  We present an algorithm for the statistical learning setting with a bounded
exp-concave loss in $d$ dimensions that obtains excess risk $O(d
\log(1/\delta)/n)$ with probability at least $1 - \delta$. The core technique
is to boost the confidence of recent in-expectation $O(d/n)$ excess risk bounds
for empirical risk minimization (ERM), without sacrificing the rate, by
leveraging a Bernstein condition which holds due to exp-concavity. We also show
that with probability $1 - \delta$ the standard ERM method obtains excess risk
$O(d (\log(n) + \log(1/\delta))/n)$. We further show that a regret bound for
any online learner in this setting translates to a high probability excess risk
bound for the corresponding online-to-batch conversion of the online learner.
Lastly, we present two high probability bounds for the exp-concave model
selection aggregation problem that are quantile-adaptive in a certain sense.
The first bound is a purely exponential weights type algorithm, obtains a
nearly optimal rate, and has no explicit dependence on the Lipschitz continuity
of the loss. The second bound requires Lipschitz continuity but obtains the
optimal rate.
