Carlo Fischione

Federico Boccardi, Hossein Shokri-Ghadikolaei, Gabor Fodor et al.

Motivated by the intrinsic characteristics of mmWave technologies, we discuss the possibility of an authorization regime that allows spectrum sharing between multiple operators, also referred to as spectrum pooling. In particular, considering user rate as the performance measure, we assess the benefit of coordination among the networks of different operators, study the impact of beamforming both at the base stations and at the user terminals, and analyze the pooling performance at different frequency carriers. We also discuss the enabling spectrum mechanisms, architectures, and protocols required to make spectrum pooling work in real networks. Our initial results show that, from a technical perspective, spectrum pooling at mmWave has the potential for a more efficient spectrum use than a traditional exclusive spectrum allocation to a single operator. However, further studies are needed in order to reach a thorough understanding of this matter, and we hope that this paper will help stimulate further research in this area.

Sindri Magnusson, Chinwendu Enyioha, Kathryn Heal et al.

Typical coordination schemes for future power grids require two-way communications. Since the number of end power-consuming devices is large, the bandwidth requirements for such two-way communication schemes may be prohibitive. Motivated by this observation, we study distributed coordination schemes that require only one-way limited communications. In particular, we investigate how dual descent distributed optimization algorithm can be employed in power networks using one-way communication. In this iterative algorithm, system coordinators broadcast coordinating (or pricing) signals to the users/devices who update power consumption based on the received signal. Then system coordinators update the coordinating signals based on the physical measurement of the aggregate power usage. We provide conditions to guarantee the feasibility of the aggregated power usage at each iteration so as to avoid blackout. Furthermore, we prove the convergence of algorithms under these conditions, and establish its rate of convergence. We illustrate the performance of our algorithms using numerical simulations. These results show that one-way limited communication may be viable for coordinating/operating the future smart grids.

Oscar Stenhammar, Gabor Fodor, Carlo Fischione

The performance of modern wireless communications systems depends critically on the quality of the available channel state information (CSI) at the transmitter and receiver. Several previous works have proposed concepts and algorithms that help maintain high quality CSI even in the presence of high mobility and channel aging, such as temporal prediction schemes that employ neural networks. However, it is still unclear which neural network-based scheme provides the best performance in terms of prediction quality, training complexity and practical feasibility. To investigate such a question, this paper first provides an overview of state-of-the-art neural networks applicable to channel prediction and compares their performance in terms of prediction quality. Next, a new comparative analysis is proposed for four promising neural networks with different prediction horizons. The well-known tapped delay channel model recommended by the Third Generation Partnership Program is used for a standardized comparison among the neural networks. Based on this comparative evaluation, the advantages and disadvantages of each neural network are discussed and guidelines for selecting the best-suited neural network in channel prediction applications are given.

Chinwendu Enyioha, Sindri Magnusson, Kathryn Heal et al.

As independent service providers shift from conventional energy to renewable energy sources, the power distribution system will likely experience increasingly significant fluctuation in supply, given the uncertain and intermittent nature of renewable sources like wind and solar energy. These fluctuations in power generation, coupled with time-varying consumer demands of electricity and the massive scale of power distribution networks present the need to not only design real-time decentralized power allocation algorithms, but also characterize how effective they are given fast-changing consumer demands and power generation capacities. In this paper, we present an Online Decentralized Dual Descent (OD3) power allocation algorithm and determine (in the worst case) how much of observed social welfare and price volatility can be explained by fluctuations in generation capacity and consumer demand. Convergence properties and performance guarantees of the OD3 algorithm are analyzed by characterizing the difference between the online decision and the optimal decision. The theoretical results in the paper are validated and illustrated by numerical experiments using real data.

Saeed Razavikia, Jaume Anguera Peris, Jose Mairton B. da Silva et al.

Federated Edge Learning (FEEL) is a distributed machine learning technique where each device contributes to training a global inference model by independently performing local computations with their data. More recently, FEEL has been merged with over-the-air computation (OAC), where the global model is calculated over the air by leveraging the superposition of analog signals. However, when implementing FEEL with OAC, there is the challenge on how to precode the analog signals to overcome any time misalignment at the receiver. In this work, we propose a novel synchronization-free method to recover the parameters of the global model over the air without requiring any prior information about the time misalignments. For that, we construct a convex optimization based on the norm minimization problem to directly recover the global model by solving a convex semi-definite program. The performance of the proposed method is evaluated in terms of accuracy and convergence via numerical experiments. We show that our proposed algorithm is close to the ideal synchronized scenario by $10\%$, and performs $4\times$ better than the simple case where no recovering method is used.

