Angelia Nedich

Soomin Lee, Angelia Nedich

Random projection algorithm is an iterative gradient method with random projections. Such an algorithm is of interest for constrained optimization when the constraint set is not known in advance or the projection operation on the whole constraint set is computationally prohibitive. This paper presents a distributed random projection (DRP) algorithm for fully distributed constrained convex optimization problems that can be used by multiple agents connected over a time-varying network, where each agent has its own objective function and its own constrained set. With reasonable assumptions, we prove that the iterates of all agents converge to the same point in the optimal set almost surely. In addition, we consider a variant of the method that uses a mini-batch of consecutive random projections and establish its convergence in almost sure sense. Experiments on distributed support vector machines demonstrate fast convergence of the algorithm. It actually shows that the number of iteration required until convergence is much smaller than scanning over all training samples just once.

Soomin Lee, Angelia Nedich

We consider a fully distributed constrained convex optimization problem over a multi-agent (no central coordinator) network. We propose an asynchronous gossip-based random projection (GRP) algorithm that solves the distributed problem using only local communications and computations. We analyze the convergence properties of the algorithm for an uncoordinated diminishing stepsize and a constant stepsize. For a diminishing stepsize, we prove that the iterates of all agents converge to the same optimal point with probability 1. For a constant stepsize, we establish an error bound on the expected distance from the iterates of the algorithm to the optimal point. We also provide simulation results on a distributed robust model predictive control problem.

Duong Thuy Anh Nguyen, Su Wang, Duong Tung Nguyen et al.

We investigate the problem of agent-to-agent interaction in decentralized (federated) learning over time-varying directed graphs, and, in doing so, propose a consensus-based algorithm called DSGTm-TV. The proposed algorithm incorporates gradient tracking and heavy-ball momentum to distributively optimize a global objective function, while preserving local data privacy. Under DSGTm-TV, agents will update local model parameters and gradient estimates using information exchange with neighboring agents enabled through row- and column-stochastic mixing matrices, which we show guarantee both consensus and optimality. Our analysis establishes that DSGTm-TV exhibits linear convergence to the exact global optimum when exact gradient information is available, and converges in expectation to a neighborhood of the global optimum when employing stochastic gradients. Moreover, in contrast to existing methods, DSGTm-TV preserves convergence for networks with uncoordinated stepsizes and momentum parameters, for which we provide explicit bounds. These results enable agents to operate in a fully decentralized manner, independently optimizing their local hyper-parameters. We demonstrate the efficacy of our approach via comparisons with state-of-the-art baselines on real-world image classification and natural language processing tasks.

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/}.

Eli Chien, Olgica Milenkovic, Angelia Nedich

The problem of estimating the support of a distribution is of great importance in many areas of machine learning, computer science, physics and biology. Most of the existing work in this domain has focused on settings that assume perfectly accurate sampling approaches, which is seldom true in practical data science. Here we introduce the first known approach to support estimation in the presence of sampling artifacts and errors where each sample is assumed to arise from a Poisson repeat channel which simultaneously captures repetitions and deletions of samples. The proposed estimator is based on regularized weighted Chebyshev approximations, with weights governed by evaluations of so-called Touchard (Bell) polynomials. The supports in the presence of sampling artifacts are calculated using discretized semi-infite programming methods. The estimation approach is tested on synthetic and textual data, as well as on GISAID data collected to address a new problem in computational biology: mutational support estimation in genes of the SARS-Cov-2 virus. In the later setting, the Poisson channel captures the fact that many individuals are tested multiple times for the presence of viral RNA, thereby leading to repeated samples, while other individual's results are not recorded due to test errors. For all experiments performed, we observed significant improvements of our integrated methods compared to those obtained through adequate modifications of state-of-the-art noiseless support estimation methods.