A Simple Parallel Algorithm with an $O(1/t)$ Convergence Rate for General Convex Programs

Yu, Hao; Neely, Michael J.

doi:10.1137/16m1059011

Cited by 51 publications

(55 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To achieve an ε-optimal solution, compared to our results, their iteration complexity is O(ε −1 ) times worse for the convex problems and O(ε − 1 2 ) worse for the strongly convex problems. Assuming Lipschitz continuity of ∇f i for every i ∈ [m], [39] proposes a new primal-dual type algorithm for nonlinearly constrained convex programs. Every iteration, it minimizes a proximal Lagrangian function and updates the multiplier in a novel way.…”

Section: General Convex Problemsmentioning

confidence: 99%

“…With sufficiently large proximal parameter that depends on the Lipschitz constants of f i 's, the algorithm converges in O(1/k) ergodic rate. The follow-up paper [38] focuses on smooth constrained convex problems and proposes a linearized variant of the algorithm in [39]. Assuming compactness of the set X , it also establishes O(1/k) ergodic convergence of the linearized method.…”

Section: General Convex Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Iteration complexity of inexact augmented Lagrangian methods for constrained convex programming

2019

Math. Program.

View full text Add to dashboard Cite

Augmented Lagrangian method (ALM) has been popularly used for solving constrained optimization problems. Practically, subproblems for updating primal variables in the framework of ALM usually can only be solved inexactly. The convergence and local convergence speed of ALM have been extensively studied. However, the global convergence rate of inexact ALM is still open for problems with nonlinear inequality constraints. In this paper, we work on general convex programs with both equality and inequality constraints. For these problems, we establish the global convergence rate of inexact ALM and estimate its iteration complexity in terms of the number of gradient evaluations to produce a solution with a specified accuracy.We first establish an ergodic convergence rate result of inexact ALM that uses constant penalty parameters or geometrically increasing penalty parameters. Based on the convergence rate result, we apply Nesterov's optimal first-order method on each primal subproblem and estimate the iteration complexity of the inexact ALM. We show that if the objective is convex, then O(ε −1 ) gradient evaluations are sufficient to guarantee an ε-optimal solution in terms of both primal objective and feasibility violation. If the objective is strongly convex, the result can be improved to O(ε − 1 2 | log ε|). Finally, by relating to the inexact proximal point algorithm, we establish a nonergodic convergence rate result of inexact ALM that uses geometrically increasing penalty parameters. We show that the nonergodic iteration complexity result is in the same order as that for the ergodic result. Numerical experiments on quadratically constrained quadratic programming are conducted to compare the performance of the inexact ALM with different settings.

show abstract

Section: General Convex Problemsmentioning

confidence: 99%

Section: General Convex Problemsmentioning

confidence: 99%

Iteration complexity of inexact augmented Lagrangian methods for constrained convex programming

2019

Math. Program.

View full text Add to dashboard Cite

show abstract

“…The conventional primal-dual subgradient method, also known as the Arrow-Hurwicz-Uzawa subgradient method, is a low complexity algorithm with an O(1/ 2 ) convergence time. Recently, a new Lagrangian dual type algorithm with a faster O(1/ ) convergence time is proposed in Yu and Neely (2017). However, if the objective or constraint functions are not separable, each iteration of the Lagrangian dual type method in Yu and Neely (2017) requires to solve a unconstrained convex program, which can have huge complexity.…”

mentioning

confidence: 99%

A primal-dual type algorithm with the O(1/t) convergence rate for large scale constrained convex programs

Neely

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

Self Cite

View full text Add to dashboard Cite

This paper considers large scale constrained convex programs, which are usually not solvable by interior point methods or other Newton-type methods due to the prohibitive computation and storage complexity for Hessians and matrix inversions. Instead, large scale constrained convex programs are often solved by gradient based methods or decomposition based methods. The conventional primal-dual subgradient method, aka, Arrow-Hurwicz-Uzawa subgradient method, is a low complexity algorithm with the O(1/ √ t) convergence rate, where t is the number of iterations. If the objective and constraint functions are separable, the Lagrangian dual type method can decompose a large scale convex program into multiple parallel small scale convex programs. The classical dual gradient algorithm is an example of Lagrangian dual type methods and has convergence rate O(1/ √ t). Recently, a new Lagrangian dual type algorithm with faster O(1/t) convergence is proposed in [1]. However, if the objective or constraint functions are not separable, each iteration of the Lagrangian dual type method in [1] requires to solve a large scale unconstrained convex program, which can have huge complexity. This paper proposes a new primal-dual type algorithm, which only involves simple gradient updates at each iteration and has the O(1/t) convergence rate.

show abstract

“…It remains to bound the first term. Since (d 0 , Π) ∈ G, by Lemma 5.2, the corresponding state-action probabilities {θ (k) } K k=1 of Π satisfies K k=1 E(g i,t ), θ (k) ≤ C 1 KΨ/T and {θ (k) } K k=1 is feasible for (32)- (33). Since {θ…”

Section: A Relaxed Constraint Setsmentioning

confidence: 99%

“…It is easy to see that if q : X → R is convex, c > 0 and b ∈ R n , the function q(x) + c 2 x − b 2 2 is c-strongly convex. Furthermore, if the function h is c-strongly convex that is minimized at a point x min ∈ X , then (see, e.g., Corollary 1 in [33]):…”

Section: Sample-path Analysismentioning

confidence: 99%

Online Learning in Weakly Coupled Markov Decision Processes

Wei

Neely

2018

Proc. ACM Meas. Anal. Comput. Syst.

Self Cite

View full text Add to dashboard Cite

We consider multiple parallel Markov decision processes (MDPs) coupled by global constraints, where the time varying objective and constraint functions can only be observed after the decision is made. Special attention is given to how well the decision maker can perform in T slots, starting from any state, compared to the best feasible randomized stationary policy in hindsight. We develop a new distributed online algorithm where each MDP makes its own decision each slot after observing a multiplier computed from past information. While the scenario is significantly more challenging than the classical online learning context, the algorithm is shown to have a tight O( √ T ) regret and constraint violations simultaneously. To obtain such a bound, we combine several new ingredients including ergodicity and mixing time bound in weakly coupled MDPs, a new regret analysis for online constrained optimization, a drift analysis for queue processes, and a perturbation analysis based on Farkas' Lemma.

show abstract

A Simple Parallel Algorithm with an $O(1/t)$ Convergence Rate for General Convex Programs

Cited by 51 publications

References 22 publications

Iteration complexity of inexact augmented Lagrangian methods for constrained convex programming

Iteration complexity of inexact augmented Lagrangian methods for constrained convex programming

A primal-dual type algorithm with the O(1/t) convergence rate for large scale constrained convex programs

Online Learning in Weakly Coupled Markov Decision Processes

Contact Info

Product

Resources

About