Remote Estimation Games with Random Walk Processes: Stackelberg Equilibrium

Remote estimation is a crucial element of real time monitoring of a stochastic process. While most of the existing works have concentrated on obtaining optimal sampling strategies, motivated by malicious attacks on cyber-physical systems, we model sensing under surveillance as a game between an attacker and a defender. This introduces strategic elements to conventional remote estimation problems. Additionally, inspired by increasing detection capabilities, we model an element of information leakage for each player. Parameterizing the game in terms of uncertainty on each side, information leakage, and cost of sampling, we consider the Stackelberg Equilibrium (SE) concept where one of the players acts as the leader and the other one as the follower. By focusing our attention on stationary probabilistic sampling policies, we characterize the SE of this game and provide simulations to show the efficacy of our results.

Optimizing Profitability in Timely Gossip Networks

We consider a communication system where a group of users, interconnected in a bidirectional gossip network, wishes to follow a time-varying source, e.g., updates on an event, in real-time. The users wish to maintain their expected version ages below a threshold, and can either rely on gossip from their neighbors or directly subscribe to a server publishing about the event, if the former option does not meet the timeliness requirements. The server wishes to maximize its profit by increasing subscriptions from users and minimizing event sampling frequency to reduce costs. This leads to a Stackelberg game between the server and the users where the sender is the leader deciding its sampling frequency and the users are the followers deciding their subscription strategies. We investigate equilibrium strategies for low-connectivity and high-connectivity topologies.

How to Make Money From Fresh Data: Subscription Strategies in Age-Based Systems

We consider a communication system consisting of a server that tracks and publishes updates about a time-varying data source or event, and a gossip network of users interested in closely tracking the event. The timeliness of the information is measured through the version age of information. The users wish to have their expected version ages remain below a threshold, and have the option to either rely on gossip from their neighbors or subscribe to the server directly to follow updates about the event if the former option does not meet the timeliness requirements. The server wishes to maximize its profit by increasing the number of subscribers and reducing costs associated with the frequent sampling of the event. We model the problem setup as a Stackelberg game between the server and the users, where the server commits to a frequency of sampling the event, and the users make decisions on whether to subscribe or not. As an initial work, we focus on directed networks with unidirectional flow of information and obtain the optimal equilibrium strategies for all the players. We provide simulation results to confirm the theoretical findings and provide additional insights.

Policy Optimization finds Nash Equilibrium in Regularized General-Sum LQ Games

In this paper, we investigate the impact of introducing relative entropy regularization on the Nash Equilibria (NE) of General-Sum $N$-agent games, revealing the fact that the NE of such games conform to linear Gaussian policies. Moreover, it delineates sufficient conditions, contingent upon the adequacy of entropy regularization, for the uniqueness of the NE within the game. As Policy Optimization serves as a foundational approach for Reinforcement Learning (RL) techniques aimed at finding the NE, in this work we prove the linear convergence of a policy optimization algorithm which (subject to the adequacy of entropy regularization) is capable of provably attaining the NE. Furthermore, in scenarios where the entropy regularization proves insufficient, we present a $\delta$-augmentation technique, which facilitates the achievement of an $\epsilon$-NE within the game.

Learning How to Strategically Disclose Information

Strategic information disclosure, in its simplest form, considers a game between an information provider (sender) who has access to some private information that an information receiver is interested in. While the receiver takes an action that affects the utilities of both players, the sender can design information (or modify beliefs) of the receiver through signal commitment, hence posing a Stackelberg game. However, obtaining a Stackelberg equilibrium for this game traditionally requires the sender to have access to the receiver's objective. In this work, we consider an online version of information design where a sender interacts with a receiver of an unknown type who is adversarially chosen at each round. Restricting attention to Gaussian prior and quadratic costs for the sender and the receiver, we show that $\mathcal{O}(\sqrt{T})$ regret is achievable with full information feedback, where $T$ is the total number of interactions between the sender and the receiver. Further, we propose a novel parametrization that allows the sender to achieve $\mathcal{O}(\sqrt{T})$ regret for a general convex utility function. We then consider the Bayesian Persuasion problem with an additional cost term in the objective function, which penalizes signaling policies that are more informative and obtain $\mathcal{O}(\log(T))$ regret. Finally, we establish a sublinear regret bound for the partial information feedback setting and provide simulations to support our theoretical results.

Using Timeliness in Tracking Infections

We consider real-time timely tracking of infection status (e.g., covid-19) of individuals in a population. In this work, a health care provider wants to detect infected people as well as people who have recovered from the disease as quickly as possible. In order to measure the timeliness of the tracking process, we use the long-term average difference between the actual infection status of the people and their real-time estimate by the health care provider based on the most recent test results. We first find an analytical expression for this average difference for given test rates, infection rates and recovery rates of people. Next, we propose an alternating minimization based algorithm to find the test rates that minimize the average difference. We observe that if the total test rate is limited, instead of testing all members of the population equally, only a portion of the population may be tested in unequal rates calculated based on their infection and recovery rates. Next, we characterize the average difference when the test measurements are erroneous (i.e., noisy). Further, we consider the case where the infection status of individuals may be dependent, which happens when an infected person spreads the disease to another person if they are not detected and isolated by the health care provider. Then, we consider an age of incorrect information based error metric where the staleness metric increases linearly over time as long as the health care provider does not detect the changes in the infection status of the people. In numerical results, we observe that an increased population size increases diversity of people with different infection and recovery rates which may be exploited to spend testing capacity more efficiently. Depending on the health care provider's preferences, test rate allocation can be adjusted to detect either the infected people or the recovered people more quickly.

