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Creators/Authors contains: "Varshney, Pramod K."

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  1. In this work, we consider a distributed online convex optimization problem, with time-varying (potentially adversarial) constraints. A set of nodes, jointly aim to minimize a global objective function, which is the sum of local convex functions. The objective and constraint functions are revealed locally to the nodes, at each time, after taking an action. Naturally, the constraints cannot be instantaneously satisfied. Therefore, we reformulate the problem to satisfy these constraints in the long term. To this end, we propose a distributed primal-dual mirror descent-based algorithm, in which the primal and dual updates are carried out locally at all the nodes. This is followed by sharing and mixing of the primal variables by the local nodes via communication with the immediate neighbors. To quantify the performance of the proposed algorithm, we utilize the challenging, but more realistic metrics of dynamic regret and fit. Dynamic regret measures the cumulative loss incurred by the algorithm compared to the best dynamic strategy, while fit measures the long term cumulative constraint violations. Without assuming the restrictive Slater’s conditions, we show that the proposed algorithm achieves sublinear regret and fit under mild, commonly used assumptions.
    Free, publicly-accessible full text available January 1, 2023
  2. We investigate the nonparametric, composite hypothesis testing problem for arbitrary unknown distributions in the asymptotic regime where both the sample size and the number of hypothesis grow exponentially large. Such asymptotic analysis is important in many practical problems, where the number of variations that can exist within a family of distributions can be countably infinite. We introduce the notion of discrimination capacity , which captures the largest exponential growth rate of the number of hypothesis relative to the sample size so that there exists a test with asymptotically vanishing probability of error. Our approach is based on various distributional distance metrics in order to incorporate the generative model of the data. We provide analyses of the error exponent using the maximum mean discrepancy and Kolmogorov–Smirnov distance and characterize the corresponding discrimination rates, i.e., lower bounds on the discrimination capacity, for these tests. Finally, an upper bound on the discrimination capacity based on Fano's inequality is developed. Numerical results are presented to validate the theoretical results.
  3. In this paper, we propose a new approach for robust compressive sensing (CS) using memristor crossbars that are constructed by recently invented memristor devices. The exciting features of a memristor crossbar, such as high density, low power and great scalability, make it a promising candidate to perform large-scale matrix operations. To apply memristor crossbars to solve a robust CS problem, the alternating directions method of multipliers (ADMM) is employed to split the original problem into subproblems that involve the solution of systems of linear equations. A system of linear equations can then be solved using memristor crossbars with astonishing O(1) time complexity. We also study the impact of hardware variations on the memristor crossbar based CS solver from both theoretical and practical points of view. The resulting overall complexity is given by O(n), which achieves O(n2.5) speed-up compared to the state-of-the-art software approach. Numerical results are provided to illustrate the effectiveness of the proposed CS solver.
  4. A memristor crossbar, which is constructed with memristor devices, has the unique ability to change and memorize the state of each of its memristor elements. It also has other highly desirable features such as high density, low power operation and excellent scalability. Hence the memristor crossbar technology can potentially be utilized for developing low-complexity and high-scalability solution frameworks for solving a large class of convex optimization problems, which involve extensive matrix operations and have critical applications in multiple disciplines. This paper, as the first attempt towards this direction, proposes a novel memristor crossbar-based framework for solving two important convex optimization problems, i.e., second-order cone programming (SOCP) and homogeneous quadratically constrained quadratic programming (QCQP) problems. In this paper, the alternating direction method of multipliers (ADMM) is adopted. It splits the SOCP and homogeneous QCQP problems into sub-problems that involve the solution of linear systems, which could be effectively solved using the memristor crossbar in O(1) time complexity. The proposed algorithm is an iterative procedure that iterates a constant number of times. Therefore, algorithms to solve SOCP and homogeneous QCQP problems have pseudo-O(N) complexity, which is a significant reduction compared to the state-of-the-art software solvers (O(N3.5)-O(N4)).