Best of Many Worlds Guarantees for Online Learning with Knapsacks

TL;DR

This paper introduces a versatile online learning framework with strong theoretical guarantees for resource-constrained decision making, applicable to stochastic, adversarial, and non-stationary environments, and handles non-convex functions.

Contribution

It develops the first best-of-many-worlds framework for online learning with knapsacks, achieving no-regret guarantees and improved competitive ratios, even with non-convex reward and cost functions.

Findings

Achieves no-regret guarantees in stochastic, adversarial, and non-stationary settings.

Provides a constant competitive ratio when budgets grow linearly with time.

Enables implementation of budget-pacing in repeated auctions.

Abstract

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.