(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing

Abstract

Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p e . Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size.The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive-decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values.Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between Opk log kq and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [GNS17].

Adaptive Influence Maximization with Myopic Feedback

“…Second, our result suggests that adaptive greedy may not bring much benefit under the myopic feedback model, so are there other adaptive algorithms that could do much better? Third, for the IC model with full-adoption feedback, because the feedback on different seed nodes may be correlated, existing adaptivity gap results in [1,4] cannot be applied, and thus its adaptivity gap is still open even though it is adaptive submodular. One may also explore beyond the IC model, and study adaptive solutions for other models such as the linear threshold model, general threshold model etc.…”

“…We remark that our introduction of the hybrid policy π is inspired by the analysis in [4], which shows that the adaptivity gap for the stochastic multi-value probing (SMP) problem is at most 2. However, our analysis is more complicated than theirs and thus is novel in several aspects.…”

“…The first line of work utilizes multilinear extension and adaptive submodularity to study adaptivity gaps for the class of stochastic submodular maximization problems and give a e/(e − 1) upper bound for matroid constraints [2,1]. The second line of work [8,9,4] studies the stochastic probing problem and proposes the idea of randomwalk non-adaptive policy on the decision tree, which partially inspires our analysis. However, their analysis also implicitly depends on adaptive submodularity.…”

Delegated Stochastic Probing

“…Moreover, unlike in the oft-studied "probe and commit" models, we also allow our algorithm to select its solution after all probes are complete. In both these respects, our model is akin to the stochastic multi-value probing model of [5]. As for our game-theoretic modeling, we make the following standard and natural assumptions: We assume that the utility distributions, as well as the inner and outer constraints, are common knowledge to both the principal and the agent.…”

“…Building on recent progress in the literature on stochastic optimization, our results reduce delegated stochastic probing to generalized prophet inequalities of a particular "greedy" form, as well as to the notion of adaptivity gap (e.g. [4,5]).…”