Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)


Electrical Engineering and Computer Science


Pramod K. Varshney


Byzantines, Corrupted Data, Distributed Inference, Distributed Learning

Subject Categories



We are living in an increasingly networked world with sensing networks of varying shapes and sizes: the network often comprises of several tiny devices (or nodes) communicating with each other via different topologies. To make the problem even more complicated, the nodes in the network can be unreliable due to a variety of reasons: noise, faults and attacks, thus, providing

corrupted data. Although the area of statistical inference has been an active area of research in the

past, distributed learning and inference in a networked setup with potentially unreliable components

has only gained attention recently. The emergence of big and dirty data era demands new

distributed learning and inference solutions to tackle the problem of inference with corrupted data.

Distributed inference networks (DINs) consist of a group of networked entities which acquire

observations regarding a phenomenon of interest (POI), collaborate with other entities in the network

by sharing their inference via different topologies to make a global inference. The central

goal of this thesis is to analyze the effect of corrupted (or falsified) data on the inference performance

of DINs and design robust strategies to ensure reliable overall performance for several

practical network architectures. Specifically, the inference (or learning) process can be that of detection

or estimation or classification, and the topology of the system can be parallel, hierarchical

or fully decentralized (peer to peer).

Note that, the corrupted data model may seem similar to the scenario where local decisions

are transmitted over a Binary Symmetric Channel (BSC) with a certain cross over probability,

however, there are fundamental differences. Over the last three decades, research community

has extensively studied the impact of transmission channels or faults on the distributed detection

system and related problems due to its importance in several applications. However, corrupted

(Byzantine) data models considered in this thesis, are philosophically different from the BSC or

the faulty sensor cases. Byzantines are intentional and intelligent, therefore, they can optimize

over the data corruption parameters. Thus, in contrast to channel aware detection, both the FC and

the Byzantines can optimize their utility by choosing their actions based on the knowledge of their

opponent’s behavior. Study of these practically motivated scenarios in the presence of Byzantines

is of utmost importance, and is missing from the channel aware detection and fault tolerant detection

literature. This thesis advances the distributed inference literature by providing fundamental

limits of distributed inference with Byzantine data and provides optimal counter-measures (using

the insights provided by these fundamental limits) from a network designer’s perspective. Note

that, the analysis of problems related to strategical interaction between Byzantines and network

designed is very challenging (NP-hard is many cases). However, we show that by utilizing the

properties of the network architecture, efficient solutions can be obtained. Specifically, we found

that several problems related to the design of optimal counter-measures in the inference context

are, in fact, special cases of these NP-hard problems which can be solved in polynomial time.

First, we consider the problem of distributed Bayesian detection in the presence of data falsification

(or Byzantine) attacks in the parallel topology. Byzantines considered in this thesis are those

nodes that are compromised and reprogrammed by an adversary to transmit false information to

a centralized fusion center (FC) to degrade detection performance. We show that above a certain

fraction of Byzantine attackers in the network, the detection scheme becomes completely incapable

(or blind) of utilizing the sensor data for detection. When the fraction of Byzantines is not

sufficient to blind the FC, we also provide closed form expressions for the optimal attacking strategies

for the Byzantines that most degrade the detection performance. Optimal attacking strategies

in certain cases have the minimax property and, therefore, the knowledge of these strategies has

practical significance and can be used to implement a robust detector at the FC.

In several practical situations, parallel topology cannot be implemented due to limiting factors,

such as, the FC being outside the communication range of the nodes and limited energy budget of

the nodes. In such scenarios, a multi-hop network is employed, where nodes are organized hierarchically

into multiple levels (tree networks). Next, we study the problem of distributed inference

in tree topologies in the presence of Byzantines under several practical scenarios. We analytically

characterize the effect of Byzantines on the inference performance of the system. We also look at

the possible counter-measures from the FC’s perspective to protect the network from these Byzantines.

These counter-measures are of two kinds: Byzantine identification schemes and Byzantine

tolerant schemes. Using learning based techniques, Byzantine identification schemes are designed

that learn the identity of Byzantines in the network and use this information to improve system

performance. For scenarios where this is not possible, Byzantine tolerant schemes, which use

game theory and error-correcting codes, are developed that tolerate the effect of Byzantines while

maintaining a reasonably good inference performance in the network.

Going a step further, we also consider scenarios where a centralized FC is not available. In

such scenarios, a solution is to employ detection approaches which are based on fully distributed

consensus algorithms, where all of the nodes exchange information only with their neighbors. For

such networks, we analytically characterize the negative effect of Byzantines on the steady-state

and transient detection performance of conventional consensus-based detection schemes. To avoid

performance deterioration, we propose a distributed weighted average consensus algorithm that is

robust to Byzantine attacks. Next, we exploit the statistical distribution of the nodes’ data to devise

techniques for mitigating the influence of data falsifying Byzantines on the distributed detection

system. Since some parameters of the statistical distribution of the nodes’ data might not be known

a priori, we propose learning based techniques to enable an adaptive design of the local fusion or

update rules.

The above considerations highlight the negative effect of the corrupted data on the inference

performance. However, it is possible for a system designer to utilize the corrupted data for network’s

benefit. Finally, we consider the problem of detecting a high dimensional signal based on

compressed measurements with secrecy guarantees. We consider a scenario where the network

operates in the presence of an eavesdropper who wants to discover the state of the nature being

monitored by the system. To keep the data secret from the eavesdropper, we propose to use cooperating

trustworthy nodes that assist the FC by injecting corrupted data in the system to deceive the

eavesdropper. We also design the system by determining the optimal values of parameters which

maximize the detection performance at the FC while ensuring perfect secrecy at the eavesdropper.


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