We consider online allocation problems with concave revenue functions and resource constraints, which are central problems in revenue management and online advertising. In these settings, requests arrive sequentially during a finite horizon and, for each request, a decision maker needs to choose an action that consumes a certain amount of resources and generates revenue. The revenue function and resource consumption of each request are drawn independently and at random from a probability distribution that is unknown to the decision maker. The objective is to maximize cumulative revenues subject to a constraint on the total consumption of resources.
We design a general class of algorithms that achieve sub-linear expected regret compared to the hindsight optimal allocation. Our algorithms operate in the Lagrangian dual space: they maintain a dual multiplier for each resource that is updated using online mirror descent. By choosing the reference function accordingly, we recover dual sub-gradient descent and dual exponential weights algorithm. The resulting algorithms are simple, efficient, and shown to attain the optimal order of regret when the length of the horizon and the initial number of resources are scaled proportionally. We discuss applications to online bidding in repeated auctions with budget constraints and online proportional matching with high entropy.