Convergence Rates Analysis of The Quadratic Penalty Method and Its Applications to Decentralized Distributed Optimization
For researchers in optimization and distributed systems, this work provides theoretical justification and improved convergence rates for widely used penalty methods, though the improvements are incremental over existing analyses.
The paper provides convergence rate analysis for a variant of the quadratic penalty method with increasing penalty and inexact minimization, showing rates of O(1/√K) for convex and O(1/K) for strongly convex problems, improved to O(1/K) and O(1/K^2) with Nesterov's acceleration. Applied to decentralized optimization, it improves existing rates and extends to a communication-efficient version.
In this paper, we study a variant of the quadratic penalty method for linearly constrained convex problems, which has already been widely used but actually lacks theoretical justification. Namely, the penalty parameter steadily increases and the penalized objective function is minimized inexactly rather than exactly, e.g., with only one step of the proximal gradient descent. For such a variant of the quadratic penalty method, we give counterexamples to show that it may not give a solution to the original constrained problem. By choosing special penalty parameters, we ensure the convergence and further establish the convergence rates of $O\left(\frac{1}{\sqrt{K}}\right)$ for the generally convex problems and $O\left(\frac{1}{K}\right)$ for strongly convex ones, where $K$ is the number of iterations. Furthermore, by adopting Nesterov's extrapolation we show that the convergence rates can be improved to $O\left(\frac{1}{K}\right)$ for the generally convex problems and $O\left(\frac{1}{K^2}\right)$ for strongly convex ones. When applied to the decentralized distributed optimization, the penalty methods studied in this paper become the widely used distributed gradient method and the fast distributed gradient method. However, due to the totally different analysis framework, we can improve their $O\left(\frac{\log K}{\sqrt{K}}\right)$ and $O\left(\frac{\log K}{K}\right)$ convergence rates to $O\left(\frac{1}{\sqrt{K}}\right)$ and $O\left(\frac{1}{K}\right)$ with fewer assumptions on the network topology for general convex problems. Using our analysis framework, we also extend the fast distributed gradient method to a communication efficient version, i.e., finding an $\varepsilon$ solution in $O\left(\frac{1}{\varepsilon}\right)$ communications and $O\left(\frac{1}{\varepsilon^{2+δ}}\right)$ computations for the non-smooth problems, where $δ$ is a small constant.