Afsaneh Mahmoudi, Hossein S. Ghadikolaei, José Mairton Barros Da Silva Júnior et al.

This paper investigates efficient distributed training of a Federated Learning~(FL) model over a wireless network of wireless devices. The communication iterations of the distributed training algorithm may be substantially deteriorated or even blocked by the effects of the devices' background traffic, packet losses, congestion, or latency. We abstract the communication-computation impacts as an `iteration cost' and propose a cost-aware causal FL algorithm~(FedCau) to tackle this problem. We propose an iteration-termination method that trade-offs the training performance and networking costs. We apply our approach when clients use the slotted-ALOHA, the carrier-sense multiple access with collision avoidance~(CSMA/CA), and the orthogonal frequency-division multiple access~(OFDMA) protocols. We show that, given a total cost budget, the training performance degrades as either the background communication traffic or the dimension of the training problem increases. Our results demonstrate the importance of proactively designing optimal cost-efficient stopping criteria to avoid unnecessary communication-computation costs to achieve only a marginal FL training improvement. We validate our method by training and testing FL over the MNIST dataset. Finally, we apply our approach to existing communication efficient FL methods from the literature, achieving further efficiency. We conclude that cost-efficient stopping criteria are essential for the success of practical FL over wireless networks.

Shashi Kant, José Mairton B. da Silva, Gabor Fodor et al.

Federated learning (FL) has emerged as an instance of distributed machine learning paradigm that avoids the transmission of data generated on the users' side. Although data are not transmitted, edge devices have to deal with limited communication bandwidths, data heterogeneity, and straggler effects due to the limited computational resources of users' devices. A prominent approach to overcome such difficulties is FedADMM, which is based on the classical two-operator consensus alternating direction method of multipliers (ADMM). The common assumption of FL algorithms, including FedADMM, is that they learn a global model using data only on the users' side and not on the edge server. However, in edge learning, the server is expected to be near the base station and have direct access to rich datasets. In this paper, we argue that leveraging the rich data on the edge server is much more beneficial than utilizing only user datasets. Specifically, we show that the mere application of FL with an additional virtual user node representing the data on the edge server is inefficient. We propose FedTOP-ADMM, which generalizes FedADMM and is based on a three-operator ADMM-type technique that exploits a smooth cost function on the edge server to learn a global model parallel to the edge devices. Our numerical experiments indicate that FedTOP-ADMM has substantial gain up to 33\% in communication efficiency to reach a desired test accuracy with respect to FedADMM, including a virtual user on the edge server.

Afsaneh Mahmoudi, José Mairton Barros Da Silva Júnior, Hossein S. Ghadikolaei et al.

Federated Learning (FL) plays a prominent role in solving machine learning problems with data distributed across clients. In FL, to reduce the communication overhead of data between clients and the server, each client communicates the local FL parameters instead of the local data. However, when a wireless network connects clients and the server, the communication resource limitations of the clients may prevent completing the training of the FL iterations. Therefore, communication-efficient variants of FL have been widely investigated. Lazily Aggregated Quantized Gradient (LAQ) is one of the promising communication-efficient approaches to lower resource usage in FL. However, LAQ assigns a fixed number of bits for all iterations, which may be communication-inefficient when the number of iterations is medium to high or convergence is approaching. This paper proposes Adaptive Lazily Aggregated Quantized Gradient (A-LAQ), which is a method that significantly extends LAQ by assigning an adaptive number of communication bits during the FL iterations. We train FL in an energy-constraint condition and investigate the convergence analysis for A-LAQ. The experimental results highlight that A-LAQ outperforms LAQ by up to a $50$% reduction in spent communication energy and an $11$% increase in test accuracy.

