Algorithmic Regularization in Over-parameterized Matrix Sensing and Neural Networks with Quadratic Activations
This solves a conjecture in matrix sensing and extends to neural networks with quadratic activations, providing theoretical insights into implicit regularization in over-parameterized learning, which is incremental but addresses a known bottleneck.
The paper tackles the problem of recovering a low-rank matrix from linear measurements using over-parameterized models, showing that gradient descent with small initialization implicitly regularizes the solution to recover the true matrix in approximately O(√r) iterations given O(dr²) measurements, even when r is much smaller than d.
We show that the gradient descent algorithm provides an implicit regularization effect in the learning of over-parameterized matrix factorization models and one-hidden-layer neural networks with quadratic activations. Concretely, we show that given $\tilde{O}(dr^{2})$ random linear measurements of a rank $r$ positive semidefinite matrix $X^{\star}$, we can recover $X^{\star}$ by parameterizing it by $UU^\top$ with $U\in \mathbb R^{d\times d}$ and minimizing the squared loss, even if $r \ll d$. We prove that starting from a small initialization, gradient descent recovers $X^{\star}$ in $\tilde{O}(\sqrt{r})$ iterations approximately. The results solve the conjecture of Gunasekar et al.'17 under the restricted isometry property. The technique can be applied to analyzing neural networks with one-hidden-layer quadratic activations with some technical modifications.