An LP for Sequential Learning Under Budgets

We present a convex framework to learn sequential decisions and apply this to the problem of learning under a budget. We consider the structure proposed [1], where sensor measurements are acquired in a sequence. The goal after acquiring each new measurement is to make a decision whether to stop and classify or to pay the cost of using the next sensor in the sequence. We introduce a novel formulation of an empirical risk objective for the multi stage sequential decision problem. This objective naturally lends itself to a non-convex multilinear formulation. Nevertheless, we derive a novel perspective that leads to a tight convex objective. This is accomplished by expressing the empirical risk in terms of linear superposition of indicator functions. We then derive an LP formulation by utilizing hinge loss surrogates. Our LP achieves or exceeds the empirical performance as the non-convex alternating algorithm that requires a large number of random initializations. Consequently, the LP has the advantage of guaranteed convergence, global optimality, repeatability and computation efficiency.