Adaptive Polyak Stepsize with Level-value Adjustment for Distributed Optimization

TL;DR

This paper introduces a distributed adaptive Polyak stepsize algorithm with level-value adjustment that eliminates the need for global optimal values, ensuring fast convergence and network consensus in distributed optimization.

Contribution

The paper proposes DPS-LA, a novel distributed adaptive Polyak stepsize method that only requires solving a simple linear feasibility problem, removing the dependency on global optimal function values.

Findings

Guarantees network consensus.

Achieves linear speedup convergence rate of O(1/√(nT)).

Numerical results confirm efficiency.

Abstract

Stepsize selection remains a critical challenge in the practical implementation of distributed optimization. Existing distributed algorithms often rely on restrictive prior knowledge of global objective functions, such as Lipschitz constants. While centralized Polyak stepsizes have recently gained attention for their parameter-free adaptability and fast convergence. However, their extension to distributed settings is hindered by the requirement for local function values at the global optimum, which are typically unavailable to individual agents. To bridge this gap, we design a novel distributed adaptive Polyak stepsize algorithm with level-value adjustment (DPS-LA), where each agent only needs to solve a computationally efficient linear feasibility problem, thereby eliminating the dependency on global optimal values. Theoretical analysis proves that DPS-LA guarantees network consensus and achieves a linear speedup convergence rate of $\mathcal{O}(1/\sqrt{nT})$. Numerical results confirm the efficiency of the proposed algorithm.