Hardness of Agnostically Learning Halfspaces from Worst-Case Lattice Problems
release_5nfuqvt5bbcm5f7bsearuxkhey
by
Stefan Tiegel
2022
Abstract
We show hardness of improperly learning halfspaces in the agnostic model
based on worst-case lattice problems, e.g., approximating shortest vectors
within polynomial factors. In particular, we show that under this assumption
there is no efficient algorithm that outputs any binary hypothesis, not
necessarily a halfspace, achieving misclassfication error better than 1/2 - ϵ even if the optimal misclassification error is as small is as
small as δ. Here, ϵ can be smaller than the inverse of any
polynomial in the dimension and δ as small as
exp(-Ω(log^1-c(d))), where 0 < c < 1
is an arbitrary constant and d is the dimension.
Previous hardness results [Daniely16] of this problem were based on
average-case complexity assumptions, specifically, variants of Feige's random
3SAT hypothesis. Our work gives the first hardness for this problem based on a
worst-case complexity assumption. It is inspired by a sequence of recent works
showing hardness of learning well-separated Gaussian mixtures based on
worst-case lattice problems.
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