First-order Methods for Geodesically Convex Optimization (1602.06053v1)

Abstract: Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for several first-order algorithms on Hadamard manifolds. Specifically, we prove upper bounds for the global complexity of deterministic and stochastic (sub)gradient methods for optimizing smooth and nonsmooth g-convex functions, both with and without strong g-convexity. Our analysis also reveals how the manifold geometry, especially \emph{sectional curvature}, impacts convergence rates. To the best of our knowledge, our work is the first to provide global complexity analysis for first-order algorithms for general g-convex optimization.

• The paper establishes global iteration complexity bounds for deterministic and stochastic first-order methods on g-convex functions over Hadamard manifolds.
• It develops novel geometric trigonometric inequalities for Alexandrov spaces that extend Euclidean optimization techniques to nonlinear geometries.
• Empirical validations on matrix Karcher mean tasks demonstrate linear convergence for full gradient methods and sublinear rates for stochastic approaches.

Summary of "First-order Methods for Geodesically Convex Optimization"

The paper by Hongyi Zhang and Suvrit Sra addresses the optimization of geodesically convex (g-convex) functions over nonlinear spaces, particularly focusing on Hadamard manifolds. The authors contribute to the relatively underdeveloped field of g-convex optimization by analyzing the iteration complexity of several first-order methods. This work provides both theoretical insights and practical tools for optimization in Riemannian geometry, a generalization of Euclidean geometry that offers richer settings for many applications, especially those involving manifold-valued data.

Key Contributions

1. Complexity Analysis for First-order Methods: The paper provides global complexity bounds for deterministic and stochastic first-order methods applied to smooth and nonsmooth g-convex functions. These bounds incorporate the manifold geometry, particularly through sectional curvature, highlighting how curvature impacts convergence rates.
2. Geometric Trigonometric Inequalities: A significant contribution is the development of new trigonometric bounds applicable to Alexandrov spaces, a broad class of metric spaces with bounded curvature. These bounds are crucial for extending Euclidean optimization methodologies to more complex geometries like those found in manifold optimization.
3. Algorithmic Implications: The authors present iteration complexity results for g-convex optimization analogous to classical Euclidean results. For instance, for g-convex functions, the subgradient method achieves a convergence rate of $O(1/\sqrt{t})$, while for g-strongly convex functions, the established rate is $O(1/t)$.

Numerical Results and Claims

The research includes empirical validation of the theoretical results by comparing the performance of various first-order methods on g-convex optimization tasks, specifically matrix Karcher mean problems. The results demonstrate that the full gradient method achieves linear convergence, whereas the stochastic gradient methods exhibit sublinear convergence but are advantageous in initial iterations.

Theoretical and Practical Implications

The findings pave the way for more efficient optimization algorithms in non-Euclidean spaces, with potential applications in areas such as diffusion tensor imaging, machine learning involving Riemannian geometry, and robust statistical estimation. The development of techniques to handle manifold curvature explicitly in optimization processes is paramount given the increasing emergence of manifold-valued data in practical problems.

Future Directions

The paper suggests several avenues for future research:

• Acceleration Techniques:

Investigating the possibility of extending acceleration techniques like Nesterov’s method to nonlinear spaces remains open. This involves finding potential analogs to linear projections in manifold settings.

• Variance Reduction in Stochastic Methods:

As variance reduction methods have greatly improved stochastic gradient methods in Euclidean spaces, analogous improvements in Riemannian optimization could be highly impactful.

• Retractions vs. Exponential Maps:

Substituting exponential maps with computationally simpler retractions while preserving convergence properties is an important area for theoretical exploration and practical implementation.

This work represents a pivotal step toward embracing the complexities of manifold optimization, inviting further research to harness the full potential of geodesically convex methodologies in both theoretical and applied settings.