Private Matrix Approximation and Geometry of Unitary Orbits

Consider the following optimization problem: Given $n \times n$ matrices $A$ and $\Lambda$, maximize $\langle A, U\Lambda U^*\rangle$ where $U$ varies over the unitary group $\mathrm{U}(n)$. This problem seeks to approximate $A$ by a matrix whose spectrum is the same as $\Lambda$ and, by setting $\Lambda$ to be appropriate diagonal matrices, one can recover matrix approximation problems such as PCA and rank-$k$ approximation. We study the problem of designing differentially private algorithms for this optimization problem in settings where the matrix $A$ is constructed using users' private data. We give efficient and private algorithms that come with upper and lower bounds on the approximation error. Our results unify and improve upon several prior works on private matrix approximation problems. They rely on extensions of packing/covering number bounds for Grassmannians to unitary orbits which should be of independent interest.

Dissecting Hessian: Understanding Common Structure of Hessian in Neural Networks

Hessian captures important properties of the deep neural network loss landscape. We observe that eigenvectors and eigenspaces of the layer-wise Hessian for neural network objective have several interesting structures -- top eigenspaces for different models have high overlap, and top eigenvectors form low rank matrices when they are reshaped into the same shape as the corresponding weight matrix. These structures, as well as the low rank structure of the Hessian observed in previous studies, can be explained by approximating the Hessian using Kronecker factorization. Our new understanding can also explain why some of these structures become weaker when the network is trained with batch normalization. Finally, we show that the Kronecker factorization can be combined with PAC-Bayes techniques to get better explicit generalization bounds.