Stochastic Direct Search Method for Blind Resource Allocation

Motivated by programmatic advertising optimization, we consider the task of sequentially allocating budget across a set of resources. At every time step, a feasible allocation is chosen and only a corresponding random return is observed. The goal is to maximize the cumulative expected sum of returns. This is a realistic model for budget allocation across subdivisions of marketing campaigns, with the objective of maximizing the number of conversions. We study direct search (also known as pattern search) methods for linearly constrained and derivative-free optimization in the presence of noise, which apply in particular to sequential budget allocation. These algorithms, which do not rely on hierarchical partitioning of the resource space, are easy to implement; they respect the operational constraints of resource allocation by avoiding evaluation outside of the feasible domain; and they are also compatible with warm start by being (approximate) descent algorithms. However, they have not yet been analyzed from the perspective of cumulative regret. We show that direct search methods achieves finite regret in the deterministic and unconstrained case. In the presence of evaluation noise and linear constraints, we propose a simple extension of direct search that achieves a regret upper-bound of the order of $T^{2/3}$. We also propose an accelerated version of the algorithm, relying on repeated sequential testing, that significantly improves the practical behavior of the approach.