Adaptive Sampling for Convex Regression
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In this paper, we introduce the first principled adaptive-sampling procedure for learning a convex function in the L_∞ norm, a problem that arises often in economics, psychology, and the social sciences. We present a function-specific measure of complexity and use it to prove that our algorithm is information-theoretically near-optimal in a strong, function-specific sense. We also corroborate our theoretical contributions with extensive numerical experiments, finding that our method substantially outperforms passive, uniform sampling for favorable synthetic and data-derived functions in low-noise settings with large sampling budgets. Our results also suggest an idealized `oracle strategy', which we use to gauge the potential for deploying the adaptive-sampling strategy on any function in any particular setting.
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