[Paper Review] Decentralized Riemannian Gradient Descent on the Stiefel Manifold

Shixiang Chen, Alfredo García+2 more|arXiv (Cornell University)|Feb 14, 2021

Distributed Control Multi-Agent Systems45 references4 citations

TL;DR

This paper proposes two decentralized Riemannian optimization algorithms—DRSGD and DRGTA—for solving non-convex distributed optimization on the Stiefel manifold, where agents minimize a global objective via local computation and communication over a network. DRGTA achieves exact convergence with a constant stepsize and an O(1/K) convergence rate to a stationary point, making it the first such algorithm for this setting.

ABSTRACT

We consider a distributed non-convex optimization where a network of agents aims at minimizing a global function over the Stiefel manifold. The global function is represented as a finite sum of smooth local functions, where each local function is associated with one agent and agents communicate with each other over an undirected connected graph. The problem is non-convex as local functions are possibly non-convex (but smooth) and the Steifel manifold is a non-convex set. We present a decentralized Riemannian stochastic gradient method (DRSGD) with the convergence rate of $\mathcal{O}(1/\sqrt{K})$ to a stationary point. To have exact convergence with constant stepsize, we also propose a decentralized Riemannian gradient tracking algorithm (DRGTA) with the convergence rate of $\mathcal{O}(1/K)$ to a stationary point. We use multi-step consensus to preserve the iteration in the local (consensus) region. DRGTA is the first decentralized algorithm with exact convergence for distributed optimization on Stiefel manifold.

Motivation & Objective

• Address the challenge of distributed non-convex optimization over the Stiefel manifold, where the constraint set is non-convex and consensus among agents is required.
• Design a decentralized algorithm that maintains convergence to a stationary point without requiring diminishing stepsize, overcoming limitations of prior methods.
• Develop a computationally efficient method that avoids vector transport and leverages multi-step consensus to preserve local iterates within the consensus region.
• Provide theoretical convergence guarantees for both stochastic and deterministic variants under smoothness and network connectivity assumptions.
• Establish the first exact convergence result with constant stepsize for decentralized optimization on the Stiefel manifold, filling a critical gap in distributed Riemannian optimization.

Proposed method

• Propose a decentralized Riemannian stochastic gradient method (DRSGD) using retraction-based updates and multi-step consensus to maintain iterates in the local consensus region.
• Introduce a novel decentralized Riemannian gradient tracking algorithm (DRGTA) that tracks the global gradient using a dual variable, enabling exact convergence with constant stepsize.
• Use a primal-dual framework to analyze convergence, leveraging gradient tracking to eliminate the O(β/(1−σ₂)) neighborhood error common in standard decentralized methods.
• Apply Riemannian optimization tools on the Stiefel manifold, including the Riemannian gradient and retraction, to handle the non-convex manifold constraint.
• Employ multi-step consensus to reduce consensus error and ensure that local iterates remain close to the average, preserving convergence properties.
• Use matrix F-norm and network parameter σ₂ to bound gradient and consensus errors, enabling tight convergence analysis.

Experimental results

Research questions

• RQ1 Can a decentralized Riemannian optimization algorithm achieve exact convergence to a stationary point on the Stiefel manifold with a constant stepsize?
• RQ2 What is the convergence rate of decentralized Riemannian methods on the Stiefel manifold when local functions are smooth but non-convex?
• RQ3 How can gradient tracking be adapted to Riemannian optimization on non-convex manifolds like the Stiefel manifold to eliminate the constant stepsize error?
• RQ4 What role does multi-step consensus play in maintaining local iterates within the consensus region and improving convergence in decentralized Riemannian settings?
• RQ5 Is it possible to design a vector transport-free, retraction-based algorithm that achieves O(1/K) convergence for distributed non-convex optimization on the Stiefel manifold?

Key findings

• DRGTA achieves exact convergence to a stationary point with a constant stepsize, with an O(1/K) convergence rate, making it the first such algorithm for distributed optimization on the Stiefel manifold.
• DRSGD achieves an O(1/√K) convergence rate to a stationary point, with iteration complexity O(1/ε²) for ε-stationary solutions.
• The algorithm avoids vector transport, reducing computational cost and simplifying implementation compared to prior methods.
• The convergence analysis relies on a primal-dual framework and bounds the gradient tracking error using network parameters and consensus error.
• The method ensures that local iterates remain within the consensus region through multi-step consensus, which helps control the deviation from the average iterate.
• Theoretical bounds show that the minimum Riemannian gradient norm decreases as O(1/√K) for DRSGD and O(1/K) for DRGTA, with explicit dependence on network connectivity and problem parameters.