  This study considers the partial monitoring problem with $k$-actions and
$d$-outcomes and provides the first best-of-both-worlds algorithms, whose
regrets are favorably bounded both in the stochastic and adversarial regimes.
In particular, we show that for non-degenerate locally observable games, the
regret is $O(m^2 k^4 \log(T) \log(k_{\Pi} T) / \Delta_{\min})$ in the
stochastic regime and $O(m k^{2/3} \sqrt{T \log(T) \log k_{\Pi}})$ in the
adversarial regime, where $T$ is the number of rounds, $m$ is the maximum
number of distinct observations per action, $\Delta_{\min}$ is the minimum
suboptimality gap, and $k_{\Pi}$ is the number of Pareto optimal actions.
Moreover, we show that for globally observable games, the regret is
$O(c_{\mathcal{G}}^2 \log(T) \log(k_{\Pi} T) / \Delta_{\min}^2)$ in the
stochastic regime and $O((c_{\mathcal{G}}^2 \log(T) \log(k_{\Pi} T))^{1/3}
T^{2/3})$ in the adversarial regime, where $c_{\mathcal{G}}$ is a
game-dependent constant. We also provide regret bounds for a stochastic regime
with adversarial corruptions. Our algorithms are based on the
follow-the-regularized-leader framework and are inspired by the approach of
exploration by optimization and the adaptive learning rate in the field of
online learning with feedback graphs.
