Zhenliang Zhang

Error Probability Bounds for Balanced Binary Relay Trees

We study the detection error probability associated with a balanced binary relay tree, where the leaves of the tree correspond to N identical and independent sensors. The root of the tree represents a fusion center that makes the overall detection decision. Each of the other nodes in the tree is a relay node that combines two binary messages to form a single output binary message. Only the leaves are sensors. In this way, the information from the sensors is aggregated into the fusion center via the relay nodes. In this context, we describe the evolution of the Type I and Type II error probabilities of the binary data as it propagates from the leaves toward the root. Tight upper and lower bounds for the total error probability at the fusion center as functions of N are derived. These characterize how fast the total error probability converges to 0 with respect to N , even if the individual sensors have error probabilities that converge to 1/2.

String Submodular Functions With Curvature Constraints

Consider the problem of choosing a string of actions to optimize an objective function that is string submodular. It was shown in previous papers that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function, achieves at least a (1 - e-1)-approximation to the optimal strategy. This paper improves this approximation by introducing additional constraints on curvature, namely, total backward curvature, totalforward curvature, and elemental forward curvature. We show that if the objective function has total backward curvature ϵ, then the greedy strategy achieves at least a (1/σ)(1 - e-σ)-approximation of the optimal strategy. If the objective function has total forward curvature e, then the greedy strategy achieves at least a (1 - ϵ)-approximation of the optimal strategy. Moreover, we consider a generalization of the diminishing-return property by defining the elemental forward curvature. We also introduce the notion of string-matroid and consider the problem of maximizing the objective function subject to a string-matroid constraint. We investigate two applications of string submodular functions with curvature constraints: 1) choosing a string of actions to maximize the expected fraction of accomplished tasks; and 2) designing a string of measurement matrices such that the information gain is maximized.

Near optimality of greedy strategies for string submodular functions with forward and backward curvature constraints

The problem of choosing a string of actions to optimize an objective function that is string submodular has been considered in [1]. There it is shown that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function, achieves at least a (1-e-1)-approximation to the optimal strategy. This paper improves this approximation by introducing additional constraints on curvature, namely, total backward curvature, total forward curvature, and elemental forward curvature. We show that if the objective function has total backward curvature σ, then the greedy strategy achieves at least a 1/σ(1 - e-σ)-approximation of the optimal strategy. If the objective function has total forward curvature c, then the greedy strategy achieves at least a (1 - ε)-approximation of the optimal strategy. Moreover, we consider a generalization of the diminishing-return property by defining the elemental forward curvature. We also introduce the notion of string-matroid and consider the problem of maximizing the objective function subject to a string-matroid constraint.

Submodularity and optimality of fusion rules in balanced binary relay trees

We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes the overall decision. Every other node in the tree is a fusion node that fuses two binary messages from its child nodes into a new binary message and sends it to the parent node at the next level. We assume that the fusion nodes at the same level use the same fusion rule. We call a string of fusion rules used at different levels a fusion strategy. We consider the problem of finding a fusion strategy that maximizes the reduction in the total error probability between the sensors and the fusion center. We formulate this problem as a deterministic dynamic program and express the solution in terms of Bellmans equations. We introduce the notion of string-submodularity and show that the reduction in the total error probability is a string-submodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, is within a factor (1 - e-1) of the optimal strategy in terms of reduction in the total error probability.

Hypothesis Testing in Feedforward Networks With Broadcast Failures

Consider a large number of nodes, which sequentially make decisions between two given hypotheses. Each node takes a measurement of the underlying truth, observes the decisions from some immediate predecessors, and makes a decision between the given hypotheses. We consider two classes of broadcast failures: 1) each node broadcasts a decision to the other nodes, subject to random erasure in the form of a binary erasure channel; 2) each node broadcasts a randomly flipped decision to the other nodes in the form of a binary symmetric channel. We are interested in conditions under which there does (or does not) exist a decision strategy consisting of a sequence of likelihood ratio tests such that the node decisions converge in probability to the underlying truth, as the number of nodes goes to infinity. In both cases, we show that if each node only learns from a bounded number of immediate predecessors, then there does not exist a decision strategy such that the decisions converge in probability to the underlying truth. However, in case 1, we show that if each node learns from an unboundedly growing number of predecessors, then there exists a decision strategy such that the decisions converge in probability to the underlying truth, even when the erasure probabilities converge to 1. We show that a locally optimal strategy, consisting of a sequence of Bayesian likelihood ratio tests, is such a strategy, and we derive the convergence rate of the error probability for this strategy. In case 2, we show that if each node learns from all of its previous predecessors, then there exists a decision strategy such that the decisions converge in probability to the underlying truth when the flipping probabilities of the binary symmetric channels are bounded away from 1/2. Again, we show that a locally optimal strategy achieves this, and we derive the convergence rate of the error probability for it. In the case where the flipping probabilities converge to 1/2, we derive a necessary condition on the convergence rate of the flipping probabilities such that the decisions based on the locally optimal strategy still converge to the underlying truth. We also explicitly characterize the relationship between the convergence rate of the error probability and the convergence rate of the flipping probabilities.

