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PAC-learning usually aims to compute a small subset($\varepsilon$-sample/net) from $n$ items, that provably approximates a givenloss function for every query (model, classifier, hypothesis) from a given setof queries, up to an additive error $\varepsilon\in(0,1)$. Coresets generalizethis idea to support multiplicative error $1\pm\varepsilon$.  Inspired by smoothed analysis, we suggest a natural generalization:approximate the \emph{average} (instead of the worst-case) error over thequeries, in the hope of getting smaller subsets. The dependency between errorsof different queries implies that we may no longer apply the Chernoff-Hoeffdinginequality for a fixed query, and then use the VC-dimension or union bound.  This paper provides deterministic and randomized algorithms for computingsuch coresets and $\varepsilon$-samples of size independent of $n$, for anyfinite set of queries and loss function. Example applications include new andimproved coreset constructions for e.g. streaming vector summarization[ICML'17] and $k$-PCA [NIPS'16]. Experimental results with open source code areprovided.

Faster PAC Learning and Smaller Coresets via Smoothed Analysis