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Motivated by programmatic advertising optimization, we consider the task ofsequentially allocating budget across a set of resources. At every time step, afeasible allocation is chosen and only a corresponding random return isobserved. The goal is to maximize the cumulative expected sum of returns. Thisis a realistic model for budget allocation across subdivisions of marketingcampaigns, when the objective is to maximize the number of conversions. Westudy direct search (aka pattern search) methods for linearly constrained andderivative-free optimization in the presence of noise. Those algorithms areeasy to implement and particularly suited to constrained optimization. Theyhave not yet been analyzed from the perspective of cumulative regret. Weprovide a regret upper-bound of the order of T 2/3 in the general case. Ourmathematical analysis also establishes, as a by-product, time-independentregret bounds in the deterministic, unconstrained case. We also propose animproved version of the method relying on sequential tests to accelerate theidentification of descent directions.

Regret Analysis of the Stochastic Direct Search Method for Blind Resource Allocation