2018/09/27 by Dar Gilboa, Sam Buchanan, Gilboa, Dar +3 · 3 citations
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1809.10313
openalex publication_date 2018/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Randomly initialized first-order optimization algorithms are the method of choice for solving many high-dimensional nonconvex problems in machine learning, yet general theoretical guarantees cannot rule out convergence to critical points of poor objective value. For some highly structured nonconvex problems however, the success of gradient descent can be understood by studying the geometry of the objective. We study one such problem -- complete orthogonal dictionary learning, and provide converge guarantees for randomly initialized gradient descent to the neighborhood of a global optimum. The resulting rates scale as low order polynomials in the dimension even though the objective possesses an exponential number of saddle points. This efficient convergence can be viewed as a consequence of negative curvature normal to the stable manifolds associated with saddle points, and we provide evidence that this feature is shared by other nonconvex problems of importance as well.