Minimax Regret of Switching-Constrained Online Convex Optimization: No Phase Transition

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.