Best of Many Worlds Guarantees for Online Learning with Knapsacks
This work addresses resource-constrained online decision-making for applications like budget pacing in auctions, offering a flexible and improved theoretical framework, though it appears incremental in building on prior no-regret guarantees.
The paper tackles the problem of online learning with knapsack constraints under stochastic, adversarial, and non-stationary inputs, providing a framework that achieves no-regret guarantees, a constant competitive ratio in adversarial cases (improving over prior O(log m log T) bounds), and handles non-convex reward and cost functions.
We study online learning problems in which a decision maker wants to maximize their expected reward without violating a finite set of $m$ resource constraints. By casting the learning process over a suitably defined space of strategy mixtures, we recover strong duality on a Lagrangian relaxation of the underlying optimization problem, even for general settings with non-convex reward and resource-consumption functions. Then, we provide the first best-of-many-worlds type framework for this setting, with no-regret guarantees under stochastic, adversarial, and non-stationary inputs. Our framework yields the same regret guarantees of prior work in the stochastic case. On the other hand, when budgets grow at least linearly in the time horizon, it allows us to provide a constant competitive ratio in the adversarial case, which improves over the best known upper bound bound of $O(\log m \log T)$. Moreover, our framework allows the decision maker to handle non-convex reward and cost functions. We provide two game-theoretic applications of our framework to give further evidence of its flexibility. In doing so, we show that it can be employed to implement budget-pacing mechanisms in repeated first-price auctions.