Algorithmic Regularization in Tensor Optimization: Towards a Lifted Approach in Matrix Sensing

TL;DR

This paper investigates how gradient descent induces implicit regularization in tensor optimization within the lifted matrix sensing framework, leading to solutions close to rank-1 tensors and aiding in global optimality.

Contribution

It demonstrates that small initialization in gradient descent on lifted tensor problems promotes approximate rank-1 solutions and critical points with escape directions, highlighting the importance of tensor parametrization.

Findings

GD with small initialization yields approximate rank-1 tensors

Lifted tensor parametrization helps escape spurious solutions

First-order methods can achieve global optimality in tensor sensing

Abstract

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.