We propose computationally efficient algorithms foronline linear optimization with bandit feedback, in which a player chooses anaction vectorfrom a given (possibly infinite) setA Rd, and then suffers a loss that can be expressed as a linear function in action vectors.

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In this paper, we propose an online convex optimization approach with two different levels of adaptivity. On a higher level, our approach is agnostic to the unknown types and curvatures of the online functions, while at a lower level, it can exploit the unknown niceness of the environments and attain problem-dependent guarantees.

In practice, however, such biased training data are commonly encountered.

In this paper we consider the dynamic assortment selection problem under an uncapacitated multinomial-logit (MNL) model. By carefully analyzing a revenue potential function, we show that a trisection based algorithm achieves an item-independent regret bound of Op? T log log T q, which matches information theoretical lower bounds up to iterated logarithmic terms. Our proof technique draws tools from the unimodal/convex bandit literature as well as adaptive confidence parameters in minimax multi-armed bandit problems.

While conceptually simple, we show that in practice, this reward modeling requires careful algorithmic considerations around model architecture and reward scaling.