We study resilient quantized consensus in multi-agent systems, where some agents may malfunction. The network consists of agents taking integer-valued states, and the agents' communication is subject to asynchronous updates and time delays. We utilize the quantized weighted mean subsequence reduced algorithm where agents communicate with others through multi-hop relays. We prove necessary and sufficient conditions for our algorithm to achieve the objective under the malicious and Byzantine attack models. Our approach has tighter graph conditions compared to the one-hop algorithm and the flooding-based algorithms for binary consensus. Numerical examples verify the efficacy of our algorithm.

We study resilient leader-follower consensus of multi-agent systems (MASs) in the presence of adversarial agents, where agents' communication is modeled by time-varying topologies. The objective is to develop distributed algorithms for the nonfaulty/normal followers to track an arbitrary reference value propagated by a set of leaders while they are in interaction with the unknown adversarial agents. Our approaches are based on the weighted mean subsequence reduced (W-MSR) algorithms with agents being capable to communicate with multi-hop neighbors. Our algorithms can handle agents possessing first-order and second-order dynamics. Moreover, we characterize necessary and sufficient graph conditions for our algorithms to succeed by the novel notion of jointly robust following graphs. Our graph condition is tighter than the sufficient conditions in the literature when agents use only one-hop communication (without relays). Using multi-hop relays, we can enhance robustness of leader-follower networks without increasing communication links and obtain further relaxed graph requirements for our algorithms to succeed. Numerical examples are given to verify the efficacy of our algorithms.

We study the problem of resilient average consensus in multi-agent systems where some of the agents are subject to failures or attacks. The objective of resilient average consensus is for non-faulty/normal agents to converge to the average of their initial values despite the erroneous effects from malicious agents. To this end, we propose a successful distributed iterative resilient average consensus algorithm for the multi-agent networks with general directed topologies. The proposed algorithm has two parts at each iteration: detection and averaging. For the detection part, we propose two distributed algorithms and one of them can detect malicious agents with only the information from direct in-neighbors. For the averaging part, we extend the applicability of an existing averaging algorithm where normal agents can remove the effects from malicious agents so far, after they are detected. Another important feature of our method is that it can handle the case where malicious agents are neighboring and collaborating with each other to mislead the normal ones from averaging. This case cannot be solved by existing detection approaches in related literature. Moreover, our algorithm is efficient in storage usage especially for large-scale networks as each agent only requires the values of neighbors within two hops. Lastly, numerical examples are given to verify the efficacy of the proposed algorithms.

We study a multi-agent resilient consensus problem, where some agents are of the Byzantine type and try to prevent the normal ones from reaching consensus. In our setting, normal agents communicate with each other asynchronously over multi-hop relay channels with delays. To solve this asynchronous Byzantine consensus problem, we develop the multi-hop weighted mean subsequence reduced (MW-MSR) algorithm. The main contribution is that we characterize a tight graph condition for our algorithm to achieve Byzantine consensus, which is expressed in the novel notion of strictly robust graphs. We show that the multi-hop communication is effective for enhancing the network's resilience against Byzantine agents. As a result, we also obtain novel conditions for resilient consensus under the malicious attack model, which are tighter than those known in the literature. Furthermore, the proposed algorithm can be viewed as a generalization of the conventional flooding-based algorithms, with less computational complexity. Lastly, we provide numerical examples to show the effectiveness of the proposed algorithm.

This paper deals with dynamic quantized consensus of dynamical agents in a general form under packet losses induced by Denial-of-Service (DoS) attacks. The communication channel has limited bandwidth and hence the transmitted signals over the network are subject to quantization. To deal with agent's output, an observer is implemented at each node. The state of the observer is quantized by a finite-level quantizer and then transmitted over the network. To solve the problem of quantizer overflow under malicious packet losses, a zooming-in and out dynamic quantization mechanism is designed. By the new quantized controller proposed in the paper, the zooming-out factor is lower bounded by the spectral radius of the agent's dynamic matrix. A sufficient condition of quantization range is provided under which the finite-level quantizer is free of overflow. A sufficient condition of tolerable DoS attacks for achieving consensus is also provided. At last, we study scalar dynamical agents as a special case and further tighten the zooming-out factor to a value smaller than the agent's dynamic parameter. Under such a zooming-out factor, it is possible to recover the level of tolerable DoS attacks to that of unquantized consensus, and the quantizer is free of overflow.

