Global Optimality of Local Search for Low Rank Matrix Recovery (1605.07221v2)

Abstract: We show that there are no spurious local minima in the non-convex factorized parametrization of low-rank matrix recovery from incoherent linear measurements. With noisy measurements we show all local minima are very close to a global optimum. Together with a curvature bound at saddle points, this yields a polynomial time global convergence guarantee for stochastic gradient descent {\em from random initialization}.

• The paper proves that non-convex formulations for low-rank matrix recovery have no spurious local minima under noiseless measurements.
• It establishes that a suitable RIP condition on the measurement operator guarantees every local minimum is global, validating the use of SGD.
• The analysis extends to noisy scenarios, demonstrating that local search methods converge near the global optimum with robust performance.

Global Optimality of Local Search for Low Rank Matrix Recovery

The paper "Global Optimality of Local Search for Low Rank Matrix Recovery" by Bhojanapalli, Neyshabur, and Srebro rigorously examines the optimization landscape of non-convex formulations used in low-rank matrix recovery. The authors focus on the role of local search methods, particularly stochastic gradient descent (SGD), in solving the matrix sensing problem, which involves recovering a low-rank matrix from linear measurements, potentially corrupted by noise.

Key Contributions and Results

The central contribution of the paper is the demonstration of the absence of spurious local minima in the non-convex factorized parameterization of the low-rank matrix recovery problem. Specifically, the authors prove that under a suitable incoherence condition on the measurement operator, all local minima of the optimization problem correspond to global minima, provided the measurements are noiseless and the matrix to be recovered is indeed low-rank.

1. Structure of Local Minima: Through rigorous arguments, it is shown that for the matrix sensing problem with noiseless measurements and low-rank target matrices, there are no spurious local minima. This result provides a strong theoretical justification for the successful empirical performance of local search methods.
2. Role of Measurement Operator: The paper emphasizes the role of the measurement operator's Restricted Isometry Property (RIP) as a critical condition for guaranteeing no spurious local minima. Specifically, a $(2r, \delta_{2r})$-RIP condition with $\delta_{2r}<1/5$ ensures the optimization landscape's favorable structure.
3. Convergence Under Noise: In realistic scenarios with noisy measurements, the authors extend their analysis to show that all local minima are close to the global minimum. This robustness to noise implies that local search methods remain effective in practical situations.
4. Saddle Points and SGD Convergence: The paper further introduces a curvature condition at saddle points, establishing a polynomial time convergence guarantee for SGD starting from random initialization. This result bridges the gap between theoretical guarantees and practical algorithms used in machine learning and signal processing applications.

Implications and Future Directions

Theoretical Implications

The paper's findings have profound implications for the theoretical understanding of non-convex optimization problems, bridging an essential gap between theory and empirical success. By demonstrating that local search methods can achieve global optimality without the requirement for sophisticated initialization schemes, the paper suggests that non-convex machine learning problems might be inherently more tractable than previously thought, provided the problem satisfies specific structural conditions like RIP.

Practical Implications

On a practical front, these results advocate for the broader adoption of local search methods without reliance on sophisticated initialization schemes, such as those based on singular value decomposition (SVD). The results suggest that for large-scale matrix recovery tasks, simple SGD starting from random initialization could be both efficient and effective, alleviating computational burdens associated with traditional convex relaxation approaches.

Future Work

Future research could explore the extension of these results to broader classes of non-convex problems beyond low-rank matrix recovery. Given the parallels between matrix factorization and neural network training, similar analyses might yield insights into the optimization landscapes of deep learning models. Investigating the applicability of these results to other domains such as tensor decompositions or broader classes of inverse problems might unlock new strategies for efficient large-scale data analysis.

Overall, the investigation carried out in this paper provides a solid foundation for enhancing our understanding of non-convex optimization methods, offering both theoretical insights and practical guidance for leveraging local search in matrix recovery and beyond.