Improving Convergence Guarantees of Random Subspace Second-order Algorithm for Nonconvex Optimization

Rei Higuchi, Pierre-Louis Poirion, Akiko Takeda

Keywords: random projection, trust region method, non-convex optimization, second-order stationary point, local convergence

TL;DR: This paper proposes the Random Subspace Homogenized Trust Region (RSHTR) method with the best theoretical guarantees among random subspace algorithms for non-convex optimization.

Abstract: In recent years, random subspace methods have been actively studied for large-dimensional nonconvex problems. Recent subspace methods have improved theoretical guarantees such as iteration complexity and local convergence rate while reducing computational costs by deriving descent directions in randomly selected low-dimensional subspaces. This paper proposes the Random Subspace Homogenized Trust Region (RSHTR) method with the best theoretical guarantees among random subspace algorithms for nonconvex optimization. RSHTR achieves an $\varepsilon$-approximate first-order stationary point in $O(\varepsilon^{-3/2})$ iterations, converging locally at a linear rate. Furthermore, under rank-deficient conditions, RSHTR satisfies $\varepsilon$-approximate second-order necessary conditions in $O(\varepsilon^{-3/2})$ iterations and exhibits a local quadratic convergence. Experiments on real-world datasets verify the benefits of RSHTR.

Primary Area: optimization

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Submission Number: 6508