Saeed Razavikia, José Mairton Barros Da Silva Júnior, Carlo Fischione

Communication and computation are traditionally treated as separate entities, allowing for individual optimizations. However, many applications focus on local information's functionality rather than the information itself. For such cases, harnessing interference for computation in a multiple access channel through digital over-the-air computation can notably increase the computation, as established by the ChannelComp method. However, the coding scheme originally proposed in ChannelComp may suffer from high computational complexity because it is general and is not optimized for specific modulation categories. Therefore, this study considers a specific category of digital modulations for over-the-air computations, QAM and PAM, for which we introduce a novel coding scheme called SumComp. Furthermore, we derive an MSE analysis for SumComp coding in the computation of the arithmetic mean function and establish an upper bound on the MAE for a set of nomographic functions. Simulation results affirm the superior performance of SumComp coding compared to traditional analog over-the-air computation and the original coding in ChannelComp approaches regarding both MSE and MAE over a noisy multiple access channel. Specifically, SumComp coding shows approximately $10$ dB improvements for computing arithmetic and geometric mean on the normalized MSE for low noise scenarios.

Oscar Stenhammar, Sundeep Rangan, Gábor Fodor et al.

The integration of accurate and reproducible wireless network simulations is a key enabler for research on open, virtualized, and intelligent communication systems. Network Digital Twins (NDTs) provide a scalable alternative to costly and time-consuming measurement campaigns, while enabling controlled experimentation and data generation for data-driven network design. In this paper, we present an open and user-friendly NDT framework that integrates controllable vehicular mobility with the site-specific ray tracer Sionna and the discrete-event ns-3 network simulator, enabling virtualized end-to-end modeling of wireless networks across the radio, network, and application layers. The proposed framework is particularly well-suited for dynamic vehicular networks and urban deployments, supporting realistic mobility, traffic dynamics, and the extraction of cross-layer metrics. To promote open-source initiatives, we release both the NDT implementation and a representative dataset generated from realistic vehicular and urban scenarios. The framework and dataset facilitate reproducible experimentation and benchmarking of machine learning-based quality of service prediction, network optimization, and intelligent network management algorithms, lowering the entry barrier for research on virtual and open wireless network services.

Oscar Stenhammar, Gábor Fodor, Carlo Fischione

Machine learning models are increasingly deployed in wireless networks with stringent performance requirements. However, dynamic propagation environments and fluctuating traffic densities introduce concept drift, which complicates the ability to maintain accurate predictive machine learning models. We propose a distributed optimization framework that jointly clusters cells and trains cluster-level predictive models, enabling nodes to cooperatively predict quality of service (QoS) distributions under communication constraints. The proposed method models QoS as a multivariate Gaussian/lognormal distribution and uses a novel clustering mechanism that groups cells with similar network conditions, allowing each cell to select the most appropriate predictor without retraining new models for each cell. By leveraging block coordinate descent, our solution efficiently clusters the cells and updates the predictive models to mitigate concept drift, while maintaining a compact model set to minimize computation overhead. Evaluation using data from realistic simulations with the Sionna ray-tracer and the ns-3 simulator shows that the method converges and yields cluster constellations that adapt to changes in the network that cause concept drift. The experimental evaluation focuses on providing a prediction of the distribution latency, jitter, and RSRP over a one-hour prediction horizon. The proposed method significantly outperforms the traditional single global predictive model approach and reduces the mean absolute error by 9-27% compared to local cell-level predictors. This demonstrates that the proposed method effectively captures local variability using far fewer models through scalable distributed clustering.

Saeed Razavikia, José Mairton Barros Da Silva Júnior, Carlo Fischione

In this work, we investigate federated edge learning over a fading multiple access channel. To alleviate the communication burden between the edge devices and the access point, we introduce a pioneering digital over-the-air computation strategy employing q-ary quadrature amplitude modulation, culminating in a low latency communication scheme. Indeed, we propose a new federated edge learning framework in which edge devices use digital modulation for over-the-air uplink transmission to the edge server while they have no access to the channel state information. Furthermore, we incorporate multiple antennas at the edge server to overcome the fading inherent in wireless communication. We analyze the number of antennas required to mitigate the fading impact effectively. We prove a non-asymptotic upper bound for the mean squared error for the proposed federated learning with digital over-the-air uplink transmissions under both noisy and fading conditions. Leveraging the derived upper bound, we characterize the convergence rate of the learning process of a non-convex loss function in terms of the mean square error of gradients due to the fading channel. Furthermore, we substantiate the theoretical assurances through numerical experiments concerning mean square error and the convergence efficacy of the digital federated edge learning framework. Notably, the results demonstrate that augmenting the number of antennas at the edge server and adopting higher-order modulations improve the model accuracy up to 60\%.

