The Price of Partial Information with Switching Costs: A Complete Characterization for Feedback Graphs

Abstract

We study how switching costs affect minimax regret under partial feedback. Which features of the feedback preserve the $\sqrt T$ rate, and which force the $T^{2/3}$ rate? We address this question in the framework of feedback graphs, which interpolate between full information and bandits: playing an action reveals the losses of its out-neighbors in a known directed graph. Prior work on feedback graphs with switching costs gave algorithms with regret of order $T^{2/3}$, but matching lower bounds only for restricted classes of graphs, leaving the question open. We establish a complete characterization of the minimax regret with switching costs for all feedback graphs: $\widetilde\Theta(\sqrt T)$ if the graph is strongly revealing, meaning that every pair of distinct actions has an endpoint that observes the losses of both; $\widetilde\Theta(\gamma^{1/3}T^{2/3})$ for every other observable graph, where $\gamma$ is the domination number; and linear otherwise. In particular, tournaments, in which every action observes its own loss and each pair of actions is joined by one directed edge, sit at the phase transition: they behave like full information, but deleting the edge of any pair forces the rate $T^{2/3}$. Hence an action that observes all losses is neither necessary nor sufficient for the rate $\sqrt T$. Our algorithm TournamentSteering for the strongly revealing case rests on a new principle: each observed comparison of two actions serves both to estimate losses and to steer play, and the loss reduction from steering pays for the variance of the estimates. For every strongly revealing graph, distributing the comparisons among observing actions keeps this variance, and hence the regret, polylogarithmic in the number of actions, by an argument similar to the greedy construction of logarithmic dominating sets in tournaments.