arXiv subjects

Nitin H. Vaidya

Publications and source records attributed to Nitin H. Vaidya.

At least 19 recordsLinked to original sources

arXiv2026

Byzantine Consensus in Directed Graphs with Message Authentication

We consider the problem of reaching consensus in communication networks that are modeled by directed graphs. We assume the existence of a message authentication mechanism (such as digital signatures) to verify the integrity of messages. We identify the necessary and sufficient conditions on the directed communication graph for the following problems to be solvable: (i) exact consensus in synchronous systems; and (ii) approximate consensus in asynchronous systems.

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arXiv2024

Approximate Byzantine Fault-Tolerance in Distributed Optimization

This paper considers the problem of Byzantine fault-tolerance in distributed multi-agent optimization. In this problem, each agent has a local cost function, and in the fault-free case, the goal is to design a distributed algorithm that allows all the agents to find a minimum point of all the agents' aggregate cost function. We consider a scenario where some agents might be Byzantine faulty that renders the original goal of computing a minimum point of all the agents' aggregate cost vacuous. A more reasonable objective for an algorithm in this scenario is to allow all the non-faulty agents to compute the minimum point of only the non-faulty agents' aggregate cost. Prior work shows that if there are up to $f$ (out of $n$) Byzantine agents then a minimum point of the non-faulty agents' aggregate cost can be computed exactly if and only if the non-faulty agents' costs satisfy a certain redundancy property called $2f$-redundancy. However, $2f$-redundancy is an ideal property that can be satisfied only in systems free from noise or uncertainties, which can make the goal of exact fault-tolerance unachievable in some applications. Thus, we introduce the notion of $(f,ε)$-resilience, a generalization of exact fault-tolerance wherein the objective is to find an approximate minimum point of the non-faulty aggregate cost, with $ε$ accuracy. This approximate fault-tolerance can be achieved under a weaker condition that is easier to satisfy in practice, compared to $2f$-redundancy. We obtain necessary and sufficient conditions for achieving $(f,ε)$-resilience characterizing the correlation between relaxation in redundancy and approximation in resilience. In case when the agents' cost functions are differentiable, we obtain conditions for $(f,ε)$-resilience of the distributed gradient-descent method when equipped with robust gradient aggregation.

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arXiv2024

Byzantine fault-tolerant distributed set intersection with redundancy

In this report, we study the problem of Byzantine fault-tolerant distributed set intersection and the importance of redundancy in solving this problem. Specifically, consider a distributed system with $n$ agents, each of which has a local set. There are up to $f$ agents that are Byzantine faulty. The goal is to find the intersection of the sets of the non-faulty agents. We derive the Byzantine set intersection problem from the Byzantine optimization problem. We present the definition of $2f$-redundancy, and identify the necessary and sufficient condition if the Byzantine set intersection problem can be solved if a certain redundancy property is satisfied, and then present an equivalent condition. We further extend our results to arbitrary communication graphs in a decentralized setting. Finally, we present solvability results for the Byzantine optimization problem, inspired by our findings on Byzantine set intersection. The results we provide are for synchronous and asynchronous systems both.

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arXiv2023

Impact of Redundancy on Resilience in Distributed Optimization and Learning

This report considers the problem of resilient distributed optimization and stochastic learning in a server-based architecture. The system comprises a server and multiple agents, where each agent has its own local cost function. The agents collaborate with the server to find a minimum of the aggregate of the local cost functions. In the context of stochastic learning, the local cost of an agent is the loss function computed over the data at that agent. In this report, we consider this problem in a system wherein some of the agents may be Byzantine faulty and some of the agents may be slow (also called stragglers). In this setting, we investigate the conditions under which it is possible to obtain an "approximate" solution to the above problem. In particular, we introduce the notion of $(f, r; ε)$-resilience to characterize how well the true solution is approximated in the presence of up to $f$ Byzantine faulty agents, and up to $r$ slow agents (or stragglers) -- smaller $ε$ represents a better approximation. We also introduce a measure named $(f, r; ε)$-redundancy to characterize the redundancy in the cost functions of the agents. Greater redundancy allows for a better approximation when solving the problem of aggregate cost minimization. In this report, we constructively show (both theoretically and empirically) that $(f, r; \mathcal{O}(ε))$-resilience can indeed be achieved in practice, given that the local cost functions are sufficiently redundant.

