Efficient Riemannian Inexact Augmented Lagrangian Method for Nonsmooth Composite Optimization on Manifolds

Concepts de base

The proposed Riemannian inexact augmented Lagrangian (RiAL) method can find an ε-stationary point of a class of Riemannian nonsmooth composite problems with O(ε^-3) first-order oracle calls, achieving the best known oracle complexity.

Résumé

This paper introduces the RiAL method for solving a class of Riemannian nonsmooth composite optimization problems. The key features are:

1. Problem Formulation: The problem is formulated as minimizing the sum of a smooth function f(x) and a nonsmooth convex function h(A(x)), where x lies on a Riemannian manifold M.

2. RiAL Algorithm: The RiAL method solves the problem by iteratively updating the primal variable x using the Riemannian gradient descent (RGD) method and the dual variable z using the classical dual update. The inner RGD subproblem is solved to a specified accuracy.

3. Oracle Complexity Analysis: The authors establish that the RiAL method can find an ε-stationary point of the problem with O(ε^-3) first-order oracle calls, which matches the best known complexity for this class of problems.

4. Numerical Results: Experiments on sparse PCA and sparse CCA problems demonstrate that the RiAL method outperforms the existing ManIAL method in terms of solution quality and computational efficiency, highlighting the benefits of using the classical dual update.

The key insights are that the classical dual update, without the need for additional projection or damping, is sufficient to achieve the optimal oracle complexity, and the RiAL method can handle general nonlinear mappings A, going beyond the previous results that focused on linear A.

Stats

The paper does not provide specific numerical values or statistics to support the key claims. The analysis focuses on establishing the theoretical oracle complexity bounds.

Citations

None.

Idées clés tirées de

On the Oracle Complexity of a Riemannian Inexact Augmented Lagrangian Method for Riemannian Nonsmooth Composite Problems

by Meng Xu, Bo ... à arxiv.org 10-02-2024

https://arxiv.org/pdf/2410.00482.pdf

Questions plus approfondies

What are the potential applications of the RiAL method beyond the sparse PCA and sparse CCA problems considered in the paper?

The Riemannian Inexact Augmented Lagrangian (RiAL) method has a broad range of potential applications beyond sparse Principal Component Analysis (PCA) and sparse Canonical Correlation Analysis (CCA). Given its formulation for solving Riemannian nonsmooth composite problems, the RiAL method can be effectively applied in various fields, including:

Machine Learning: The RiAL method can be utilized in training models that involve manifold constraints, such as in deep learning architectures where weight matrices are constrained to lie on specific manifolds (e.g., orthogonal or unitary matrices).

Computer Vision: Applications in image processing, such as image registration and shape analysis, can benefit from the RiAL method, particularly when dealing with nonsmooth objectives that arise from geometric constraints.

Signal Processing: The method can be applied in problems like beamforming and array processing, where the optimization involves Riemannian manifolds due to the constraints on the signal vectors.

Robotics: In motion planning and control, the RiAL method can be used to optimize trajectories that are constrained to lie on manifolds, such as the configuration space of robotic arms.

Statistics: The RiAL method can be extended to handle statistical models that require optimization over manifolds, such as in the estimation of covariance matrices or in the context of manifold-valued data.

Network Analysis: In the context of network optimization, the RiAL method can be applied to problems involving the optimization of network flows or structures that are subject to manifold constraints.

These applications highlight the versatility of the RiAL method in addressing complex optimization problems across various domains, leveraging its ability to efficiently find ε-stationary points in Riemannian nonsmooth composite settings.

How can the RiAL method be extended to handle additional constraints or regularizers beyond the nonsmooth composite structure considered here?

The RiAL method can be extended to accommodate additional constraints or regularizers by modifying the augmented Lagrangian formulation and the optimization framework. Here are several strategies to achieve this:

Incorporating Additional Constraints: The RiAL method can be adapted to handle more complex constraints by augmenting the Lagrangian function with additional terms that represent these constraints. For instance, if there are equality or inequality constraints, they can be incorporated into the Lagrangian using Lagrange multipliers, similar to how the existing constraints are handled.

Regularization Terms: To include regularizers, such as L1 or L2 penalties, the objective function can be modified to include these terms. The augmented Lagrangian can then be adjusted to account for the regularization, ensuring that the optimization process balances the original objective with the regularization effect.

Proximal Operators: The use of proximal operators can be integrated into the RiAL method to handle nonsmooth regularizers. By defining proximal mappings for the regularization terms, the optimization can be performed in a way that respects the additional structure imposed by these regularizers.

Adaptive Penalty Parameters: The penalty parameter σ in the RiAL method can be adapted dynamically based on the convergence behavior of the algorithm. This allows for better handling of additional constraints, as the algorithm can adjust its sensitivity to the constraints as it progresses.

Multi-Objective Optimization: The RiAL method can be extended to multi-objective optimization problems by formulating a composite objective that combines multiple objectives with appropriate weights. The augmented Lagrangian can then be designed to reflect the trade-offs between these objectives.

By implementing these strategies, the RiAL method can be effectively tailored to solve a wider variety of optimization problems that involve additional constraints or regularizers, enhancing its applicability in real-world scenarios.

Are there any other Riemannian optimization problems where the classical dual update can be shown to be sufficient for achieving optimal complexity, without the need for additional techniques like projection or damping?

Yes, there are several Riemannian optimization problems where the classical dual update can be sufficient for achieving optimal complexity without requiring additional techniques such as projection or damping. Some notable examples include:

Riemannian Least Squares Problems: In problems where the objective is to minimize a least squares loss over a Riemannian manifold, the classical dual update can be effective. The structure of the problem often allows for direct updates without the need for projections, especially when the manifold is well-defined and the loss function is smooth.

Riemannian Trust-Region Methods: These methods often utilize classical dual updates to manage constraints effectively. The trust-region framework can be designed to ensure that the updates remain within the feasible region of the manifold, thus eliminating the need for additional projection steps.

Riemannian Optimization with Smooth Objectives: For problems where the objective function is smooth and the constraints are simple (e.g., linear constraints), the classical dual update can suffice. The smoothness of the objective allows for efficient gradient-based updates, and the constraints can be handled directly through the dual formulation.

Matrix Completion Problems: In scenarios where the goal is to complete a low-rank matrix subject to Riemannian constraints, classical dual updates can be employed effectively. The structure of the problem often allows for efficient updates without the need for damping or projection techniques.

Geometric Optimization Problems: Problems that involve optimizing geometric quantities, such as distances or angles on manifolds, can often be solved using classical dual updates. The geometric nature of these problems typically allows for straightforward updates that respect the manifold structure.

In these cases, the classical dual update provides a robust framework for achieving optimal complexity, demonstrating that it can be sufficient for a variety of Riemannian optimization problems without the complications introduced by additional techniques.