Homotopy Analysis for Tensor PCA
This work addresses the problem of ensuring theoretical guarantees for nonconvex optimization in machine learning, particularly for tensor PCA, and is incremental as it builds on existing homotopy methods with specific adaptations.
The paper tackles the challenge of developing efficient nonconvex algorithms with theoretical guarantees by analyzing homotopy methods for global optimization, specifically applying them to tensor PCA to prove global convergence in high-noise regimes with tight signal-to-noise requirements matching the best degree-4 sum-of-squares algorithm.
Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimization of nonconvex functions. These methods start from an objective function that is efficient to optimize (e.g. convex), and progressively modify it to obtain the required objective, and the solutions are passed along the homotopy path. For the challenging problem of tensor PCA, we prove global convergence of the homotopy method in the "high noise" regime. The signal-to-noise requirement for our algorithm is tight in the sense that it matches the recovery guarantee for the best degree-4 sum-of-squares algorithm. In addition, we prove a phase transition along the homotopy path for tensor PCA. This allows to simplify the homotopy method to a local search algorithm, viz., tensor power iterations, with a specific initialization and a noise injection procedure, while retaining the theoretical guarantees.