Wei Shi

Hoi-To Wai, Wei Shi, Cesar A. Uribe et al.

This paper studies an acceleration technique for incremental aggregated gradient ({\sf IAG}) method through the use of \emph{curvature} information for solving strongly convex finite sum optimization problems. These optimization problems of interest arise in large-scale learning applications. Our technique utilizes a curvature-aided gradient tracking step to produce accurate gradient estimates incrementally using Hessian information. We propose and analyze two methods utilizing the new technique, the curvature-aided IAG ({\sf CIAG}) method and the accelerated CIAG ({\sf A-CIAG}) method, which are analogous to gradient method and Nesterov's accelerated gradient method, respectively. Setting $κ$ to be the condition number of the objective function, we prove the $R$ linear convergence rates of $1 - \frac{4c_0 κ}{(κ+1)^2}$ for the {\sf CIAG} method, and $1 - \sqrt{\frac{c_1}{2κ}}$ for the {\sf A-CIAG} method, where $c_0,c_1 \leq 1$ are constants inversely proportional to the distance between the initial point and the optimal solution. When the initial iterate is close to the optimal solution, the $R$ linear convergence rates match with the gradient and accelerated gradient method, albeit {\sf CIAG} and {\sf A-CIAG} operate in an incremental setting with strictly lower computation complexity. Numerical experiments confirm our findings. The source codes used for this paper can be found on \url{http://github.com/hoitowai/ciag/}.

Hoi-To Wai, Wei Shi, Angelia Nedic et al.

We propose a new algorithm for finite sum optimization which we call the curvature-aided incremental aggregated gradient (CIAG) method. Motivated by the problem of training a classifier for a d-dimensional problem, where the number of training data is $m$ and $m \gg d \gg 1$, the CIAG method seeks to accelerate incremental aggregated gradient (IAG) methods using aids from the curvature (or Hessian) information, while avoiding the evaluation of matrix inverses required by the incremental Newton (IN) method. Specifically, our idea is to exploit the incrementally aggregated Hessian matrix to trace the full gradient vector at every incremental step, therefore achieving an improved linear convergence rate over the state-of-the-art IAG methods. For strongly convex problems, the fast linear convergence rate requires the objective function to be close to quadratic, or the initial point to be close to optimal solution. Importantly, we show that running one iteration of the CIAG method yields the same improvement to the optimality gap as running one iteration of the full gradient method, while the complexity is $O(d^2)$ for CIAG and $O(md)$ for the full gradient. Overall, the CIAG method strikes a balance between the high computation complexity incremental Newton-type methods and the slow IAG method. Our numerical results support the theoretical findings and show that the CIAG method often converges with much fewer iterations than IAG, and requires much shorter running time than IN when the problem dimension is high.

Zhi Li, Wei Shi, Ming Yan

This paper proposes a novel proximal-gradient algorithm for a decentralized optimization problem with a composite objective containing smooth and non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with by gradient and proximal updates, respectively. The proposed algorithm is closely related to a previous algorithm, PG-EXTRA \cite{shi2015proximal}, but has a few advantages. First of all, agents use uncoordinated step-sizes, and the stable upper bounds on step-sizes are independent of network topologies. The step-sizes depend on local objective functions, and they can be as large as those of the gradient descent. Secondly, for the special case without non-smooth terms, linear convergence can be achieved under the strong convexity assumption. The dependence of the convergence rate on the objective functions and the network are separated, and the convergence rate of the new algorithm is as good as one of the two convergence rates that match the typical rates for the general gradient descent and the consensus averaging. We provide numerical experiments to demonstrate the efficacy of the introduced algorithm and validate our theoretical discoveries.