[Paper Review] Proximal Point Approximations Achieving a Convergence Rate of O(1/k) for Smooth Convex-Concave Saddle Point Problems: Optimistic Gradient and Extra-gradient Methods.

Aryan Mokhtari, Asuman Ozdaglar+1 more|arXiv (Cornell University)|Jun 3, 2019

Stochastic Gradient Optimization Techniques15 citations

TL;DR

This paper establishes the first O(1/k) convergence rate for the optimistic gradient descent-ascent (OGDA) method in smooth convex-concave saddle point problems by interpreting OGDA and the extra-gradient (EG) method as approximations of the proximal point method. It proves both algorithms generate bounded iterates and achieve O(1/k) convergence for the primal-dual gap using averaged iterates, offering a simplified convergence analysis without compactness assumptions.

ABSTRACT

We study the iteration complexity of the optimistic gradient descent-ascent (OGDA) method and the extra-gradient (EG) method for finding a saddle point of a convex-concave unconstrained min-max problem. To do so, we first show that both OGDA and EG can be interpreted as approximate variants of the proximal point method. This is similar to the approach taken in [Nemirovski, 2004] which analyzes EG as an approximation of the `conceptual mirror prox'. In this paper, we highlight how gradients used in OGDA and EG try to approximate the gradient of the Proximal Point method. We then exploit this interpretation to show that both algorithms produce iterates that remain within a bounded set. We further show that the primal dual gap of the averaged iterates generated by both of these algorithms converge with a rate of $\mathcal{O}(1/k)$. Our theoretical analysis is of interest as it provides a the first convergence rate estimate for OGDA in the general convex-concave setting. Moreover, it provides a simple convergence analysis for the EG algorithm in terms of function value without using compactness assumption.

Motivation & Objective

• To establish the convergence rate of the optimistic gradient descent-ascent (OGDA) method in smooth convex-concave min-max problems.
• To show that both OGDA and extra-gradient (EG) methods can be interpreted as approximations of the proximal point method.
• To prove that the primal-dual gap of averaged iterates converges at a rate of O(1/k) for both algorithms.
• To provide a simplified convergence analysis for EG without requiring compactness assumptions.
• To demonstrate that iterates generated by OGDA and EG remain within a bounded set under the given conditions.

Proposed method

• Interpreting OGDA and EG as approximate variants of the proximal point method, leveraging gradient approximation to mirror the proximal point's ideal update.
• Using the proximal point interpretation to establish boundedness of iterates for both OGDA and EG under smooth convex-concave assumptions.
• Analyzing the primal-dual gap of averaged iterates to derive the O(1/k) convergence rate.
• Applying a novel analysis framework that avoids compactness assumptions, simplifying convergence proofs for EG.
• Establishing that the gradient steps in OGDA and EG effectively approximate the proximal point gradient, enabling convergence guarantees.

Experimental results

Research questions

• RQ1 Can the optimistic gradient descent-ascent (OGDA) method achieve a convergence rate of O(1/k) in smooth convex-concave saddle point problems?
• RQ2 How do OGDA and extra-gradient (EG) methods relate to the proximal point method in terms of algorithmic structure and convergence?
• RQ3 Does the primal-dual gap of averaged iterates converge at O(1/k) for both OGDA and EG under convex-concave conditions?
• RQ4 Can the convergence analysis of EG be simplified without relying on compactness assumptions?
• RQ5 Do OGDA and EG generate bounded iterates in the general convex-concave setting?

Key findings

• The OGDA method achieves a convergence rate of O(1/k) for the primal-dual gap of averaged iterates in smooth convex-concave saddle point problems, marking the first such rate estimate for OGDA in this setting.
• Both OGDA and EG generate bounded iterates under the smooth convex-concave assumption, ensuring stability of the iterates during optimization.
• The extra-gradient method achieves O(1/k) convergence for the primal-dual gap without requiring a compactness assumption, simplifying the theoretical analysis.
• The proximal point interpretation provides a unifying framework that explains the convergence behavior of both OGDA and EG by approximating the ideal proximal update.
• The analysis reveals that the gradient steps in OGDA and EG effectively emulate the proximal point method’s gradient, enabling convergence guarantees through this approximation.