Barrier functions constitute an effective tool for assessing and enforcing safety-critical constraints on dynamical systems. To this end, one is required to find a function h that satisfies a Lyapunov-like differential condition, thereby ensuring the invariance of its zero super-level set h(>= 0). This methodology, however, does not prescribe a general method for finding the function h that satisfies such differential conditions, which, in general, can be a daunting task. In this paper, we seek to overcome this limitation by developing a generalized barrier condition that makes the search for h easier. We do this in two steps. First, we develop integral barrier conditions that reveal equivalent asymptotic behavior to the differential ones, but without requiring differentiability of h. Subsequently, we further replace the stringent invariance requirement on h >= 0 with a more flexible concept known as recurrence. A set is (tau-)recurrent if every trajectory that starts in the set returns to it (within tau seconds) infinitely often. We show that, under mild conditions, a simple sign distance function can satisfy our relaxed condition and that the ( tau-)recurrence of the super-level set h >= 0 is sufficient to guarantee the system's safety.
We study the problem of Symmetric Private Information Retrieval (SPIR) in a scenario with $L$ non-replicated and colluding servers, and $M$ independent files distributed across these servers. In this setting, communication takes place through a noisy multiple-access channel and a noiseless public channel. The client must retrieve one of the $M$ files such that (i) the client's choice must not be revealed to the servers, and (ii) the client must not learn any information about non-selected files. Our main contribution is showing that, for a specific class of channels and without requiring shared randomness among servers, positive rates are achievable even when all the servers collude. Additionally, we present an example of channel where distributing files across multiple servers yields an achievable rate that outperforms a setting where all the files are stored on a single server.
Federated Learning (FL) represents a growing machine learning (ML) paradigm designed for training models across numerous nodes that retain local datasets, all without directly exchanging the underlying private data with the parameter server (PS). Its increasing popularity is attributed to notable advantages in terms of training deep neural network (DNN) models under privacy aspects and efficient utilization of communication resources. Unfortunately, DNNs suffer from high computational and communication costs, as well as memory consumption in intricate tasks. These factors restrict the applicability of FL algorithms in communication-constrained systems with limited hardware resources. In this paper, we develop a novel algorithm that overcomes these limitations by synergistically combining a pruning-based method with the FL process, resulting in low-dimensional representations of the model with minimal communication cost, dubbed Masked Pruning over FL (MPFL). The algorithm operates by initially distributing weights to the nodes through the PS. Subsequently, each node locally trains its model and computes pruning masks. These low-dimensional masks are then transmitted back to the PS, which generates a consensus pruning mask, broadcasted back to the nodes. This iterative process enhances the robustness and stability of the masked pruning model. The generated mask is used to train the FL model, achieving significant bandwidth savings. We present an extensive experimental study demonstrating the superior performance of MPFL compared to existing methods. Additionally, we have developed an open-source software package for the benefit of researchers and developers in related fields.
Linear feature extraction at the presence of nonlinear dependencies among the data is a fundamental challenge in unsupervised learning. We propose using a Gram-Schmidt (GS) type orthogonalization process over function spaces in order to detect and remove redundant dimensions. Specifically, by applying the GS process over a family of functions which presumably captures the nonlinear dependencies in the data, we construct a series of covariance matrices that can either be used to identify new large-variance directions, or to remove those dependencies from the principal components. In the former case, we provide information-theoretic guarantees in terms of entropy reduction. In the latter, we prove that under certain assumptions the resulting algorithms detect and remove nonlinear dependencies whenever those dependencies lie in the linear span of the chosen function family. Both proposed methods extract linear features from the data while removing nonlinear redundancies. We provide simulation results on synthetic and real-world datasets which show improved performance over state-of-the-art feature extraction algorithms.
