Sequential Testing for Sparse Recovery

This paper studies sequential methods for recovery of sparse signals in high dimensions. When compared to fixed sample size procedures, in the sparse setting, sequential methods can result in a large reduction in the number of samples needed for reliable signal support recovery. Starting with a lower bound, we show any coordinate-wise sequential sampling procedure fails in the high dimensional limit provided the average number of measurements per dimension is less then log s/D(P_0||P_1) where s is the level of sparsity and D(P_0||P_1) the Kullback-Leibler divergence between the underlying distributions. A series of Sequential Probability Ratio Tests (SPRT) which require complete knowledge of the underlying distributions is shown to achieve this bound. Motivated by real world experiments and recent work in adaptive sensing, we introduce a simple procedure termed Sequential Thresholding which can be implemented when the underlying testing problem satisfies a monotone likelihood ratio assumption. Sequential Thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than log s/ D(P_0||P_1), achieving the lower bound. For comparison, we show any non-sequential procedure fails provided the number of measurements grows at a rate less than log n/D(P_1||P_0), where n is the total dimension of the problem

S2: An Efficient Graph Based Active Learning Algorithm with Application to Nonparametric Classification

This paper investigates the problem of active learning for binary label prediction on a graph. We introduce a simple and label-efficient algorithm called S2 for this task. At each step, S2 selects the vertex to be labeled based on the structure of the graph and all previously gathered labels. Specifically, S2 queries for the label of the vertex that bisects the *shortest shortest* path between any pair of oppositely labeled vertices. We present a theoretical estimate of the number of queries S2 needs in terms of a novel parametrization of the complexity of binary functions on graphs. We also present experimental results demonstrating the performance of S2 on both real and synthetic data. While other graph-based active learning algorithms have shown promise in practice, our algorithm is the first with both good performance and theoretical guarantees. Finally, we demonstrate the implications of the S2 algorithm to the theory of nonparametric active learning. In particular, we show that S2 achieves near minimax optimal excess risk for an important class of nonparametric classification problems.Comment: A version of this paper appears in the Conference on Learning Theory (COLT) 201

Network Inference from Co-Occurrences

The recovery of network structure from experimental data is a basic and fundamental problem. Unfortunately, experimental data often do not directly reveal structure due to inherent limitations such as imprecision in timing or other observation mechanisms. We consider the problem of inferring network structure in the form of a directed graph from co-occurrence observations. Each observation arises from a transmission made over the network and indicates which vertices carry the transmission without explicitly conveying their order in the path. Without order information, there are an exponential number of feasible graphs which agree with the observed data equally well. Yet, the basic physical principles underlying most networks strongly suggest that all feasible graphs are not equally likely. In particular, vertices that co-occur in many observations are probably closely connected. Previous approaches to this problem are based on ad hoc heuristics. We model the experimental observations as independent realizations of a random walk on the underlying graph, subjected to a random permutation which accounts for the lack of order information. Treating the permutations as missing data, we derive an exact expectation-maximization (EM) algorithm for estimating the random walk parameters. For long transmission paths the exact E-step may be computationally intractable, so we also describe an efficient Monte Carlo EM (MCEM) algorithm and derive conditions which ensure convergence of the MCEM algorithm with high probability. Simulations and experiments with Internet measurements demonstrate the promise of this approach.Comment: Submitted to IEEE Transactions on Information Theory. An extended version is available as University of Wisconsin Technical Report ECE-06-

High-Dimensional Matched Subspace Detection When Data are Missing

We consider the problem of deciding whether a highly incomplete signal lies within a given subspace. This problem, Matched Subspace Detection, is a classical, well-studied problem when the signal is completely observed. High- dimensional testing problems in which it may be prohibitive or impossible to obtain a complete observation motivate this work. The signal is represented as a vector in R^n, but we only observe m << n of its elements. We show that reliable detection is possible, under mild incoherence conditions, as long as m is slightly greater than the dimension of the subspace in question

Linear Bandits with Feature Feedback

This paper explores a new form of the linear bandit problem in which the algorithm receives the usual stochastic rewards as well as stochastic feedback about which features are relevant to the rewards, the latter feedback being the novel aspect. The focus of this paper is the development of new theory and algorithms for linear bandits with feature feedback. We show that linear bandits with feature feedback can achieve regret over time horizon $T$ that scales like $k\sqrt{T}$, without prior knowledge of which features are relevant nor the number $k$ of relevant features. In comparison, the regret of traditional linear bandits is $d\sqrt{T}$, where $d$ is the total number of (relevant and irrelevant) features, so the improvement can be dramatic if $k\ll d$. The computational complexity of the new algorithm is proportional to $k$ rather than $d$, making it much more suitable for real-world applications compared to traditional linear bandits. We demonstrate the performance of the new algorithm with synthetic and real human-labeled data