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arXiv2021

Byzantine Consensus in Directed Hypergraphs

Byzantine consensus is a classical problem in distributed computing. Each node in a synchronous system starts with a binary input. The goal is to reach agreement in the presence of Byzantine faulty nodes. We consider the setting where communication between nodes is modelled via a directed hypergraph. In the classical point-to-point communication model, the communication between nodes is modelled as a simple graph where all messages sent on an edge are private between the two endpoints of the edge. This allows a faulty node to equivocate, i.e., lie differently to its different neighbors. Different models have been proposed in the literature that weaken equivocation. In the local broadcast model, every message transmitted by a node is received identically and correctly by all of its neighbors. In the hypergraph model, every message transmitted by a node on a hyperedge is received identically and correctly by all nodes on the hyperedge. Tight network conditions are known for each of the three cases for undirected (hyper)graphs. For the directed models, tight conditions are known for the point-to-point and local broadcast models. In this work, we consider the directed hypergraph model that encompasses all the models above. Each directed hyperedge consists of a single head (sender) and at least one tail (receiver), This models a local multicast channel where messages transmitted by the sender are received identically by all the receivers in the hyperedge. For this model, we identify tight network conditions for consensus. We observe how the directed hypergraph model reduces to each of the three models above under specific conditions. In each case, we relate our network condition to the corresponding known tight conditions. The directed hypergraph model also encompasses other practical network models of interest that have not been explored previously, as elaborated in the paper.

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arXiv2021

Asynchronous Distributed Optimization with Redundancy in Cost Functions

This paper considers the problem of asynchronous distributed multi-agent optimization on server-based system architecture. In this problem, each agent has a local cost, and the goal for the agents is to collectively find a minimum of their aggregate cost. A standard algorithm to solve this problem is the iterative distributed gradient-descent (DGD) method being implemented collaboratively by the server and the agents. In the synchronous setting, the algorithm proceeds from one iteration to the next only after all the agents complete their expected communication with the server. However, such synchrony can be expensive and even infeasible in real-world applications. We show that waiting for all the agents is unnecessary in many applications of distributed optimization, including distributed machine learning, due to redundancy in the cost functions (or {\em data}). Specifically, we consider a generic notion of redundancy named $(r,ε)$-redundancy implying solvability of the original multi-agent optimization problem with $ε$ accuracy, despite the removal of up to $r$ (out of total $n$) agents from the system. We present an asynchronous DGD algorithm where in each iteration the server only waits for (any) $n-r$ agents, instead of all the $n$ agents. Assuming $(r,ε)$-redundancy, we show that our asynchronous algorithm converges to an approximate solution with error that is linear in $ε$ and $r$. Moreover, we also present a generalization of our algorithm to tolerate some Byzantine faulty agents in the system. Finally, we demonstrate the improved communication efficiency of our algorithm through experiments on MNIST and Fashion-MNIST using the benchmark neural network LeNet.

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arXiv2021

Byzantine Fault-Tolerant Distributed Machine Learning Using Stochastic Gradient Descent (SGD) and Norm-Based Comparative Gradient Elimination (CGE)

This paper considers the Byzantine fault-tolerance problem in distributed stochastic gradient descent (D-SGD) method - a popular algorithm for distributed multi-agent machine learning. In this problem, each agent samples data points independently from a certain data-generating distribution. In the fault-free case, the D-SGD method allows all the agents to learn a mathematical model best fitting the data collectively sampled by all agents. We consider the case when a fraction of agents may be Byzantine faulty. Such faulty agents may not follow a prescribed algorithm correctly, and may render traditional D-SGD method ineffective by sharing arbitrary incorrect stochastic gradients. We propose a norm-based gradient-filter, named comparative gradient elimination (CGE), that robustifies the D-SGD method against Byzantine agents. We show that the CGE gradient-filter guarantees fault-tolerance against a bounded fraction of Byzantine agents under standard stochastic assumptions, and is computationally simpler compared to many existing gradient-filters such as multi-KRUM, geometric median-of-means, and the spectral filters. We empirically show, by simulating distributed learning on neural networks, that the fault-tolerance of CGE is comparable to that of existing gradient-filters. We also empirically show that exponential averaging of stochastic gradients improves the fault-tolerance of a generic gradient-filter.