Detection Performance in Balanced Binary Relay Trees With Node and Link Failures

We study the distributed detection problem in the context of a balanced binary relay tree, where the leaves of the tree correspond to N identical and independent sensors generating binary messages. The root of the tree is a fusion center making an overall decision. Every other node is a relay node that aggregates the messages received from its child nodes into a new message and sends it up toward the fusion center. We derive upper and lower bounds for the total error probability PN as explicit functions of N in the case where nodes and links fail with certain probabilities. These characterize the asymptotic decay rate of the total error probability as N goes to infinity. Naturally, this decay rate is not larger than that in the non-failure case, which is √N . However, we derive an explicit necessary and sufficient condition on the decay rate of the local failure probabilities pk (combination of node and link failure probabilities at each level) such that the decay rate of the total error probability in the failure case is the same as that of the non-failure case. More precisely, we show that logPN-1=Θ(√N) if and only if logpk-1=Ω(2k/2) .

Error probability bounds for binary relay trees with unreliable communication links

We study the decentralized detection problem in the context of balanced binary relay trees. We assume that the communication links in the tree network fail with certain probabilities. Not surprisingly, the step-wise reduction of the total detection error probability is slower than the case where the network has no communication link failures. We show that, under the assumption of identical communication link failure probability in the tree, the exponent of the total error probability at the fusion center is o(√N) in the asymptotic regime. In addition, if the given communication link failure probabilities decrease to 0 as communications get closer to the fusion center, then the decay exponent of the total error probability is Θ(√N), provided that the decay of the failure probabilities is sufficiently fast.

Detection performance of M-ary relay trees with non-binary message alphabets

We study the detection performance of M-ary relay trees, where only the leaves of the tree represent sensors making measurements. The root of the tree represents the fusion center which makes an overall detection decision. Each of the other nodes is a relay node which aggregates M messages sent by its child nodes into a new compressed message and sends the message to its parent node. Building on previous work on the detection performance of M-ary relay trees with binary messages, in this paper we study the case of non-binary relay message alphabets. We characterize the exponent of the error probability with respect to the message alphabet size D, showing how the detection performance increases with D. Our method involves reducing a tree with non-binary relay messages into an equivalent higher-degree tree with only binary messages.

Performance bounds for the k-batch greedy strategy in optimization problems with curvature

The k-batch greedy strategy is an approximate algorithm to solve optimization problems where the optimal solution is hard to obtain. Starting with the empty set, the k-batch greedy strategy adds a batch of k elements to the current solution set with the largest gain in the objective function while satisfying the constraints. In this paper, we bound the performance of the k-batch greedy strategy with respect to the optimal strategy by defining the total curvature αk. We show that when the objective function is nondecreasing and submodular, the k-batch greedy strategy satisfies a harmonic bound 1/(1 + αk) for a general matroid constraint and an exponential bound (1 - (1 - αk/t)t/αk for a uniform matroid constraint, where k divides the cardinality of the maximal set in the general matroid, t = K/k is an integer, and K is the rank of the uniform matroid. We also compare the performance of the k-batch greedy strategy with that of the k1-batch greedy strategy when k1 divides k. Specifically, we prove that when the objective function is nondecreasing and submodular, the k-batch greedy strategy has better harmonic and exponential bounds in terms of the total curvature. Finally, we illustrate our results by considering a task-assignment problem.

Rate of learning in hierarchical social networks

We study a social network consisting of agents organized as a hierarchical M-ary rooted tree, common in enterprise and military organizational structures. The goal is to aggregate information to solve a binary hypothesis testing problem. Each agent at a leaf of the tree, and only such an agent, makes a direct measurement of the underlying true hypothesis. The leaf agent then generates a binary message and sends it to its supervising agent, at the next level of the tree. Each supervising agent aggregates the messages from the M members of its group, produces a summary message, and sends it to its supervisor at the next level, and so on. Ultimately, the agent at the root of the tree makes an overall decision. We derive upper and lower bounds for the Type I and Type II error probabilities associated with this decision with respect to the number of leaf agents, which in turn characterize the converge rates of the Type I, Type II, and total error probabilities.