The expected improvement (EI) is a popular technique to handle the tradeoff between exploration andexploitation underuncertainty. Thistechnique hasbeen widely used in Bayesian optimization but it is not applicable for the contextual bandit problem which is a generalization of the standard bandit and Bayesian optimization.

In the introduction we discuss exact oracles for simplicity, but our results account for inexactness. Our results hold for any weighted Euclidean (semi)norm, i.e.,

We reveal which problems in unbalanced optimal transport can/cannot be solved efficiently.

We demonstrate this by deriving an upper bound on theRademacher Complexitythatdepends ontwokeyquantities: (i)theintrinsic dimension, which is a measure of isotropy, and (ii) the largest eigenvalue of the second moment (covariance) matrix ofthe distribution.