A Bayesian Approach to Low-Discrepancy Subset Selection

The L infinity star discrepancy is a measure for how uniformly a point set is distributed in a given space. Point sets of low star discrepancy are used as designs of experiments, as initial designs for Bayesian optimization algorithms, for quasi-Monte Carlo integration methods, and many other applications. Recent work has shown that classical constructions such as Sobol', Halton, or Hammersley sequences can be outperformed by large margins when considering point sets of fixed sizes rather than their convergence behavior. These results, highly relevant to the aforementioned applications, raise the question of how much existing constructions can be improved through size-specific optimization. In this work, we study this question for the so-called Kronecker construction. Focusing on the 3-dimensional setting, we show that optimizing the two configurable parameters of its construction yields point sets outperforming the state-of-the-art value for sets of at least 500 points. Using the algorithm configuration technique irace, we then derive parameters that yield new state-of-the-art discrepancy values for whole ranges of set sizes.

When, in terms of the number of data points, the size of a dataset exceeds available computing resources, or when labeling is expensive, an attractive solution consists of selecting only some of the data points (subdata) for further consideration. A central question for selecting subdata of size $n$ from $N$ available data points is which $n$ points to select. While an answer to this question depends on the objective, one approach for a parametric model and a focus on parameter estimation is to select subdata that retains maximal information. Identifying such subdata is a classical NP-hard problem due to its inherent discreteness. Based on optimal approximate design theory, we develop a new methodology for information-based subdata selection, resulting in subdata that approaches the optimal solution. To achieve this, we develop a novel algorithm that applies to a general model, accommodates arbitrary choices of $N$ and $n$, and supports multiple optimality criteria, and we prove its convergence. Moreover, the new methodology facilitates an assessment of the efficiency of subdata selected by any method by obtaining tight lower and upper bounds for the efficiency. We show that the subdata obtained through the new methodology is highly efficient and outperforms all existing methods.

This paper considers the Bayesian top-$k$ identification problem, wherein a decision maker is tasked with selecting $k$ items from a pool of items of unknown quality. To assess the items, the decision maker adaptively allocates a fixed sampling budget, where each sample of an item yields a noisy observation of its quality, with the objective of minimizing the expected opportunity cost of the selected set. We propose an approach that allocates each batch of samples by minimizing a surrogate objective derived from an upper bound on the expected opportunity cost. The approach is general: samples can be allocated in batches, the selection size $k$ is arbitrary, and it applies to many sampling distributions. It is also simple and intuitive. We prove that the method's posterior expected opportunity cost decays at an exponential rate of at least half the best rate achievable by any sampling rule. In simulations across a wide range of settings, the method is competitive with, and often improves upon, the best existing algorithm in each setting.