Stochastic gradient descent on Riemannian manifolds (1111.5280v4)

Abstract: Stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise. In this paper, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the gradient descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and is illustrated here by four examples. In particular a novel gossip algorithm on the set of covariance matrices is derived and tested numerically.

• The paper establishes that SGD converges almost surely on Riemannian manifolds under specific compactness and step size conditions.
• It leverages intrinsic geometrical tools such as the exponential map and retractions to maintain constraints during optimization.
• Applications demonstrated include online PCA, intrinsic mean estimation in hyperbolic space, fixed-rank matrix identification, and covariance consensus.

Stochastic Gradient Descent on Riemannian Manifolds

The paper presents a comprehensive paper on extending stochastic gradient descent (SGD) to Riemannian manifolds, aiming to tackle optimization problems where the cost function evaluations are noisy. This work builds on the established results in Euclidean spaces, proving convergence guarantees under specific assumptions, now applied to the non-Euclidean setting of Riemannian manifolds. The framework not only generalizes existing techniques but also illustrates potential applications through several examples.

Convergence on Riemannian Manifolds

A significant contribution is the proof of almost sure convergence of the SGD algorithm on Riemannian manifolds. The authors demonstrate that under certain conditions—such as the presence of a compact set and appropriate step size sequences—convergence to a critical point of the cost function is ensured. This extends the convergence results from Euclidean spaces to a broader set of spaces defined by Riemannian metrics.

Algorithmic Framework

The proposed algorithm utilizes Riemannian geometry's intrinsic properties, leveraging the exponential map and retractions for efficient navigation on manifolds. The exponential map ensures that the optimization updates remain on the manifold, a necessity when dealing with nonlinear constraints inherent to these spaces.

Illustrative Examples

The paper includes four examples to highlight the applicability and performance of the proposed methodology:

1. Online PCA Using the Oja Algorithm: The Oja algorithm, reformulated in the Riemannian context, allows for a robust convergence analysis. The paper shows how online PCA can be performed efficiently with guaranteed convergence.
2. Intrinsic Mean on Hyperbolic Space: Utilizing hyperbolic geometry, the authors compute intrinsic means, demonstrating the method’s utility in domains where traditional Euclidean assumptions fail.
3. Fixed-rank Positive Semi-definite Matrix Identification: The paper addresses challenges in estimating low-rank matrices, a problem prevalent in machine learning applications like Mahalanobis distance learning. The extension of the algorithm's applicability is emphasized, showing convergence where traditional methods may lack feasibility or guarantees.
4. Consensus on Covariance Matrices: Introducing a novel gossip algorithm, the work showcases how the proposed SGD method can outperform standard approaches in reaching consensus faster due to the geometric properties utilized.

Practical Implications and Future Directions

The implications of this research are profound, offering a versatile approach for solving manifold-structured optimization problems relevant in many fields from machine learning to control systems. By providing convergence guarantees and applicability through retractions, the framework paves the way for developing more efficient algorithms for handling non-Euclidean data structures.

Future research could expand upon this work by addressing open questions in matrix completion or enhancing consensus algorithms on non-linear spaces, potentially leading to improved convergence rates and robustness in uncertain environments.

This paper lays foundational work for the application of Riemannian manifold theory within stochastic optimization, opening avenues for further exploration and practical exploitation of the rich geometric structures present in complex data.