Time-varying graphical LASSO (TVGL) aims to infer a sequence of graphs from time series data and has been widely used in many statistical inference problems. The existing algorithms usually suffer from high computational cost when solving large-scale TVGL problems. In this paper, we develop an efficient and scalable hierarchical block coordinate descent (HBCD) method for solving TVGL with smooth temporal difference prior. The proposed HBCD method contains both outer-loop and inner-loop BCD iterations. The outer loops seperate the original TVGL problem into a sequence of subproblems, which are variants of the static graphical LASSO problems. Then, we propose an efficient BCD method to solve the inner-loop subproblems. We provide theoretical analysis that indicates the linear convergence of our proposed method. Furthermore, numerical experiments on both synthetic and real datasets show that our method significantly outperforms the state-of-the-art algorithm in terms of both the required iterations and CPU time to reach the target precision.

Decentralized optimization aims to cooperatively solve a global finite-sum loss function, where each agent only possesses knowledge of its own local function. Real-world applications introduce challenges such as unstable channels and differential privacy concerns, necessitating the development of more robust algorithms. This paper proposes a framework that explores second-order information using two approaches: global Newton tracking and local Newton preconditioning. Furthermore, we adapt a generic convergence result for gradient tracking methods to demonstrate the almost sure convergence property of both schemes under specific conditions. To validate the effectiveness of the proposed algorithm, numerical experiments are conducted on synthetic and real datasets, showcasing its robustness and superior accuracy compared to existing first-order methods.

Optimization problems with access to only zeroth-order information of the objective function on Riemannian manifolds arise in various applications, spanning from statistical learning to robot learning. While various zeroth-order algorithms have been proposed in Euclidean space, they are not inherently designed to handle the challenging constraints imposed by Riemannian manifolds. The proper adaptation of zeroth-order techniques to Riemannian manifolds remained unknown until the pioneering work of (Li et al., 2023a). However, zeroth-order algorithms are widely observed to converge slowly and be unstable in practice. To alleviate these issues, we propose a Riemannian accelerated zeroth-order algorithm with improved robustness. Regarding efficiency, our accelerated algorithm has the function query complexity of $\mathcal{O}(\epsilon^{-7/4}d)$ for finding an $\epsilon$-approximate first-order stationary point. By introducing a small perturbation, it exhibits a function query complexity of $\tilde{\mathcal{O}}(\epsilon^{-7/4}d)$ for seeking a second-order stationary point with a high probability, matching state-of-the-art result in Euclidean space. Moreover, we further establish the almost sure convergence in the asymptotic sense through the Stable Manifold Theorem. Regarding robustness, our algorithm requires larger smoothing parameters in the order of $\tilde{\mathcal{O}}(\epsilon^{7/8}d^{-1/2})$, improving the existing result by a factor of $\tilde{\mathcal{O}}(\epsilon^{3/4})$.