[Paper Review] Globally Convergent Stochastic Optimization for Canonical Correlation Analysis.

Weiran Wang, Jialei Wang+1 more|arXiv (Cornell University)|Apr 7, 2016

Stochastic Gradient Optimization Techniques20 references6 citations

TL;DR

This paper proposes a globally convergent stochastic optimization algorithm for canonical correlation analysis (CCA), leveraging an alternating least squares formulation and approximate stochastic gradient methods to solve nonconvex, non-decoupled objectives. The method achieves significantly improved time complexity and superior empirical performance over prior approaches.

ABSTRACT

We study the stochastic optimization of canonical correlation analysis (CCA), whose objective is nonconvex and does not decouple over training samples. Although several stochastic optimization algorithms have been recently proposed to solve this problem, no global convergence guarantee was provided by any of them. Based on the alternating least squares formulation of CCA, we propose a globally convergent stochastic algorithm, which solves the resulting least squares problems approximately to sufficient accuracy with state-of-the-art stochastic gradient methods for convex optimization. We provide the overall time complexity of our algorithm which significantly improves upon that of previous work. Experimental results demonstrate the superior performance of our algorithm.

Motivation & Objective

• To address the lack of global convergence guarantees in existing stochastic optimization methods for CCA.
• To develop a globally convergent algorithm for CCA, which involves a nonconvex, non-decomposable objective function over training samples.
• To improve time complexity compared to prior stochastic CCA methods through efficient approximate solutions to least squares subproblems.
• To provide theoretical convergence guarantees while maintaining practical efficiency on real-world datasets.

Proposed method

• The method reformulates CCA as an alternating least squares problem to enable iterative optimization.
• It applies state-of-the-art stochastic gradient methods to approximately solve the resulting convex subproblems with sufficient accuracy.
• The algorithm alternates between optimizing over the two sets of canonical vectors, ensuring convergence to a critical point.
• Theoretical analysis establishes global convergence under mild conditions, leveraging the structure of the alternating least squares formulation.
• Time complexity is analyzed and shown to be significantly better than previous stochastic CCA methods.

Experimental results

Research questions

• RQ1 Can a stochastic optimization algorithm for CCA achieve global convergence despite the nonconvex and non-decomposable nature of the objective?
• RQ2 What is the optimal level of accuracy required for solving the subproblems to ensure global convergence?
• RQ3 How does the proposed algorithm compare in time complexity to existing stochastic CCA methods?
• RQ4 What empirical performance gains does the algorithm achieve on real-world datasets?

Key findings

• The proposed algorithm achieves global convergence for CCA, resolving a key limitation of prior stochastic methods.
• The time complexity of the algorithm is significantly lower than that of previous stochastic CCA approaches.
• Experimental results show superior performance in terms of convergence speed and solution quality on benchmark datasets.
• The method maintains theoretical convergence guarantees while achieving practical efficiency through approximate subproblem solving.