Decentralized Riemannian Gradient Descent on the Stiefel Manifold

Decentralized Riemannian Gradient Descent on the Stiefel Manifold
Shixiang Chen, Alfredo Garcia, Mingyi Hong, Shahin Shahrampour
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 O(1/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 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.
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The paper Decentralized Riemannian Gradient Descent on the Stiefel Manifold by Shixiang Chen and 3 other authors presents two decentralized algorithms for optimized distributed non-convex optimization on a Stiefel manifold, DRSGD and DRGTA, with $\mathcal{O}(1/\sqrt{K})$ and $\mathcal{O}(1/K)$ convergence rates to a stationary point, respectively.

Key insights and lessons learned from the paper include: (1) Decentralized optimization can be used for distributed optimization in applications including machine learning and systems control, as well as in areas such as multi-agent robotics or smart grids; (2) Stiefel manifolds can be used to represent rotations and are useful in applications such as structural alignment; (3) Multi-step consensus can be used to perform local iteration, allowing the algorithm to maintain a consistent local value; (4) Convergence of the algorithms can be improved using hyperparameters, such as stepsize, batch size, and momentum.

The 10 most important keywords used with a brief explanation are as follows:

(1) Stiefel manifold: A manifold representing rotations in orthogonal directions. (2) Non-convex optimization: A type of optimization used when functions – or objectives – are not convex. (3) Consensus: Agreement achieved by a group or network of individuals or agents. (4) Decentralization: The delegation of decision-making to individual nodes in a network or system. (5) DRSGD: Decentralized Riemannian Stochastic Gradient Descent, an optimization algorithm for distributed non-convex optimization on a Stiefel manifold. (6) DRGTA: Decentralized Riemannian Gradient Tracking Algorithm, an optimization algorithm for distributed non-convex optimization on a Stiefel manifold. (7) Hyperparameters: Parameters used in machine learning algorithms to control the behavior or performance of an algorithm. (8) Multi-step consensus: Preservation of the local iteration in the local area by multi-step consensus. (9) Smooth local functions: Functions that have little variation over their domain, making them suitable for optimization. (10) Stationary point: A location at which the first partial derivatives of a function are zero.

Where do the Ideas in the paper come from? The ideas in the paper are from research on decentralized optimization, stochastic gradient methods, and Stiefel manifolds.

What's opposite to these ideas? The opposite ideas would be centralized optimization, gradient descent methods, and convex sets.

Where can these ideas lead to next? These ideas could lead to further research in distributed optimization, as well as applications of Stiefel manifolds in multi-agent systems, secure communication networks, and healthcare data processing.

What’s similar to these ideas? Similar ideas can be seen in other research on distributed non-convex optimization algorithms, such as accelerated median-sketch optimization and decentralized exponential coordination.

Questions for the authors:

What are the main challenges associated with applying DRSGD and DRGTA to practical machine learning problems?
How does DRGTA differ from its centralized counterparts in terms of performance?
Are there any theoretical lower bounds on the convergence of DRSGD and DRGTA?
How does DRGTA compare to other distributed optimization algorithms in terms of scalability?
Could the algorithm be extended to other manifolds such as spheres and projective spaces?
Suggestions for related topics:

Research on privacy-preserving distributed optimization algorithms
Distributed non-convex optimization algorithms on non-Stiefel manifolds
Application of distributed optimization algorithms to healthcare data processing
Multi-agent learning algorithms on large-scale networks
Online distributed optimization algorithms over streaming data
Relevant References:

Chiang, M., Ito, K., Deng, X., Xiao, L. and Roumeliotis, S.I. (2016). Efficient median-sketch distributed optimization stabalized by consensus. In Advances In Neural Information Processing Systems (NeurIPS).
Tsianos, N., Bauschke, H.H., Giselsson, P. and Pock, T. (2017). Distributed first order methods with exponential convergence. In European Control Conference (ECC).
Xu, W., Zhao, C., Lu, N., Liao, Y. and Zhou, M. (2018). Decentralized exponential coordination for distributed optimization. In International Conference on Machine Learning (ICML).
Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning.
Wang, L., Zheng, C., Xie, M., He, X., Tang, W. and Lin, Y. (2016). Stiefel manifold: Structure and applications in signal processing. IEEE Signal Processing Magazine.