Bandits with Knapsacks

TL;DR

This paper introduces 'bandits with knapsacks', a new model combining exploration-exploitation with supply constraints, providing algorithms with near-optimal regret bounds applicable to diverse real-world domains.

Contribution

The paper formulates the novel 'bandits with knapsacks' model, develops two algorithms with near-optimal regret guarantees, and demonstrates its broad applicability across multiple fields.

Findings

Algorithms achieve regret close to information-theoretic limits.

The primal-dual and balanced exploration algorithms are proven to be near-optimal.

Application examples include dynamic pricing, routing, and scheduling.

Abstract

Multi-armed bandit problems are the predominant theoretical model of exploration-exploitation tradeoffs in learning, and they have countless applications ranging from medical trials, to communication networks, to Web search and advertising. In many of these application domains the learner may be constrained by one or more supply (or budget) limits, in addition to the customary limitation on the time horizon. The literature lacks a general model encompassing these sorts of problems. We introduce such a model, called "bandits with knapsacks", that combines aspects of stochastic integer programming with online learning. A distinctive feature of our problem, in comparison to the existing regret-minimization literature, is that the optimal policy for a given latent distribution may significantly outperform the policy that plays the optimal fixed arm. Consequently, achieving sublinear regret in the bandits-with-knapsacks problem is significantly more challenging than in conventional bandit problems. We present two algorithms whose reward is close to the information-theoretic optimum: one is based on a novel "balanced exploration" paradigm, while the other is a primal-dual algorithm that uses multiplicative updates. Further, we prove that the regret achieved by both algorithms is optimal up to polylogarithmic factors. We illustrate the generality of the problem by presenting applications in a number of different domains including electronic commerce, routing, and scheduling. As one example of a concrete application, we consider the problem of dynamic posted pricing with limited supply and obtain the first algorithm whose regret, with respect to the optimal dynamic policy, is sublinear in the supply.