Algorithmic Regularization in Tensor Optimization: Towards a Lifted Approach in Matrix Sensing
This addresses the matrix sensing problem for machine learning practitioners by providing theoretical insights into implicit regularization, though it appears incremental as it builds on existing lifted frameworks.
The paper tackled the problem of achieving global optimality in non-convex matrix sensing by analyzing gradient descent in a lifted tensor optimization framework, showing that with small initialization, it yields approximate rank-1 tensors and critical points with escape directions.
Gradient descent (GD) is crucial for generalization in machine learning models, as it induces implicit regularization, promoting compact representations. In this work, we examine the role of GD in inducing implicit regularization for tensor optimization, particularly within the context of the lifted matrix sensing framework. This framework has been recently proposed to address the non-convex matrix sensing problem by transforming spurious solutions into strict saddles when optimizing over symmetric, rank-1 tensors. We show that, with sufficiently small initialization scale, GD applied to this lifted problem results in approximate rank-1 tensors and critical points with escape directions. Our findings underscore the significance of the tensor parametrization of matrix sensing, in combination with first-order methods, in achieving global optimality in such problems.