Optimal Design Emulators: A Point Process Approach
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Design of experiments is a fundamental topic in applied statistics with a long history. Yet its application is often limited by the complexity and costliness of constructing experimental designs in the first place. For example, in optimal design, constructing an experimental design involves searching the high-dimensional input space -- a computationally expensive procedure that only guarantees a locally optimal solution. This is a difficult problem that, typically, can only be "solved" by changing the optimality criterion to be based on a simplified model. Such approximations are sometimes justifiable but rarely broadly desirable. In this work, we introduce a novel approach to the challenging design problem. We will take a probabilistic view of the problem by representing the optimal design as being one element (or a subset of elements) of a probability space. Given a suitable distribution on this space, a generative process can be specified from which stochastic design realizations can be drawn. We describe a scenario where the classical (point estimate) optimal design solution coincides with the mode of the generative process we specify. We conclude with outlining an algorithm for drawing such design realizations, its extension to sequential design, and applying the techniques developed to constructing space-filling designs for Stochastic Gradient Descent and entropy-optimal designs for Gaussian process regression.
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