  We investigate quantum algorithms for classification, a fundamental problem
in machine learning, with provable guarantees. Given $n$ $d$-dimensional data
points, the state-of-the-art (and optimal) classical algorithm for training
classifiers with constant margin runs in $\tilde{O}(n+d)$ time. We design
sublinear quantum algorithms for the same task running in $\tilde{O}(\sqrt{n}
+\sqrt{d})$ time, a quadratic improvement in both $n$ and $d$. Moreover, our
algorithms use the standard quantization of the classical input and generate
the same classical output, suggesting minimal overheads when used as
subroutines for end-to-end applications. We also demonstrate a tight lower
bound (up to poly-log factors) and discuss the possibility of implementation on
near-term quantum machines. As a side result, we also give sublinear quantum
algorithms for approximating the equilibria of $n$-dimensional matrix zero-sum
games with optimal complexity $\tilde{\Theta}(\sqrt{n})$.