Seyed Mohammad Azimi-Abarghouyi, Carlo Fischione, Kaibin Huang

Over-the-Air Federated Learning (AirFL) is an emerging paradigm that tightly integrates wireless signal processing and distributed machine learning to enable scalable AI at the network edge. By leveraging the superposition property of wireless signals, AirFL performs communication and model aggregation of the learning process simultaneously, significantly reducing latency, bandwidth, and energy consumption. This article offers a tutorial treatment of AirFL, presenting a novel classification into three design approaches: CSIT-aware, blind, and weighted AirFL. We provide a comprehensive guide to theoretical foundations, performance analysis, complexity considerations, practical limitations, and prospective research directions.

Krishnendu S. Tharakan, Carlo Fischione

Wireless connectivity promises to unshackle virtual reality (VR) experiences, allowing users to engage from anywhere, anytime. However, delivering seamless, high-quality, real-time VR video wirelessly is challenging due to the stringent quality of experience requirements, low latency constraints, and limited VR device capabilities. This paper addresses these challenges by introducing a novel decentralized multi task fair federated learning (DMTFL) based caching that caches and prefetches each VR user's field of view (FOV) at base stations (BSs) based on the caching strategies tailored to each BS. In federated learning (FL) in its naive form, often biases toward certain users, and a single global model fails to capture the statistical heterogeneity across users and BSs. In contrast, the proposed DMTFL algorithm personalizes content delivery by learning individual caching models at each BS. These models are further optimized to perform well under any target distribution, while providing theoretical guarantees via Rademacher complexity and a probably approximately correct (PAC) bound on the loss. Using a realistic VR head-tracking dataset, our simulations demonstrate the superiority of our proposed DMTFL algorithm compared to baseline algorithms.

Adnan Shahid, Adrian Kliks, Ahmed Al-Tahmeesschi et al.

This white paper discusses the role of large-scale AI in the telecommunications industry, with a specific focus on the potential of generative AI to revolutionize network functions and user experiences, especially in the context of 6G systems. It highlights the development and deployment of Large Telecom Models (LTMs), which are tailored AI models designed to address the complex challenges faced by modern telecom networks. The paper covers a wide range of topics, from the architecture and deployment strategies of LTMs to their applications in network management, resource allocation, and optimization. It also explores the regulatory, ethical, and standardization considerations for LTMs, offering insights into their future integration into telecom infrastructure. The goal is to provide a comprehensive roadmap for the adoption of LTMs to enhance scalability, performance, and user-centric innovation in telecom networks.

Henrik Hellström, Viktoria Fodor, Carlo Fischione

Motivated by increasing computational capabilities of wireless devices, as well as unprecedented levels of user- and device-generated data, new distributed machine learning (ML) methods have emerged. In the wireless community, Federated Learning (FL) is of particular interest due to its communication efficiency and its ability to deal with the problem of non-IID data. FL training can be accelerated by a wireless communication method called Over-the-Air Computation (AirComp) which harnesses the interference of simultaneous uplink transmissions to efficiently aggregate model updates. However, since AirComp utilizes analog communication, it introduces inevitable estimation errors. In this paper, we study the impact of such estimation errors on the convergence of FL and propose retransmissions as a method to improve FL convergence over resource-constrained wireless networks. First, we derive the optimal AirComp power control scheme with retransmissions over static channels. Then, we investigate the performance of Over-the-Air FL with retransmissions and find two upper bounds on the FL loss function. Finally, we propose a heuristic for selecting the optimal number of retransmissions, which can be calculated before training the ML model. Numerical results demonstrate that the introduction of retransmissions can lead to improved ML performance, without incurring extra costs in terms of communication or computation. Additionally, we provide simulation results on our heuristic which indicate that it can correctly identify the optimal number of retransmissions for different wireless network setups and machine learning problems.

Yeongwoo Kim, Ezeddin Al Hakim, Johan Haraldson et al.

