Contextual Recommendations and Low-Regret Cutting-Plane Algorithms

TL;DR

This paper introduces novel algorithms for contextual linear bandits with low regret, leveraging cutting-plane methods and convex geometry techniques, applicable to routing and recommendation systems.

Contribution

It presents new low-regret algorithms for contextual bandits using cutting-plane methods and convex geometry, including variants with list recommendations and nearly tight bounds.

Findings

Achieves regret $O(d\,\log T)$ and $\,\exp(O(d \log d))$

Provides algorithms with $O(d^2 \log d)$ regret and polynomial list size

Develops nearly tight algorithms for weaker feedback models

Abstract

We consider the following variant of contextual linear bandits motivated by routing applications in navigational engines and recommendation systems. We wish to learn a hidden $d$-dimensional value $w^*$. Every round, we are presented with a subset $\mathcal{X}_t \subseteq \mathbb{R}^d$ of possible actions. If we choose (i.e. recommend to the user) action $x_t$, we obtain utility $\langle x_t, w^* \rangle$ but only learn the identity of the best action $\arg\max_{x \in \mathcal{X}_t} \langle x, w^* \rangle$. We design algorithms for this problem which achieve regret $O(d\log T)$ and $\exp(O(d \log d))$. To accomplish this, we design novel cutting-plane algorithms with low "regret" -- the total distance between the true point $w^*$ and the hyperplanes the separation oracle returns. We also consider the variant where we are allowed to provide a list of several recommendations. In this variant, we give an algorithm with $O(d^2 \log d)$ regret and list size $\mathrm{poly}(d)$. Finally, we construct nearly tight algorithms for a weaker variant of this problem where the learner only learns the identity of an action that is better than the recommendation. Our results rely on new algorithmic techniques in convex geometry (including a variant of Steiner's formula for the centroid of a convex set) which may be of independent interest.