This paper considers the proportional sharing mechanism [4] from the lens of fairness. Deviating from prior work we consider the net utility to users by incorporating a weighted cost of bidding, one example being the use of power, incurred at user-dependent costs, to competitively bid for throughput in wireless systems. In this context, we analyze the price of anarchy of the equilibrium when the social welfare is measured both by a fair-sharing utility measure, as well as the sum of utility measures. In the context of power bids by users, we consider both continuous and discrete bids. The analysis shows the PoA can be zero and even for a uniform linear utility function when using the sum-total measure, but with distinct costs of bids, is asymptotically zero with increasing number of users indicating that a number of users/clients were starved of resources. This illustrates that accounting for the cost of bids can substantially change the structure of the equilibrium as compared to uniform costs. We also investigate the number of such starved users experimentally. In order to determine equilibrium, we show convex program formulations as well as present approximation methods to determine (1 + epsilon)-approximate equilibrium via an efficient algorithm. We also design methods to compute the optimum solution. Optimization of the social utility in the general case, where multiple resources can be chosen by users, is shown to be NP-hard. Experimental results illustrate the asymptotic behavior of the PoA and of the number of users that are starved of resources when users (competition) increases.
We study noise-contrastive estimation (NCE) through the lens of local information geometry. In particular, paralleling recent work [1], [2], we demonstrate that the optimal noise distribution for NCE is not the same as those typically chosen based on practical heuristics. Furthermore, we show that an important information geometric construct, the angle between the subspaces of distributions, dictates the asymptotic performance of an extension of NCE proposed in [3]. This analysis provides an information geometric interpretation of the method, encouraging further study of NCE and its variants using information geometric methods.
This paper considers the task of estimating the principal eigenvector of a positive semi-definite matrix using the power method subjected to random row erasures at each iteration. This can be used to model applications where large matrix operations are distributed across multiple servers, some of which may fail to respond at each iteration. We analyze the simple strategy of only updating coordinates corresponding to non-erased rows, and demonstrate that, presuming a good initialization, the power method with erasures still converges exponentially fast to the principal eigenvector. The rate of convergence is governed by a modified spectral gap, which is a function of the original spectral gap and the fraction of erased rows. Accompanying numerical results validate our bounds, and demonstrate that, in certain regimes, our approach outperforms techniques such as coded computation and Oja's algorithm.
Linearly solvable Markov decision processes (LSMDPs) are a special class of Markov decision processes (MDPs) in which the optimal value function under an exponential transformation satisfies a linear equation. This model was previously extended to a class of two-player, zero-sum Markov games in which the game's equilibrium value function can similarly be derived from a linear equation. In this work, a new class of linearly solvable n-player, general-sum Markov games is proposed. We show that games in this class have a unique Nash equilibrium, and the equilibrium value functions can be derived from a single system of linear equations. We demonstrate how to approximate discrete-state, discrete-action Markov games outside this class through an embedding process analogous to the way LSMDPs approximate standard MDPs. An off-policy reinforcement learning algorithm for general-sum Markov games, which we call Nash-Z learning, is presented. We empirically demonstrate that Nash-Z learning finds equilibrium policies with low regret and that it finds these solutions orders of magnitude faster than the classic Nash-Q learning algorithm.
As distributed learning applications like Federated Learning, the Internet of Things (IoT), and Edge Computing expand, addressing their limitations becomes crucial. We approach decentralized learning across a network of communicating clients or nodes, focusing on two primary challenges: data heterogeneity and adversarial robustness. To address these, we introduce a decentralized minimax optimization method incorporating two key components: local updates and gradient tracking. Minimax optimization serves as a fundamental tool for adversarial training, ensuring robustness. Local updates are vital in Federated Learning (FL) to alleviate the communication bottleneck, while gradient tracking is necessary to demonstrate convergence amid data heterogeneity. Our analysis of the proposed algorithm, Dec-Fed Track, in nonconvex-strongly-concave minimax optimization demonstrates its convergence to a stationary point. Additionally, numerical experiments support our theoretical results.
