Relaxed Proximal Point Algorithm: Tight Complexity Bounds and Acceleration without Momentum

TL;DR

This paper analyzes the relaxed proximal point algorithm (RPPA) with various relaxation schedules, establishing tight convergence bounds and accelerated rates, thereby advancing understanding of optimization methods without momentum.

Contribution

The paper extends convergence analysis and acceleration results of RPPA for different relaxation schedules, including new modifications, confirming conjectures and improving rates.

Findings

Type (i) achieves tight $O(1/N)$ convergence.

Type (ii) improves convergence rate by approximately $ oot2 elax$ over type (i).

Type (iii) attains $O(1/N^{1.2716})$ accelerated convergence.

Abstract

In this paper, we focus on the relaxed proximal point algorithm (RPPA) for solving convex (possibly nonsmooth) optimization problems. We conduct a comprehensive study on three types of relaxation schedules: (i) constant schedule with relaxation parameter $\alpha_k\equiv \alpha \in (0, \sqrt{2}]$, (ii) the dynamic schedule put forward by Teboulle and Vaisbourd [TV23], and (iii) the silver stepsize schedule proposed by Altschuler and Parrilo [AP23b]. The latter two schedules were initially investigated for the gradient descent (GD) method and are extended to the RPPA in this paper. For type (i), we establish tight non-ergodic $O(1/N)$ convergence rate results measured by function value residual and subgradient norm, where $N$ denotes the iteration counter. For type (ii), we establish a convergence rate that is tight and approximately $\sqrt{2}$ times better than the constant schedule of type (i). For type (iii), aside from the original silver stepsize schedule put forward by Altschuler and Parrilo, we propose two new modified silver stepsize schedules, and for all the three silver stepsize schedules, $O(1/N^{1.2716})$ accelerated convergence rate results with respect to three different performance metrics are established. Furthermore, our research affirms the conjecture in [LG24][Conjecture 3.2] on GD method with the original silver stepsize schedule.