cs.LG↗

arXiv2021

Byzantine Fault-Tolerance in Peer-to-Peer Distributed Gradient-Descent

We consider the problem of Byzantine fault-tolerance in the peer-to-peer (P2P) distributed gradient-descent method -- a prominent algorithm for distributed optimization in a P2P system. In this problem, the system comprises of multiple agents, and each agent has a local cost function. In the fault-free case, when all the agents are honest, the P2P distributed gradient-descent method allows all the agents to reach a consensus on a solution that minimizes their aggregate cost. However, we consider a scenario where a certain number of agents may be Byzantine faulty. Such faulty agents may not follow an algorithm correctly, and may share arbitrary incorrect information to prevent other non-faulty agents from solving the optimization problem. In the presence of Byzantine faulty agents, a more reasonable goal is to allow all the non-faulty agents to reach a consensus on a solution that minimizes the aggregate cost of all the non-faulty agents. We refer to this fault-tolerance goal as $f$-resilience where $f$ is the maximum number of Byzantine faulty agents in a system of $n$ agents, with $f < n$. Most prior work on fault-tolerance in P2P distributed optimization only consider approximate fault-tolerance wherein, unlike $f$-resilience, all the non-faulty agents' compute a minimum point of a non-uniformly weighted aggregate of their cost functions. We propose a fault-tolerance mechanism that confers provable $f$-resilience to the P2P distributed gradient-descent method, provided the non-faulty agents satisfy the necessary condition of $2f$-redundancy, defined later in the paper. Moreover, compared to prior work, our algorithm is applicable to a larger class of high-dimensional convex distributed optimization problems.

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arXiv2020

Preserving Statistical Privacy in Distributed Optimization

We present a distributed optimization protocol that preserves statistical privacy of agents' local cost functions against a passive adversary that corrupts some agents in the network. The protocol is a composition of a distributed ``{\em zero-sum}" obfuscation protocol that obfuscates the agents' local cost functions, and a standard non-private distributed optimization method. We show that our protocol protects the statistical privacy of the agents' local cost functions against a passive adversary that corrupts up to $t$ arbitrary agents as long as the communication network has $(t+1)$-vertex connectivity. The ``{\em zero-sum}" obfuscation protocol preserves the sum of the agents' local cost functions and therefore ensures accuracy of the computed solution.

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arXiv2020

Improved Extension Protocols for Byzantine Broadcast and Agreement

Byzantine broadcast (BB) and Byzantine agreement (BA) are two most fundamental problems and essential building blocks in distributed computing, and improving their efficiency is of interest to both theoreticians and practitioners. In this paper, we study extension protocols of BB and BA, i.e., protocols that solve BB/BA with long inputs of $l$ bits using lower costs than $l$ single-bit instances. We present new protocols with improved communication complexity in almost all settings: authenticated BA/BB with $t<n/2$, authenticated BB with $t<(1-ε)n$, unauthenticated BA/BB with $t<n/3$, and asynchronous reliable broadcast and BA with $t<n/3$. The new protocols are advantageous and significant in several aspects. First, they achieve the best-possible communication complexity of $Θ(nl)$ for wider ranges of input sizes compared to prior results. Second, the authenticated extension protocols achieve optimal communication complexity given the current best available BB/BA protocols for short messages. Third, to the best of our knowledge, our asynchronous and authenticated protocols in the setting are the first extension protocols in that setting.

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arXiv2020

Byzantine Fault-Tolerance in Decentralized Optimization under Minimal Redundancy

This paper considers the problem of Byzantine fault-tolerance in multi-agent decentralized optimization. In this problem, each agent has a local cost function. The goal of a decentralized optimization algorithm is to allow the agents to cooperatively compute a common minimum point of their aggregate cost function. We consider the case when a certain number of agents may be Byzantine faulty. Such faulty agents may not follow a prescribed algorithm, and they may share arbitrary or incorrect information with other non-faulty agents. Presence of such Byzantine agents renders a typical decentralized optimization algorithm ineffective. We propose a decentralized optimization algorithm with provable exact fault-tolerance against a bounded number of Byzantine agents, provided the non-faulty agents have a minimal redundancy.

