A Proximal-Gradient Homotopy Method for the Sparse Least-Squares Problem

We consider solving the $\ell_1$-regularized least-squares ($\ell_1$-LS) problem in the context of sparse recovery, for applications such as compressed sensing. The standard proximal gradient method, also known as iterative soft-thresholding when applied to this problem, has low computational cost per iteration but a rather slow convergence rate. Nevertheless, when the solution is sparse, it often exhibits fast linear convergence in the final stage. We exploit the local linear convergence using a homotopy continuation strategy, i.e., we solve the $\ell_1$-LS problem for a sequence of decreasing values of the regularization parameter, and use an approximate solution at the end of each stage to warm start the next stage. Although similar strategies have been studied in the literature, there have been no theoretical analysis of their global iteration complexity. This paper shows that under suitable assumptions for sparse recovery, the proposed homotopy strategy ensures that all iterates along the homotopy solution path are sparse. Therefore the objective function is effectively strongly convex along the solution path, and geometric convergence at each stage can be established. As a result, the overall iteration complexity of our method is $O(\log(1/\epsilon))$ for finding an $\epsilon$-optimal solution, which can be interpreted as global geometric rate of convergence. We also present empirical results to support our theoretical analysis

A Proximal Stochastic Gradient Method with Progressive Variance Reduction

We consider the problem of minimizing the sum of two convex functions: one is the average of a large number of smooth component functions, and the other is a general convex function that admits a simple proximal mapping. We assume the whole objective function is strongly convex. Such problems often arise in machine learning, known as regularized empirical risk minimization. We propose and analyze a new proximal stochastic gradient method, which uses a multi-stage scheme to progressively reduce the variance of the stochastic gradient. While each iteration of this algorithm has similar cost as the classical stochastic gradient method (or incremental gradient method), we show that the expected objective value converges to the optimum at a geometric rate. The overall complexity of this method is much lower than both the proximal full gradient method and the standard proximal stochastic gradient method

Near-optimal ground state preparation

Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem

Feasibility studies of a converter-free grid-connected offshore hydrostatic wind turbine

Owing to the increasing penetration of renewable power generation, the modern power system faces great challenges in frequency regulations and reduced system inertia. Hence, renewable energy is expected to take over part of the frequency regulation responsibilities from the gas or hydro plants and contribute to the system inertia. In this article, we investigate the feasibility of frequency regulation by the offshore hydrostatic wind turbine (HWT). The simulation model is transformed from NREL (National Renewable Energy Laboratory) 5-MW gearbox-equipped wind turbine model within FAST (fatigue, aerodynamics, structures, and turbulence) code. With proposed coordinated control scheme and the hydrostatic transmission configuration of the HWT, the `continuously variable gearbox ratio' in turbulent wind conditions can be realised to maintain the constant generator speed, so that the HWT can be connected to the grid without power converters in-between. To test the performances of the control scheme, the HWT is connected to a 5-bus grid model and operates with different frequency events. The simulation results indicate that the proposed control scheme is a promising solution for offshore HWT to participated in frequency response in the modern power system