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We study online learning problems in which a decision maker wants to maximizetheir expected reward without violating a finite set of $m$ resourceconstraints. By casting the learning process over a suitably defined space ofstrategy mixtures, we recover strong duality on a Lagrangian relaxation of theunderlying optimization problem, even for general settings with non-convexreward and resource-consumption functions. Then, we provide the firstbest-of-many-worlds type framework for this setting, with no-regret guaranteesunder stochastic, adversarial, and non-stationary inputs. Our framework yieldsthe same regret guarantees of prior work in the stochastic case. On the otherhand, when budgets grow at least linearly in the time horizon, it allows us toprovide a constant competitive ratio in the adversarial case, which improvesover the best known upper bound bound of $O(\log m \log T)$. Moreover, ourframework allows the decision maker to handle non-convex reward and costfunctions. We provide two game-theoretic applications of our framework to givefurther evidence of its flexibility. In doing so, we show that it can beemployed to implement budget-pacing mechanisms in repeated first-priceauctions.

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