  In this paper, we study the role of feedback in online learning with
switching costs. It has been shown that the minimax regret is
$\widetilde{\Theta}(T^{2/3})$ under bandit feedback and improves to
$\widetilde{\Theta}(\sqrt{T})$ under full-information feedback, where $T$ is
the length of the time horizon. However, it remains largely unknown how the
amount and type of feedback generally impact regret. To this end, we first
consider the setting of bandit learning with extra observations; that is, in
addition to the typical bandit feedback, the learner can freely make a total of
$B_{\mathrm{ex}}$ extra observations. We fully characterize the minimax regret
in this setting, which exhibits an interesting phase-transition phenomenon:
when $B_{\mathrm{ex}} = O(T^{2/3})$, the regret remains
$\widetilde{\Theta}(T^{2/3})$, but when $B_{\mathrm{ex}} = \Omega(T^{2/3})$, it
becomes $\widetilde{\Theta}(T/\sqrt{B_{\mathrm{ex}}})$, which improves as the
budget $B_{\mathrm{ex}}$ increases. To design algorithms that can achieve the
minimax regret, it is instructive to consider a more general setting where the
learner has a budget of $B$ total observations. We fully characterize the
minimax regret in this setting as well and show that it is
$\widetilde{\Theta}(T/\sqrt{B})$, which scales smoothly with the total budget
$B$. Furthermore, we propose a generic algorithmic framework, which enables us
to design different learning algorithms that can achieve matching upper bounds
for both settings based on the amount and type of feedback. One interesting
finding is that while bandit feedback can still guarantee optimal regret when
the budget is relatively limited, it no longer suffices to achieve optimal
regret when the budget is relatively large.
