  We provide nearly optimal algorithms for online facility location (OFL) with
predictions. In OFL, $n$ demand points arrive in order and the algorithm must
irrevocably assign each demand point to an open facility upon its arrival. The
objective is to minimize the total connection costs from demand points to
assigned facilities plus the facility opening cost. We further assume the
algorithm is additionally given for each demand point $x_i$ a natural
prediction $f_{x_i}^{\mathrm{pred}}$ which is supposed to be the facility
$f_{x_i}^{\mathrm{opt}}$ that serves $x_i$ in the offline optimal solution.
  Our main result is an $O(\min\{\log {\frac{n\eta_\infty}{\mathrm{OPT}}},
\log{n} \})$-competitive algorithm where $\eta_\infty$ is the maximum
prediction error (i.e., the distance between $f_{x_i}^{\mathrm{pred}}$ and
$f_{x_i}^{\mathrm{opt}}$). Our algorithm overcomes the fundamental
$\Omega(\frac{\log n}{\log \log n})$ lower bound of OFL (without predictions)
when $\eta_\infty$ is small, and it still maintains $O(\log n)$ ratio even when
$\eta_\infty$ is unbounded. Furthermore, our theoretical analysis is supported
by empirical evaluations for the tradeoffs between $\eta_\infty$ and the
competitive ratio on various real datasets of different types.
