The second identity can be proved in a similar manner.

In particular, we express generalization gap in terms of the power spectra of the sample design and that of the function to be learned.

A common challenge in machine learning and related fields is the need to efficiently explore high dimensional parameter spaces using small numbers of samples. Typical examples are hyper-parameter optimization in deep learning and sample mining in predictive modeling tasks. All such problems trade-off exploration, which samples the space without knowledge of the target function, and exploitation where information from previous evaluations is used in an adaptive feedback loop. Much of the recent focus has been on the exploitation while exploration is done with simple designs such as Latin hypercube or even uniform random sampling. In this paper, we introduce optimal space-filling sample designs for effective exploration of high dimensional spaces. Specifically, we propose a new parameterized family of sample designs called space-filling spectral designs, and introduce a framework to choose optimal designs for a given sample size and dimension. Furthermore, we present an efficient algorithm to synthesize a given spectral design. Finally, we evaluate the performance of spectral designs in both data space and model space applications. The data space exploration is targeted at recovering complex regression functions in high dimensional spaces. The model space exploration focuses on selecting hyper-parameters for a given neural network architecture. Our empirical studies demonstrate that the proposed approach consistently outperforms state-of-the-art techniques, particularly with smaller design sizes.

This paper proposes a new approach to construct high quality space-filling sample designs. First, we propose a novel technique to quantify the space-filling property and optimally trade-off uniformity and randomness in sample designs in arbitrary dimensions. Second, we connect the proposed metric (defined in the spatial domain) to the objective measure of the design performance (defined in the spectral domain). This connection serves as an analytic framework for evaluating the qualitative properties of space-filling designs in general. Using the theoretical insights provided by this spatial-spectral analysis, we derive the notion of optimal space-filling designs, which we refer to as space-filling spectral designs. Third, we propose an efficient estimator to evaluate the space-filling properties of sample designs in arbitrary dimensions and use it to develop an optimization framework to generate high quality space-filling designs. Finally, we carry out a detailed performance comparison on two different applications in 2 to 6 dimensions: a) image reconstruction and b) surrogate modeling on several benchmark optimization functions and an inertial confinement fusion (ICF) simulation code. We demonstrate that the propose spectral designs significantly outperform existing approaches especially in high dimensions.