In this section, we examine our theoretical results with controlled experiments via synthetic data. We do not have a complete explanation for such spikes. At first glance, overfitting could happen when the number of linear measurements is less than the size of the groundtruth matrix. Moreover, when the measurements satisfy RIP, Li et al. Soltanolkotabi [ 45 ] show that GD exactly recovers the ground truth. To our best knowledge, most existing generalization analysis for flat regularization are for two-layer models, e.g., Li et al.

This generalization ability cannot be explained by the capacity of the explicitly specified model class (namely, the functions representable in the chosen architecture). Instead, it seems that the optimization algorithm biases us toward a "simple" model, minimizing

We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix X with gradient descent on a factorization of X. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.

We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of X. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.

We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of $X$. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.