Budget Feasible Mechanisms for Experimental Design

In the classical experimental design setting, an experimenter E has access to a population of $n$ potential experiment subjects $i\in \{1,...,n\}$, each associated with a vector of features $x_i\in R^d$. Conducting an experiment with subject $i$ reveals an unknown value $y_i\in R$ to E. E typically assumes some hypothetical relationship between $x_i$'s and $y_i$'s, e.g., $y_i \approx \beta x_i$, and estimates $\beta$ from experiments, e.g., through linear regression. As a proxy for various practical constraints, E may select only a subset of subjects on which to conduct the experiment. We initiate the study of budgeted mechanisms for experimental design. In this setting, E has a budget $B$. Each subject $i$ declares an associated cost $c_i >0$ to be part of the experiment, and must be paid at least her cost. In particular, the Experimental Design Problem (EDP) is to find a set $S$ of subjects for the experiment that maximizes V(S) = \log\det(I_d+\sum_{i\in S}x_i\T{x_i}) under the constraint $\sum_{i\in S}c_i\leq B$; our objective function corresponds to the information gain in parameter $\beta$ that is learned through linear regression methods, and is related to the so-called $D$-optimality criterion. Further, the subjects are strategic and may lie about their costs. We present a deterministic, polynomial time, budget feasible mechanism scheme, that is approximately truthful and yields a constant factor approximation to EDP. In particular, for any small $\delta > 0$ and $\epsilon > 0$, we can construct a (12.98, $\epsilon$)-approximate mechanism that is $\delta$-truthful and runs in polynomial time in both $n$ and $\log\log\frac{B}{\epsilon\delta}$. We also establish that no truthful, budget-feasible algorithms is possible within a factor 2 approximation, and show how to generalize our approach to a wide class of learning problems, beyond linear regression

Attention and Anticipation in Fast Visual-Inertial Navigation

We study a Visual-Inertial Navigation (VIN) problem in which a robot needs to estimate its state using an on-board camera and an inertial sensor, without any prior knowledge of the external environment. We consider the case in which the robot can allocate limited resources to VIN, due to tight computational constraints. Therefore, we answer the following question: under limited resources, what are the most relevant visual cues to maximize the performance of visual-inertial navigation? Our approach has four key ingredients. First, it is task-driven, in that the selection of the visual cues is guided by a metric quantifying the VIN performance. Second, it exploits the notion of anticipation, since it uses a simplified model for forward-simulation of robot dynamics, predicting the utility of a set of visual cues over a future time horizon. Third, it is efficient and easy to implement, since it leads to a greedy algorithm for the selection of the most relevant visual cues. Fourth, it provides formal performance guarantees: we leverage submodularity to prove that the greedy selection cannot be far from the optimal (combinatorial) selection. Simulations and real experiments on agile drones show that our approach ensures state-of-the-art VIN performance while maintaining a lean processing time. In the easy scenarios, our approach outperforms appearance-based feature selection in terms of localization errors. In the most challenging scenarios, it enables accurate visual-inertial navigation while appearance-based feature selection fails to track robot's motion during aggressive maneuvers.Comment: 20 pages, 7 figures, 2 table

Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets

Consider the following problem: given a set system (U,I) and an edge-weighted graph G = (U, E) on the same universe U, find the set A in I such that the Steiner tree cost with terminals A is as large as possible: "which set in I is the most difficult to connect up?" This is an example of a max-min problem: find the set A in I such that the value of some minimization (covering) problem is as large as possible. In this paper, we show that for certain covering problems which admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system I is given by a p-system or q-knapsacks or both. This result is similar to results for constrained maximization of submodular functions. Although many natural covering problems are not even approximately submodular, we show that one can use properties of the online algorithm as a surrogate for submodularity. Moreover, we give stronger connections between max-min optimization and two-stage robust optimization, and hence give improved algorithms for robust versions of various covering problems, for cases where the uncertainty sets are given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and http://arxiv.org/abs/0912.1045 appeared in ICALP 201

Sample Complexity Bounds for Influence Maximization

Influence maximization (IM) is the problem of finding for a given s ? 1 a set S of |S|=s nodes in a network with maximum influence. With stochastic diffusion models, the influence of a set S of seed nodes is defined as the expectation of its reachability over simulations, where each simulation specifies a deterministic reachability function. Two well-studied special cases are the Independent Cascade (IC) and the Linear Threshold (LT) models of Kempe, Kleinberg, and Tardos [Kempe et al., 2003]. The influence function in stochastic diffusion is unbiasedly estimated by averaging reachability values over i.i.d. simulations. We study the IM sample complexity: the number of simulations needed to determine a (1-?)-approximate maximizer with confidence 1-?. Our main result is a surprising upper bound of O(s ? ?^{-2} ln (n/?)) for a broad class of models that includes IC and LT models and their mixtures, where n is the number of nodes and ? is the number of diffusion steps. Generally ? ? n, so this significantly improves over the generic upper bound of O(s n ?^{-2} ln (n/?)). Our sample complexity bounds are derived from novel upper bounds on the variance of the reachability that allow for small relative error for influential sets and additive error when influence is small. Moreover, we provide a data-adaptive method that can detect and utilize fewer simulations on models where it suffices. Finally, we provide an efficient greedy design that computes an (1-1/e-?)-approximate maximizer from simulations and applies to any submodular stochastic diffusion model that satisfies the variance bounds