  Logistic regression remains one of the most widely used tools in applied
statistics, machine learning and data science. Practical datasets often have a
substantial number of features $d$ relative to the sample size $n$. In these
cases, the logistic regression maximum likelihood estimator (MLE) is biased,
and its standard large-sample approximation is poor. In this paper, we develop
an improved method for debiasing predictions and estimating frequentist
uncertainty for such datasets. We build on recent work characterizing the
asymptotic statistical behavior of the MLE in the regime where the aspect ratio
$d / n$, instead of the number of features $d$, remains fixed as $n$ grows. In
principle, this approximation facilitates bias and uncertainty corrections, but
in practice, these corrections require an estimate of the signal strength of
the predictors. Our main contribution is SLOE, an estimator of the signal
strength with convergence guarantees that reduces the computation time of
estimation and inference by orders of magnitude. The bias correction that this
facilitates also reduces the variance of the predictions, yielding narrower
confidence intervals with higher (valid) coverage of the true underlying
probabilities and parameters. We provide an open source package for this
method, available at https://github.com/google-research/sloe-logistic.