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arXiv2020

Asynchronous Byzantine Approximate Consensus in Directed Networks

In this work, we study the approximate consensus problem in asynchronous message-passing networks where some nodes may become Byzantine faulty. We answer an open problem raised by Tseng and Vaidya, 2012, proposing the first algorithm of optimal resilience for directed networks. Interestingly, our results show that the tight condition on the underlying communication networks for asynchronous Byzantine approximate consensus coincides with the tight condition for synchronous Byzantine exact consensus. Our results can be viewed as a non-trivial generalization of the algorithm by Abraham et al., 2004, which applies to the special case of complete networks. The tight condition and techniques identified in the paper shed light on the fundamental properties for solving approximate consensus in asynchronous directed networks.

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arXiv2020

A Private and Finite-Time Algorithm for Solving a Distributed System of Linear Equations

This paper studies a system of linear equations, denoted as $Ax = b$, which is horizontally partitioned (rows in $A$ and $b$) and stored over a network of $m$ devices connected in a fixed directed graph. We design a fast distributed algorithm for solving such a partitioned system of linear equations, that additionally, protects the privacy of local data against an honest-but-curious adversary that corrupts at most $τ$ nodes in the network. First, we present TITAN, privaTe fInite Time Average coNsensus algorithm, for solving a general average consensus problem over directed graphs, while protecting statistical privacy of private local data against an honest-but-curious adversary. Second, we propose a distributed linear system solver that involves each agent/devices computing an update based on local private data, followed by private aggregation using TITAN. Finally, we show convergence of our solver to the least squares solution in finite rounds along with statistical privacy of local linear equations against an honest-but-curious adversary provided the graph has weak vertex-connectivity of at least $τ+1$. We perform numerical experiments to validate our claims and compare our solution to the state-of-the-art methods by comparing computation, communication and memory costs.

math.OC↗

arXiv2020

Resilience in Collaborative Optimization: Redundant and Independent Cost Functions

This report considers the problem of Byzantine fault-tolerance in multi-agent collaborative optimization. In this problem, each agent has a local cost function. The goal of a collaborative optimization algorithm is to compute a minimum of the aggregate of the agents' cost functions. We consider the case when a certain number of agents may be Byzantine faulty. Such faulty agents may not follow a prescribed algorithm, and they may send arbitrary or incorrect information regarding their local cost functions. A reasonable goal in presence of such faulty agents is to minimize the aggregate cost of the non-faulty agents. In this report, we show that this goal can be achieved if and only if the cost functions of the non-faulty agents have a minimal redundancy property. We present different algorithms that achieve such tolerance against faulty agents, and demonstrate a trade-off between the complexity of an algorithm and the properties of the agents' cost functions. Further, we also consider the case when the cost functions are independent or do not satisfy the minimal redundancy property. In that case, we quantify the tolerance against faulty agents by introducing a metric called weak resilience. We present an algorithm that attains weak resilience when the faulty agents are in the minority and the cost functions are non-negative.

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arXiv2019

Randomized Reactive Redundancy for Byzantine Fault-Tolerance in Parallelized Learning

This report considers the problem of Byzantine fault-tolerance in synchronous parallelized learning that is founded on the parallelized stochastic gradient descent (parallelized-SGD) algorithm. The system comprises a master, and $n$ workers, where up to $f$ of the workers are Byzantine faulty. Byzantine workers need not follow the master's instructions correctly, and might send malicious incorrect (or faulty) information. The identity of the Byzantine workers remains fixed throughout the learning process, and is unknown a priori to the master. We propose two coding schemes, a deterministic scheme and a randomized scheme, for guaranteeing exact fault-tolerance if $2f < n$. The coding schemes use the concept of reactive redundancy for isolating Byzantine workers that eventually send faulty information. We note that the computation efficiencies of the schemes compare favorably with other (deterministic or randomized) coding schemes, for exact fault-tolerance.

