No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis
This provides a unified theoretical foundation for understanding optimization landscapes in machine learning, addressing a key challenge for researchers and practitioners working with non-convex low-rank matrix problems.
The paper tackles the problem of non-convex optimization in low-rank matrix problems like matrix sensing, matrix completion, and robust PCA, showing that all local minima are globally optimal and no high-order saddle points exist, which explains the global convergence of simple algorithms like stochastic gradient descent.
In this paper we develop a new framework that captures the common landscape underlying the common non-convex low-rank matrix problems including matrix sensing, matrix completion and robust PCA. In particular, we show for all above problems (including asymmetric cases): 1) all local minima are also globally optimal; 2) no high-order saddle points exists. These results explain why simple algorithms such as stochastic gradient descent have global converge, and efficiently optimize these non-convex objective functions in practice. Our framework connects and simplifies the existing analyses on optimization landscapes for matrix sensing and symmetric matrix completion. The framework naturally leads to new results for asymmetric matrix completion and robust PCA.