In this paper, we establish new pointwise bounds on the prediction error of GP under the frequentist setting with Gaussian noise. Consequently, we prove improved convergence rates of cumulative regret bound for both GP-UCB and GP-TS. Of note, the new prediction error bound under Gaussian noise can be applied to general BO algorithms and convergence analysis, e.g., the asymptotic convergence of expected improvement (EI) with noise.

Jingyi Wang, Haowei Wang, Szu Hui Ng et al.

Expected improvement (EI) is one of the most widely used acquisition functions in Bayesian optimization (BO). Despite its proven empirical success in applications, the cumulative regret upper bound of EI remains an open question. In this paper, we analyze the classic noisy Gaussian process expected improvement (GP-EI) algorithm. We consider the Bayesian setting, where the objective is a sample from a GP. Three commonly used incumbents, namely the best posterior mean incumbent (BPMI), the best sampled posterior mean incumbent (BSPMI), and the best observation incumbent (BOI) are considered as the choices of the current best value in GP-EI. We present for the first time the cumulative regret upper bounds of GP-EI with BPMI and BSPMI. Importantly, we show that in both cases, GP-EI is a no-regret algorithm for both squared exponential (SE) and Matérn kernels. Further, we present for the first time that GP-EI with BOI either achieves a sublinear cumulative regret upper bound or has a fast converging noisy simple regret bound for SE and Matérn kernels. Our results provide theoretical guidance to the choice of incumbent when practitioners apply GP-EI in the noisy setting. Numerical experiments are conducted to validate our findings.