Small Total-Cost Constraints in Contextual Bandits with Knapsacks, with Application to Fairness
This work addresses fairness constraints in resource allocation for machine learning systems, though it is incremental as it builds on existing CBwK methods to handle smaller budgets.
The paper tackles the problem of handling small total-cost constraints in contextual bandits with knapsacks, specifically enabling constraints of order √T (up to poly-log terms) to support fairness applications like equalized average costs between groups, achieving this through a dual strategy with projected-gradient-descent updates and adaptive step-size tuning.
We consider contextual bandit problems with knapsacks [CBwK], a problem where at each round, a scalar reward is obtained and vector-valued costs are suffered. The learner aims to maximize the cumulative rewards while ensuring that the cumulative costs are lower than some predetermined cost constraints. We assume that contexts come from a continuous set, that costs can be signed, and that the expected reward and cost functions, while unknown, may be uniformly estimated -- a typical assumption in the literature. In this setting, total cost constraints had so far to be at least of order $T^{3/4}$, where $T$ is the number of rounds, and were even typically assumed to depend linearly on $T$. We are however motivated to use CBwK to impose a fairness constraint of equalized average costs between groups: the budget associated with the corresponding cost constraints should be as close as possible to the natural deviations, of order $\sqrt{T}$. To that end, we introduce a dual strategy based on projected-gradient-descent updates, that is able to deal with total-cost constraints of the order of $\sqrt{T}$ up to poly-logarithmic terms. This strategy is more direct and simpler than existing strategies in the literature. It relies on a careful, adaptive, tuning of the step size.