Accelerated linearized alternating direction method of multipliers with Nesterov extrapolation

The alternating direction method of multipliers (ADMMs) has found widespread use in solving separable convex optimization problems. In this paper, by employing Nesterov extrapolation technique, we propose two families of accelerated linearized ADMMsfor addressing two-block linearly constrained separable convex optimization problems where each block of the objective function exhibits a `nonsmooth' plus `smooth' composite structure. Our proposed accelerated linearized ADMMs extend two classical Nesterov acceleration methods designed for unconstrained composite optimization problems to linearly constrained problems. These methods are capable of achieving non-ergodic convergence rates of O(1/k^2) provided that one block of the objective function exhibits strong convexity and the gradients of smooth terms are Lipschitz continuous. We show that the proposed methods can reduce to accelerated linearized augmented Lagrangian methods (ALMs) with non-ergodic O(1/k2 ) convergence rates for solving one-block linearly constrained convex optimization problems. Furthermore, we explore hybrid versions of the proposed linearized accelerated ADMMs, which also demonstrate non-ergodic convergence rates of O(1/k2) under the same assumption. We also investigate inexact versions of the proposed methods. Under mild assumptions regarding error sequences, these methods still maintain O(1/k2) non-ergodic convergence rates. By choosing different extrapolation parameters, we explore the relationship between the proposed methods and existing accelerated methods

Randomized Smoothing for Stochastic Optimization

We analyze convergence rates of stochastic optimization procedures for non-smooth convex optimization problems. By combining randomized smoothing techniques with accelerated gradient methods, we obtain convergence rates of stochastic optimization procedures, both in expectation and with high probability, that have optimal dependence on the variance of the gradient estimates. To the best of our knowledge, these are the first variance-based rates for non-smooth optimization. We give several applications of our results to statistical estimation problems, and provide experimental results that demonstrate the effectiveness of the proposed algorithms. We also describe how a combination of our algorithm with recent work on decentralized optimization yields a distributed stochastic optimization algorithm that is order-optimal.Comment: 39 pages, 3 figure