[Paper Review] Distributed Saddle-Point Problems Under Similarity

Aleksandr Beznosikov, Gesualdo Scutari+2 more|arXiv (Cornell University)|Jul 22, 2021

Stochastic Gradient Optimization Techniques42 references11 citations

TL;DR

This paper proposes communication-efficient distributed algorithms for solving (strongly-)convex-(strongly-)concave saddle-point problems under function similarity, where local functions across nodes are statistically or structurally similar. It establishes tight lower bounds on communication complexity that scale favorably with the similarity parameter δ, and introduces algorithms matching these bounds up to logarithmic factors, significantly improving efficiency over non-similar methods in ill-conditioned settings like robust regression.

ABSTRACT

We study solution methods for (strongly-)convex-(strongly)-concave Saddle-Point Problems (SPPs) over networks of two type - master/workers (thus centralized) architectures and meshed (thus decentralized) networks. The local functions at each node are assumed to be similar, due to statistical data similarity or otherwise. We establish lower complexity bounds for a fairly general class of algorithms solving the SPP. We show that a given suboptimality $ε>0$ is achieved over master/workers networks in $Ω\big(Δ\cdot δ/μ\cdot \log (1/\varepsilon)\big)$ rounds of communications, where $δ>0$ measures the degree of similarity of the local functions, $μ$ is their strong convexity constant, and $Δ$ is the diameter of the network. The lower communication complexity bound over meshed networks reads $Ω\big(1/{\sqrtρ} \cdot δ/μ\cdot\log (1/\varepsilon)\big)$, where $ρ$ is the (normalized) eigengap of the gossip matrix used for the communication between neighbouring nodes. We then propose algorithms matching the lower bounds over either types of networks (up to log-factors). We assess the effectiveness of the proposed algorithms on a robust logistic regression problem.

Motivation & Objective

• Address the lack of theoretical and algorithmic frameworks that exploit function similarity in distributed saddle-point problems (SPPs), particularly in decentralized and master-worker networks.
• Establish tight lower bounds on communication complexity for solving SPPs when local functions are similar, measured by a parameter δ.
• Design provably optimal algorithms for both master-worker and mesh network architectures that match the derived lower bounds up to logarithmic factors.
• Demonstrate the practical advantage of similarity-aware design in real-world problems such as distributed robust regression.
• Provide a theoretical foundation showing that similarity reduces communication cost compared to standard SPP solvers that ignore function similarity.

Proposed method

• Define δ-related SPPs where local functions differ by at most δ in their Hessian or gradient norms, capturing statistical similarity in data across nodes.
• Derive lower communication complexity bounds for master-worker networks: Ω(Δ·δ/μ·log(1/ε)), where Δ is network diameter, μ is strong convexity parameter, and ε is target suboptimality.
• Establish a lower bound for mesh networks: Ω(1/√ρ · δ/μ · log(1/ε)), where ρ is the normalized eigengap of the gossip matrix.
• Propose novel distributed algorithms for both network types that achieve the derived lower bounds up to logarithmic factors, using similarity-aware preconditioning.
• Integrate local similarity information into the algorithmic oracle to reduce communication rounds by leveraging the fact that δ ≪ L (Lipschitz constant) in many learning problems.
• Validate the approach on a distributed robust logistic regression problem, showing faster convergence and reduced communication compared to baseline methods.

Experimental results

Research questions

• RQ1 What is the fundamental communication complexity lower bound for solving distributed SPPs when local functions are similar (δ-related)?
• RQ2 How does function similarity (measured by δ) affect the communication efficiency of distributed SPP solvers compared to standard methods that ignore similarity?
• RQ3 Can we design distributed algorithms that achieve communication complexity matching the derived lower bounds for both master-worker and mesh networks?
• RQ4 To what extent does similarity-aware preconditioning improve convergence in ill-conditioned problems such as robust regression?
• RQ5 How do the communication complexity bounds scale with key parameters like δ, μ, Δ, and ρ in different network topologies?

Key findings

• The communication complexity lower bound for master-worker networks is Ω(Δ·δ/μ·log(1/ε)), showing a favorable dependence on δ/μ rather than L/μ.
• For mesh networks, the lower bound is Ω(1/√ρ · δ/μ · log(1/ε)), which improves over prior bounds that scale with L/μ when δ/μ ≪ L/μ.
• The proposed algorithms match the derived lower bounds up to logarithmic factors, proving their optimality in terms of communication efficiency.
• In robust regression experiments, the similarity-aware algorithms converge faster and require fewer communication rounds than standard distributed SPP solvers.
• The improvement is most pronounced in ill-conditioned problems where δ/μ is small but L/μ is large, making standard methods impractical.
• The analysis shows that function similarity can be exploited to reduce communication cost without sacrificing convergence guarantees, even in decentralized settings.