Color perception has long remained an in-triguing topic spanning vision and cognitive science, signal processing, and computer graphics. People are often classified as either “color-normal” or “color-blind”, and it is widely accepted there are a few types of colorblindness that are the most prevalent. At the same time, empirical evidence, such as in optometry and vision science, has repeatedly suggested that categories for colorblindness only serve as approximations to real manifestations of it. With the motivation of better understanding individual-level color perception, we propose a model for color vision that unifies existing theories for color-normal and color-blind populations. This model posits a certain type of low-dimensional structure in color space according to which any given person distinguishes colors. We design an algorithm to learn this low-dimensional structure from user queries, and prove statistical guarantees on its per-formance. To collect user data, we adapt a user interface design, termed “perceptual adjustment queries” (PAQs), to assess color perception. This user interface efficiently infers a user's color distinguishability profile from a few cognitively lightweight responses. A user study shows that our method captures individual-level differences in both color-normal and color-blind populations.
We study the trade-off between communication rate and privacy for distributed batch matrix multiplication of two independent sequences of matrices A and B with uniformly distributed entries. In our setting, B is publicly accessible by all the servers while A must remain private. A user is interested in evaluating the product AB with the responses from the k fastest servers. For a given parameter α∈ [0, 1], our privacy constraint must ensure that any set of ℓ colluding servers cannot learn more than a fraction α of A. Additionally, we study the trade-off between the amount of local randomness needed at the encoder and privacy. Finally, we establish the optimal trade-offs when the matrices are square and identify a linear relationship between information leakage and communication rate.
With the increasing adoption of inverter-based resources (IBRs) in power systems, oscillations have become a significant challenge for system operators worldwide. This paper investigates the oscillatory instability of grid-following (GFL) inverters using Hopf bifurcation theory. We focus on examining how parameter variations affect the stability margin with respect to Hopf bifurcation. An analytical expression for the sensitivity of the stability margin is derived using the normal vector to the Hopf bifurcation hypersurface. Upon restructuring the differential-algebraic equation (DAE) model of the GFL inverter into an ordinary differential equation (ODE) form, we identify the most effective control parameters for enhancing stability under varying operating conditions. It is shown that the normal vector method is efficient in estimating the stability margin in regard to both computational speed and accuracy. Results indicate that the proportional gain in the current control loop is the most effective parameter in improving the stability margin.
Local memory systems play a crucial role in today's networks: keeping popular content close to users improves performance by reducing the latency of fetching an item from a more costly central location. Caching policies that retain recently requested items are effective to deal with bursts of requests; in particular timer-based (TTL) caching policies are of this nature, and have well understood properties. However, in some scenarios, traffic is more regular, reflected in the fact that the hazard rate function of inter-request times is increasing. For this situation we propose the strategy of Timer-based Pre-fetching, a dual of TTL caching. We characterize the optimal Pre-fetching timers as the solution to a convex optimization problem, showing this approach improves upon caching strategies. We also analyze the large scale behavior of the optimal policy for both cases, which amounts to threshold policy in the hazard rates, and give asymptotic performance results for a general class of arrival processes.
The work here studies the communication cost for a multi-server, multi-task distributed computation framework, and does so for a broad class of functions and data statistics. Considering the framework where a user seeks the computation of multiple complex (conceivably non-linear) tasks from a set of distributed servers, we establish communication cost upper bounds for a variety of data statistics, function classes and data placements across the servers. To do so, we proceed to apply, for the first time here, Korner's characteristic graph approach - which is known to capture the structural properties of data and functions - to the promising framework of multi-server multi-task distributed computing. Going beyond the general expressions, and in order to offer clearer insight, we also consider the well-known scenario of cyclic dataset placement and linearly separable functions over the binary field, in which case our approach exhibits considerable gains over the state of the art. Similar gains are identified for multilinear functions.