The Role of Gossiping for Information Dissemination over Networked Agents

We consider information dissemination over a network of gossiping agents (nodes). In this model, a source keeps the most up-to-date information about a time-varying binary state of the world, and $n$ receiver nodes want to follow the information at the source as accurately as possible. When the information at the source changes, the source first sends updates to a subset of $m\leq n$ nodes. After that, the nodes share their local information during the gossiping period to disseminate the information further. The nodes then estimate the information at the source using the majority rule at the end of the gossiping period. To analyze information dissemination, we introduce a new error metric to find the average percentage of nodes that can accurately obtain the most up-to-date information at the source. We characterize the equations necessary to obtain the steady-state distribution for the average error and then analyze the system behavior under both high and low gossip rates. In the high gossip rate, in which each node can access other nodes' information more frequently, we show that the nodes update their information based on the majority of the information in the network. In the low gossip rate, we introduce and analyze the gossip gain, which is the reduction at the average error due to gossiping. In particular, we develop an adaptive policy that the source can use to determine its current transmission capacity $m$ based on its past transmission rates and the accuracy of the information at the nodes. In numerical results, we show that when the source's transmission capacity $m$ is limited, gossiping can be harmful as it causes incorrect information to disseminate. We then find the optimal gossip rates to minimize the average error for a fixed $m$. Finally, we illustrate the outperformance of our adaptive policy compared to the constant $m$-selection policy even for the high gossip rates.

Version Age of Information in Clustered Gossip Networks

We consider a network consisting of a single source and $n$ receiver nodes that are grouped into equal-sized clusters. We use cluster heads in each cluster to facilitate communication between the source and the nodes within that cluster. Inside clusters, nodes are connected to each other according to a given network topology. Based on the connectivity among the nodes, each node relays its current stored version of the source update to its neighboring nodes by $local$ $gossiping$. We use the $version$ $age$ metric to assess information freshness at the nodes. We consider disconnected, ring, and fully connected network topologies for each cluster. For each network topology, we characterize the average version age at each node and find the average version age scaling as a function of the network size $n$. Our results indicate that per node average version age scalings of $O(\sqrt{n})$, $O(n^{\frac{1}{3}})$, and $O(\log n)$ are achievable in disconnected, ring, and fully connected cluster models, respectively. Next, we increase connectivity in the network and allow gossiping among the cluster heads to improve version age at the nodes. With that, we show that when the cluster heads form a ring network among themselves, we obtain per node average version age scalings of $O(n^{\frac{1}{3}})$, $O(n^{\frac{1}{4}})$, and $O(\log n)$ in disconnected, ring, and fully connected cluster models, respectively. Next, focusing on a ring network topology in each cluster, we introduce hierarchy to the considered clustered gossip network model and show that when we employ two levels of hierarchy, we can achieve the same $O(n^{\frac{1}{4}})$ scaling without using dedicated cluster heads. We generalize this result for $h$ levels of hierarchy and show that per user average version age scaling of $O(n^{\frac{1}{2h}})$ is achievable in the case of a ring network in each cluster across all hierarchy levels.

Gossiping with Binary Freshness Metric

We consider the binary freshness metric for gossip networks that consist of a single source and $n$ end-nodes, where the end-nodes are allowed to share their stored versions of the source information with the other nodes. We develop recursive equations that characterize binary freshness in arbitrarily connected gossip networks using the stochastic hybrid systems (SHS) approach. Next, we study binary freshness in several structured gossip networks, namely disconnected, ring and fully connected networks. We show that for both disconnected and ring network topologies, when the number of nodes gets large, the binary freshness of a node decreases down to 0 as $n^{-1}$, but the freshness is strictly larger for the ring topology. We also show that for the fully connected topology, the rate of decrease to 0 is slower, and it takes the form of $n^{-\rho}$ for a $\rho$ smaller than 1, when the update rates of the source and the end-nodes are sufficiently large. Finally, we study the binary freshness metric for clustered gossip networks, where multiple clusters of structured gossip networks are connected to the source node through designated access nodes, i.e., cluster heads. We characterize the binary freshness in such networks and numerically study how the optimal cluster sizes change with respect to the update rates in the system.

Freshness Based Cache Updating in Parallel Relay Networks

We consider a system consisting of a server, which receives updates for $N$ files according to independent Poisson processes. The goal of the server is to deliver the latest version of the files to the user through a parallel network of $K$ caches. We consider an update received by the user successful, if the user receives the same file version that is currently prevailing at the server. We derive an analytical expression for information freshness at the user. We observe that freshness for a file increases with increase in consolidation of rates across caches. To solve the multi-cache problem, we first solve the auxiliary problem of a single-cache system. We then rework this auxiliary solution to our parallel-cache network by consolidating rates to single routes as much as possible. This yields an approximate (sub-optimal) solution for the original problem. We provide an upper bound on the gap between the sub-optimal solution and the optimal solution. Numerical results show that the sub-optimal policy closely approximates the optimal policy.