8 research outputs found
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 Bellman's equations. We introduce the notion
of stringsubmodularity and show that the reduction in the total error
probability is a stringsubmodular function. Consequentially, we show that the
greedy strategy, which only maximizes the level-wise reduction in the total
error probability, is within a factor of the optimal strategy in terms of
reduction in the total error probability
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 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 as explicit functions of in the case where
nodes and links fail with certain probabilities. These characterize the
asymptotic decay rate of the total error probability as goes to infinity.
Naturally, this decay rate is not larger than that in the non-failure case,
which is . However, we derive an explicit necessary and sufficient
condition on the decay rate of the local failure probabilities
(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 if and only if
Beliefs in Decision-Making Cascades
This work explores a social learning problem with agents having nonidentical
noise variances and mismatched beliefs. We consider an -agent binary
hypothesis test in which each agent sequentially makes a decision based not
only on a private observation, but also on preceding agents' decisions. In
addition, the agents have their own beliefs instead of the true prior, and have
nonidentical noise variances in the private signal. We focus on the Bayes risk
of the last agent, where preceding agents are selfish.
We first derive the optimal decision rule by recursive belief update and
conclude, counterintuitively, that beliefs deviating from the true prior could
be optimal in this setting. The effect of nonidentical noise levels in the
two-agent case is also considered and analytical properties of the optimal
belief curves are given. Next, we consider a predecessor selection problem
wherein the subsequent agent of a certain belief chooses a predecessor from a
set of candidates with varying beliefs. We characterize the decision region for
choosing such a predecessor and argue that a subsequent agent with beliefs
varying from the true prior often ends up selecting a suboptimal predecessor,
indicating the need for a social planner. Lastly, we discuss an augmented
intelligence design problem that uses a model of human behavior from cumulative
prospect theory and investigate its near-optimality and suboptimality.Comment: final version, to appear in IEEE Transactions on Signal Processin
Quantized Consensus by the Alternating Direction Method of Multipliers: Algorithms and Applications
Collaborative in-network processing is a major tenet in the fields of control, signal processing, information theory, and computer science. Agents operating in a coordinated fashion can gain greater efficiency and operational capability than those perform solo missions. In many such applications the central task is to compute the global average of agents\u27 data in a distributed manner. Much recent attention has been devoted to quantized consensus, where, due to practical constraints, only quantized communications are allowed between neighboring nodes in order to achieve the average consensus. This dissertation aims to develop efficient quantized consensus algorithms based on the alternating direction method of multipliers (ADMM) for networked applications, and in particular, consensus based detection in large scale sensor networks.
We study the effects of two commonly used uniform quantization schemes, dithered and deterministic quantizations, on an ADMM based distributed averaging algorithm. With dithered quantization, this algorithm yields linear convergence to the desired average in the mean sense with a bounded variance. When deterministic quantization is employed, the distributed ADMM either converges to a consensus or cycles with a finite period after a finite-time iteration. In the cyclic case, local quantized variables have the same sample mean over one period and hence each node can also reach a consensus. We then obtain an upper bound on the consensus error, which depends only on the quantization resolution and the average degree of the network. This is preferred in large scale networks where the range of agents\u27 data and the size of network may be large.
Noticing that existing quantized consensus algorithms, including the above two, adopt infinite-bit quantizers unless a bound on agents\u27 data is known a priori, we further develop an ADMM based quantized consensus algorithm using finite-bit bounded quantizers for possibly unbounded agents\u27 data. By picking a small enough ADMM step size, this algorithm can obtain the same consensus result as using the unbounded deterministic quantizer. We then apply this algorithm to distributed detection in connected sensor networks where each node can only exchange information with its direct neighbors. We establish that, with each node employing an identical one-bit quantizer for local information exchange, our approach achieves the optimal asymptotic performance of centralized detection. The statement is true under three different detection frameworks: the Bayesian criterion where the maximum a posteriori detector is optimal, the Neyman-Pearson criterion with a constant type-I error constraint, and the Neyman-Pearson criterion with an exponential type-I error constraint. The key to achieving optimal asymptotic performance is the use of a one-bit deterministic quantizer with controllable threshold that results in desired consensus error bounds
The Unlucky broker
2010 - 2011This dissertation collects results of the work on the interpretation, characteri-
zation and quanti cation of a novel topic in the eld of detection theory -the
Unlucky Broker problem-, and its asymptotic extension. The same problem can be also applied to the context of Wireless Sensor
Networks (WSNs). Suppose that a WSN is engaged in a binary detection task.
Each node of the system collects measurements about the state of the nature
(H0 or H1) to be discovered. A common fusion center receives the observations
from the sensors and implements an optimal test (for example in the Bayesian
sense), exploiting its knowledge of the a-priori probabilities of the hypotheses.
Later, the priors used in the test are revealed to be inaccurate and a rened pair
is made available. Unfortunately, at that time, only a subset of the original data
is still available, along with the original decision. In the thesis, we formulate the problem in statistical terms and we consider
a system made of n sensors engaged in a binary detection task. A successive
reduction of data set's cardinality occurs and multiple re nements are required.
The sensors are devices programmed to take the decision from the previous
node in the chain and the available data, implement some simple test to decide
between the hypotheses, and forward the resulting decision to the next node.
The rst part of the thesis shows that the optimal test is very di cult to be
implemented even with only two nodes (the unlucky broker problem), because
of the strong correlation between the available data and the decision coming
from the previous node. Then, to make the designed detector implementable
in practice and to ensure analytical tractability, we consider suboptimal local
tests.
We choose a simple local decision strategy, following the rationale ruling the
optimal detector solving the unlucky broker problem: A decision in favor of H0
is always retained by the current node, while when the decision of the previous
node is in favor of H1, a local log-likelihood based test is implemented.
The main result is that, asymptotically, if we set the false alarm probability
of the rst node (the one observing the full data set) the false alarm probability
decreases along the chain and it is non zero at the last stage. Moreover, very
surprisingly, the miss detection probability decays exponentially fast with the
root square of the number of nodes and we provide its closed-form exponent, by
exploiting tools from random processes and information theory. [edited by the author]X n.s