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1. An Explicit Convergence Rate for Nesterov’s Method from SDP

   Safavi, Sam; Joshi, Bikash; França, Guilherme; Bento, José (June 2018, 2016 IEEE International Symposium on Information Theory)

   The framework of Integral Quadratic Constraints (IQC) introduced by Lessard et al. (2014) reduces the com- putation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). In particular, this technique was applied to Nesterov’s accelerated method (NAM). For quadratic functions, this SDP was explicitly solved leading to a new bound on the convergence rate of NAM, and for arbitrary strongly convex functions it was shown numerically that IQC can improve bounds from Nesterov (2004). Unfortunately, an explicit analytic solution to the SDP was not provided. In this paper, we provide such an analytical solution, obtaining a new general and explicit upper bound on the convergence rate of NAM, which we further optimize over its parameters. To the best of our knowledge, this is the best, and explicit, upper bound on the convergence rate of NAM for strongly convex functions. 

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2. An Explicit Convergence Rate for Nesterov’s Method from SDP

   Safavi, Sam; Joshi, Bikash; França, Guilherme; Bento, José (June 2018, 2018 IEEE International Symposium on In Information Theory)

   The framework of Integral Quadratic Constraints (IQC) introduced by Lessard et al. (2014) reduces the com- putation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). In particular, this technique was applied to Nesterov’s accelerated method (NAM). For quadratic functions, this SDP was explicitly solved leading to a new bound on the convergence rate of NAM, and for arbitrary strongly convex functions it was shown numerically that IQC can improve bounds from Nesterov (2004). Unfortunately, an explicit analytic solution to the SDP was not provided. In this paper, we provide such an analytical solution, obtaining a new general and explicit upper bound on the convergence rate of NAM, which we further optimize over its parameters. To the best of our knowledge, this is the best, and explicit, upper bound on the convergence rate of NAM for strongly convex functions. 

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3. Nesterov Accelerated Shuffling Gradient Method for Convex Optimization

   Trang H. Tran, Katya Scheinberg (July 2022, Proceedings of Machine Learning Research)

   In this paper, we propose Nesterov Accelerated Shuffling Gradient (NASG), a new algorithm for the convex finite-sum minimization problems. Our method integrates the traditional Nesterov’s acceleration momentum with different shuffling sampling schemes. We show that our algorithm has an improved rate of O(1/T ) using unified shuffling schemes, where T is the number of epochs. This rate is better than that of any other shuffling gradient methods in convex regime. Our convergence analysis does not require an assumption on bounded domain or a bounded gradient condition. For randomized shuffling schemes, we improve the convergence bound further. When employing some initial condition, we show that our method converges faster near the small neighborhood of the solution. Numerical simulations demonstrate the efficiency of our algorithm. 

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4. Distributed Learning-based Stability Assessment for Large Scale Networks of Dissipative Systems

   https://doi.org/10.1109/CDC45484.2021.9683774  

   Jena, Amit; Huang, Tong; Sivaranjani, S.; Kalathil, Dileep; Xie, Le (December 2021, IEEE Conference on Decision and Control (CDC))

   We propose a new distributed learning-based framework for stability assessment of a class of networked nonlinear systems, where each subsystem is dissipative. The aim is to learn, in a distributed manner, a Lyapunov function and associated region of attraction for the networked system. We begin by using a neural network function approximation to learn a storage function for each subsystem such that the subsystem satisfies a local dissipativity property. We next use a satisfiability modulo theories (SMT) solver based falsifier that verifies the local dissipativity of each subsystem by deter- mining an absence of counterexamples that violate the local dissipativity property, as established by the neural network approximation. Finally, we verify network-level stability by using an alternating direction method of multipliers (ADMM) approach to update the storage function of each subsystem in a distributed manner until a global stability condition for the network of dissipative subsystems is satisfied. This step also leads to a network-level Lyapunov function that we then use to estimate a region of attraction. We illustrate the proposed algorithm and its advantages on a microgrid interconnection with power electronics interfaces. 

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5. Dissipativity Theory for Accelerating Stochastic Variance Reduction: A Unified Analysis of SVRG and Katyusha Using Semidefinite Programs

   Hu, B; Wright, S; Lessard, L (July 2018, Proceedings of Machine Learning Research)

   Techniques for reducing the variance of gradient estimates used in stochastic programming algorithms for convex finite-sum problems have received a great deal of attention in recent years. By leveraging dissipativity theory from control, we provide a new perspective on two important variance-reduction algorithms: SVRG and its direct accelerated variant Katyusha. Our perspective provides a physically intuitive understanding of the behavior of SVRG-like methods via a principle of energy conservation. The tools discussed here allow us to automate the convergence analysis of SVRG-like methods by capturing their essential properties in small semidefinite programs amenable to standard analysis and computational techniques. Our approach recovers existing convergence results for SVRG and Katyusha and generalizes the theory to alternative parameter choices. We also discuss how our approach complements the linear coupling technique. Our combination of perspectives leads to a better understanding of accelerated variance-reduced stochastic methods for finite-sum problems. 

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