In the resource management of wireless networks, Federated Learning has been used to predict handovers. However, non-independent and identically distributed data degrade the accuracy performance of such predictions. To overcome the problem, Federated Learning can leverage data clustering algorithms and build a machine learning model for each cluster. However, traditional data clustering algorithms, when applied to the handover prediction, exhibit three main limitations: the risk of data privacy breach, the fixed shape of clusters, and the non-adaptive number of clusters. To overcome these limitations, in this paper, we propose a three-phased data clustering algorithm, namely: generative adversarial network-based clustering, cluster calibration, and cluster division. We show that the generative adversarial network-based clustering preserves privacy. The cluster calibration deals with dynamic environments by modifying clusters. Moreover, the divisive clustering explores the different number of clusters by repeatedly selecting and dividing a cluster into multiple clusters. A baseline algorithm and our algorithm are tested on a time series forecasting task. We show that our algorithm improves the performance of forecasting models, including cellular network handover, by 43%.

Rong Du, Sindri Magnússon, Carlo Fischione

An important task in the Internet of Things (IoT) is field monitoring, where multiple IoT nodes take measurements and communicate them to the base station or the cloud for processing, inference, and analysis. This communication becomes costly when the measurements are high-dimensional (e.g., videos or time-series data). The IoT networks with limited bandwidth and low power devices may not be able to support such frequent transmissions with high data rates. To ensure communication efficiency, this article proposes to model the measurement compression at IoT nodes and the inference at the base station or cloud as a deep neural network (DNN). We propose a new framework where the data to be transmitted from nodes are the intermediate outputs of a layer of the DNN. We show how to learn the model parameters of the DNN and study the trade-off between the communication rate and the inference accuracy. The experimental results show that we can save approximately 96% transmissions with only a degradation of 2.5% in inference accuracy. Our findings have the potentiality to enable many new IoT data analysis applications generating large amount of measurements.

Hossein S. Ghadikolaei, Hadi Ghauch, Gabor Fodor et al.

Inter-operator spectrum sharing in millimeter-wave bands has the potential of substantially increasing the spectrum utilization and providing a larger bandwidth to individual user equipment at the expense of increasing inter-operator interference. Unfortunately, traditional model-based spectrum sharing schemes make idealistic assumptions about inter-operator coordination mechanisms in terms of latency and protocol overhead, while being sensitive to missing channel state information. In this paper, we propose hybrid model-based and data-driven multi-operator spectrum sharing mechanisms, which incorporate model-based beamforming and user association complemented by data-driven model refinements. Our solution has the same computational complexity as a model-based approach but has the major advantage of having substantially less signaling overhead. We discuss how limited channel state information and quantized codebook-based beamforming affect the learning and the spectrum sharing performance. We show that the proposed hybrid sharing scheme significantly improves spectrum utilization under realistic assumptions on inter-operator coordination and channel state information acquisition.

Othmane Mazhar, Cristian R. Rojas, Carlo Fischione et al.

We address the new problem of estimating a piece-wise constant signal with the purpose of detecting its change points and the levels of clusters. Our approach is to model it as a nonparametric penalized least square model selection on a family of models indexed over the collection of partitions of the design points and propose a computationally efficient algorithm to approximately solve it. Statistically, minimizing such a penalized criterion yields an approximation to the maximum a posteriori probability (MAP) estimator. The criterion is then analyzed and an oracle inequality is derived using a Gaussian concentration inequality. The oracle inequality is used to derive on one hand conditions for consistency and on the other hand an adaptive upper bound on the expected square risk of the estimator, which statistically motivates our approximation. Finally, we apply our algorithm to simulated data to experimentally validate the statistical guarantees and illustrate its behavior.

Hadi Ghauch, Mikael Skoglund, Hossein Shokri-Ghadikolaei et al.

We summarize our recent findings, where we proposed a framework for learning a Kolmogorov model, for a collection of binary random variables. More specifically, we derive conditions that link outcomes of specific random variables, and extract valuable relations from the data. We also propose an algorithm for computing the model and show its first-order optimality, despite the combinatorial nature of the learning problem. We apply the proposed algorithm to recommendation systems, although it is applicable to other scenarios. We believe that the work is a significant step toward interpretable machine learning.

Hadi Ghauch, Hossein Shokri-Ghadikolaei, Carlo Fischione et al.

The lack of mathematical tractability of Deep Neural Networks (DNNs) has hindered progress towards having a unified convergence analysis of training algorithms, in the general setting. We propose a unified optimization framework for training different types of DNNs, and establish its convergence for arbitrary loss, activation, and regularization functions, assumed to be smooth. We show that framework generalizes well-known first- and second-order training methods, and thus allows us to show the convergence of these methods for various DNN architectures and learning tasks, as a special case of our approach. We discuss some of its applications in training various DNN architectures (e.g., feed-forward, convolutional, linear networks), to regression and classification tasks.

