  PAC-learning usually aims to compute a small subset
($\varepsilon$-sample/net) from $n$ items, that provably approximates a given
loss function for every query (model, classifier, hypothesis) from a given set
of queries, up to an additive error $\varepsilon\in(0,1)$. Coresets generalize
this 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 the
queries, in the hope of getting smaller subsets. The dependency between errors
of different queries implies that we may no longer apply the Chernoff-Hoeffding
inequality for a fixed query, and then use the VC-dimension or union bound.
  This paper provides deterministic and randomized algorithms for computing
such coresets and $\varepsilon$-samples of size independent of $n$, for any
finite set of queries and loss function. Example applications include new and
improved coreset constructions for e.g. streaming vector summarization
[ICML'17] and $k$-PCA [NIPS'16]. Experimental results with open source code are
provided.