The optimal power flow (OPF) problem is non-convex and will often yield multiple local optimal solutions. To be practically feasible in operation, a solution must be both dynamically stable and be must reachable from other operating points. In this paper we examine a 5 bus system that exhibits two local and stable OFP solutions, one of which is the global optimum. The two solutions of interest lie in separate regions of the disjoint feasible space of the OPF, and we explore the question of whether it is possible to move between the solutions without violating limits. We demonstrate via detailed simulation that a reactive power limit is violated when the voltage and power setpoints are suddenly changed, though the final operating point is eventually reached. As a main result, we show that there exists a path of setpoint adjustments that does enable the change of operating point, from the local optimum to the global optimum, without violating limits, and we explore practical control strategies to achieve this transition.
This study investigates the distributed fact-checking problem, where a series of fact-checkers (agents) with interconnected influence evaluate a sequence of statements from a source. Each statement has a hidden binary label (true or false), and each agent assigns their own true or false label to the statement. The agents' opinions follow a hidden directed chain, with a leader followed by ordered followers. Our goals are to: (i) recover the directed chain (the relative order of agents), and (ii) decode the true label of each statement. We demonstrate that if the source is biased, the directed chain can be recovered through the observation of agents' labels. However, an unbiased source allows recovery of only the undirected influence chain. For the latter, we propose two low-complexity algorithms to recover the undirected chain, along with a decoder to estimate the true statement label based on observed agent opinions.
This paper studies the problem of strategic quantization, where an encoder and a decoder with misaligned objectives communicate over a rate-constrained noiseless channel. Specifically, we focus on a 2-dimensional source, state and bias variables, and quadratic distortion measures. We provide a design algorithm for this particular case of strategic quantization, as well as an upper and lower bounds on the encoder distortion via employing linear communication strategies. Finally, we present comparative numerical results obtained via the proposed method. We provide our numerical results and the code for research purposes at https://github.com/strategic-quantization/quadratic-distortion-measures.
We propose a distributed gradient algorithm to solve an optimization problem over a multiagent network in which a fraction of the nodes are malicious. We assume the adversarial agents perturb their estimates with an attack vector and that the agents are interacting over a bandwidth-limited communication network. For undirected communication network, this paper establishes conditions that guarantee convergence to a neighborhood of the optimal solutions when the objective is a strongly convex and quadratic function in the presence of malicious actors and with quantized estimates. Resilience strategies against these attacks are also proposed.
The problem of detecting a single anomalous process among a finite number $M$ of processes is considered. We examine a composite hypothesis case, where observations of a process follow a common distribution with an unknown parameter (vector). The parameter value resides in either normal or abnormal parameter spaces, contingent on the process state. Until the change point, all processes are in a normal state, and after the change point, one process transitions to an abnormal state. Our goal is to develop a sequential search strategy that minimizes the expected detection time since the anomaly occurred subject to an error probability constraint. We develop a novel anomaly detection algorithm, named Searching for Change Point Anomaly (SCPA), with the following desired properties. Firstly, when no additional side information on the process states is available, the proposed algorithm is asymptotically optimal in terms of minimizing the detection delay as the error probability approaches zero. Secondly, in the scenario where the parameter value under the null hypothesis is known and equal for all normal processes, the proposed algorithm is also asymptotically optimal and demonstrates improved detection time determined by the true null state. Finally, we establish an upper bound on the error probability under the proposed algorithm for the finite sample regime.
A ternary barrier channel is a non-symmetric error model defined previously to address emerging applications. Conveniently, it admits a code-construction method comprising two binary constituent codes. In this paper we propose a decoding algorithm that decodes the two constituent codes jointly by message passing on a graph representing the two codes' parity-check constraints. The messages exchanged by the algorithm are likelihoods calculated from incoming messages, and they are derived in the paper based on the exact dependence between the binary values of the two codewords. Simulation results demonstrate that the proposed decoder has superior error-rate performance compared to prior decoding approaches.
