Geodesic Gradient Descent: A Generic and Learning-rate-free Optimizer on Objective Function-induced Manifolds
arXiv:2603.06651v1 Announce Type: new Abstract: Euclidean gradient descent algorithms barely capture the geometry of objective function-induced hypersurfaces and risk driving update trajectories off the hypersurfaces. Riemannian gradient descent algorithms address these issues but fail to represent complex hypersurfaces via a single classic manifold. We propose geodesic gradient descent (GGD), a generic and learning-rate-free Riemannian gradient descent algorithm. At each iteration, GGD uses an n-dimensional sphere to approximate a local neighborhood on the objective function-induced hypersurface, adapting to arbitrarily complex geometries. A tangent vector derived from the Euclidean gradient is projected onto the sphere to form a geodesic, ensuring the update trajectory stays on the hypersurface. Parameter updates are performed using the endpoint of the geodesic. The maximum step size of the gradient in GGD is equal to a quarter of the arc length on the n-dimensional sphere, thus eli
arXiv:2603.06651v1 Announce Type: new Abstract: Euclidean gradient descent algorithms barely capture the geometry of objective function-induced hypersurfaces and risk driving update trajectories off the hypersurfaces. Riemannian gradient descent algorithms address these issues but fail to represent complex hypersurfaces via a single classic manifold. We propose geodesic gradient descent (GGD), a generic and learning-rate-free Riemannian gradient descent algorithm. At each iteration, GGD uses an n-dimensional sphere to approximate a local neighborhood on the objective function-induced hypersurface, adapting to arbitrarily complex geometries. A tangent vector derived from the Euclidean gradient is projected onto the sphere to form a geodesic, ensuring the update trajectory stays on the hypersurface. Parameter updates are performed using the endpoint of the geodesic. The maximum step size of the gradient in GGD is equal to a quarter of the arc length on the n-dimensional sphere, thus eliminating the need for a learning rate. Experimental results show that compared with the classic Adam algorithm, GGD achieves test MSE reductions ranging from 35.79% to 48.76% for fully connected networks on the Burgers' dataset, and cross-entropy loss reductions ranging from 3.14% to 11.59% for convolutional neural networks on the MNIST dataset.
Executive Summary
The article introduces Geodesic Gradient Descent (GGD), a generic and learning-rate-free optimizer for objective function-induced manifolds. GGD approximates a local neighborhood on the hypersurface using an n-dimensional sphere, ensuring the update trajectory stays on the hypersurface. The algorithm eliminates the need for a learning rate, relying on the maximum step size of the gradient. Experimental results demonstrate GGD's efficacy, achieving significant reductions in test MSE and cross-entropy loss compared to the classic Adam algorithm. The proposed algorithm addresses the limitations of Euclidean gradient descent and Riemannian gradient descent algorithms, offering a novel approach to optimize complex hypersurfaces. This innovation has the potential to improve the performance of deep learning models and inspire further research in geometric optimization.
Key Points
- ▸ GGD approximates a local neighborhood on the hypersurface using an n-dimensional sphere.
- ▸ GGD eliminates the need for a learning rate, relying on the maximum step size of the gradient.
- ▸ Experimental results demonstrate GGD's efficacy in reducing test MSE and cross-entropy loss.
Merits
Strength in Geometric Representation
GGD effectively approximates complex hypersurfaces using an n-dimensional sphere, enabling the optimization of arbitrarily complex geometries.
Learning-rate-free Optimization
GGD eliminates the need for a learning rate, reducing the risk of hyperparameter tuning and improving the stability of the optimization process.
Demerits
Computational Complexity
GGD's reliance on n-dimensional sphere approximations may increase computational complexity, particularly for large-dimensional spaces.
Limited Scalability
GGD's performance on large-scale optimization problems remains to be explored, and its scalability may be limited by the computational complexity of the algorithm.
Expert Commentary
The article presents a promising approach to geometric optimization, building upon the foundation of Riemannian gradient descent algorithms. GGD's learning-rate-free optimization and geometric representation demonstrate its potential to improve the performance of deep learning models. However, further research is necessary to explore the computational complexity and scalability of the algorithm. The implications of GGD are far-reaching, with potential applications in various fields, including deep learning, computer vision, and machine learning.
Recommendations
- ✓ Further investigation into the computational complexity and scalability of GGD is necessary to ensure its practical applicability.
- ✓ The development of GGD variants and extensions to explore its performance on large-scale optimization problems and its applicability in various fields is recommended.