Online Learning over a Finite Action Set with Limited Switching

This paper studies the value of switching actions in the Prediction From Experts problem (PFE) and Adversarial Multiarmed Bandits problem (MAB). First, we revisit the well-studied and practically motivated setting of PFE with switching costs. Many algorithms achieve the minimax optimal order for both regret and switches in expectation ; however, high probability guarantees are an open problem. We present the first algorithms that achieve this optimal order for both quantities with high probability. This also implies the first high probability guarantees for several other problems, and, in particular, is efficiently adaptable to online combinatorial optimization with limited switching. Next, to investigate the value of switching actions more granularly, we introduce the switching budget setting, which limits algorithms to a fixed number of (costless) switches. Using this 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 PFE and MAB, for all switching budgets, and for both expectation and high probability guarantees. Interestingly, as the switching budget decreases, the minimax regret rate admits a phase transition for PFE but not for MAB. These results recover and generalize the known minimax rates for the (arbitrary) switching cost setting.