  Tree-form sequential decision making (TFSDM) extends classical one-shot
decision making by modeling tree-form interactions between an agent and a
potentially adversarial environment. It captures the online decision-making
problems that each player faces in an extensive-form game, as well as Markov
decision processes and partially-observable Markov decision processes where the
agent conditions on observed history. Over the past decade, there has been
considerable effort into designing online optimization methods for TFSDM.
Virtually all of that work has been in the full-feedback setting, where the
agent has access to counterfactuals, that is, information on what would have
happened had the agent chosen a different action at any decision node. Little
is known about the bandit setting, where that assumption is reversed (no
counterfactual information is available), despite this latter setting being
well understood for almost 20 years in one-shot decision making. In this paper,
we give the first algorithm for the bandit linear optimization problem for
TFSDM that offers both (i) linear-time iterations (in the size of the decision
tree) and (ii) $O(\sqrt{T})$ cumulative regret in expectation compared to any
fixed strategy, at all times $T$. This is made possible by new results that we
derive, which may have independent uses as well: 1) geometry of the dilated
entropy regularizer, 2) autocorrelation matrix of the natural sampling scheme
for sequence-form strategies, 3) construction of an unbiased estimator for
linear losses for sequence-form strategies, and 4) a refined regret analysis
for mirror descent when using the dilated entropy regularizer.
