A Block Coordinate Descent Method for Nonsmooth Composite Optimization under Orthogonality Constraints

ICLR 2025 Conference Submission6133 Authors

Keywords: Orthogonality Constraints, Nonconvex Optimization, Nonsmooth Composite Optimization, Block Coordinate Descent, Convergence Analysis

Abstract: Nonsmooth composite optimization with orthogonality constraints is crucial in statistical learning and data science, but it presents challenges due to its nonsmooth objective and computationally expensive, non-convex constraints. In this paper, we propose a new approach called \textbf{OBCD}, which leverages Block Coordinate Descent (BCD) to address these challenges. \textbf{OBCD} is a feasible method with a small computational footprint. In each iteration, it updates $k$ rows of the solution matrix, where $k \geq 2$, while globally solving a small nonsmooth optimization problem under orthogonality constraints. We prove that the limiting points of \textbf{OBCD}, referred to as (global) block-$k$ stationary points, offer stronger optimality than standard critical points. Furthermore, we show that \textbf{OBCD} converges to $\epsilon$-block-$k$ stationary points with an ergodic convergence rate of $\mathcal{O}(1/\epsilon)$. Additionally, under the Kurdyka-Lojasiewicz (KL) inequality, we establish the non-ergodic convergence rate of \textbf{OBCD}. We also extend \textbf{OBCD} with breakpoint searching methods for subproblem solving and greedy strategies for working set selection. Comprehensive experiments demonstrate the superior performance of our approach across various tasks.

Primary Area: optimization

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Submission Number: 6133