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arXiv2019

Exact Byzantine Consensus on Arbitrary Directed Graphs under Local Broadcast Model

We consider Byzantine consensus in a synchronous system where nodes are connected by a network modeled as a directed graph, i.e., communication links between neighboring nodes are not necessarily bi-directional. The directed graph model is motivated by wireless networks wherein asymmetric communication links can occur. In the classical point-to-point communication model, a message sent on a communication link is private between the two nodes on the link. This allows a Byzantine faulty node to equivocate, i.e., send inconsistent information to its neighbors. This paper considers the local broadcast model of communication, wherein transmission by a node is received identically by all of its outgoing neighbors. This allows such neighbors to detect a faulty node's attempt to equivocate, effectively depriving the faulty nodes of the ability to send conflicting information to different neighbors. Prior work has obtained sufficient and necessary conditions on undirected graphs to be able to achieve Byzantine consensus under the local broadcast model. In this paper, we obtain tight conditions on directed graphs to be able to achieve Byzantine consensus with binary inputs under the local broadcast model. The results obtained in the paper provide insights into the trade-off between directionality of communication and the ability to achieve consensus.

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arXiv2019

Partially Replicated Causally Consistent Shared Memory: Lower Bounds and An Algorithm

The focus of this paper is on causal consistency in a {\em partially replicated} distributed shared memory (DSM) system that provides the abstraction of shared read/write registers. Maintaining causal consistency in distributed shared memory systems has received significant attention in the past, mostly on {\em full replication} wherein each replica stores a copy of all the registers in the shared memory. To ensure causal consistency, all causally preceding updates must be performed before an update is performed at any given replica. Therefore, some mechanism for tracking causal dependencies is required, such as vector timestamps with the number of vector elements being equal to the number of replicas in the context of full replication. In this paper, we investigate causal consistency in {\em partially replicated systems}, wherein each replica may store only a subset of the shared registers. Building on the past work, this paper makes three key contributions: 1. We present a necessary condition on the metadata (which we refer as a {\em timestamp}) that must be maintained by each replica to be able to track causality accurately. The necessary condition identifies a set of directed edges in a {\em share graph} that a replica's timestamp must keep track of. 2. We present an algorithm for achieving causal consistency using a timestamp that matches the above necessary condition, thus showing that the condition is necessary and sufficient. 3. We define a measurement of timestamp space size and present a lower bound (in bits) on the size of the timestamps. The lower bound matches our algorithm in several special cases.

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arXiv2019

Exact Byzantine Consensus on Undirected Graphs under Local Broadcast Model

This paper considers the Byzantine consensus problem for nodes with binary inputs. The nodes are interconnected by a network represented as an undirected graph, and the system is assumed to be synchronous. Under the classical point-to-point communication model, it is well-known [7] that the following two conditions are both necessary and sufficient to achieve Byzantine consensus among $n$ nodes in the presence of up to $f$ Byzantine faulty nodes: $n \ge 3f+1$ and vertex connectivity at least $2f+1$. In the classical point-to-point communication model, it is possible for a faulty node to equivocate, i.e., transmit conflicting information to different neighbors. Such equivocation is possible because messages sent by a node to one of its neighbors are not overheard by other neighbors. This paper considers the local broadcast model. In contrast to the point-to-point communication model, in the local broadcast model, messages sent by a node are received identically by all of its neighbors. Thus, under the local broadcast model, attempts by a node to send conflicting information can be detected by its neighbors. Under this model, we show that the following two conditions are both necessary and sufficient for Byzantine consensus: vertex connectivity at least $\lfloor 3f/2 \rfloor + 1$ and minimum node degree at least $2f$. Observe that the local broadcast model results in a lower requirement for connectivity and the number of nodes $n$, as compared to the point-to-point communication model. We extend the above results to a hybrid model that allows some of the Byzantine faulty nodes to equivocate. The hybrid model bridges the gap between the point-to-point and local broadcast models, and helps to precisely characterize the trade-off between equivocation and network requirements.

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