  We study the problem of switching-constrained online convex optimization
(OCO), where the player has a limited number of opportunities to change her
action. While the discrete analog of this online learning task has been studied
extensively, previous work in the continuous setting has neither established
the minimax rate nor algorithmically achieved it. In this paper, we show that $
T $-round switching-constrained OCO with fewer than $ K $ switches has a
minimax regret of $ \Theta(\frac{T}{\sqrt{K}}) $. In particular, it is at least
$ \frac{T}{\sqrt{2K}} $ for one dimension and at least $ \frac{T}{\sqrt{K}} $
for higher dimensions. The lower bound in higher dimensions is attained by an
orthogonal subspace argument. In one dimension, a novel adversarial strategy
yields the lower bound of $O(\frac{T}{\sqrt{K}})$, but a precise minimax
analysis including constants is more involved. To establish the tighter
one-dimensional result, we introduce the \emph{fugal game} relaxation, whose
minimax regret lower bounds that of switching-constrained OCO. We show that the
minimax regret of the fugal game is at least $ \frac{T}{\sqrt{2K}} $ and
thereby establish the optimal minimax lower bound in one dimension. To
establish the dimension-independent upper bound, we next show that a
mini-batching algorithm provides an $ O(\frac{T}{\sqrt{K}}) $ upper bound, and
therefore conclude that the minimax regret of switching-constrained OCO is $
\Theta(\frac{T}{\sqrt{K}}) $ for any $K$. This is in sharp contrast to its
discrete counterpart, the switching-constrained prediction-from-experts
problem, which exhibits a phase transition in minimax regret between the
low-switching and high-switching regimes.