Nishal Silva, Chathuranga Weeraddana, Carlo Fischione

Musical onset detection is a key component in any beat tracking system. Existing onset detection methods are based on temporal/spectral analysis, or methods that integrate temporal and spectral information together with statistical estimation and machine learning models. In this paper, we propose a method to localize onset components in music by using the S-transform, and thus, the method is purely based on temporal/spectral data. Unlike the other methods based on temporal/spectral data, which usually rely short time Fourier transform (STFT), our method enables effective isolation of crucial frequency subbands due to the frequency dependent resolution of S-transform. Moreover, numerical results show, even with less computationally intensive steps, the proposed method can closely resemble the performance of more resource intensive statistical estimation based approaches.

Pangun Park, Sinem Coleri Ergen, Carlo Fischione et al.

Wireless networked control systems (WNCS) are composed of spatially distributed sensors, actuators, and con- trollers communicating through wireless networks instead of conventional point-to-point wired connections. Due to their main benefits in the reduction of deployment and maintenance costs, large flexibility and possible enhancement of safety, WNCS are becoming a fundamental infrastructure technology for critical control systems in automotive electrical systems, avionics control systems, building management systems, and industrial automation systems. The main challenge in WNCS is to jointly design the communication and control systems considering their tight interaction to improve the control performance and the network lifetime. In this survey, we make an exhaustive review of the literature on wireless network design and optimization for WNCS. First, we discuss what we call the critical interactive variables including sampling period, message delay, message dropout, and network energy consumption. The mutual effects of these communication and control variables motivate their joint tuning. We discuss the effect of controllable wireless network parameters at all layers of the communication protocols on the probability distribution of these interactive variables. We also review the current wireless network standardization for WNCS and their corresponding methodology for adapting the network parameters. Moreover, we discuss the analysis and design of control systems taking into account the effect of the interactive variables on the control system performance. Finally, we present the state-of-the-art wireless network design and optimization for WNCS, while highlighting the tradeoff between the achievable performance and complexity of various approaches. We conclude the survey by highlighting major research issues and identifying future research directions.

Riccardo Bonetto, Michele Rossi, Stefano Tomasin et al.

In electrical distribution grids, the constantly increasing number of power generation devices based on renewables demands a transition from a centralized to a distributed generation paradigm. In fact, power injection from Distributed Energy Resources (DERs) can be selectively controlled to achieve other objectives beyond supporting loads, such as the minimization of the power losses along the distribution lines and the subsequent increase of the grid hosting capacity. However, these technical achievements are only possible if alongside electrical optimization schemes, a suitable market model is set up to promote cooperation from the end users. In contrast with the existing literature, where energy trading and electrical optimization of the grid are often treated separately or the trading strategy is tailored to a specific electrical optimization objective, in this work we consider their joint optimization. Specifically, we present a multi-objective optimization problem accounting for energy trading, where: 1) DERs try to maximize their profit, resulting from selling their surplus energy, 2) the loads try to minimize their expense, and 3) the main power supplier aims at maximizing the electrical grid efficiency through a suitable discount policy. This optimization problem is proved to be non convex, and an equivalent convex formulation is derived. Centralized solutions are discussed first, and are subsequently distributed. Numerical results to demonstrate the effectiveness of the so obtained optimal policies are then presented.

Pradeep Chathuranga Weeraddana, George Athanasiou, Martin Jakobsson et al.

Ensuring privacy of sensitive data is essential in many contexts, such as healthcare data, banks, e-commerce, wireless sensor networks, and social networks. It is common that different entities coordinate or want to rely on a third party to solve a specific problem. At the same time, no entity wants to publish its problem data during the solution procedure unless there is a privacy guarantee. Unlike cryptography and differential privacy based approaches, the methods based on optimization lack a quantification of the privacy they can provide. The main contribution of this paper is to provide a mechanism to quantify the privacy of a broad class of optimization approaches. In particular, we formally define a one-to-many relation, which relates a given adversarial observed message to an uncertainty set of the problem data. This relation quantifies the potential ambiguity on problem data due to the employed optimization approaches. The privacy definitions are then formalized based on the uncertainty sets. The properties of the proposed privacy measure is analyzed. The key ideas are illustrated with examples, including localization, average consensus, among others.