  In this paper we introduce the transductive linear bandit problem: given a
set of measurement vectors $\mathcal{X}\subset \mathbb{R}^d$, a set of items
$\mathcal{Z}\subset \mathbb{R}^d$, a fixed confidence $\delta$, and an unknown
vector $\theta^{\ast}\in \mathbb{R}^d$, the goal is to infer
$\text{argmax}_{z\in \mathcal{Z}} z^\top\theta^\ast$ with probability
$1-\delta$ by making as few sequentially chosen noisy measurements of the form
$x^\top\theta^{\ast}$ as possible. When $\mathcal{X}=\mathcal{Z}$, this setting
generalizes linear bandits, and when $\mathcal{X}$ is the standard basis
vectors and $\mathcal{Z}\subset \{0,1\}^d$, combinatorial bandits. Such a
transductive setting naturally arises when the set of measurement vectors is
limited due to factors such as availability or cost. As an example, in drug
discovery the compounds and dosages $\mathcal{X}$ a practitioner may be willing
to evaluate in the lab in vitro due to cost or safety reasons may differ vastly
from those compounds and dosages $\mathcal{Z}$ that can be safely administered
to patients in vivo. Alternatively, in recommender systems for books, the set
of books $\mathcal{X}$ a user is queried about may be restricted to well known
best-sellers even though the goal might be to recommend more esoteric titles
$\mathcal{Z}$. In this paper, we provide instance-dependent lower bounds for
the transductive setting, an algorithm that matches these up to logarithmic
factors, and an evaluation. In particular, we provide the first non-asymptotic
algorithm for linear bandits that nearly achieves the information theoretic
lower bound.
