Covering Numbers for Semicontinuous Functions
Naval Postgraduate School Monterey United States
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Considering the metric space of extended real-valued lower semicontinuous functions under the epi-distance, the paper gives an upper bound on the covering numbers of bounded subsets of such functions. No assumptions about continuity, smoothness, variation, and even niteness of the functions are needed. The bound is shown to be nearly sharp through the construction of a set of functions with covering numbers deviating from the upper bound only by a logarithmic factor. The analogy between lower and upper semicontinuous functions implies that identical covering numbers hold for bounded sets of the latter class of functions as well, but now under the hypo-distance metric.
- Numerical Mathematics