  In the model of online caching with machine learned advice, introduced by
Lykouris and Vassilvitskii, the goal is to solve the caching problem with an
online algorithm that has access to next-arrival predictions: when each input
element arrives, the algorithm is given a prediction of the next time when the
element will reappear. The traditional model for online caching suffers from an
$\Omega(\log k)$ competitive ratio lower bound (on a cache of size $k$). In
contrast, the augmented model admits algorithms which beat this lower bound
when the predictions have low error, and asymptotically match the lower bound
when the predictions have high error, even if the algorithms are oblivious to
the prediction error. In particular, Lykouris and Vassilvitskii showed that
there is a prediction-augmented caching algorithm with a competitive ratio of
$O(1+\min(\sqrt{\eta/OPT}, \log k))$ when the overall $\ell_1$ prediction error
is bounded by $\eta$, and $OPT$ is the cost of the optimal offline algorithm.
  The dependence on $k$ in the competitive ratio is optimal, but the dependence
on $\eta/OPT$ may be far from optimal. In this work, we make progress towards
closing this gap. Our contributions are twofold. First, we provide an improved
algorithm with a competitive ratio of $O(1 + \min((\eta/OPT)/k, 1) \log k)$.
Second, we provide a lower bound of $\Omega(\log \min((\eta/OPT)/(k \log k),
k))$.
