  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.