Consensus and cluster forming of multiagent systems in the face of jamming attacks along with reactive recovery actions by a defender are discussed. The attacker is capable to disable some of the edges of the network with the objective to divide the agents into a smaller size of clusters while, in response, the defender recovers some of the edges by increasing the transmission power. We consider repeated games where the resulting optimal strategies for the two players are derived in a rolling horizon fashion. The attacker and the defender possess different computational abilities to calculate their strategies. This aspect is represented by the non-uniform values of the horizon lengths and the game periods. Theoretical and simulation based results demonstrate the effects of the horizon lengths and the game periods on the agents' states.

A two-player game-theoretic problem on resilient graphs in a multiagent consensus setting is formulated. An attacker is capable to disable some of the edges of the network with the objective to divide the agents into clusters by emitting jamming signals while, in response, the defender recovers some of the edges by increasing the transmission power for the communication signals. Specifically, we consider repeated games between the attacker and the defender where the optimal strategies for the two players are derived in a rolling horizon fashion based on utility functions that take both the agents' states and the sizes of clusters (known as network effect) into account. The players' actions at each discrete-time step are constrained by their energy for transmissions of the signals, with a less strict constraint for the attacker. Necessary conditions and sufficient conditions of agent consensus are derived, which are influenced by the energy constraints. The number of clusters of agents at infinite time in the face of attacks and recoveries are also characterized. Simulation results are provided to demonstrate the effects of players' actions on the cluster forming and to illustrate the players' performance for different horizon parameters.

In this paper, we consider a resilient consensus problem for the multi-agent network where some of the agents are subject to Byzantine attacks and may transmit erroneous state values to their neighbors. In particular, we develop an event-triggered update rule to tackle this problem as well as reduce the communication for each agent. Our approach is based on the mean subsequence reduced (MSR) algorithm with agents being capable to communicate with multi-hop neighbors. Since delays are critical in such an environment, we provide necessary graph conditions for the proposed algorithm to perform well with delays in the communication. We highlight that through multi-hop communication, the network connectivity can be reduced especially in comparison with the common onehop communication case. Lastly, we show the effectiveness of the proposed algorithm by a numerical example.

With the proliferation of training data, distributed machine learning (DML) is becoming more competent for large-scale learning tasks. However, privacy concern has to be attached prior importance in DML, since training data may contain sensitive information of users. Most existing privacy-aware schemes are established based on an assumption that the users trust the server collecting their data, and are limited to provide the same privacy guarantee for the entire data sample. In this paper, we remove the trustworthy servers assumption, and propose a privacy-preserving ADMM-based DML framework that preserves heterogeneous privacy for users' data. The new challenging issue is to reduce the accumulation of privacy losses over ADMM iterations as much as possible. In the proposed privacy-aware DML framework, a local randomization approach, which is proved to be differentially private, is adopted to provide users with self-controlled privacy guarantee for the most sensitive information. Further, the ADMM algorithm is perturbed through a combined noise-adding method, which simultaneously preserves privacy for users' less sensitive information and strengthens the privacy protection of the most sensitive information. Also, we analyze the performance of the trained model according to its generalization error. Finally, we conduct extensive experiments using synthetic and real-world datasets to validate the theoretical results and evaluate the classification performance of the proposed framework.

In this paper, we obtain generic bounds on the variances of estimation and prediction errors in time series analysis via an information-theoretic approach. It is seen in general that the error bounds are determined by the conditional entropy of the data point to be estimated or predicted given the side information or past observations. Additionally, we discover that in order to achieve the prediction error bounds asymptotically, the necessary and sufficient condition is that the "innovation" is asymptotically white Gaussian. When restricted to Gaussian processes and 1-step prediction, our bounds are shown to reduce to the Kolmogorov-Szeg\"o formula and Wiener-Masani formula known from linear prediction theory.