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# Online learning over a finite action set with limited switching

*Proceedings of the 31st Conference On Learning Theory*, PMLR 75:1569-1573, 2018.

#### Abstract

We study the value of switching actions in the Prediction From Experts (PFE) problem and Adversarial Multi-Armed Bandits (MAB) problem. First, we revisit the well-studied and practically motivated setting of PFE with switching costs. Many algorithms are known to achieve the minimax optimal order of $O(\sqrt{T \log n})$ in \textit{expectation} for both regret and number of switches, where $T$ is the number of iterations and $n$ the number of actions. However, no \textit{high probability} guarantees are known. Our main technical contribution is the first algorithms which with high probability achieve this optimal order for both regret and number of switches. This settles an open problem of [Devroye et al., 2015], directly implies the first high probability guarantees for several problems of interest, and is efficiently adaptable to the related problem of online combinatorial optimization with limited switching. \par Next, to investigate the value of switching actions at a more granular level, we introduce the setting of \textit{switching budgets}, in which the algorithm is limited to $S \leq T$ switches between actions. This entails a limited number of free switches, in contrast to the unlimited number of expensive switches allowed in the switching cost setting. Using the above result and several reductions, we unify previous work and completely characterize the complexity of this switching budget setting up to small polylogarithmic factors: for both the PFE and MAB problems, for all switching budgets $S \leq T$, and for both expectation and high probability guarantees. For PFE, we show that the optimal rate is of order $\tilde{\Theta}(\sqrt{T\log n})$ for $S = \Omega(\sqrt{T\log n})$, and $\min(\tilde{\Theta}(\tfrac{T\log n}{S}), T)$ for $S = O(\sqrt{T \log n})$. Interestingly, the bandit setting does not exhibit such a phase transition; instead we show the minimax rate decays steadily as $\min(\tilde{\Theta}(\tfrac{T\sqrt{n}}{\sqrt{S}}), T)$ for all ranges of $S \leq T$. These results recover and generalize the known minimax rates for the (arbitrary) switching cost setting.